A COMPUTER PROGRAM FOR DESIGNING CHARGED PARTICLE BEAM TRANSPORT SYSTEMS*

A COMPUTER PROGRAM FOR DESIGNING CHARGED PARTICLE BEAM TRANSPORT SYSTEMS*

I

TRANSPORT

SLAC-Ql, Rev. 3

·uC-28

(I/A)

K. L. BROWN AND F. ROTHACKER

Stanford Linear Accelerator Center

Stanford Univeraitu, Stanford, California 94905

D.C.CAREY

Fermi National Accelerator Laboratoru

Batavia, Illinois 60510

CH. ISELIN

CERN, Geneva, Switzerland

May 1Q83

Prepared for the Department of Energy

under contract number DE-AC03-76SF00515

Printed in the United States of America. Available from the National Technical Information Service, U.S. Department of Commerce, 5285 Port Royal Road, Springfield,

VA 22161. Price: Printed Copy A , Microfiche AOL

• This report is also issued as CERN-80-04 and NAL-Ql. 

TABLE OF CONTENTS

INTRODUCTION

MATHEMATICAL FORMULATION OF "TRANSPORT"

General Conventions

The Transfer Matrix R

The Beam Matrix u

Fitting

INPUT FORMAT FOR TRANSPORT

An Example of a TRANSPORT Input Deck

The Use of Labels

Using the Program

Table I: Summary of TRANSPORT Type Codes

OUTPUT FORMAT

General Appearance

Initial Listing

Listing During the Calculation

Calculated Quantities

TITLE CARD

INDICATOR CARD (0, 1, or 2}

COMMENT CARDS

Example of the Use of Comment Cards in a Data Set

LISTING OF AVAILABLE TRANSPORT TYPE CODE ENTRIES

ii

Page 

I

INPUT BEAM: Type Code 1.0

The Pha.se Ellipse Beam Matrix

r.m.s. Addition to the BEAM

FRINGING FIELDS AND POLE-FACE ROTATIONS FOR

BENDING MAGNETS: Type Code 2.0

Pole-Face Rotation Matrix

DRIFT: Type Code 3.0

DRIFT Space Matrix

WEDGE BENDING MAGNET: Type Code 4.0

First-Order Wedge Bending Magnet Matrix

QUADRUPOLE: Type Code 5.0

First-Order Quadrupole Matrix

TRANSFORM 1 UPDATE: Type Code 6.0.1.

AUXILIARY TRANSFORMATION MATRIX (R2): Type Code 6.0.2.

SJilFT IN THE BEAM CENTROID: Type Code 7 .0

MAGNET ALIGNMENT TOLERANCES: Type Code 8.0

Example No. 1: A Bending Magnet with a Known Misalignment

Example No. 2: A Bending Magnet with an Uncertain position

Example No. 3: A Misaligned Quadrupole Triplet

Example No. 4: Misaligned Quadrupoles in a Triplet

Input for a Misalignment Table

Example of a Misalignment Table

iii 

REPETITION: Type Code Q.O

Example of Nesting

VARY CODES AND FITTING CONSTRAINTS: Type Code 10.0

Vary Codes

First-Order Vary Codes

Second-Order Vary Codes

Coupled Vary Codes

Possible Fitting Constraints

First-order constraints

Second-order constraints

RI matrix fitting constraints

R2 matrix fitting constraints

u(BEAM) matrix fitting constraints

Beam correlation matrix (r) fitting constraints

First moment constraint

System length constraint

AGS machine constraint

Floor coordinate fitting constraint

Tl matrix fitting constraints

T2 matrix fitting constraints

Second-order u(BEAM) matrix fitting constraint

Sextupole strength constraints

Internal constraints

Corrections and covariance. matrix

iv 

ACCELERATION: Type Code 11.0

Accelerator Section Matrix

BEAM (ROTATED ELLIPSE): Type Code 12.0

OUTPUT PRINT CONTROL INSTRUCTIONS: Type Code 13.0

Beam Matrix Print Controls 1., 2., 3.

Transformation Matrix Print Controls 4., 5., 6., 24.

Misalignment Table Print Control 8.

Coordinate Layout Control 12.

General Output Format Controls 17., 18., 19.

Punched Output Controls 29., 30., 31., 32., 33., 34., 35., 36.

ARBITRARY TRANSFORMATION INPUT: Type Code 14.0

INPUT-OUTPUT UNITS: Type code 15.0

SPECIAL INPUT PARAMETERS: Type Code 16.0

Code Digits for Special Parameters

Second-order term in the midplane field expansion of

bending magnets

Particle mass

Horizontal half-aperture of bending magnet

Vertical half-aperture of bending magnet

Cumulative length of the system

K 1 fringing-field integral

K2 fringing-field integral 

Curved entrance face of a bending magnet

Curved exit face of a bending magnet

Tilt-To-Focal Plane Element

Initial Beam Line Coordinate and Direction

SECOND-ORDER CALCULATION: Type Code 17.0

SEXTUPOLE: Type Code 18.0

SOLENOID: Type Code lQ.O

First-Order Solenoid Matrix

COORDINATE ROTATION: Type Code 20.0

Examples

STRAY MAGNETIC FIELD: Type Code 21.0

SENTINEL

REFERENCES

vi 

INTRODUCTION

TRANSPORT is a first- and second-order matrix multiplication computer program

intended for the design of static-magnetic beam transport systems. It has been in

existence in various evolutionary versions since 1963. The present version, described

in this manual, includes both first- and second-order fitting capabilities.

Many people Crom various laboratories around the world have contributed either directly or indirectly to the development of TRANSPORI'. The first-order matrix methods were developed by the AGS machine theorists1 followed by a paper by Penner.2

The extension of the first-order matrix methods to include second and higher orders

was conceived and developed by Brown, Belbeoch and Bounin3 in Orsay, France, in

1958-59. The original first-order TRANSPORT computer program was written in

BALGOL by C.H. Moore at SLAC in collaboration with H.S. Butler and S.K. Howry

in 1963. The second-order portion of the program was developed and debugged by

Howry and Brown,4 also in BALGOL. The resulting BALGOL version was translated

into FORI'RAN by S. Kowalski at MIT and later debugged and improved by Kear,

Howry and Brown at SLAC.5 In 1971-72, D. Carey at FNAL completely rewrote the

program and developed an efficient second-order fitting routine using the coupling

coefficients (partial derivatives) of multipole components to the optics as derived by

Brown.6 This version was implemented at SLAC by F. Rothacker in the early spring

of 1972 and subsequently carried to CERN in April, 1972, by K.L. Brown. C. Iselin

of CERN made further contributions to the program structure and improved the convergence capabilities of the first-order fitting routines.

A standard version of the resulting program has now been adopted at SLAC,

FNAL, and CERN. This manual describes the use of this standard version and;, not

neceaaarilg applicable to other veraiona of TRANSPORT. Copies of this manual may

also be obtained from

1. The Reports Office, CERN, 23 Geneva, Switzerland (Ref. CERN 73-16}.

2. The Reports Office, Fermi National Accelerator Laboratory, P. 0. Box 500,

Batavia, IL 60510 (Ref. NAL-91}.

A standard version of TRANSPORI' has been implemented on the IBM 360, using

the FORI'RAN H compiler. It requires approximately 270,000 bytes of memory for

1 

operation. It is normally distributed on g track magnetic tape at 800 BPI having a

standard IBM label. This program may be obtained from:

Frank Rothacker

TRANSPORT Program Librarian

Mail Bin 88

Stanford Linear Accelerator Center

P. 0. Box 4349

Stanford, California 94305

European users are encouraged to obtain the program from:

Ch. Iselin

CH-1211

CERN

23 Geneva, Switzerland

PDP 10, IBM and CDC users may also obtain the program from:

D. C. Carey

Fermi National Accelerator Laboratory

P. 0. Box 500

Batavia, IL 60510

Using the Program at SLAC

All input data must be in the form of 80 byte records. Columns 1-72 are processed

by the program, and columns 73-80 are treated as comments. Usually columns 73-80

are used as the line number identification field. The job control language deck setup

for operating the program at SLAC is as follows:

/ / job card information

//STEPl EXEC PGM = TRANS, REGION = 280 K

//STEPLIB DD DSN = WYL.CG.FXR.L, DISP = SHR

/ /FT04F001 DD SYSOUT = B

//FT06F001 DD SYSOUT = A

I /FT05F001 DD *

transport data deck goes here.

2 

The present authors assume responsibility for the contents of this manual, but in

no way imply that they are solely responsible for the entire evolution of the program.

3 

MATHEMATICAL FORMULATION OF "TRANSPORT"

General Conventions

A beam line is comprised of a set of magnetic elements placed sequentially at

intervals along an assumed reference trajectory. The reference trajectory is here taken

to be a path of a charged particle passing through idealized magnets (no fringing fields)

and having the central design momentum of the beam line.

In TRANSPORT,• a beam line is described as a sequence of elements. Such

elements may consist not only of magnets and the intervals between them, but also of

specifications of the input beam, calculations to be done, or special configurations of

the magnets. A certain relation, described below, of the magnets and their fields to

the assumed reference trajectory is considered normal. Alternative configurations can

be described by means of elements provided for such purposes.

The two coordinates transverse to the initial reference trajectory are labeled as

horizontal and vertical. A bending magnet will normally bend in the horizontal plane.

to allow for other possibilities a coordinate rotation element is provided. Because of

such other possibilities, when describing bending magnets we shall often speak of the

bend and nonbend planes. The transverse coordinates will also often be labeled x and

y, while the longitudinal coordinate will be labeled z.

All magnets are normally considered "aligned" on the central trajectory. A particle

following the central trajectory through a magnet experiences a uniform field which

begins and ends abruptly at the entrance and exit faces of the idealized magnet.

Therefore, through a bending magnet the reference trajectory is the arc of a circle,

while through all other magnetic elements it is a straight line. To accomodate a more

gradual variation of field at the ends of a bending magnet a fringing field element is

provided. In order to represent an orientation with respect to the reference trajectory

other than normal of a magnet or section of a beam line, a misalignment element also

exists.

The magnetic field of any magnet, except a solenoid, is assumed to have midplane

symmetry. This means that the scalar potential expanded in transverse coordinates

•For a more complete description of the mathematical basis of TRANSPORT, ref er

to SLAC-75,4 and to other references listed at the end of this manual.

4 

about the reference trajectory is taken to be an odd function of the vertical coordinate.

Ir a coordinate rotation is included, then the potential is odd in the coordinate to which

the vertical has been rotated. For a bending magnet this will always be in the nonbend

plane.

The program TRANSPORT will step through the beam line, element by element,

calculating the properties or the beam or other quantities, described below' where

requested. Therefore one of the first elements is a specification of the phase space

region occupied by the beam entering the system. Magnets and intervening spaces

and other elements then follow in the sequence in which they occur in the beam line.

Specifications of calculations to be done or of configurations other than normal are

placed in the same sequence, at the point where their effect is to be made.

The Transfer Matrix R

The following of a charged particle through a system of magnetic lenses may be

reduced to a process of matrix multiplication. At any specified position in the system

an arbitrary charged particle is represented by a vector (single column matrix) X, whose

components are the positions, angles, and momentum of the particle with respect to

the reference trajectory, i.e.

Definitions:

9

X= y

"' l

6

x = the horizontal displacement of the arbitrary ray with respect to the

assumed central trajectory.

0 - the angle this ray makes in the horizontal plane with respect to the

assumed central trajectory.

y - the vertical displacement of the ray with respect to the assumed

central trajectory.

5 

¢> - the vertical angle of the ray with respect to the assumed

central trajectory.

l - the path length difference between the arbitrary ray and the

central trajectory.

6 - ap = p is the fractional momentum deviation of the ray from the

assumed central trajectory.

This vector, for a given particle, will henceforth be ref erred to as a ray. The

magnetic lens is represented to first order by a square matrix R, which describes the

action of the magnet on the particle coordinates. Thus the passage of a charged

particle through the system may be represented by the equation

X(l) = RX(O) , (1)

where X(O) is the initial coordinate vector and X(l) is the final coordinate vector of

the. particle under consideration. The same transformation matrix R is used for all

such particles traversing a given magnet [one particle differing from another only by

its initial coordinate vector X(O)].

The traversing of several magnets and interspersing drift spaces is described by the

same basic equation, but with R now being replaced by the product matrix R( t) =

R(n) ... R(3)R(2)R(l) of the individual matrices of the system elements. This cumulative transfer matrix is automatically calculated by the program and is called

TRANSFORM 1. It may be printed where desired, as described in later sections.

This formalism may be extended to second order by the addition of another term. 4

The components of the final coordinate vector, in terms of the original, are now given

as

Xi(l) = E Rij Xj(O) + E Tijk Xj(O) Xk(O) '

j jk

where T is the second-order transfer matrix. It too is accumulated by the program as

one traverses a series of elements. At each point the series is again truncated to second

order. Normally the program will calculate only the first-order terms and their effect.

If it is desired to include second-order effects in a beam line, an element is provided

which specifies that a second-order calculation is to be done. For more information on

the T matrix, see the references at the end of the manual.

6 

The following of a charged particle via TRANSPORT through a system of magnets

is thus analogous to tracing rays through a system of optical lenses. The difference is

that TRANSPORT is a matrix calculation which truncates the problem to either firstor second-order in a Taylor's expansion about a central trajectory. For atud11ing beam

optita to greater preciaion than a aecond-order TRANSPORT calculation permita, ra11-

tracing programa which directl11 integrate the baait differential equation of motion are

recommended.1

The Beam Matrix u

In accelerator and beam transport systems, the behavior of an individual particle is

often of less concern than is the behavior of a bundle of particles (the beam), of which

an individual particle is a member. An extension of the matrix algebra of Eq. (1)

provides a convenient means for defining and manipulating this beam. TRANSPORT

assumes that the beam may be correctly represented in phase space by an ellipsoid

in the six-dimensional coordinate system described above. Particles in a beam are

assumed to occupy the volume enclosed by the ellipsoid, each point representing a

possible ray. The sum total of all phase points, the phase space volume, is commonly

ref erred to as the "phase space" occupied by the beam.

The validity and interpretation of this phase ellipse formalism must be ascertained

for each system being designed. However, in general, for charged particle beams in, or

emanating from accelerators, the first-order phase ellipse formalism of TRANSPORT is

a reasonable representation of physical reality. For other applications, such as charged

particle spectrometers, caution is in order in its use and interpretation.

The equation of an n-dimensional ellipsoid may be written in matrix form as follows:

X(o)T u(o)-1 X(O)=1 ' (2)

where X(Of is the transpose of the coordinate vector X(O), and u(O) is a real, positive

definite, symmetric matrix.

The volume of the n-dimensional ellipsoid defined by sigma is [7rm/2 /f(n/2 + 1))

(det u)112. The area of the projection in one plane is A= 7r(det u1)1

12, where u1 is

7 

the submatrix corresponding to the given plane. This is the "phase space" occupied

by the beam.

As a particle passes through a system of magnets, it undergoes the matrix transformation of Eq. (1). Combining this transformation with the equation of the initial

ellipsoid, and using the identity nn-1 =I (the unity matrix), it follows that

from which we derive

(3)

The equation of the new ellipsoid after the transformation becomes

X(lf u{l)-1 X(l) = 1 , (4)

where

u(l) = Ru(O) RT . (5)

It can readily be shown that the square roots of the diagonal terms of the sigma

matrix are a measure of the "beam size" in each coordinate. The off-diagonal terms

determine the orientation of the ellipsoid in n-dimensional space (for TRANSPORT

n = 6). • Thus, we may specify the beam at any point in the system via Eq. (5),

given the initial "phase space" represented by the matrix elements of u(O).

The initial beam is specified by the user as one of the first elements of the beam

line. Normally it is taken to be an upright ellipse centered on the reference trajectory;

that is, there are no correlations between coordinates. Both correlations and centroid

displacements may be introduced via additional elements.

The phase ellipse may be printed wherever desired. For an interpretation of the

parameters printed see the section under Type Code 1.0.

• See the Appendix of this report (to be published under a separate cover), or the

Appendix of Ref. 5, for a derivation of these statements.

8 

When a second-order calculation is specified, the second-order matrix elements are

included in the beam matrix.

Fitting

Several types of physical elements have been incorporated in the program to facilitate the design of very general beam transport systems. Included are an arbitrary

drift distance, bending magnets, quadrupoles, sextupoles, solenoids, and an accelerator section (to first-order only). Provision is made in the program to vary some of the

physical parameters of the elements comprising the system and to impose various constraints on the beam design. In a first-order run one may fit either the TRANSFORM

(R) matrix representing the transformation of an arbitrary ray through the system

and/or the phase ellipse (sigma) matrix representing a bundle of rays by the system as

transformed. In a second-order run one may fit either the second-order TRANSFORM

(T) .matrix or minimize the net contribution of second-order terms to the beam (sigma)

matrix.

The program will normally make a run through the beam line using values for the

physical parameters as specified by the user and printing the results. If constraints

and parameters to be varied are indicated, it will attempt to fit. To do this it will

make an additional series of runs through the beam line. Each time it will calculate

corrections to be made from the previous step to the varied parameters to try to satisfy

the indicated constraints. When the constraints are satisfied (or the fitting procedure

has failed) the program will make a final run through the beam line again printing the

results. In this final run the values of the physical parameters used are those which

are the result of the fitting procedure.

Thus, in principle, the program is capable of searching for and finding the firstor second-order solution to any physically realizable problem. In practice, life is not

quite so simple. The user will find that an adequate knowledge of geometric magnetic

optics principles is a necessary prerequisite to the successful use of TRANSPORT. He

(or she) should possess a thorough understanding of the first-order matrix algebra of

beam transport optics and of the physical interpretation of the various matrix elements.

g 

In other words, the program is superb at doing the numerical calculations for the

problem but not the physics. The user must provide a reasonable physical input if

he (or she) expects complete satisfaction from the program. For this reason a list of

pertinent reprints and references is included at the end of this manual. They should

provide assistance to the inexperienced as well as the experienced user.

10 

INPUT FORMAT FOR TRANSPORT

By the TRANSPORT input DATA SET is meant the totality of data read by the

program in a single job. A DATA SET may consist of one or more problems placed

sequentially. A problem specifies a calculation or set of calculations to be done on a

given beam line.

A problem, in turn, may consist of one or more problem steps. The data in the

first step of a problem specify the beam line and the calculations to be made. The

data in succeeding steps of the same problem specify only changes to the data given

in the first step.

A common example of a problem with several steps is sequential fitting. In the

first step one may specify that certain parameters are to be varied to satisfy certain

constraints. Once the desired fit has been achieved the program will then proceed to

the next step. The data in this step now need specify only which new parameters to

vary, or old ones no longer to vary, or which constraints to add or delete. The values

of the varied parameters that are passed from one step to the next one are those that

are the result of the fitting procedure.

A problem step contains three kinds of DATA cards: the TITLE card, the INDICATOR card, and the ELEMENT cards.

The TITLE card contains a string of characters and blanks enclosed by single

quotes. Whatever is between the quotes will be used as a heading in the output of a

TRANSPORT run.

The second card of the input is the INDICATOR card. If the data which follow

describe a new problem, a zero (0) is punched in any column on the card. If the data

which follow describe changes to be made in the previous problem step, a one (1)

or two (2) is punched in any column on the card. For further explanation read the

Indicaator Card section of this manual.

The remaining cards in the deck for a given problem step contain the DATA

describing the beam line and the calculations to be done. The DATA consist of a

sequence of elements whose order is the same as encountered as one proceeds down

the beam line. Each element specifies a magnet or portion thereof or other piece of

equipment, a drift space, the initial beam phase space, a calculation to be done, or a

11 

print instruction. Calculation specifications, such as misalignments and constraints,

are placed in sequence with the other beam line elements where their effect is to take

place. The input format of the cards is "free-field," which is described below. The

data for a given problem step are terminated by the word SENTINEL, which need

not be punched on a separate card.

Each element, in turn, is given by a sequence of items (mostly numbers), separated

by spaces and terminated by a semicolon. The items, in order, are a type code number,

a very field, the physical parameters, and an optional label.

The type code number identifies the element, indicating what sort of entity (such

as a magnet, drift space, constraint, etc.) is represented. It is an integer (number)

followed by a decimal point. The interpretation of the physical parameters which

follow is therefore dependent on the type code number. The type code numbers and

their meanings are summarized in Table I. Ir the type code number is negative, the

element will be ignored in the given problem step. However, storage for that element

will be allocated by the program, so that the element may be introduced in a later

step of the same problem. Storage apace for anu element in anu problem atep muat be

allocated in the firat atep of the problem.

The vary field indicates which physical parameters of the element are to be adjusted

if there is to be any fitting. It is punched immediately (no intervening blanks) to the

right of the decimal point of the type code number. See the section under type code

10.0 for an explanation or the use or vary codes.

The physical parameters are the quantities which describe the physical element

represented. Such parameters may be lengths, magnetic fields, apertures, rotation

angles, beam dimensions, or other quantities, depending on the type code number.

The meanings for the physical parameters for each type code are described thoroughly

in the section for that type code. A summary, indicating the order in which the

physical parameters should be punched, is given in Table I. For any element the first

physical parameter is the second entry in Table I or the second parameter in the section

describing a given element. In some cases the parameters of an element do not really

refer to physical quantities, but will nevertheless be referred to as such in this manual.

12 

The label, if present, contains one to four characters and is enclosed by single

quotes, slashes, or equal signs. During the calculation the elements will be printed in

sequence and the label for a given element will be printed with that element. Labels

are useful in problems with many elements and/or when sequential fitting is used.

They must be used to identify any element to be changed in succeeding steps of a

given problem.

Provision has been made in the program to allow the user to introduce comments

before any type code entry in the data deck. This is accomplished by enclosing the

comments made on each card within single parentheses.

Each element mud be terminated bv a aemitolon (;). Optionally a semicolon may

be replaced by an asterisk(*) or a dollar sign($). Spaces before and after the semicolon

are allowed but not required. If the program encounters a semicolon, dollar sign, or

asterisk before the expected number of parameters has been read in and if the indicator

card was a zero (0), the remaining parameters are set to zero. If the indicator card

was a one (1) or two (2), then the numbers indicated on the card are substituted for

the numbers remaining from the previous solution; all other numbers are unchanged.

The "free-field" input format of the data cards makes it considerably easier to

prepare input than the standard fixed-field formats of FORTRAN. Numbers may be

punched anywhere on the card and must simply be in the proper order. They must be

separated by one or more blanks. Several elements may be included on the same card

and a single element may continue from one card to the next. A single number must be

all on one card; it may not continue from one card to the next. The program storage

is limited to a total of 2,000 locations (including type codes and those parameters not

punched but implied equal to zero) and 500 elements.

A decimal number (e.g., 2.47} may be represented in any of the following ways:

2.47

.00247+3

.0247E+02

247E-2

247000-5

The sample problem below contains two problem steps, each beginning with title

13 

and indicator cards and terminating with a SENTINEL. The first step causes TRANSPORT to do a first-order calculation with fitting. The second mitiates a second-order

calculation with the data that is a result of that fitting. Corresponding elements between the two steps are identified by having the same label.

The type ten element which specifies the fitting condition is labeled FITI. It is

active for the first-order calculation, but is turned off for the second-order calculation.

The vary codes for elements DRl are set to zero for the second-order problem. The

second-order element, SECl, is ineffective during the fitting, but causes the program

to compute the second-order matrices in the second calculation.

An Example of a TRANSPORT Input Deck

'FORTRAN H CHECK ON BETA FIT' Title eard

0

1. .5 1 .5 1 .5 1 1

-17. 'SECl' ; 13 3 ;

3.3 2. 7 45 'DRI' ;

2 0 ; 4 g.87Q 10 .5 2 0

3.3 2.745 'DRI' ;

13 4 ;

10 -1 2 o .0001 'FITl'

SENTINEL

'SECOND ORDER'

1

17 'SECl' l 3 'DRI' ;

-10 'FITl'

SENTINEL

)

Indicator eard

Elementa

Title eard

Indicator eard

Elementa to be

changed 1

SENTINEL Second aentinel aignifiea end of run.

Firat pro61em atep

Second problem step

As many problems and problem steps as one wishes may be stacked in one job.

Note that in previous versions of TRANSPORT a decimal point was required with

everv numerical entrv ezeept the indicator eard (whieh must not have a decimal point

in anv version of TRANSPORT).

14 

The Use of Labels

The use of labels is available for identification of individual elements. When inserted for the user's convenience, the association of a label with a given element is

optional. If the parameters of an element are to be changed between steps of a given

problem, a label is required. The label identifies the element in the earlier step to

which the changes specified in the latter step are to apply.

The label may be placed anywhere among the parameters of a given element. It

should be enclosed in quotes, slashes, or equal signs. Blanks within a label are ignored.

The maximum length of a label is four non-blank characters.

As an example, the following all denote the same drift space:

'DRF'

3/DRF/1&-h

3.

3. 1.5

.15El =DRF = $

On a 15.0 type code element the label may not be the third item. This is to avoid

ambiguities with the unit name. Thus the following are not equivalent:

J This

I

15. 1. 'FT' 'CM'

15. 1. 'CM' 'FT'

entry is used as the units symbol.

This entry is used as a label.

If the parameters describing an element are to differ in succeeding steps of a given

problem the element must be included in both steps, having the same label each time.

All elements which appear in a problem must be included in the first step (indicator

card 0) of that problem. Only those to be changed in later steps need to be labeled.

In later steps (indicated card 1) of a problem only those elements to be changed are

specified. The elements to be changed are identified by their labels.

If the type code number of an element is negative in a given step of a problem,

that element will be ignored when the calculation is performed. However, storage

space in the computer will be allocated for the element for possible activation in later

steps of the problem. In the later step, only those parameters to be changed need

to be specified. The storage space allocated for the parameters of a given element is

15 

determined only by the type code. The sole exceptions are the continuation codes for

type codes 1.0 and 14.0.

For example, if a fitting constraint is to be ignored in the first step of a problem,

but activated in a latter step, it should be indicated in both steps. In the first step

such an element might appear as

-10. 'FIT' ;

In the later step one would then insert

10. 1. 2. 0.0 .001 'FIT'

causing a waist constraint to be imposed on the beam. Alternatively one can specify

the physical parameters in the first step and then, in the later step, merely indicate

that the element is now to be activated. The above procedure is therefore equivalent

to placing the element:

-10. 1. 2. 0.0 .001 'FIT' ;

in the first problem step, and the element

10. 'FIT'

in the later step.

Vary codes may also be inserted or removed in passing from one problem step to

the next. For instance, one might wish to vary the field of a quadrupole in one step of

a problem and then use the fitted value as data in the following step. The first step

might then contain the element:

5.01 5.0 10.0 5.0 'QUAD'

and the following step would contain the element

5. 'QUAD' ;

Since, in the second step, the first item on the card contains no vary code the vary

code is deleted. All other parameters, not being respecified, are left unchanged.

Several elements may have the same label. IC, as in the above example, one wished

to vary the field of several quadrupoles in one step, then pass the final values to the next

step, one could give all such elements the same label. There might be four quadrupoles,

all labeled 'QUAD', being varied simultaneously. IC the data for the next step contain

the single element

16 

5. 'QUAD' ;

the vary code on all elements labeled 'QUAD' will be deleted.

The physical parameters of an element may be changed between steps of a problem.

In the first step a bending magnet may be given a length of 5 meters.

4. 5.0 10.0 0.0 'BEND' ;

In a succeeding step, its length could be increased to 10 meters by inserting the element

4. 10.0 'BEND' ;

All parameters, up to and including the one to be changed, must be specified. the

remaining, if deleted, will be left unchanged from the previous step.

17 

Table 1: Summary of TRANSPORT Type Codes

PHYSICAL EI»Err 1YPE 2nd 3rd 4th 5th 6th 7th 8th 9th CD!E ENTRY ENTRY ENTRY ENfRY ENfRY ENJ'RY ENTRY ENTRY

BEAM l.vvvvvvO x(cm) e(mr) y(cm) HmrJ 1(cm) 6(percent) Po

r.m.s. ADDITIOO TO l.vvvvvvOO Ax(cm) 60(mr) dy(cm) ~(mr) dl(aqj M(percent) dP 0 BEAM ENVELOPE (GeV/c)

POLE FACE roTATIOO 2.v ANGLE-OF

lllTATIOO (degrees)

!IUFI' 3.v LENGIH (metres)

BENDING MAGNET 4.vvv l.ENGlH (metres) FIELD (kG) FIELD GRADIENJ' (n-value)

QUADR!l'OLE s.vvo l.ENGlH (metres) FIELD (kG) llALF-APERTIJij; (an)

TRAllSRJRM 1 UPDATE 6.0 o.o 1.0

TRAllSRJRM 2 lPDATE 6.0 o.o 2.0

BEAM CENTRJ!D SHIFI' 7.VVVVVV SHIFI' (x)(cm) SHIFI' (0) (mr) SHIFI' (y)(cm) SHIFI' (4>)(mr) SHIFI' (1)(cm) SHIFI' (6 percent)

ALI QMlm' TOLERANCE 8.vvvvvvO DISPLACD£NI' (x)(cm) roTATIOO (e)(mr) DISPLACBIElll' (y)(cm) roTATIOO (4>)(mr) DISPLACEMENT (z)(cm roTATIOO ( Q )(mr) !DIE

NlM!llt

REPEAT aJNTR>L 9.0 Nl.MIER OF REPEATS

FITTING CXHITRAINTS 10.0 ±I J !ESIRED VALUE OF AC£IJRACT OF (I ,J) MATRIX FIT El..Efo£m'S

Note: +I is used for fitting a beam (a) matrix element. -I is used for fitting an Rl matrix element.

... (I + 20) is used for fitting an R2 matrix element.

ACCELERATOR 11.0 LENGIH (metres) I E (energy gain) 1 • (phase lag) I (WAVELENG!H) (cm) I I (GeV) (degrees)

BEAM

(Rotated Ellipse) 12.0 1l!E FIFl'EEN CDRRELATIOOS AIOIG 1l!E SIX ELIMNJ'S (This entry nust be preceded by a type code 1.0 entry.)

INPUT /OUl'P\Jf . 13.0 ClJNTR>L OPTIONS !DIE Nl.MBER

ARBITRARY R MATRIX 14.vvvvvvO R(J,l) R(J,2) R(J,3) R(J,4) R(J,S) R(J,6) J

ltlITS ClJNTR>L (Transport 15.0 !DIE ~IT SYl8JL SCALE FACTOR

Dimensions) (if required)

QUADRATIC TERM 16.0v 1.0 £(1J • scl;i' Po in units of transverse lenath {an) OF BENDING FIELD I MASS OF PARTICLES IN BEAM 16.0 3.0 11/11 (dimensionless) m • uss of electron

HALF-APEIUl.IRE OF BENDING MAGNET 16.0 4.0 w/2 (an)

IN x-PLANE

HALF-APERTIJRE OF

BENDING MAGNET 16.0 s.o g/2 (ca) IN y-PLANE (gap)

LENGlH OF SYST!M 16.0 6.0 L ( .. ties)

FRINGE FIELD CDR- 16.0 7.0 K1 (dimensionless) RECTIOO CDEFFICIENJ'

FRINGE FIELD CDR- 16.0 8.0 K, (dimensionless) RECTI<l" CDEFFI CIENJ'

ClRVA11JRE OF Em-RANCE FACE OF 16.0V 12.0 (l/Ri) (l/ .. tres)

BENDING MAGNET

OJRVA11JRE OF EXIT FACE OF 16.0V 13.0 (l/Ra) Cl/metres) BENDING MAGNET

RJCAL PLANE 16.0 15.0 Angle of focal plane rotatim (degrees).

roTATIOO See type code 16. 0 for details.

INITIAL BEAM LINE 16.0V 16.0 Xo x-CDORDINATE

INITIAL BEAM LINE 16.0V 17.0 Yo y-COORDINATE

INITIAL BEAM LINE 16.0V 18.0 z-CDORDINATE •• -

INITIAL BEAM LINE 16.0V 19.0 e, llJRIZOOTAL ANGLE

INITIAL BEAM LINE 16.0V 20.0 VERTICAL ANGLE •• SE~ 17.0 CALCULATIONS

SEXTlPOLE 18.0V l.ENGlH (metres) FIELD (kG) HALF-APERI\JRE (cm)

SOLENOID 19.w LENGlH (metres) FIELD (kG)

BEAM lllTATIOO 20.v ANGLE OF roTATIOO (degrees)

STRAY FIELD 21.0 See later section of report.

Note: The v•s followins: the type codes indicate the parameters which may be varied. see sectim under type code 10.0 for a detailed explanation of Vary Codes. The 1.11its are standard TRANSPOITT 1.11its (as shown) 1.11less changed via type code 15.0 entries.

18 

OUTPUT FORMAT

General Appearance

Here we give a brief description of the general appearance of the output and its

meaning. The user may refer to the sample output shown on pages 22 through 27.

It is the printed output resulting from the sample data shown in the section on input

format. In a simple example it is not possible to show each of the different type codes.

Several of the type codes produce output which is not characteristic of all other type

codes. We therefore refer the user to the sections on the various type codes for an

explanation of any features peculiar to a. given type code.

The output for ea.ch step of a given problem is printed separately. The printing for

one step is completed before that for the next step is begun. Therefore we will describe

the output for a. single problem step. The output shown below is from a. problem with

two steps.

Initial Listing

For each problem step, the program begins by printing out the user's input records.

Listing During the Calculation

The program now begins the calculation. H there is no fitting, one listing of the

beam line will be made. H there is fitting there will normally be two listings. The

first will represent the beam line before any fitting has occurred. The second will be

based on the new values of the physical parameters which were altered by the fitting

process. If sequential fitting is employed and an indicator card of two (2) is used the

first run will be omitted. The user should read the section describing the indicator

card for further explanation.

In any listing the elements are printed in order with their labels and physical

parameters. Elements with negative type code numbers are ignored. Each listed element is preceded by the name of that type of element, enclosed in asterisks. All

physical elements are listed in this way. Some of the other elements are not explicitly

listed but produce their effect in either the calculated quantities or the listing of the

19 

beam line. For descriptions or individual cases, the reader should consult the sections

on the type codes.

Calculated quantities appear in the listing as requested in the input data. Important cases will be described in greater detail below. The physical parameters Cor

each element are printed with the appropriate units. For some elements a calculated

quantity, not in the input data, will appear, enclosed in parenthesis. Such quantities

are explained in the sections under the individual type codes.

Calculated Quantities

The important cases or calculated quantities which appear in the output are the

transfer matrices, the beam matrix, the layout coordinates, and the results or the

fitting procedure. The transfer and beam matrices and layout coordinates appear as

requested in the listing or the beam line. The results or the fitting procedure appear

between the two listings. All these quantities are explained in greater detail below.

The transfer and beam matrices appear only where requested. A request for printing of layout coordinates should be made at the beginning of the beam line. The

coordinates will then be printed after each physical element. In all cases the quantities

printed are the values of the interface between two elements. They are evaluated at

a point after the element listed above them and before the element listed below. For

further explanation of calculated quantities the user should read the section on the

mathematical formulation of TRANSPORT and the sections on the appropriate type

codes. For the transfer matrix the appropriate type code is thirteen; for the beam

matrix it is one, and for the coordinte layout it is again thirteen.

Quantities relevant to the fitting appear between the two listings of the beam line.

At each iteration of the fitting procedure a line is printed containing the value of the

relaxation factor used, the value or chi-squared before the iteration was made, and the

corrections made to each of the varied parameters. Once the fitting is complete the

final chi-squared and the covariance matrix are printed. For further details the user

should read the section on type code 10.0.

20 

TRANSPORT UPDATED JULY rn7 4

NIU HISALlCNHENT FEATURES HAY I! ACTIVATED BY ttEANS OF COD! ON ttlSALtCNttENT CARD

A IN THE Tl!NS POSITION CAUSES THI! RESULTS OF THI! llISALICNllENT TO Bl! PLACED IN A llISALIClllll!NT TAllLE,

UHtCH IS PRINTED UHl!REVER AN 13. 8. CARD IS INSERTED

A 4 IH THI! ONES POSITION MISALIGNS ALL SUBSEQUENT BEND MAGNETS, UP TO TEN

A 5 IN THE ONES POSITION HISALICHS ALL SUllSEQUEHT QUADRUPOLES, UP TO TEN

A 3 lN THF. ON!& POSITION MISALIGNS ALL SUBSEQUENT BENDS OR QUADS, UP TO TEN

THE UNITS FOR BEND ANGLE, POLE FACE ROTATION ANGLE, AND LAYOUT ANGLE MAY RE CHANCED VIA A 15. 7. ENTRY

THI! ANGLES or THF. REFERENCE TRAJECTORY MOU APPEAR IN THE COORDINATE LAYOUT

THE PRINTING or PHYSICAL PARAMETERS FOR ELEMENTS HAY BE SUPPRESSED BY A 13. 17. CARD

ONLY VARIED ELEMENTS OR CONSTRAINTS VILL RE PRINTED IF A 13. 18. CARD IS INSERTED

THE TRANSFER AND BEAtt MATRICES UILL OCCUPY A SINGLE LINE IF A 13. 19. CARD IS INSERTED

THE INITIAL LISTING OF THE DECK VILL BE DELETED IF 10 IS ADOED TO THE INDICATOR

THE REAtt MATRIX IS PRINTED AFTER EACH ELEMENT ONLY IF ACTIVATED VIA A 13. 3. CARD

THE TRANSFER ttATRIX MAY Bl! PRINTl!D AFTER EACH ELl!llEHT BY USING A 13. 6. CARD

THI! lll!AH CENTROID HAY RE VARIED

"FORTRAN H CHECK ON BETA FIT

0

1.000000 o.50000 1.00000 o.50000 1.00000 o.50000 1.00000

•17. "S!CI"

13. 3.00000

3.3 "ORI " 2.74500

2.0 o.o 4.000 9.87900 10.00000 o.50000;

2.0 o.o 3.3 "ORI " 2.74500

13. 4.00000

10.0 "FIT " -1.00000 2 .0000.0 o.o 0.00010;

SENTINEL

21

1.00000; 

TRANSPORT UPDATED JULY 1974 (Continued)

POUllAll ff CRF.Cll'. ON UTA Pit

•11A11• •• 1000000 CIY o.o II o.o 0.500 CH o.o 1.000 llR o.o o.o 0.500 Cit o.o o.o o.o 1.000 HR o.o o.o o.o o.o 0.500 CH o.o o.o o.o o.o o.o 1.000 PC o.o o.o o.o o.o o.o •ourT• ], •1111 . 2.14500 H

VARY COP~ • ]

2, 145 II o.o 0.510 CH o.o 1.000 llR o.411 o.o 0.510 CH o.o o.o o.o 1.000 llR o.o o.o o.411 o.o o.5oo CH o.o o.o o.o o.o o.o 1 •. 000 PC o.o o.o o.o o.o o.o •ROTAT* 2. o.o DIG

2.145 H o.o 0.510 Cll o.o 1.000 Ill o.411 o.o o.510 CH o.o o.o o.o 1.000 llR o.o o.o o.411 o.o 0.500 CH o.o o.o o.o o.o o.o 1.000 PC o.o o.o o.o o.o o.o *llftP* 4. 9,11900 II 10,00000 ltG o.5onoo ( 3,336 II . 169,690 DEG )

12,624 II o.o 10.0ll Cl! o.o 12,324 llR 0.993 o.o 0.369 Cl! o.o o.o o.o 1,360 llR o.o o.o -0.016 o.o 11.690 Cl! -o.992 -o.972 o.o o.o o.o 1.000 PC 0.999 0.994 o.o o.o -0.991

•ROTAT* 2. o.o DEG

12. 624 II o.o 10.oll CH o.o 12. 324 llR 0.993 o.o 0.369 CH o.o o.n o.o 1.360 llR o.o o.o -0.086 o.o 11. 690 CH -o.992 -o.972 o.o o.o o.o 1.000 PC 0.999 0.994 o.o o.o . :0.991 •nurr• 3. "Dll " 2.14500 "

VARY COii! • 3

15, 369 H o.o ll.311 CH o.o 12. 324 llR o.996 o.o 0.502 Cit o.o o.o o.o 1.360 llR o.o o.o 0.681 o.o 11.690 en -0.918 •0.912 o.o o.o o.o 1.000 PC o.999 o.994 o.o o.o -0.991

*TRANS FOUi •• -1.00383 -0.00418 o.o o.o o.o 13,36808

-1.13605 -1.00383 o.o o.o o.o 12.24819

o.o o.o -1.00383 -0.00411 o.n o.o o.o o.o -1.83605 -1.00313 o.o o.o -1. 22418 -1.33681 o.o o.o 1.00000 -11. 58649

o.o o.o o.o o.o o.o 1.00000 •rn• 10.0 "rtt " -1. 2. o.o 111.00010 -o. 00418 )

•LENGTH* 15. 36899 H

*CORRF.CTIOPS*

0.100001+01 ( 0.11416F.+04) -0.0201

O. IOOOOE+OI ( 0.63241£-02) -0.0000

*COVOIANCF. (FIT o.39529P.-05 )

n.onn

22 

TRANSPORT UPDATED JULY 1974 (Continued)

Fl•RT~~·: II C:llf:C:K "'' ftf:TA r 1 ·r

* ~ F.Atl* I. 1.00000 C:F.V

11.0 ti o.o o. 500 Cfl

o.o 1.000 llR o.o o.o o. 500 Ctl ·o.o o.o 1),0 1.000 llR o.o o.o o.o o.o 0.500 Cll o.o o.o o.o o.o o.o l.ooo PC o.o o.o o.o o.o o.o *ORIFT* 3. "ORI .. 2.72414 II

VAPY CllOF. • l

2.724 rt o.o 0.569 Cll

o.o 1.000 llR 0.478 o.o o.569 en o.o o.o o.o l.000 llR o.o o.o 0.478 o.o o. 500 Cit o.o o.o o.o o.o o.o 1.000 PC o.o o.o o.o o.o o.o *ROTAT* 2. o.o DEG

2.724 ti o.o o. 569 en o.o 1.000 llR 0.478 o.o o.569 Cit o.o o.o o.o 1.000 llR o.o o.o o.478 o.o 0.500 Cit o.o o.o o.o o.o o.o 1.000 PC o.o o.o o.o o.o o.o * R F.tlO* 4. 9.87900 II 10.00000 KG 0.50000 ( l. 336 ti . 169.690 DEG )

12.601 rt o.o 10.013 Cll

o.o 12.124 llR o.991 o.o 0. 370 Cit o.o o.o o.o 1.357 llR o.o o.o -0.086 o.o 11. 690 Cit -0.992 -0.972 o.o o.o o.o l.ooo PC o. 999 0.994 o.o o.o -o.991 *ROTAT* 2. o.o DEG

12,603 n o.o 10.013 Cit

o.o 12.324 llR o. 993 o.o 0. 370 Cit o.o o.o o.o 1,357 llR o.o o.o -0.086 o.o 11. 690 Cit -o. 992 -0.972 o.o o.o o.o 1.000 PC o.999 o.994 o.o o.o -0.991 *DRIFT* 3, "ORl " 2.72414 II VARY GOOF. • 3

15, 32 7 II o.o 13.352 CH o.o 12. 324 llR o.996 o.o o. 500 CH o.o o.o o.o 1,357 !IR o.o o.o o.676 o.o 11.690 C!I -o.989 -o.972 o.o o.o o.o 1.000 PC o.999 0.994 o.o o.o -0.991 *TRANS FOR~! 1•

-1.00000 -0.00000 o.o o.o o.o 13. 34253

-1.83605 -1.00000 o.o o.o o.o 12.24879

n.o o.o -1. 00000 0.00000 o.o o.o o.o o.o -1. 83605 -1. 00000 o.o o.o -1.22488 -1.33425 o.o o.o 1.00000 -11.58649

o.o o.o o.o o.o o.o I. 00000 wrrr• 10.0 "Flt " -1. 2. o.o /0.00.010 -0.00000 )

•Lf:MGT~* 15,32727 II

23 

TRANSPORT UPDATED JULY 1974 (Continued)

"!!;f.CO!lh llRhF.R

I

1.000000 o. soooo 1.oonoo o. 50000 1.00000 o. soooo 1.00000 t .00000;

17. "SF.C 1"

11. l. 00000;

1. n "THU " Z.7Z414; z .o o.o ;

4.ooo •• 87900 10.00000 o. soooo; 2 .o o.o I

1.n "nRt " Z.72414;

tl. 4.00000; . -10. "rrr " -1.00000 z. 00000 o.o 0.00010; sr.~n ra:1. SF.CIH!ll OR1'F.R

*BF.Aft* I. I. 00000 GEY

o.o !I o.o o. soo CH

o.o 1.000 llR o.o

o.o o. soo en o.o o.o

o.o 1.000 HR o.o o.o o.o o.o o.soo en o.o o.o o.o o.o

o.o 1.000 PC o.o o.o o.o o.o o.o

*2Nn ORJ1ER* 17. "IF.Cl" CAUSSIA N DISTRIBUTION

*hRIFT* l. "DIH " 2. 7Z414 II 2. 724 H o.o O. S69 CH

o.o 1.000 llR 0.478

o.o O. S69 en o.o o.o o.o 1.000 llR o.o o.o 0.478

-o.ooo o. soo CH o.o o.o o.o o.o

o.o 1.000 PC o.o o.o o.o o.o o.o *RtlTI\ T* z. o.o nr.c 2. 724 II o.ooo O. S69 Cll

o.o 1.000 HR 0.478 o.o o.S69 CH o.o o.o

o.o 1.000 llR o.o o.o 0.478

-o.ooo o. soo CH ,o.ooo o.o o.o o.o

o.o I. 000 PC o.o o.o o.o o.o o.o

•11r.t1n• 4. 9.87900 II 10. 00000 KG o. soooo ( l. ll6 II 169. 690 DEG l

12. 601 n 0.068 10.0ll CM

0.011 IZ .124 llR 0,99]

o.o 0.170 Cll o.o o.o

o.o 1.1s8 llR o.o o.o -0.086

-o.ooz 11.690 CH -0.992 -0.97Z o.o o.o

o.o 1.000 PC o.999 0.994 o.o o.o -0.991

*ROT AT* z. o.o OF.G

I Z. 601 II 0.068 10.0ll en 0.037 I Z.124 HR 0.991

o.o 0.110 c" o.o o.o

o.o 1. lS8 llR o.o o.o -0.086

-0.002 11.690 Cll -o. 992 -o. 9 72 o.o o.o

o.o 1.000 PC 0.999 0.994 o.o o.o -0.991

•DR1''T* l. "1HU " Z.7Z414

IS.127 " ,, 0.078 ll.lSZ CH

0.011 IZ. lZ4 llR o.996

o.o o. soo Cll o.o o.o

o.o I. ]SB llR o.o o.o o.676

-0.021 11.690 Cll -o. 988 -o. 97 2 o.o o.o

o.o 1.000 PC 0.999 0.994 o.o o.o -0.991

*TRANsrt•R~I 1 •

-1.00000 -0.00000 o.o o.o o.o I l. l4ZSl

-1.8l60S -1.00000 o.o o.o o.o IZ. Z4S79

o.o o.o -1.00000 0.00000 o.o o.o

o.o o.o -1.8160S -1.00000 o.o o.o

-1.2Z488 -1. ll4B o.o o.o 1.00000 -.ll.S864'

o.o o.o o.o o.o o.o 1.00000

24 

TRANSPORT UPDATED JULY 1974 (Continued)

*2ND ORDER TRANSFORtt*

1 11 l.124E-03

1 12 l.225E-03 22 6.673E-04

1 13 o.o 23 o.o 33 -4.871E-03

1 14 o.o 24 o.o 34 -2.042E-03 44 -l.112E-03

1 15 o.o 25 o.o 35 o.o 45 o.o 55 o.o

l 16 2.065E-02 26 3.066E-02 36 o.o 46 o.o 56 o.o 1 66 7.908E-02

2 11 6.046E-07

2 12 1. 316E-06 2 22 6.126E-04

2 13 o.o 2 23 o.o 2 33 -5.504E-03

2 14 o.o 2 24 o.o 2 34 -2.999E-03 2 44 -l.021E-03

2 15 o.o 2 25 o.o 2 35 o.o 2 45 o.o 2 55 o.o

2 16 3.097E-02 2 26 3.564E-02 2 36 o.o 2 46 o.o 2 56 o.o 2 66 3.835E-02

3 11 o.o

3 12 o.o 3 22 o.o

3 13 -7.500E-04 3 23 -2.042E-03 3 33 o.o

3 14 -2.042E-03 3 24 -2.224E-03 3 34 o.o 3 44 o.o

3 15 o.o 3 25 o.o 3 35 o.o 3 45 o.o 3 55 o.o

3 16 o.o 3 26 o.o 3 36 -l.064E-02 3 46 -3.414E-03 3 56 o.o 3 66 o.o

4 11 o.o

4 12 o.o 4 22 o.o

4 13 5.503E-03 4 23 5.994E-03 4 33 o.o

4 14 -2.99'9E-03 4 24 -2.042E-03 4 34 o.o 4 44 o.o

4 15 o.o 4 25 o.o 4 35 o.o 4 45 o.o 4 55 o.o

4 16 o.o 4 26 o.o 4 36 5.756E-03 4 46 4.367E-03 4 56 o.o 4 66 o.o

*LENGTH* 15.32727 II

25 

This Page Left Blank

26 

F .. ZJMJ g e t a sue•t< ·

TITLE CARD

The title card is the first card in every problem step of a TRANSPORT data set.

The title tard ia alwa111 required and muat 6e /ollowed 611 the indicator tard (see next

section) to indicate whether the data to follow is new (0 card) or a continuation of a

previous data set (a I card or a 2 card).

The title muat 6e entloaed within either quotation marka {'}, alaahea (/}, or equal

aigna (-) on a aingle card. The string may begin and end in any column (free field

format), for example

'SLAC 20 GeV/c SPECTROMETER'

or

/SLAC 20 GeV SPECTROMETER/

Note that whichever character is used to enclose the title muat not 6e used again

within the title itself.

Example of a DATA SET for a Single Problem Step

SENTINEL (need not be on separate card)

Elements

O or 1 or 2

Title card

27 

INDICATOR CARD (O, 1, or 2)

The second card of the input for each step of a problem is the indicator card. If

the data which follow describe a new problem, a zero (0) is punched in any column on

the card. If the data which follow describe changes to be made in the previous step of

a given problem, a one (1) or two (2) is punched in any column on the card.

If a given problem step involves fitting, the program will normally list the beam

line twice, printing each time the sequence of elements along with transfer or beam

matrices where specified. The first listing uses the parameters of each element before

any fitting has taken place. The second shows the results of the fitting. If a problem

involving fitting has several steps, the second run of a given step often differs little

from the first run of the following step.

If the second or subsequent step of a problem involves fitting and one wishes to

print both runs through the beam line, a one ( 1) is punched on the indicator card. If

the· first listing is to be suppressed a two (2) is punched. If no fitting is involved, the

program will ignore the 2 and will do one single run through the system.

If the initial listing is to be deleted, 10 is added to the indicator to give 10, 11 or

12. In order to be consistent with earlier versions of TRANSPORT, an indicator of

minus one (-1) is interpreted as a two (2), but nine {9} ia not interpreted aa twelve {12}.

The sample problem input shown on page 14 causes TRANSPORT to do a firstorder calculation with fitting (0 indicator card) and then to do a second-order calculation (1 indicator card) with the data that is the result of the fitting.

28 

COMMENT CARDS

Comment cards may be introduced anywhere in the deck where an element would

be allowed by enclosing the comments made on each card within single parentheses. No

parentheses are allowed within the parentheses or any comment card. The comments

are not stored, but appear only in the initial listing or the given problem step.

Example of the Use or Comment Cards in a Data Set

'Title Card'

0

(TIIlS IS A TEST PROBLEM TO ILLUSTRATE THE)

(USE OF COMMENT CARDS)

elements

(COMMENTS MAY ALSO BE MADE BETWEEN)

(TYPE CODE ENTRIES)

elements

SENTINEL

29 

LISTING OF AVAILABLE TRANSPORT TYPE CODE ENTRIES

INPUT BEAM : Type Code 1.0

The phase space and the average momentum of the input beam for a TRANSPORT

calculation are specified by this element. The input is given in terms of the semi-axes

of a six-dimenational erect• beam ellipsoid representing the phase space variables

x, 0, y, </>, l, and 6. Each of these six parameters is entered as a positive quantity,

but should be thought of as ±z, ±0, etc.; hence, the total beam width is 2z, the total

horizontal beam divergence is 20 and so forth.

Usually the BEAM card is the third card in the deck. H other than standard

TRANSPORT units are to be used, the units specification cards (Type Code 15.0)

should precede the BEAM card. Standard TRANSPORT units for x, 0, y, </>, l, and

6 are cm, mr, cm, mr, cm and percent. The standard unit for the momentum p(O)

is ~eV /c. Also if a beam line coordiante layout is desired, the card specifying that a

layout is to be made (a 13.0 12.0 element), and any initial coordinates (see Type Code

16.0) all precede the BEAM card.

There are eight entries (all positive) to be made on the BEAM card.

1 - The Type Code 1.0 (specifies a BEAM entry follows).

2 - One-half the horizontal beam extent (x) (cm in standard units).

3 - One-half the horizontal beam divergence (0) (mr).

4 - One-half the vertical beam extent (y) (cm).

5 - One-half the vertical beam divergence(</>) (mr).

6 - One-half the longitudinal beam extent (l) (cm).

7 - One-half the momentum spread (6) (in units of percent lip/p).

8 - The momentum of the central trajectory [p(O)] (GeV /c).

All eight entries must be made even if they are zero (0). As for all other type

codes, the last entry must be followed by a aemicolon, dollar aign, or aateriak. Thus a

typical BEAM entry might be

•For a rotated (non-erect) phase ellipsoid input, see Type Code 12.0.

30 

1 La_bel (if desired)

1. 0.5 2. 1.3 2.5 0. 1.5 10. ' '

meaning, x = ±0.5 cm, fJ = ±2.0 mr, 1J = ±1.3 cm, </> = ±2.5 mr, l =

±0.0 cm, 6 = ±1.5 percent Ap/p, and the central momentum p(O) = 10.0 GeV/c.

The units of the tabulated matrix elements in either the first-order R or sigma

matrix or second order T matrix of a TRANSPORr print-out will correspond to the

units chosen for the BEAM card. For the above example, the R{l2) = (z/fJ) matrix

element will have the dimensions of cm/mr; and the T(236) = (fJ/y6) matrix element

will have the dimensions mr /(cm · percent) and so forth.

The longitudinal extent l is useful for pulsed beams. It inditatea the apread in

length of partidea in a pulae. It does not interact with any other component and may

be set to zero if the pulse length is not important.

·The phase ellipse (sigma matrix) beam parameters may be printed as output after

every physical element if activated by a {13. 3. ;) element. Alternatively, individual

printouts may be activated by a (13. 1. ;) element. The projection of the semi-axes

of the ellipsoid upon each of its six coordinates axes is printed in a vertical array, and

the correlations among these components indicating the phase ellipse orientations are

printed in a triangular array (see the following pages).

The phase ellipse beam matrix

The beam matrix carried in the computer has the following construction:

x (J l

x u(ll)

(J u(21) u(22)

11 u{31) u(32) u(33)

(J u(41) u(42) u( 43) u(44)

l u(51) u(52) u(53) u(54) u(55)

6 u(61) u(62) u(63) u(64) u(65) u(66)

The matrix is symmetric so that only a triangle of elements is needed.

31 

In the printed out this matrix has a somewhat different format for ease of interpretation:

z (J l

z fo (11) CM

(J fo (22) MR r(22)

11 fo (33) CM r(31) r(32)

<P fo(44) MR r(41) r(42) r(43)

l fo(55) CM r(51) r(52) r(53) r(54)

6 fo(66) PC r(61) r(62) r(63) r(64) r(65)

where

r( .. ) u(ij)

•J = [u(ii) u(jj))l/2

As a result of the fact that the u matrix is positive definite, the r(ij) satisfy the

relation

lr(ii)I < 1 .

The Cull significance of the u( ij) and the r( ij) are discussed in detail in the Appendix ("Description of Beam Matrix"). The units are always printed with the matrix.

In brief, the meaning of the fo (ii) is as follows:

fo (11) - Xmaz - the maximum (haH)-width of the beam envelope in the

x(bend)-plane at the point or the print-out.

fo(22) - 8ma.z - the maximum (haH)-angular divergence of the beam

envelope in the x(bend)-plane.

fo (33) - 1/maz - the maximum (haH)-height of the beam envelope.

fo(44) - tPma.z - the maximum (haH)-angular divergence of the beam

envelope in the y(non-bend)-plane.

fo(55) - lma.z - one-hall the longitudinal extent of the bunch of

particles.

fo(66) - 6 - the half-width (~Ap/p) of the momentum interval

being transmitted by the system.

32 

The units appearing next to the Vii (ii) in the TRANSPORT print-out are the

units chosen for coordinates z, 6, 11, t/>, land 6 = Ap/p, respectively.

To the immediate left of the listing of the beam envelope size in a TRANSPORT

print-out, there appears a column of numbers whose values will normally be zero.

These numbers are the coordinates of the centroid of the beam phase ellipse (with

respect to the initially assumed central trajectory of the system). They may become

nonzero under one of three circumstances:

1. when the misalignment (Type Code 8.0) is used,

2. when a beam centroid shift (Type Code 7.0) is used, or

3. when a second-order claculation (Type Code 17.0) is used.

To aid in the interpretation of the phase ellipse parameters listed above, an example

of an (z, 6) plane ellipse is illustrated below. For further details the reader should refer

to ~he Appendix of this report.

e

CENTROID 1358A1

A TWO-DIMENSIONAL BEAM PHASE ELLIPSE

33 

The area of the ellipse is given by:

A= n-(det a)1

12 = n'Zmaz (Jint = 'Jr'Zint 9maz = 1rl

The equation of the ellipse is:

where

a = [ ::: :: ] = < [ _~ -~ ]

and

Q Q

r21 = r12 =- =--- ,/1 + cr2 ../Ft

r.m.s. addition to the BEAM

To allow for physical phenomena such as multiple scattering, provision has been

made in the program to permit an r .m.s. addition to the beam envelope. There are

nine entries to be included:

1 - Type Code 1.0 (specifying a BEAM entry follows).

2 - The r.m.s. addition to the horizontal beam extent (Az) (cm).

3 - The r.m.s. addition to the horizontal beam divergence (AO) (mr).

4 - The r.m.s. addition to the vertical beam extent (Ay) (cm).

5 - The r.m.s. addition to the vertical beam divergence (At/>) (mr).

6 - The r.m.s. longitudinal beam extent (Al) (cm).

7 - The r.m.s. momentum spread (A6) (in percent Ap/p).

8 - The momentum change in the central trajectory (Ap(O)] in (GeV /c).

g - The code digit 0. indicating an r.m.s. addition to the BEAM is

being made.

34 

The units for the r .m.s. addition are the same as those selected for a regular BEAM

Type Code 1.0 entry. Thus a typical r.m.s. addition to the BEAM would appear as

follows:

1. .1 .2 .15 .3 0. .13 -0.1 0.

where the last entry (0.) preceding the semicolon signifies an r.m.s. addition to the

BEAM is being made and the next to the last entry indicates a central momentum

change of -0.1 GeV /c.

35 

FRINGING FIELDS AND POLE-FACE ROTRATIONS FOR

BENDING MAGNETS: Type Code 2.0

To provide for fringing fields and/or pole-face rotations on bending magnets, the

Type Code 2.0 element is used.

There are two parameters:

1 - Type Code 2.0.

2 - Angle of pole-face rotation (degrees).

The Type Code 2.0 element must either immediately precede a bending magnet

(Type Code 4.0) element (in which case it indicates an entrance fringing field and

pole-face rotation) or immediately follow a Type Code 4.0 element (exit fringing field

and pole-face rotation) with no other data entries between.• A positive sign of the

angle on either entrance or exit pole-faces corresponds to a non-bend plane focusing

act~on and bend plane defocusing action.

For example, a symmetrically oriented rectangular bending magnet whose total

bend is 10 degrees would be represented by the three entries

2. 5. 4. 2. 5.

The angle of rotation may be varied. For example, the element 2.1 5. ; would

allow the angle to vary from an initial guess of 5 degrees to a final value which would,

say, satisfy a vertical focus constraint imposed upon the system. See the Type Code

10.0 section for a complete discussion of vary codes.

Even if the pole-face rotation angle is zero, 2. 0. ; entries must be included in

the data set before and after a Type Code 4.0 entry if fringing-field effects are to be

calculated.

A single Type Code 2.0 entry that follows one bending magnet and precedes another

will be associated with the latter.

• It ia eztremelp important that no data entries 6e made between a Tupe Code 2.0

and a Tupe Code 4.0 entry. If this occurs, it may result in an incorrect matrix

multiplication in the program.and hence an incorrect physical answer. If this rule is

violated, an error message will be printed.

36 

Should it be desired to misalign such a magnet, an update element must be inserted immediately before the first type 2.0 code entry and the convention appropriate

to misalignment or a set or elements applied, since, indeed, three separate transformations are involved. See section under Type Code 8.0 for a discussion of misalignment

calculations and the section under Type Code 6.0 for a discussion of updates.

The type code signifying a rotated pole-face is 2.0. The input format is:

{Label (if desired)

2. {3. 'RO'

The units for f3 are degrees.

Pole - Face Rotation Matrix

_The first-order R matrix for a pole-face rotation used in a TRANSPORI' calculation is as follows:

l 0 0 0 0 0

tan{J l 0 0 0 0 PO

R= 0 0 l 0 0 0

0 0 _ tan{fJ-'1} l 0 0 PO

0 0 0 0 l 0

0 0 0 0 0 l

Definitions:

{J = Angle or rotation or pole-face (see figure on following page for

sign convention or {J).

Po - Bending radius of central trajectory.

g - Total gap of magnet.

t/J - Correction term resulting from spatial extent of fringing fields.•

• See SLAC-754 (page 74) for a discussion of¢.

37 

where

X1

FIELD BOUNDARIES FOR BENDING MAGNETS

CENTRAL

TRAJECTORY

The TRANSPORT sign conventions for x, {J, R and h are all positive as shown in

the figure. The positive y direction is out of the paper. Positive {J's imply transverse

focusing. Positive R's (convex curvatures) represent negative sextupole components of

strength S = (-h/2R) sec3{J. (See SLAC-75, page 71.)

• See Type Code 16.0 for input formats for g, Ki, K2 TRANSPORT entries.

38 

DRIFT: Type Code 3.0

A drift space is a field-free region through which the beam passes. There are two

parameters:

1 - Type Code 3.0 (specifying a drift length).

2 - (Effective) drift length (meters). The length of a drift space

may be varied in either first- or second-order fitting.

Typical input format for a DRIFT:

DRIFT Space Matrix

I Label (if desired)

(not to exceed four

spaces between quotes).

3. 6. 'DI'

The first-order R matrix for a drift space is as follows:

l L 0 0 0 0

0 l 0 0 0 0

0 0 l L 0 0

0 0 0 l 0 0

0 0 0 0 l 0

0 0 0 0 0 l

where

L = the length of the drift space .

The dimensions of L. are those chosen for longitudinal length via a r-= units symbol

1

J scale factor (if needed)

15. 8. ' ' ; type code entry (if used) preceding the BEAM (Type Code 1.0) card.

If no 15. 8. entry is made, the units of L. will automatically be in meters (standard

TRANSPORT units).

39 

WEDGE BENDING MAGNET: Type Code 4.0

A wedge bending magnet implies that the central trajectory of the beam enters

and exists perpendicularly to the pole-face boundaries (to include fringing-field effects

and non-perpendicular entrance or exit boundaries - see Type Codes 2.0 and 16.0).

There are four first-order parameter to be specified for the wedge magnet via Type

Code 4.0:

1 - Type Code 4.0 (specifying a wedge bending magnet).

2 - The (effective) length L of the central trajectory in meters.

3 - The central field strength B(O) in kG,

B(O) = 33.356 (p/ Po) ,

where pis the momentum in GeV /c and Po is the bending

radius of the central trajectory in meters.

4 - The field gradient (n-value, dimensionless); where n is

defined by the equation

By(x, 0, t) = By(O, O, t)(l - nhx + ... ) ,

where

h = 1/po. See SLAC-75 (page 31).4

The quantities L, B{O), and n may be varied for first-order fitting (see Type Code

10.0 for a discussion of vary codes).

The bend radius in meters and the bend angle in degrees are printed in the output.

A typical first-order TRANSPORT input for a wedge magnet is

4. L. B. n.

r-- Label (not to

! exceed four spaces)

I I •

'

If fringing field effects are to be included, a Type Code 2.0 entry must immediately

precede and follow the pertinent Type Code 4.0 entry (even if there are no pole-face

rotations). Thus a typical TRANSPORT input for a bending magnet including fringing

fields might be:

40 

1

Labels (not to exceed

four spaces) if de5ired

2. 0. ' '

4. L. B(O). n. ' '

2. 0. ' ' '

For non-zero pole-face rotations a typical data input might be

2. 10. ; 4. L. B(O). n. ; 2. 20. ;

Note that the use of labels is optional and that all data entries may be made on one

line if desired.

The sign conventions for bending magnet entries are illustrated in the following

figure. For TRANSPORT a positive bend is to the right looking in the direction of

particle travel. To represent a bend in another sense, the coordinate rotation matrix

(Type Code 20.0) should be used as follows:

A bend up is represented by rotation the (x,y) coordinates by -QO.O degrees about

the (z) beam axis as follows:

Labels (not to exceed four spaces) if desired

20. -QO. ' '

2. ,8(1). ' '

4. L. B. n. ' '

2. ,8(2). ' '

20. +QO. ' ' (returns coordinates to their initial orientation)

A bend down is accomplished via:

20. +QO. ' ' '

2.

4.

2.

20. -QO. ' '

41 

A bend to the left (looking in the direction of beam travel) is accomplished by

rotating the x,y coordinates by 180 degrees, e.g.

20.

2.

4.

2.

20.

+180. ' ' '

-180. ' '

FIELD BOUNDARIES FOR BENDING MAGNETS

CENTRAL

TRAJECTORY

The TRANSPORT sign conventions for x, /3, R and h are all positive as shown in

the figure. The positive y direction is out of the paper. Positive /3's imply transverse

focusing. Positive R's (convex curvatures) represent negative sextupole components of

strength S = (- h/2R) sec2{3. (See SLAC-75, page 71.)

42 

First - Order Wedge Bending Magnet Matrix

cos kxL -J; sin kxL 0 0

-kx sin kxL cos kxL 0 0

0 0 cos kyL #;sin kyL ,

0 0 -ky sin kyL cos kyL

-#;sin kxL - ~ ( 1 - cos kxL) 0 0

0 0 0 0

Definitions: h - 1/po, ki = (1- n)h2, ki = nh2•

a - hL = the angle of bend.

L - path length of the central trajectory.

0 ~ ( 1 - cos kxL)

0 #;sin kxL

0 0

0 0

1 -~(kxL-sin kxL)

0 1

The field expansion for the midplane of a bending magnet is taken from Eq. (18)

page of 31 of SLAC-75, thereby defining the dimensionless quantities n and f3 as follows:

By(x, 0, t) = By(O, 0, t) (1- nhx + /3h 2x2 +1h3x3 + ... )

The type code signifying a BEND is 4.0. The input format for a TRANSPORT

calculation is:

l Label (not to exceed four spaces)

4. L. B. n. ' '

If n is not included in the data entry, the program assumes it to be zero. A f3 entry

for a second-order calculation is made via the 16.0 1.0 element. (Do not confuse this

/3 with a pole-face rotation.}

The standard units for L and Bare meters and kG. If desired, these units may be

changed by 15.0 8.0 and 15.0 g_o type code entries preceding the BEAM Card.

43 

LJ N

y

t ty .. 0 x 0 x

w--.

s N s

DIPOLE QUADRUPOLE SEXTUPOLE

Illustration of the mangetic midplane (x axis) for dipole, quadrupole and sextupole

elements. The magnet polarities indicate multipole elements that are positive with

respect to each other.

44 

QUADRUPOLE: Type Code 5.0

A quadrupole provides focusing in one transverse plane and defocusing in the other.

There are four parameters to be specified for a TRANSPORT calculation:

I - Type Code 5.0 (specifying a quadrupole).

2 - (Effective) magnet length L (in meters).

3 - Field at pole tip B (in kG). A positive field implies horizontal

focusing; a negative field, vertical focusing.

4 - Half-aperture a (in cm). Radius of the circle tangent to

the pole tips.

The length and field of a quadrupole may be varied in first-order fitting. The

aperture may not be.

The strength of the quadrupole is computed from its field, aperture and length.

The. horizontal focal length is printed in parentheses as output. A positive focal length

indicates horizontal focusing and a negative focal length indicates horizontal defocusing. The quantity actually printed is the reciprocal of the (O/x) transfer matrix element

(I/R21) for the quadrupoles. Thus two identical quadrupoles of opposite polarity will

have different horizontal focal lengths due to the difference between the sine and the

hyperbolic sine.

The type code for a QUAD is 5.0. The input format for a typical data set is:

5. L. B. a.

I Label (if desired)

not to exceed four

spaces between quotes

' '

The standard TRANSPORT units for L, B and a are meters, kG and cm, respectively.

If other units are desired they must be chosen via the appropriate 15.0 Type Code

entries preceding the BEAM (Type Code 1.0) card.

45 

'

First - Order Quadrupole Matrix

cos kqL J; sin kqL 0 0 0 0

f

-kq sin kqL cos kqL 0 0 0 0

0 0 coshkqL J; sinh kqL 0 0

f

0 0 kq sinhkqL coshkqL 0 0

0 0 0 0 1 0

0 0 0 0 0 1

These elements are for a quadrupole which focuses in the horizontal (x) plane (B

positive). A vertically (y-plane) focusing quadrupole (B negative) has the first two

diagonal submatrices interchanges.

Definitions: L - the effective length of the quadrupole.

a - the radius of the aperture.

Bo - the field at the radius a.

k~ - (Bo/a)(l/BoPo), where (Bpo) =the magnetic rigidity

(momentum) of the central trajectory.

46 

N

y

y

g 0 x 0 x

s N S

DIPOLE QUADRUPOLE SEXTUPOLE

Illustration of the magnetic midplane (x a.xis) for dipole, quadrupole and sextupole

elements. The magnet polarities indicate multipole elements that are positive with

respect to each other.

47 

TRANSFORM 1 UPDATE*: Type Code_ 6.0.1.

To re-initialize the matrix TRANSFORM I (the product of the R matrices, RI)

use Type Code 6.0. A (6. 0. 1. ;) card effects an update of the RI matrix and

initiates the accumulation of a new product matrix at the point of the update. This

facility is often useful for misaligning a set of magnets or fitting only a portion of a

system.

The matrix RI is updated by no other element. It is not used in the calculation of

the beam matrix. The beam matrix is claculated from the auxiliary transfer matrix

R2 described on the next page.

A TRANSFORM I matrix will be printed at any position in the data set where a

(I3. 4. ;) entry is inserted.

See the following section for the introduction of an auxiliary transformation matrix

R2 _(TRANSFORM 2) to avoid the need for TRANSFORM I updates.

The (6. 0. 1. ;) card also causes an update of the R2 matrix.

• By "updating" we mean initiating a new starting point for the accumulation (multiplication) of the R matrix. At the point of update the previos accumulation is

discontinued. When the next element possessing a transfer matrix is encountered, the

accumulated transfer matrix RI is set equal to the individual transfer matrix R for

that element. Accumulation is resumed thereafter.

48 

AUXILIARY TRANSFORMATION MATRIX (R2):

Type Code 6.0.2.

The RI matrix represents the accumulated transfer matrix from either the beginning of the beam line or the last explicit RI update (6. 0. 1. ;). However several

elements in TRANSPORT which affect the beam matrix cannot be represented in any

transfer matrix. To avoid update complications with RI an auxiliary transfer matrix

R2 exists. To avoid update complications with RI an auxiliary transfer matrix R2

exists. The beam matrix is then calculated from the R2 matrix and the beam matrix

at the last R2 update.

Both the RI and R2 matrices are normally available for printing. However there is

no redundancy in computer use, since, internally to the program, only R2 is calculated

at each element. The matrix RI is calculated from R2 only as needed.

The R2 matrix is updated explicitly via a (6. 0. 2. ;) entry. It may be printed by a

(I3.- 24. ;) entry. Constraints on R2 are imposed similarly to those on RI. For details

see the section describing Type Code IO.O.

The complete list of elements which update TRNASFORM 2 is:

1. A beam Type Code 1.0 entry.

2. The (6. 0. 1. ;) entry.

3. The (6. 0. 2. ;) entry.

4. A centroid shift Type Code 7 .0 entry.

5. A misalignment Type Code 8.0 entry.

6. A stray field Type Code 21.0 entry.

Please note that automatic updates of TRANSFORM 2 occur when an align element (Type Code 8.0) is inserted specifying the misalignment of all subsequent bending

magnets. These TRANSFORM 2 updates take place immediately before and immediately after any bending magnet which has either the entrance or exit fringe fields

specified via a Type Code 2 entry.

49 

SHIFT IN THE BEAM CENTROID: Type Code 7 .0

Sometimes it is convenient to redefine the beam centroid * such that it does not

coincide with the TRANSPORT reference trajectory. Provision has been made for

this possibility via Type Code 7.0. Seven parameters are required:

1 - Type Code 7.0.

(2 to 7) - The coordinates x, 0, </J, l, and o defining the shift in

the location of the beam centroid with respect to its

previous position. The units for x, </J, y, </J, l, /j are the

same as those chosen for the BEAM (Type Code 1.0

entry), normally cm, mr, cm, mr, cm, and percent.

Any or all of the six beam centroid shift parameters may be varied in first-order

fitting. The centroid position may then be constrained at any later point in the beam

line by this procedure .

. The transformation matrix R2 is updated by this element.

In order for this code to function properly, the initial BEAM entry (Type Code

1.0) must have a nonzero phase space volume, for example a

1. 0. 0. 0. 0. 0. 0. p(O).

BEAM entry is not permissible when calculating a shift in the beam centroid, whereas

a

1. 1. 1. 1. 1. 1. 1. p(O).

entry (nonzero phase volume) is acceptable.

*By "beam centroid" we mean the center of the beam ellipsoid.

50 

MAGNET ALIGNMENT TOLERANCES: Type Code 8.0

The first-order effects of the misalignment of a magnet or group of magnets are a

shift in the centroid of the beam and a change in the beam focusing characteristics.

Two varieties of misalignment are commonly encountered: ( 1) the magnet is displaced

and/or rotated by a known amount; or (2) the actual position of the magnet is

uncertain within a given tolerance. TRANSPORT has the capability of simulating

the misalignment or either single magnets or entire sections or a beam line. Any

combination or the above alternatives may be simulated through the use or the "align"

element. The results may be displayed in either the printed output of the beam (sigma)

matrix or tabulated in a special misalignment table (described below).

There are eight parameters to be specified:

1 - Type Code 8.0 (specifying a misalignment).

2 - The magnet displacement in the horizontal direction (cm).

3 - A rotation about the horizontal a.xis (mr).

4 - A displacement in the vertical direction (cm).

5 - A rotation about the vertical a.xis (mr).

6 A displacement in the beam direction (cm).

7 - A rotation about the beam direction (mr).

8 - A three-digit code number (defined below)

specifying the type or misalignment.

The three displacements and three rotations comprise the six degrees of freedom

of a rigid body and are used as the six misalignment coordinates. The coordinate

system employed is that to which the beam is referred at the point it enters the magnet.

For example, a rotation of a bending magnet about the beam direction (parameter

7 above) is referred to the direction of the beam where it enters the magnet. The

units employed are the standard TRANSPORT units shown above, unless redefined

by Type Code 15.0 entries. Ir the units are changed, the units of the misalignment

displacements are those determined by the 15. 1. type code entry; the units for the

misalignment rotations are those determined by the 15. 2. type code entry.

The misalignment of any physical element or section of a beam line may be simulated. Misaligned sections of a beam line may be nested. A beam line rotation (Type

Code 20.0) may be included in a misaligned section. Thus, for example, one can sim51 

ulate the misalignment or magnets that bend vertically. The arbitrary matrix (Type

Code LI.OJ may not be intluded in a miaaligned section. A misalignment must never

be included in a second-order run (Type code 17.0}.

A misalignment element may indicate that a single magnet or section or the beam

line is to be misaligned, or it may indicate that all subsequent magnets or a given type

(quadrupoles and/or bending magnets) are to be misaligned. The type or misalignment

is specified in the three-digit code number, and the location or the Type Code 8.0 align

element depends on the type or misalignment.

If a misalignment pertains to a single magnet or a single section or the beam

line, then the misalignment element (Type Code 8.0) must directly follow that magnet

or section of the beam line. If a misalignment element indicates that all subsequent

magnets of a given type are to be misaligned, it must precede the first of such magnets.

Further description or the available types of misalignment is given in the table below.

The results of the misalignment may be displayed in either the beam (sigma) matrix

or in a misalignment table. If the results are displayed in the beam (sigma) matrix,

then that matrix is altered by the effects of the misalignment. The effects of additional

misalignments cause further alterations, so that at any point along the beam line, the

beam (sigma) matrix will contain the combined effects of all previous misalignments.

The misalignment table can be used to show independently the effect on the beam

matrix of a misalignment in each degree of freedom of each misaligned magnet. Each

new misalignment to be entered in the table creates a new set of six duplicates of the

beam matrix. Printed for each duplicate beam matrix are the centroid displacement

and the beam half width in each or the six beam coordinates. Each of the six matrices shows the combined result of the undisturbed beam matrix and the effect of the

misalignment in a single coordinate of a single magnet or section of the beam line. In

a single TRANSPORT run the results or misaligning up to ten magnets or sections of

the beam line may be included in the misalignment table. Further requests for entry

in the misalignment table will be ignored. Examples of such a table and the input

which generated it are shown below.

When the user specifies that the actual position or the magnet(s) is uncertain

within a given tolerance, the printout will show a change in the beam (sigma) matrix

52 

resulting from the effects of the misalignment(s). Thus, if one wishes to determine the

uncertainty in the beam centroid resulting from uncertainties in the positioning of the

magnets, the initial beam dimensions should be set to zero, i.e., the beam card entry

at the beginning of the system should appear as follows:

1. 0. 0. 0. 0. 0. Q p(O).

If it is desired to know the effect of an uncertainty in position on the beam focusing

characteristics, then a nonzero initial phase space must be specified. The printout will

then show the envelope of all possible rays, including both the original beam and the

effects of the misalignment.

If the misalignment is a known amount, it may affect the beam centroid as well

as the beam dimensions. Therefore one should place on the BEAM card the actual

dimensions of the beam entering the system. For a known misalignement, the program

requires that the initial beam specified by Type Code 1.0 must be given a nonzero phase

volume, to insure a correct printout.

An align element pertaining to a single magnet or section of the beam line updates

the BEAM (sigma) matrix and the R2 matrix, but not the RI matrix. A misalignment

element which indicates misalignment of all subsequent magnets of a given type will

update the BEAM (sigma) matrix and the R2 matrix before each bending magnet with

fringe fields and after each misaligned magnet of any type.

The tolerances may be varied. Thus, type-vary code 8.111111 permits any of

the six parameters (2 through 7 above) to be adjusted to satisfy whatever BEAM

constraints may follow. For fitting, a misalignment must pertain to a single magnet or

single section of the beam line, and the results must be displayed in the beam (sigma)

matrix. (See the section under Type Code 10.0 for a discussion of the use of vary

codes.)

The meaning of the options for each digit of the three-digit code number is given

in the following table.

53 

A. The units position specifies the magnet(s) or section of the beam line to be

misaligned.

Code Number

:xx:o

:XX:I

:XX:2

Interpretation

The single magnet (type code element) immediately preceding

the align card is to be misaligned. A bending magnet with

fringe fields should be misaligned using one of the options

described below.

The last RI matrix update (the start of the beam line or a

6. 0. 1. ; type code entry) marks the beginning of the

section to be misaligned. The misalignment element itself

marks the end. The section is treated as a unit and misaligned as a whole. The misalignments of quadrupole triplets

and other combinations involving more than two quadrupoles

may be studied using this code digit.

The last R2 matrix update (see Type Code 6.0 for a list of

elements which update R2) marks the beginning of the

misaligned section. The misalignedment element marks

the end. This options makes use of the fact that R2 matrix

updates do not affect the RI matrix.

A bending magnet with fringing fields or pole face rotations

(Type Code 2.0) should be misaligned using this option. See

examples I and 2 below for an illustration of this. An array

of quadrupoles provides another example of the use of this

option. By successive application of align elements, the

' elements of a quadrupole triplet could be misaligned

relative to each other, and then the triplet as a whole

could be misaligned. See example 3 below for an illustration

of this.

54 

Code Number

xxa

XX4

XX5

Interpretation·

An array of quadrupoles provides anotehr example of the

use of this option. By successive application of align

elements, the elements of a quadrupole triplet could be

misaligned relative to each other, and then the triplet

as a whole could be misaligned. See example 3 below for

an illustration of this.

All subsequent bending magnets and quadrupoles are

independently misaligned by the amount specified. This

option is useful in conjunction with the tabular display

of the misalignment results (see below). A bending magent,

with fringing fields included, is treated as a single unit

and misaligned accordingly.

All subsequent bending magnets, including fringing fields,

are independently misaligned by the amount specified.

See XX3 above for further comments.

All subsequent quadrupoles are independently misaligned

by the amount specified. See example 4 below for an

illustration of this. See XX3 above for further comments.

55 

B. The tens position defines the mode of display of the results of the misalignment.

Code Number

xox

XIX

Inter retation

The beam matrix contains the results of the misalignment.

The beam matrix is printed wherever a 13. 1. ; card

is encountered. The beam matrix will then contain

contributions from all previous misalignments.

A table is used to store the results of misalignments.

The effect of up to ten independently misaligned magnets

may be shown in the table in a single run. The table is

printed via a 13. 8. ; card, and may be compared with

the undisturbed beam matrix (printed by a 13. 1. ; card)

at any point. An example of such a table is shown below.

C. The hundreds position distinguishes between an uncertainty in position (OXX:)

or a known displacement (lXX).

Any combination of digits may be used to define the exact circumstances intended.

Thus, code 111. indicates the deliberate displacement of a set of magnets (referred to

the point where the beam enters the set), with the results to be stored in a table.

56 

Example No. 1: A Bending Magnet with a Known Misalignment

A bending magnet (including fringe fields) misaligned by a known amount might

be represented as follows:

3. L(l).

6. 0. 2.

2. 0. 4. 1. B. n. 2. 0.

8. 0. 0. 0. 0. 0. 2. 102.

3. L(2).

This represents a known rotation of the bending magnet about the incoming beam

direction (z axis) by 2.0 mr. The result of this misalignment will be a definite shift in

the beam centroid, and a mixing of the horizontal and vertical coordinates. The use

of the 6. 0. 2. ; transform two update and the misalignment code number XX:2 is

necessary because the magnetic array (bending magnet + fringing fields) consists of

three type code elements instead of one.

57 

Example No. 2: A Bending Magnet with an Uncertain Position

A bending magnet having an uncertainty of 2 mrad in its angular positioning about

the incoming beam (z axis) would be represented as follows:

3. L(l).

6. 0. 2.

2. 0. 4. L. B. n. 2. 0.

8. 0. 0. 0. 0. 0. 2.0 002.

3. L(2).

To observe the uncertainty in the location of the outcoming beam centroid, the

input BEAM card should have zero phase space dimensions as follows:

1. 0. 0. 0. 0. 0. 0. p(O).

·If the beam dimensions on the input BEAM card are nonzero, the resultant beam

(sigma) matrix will show the envelope of possible rays, including both the input beam

and the effect of the misalignment.

58 

Example No. 3: A Misaligned Quadrupole Triplet

One typical use of both the RI and R2 matrices is to permit the misalignment of

a triplet. For example, an uncertainty in the p~itions within the following triplet:

5. 1. -8. 10.

4. 2. 7. 10.

5. 1. -8. 10.

may be induced by appropriate 8. element as noted:

6. 0. 1.

5. 1. -8. 10.

6. 0. 2.

5. 2. 7. 10.

5. 1. -8. 10.

8. 000.

8. 002.

8. 001.

The first 8. card in the list refers to the misalignment of the third magnet only.

The second 8. card refers to the misalignment of the second and third magnets as a

single unit via the R2 matrix update (the 6. 0. 2. ; entry). The last 8. card refers

to the misalignment of the whole triplet as a single unit via the Rl matrix update (the

6. 0. 1. ; entry).

The comments about the BEAM card (Type Code 1.0 entry) in Example 2 above

are applicable here also.

59 

Example No. 4: Misaligned Quadrupoles in a Triplet

Individual uncertainties in the positions of the quadrupoles in the triplet in Example 3 above may be induced by a single misalignment as follows:

8. --- --- 015. '

5. 1. -8. 10.

5. 2. 7. 10.

5. 1. -8. 10.

The effect of each misalignment coordinate on each quadrupole will be stored

separately in a table. This table is printed wherever a 13. 8. ; type code is inserted.

60 

This Page Left Blank

61 

"RECOMBINED MODE IIlGH RESOLUTION BEAM"

0

lS.

15.

16 .oo

8.000000

1.000000

J.o

20.0

4.ooo

20.0

J.o

20.0

4.ooo

20.0

3.0

5.oo

J.o

13.

13.

J.o

s.oo

3.0

13.

13.

J.o

20.0

2.0

4.ooo

2.0

20.0

3.0

20.0

2.0

4.ooo

2.0

20.0

J.o

20.0

2.0

4.ooo

2.0

20.0

13.

13.

3.o

5.oo

3.0

5.oo

3.0

13.

13.

13.

"Bl

"82

"Ql

"C2A "

"Q2

"C2B "

"Bl

"B4

"B5

"Q3

"Q4

"Cl

more beam line

1.00000

8.00000

19.00000

0.02000

0.05000

80. 74998;

180.00000;

I0.25000

-180.00000;

1.75000;

180.00000;

10.25000

-180.00000;

27.68999;

10.00000

7.70000;

1.00000;

4.ooooo;

3. 60000;

10.00000

5.ooooo;

1.00000;

4.00000;

1.50000;

180.00000;

8.53000;

20.00000

8.53000;

-180. 00000;

1.50000;

180.00000;

8.SJooo;

20.00000

8.53000;

-180.00000;

1. 50000;

180.00000;

8.53000;

20.00000

8 .SJOOO;

-180.00000;

1.00000;

4.00000;

5.50000;

10.00000

2.00000;

10.00000

289.83960;

1.00000;

"Ml

"FT

2.50000;

0 .10000

o.56200

9.18192

9.18192

-4.00060

3.45510

11.66998

18.66998

11.66998

3.51760

-3.64760

0.05730

o.3o48D

0.02000

0.0500<'

o.o

o.o

2 .54000;

2.S4000;

o.o

o.o

o.o

2.54000;

2.54000;

8. DODOO; -- print lnlltruction for

4. DODOO; mlHUgn1tent table

\ Including more quadrupolea and

miaallgnment table print instructions If desired

SEITillL

0 .10000

0.86700

0.10000

o.o

quadrupolee to be misaligned

1. DODOO

o.o 15 .00000; --mlHllgnment

200.00000; element

~ for a ~ Table. Shown la tbe input for a mlaallgnment run of

early ~ line. The mlaallgnment element apeciflea that all

subsequent quadrupolea are to be given an uncertain mlaallgnment by tbe amount apeclfled and tbe reaulta (for up to ten quadrupole&) are to be entered in a table.

The portion of tbe output produced by tbe Indicated print lutructlon la abown In

the next figure. 

548.829 FT o.o 0.087 CH o.o 0.329 Mil 0.035 o.o 0.168 CH -o.ooo o.ooo o.o o.338 Mil o.ooo o.ooo -0.648 o.o 0.209 CH -0.010 -1.000 -o.ooo o.ooo o.o o.o PC o.o o.o o.o o.o o.o *HISALIGllHENT EPPECT TABLE POil HISALIGNHEllTS OP

0.020 CH 0.100 Mil 0.020 CH 0.100 HR O.IOO CH 1.000 HR

*HISALIGNHEllT OP QI * o.o 0.140 CH o.o 0.085 CH o.o 0.085 CH o.o 0.119 CH o.o 0.085 CH o.o 0.092 CH o.o 0.330 Mil o.o o.329 HR o.o 0.329 Mil o.o o.330 HR o.o o.329 HR o.o 0.329 Ill o.o 0.168 CH o.o 0.202 Cll o.o 0.221 CH o.o o.168 Cll o.o 0.168 Cll o.o 0.112 Cll o.o 0.338 !Ill o.o o.338 !Ill o.o o.338 HR o.o 0.338 Ml o.o 0.338 HR o.o o.338 HR o.o 0.209 CH o.o 0.209 CH o.o 0.209 CH o.o 0.209 Cll o.o 0.209 Cll o.o 0 .209 Cll o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC

*llISALIGllHEllT OP Q2 • o.o 0 .163 CH o.o 0.089 CH o.o 0.089 CH o.o o.139 CH o.o o.089 Cll o.o 0.095 Cll o.o o.33o Mil o.o o.329 HR o.o o.J29 HR o.o o.330 Ml o.o o.329 HR o.o 0.329 Ill o.o o.168 CH o.o o.183 CH o.o o.193 CH o.o o.168 Cll o.o 0.168 CH o.o 0.172 CH o.o o.338 HR o.o o.338 HR o.o o.338 llR o.o o.338 HR o.o o.338 HI o.o 0.338 Ill o.o 0.209 CH o.o 0.209 CM o.o 0.209 Cll o.o 0.209 Cll o.o 0.209 CH o.o 0.209 CH o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC *HISALIGllHENT OP Ql • o.o 0 .166 CH o.o o.087 Cll o.o 0.081 CH o.o o.136 CM o.o 0.087 Cll o.o 0.095 CH o.o o.llo llR o.o o.329 HR o.o o.329 !Ill o.o 0.330 HR o.o o.329 HI o.o 0.329 MR o.o o.168 CH o.o o.184 CH o.o 0.193 CH o.o 0 .168 Cll o.o 0.168 CH o.o 0 .112 CH o.o o.338 HI o.o o.338 HI o.o o.338 HR o.o o.338 !Ill o.o o.338 Mil o.o 0.338 !Ill

C> o.o 0.209 Cll o.o 0.209 CH o.o 0.209 CH o.o 0.209 CH o.o 0.209 CH o.o 0 .209 Cll ~ o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC

*llISALIGllHEllT OF Q4 • o.o 0.150 CH o.o 0.081 Cll o.o 0.087 CH o.o o.126 CH o.o 0.081 CH o.o 0.095 Cll o.o o.llo HI o.o o.329 HR o.o o.329 Ill o.o o.llo HR o.o o.329 !Ill o.o o.329 HR o.o 0.168 CH o.o 0.190 CH o.o 0.203 CH o.o o.168 CH o.o o.168 Cll o.o 0.172 CH o.o o.338 HR o.o o.338 HI o.o o.338 HR o.o o.338 llR o.o o.338 HR o.o o.338 HR o.o 0.209 CH o.o 0.209 CH o.o 0.209 CH o.o 0.209 CH o.o 0 .209 Cll o.o 0.209 CH o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC o.o o.o PC

*TRAllSFOIH I* "Cl " -1.70691 0.00113 -0.00000 -0.00000 o.o -6.33738

-0.27910 -0.58566 -0.00000 -0.00000 o.o -o.59894 Examf.le of a Misalignment TU>le. Tiie mlaalfpm'Jnt table, the unperturbed -0.00000 -0.00000 -2.49643 0.13068 o.o 0.00000 comp ete beaiii matrix, and ilie flr.t-order tranafer matrix are all abown at -0.00000 -o .00000 -0.21183 -0.38917 o.o 0.00000 the same point 1n the beam ltne. The Dli8alfgnment element (Dot abown) hu

0.07464 0.37123 -0.00000 -0.00000 1.00000 -0.02251 Indicated an uncertain ml8allgnment, so the beam cemrotd ta umffected. Tbe o.o o.o o.o o.o o.o 1.00000 llUlpltudea of the mtaaligmnenta tn each coordinate are abown above the *LEllGTB* 548.82861 PT colUDlllll to which they partmtn. Therenltaof~•~ymtaallgnlngmcll

magnet are tncllcated by the label for that magnet. 

REPETITION: Type Code 0.0

Many systems include a set of elements that are repeated several times. To minimize the chore of input preparation, a 'repeat' facility is provided.

There are two parameters:

1 - Type Code Q.O

2 - Code digit. H nonzero, it states the number of repeitions

desired from the point if appears. H zero it marks the end

of a repeating unit.

For example, a total bend of 12 degrees composed of four 3-degree bending magnets

each separated by 0.5 metres could be represented by Q. 4. ; 4. - - - ; 3. .5 ; 9.

0. ; Those elements (in this case a bend and drift) between the 9. 4. ; and 9. 0. ;

would be employed four times.

There is no indication of the Q.O cards in the printed TRANSPORT output when

calculating except for the repeated listing of the elements they control.

Vary codes may be used within a repeating unit in the usual fashion. However all

repetitions of a given varied element will be coupled.

Repeat cards may be nested four deep. By "nesting" we mean a repeat within a

repeat. An example is given below.

64 

Example of Nesting

9. 2.

3. 10.

9. 3.

3. 20.

9. 4.

3. 50. ~Inner Block

9. o.

Next Inner Block Outer Block

9. o.

3. 1.5

Q. 0.

The total length of this sequence is:

2*(10. + 3*(20 + 4• 50) + 1.5) = 1343

65 

VARY CODE AND FITTING CONSTRAINTS: Type Code 10.0

Some (not all) of the physical parameters of the elements comprising a beam line

may be varied in order to fit selected matrix elements. In a. first-order calculation

one might fit elements of the RI or R2 transformation matrices or the beam (sigma)

matrix. In second order one might constrain a.n element of the second-order matrix Tl

or minimize the net contribution of aberrations to a given beam coordinate. Special

constraints are also available.

One may not mix orders in fitting. First order vary codes and constraints must be

inserted only in a first-order calculation, and similarly for second order.

The physical parameters to be varied are selected via 'Vary Codes' attached to

the type codes of the elements comprising the system. The fitting constraints on

matrix elements are selected via Type Code 10.0 entries placed in the system where

the constraint is to be imposed.

Vary Codes

Associated with each physical element in a system is a vary code which specifies

which physical parameters of the element may be varied. This code occupies the

fraction portion of the type code specifying the element. It has one digit for each

parameter, the digits having the same order in the code as the physical parameters

have on the card. A 'O' indicates the parameter may not be varied; a 'l' that it may

be. For instance, 3.0 is the combined type (3) and vary code (0) for a drift length

which is to remain fixed; 3.1 indicates a drift legnth that may be varied (by the virtue

of the .1). The Type Code 4.010 indicates a bending magnet with a variable magnetic

field. In punching the code 3.0, the zero need not be punched. In punching the 4.010

code, the first zero must be punched but the second zero need not be.

66 

First - Order Vary Codes

In a first-order run the following parameters marked v may be varied, those marked

0 may not be varied.

BEAM ...

R.M.S. ADDITION ...

ROTAT ...

DRIFT ...

BEND ...

QUAD ...

l.vvvvvvO - All components of the input beam

may be varied, except the momentum.

l.vvvvvvOO - All components of an r.m.s. addition

may be varied except the momentum change ~p.

2. v - The pole face angle of a bending manget

may be varied.

3.v - The drift length may be varied.

4.vvv - The length, the field, and/or the n-value

may be varied.

5.vvO - The length may be varied; the field may be,

the aperture may not be.

AXIS SIIlFT . . . 7. vvvvvv - Any of the axis shift parameters

may be varied.

ALIGN... 8.vvvvvvO -Any of the alignment parameters

may be varied.

INITIAL COORDINATES . . . 16.0v - Any of the three initial position floor

coordinates or two angle coordinates may be varied.

MATRIX... 14.vvvvvvO -Any of the first order matrix elements

may be varied.

SOLENOID . . . IQ.vv - The length and/or field may be varied.

BEAM ROTATION... 20.v - The angle of rotation may be varied.

The use of the permissive 'may' rather than the imperative 'will' in discussing

variables is meaningful. The program will choose the parameters it will vary from

among those that it may vary. In general it chooses to vary those parameters that

have the greatest influence upon the conditions to be fit.

67 

Second-Order Vary Codes

In a second-order run the following parameters may be varied:

DRIFT ...

f(l) ...

3.v - The drift length may be varied. Variation of a drift length

should be done with caution as it may affect the first-order

properties of the beam line. But inversely coupled drift spaces

straddling a sextupole will, for example, show only second-order

effects.

16.0v 1. - The normalized quadratic term (sextupole component)

in the midplane expansion for the field or a bending magnet

may be varied.

1/Rl . . . 16.0v 12. - The pole face curvature of a bending magnet entrance

may be varied.

1/R2 . . . 16.0v 13. - The pole face curvature of a bending magnet exit

may be varied.

SEXTUPOLE 18.0v - The field strength may be varied.

The special parameter cards (Type Code 16.0) once introduced apply to all subsequent magnets in a beam line until another Type Code 16.0 specifying the same

parameter is encountered. Thus if such a parameter is varied, the variation will apply simultaneously to all subsequent magnets to which it pertains. The variation will

persist until the parameter or vary code attached to the parameter is changed by the

introduction of another Type Code 16.0 card specifying the same parameter.

Coupled Vary Codes

It is possible to apply the same correction to each of several variables. This may

be done by replacing the digit 1 in the vary code with one of the digits 2 through g, or

a letter A through Z. All such variables whose vary digits are the same, regardless of

position will receive the same correction. For example, the three type-vary codes (5.0A,

5.01, 5.0A) might represent a symmetric triplet. The same correction will be made to

the first and third quadrupoles, guaranteeing that the triplet will remain symmetric.

68 

If a vary digit is immediately preceded by a minus sign, the computed correction

will be subtracted from, rather than added to, this variable. Thus parameters with the

same vary digit, one of them being preceded by a minum sign, will be inversely coupled.

For example the type-vary code sequence (3.B, 5.01, 3.-B) will allow the quadrupole to

move without altering the total system length.

Vary digits may also be immediately preceded by a plus sign without changing

their meaning. Thus 5.0A is the same as 5.0+A. For historical reasons, the vary digits

(Q and 4), (8 and 3), and (7 and 2) are also inversely coupled. Inverse coupling may

not be used with type codes 1.0 or 8.0.

The total number of independent variables in a first-order run is limited to 20 by

reasons of the mathematical method of fitting and to 10 for a second-order run. So

far as this limit is concerned, variables that are tied together count as one. Variables

within repeat elements (Type Code 0.0) also count only one.

Possible Fitting Constraints

A variety of possible constraints is available. Fitting may be done in either firstor second-order, but not in both simultaneously. The order of the constraint must

be appropriate to the order of the run. A list of constraints available is given below.

They are explained more fully on later pages.

First - order constraints

1. An element of the first-order transfer matrix RI.

2. An element of the auxiliary first-order matrix R2.

3. A u (BEAM) matrix element.

4. The correlations r in the beam coordinates.

5. The first moments of the beam.

6. The total system length.

7. An AGS machine constraint.

8. The reference trajectory ftoor coordinates.

60 

Second - order constraints

1. An element of the second-order transfer matrix Tl.

2. An element of the second-order auxiliary transfer matrix T2.

3. The net contributions of aberrations to a given corrdinate of the beam matrix

u.

4. The strength of sextupoles used in the system.

The second-order matrices are actually computed using the auxiliary matrix T2.

Therefore, when activating second-order fitting, one must not include any element

which causes an update of the R2 matrix. For a complete list of such elements see

Type Code 6.0.

The present value of the constrained quantity, as well as the desired value, is

printed in the output. In the case of transfer matrix elements this value may be

chec~ed by printing the transfer matrix itself. Certain other constrained quantities

may be checked similarly. Exceptions are noted in the explanations following.

Rl matrix fitting cosntraints

There are five parameters to be specified when imposing a constraint upon the (i,j)

element of an Rl matrix.

1 - Type Code 10.n (specifying that a fitting constraint follows).

2 - Code digit (-i).

3 - Code digit (j ).

4 - Desired value of the (i,j) matrix element.

5 - Desired accuracy of fit (standard deviation).

Note that any fitting constraint on an Rl matrix element is from the preceding

update of the Rl matrix. An Rl matrix is updated only by a (6. 0. 1. ;) entry.

The symbol (n) is normally zero or blank. If n = 1, then entry 4 is taken to be a

lower limit on the matrix element. If n = 2, entry 4 is taken to be an upper limit.

70 

Some typical Rl matrix constraints are as follows:

Desired Optional Condition Typical Fitting Constraint

Point to point imaging:

Horizontal plane R(l2) = 0 10. -1. 2. 0. .0001 'Fl' .

,

Vertical plane R(34) = 0 10. -3. 4. 0. .0001 'F2' .

,

Parallel to point focus:

Horizontal plane R(ll) = O 10. -1. 1. 0 . . 0001 'F3' '

Vertical plane R(33) = O 10. -3. 3. 0 . . 0001 'F4' .

'

Point to parallel transformation:

Horizontal plane R(22) = 0 10. -2. 2. 0. .0001 'F5' .

'

Vertical plane R(44) = O 10. -4. 4. 0. .0001 'F6' .

'

Achromatic beam:

Horizontal plane 10. -1. 6. 0. .0001 'F7' '

R( 16) = R(26) = O 10. -2. 6. 0. .0001 'F8' .

'

Zero dispersion beam:

Horizontal plane R(16) = 0 10. -1. 6. 0 . . 0001 'FQ' .

'

Simultaneous point to point and

waist to waist imaging:

Horizontal plane 10. -1. 2. 0. .0001 'FlO' '

R(12) = R(21) = 0 10. -2. 1. 0. .0001 'Fll' '

Vertical plane 10. -3. 4. 0 . . 0001 'F12' '

R(34) = R(43) = 0 10. -4. 3. 0. .0001 'F13' '

Simultaneous parallel to point and

waist to waist transformation:

Horizontal plane 10. -1. 1. 0. .0001 'F14' '

R(ll) = R(22) = 0 10. -2. 2. 0. .0001 'Fl5' .

'

Vertical plane 10. -3. 3. 0. .0001 'F16' '

R(33) = R(44) = 0 10. -4. 4. 0. .0001 'F17' '

71 

R2 matrix fitting constraints

There are five parameters to be specified when imposing a constraint upon the (i,j)

element of an R2 matrix.

1 - Type Code 10.n

2 - Code digit -(20 + i).

3 - Code digit (j ).

4 - Desired value of the (i,j) matrix element.

5 - Desired accuracy of fit (standard deviation).

Some typical R2 matrix constraints are as follows:

The symbol (n) is normally zero or blank. If n = 1, then entry 4 is taken to be a lower

limit on the matrix element. If n = 2, entry 4 is taken to be an upper limit.

Desired Optional Condition Typical Fitting Constraint

Point to point imaging:

Horizontal plane R(12) = 0 10. -21. 2. 0. .001 'Fl' '

Vertical plane R(34) = 0 10. -23. 4. 0. .001 'F2' '

Parallel to point focus:

Horizontal plane R(ll) = 0 10. -21. 1. 0. .001 'Fl' .

'

Vertical plane R(33) = o 10. -23. 3. 0. .001 'F2' '

Achromatic beam:

Horizontal plane 10. -21. 6. 0. .001 'F3' '

R(16) = R(26) = O 10. -22. 6. 0. .001 'F4' '

See Type Code 6.0 for a complete list of elements which update the R2 matrix.

72 

u(BEAM) matrix fitting constraints

There are five parameters to be specified when imposing a constraint upon the (i,j)

element of a u(BEAM) matrix.

1 - Type Code 10.n

2 - Code digit (i). (i < j)

3 - Code digit (j ).

4 - Desired value of the (i,j) matrix element.

5 - Desired accuracy of fit (standard deviation).

The symbol (n) is normally zero or blank. If n = 1, then entry 4 is taken to be a

lower limit on the matrix element. If n = 2, entry 4 is taken to be an upper limit. If i

= j, then the value inserted in entry 4 is the desired beam size [u(ii)] 112 e.g. x(max)

= [u(l1)] 112 etc.

Some typical u matrix constraints are as follows:

Desired Optional Condition Typical Fitting Constraint

Horizontal waist u(21) - 0 10. 2. 1. 0. .001 'Fl' '

Vertical Waist u(43) - 0 10. 4. 3. 0. .001 'F2' .

'

Fit beam size to x( max) - lcm 10. 1. 1. 1. .001 'F3' .

'

Fit beam size to y(max) - 2cm 10. 3. 3. 2. .001 'F4' .

'

Limit max beam size to x - 2cm 10.2 1. 1. 2 . . 01 'F5' '

Limit min beam size to y - lcm 10.1 3. 3. 1. .01 'F6' '

In general, it will be found that achieving a satisfactory 'beam' fit with TRANSPORT is more difficult than achieving an R matrix fit. When difficulties are encountered, it is suggested that the user 'help' the program by employing sequential (step

by step) fitting procedures when setting up the data for his problem. More often than

not a "failure to fit" is caused by the user requesting the program to find a physically unrealizable solution. An often encountered example is a violation of Liouville's

theorem.

73 

Beam correlation matrix (r) fitting constraints

Five parameters are needed for a constraint on the (i,j) element of the beam correlation matrix.

1 - Type Code 10.n

2 - Code digit ( 10 + i).

3 - Code digit (j ).

4 - Desired value of the (i,j) matrix element.

5 - Desired accuracy of fit (standard deviation).

TRANSPORT does not print the beam (u) matrix directly. Instead it prints the

beam half widths and represents the off-diagonal elements by the correlation matrix.

If one wishes to fit an element of this matrix to a nonzero value it is convenient to be

able to constrain the matrix element directly.

Some typical r matrix constraints are as follows:

Desired 0 tical Condition ical Fittin Constraint

Horizontal waist r(21) = 0 10. 12. 1. 0. .001 'Fl'

yy 1 correlation= r(34) = 0.2 10. 13. 4. 0.2 .001 'F2'

First moment constraint

In first order, known misalignments and centroid shifts cause the centre (centroid)

of the phase ellipsoid to be shifted from the reference trajectory, i.e., they cause the

beam to have a nonzero first moment. The first moments appear in a vertical array

to the left of the vertical array giving the Ju( ii). The units of the corresponding

quantities are the same.

It is perhaps helpful to emphasize that the origin always lies on the reference

trajectory. First moments refer to this origin. However, the ellipsoid is defined with

respect to its centre, so the covariance matrix, as printed, defines the second moment

about the mean.

First moments may be fitted. The code digits are i = 0 and j, where j is the index

of the quantity being fit. Thus 10. 0. 1. .1 .01 ; constraints the horizontal (1.)

displacement of the ellipsoid to be 0.1 ± 0.01 cm.

74 

This constraint is useful in deriving the alignment tolerances of a system or in

warning the system designer to offset the element in order to accommodate a centroid

shift.

System length constraint

A running total of the lengths of the various elements encountered is kept by the

program and may be fit. The code digits are i = 0., j = 0.

Thus the element (10. 0. 0. 150. 5. ;) would make the length of the system

prior to this element equal to 150 ± 5 metres. Presumably there would be a variable

drift length somewhere in the system. By redefining the cumulative length via the (16.

6. L. ;) element, partial system lengths may be accumulated and fit.

AGS machine constraint•

Provision has been made in the program for fitting the betatron phase shift angle

µ,associated with the usual AGS treatment of magnet systems.

In the horizontal plane: use code digits i = -11., j = 2., and specify:

A= ~ cos-1 (0.5 (Ru+ R22)] = :1r (horiz)

= freq./(No. of periods)

In the vertical plane: i = -13., j = 4., and

A= ~ cos1 (0.5 (R33 + R44)) = :1r (vert)

For example, if there are 16 identical sectors to a proposed AGS machine and

the betatron frequencies per revolution are to be 3.04 and 2.14 for the horizontal and

vertical planes respectively, then the last element of the sector should be followed by

the constraints:

• See Courant and Snyder. 1 Also note that this constraint is valid only when the

unit cell structure and the corresponding beta functions are both periodic.

75 

1.e.

10. -11. 2. .1go .001

10. -13. 4. .134 .001

3.04_0 00 -- 1 16 . and 2

·

14 =0134 16 .

For example: A typical data listing might be:

5.01

3.

5.01

3.

10. -11. 2. 0.100 .001

10. -13. 4. 0.134 .001

Floor coordinate fitting constraint

Five parameters are needed to specify a floor coordinate constraint:

1 - Type Code 10.0

2 - Code digit 8.

3 - Code digit (j ).

4 - Desired value of floor coordinate.

5 - Desired accuracy of fit (standard deviation).

The code digit (j) indicates the floor coordinate to be constrained. Its possible

values are 1 to 6 indicating the floor x, y, z, theta, phi, and psi, respectively. Theta is

the angle which the floor projection of the reference trajectory makes with the floor z

axis. Phi is the vertical pitch. Psi is a rotation about the reference trajectory. This

is also the order in which coordinates are printed in the floor layout activated by the

13. 12. ; element. Initial coordinates are given on Type Codes 16. 16. ; through

16. 20. ; and Type Code 20.

The floor coordinates are actually zero-th rather than first order properties of a

beam line. However, in TRANSPORT, they may be constrained in a first-order fitting

run, and therefore are included here.

76 

Tl matrix fitting constraints

Five parameters are needed for a constraint on the (i,j,k) element of the secondorder transfer matrix Tl.

1 - Type Code 10.0

2 - Code digit (-i).

3 - Code digit (lOj + k).

4 - Desired value of the {i,j,k) matrix element.

5 - Desired accuracy of fit (standard deviation).

Note that upper and lower limit constraints are not available for second order

fitting.

Some typical Tl matrix constraints are as follows:

Desired Optical Condition Typical Fitting Constraint

Geometric aberration T122 = 0 10. -1. 22. .0 .001 'Fl' .

'

Chromatic aberration T 346 = .5 10. -3. 46. .5 .001 'F2' .

'

There must be no updates of the R2 matrix when constraining an element of the

Tl matrix. There is no limit on the number of constraints which may be imposed.

If no drift lengths are varied the problem will be linear and the absolute size of

the tolerances will be unimportant. Only their relative magnitude will be significant.

Sometimes only a subset of the elements of the matrix Tijk which give significant

contributions to beam dimensions need be eliminated. In such cases one may wish to

minimize the effect of this subset, by weighing each matrix element according to its

importance. One does this by including a constraint for each such matrix element, and

setting its tolerance equal to the inverse of the phase space factor which the matrix

element multiplies. For a matrix element Tijk acting on an uncorrelated initial phase

space, the tolerance factor would be l/(xojXok), where Xoj and xok are the initial beam

half widths specified by the Type Code 1.0 card.

77 

T2 matrix fitting constraints

Five parameters are needed for a constraint on the (i,j,k) element of the second

order auxiliary transfer matrix T2.

1 - Type Code 10.0

2 - Code digit -(20 + i).

3 - Code digit (lOj + k).

4 - Desired value of the (i,j,k) matrix element.

5 - Desired accuracy of fit (standard deviation).

Note that upper and lower limit constraints are not available for second-order

fitting.

Some typical T2 matrix constraints are as follows:

Desired Optical Condition Typical Fitting Constraint

Geometric aberration T122 = 0 10. -21. 22. .0 .001 'Fl' .

'

Chromatic aberration T346 = .5 10. -23. 46. .5 .001 'F2' '

By using a T2 constraint the user may fit an element of the second-order transfer

matrix which pertains to any section of the beam. One causes an R2 update at the

beginning of the section with a 6. 0. 2. ; element. One then places the T2 constraint

at the end of the section. Any number of such constraints may be imposed. This is

the only second-order constraint that may be used in conjunction with an R2 update.

Ir a printing of the Tl matrix is requested via a 13. 4. ; element it will be the

second-order transfer matrix from the last Rl update. The comments about phase

space weighting, made in connection with the Tl constraint, are equally valid for the

T2 constraint, provided the phase space factors are obtained from the beam matrix at

the position of the R2 update.

78 

Second - order u(BEAM) matrix fitting constraint

Five parameters must be specified for a constraint on the second-order contributions to a beam matrix diagonal element u;;.

1 - Type Code 10.0

2 - Code digit (i).

3 - Code digit (i).

4 - The number 0.

5 - Desired accuracy of fit (standard deviation).

If, for example, one wished to minimize the net contributions of second-order

aberrations to the horizontal divergence, one would insert the following card:

10. 2. 2. .0 .01

The quantity that is minimized is the net increase due to second-order terms in

the second moment of the beam about the origin. This quantity is treated as the

chi-s.quared of the problem, so the only meaningful desired value for the fit is zero.

The square root of this quantity is printed in the output. It is computed using the R2

matrix. Therefore, once again, one must not include any element which updates the

R2 matrix. Centroid shifts must not be inserted when doing second-order fitting, even

immediately following the beam card.

The second-order image of the initial beam centroid at some later point in the

beam is not necessarily the beam centroid at the later point. The parameters printed

by TRANSPORT are the new centroid position and the beam matrix about the new

centroid. One must therefore look at both of these to observe the effects of the fitting procedure. It may even happen that an improvement in one parameter will be

accompanied by a slight deterioration in the other.

The beam profile at any point is a function of the initial beam parameters. One

may therefore impose weights on the effect of the various aberrations by the choice

of parameters on the BEAM card. One might, for example, adjust the strength of

the correction of the chromatic aberrations by the choice of the Ap/p parameter. In

particular, when using a BEAM constraint, one should not attempt to minimize or

eliminate chromatic aberrations if Ap/p is set equal to zero on the beam card (Type

Code 1.0}.

Correlations (the 12.0 card) may also be included in the initial beam specification.

79 

Sextupole strength constraints

Five parameters must be specified for a constraint on sextupole strength.

1 - Type Code 10.0

2 - Code digit 18.

3 - Code digit 0.

4 - The number 0.

5 - Desired maximum sextupole field strength.

A single sextupole constraint card applies to all sextupoles which follow. The

maximum field strength is treated as a standard deviation an may be exceeded on an

optimal fit.

One can employ this constraint to find the optimal locations for sextupoles. By

placing inversely coupled drift lengths before and after the sextupole its longitudinal

position may be varied. By constraining the field strength the sextupole can be slid

to a.position where the coupling coefficients to the aberrations will be largest. One

will need to experiment with adjusting the maximum field strength to achieve the best

configuration.

Internal constraints

A set of upper and lower bounds on the value of each type of parameter is in the

memory of the program. H a correction is computed for a parameter which would take

its value outside this range, it is reset to the limit of the range. The current limit are:

T e Code Limits

1.0 0 < input beam

2.0 -60 < pole-face rotation < 60 (deg)

3.0 0 < drift

4.0 0 < magnet length

5.0 0 < quad length

20.0 -360 < beam rotation < 360 (deg)

These limits apply only when a parameter is being varied. Fixed values that exceed

this range may be used as desired.

These constraints were included to avoid physically meaningless solutions.

80 

Corrections and covariance matrix

When the program is fitting, it makes a series of runs through the beam line. From

each run it calculates the chi-squared and the corrections to be made to the varied

parameters. For each iteration a single line is printed containing these quantities.

The program calculates the corrections to be made using a matrix inversion proced ure. However, because some problems are difficult, it proceeds with caution. The

corrections actually made are sometimes a fixed fraction of those calculated. This

fraction, used as a scaling factor, is the first item appearing on the line of printed output. The second factor is the chi-squared before the calculated corrections are made.

Following are the corrections to be made to the varied parameters. They are in the

order in which they appear in the beam line. If several parameters are coupled, they

are considered as one and their position is determined by the first to appear.

When covnergence has occured, the final value of the chi-squared and the covariance matrix are printed. The covariance matrix is symmetric, so only a triangular

matrix is shown. The diagonal elements give the change in each varied parameter

needed to produce a unit increase in the chi-squared. The off-diagonal elements give

the correlations between the varied parameters.

The appearance of the chi-squared and covariance matrix is:

*COVARIANCE (FIT x2)

rn,n-1

For more details on the mathematics of the fitting, the user should consult the

Appendix. For an example of the output of the program he (or she) should refer to

the section on output format.

81 

ACCELERATION: Type Code 11.0

An energy gain is reflected in both the divergence and the width of the beam. This

element provides a simulation of a travelling wave linear accelerator energy gain over

a field free drift length (i.e. no externally applied magnetic field).

There are five parameters:

1 - Type Code 11.0

2 - Accelerator length (metres).

3 - Energy gain (GeV).

4 - 4' (phase lag in degrees).

5 - ~ (wavelength in cm).

The new beam energy is printed as output.

The energy of the reference trajectory is assumed to increase linearly over the entire

accelerator length. Ir this is not the case, an appropriate model may be constructed by

combining separate 11.0 elements. An 11.0 element with a zero energy gain is identical

to a drift length.

None of the parameters may be varied.

Second-order matrix elements have not been incorporated in the program for the

accelerator section.

The units of parameters 2, 3 and 5 are changed by 15. 8., 15. 11., and 15. 5.

type code entries respectively.

82 

Accelerator Section Matrix

[L Eo ln (l + llE cos t) J l llE cos ¢ Eo 0 0 0 0

0 Eo 0 0 0 0

Eo + llE cos ~

0 0 1 ~ Eo ln (1 + AE cod) J llE cos ~ Eo 0 0

0 0 0 Eo 0 0 Eo + llE cos ~

0 0 0 0 1 0

0 ~ ..... ! ~t2·j Eo

0 0 0 Eo + llE cos ~ """'t"" Eo + 6E cos ~

Definitions: L - effective length of accelerator sector.

Eo - particle energy at start of sector.

~E - energy gain over sector length.

4' - phase lag or the reference particle behind the crest

of the accelerating wave, i.e. if 4' is positive then for

some l > 0 the particles having this value are riding

the crest of the wave; the units of 4' are degrees.

~ - wavelength of accelerating wave; the units of ~

are those of l (normally cm).

This matrix element assumes that Eo ::> moc2 (Cully relativistic).

83 

BEAM (ROTATED ELLIPSE): Type Code 12.0

To allow the output beam from some point in a system to become the input beam

of some succeeding system, provision has been made for re-entering the correlation

matrix which appears as a triangular matrix in the beam output. (See section under

Type Code 1.0 for definitions.)

There are 16 parameters:

1 - Type Code 12.0

2 to 16 - The 15 correlations [r(ij)) among the 6 beam components -

in the order printed (by rows).

Several cards may be used to insert the 15 correlations, if necessary.

Since this element is solely an extension of the beam input, a 12.0 element must

immediately be preceded by a 1.0 (BEAM) element entry.

The effect of this element in the printed output is shown only in the beam matrix.

If the beam matrix is printed automatically, it is not printed directly after the BEAM

element but only after the correlation matrix has been inserted.

84 

OUTPUT PRINT CONTROL INSTRUCTIONS: Type Code 13.0

A number of control codes which transmit output print instructions to the program

have been consolidated into a single type code.

There are two parameters:

1 - Type Code 13.0

2 - Code number.

The effects of the various code numbers will be described below (not in numerical

order).

Several codes are available to control various aspects of the printed output. Most

type codes produce a line of output that advertises their existence. Those that do not,

usually have an obvious effect upon the remainder of the output and thus make their

presence clear.

Beam Matrix Print Controls 1., 2., 3.

( 13. 1. ;): The current beam (O') matrix is printed by this code.

(13. 3. ;): The beam (O') matrix will be printed after every physical element which

follows this code.

(13. 2. ;): The effect of a previous (13. 3. ;) code is cancelled and the beam (O')

matrix is printed only when a (13. 1. ;) code is encountered or when another (13. 3.

;) code is inserted. The suppression of the beam matrix is the normal default.

Transformation Matrix Print Controls 4., 5., 6., 24.

(13. 4. ;): The current transformation matrix Rl (TRANSFORM 1) is printed

by this code. If the program is computing a second-order matrix, this second-order

transformation matrix will be included in the print-out. This matrix is cumulative

from the last Rl (TRANSFORM 1) update. The units of the elements of the printed

matrix are consistent with the input units associated with the Type Code 1.0 (BEAM)

entry.

85 

(I3. 6. ;): The transformation matrix RI will be printed after every physical

element which follows this code. The second order matrix win be printed automatically

only if the one-line form (code I3. rn. ;) of the transformation is selected. The second

order matrix will, however, be printed at each location of a (I3. 4. ;) element. The

first-order matrix will not be repeated.

(I3. 5. ;): The automatic printing of RI will be suppressed and RI will be printed

only when subsequently requested.

( I3. 24. ;): The TRANSFORM 2 matrix, R2, will be printed by this code. The

format and units of R2 are identical with those of RI, which is printed by the (I3. 4.

;) code. For a list of elements which update the R2 matrix, see Type Code 6.0.

The units of the tabulated matrix elements in either the first-order R or sigma

matrix or second-order T matrix of a TRANSPORT print-out will correspond to the

units chosen for the BEAM card. For example, the R(I2) = (x/O) matrix element will

normally have the dimensions-of cm/mr; and the T(236) = (O/yfJ) matrix element will

have the dimensions mr/(cm-percent tJ.p/p) and so forth.

Misalignment Table Print Control 8.

The misalignment summary table is printed wherever a (I3. 8. ;) element is

inserted. Its contents are the eflects of all previously specified misalignments whose

results were to be stored in a table. A full description of the table and its contents is

to be found in the section on the align element (Type Code 8.0).

Coordinate Layout Control I2.

One can produce a layout of a beam line in any Cartesian coordinate system one

chooses. The coordinates printed represent the x, y and z position, and the angles

theta, phi and psi, respectively, of the reference trajectory at the interface between

two elements. Theta is the angle which the floor projection of the reference trajectory

makes with the floor z axis. Phi is the vertical pitch. Psi is a rotation about the

reference trajectory. In the printed output the values given are those at the exit of

the element listed above and at the entrance of the element listed immediately below.

86 

A request for a layout is specified by placing a (I3. I2. ;) card before the beam

card. If no additional cards are inserted the reference trajectory of the beam line will

be assumed to start at the origin and proceed along the positive z-axis. The y-axis will

point up and the x-axis to the left. One can also specify other starting coordinates and

orientations by placing certain other cards before the beam card. For a description of

such cards see Type Code I6.0 (special parameters).

The calculation of the coordinates is done from the parameters of the physical

elements as given in the data. Therefore, if effective lengths are given for magnetic

elements, the coordinates printed will be those at the effective field boundary. The

effects of fringing fields in bending magnets are not taken into account.

General Output Format Controls I7. I8. 19.

(13. I7. ;): The subsequent printing of the physical parameters of all physical

elements will be suppressed. Only the type code and the label will remain. This

element is useful in conjunction with the (I3. IQ. ;) element which restricts the beam

(u) matrix and the transformation (RI) matrix each to a single row. The elements

of these matrices then appear in uninterrupted columns in the output, similar to the

TRAMP computer code used at the Rutherford Laboratory, CERN, and elsewhere.

( I3. I8. ;): Only varied elements and constraints will be printed. This element,

in conjunction with the various options on the indicator card, can produce a very

abbreviated output. The entire output of a multistep problem can now easily be

printed on a teletype or other terminal.

(I3. IQ. ;): The beam (u) and transformation (RI or R2) matrices, when printed,

will occupy a single line. Only those elements are printed which will be nonzero if

horizontal mid plane symmetry is maintained. The second-order transformation matrix

will obviously occupy several lines. This element, in conjunction with the I3. I7. ;

element and either the I3. 3. ; element or the 13. 6. ; element, will produce output in

which the printed matrix elements will occupy single uninterrupted columns. For visual

appearances it is recommended that, if both beam (u) and transformation matrices

are desired, they be printed in separate steps of a given problem.

87 

Punched Output Controls 2g., 30., 31., 32., 33., 34., 35., 36.

If the control is equal to 2g, all or the terms in the first-order matrix and the x

and y terms or the second-order matrix are punched.

If the control is equal to 30, all or the terms or the first-order matrix and all

second-order matrix elements are punched out.

If the control, n, is greater than 30, all or the first-order terms are punched and

the second-order matrix elements which correspond to (n-30.), i.e., if n = 32, the

second-order theta matrix elements are punched out. If n = 31, the second-order x

matrix elements are punched, and so forth.

88 

ARBITRARY TRANSFORMATION INPUT: Type Code 14.0

To allow for the use of empirically determined fringing fields and other specific

(perhaps nonphase-space-conserving) transformations, provision has been made for

reading in an arbitrary transformation matrix. The first-order 6 X 6 matrix is read

in row by row.

There are eight parameters for each row of a first-order matrix entry:

1 - Type Code 14.0

2 to 7 - The six numbers comprising the row. The units must

be those used to print the tr an sf er matrix; in other

words, consistent with the BEAM input/output.

8 - Row number ( 1. to 6.)

A complete matrix must be read and applied one row at a time. Rows that do not

differ from the unit transformation need not be read.

For example, ( 14. -.1 .Q 0. 0. 0. 0. 2. ;) introduces a transformation

matrix whose second row is given but which is otherwise a unit matrix. Note that

this transformation does not conserve phase space because R(22) = o.g, i.e. the

determinant of R ~ 1.

Any of the components of a row may be varied; however, there are several restrictions.

Type Code 14.0 elements that immediately follow one another will all be used to

form a single transformation matrix. If distinct matrices are desired, another element

must be inserted to separate the Type Code 14.0 cards. Several do-nothing elements

are available; for example, a zero length drift (3. 0. ; ) is a convenient one.

When the last of a sequence of Type Code 14.0 cards is read, the assembled transformation matrix will be printed in the output. Note that

O ) (au

azz O

Hence, a matrix formed by successive 14. (3. 0. ;), 14. - elements is not always equal

to the one formed by leaving out the (3. 0. ;) element.

89 

If components of a 14.0 card are to be varied it must be the last 14.0 card in its

matrix. This will force a matrix to be split into factors if more than one row has

variable components.

If it is desired to read in the second-order matrix coeficients for the ith row, then

the following 22 additional numbers may be read in.

g - continuation code 0.

10 to 30 - The 21 coefficients:

T(ill) T(i12) T(i13) T(i14) T(i15) T(i16)

T(i22) T(i23) T(i24) T(i25) T(i26) T(i33)

T(i34) T(i35) T(i36) T(i44) T(i45) T(i46)

T(i55) T(i56) T(i66)

in that order, where i is the row number.

It is necessary to read in the first-order

matrix row which corresponds to the

second-order matrix row being read in.

As in the first-order case, full rows not different from the identity matrix [i.e.,

R(ii) = 1, all other R(ij) = O, and all T(ijk) = O] need not be read in.

90 

INPUT-OUTPUT UNITS: Type Code 15.0

TRANSPORT is designed with a standard set of units that have been used throughout this manual. However, to accommodate other units conveniently, provision has

been made for redefining the units to be employed. This is accomplished by insertion

of one or more of the following elements.

There are four parameters to be specified:

1 - Type Code 15.0

2 - Code digit.

3 - The abbreviation of the unit (see examples below).

This will be printed on the output listing.

It must be enclosed in single quotes and is a

maximum of three characters long (four for

energy). The format for insertion is the same

as for labels.

4 - The scale factor (if needed).

The scale factor is the size of the new unit

relative to the standard TRANSPORT unit.

For example, if the new unit is inches and the

standard TRANSPORT unit cm, the scale factor

is (2.54).

91 

The various units that may be changed are:

Code Standard Symbols used

Digit Quantity TRANSPORT Unit in SLAC-75

1.0 horizontal and vertical transverse cm x,y

dimensions, magnet apertures,

and misalignment displacements

2.0 horizontal and vertical angles mr (), </>

and misalignment rotation angles

3.0 vertical beam extent (only r cm y

and bending magnet gap height

4.0 vertical beam divergence (only r mr </>

5.0 pulsed beam length and wave cm i

length in accelerator

6.0 momentum spread percent (PC)

7.0 bend, pole face rotation, and degrees (DEG)

coordinate layout angles

8.0 length (longitudinal) of elements, metres (M) t

layout coordinates, and bending

magnet pole face curvatures

Q.O magnetic fields kG B

10.0 mass electron mass m

11.0 momentum and GeV/c p(O)

energy gain in accelerator section GeV aE

• These codes should not be used if the coordinate rotation (20.0) type code is used

anywhere in the system.

92 

Units are normally restored at the end of a problem step. Once changed, they

remain the same for all succeeding problem steps in an input deck until a 0 indicator

card is encounter, at which time they are reset to standard TRANSPORT units. The

units may be reset to standard units by inserting a {15.0) Type Code entry.

The 15.0 elements are the first cards in a deck (immediately following the title

card and the 0 and 1 indicator card) and should not be inserted in any other location.

They produce no printed output during the calculation, their effect being visible only

in the output from other elements.

Example: To change length to feet, width to inches, and momentum to MeV /c,

add to the front of the deck the elements

15. 8. 'FT' 0.3048;

15. 1. 'IN' 2.54;

15. 11. 'MEV' 0.001;

The scale factor, 0.3048, multiplies a length expressed in the new unit, feet, to

convert it to the reference unit, metres, etc.

For the conventional units listed below, it is sufficient to stop with the unit name

(the conversion factor is automatically inserted by the program). If units other than

those listed below are desired, then the unit name and the appropriate conversion

factor must be included. If the automatic feature is used with older versions of the

program, there must be no blank spaces between the quotes and the unit name.

93 

Input-output units: Type Code 15.0

(Conversion factors for dimension changes versus code digit "and unit)

Unit

1. 2. 3.

'CM' 1. 1.

'M' 100. 100.

'IN' 2.54 2.54

'FT' 30.48 30.48

'MM' .1 .1

'R' 1000.

'MR' 1.

'PC'

'P/10'

'N'

'MeV'

'GeV'

'KG'

'G'

PC is a.n abbreviation for percent

P/10 means one-tenth of a percent

N means 100 percent

Code Digit

4. 5. 6. 7. 8. 9. 10.

1. .01

100. 1.

2.54 .0254

30.48 .3048

.1 .001

1000.

1.

1.

.1

100.

1.

.001

94

11.

.001

1. 

SPECIAL INPUT PARAMETERS: Type Code 16.0

A number of constants are used by the program which do not appear as parameters

in elements of any Type Code. A special element has been provided to allow the

designer to set their values. These special parameter entries must always precede the

physical element(s) to which they apply. Once introduced, they apply to all suceeding

elements in the beam line unless reset to zero or to new values.

There are three parameters:

1 - Type Code 16.0

2 - Code digit.

3 - Value of the constant.

A number of such constants have been defined in this manner. All have a normal value

that is initialized at the beginning of each run.

Code Digits for Special Parameters

1. £(1) - A second-order measure of magnetic field inhomogeneity in bending

magnets. If B(x) = B(O)[l - n(x/ Po)+ f3(x/ Po)2 - • • ·] is the field

expansion in the median (y = 0) plane, then £(1) is defined as

£(1) = /3(1/ Po)2 (where Po is measured in unit of horizontal beam

width - normally cm). This parameter affects second-order

calculations only. Normally the value is 0. It may be varied in

second-order fitting.

3. (M/m) - Mass of the particles comprising the beam, in units of the electron

mass; normally 0. A non-zero mass introduces the dependence of

pulse length on velocity, an important effect in low-energy pulsed

beams.

4. W /2 - Horizontal half-aperture of bending magnet, in the same units as

5. g/2

horizontal beam width, normally 0 (i.e. effect of horizontal half

aperture is ignored).

- Vertical half-aperture of bending magnet, in the same units as

vertical beam height; this parameter must be inserted if the effect

of the spatial extent of the fringing fields upon transverse focusing

is to be taken into account. (See Type Code 2.0 and 4.0 as a cross

referenc-e) normally 0.

95 

6. L - Cumulative length of system, in the same units as system length.

It is set to zero initially, then increased by the length· of each element,

and finally printed at the end of the system. This element allows the

cumulative length to be reset as desired.

7. Ki - An integral related to the extent of the fringing field of a bending

magnet. See section under Type Code 2.0 and SLAC-75 page 74

for further explanation. If the (16. 5. g/2. ;) element has been

inserted, the program inserts a default value of Ki = 1/2 unless

a (16. 7. Ki. ;) element is introduced, in which case the program

uses the Ki value selected by the user. The table below shows

typical values for various types of magnet designs.

8. K 2 - A second integral related to the extent of the fringing field.

Default value of K2 = 0 unless specified by a (16. 8. K2. ;) entry.

Typical values of Ki and K2 are gien below for four types of fringing field boundaries:

(a) a linear drop-off of the field,

(b) a clamped "Rogowski" fringing field,

(c) an undamped "Rogowski" fringing field,

(d) a "square-edged" nonsaturating magnet.

Model Ki

Linear b/6g

Clamped Rogowski 0.4

Unclamped Rogowski 0.7

Square-edged magnet 0.45

K2*

3.8

4.4

4.4

2.8

where b is the extent of the linear fringingfield. See page 98 of Reference 8.

* For most applications K2 is unimportant. If you find it is important to your result,

you should probably be making a more accurate calculation with a differential equation

ray-tracing program (~ee ~ef. 7).

96 

12. 1/ R1 - Where R1 is the radius of curvature (in units of longitudinal

length, noramlly metres) of the entrance face of bending magnets.

(See figure on p. 106.)

13. 1/ R2 - Where R2 is the radius of curvature (in units of longitudinal

length, normally metres) of the exit face of bending magnets.

(See figure on p. 106.)

The pole face curvatures (l/R1) and (l/R2) affect the system only in second-order,

creating an effective sextupole component in the neighbourhood of the magnet. If the

parameters are not specified, they are assumed to be zero, i.e. no curvature and hence

no sextupole component. Either parameter (or both) may be varied in second-order

fitting.

x,

FIELD BOUNDARIES FOR BENDING MAGNETS

CENTRAL

TRAJECTORY

The TRANSPORT sign conventions for x, /3, R and h are all positive as shown in

the figure. The positive y direction is out of the paper. Positive f3's imply transverse

focusing. Positive R's (convex curvatures) represent negative sextupole components of

strength S = (-h/2R) sec3{3. (See SLAC-75, page 71.)

97 

Tilt - to - Focal Plane (16. 15. o. ; ) Element

Very often it is desired to have a listing of the second-order aberrations along the

focal plane of a system rather than perpendicular to the optic axis, i.e. along the x

coordinate. If the focal plane makes an angle a with respect to the x axis (measured

clockwise) then provision has been made to rotate to this focal plane and print out the

second-order aberrations. This is achieved by the following procedures:

Alpha is the focal-plane tilt angle (in degrees) measured from the perpendicular to

the optic axis (o is normally zero).

The programming procedure for a tilt in the x(bend)-plane (rotation about y axis)

is:

16. 15. a.

3. 0. (a necessary do-nothing element)

13. 4.

16. 15. -a. ; (rotate back to zero)

3. 0. (a necessary do-nothing element)

16. 15. 0. ; (to turn off rotation element)

The programming procedure for a tilt in the y-plane (rotation about x-axis) is:

16. 15. a.

20. 90.

3. 0.

20. -90.

13. 4.

16. 15. -a. (rotate back to zero)

20. 90.

3. 0.

20. -90.

16. 15. 0. (to turn off rotation element)

98 

Initial Beam Line Coordinates and Direction

When requesting a beam line coordinate layout via a (13. 12. ;) element one

can employ any coordinate system one desires. The position and direction of the

beginning of the reference trajectory in this coordinate system are given on elements

16. 16. through 16. 20. Such cards should be placed before the beam card, but after

any units changes. Their meanings are as follows:

16. 16.

}

zo, Yo and zo, respectively, the coordinates of the

16. 17. initial point of the reference trajectory in the units chosen

16. 18. for longitudinal length.

16. 19. l Oo and </>o, the initial horizontal and vertical angles

16. 20. of the reference trajectory in degrees.

The angle t/J may be set by using Type Code 20.0.

When specifying the_ initial orientation of the reference trajectory via the two

angles, one must give the horizontal angle first. The meaning of the two angles is

given in the following figure. Any of the above five parameters not explicitly specified

will be taken to equal zero.

The initial coordinates may be varied in first-order fitting. Their values will affect

only the beam line floor coordinates and not any beam or transfer matrix element.

gg 

....---------------------------------------- -

REFERENCE

TRAJECTORY

SPECIFICATION OF INITIAL ANGLES 00 AND <Po FOR BEAM LINE LAYOUT.

100 

SECOND-ORDER CALCULATION: Type Code 17.0

A second-order calculation may be obtained provided no alignments are employed.

A special element instructs the program to calculate the second-order matrix elements.

It must be inserted immediately following the beam (1. element).

Only one parameter should be specified:

1 - Type Code 17.0 (signifying a second-order calculation

is to be made).

To print out the second-order Tl matrix terms at a given location in the system,

the (13. 4. ;) print control card is used. For T2, the ( 13. 24. ;) print control card

is used. The update rules are the same as those for the corresponding first-order R

matrix. See SLAC-75 for definitions of subscripts in the second order T(ijk) matrix

elements.

The values of the BEAM (sigma) matrix components may be perturbed from their

first-order value by the second-order aberrations. In a second-order TRANSPORT

calculation, the initial beam is assumed to have a Gaussian distribution. For exact

details the reader should consult the Appendix. For the beam matrix to be calculated

correctly, there should be no elements which update the R2 matrix. If a centroid shift

is present, it must immediately follow the beam (Type Code 1.0) or beam rotated

ellipse (Type Code 12.0) card.

Only second-order fitting may be done in a second-order run. See the section on

Type Code 10.0 for a list of quantities that may be constrained in a second-order run.

If a beam constraint is to be imposed in second-order, there must be no centroid shifts

present anywhere.

Second-order matrices are included in the program for quadrupoles, bending magnets (including fringing fields), the arbitrary matrix, sextupoles, and solenoids. They

have not been calculated for the acceleration (Type Code 11.0) element.

101 

SEXTUPOLE: Type Code 18.0

Sextupole (hexapole) magnets are used to modify second-order aberrations in beam

transport systems. The action of a sextupole on beam particles is a. second and higher

order effect, so in first order runs (absence of the 17.0 ca.rd) this element will act as a

drift space.

There are four parameters:

1 - Type Code 18.0

2 - Effective length (metres).

3 - Field at pole tips (kG). Both positive and negative

fields are possible (see figures below).

4 - Half-aperture (cm). Radius of circle tangent to pole tips.

Other orientations of the sextupole may be obtained using the beam rotation element (Type Code 20.0).

The pole tip field may b varied in second-order fitting. It may also be constrained

not to exceed a certain specified maximum field. (See the explanation of vary codes in

the section on Type Code 10.0.) Such a constraint allows one to take into account the

physical realities of limitations on pole tip fields.

See SLAC-75 for a tabulation of sextupole matrix elements. The TRANSPORT

input format for a typical data set is:

r-Label is desired

l (not to exceed four spaces)

18. L. b. a. ' '

102 

N

y

y

9 0 x

M s

0 x ~--

N

DIPOLE QUADRUPOLE SEXTUPOLE

Illustration of the magnetic midplane (x axis) for dipole, quadrupole and sextupole

elements. The magnet polarities indicate multipole elements that are positive with

respect to each other.

103 

SOLENOID: Type Code 19.0

The solenoid is most often used as a focusing element in systems passing low-energy

particles. Particles in a solenoidal field travel along helical trajectories. The solenoid

fringing field effects necessary to produce the focusing are included.

There are three parameters:

I - Type Code rn.o

2 - Effective length of the solenoid (meters).

3 - The field (kG). A positive field, by convention, points in

the direction of positive z for positively charged particles.

The length and the field may be varied in first-order fitting. Both first- and secondorder matrix calculations are available for the solenoid.

A typical input format is:

r--- Label is desired

! (not to exceed four spaces)

rn. L. B. ' '

First - Order Solenoid Matrix

Solenoid R matrix

Definitions: L - effective length of solenoid.

K - B(0)/(2Bpo), where B(O) is the field inside the

solenoid and (Bpo) is the (momentum) of the

central trajectory.

C - cos KL

S - sin KL

For a derivation of thi.S transformation see report SLAC-4 by R. Helm.

104 

Alternate forms of matrix representation of the solenoid:

c2 1 SC K SC 1 s2 K 0 0

-KSC c2 -KS2 SC 0 0

-SC 1 s2 c2 1 SC 0 0 R(Solenoid) = -K K

KS2 -SC -KSC c2 0 0

0 0 0 0 1 0

0 0 0 0 0 1

Rotating the transverse coordinates about the z axis by an angle= -KL decouples

the x and y first-order terms, i.e.

c 1 s K 0 0 0 0

-KS c 0 0 0 0

R(-KL) · R(Solenoid) =

0 0 c 1 s 0 0 K

0 0 -KS c 0 0

0 0 0 0 1 0

0 0 0 0 0 1

105 

COORDINATE ROTATION: Type Code 20.0

The transverse coordinates x and y may be rotated through an angle a about

the z axis (the axis tangent to the central trajectory at the point in question. •

Thus a rotated bending magnet, quadrupole, or sextupole may be inserted into a

beam transport system by preceding and following the element with the appropriate

coordinate rotation. (See examples below.) The positive sense of rotation is clockwise

about the positive z axis.

There are two parameters to be specified for a coordinate rotation:

1 - Type Code 20.0 (signifying a beam coordinate rotation).

2 - The angle of rotation a (degrees).

The angle of rotation may be varied in a first-order fitting (see Type Code 10.0).

Note:

This transformation assumes that the units of (x and y) and (0 and </>) are the

same. This is always true unless a 15.0 3.0 or a 15.0 4.0 type code has been used.

• See SLAC-75, 4 page 45 ,and page 12, Fig. 4, for definitions of x, y, and z coordinates.

106 

Examples:

For a bending magnet, the beam rotation matrix may be used to specify a rotated

magnet.

Example No. 1

A bend up is represented by rotating the x,y coordiantes by -90.0 degrees as follows:

!Label (not to exceed four spaces) is desired

20. -90. ' '

2. .8( 1 ). ' '

4. L. B. n. ' '

2. ,8(2). ' '

20. +90. ' ' (returns coordinates to their initial orientation)

A bend down is accomplished via a +90 degree rotation.

20. +90. ' ' ;

2.

4.

2.

20. -90. ' ' ;

A bend to the left (looking in the direction of beam travel) is accomplished by

rotating the x,y coordinates by 180 degrees, e.g.

2-0. +180. ' ' ;

2.

4.

2.

20. -180. ' '

107 

Example No. 2

A quadrupole rotated clockwise by 60 degrees about the positive z axis would be

specified as follows:

where

20. +60. ' I

5. L. B. a. ' '

20. -60. ' I

C - cos a,

S - SID a,

R=

Beam rotation matrix

c s

c s

-S c

-S c

1

1

a - angle of coordinate rotation about the beam axis,

blank spaces are zeros.

e.g. for a = +90 degrees, this matrix interchanges rows 1 and 2 with 3 and 4 of the

accumulated R matrix as follows:

[ :I

0 1 01 [R(U)

R(l2) R(13)

0 0 1 R(21) R(22) R(23) R(24) R(J4) l

0 0 0 X R(31) R(32) R(33) R(34)

-1 0 O R(41) R(42) R(43) R(44)

[ R(31} R(32) R(33)

R(41) R(42) R(43) R(44) R(34) l - -R(ll) -R(12) -R(13) -R(14)

-R(21) -R(22) -R(23) -R(24)

(The rest of the matrix is unchanged.)

108 

1

2 -

3 -

4 -

STRAY MAGNETIC FIELD: Type Code 21.0

Element No. 21.0

Code No. n.

(JJL}

± (uBL)

n=4:

n=2:

+:

horizontal deflection

vertical deflection

mean value of f B dz.

Gaussian random number generator;

affects beam first moment.

uncertainty in f B dz - affects beam

second moment.

Uses the misalignment element (8.) to calculate an angular deflection equal to f Bdz/Bp).

This type code is not functioning in the present version of the program.

109 

SENTINEL

Each step of every problem in a TRANSPORT data set must be terminated with

the word SENTINEL. The word SENTINEL need not be on a separate card. For a

description of the form of a TRANSPORT data set see the section on input format.

An entire run, consisting of one or several problems, is indicated by an additional card

containing the word SENTINEL. Thus, at the end of the entire data set the word SENTINEL will appear twice.

ACKNOWLEDGEMENTS

R. Helm's suggestions and criticisms at SLAC have been invaluable throughout the

deve~opment of the program and the underlying theory. R. Pordes has ably assisted

D. Carey at FNAL during the more recent developments of the program.

110 

REFERENCES

1. E. D. Courant and H. S. Snyder, "Theory of the Alternating Gradient Synchrotron," Ann. Phys. ~, 1-48 (1958).

2. S. Penner, "Calculations of Properties of Magnetic Deflection Systems," Rev.

Sci. Instrum. 32, 150-160 (1961).

3. K. L. Brown, R. Belbeoch and P. Bounin, "First- and Second-Order Magnetic

Optics Matrix Equations for the Midplane of Uniform-Field Wedge Magnets,"

Rev. Sci. Instrum. 35, 481-485 (1964).

4. K. L. Brown, "A First- and Second-Order Matrix Theory for the Design of

Beam Transport Systems and Charged Particle Spectrometers," SLAC Report

No. 75, or Advances Particle Phys. l, 71-134 (1967).

5. K. L. Brown and S. K. Howry, "TRANSPORT/360, a Computer Program

for Design Charged Particle Beam Transport Systems," SLAC Report No. 91

(1970). The present manual supersedes Reference 5.

6. K. L. Brown, "A Systematic Procedure for Designing High Resolving Power

Beam Transport Systems or Charged Particle Spectrometers," Proc. 3rd Int.

Conf. on Magnet Technology, Hamburg, Germany, May 1970, p. 348-366.

(SLAC-PUB-762, June 1970).

7. Suggested Ray-Tracing Programs to Supplement TRANSPORT:

David C. Carey, "TURTLE (Trace Unlimited Rays Through Lumped Elements," Fermilab Report No. NAL-64 (1971). This is a computer program

using TRANSPORT notation and designed to be run using the same data

cards as for a previous TRANSPORT run.

K. L. Brown and Ch. Iselin, "DECAY TURTLE (Trace Unlimited Rays

Through Lumped Elements)," CERN Report 74-2 (1974). This is an extension of TURTLE to include particle decay calculations.

H. Enge and S. Kowalski have developed a Ray-Tracing program using

essentially the same terminology as TRANSPORT. Any experienced user

of TRANSPORT should find it easy to adapt to the M.l.T. program.

8. K. G. Steffen, High·Energv Beam Optics, Interscience Monographs and Texts in

Physics and Astronomy, Vol. 17, John Wiley and Sons, New York (1965).

111 

SUGGESTED BIBLIOGRAPHY

A. P. Banford, The Transport of Charged Particle Beams, E. and F. N. Spon Ltd.,

London (1966).

K. L. Brown et al., Nucl. Instrum. Methods 141, 393-399 (1977).

112 

Table 1. Summary of TRANSPORT Type Codes

PHYS I CAL EU.l£m" 1YPE

CDIE

2nd

ENTRY

BEAM

r.11.s. AOOITI~ TO

BEAM ENVEUPE

l.vvvvvvO X(Cll)

l.vvvvvvOO t.x(cn)

P!l.E FAQ; ROTATI~ 2. v

IIUFI 3.v

BENDING M.llJlE1" 4. vvv

QUADRLl'OLE S. vvll

TRANSR>Rol 1 Ll'DATE 6.0

TRANSFORM 2 Ll'DATE 6. 0

ANGLE.OF

ROTATl<Ji (degrees)

LENG1ll (metres)

LENGIH ( .. tres)

LENG1ll (metres)

o.o

o.o

BEAM Q;N'l'R>ID SHIFI 7 .vvvvvv SHIFT (x)(ca)

3rd

ENTRY

e(mr)

6e(mr)

FIELD (kG)

FIELD (kG)

1.0

2.0

SHIFT (e)

4th 5th 6th ENTRY ENTRY ENTRY

y(an) *(11r) 1(cn)

6y(on) 6+(mr) 61(aij

FIELD GRADIENT (n-value)

HALF-APEK11.RE (en)

(mr) SlllFI (y)(cm) SHIFI <*) (11r) SlllFI (l)(an)

7th 8th ENTRY ENTRY

6(percent) Po

66 (percent) bP

(GeV/c)

SlllFI (6 percent)

ALIC»ENT TOLERANO: 8.vvvvvvO DISP!AC»ENT (x)(Cll) ROTATI~ (e)(mr) DISPJ.>.C8'1ENr (y)(an) ROTATI~ (•)(mr) DISPLACBENT (z)(aa ROTATI~ (a) (mr) ~

REPEAT o::mR>L 9.0

FITTING <Xl\ISTRAIITTS 10.0

NlMlER OF

REPEATS

IESIRED VALUE OF (I ,J) MATRIX

ELEl£tll'S

AroJRACT OF FIT

Note: +I is used for fitting a beam (o) matrix element. -I is used for fitting an Rl aatrix element.

- (I • 20) is used for fitting an R2 matrix elaaent.

ACO:LERATOR 11.0 LENGIH (metres) 17~~ergy gain) I tde~~/ag) I (WAVELENGIH) (cm) I I ~ated Ellipse) 12.0 1llE FIFTEEN CDRRELATIONS ~ 1llE SIX ELEl£tll'S (This entry llJSt be preceded by a type code 1.0 Sltry.)

INPUT /OUl'Plll'

OPTIONS 13.0

ARBITRARY R MATRIX 14.vvvvvvO R(J,l) R(J,2) R(J,3) R(J,4)

Ltl!TS o::mR:JL (Transport

Dimensions)

QUADRATIC TERol OF BENDING FIELD

MA.SS OF PART! Cl.ES IN BEAM

HALF-APEIUUUO OF

BENDING MA.G£1'

IN x-PLANE

HALF-APERl1JRE OF

BENDING MAQET

IN y-PLANE (iap)

LENG1ll Of SYSTEM

FRINGE FIELD CDRRECTI~ a:£FFICIENT

FRINGE FIELD CDRRECTI~ a:£FFIC1ENT

ClRVAnJRE OF ENTRANO: FAQ; OF

BENDING MAlMT

ClJRVATil!E OF EXIT FAQ; OF BENDING MAlMT

10CAL PLANE ROTATI~

INITIAL BEAM LINE x-<XJORDINATE

INITIAL BEAM LINE y-<XJORDINATE

INITIAL BEAM LINE z-CDOIU>INATE

INITIAL BEAM LINE lllRI ztmAL ANGLE

INITIAL BEAM LINE VEKTICAL ANGLE

SE~

~TIONS

SEXIU'OLE

SOLEIDID

BEAM ROTATl<li

STRAY FIELD

15.0

16.0v

16.0

16.0

16.0

16.0

16.0

16.0

16.0V

16.0V

16.0

16.0V

16.0V

16.0V

16.0V

16.0V

17.0

18.0V

19.w

20.v

21.0

CDIE

LO

3.0

4.0

s.o

6.0

7.0

a.o

- ~

12.0

13.0

15.0

16.0

17.0

18.0

19.0

zo.o

LENGIH (metres)

lll!T SYMIJJL SCALE FACTOR

(if required)

p0 in m.its of transverse length (cm)

I M/m (dimensionless) a • uss of electron

w/2 (Cll)

a/2 (Cll)

L (metres)

11 (diMnsionless)

11 (dimensionless)

(l/R1) (l/ .. tres)

(l/R,) (l/ .. tres)

Angle of focal plane rotation (degrees).

See type code 16.0 for details.

x,

Yo

••

e,

••

FIELD (kG)

LENG1ll (metres) FIELD (kG)

ANGLE OF

ROTATI<Ji (degrees)

See later section of report.

R(J,S) R(J,6)

Note: The v's followina the type codes indicate the par-ters which may be varied. See section lmder type code 10.0 for a detailed explanaticl) of Vary Codes. The imits are standard 'JlW6PORT U'lits (as shown) U'lless changed via type code 15.0 entries.

113

J

9th

ENTRY

0 

HEW FEATURES IN TRANSPORT*+/

David C. Carey

Fermi National Accelerator Laboratory

Batavia. Illinois 60510

K. L. Brown

SLAC-91 Addendum

February 1983

Stanford Linear Accelerator Center

Stanford University, Stanford. California 94305

The computer program TRANSPORT, used for designing charged particle

beam transport systems, has been modified to include a number of new features. Among them are the possibility of using accelerator notation to

specify the beam matrix, expanded fitting capabilities including the

ability to constrain algebraic combinations of matrix elements, and more

flexible means of specifying individual or groups of beam line elements.

This document is to be used as a supplement to the TRANSPORT manual.

It is organized by type code, as is the manual. The contents of this

report will eventually be written into a revised TRANSPORT manual. In a

few instances, the material of the manual has been rewritten, but in most

cases it has simply been expanded.

* Work supported by the Department of Energy, contract DE-AC03-76SF00515.

+ Transport Addendum.

/ This report is also issued as HAL-91. 

- 2 -

INPUT BEAM: Type Code 1.0

The beam matrix may also be specified in terms of accelerator notation

using the parameters B, a, and ~- A 13. 7. ; element placed before the

beam element (Type code 1.0) will indicate that the beam matrix input and

output and beam matrix constraints are all expressed in accelerator notation. The meaning of the accelerator parameters B and a is shown in the

illustration below. The parameters Band a may be defined for each transverse plane. The use of this option is not compatible with either misa1 ignments (type code 8.) or second order Ctype code 17.), or elements

which produce x-y coupling.

If accelerator notation_ is to be used, there are eight entries to be

made on the beam card.

The type code 1.0 (specifies a beam entry follows).

2 Bx

3 «x

4 By

5 «y

6 0.

7 0.

8 The momentum of the central trajectory [pCO>] CGeV/c).

The units for B are the quotient of the transverse length units and

the transverse angle units. Conventionally B is expressed in mm/mr or

meters. Following this convention requires a change in the transverse

distance unit to mm using the 15. type code. 

- 3 -

The parameter ~ is used to characterize the behavior of off-momentum

trajectories. The components of the n vector are specified using the

centroid shift Ctype code 7.). If no type code 7 element is present, the

fractional momentum deviation of ~ from the beam design momentum is one

unit in fractional momentum spread. Normally this is in percent dp/p,

but can also be changed via a 15. type code entry. All other coord~nates

of ~ are then taken to equal zero. A beam correlation entry Ctype code

12.) is not used with accelerator notation since «x and cr.y express the

degree of correlation between position and angle in the two transverse

planes.

Accelerat~r designers are accustomed to expressing a and ~ in meters.

These are equivalent to millimeters per milliradian and millimeters permil Cone part in a thousand S). A complete set of units change specifications to the latter set of units is

15.

15.

1.

6.

'mm'

'pm' • 1

The beam matrix, whenever printed, will be expressed in accelerator

notation. If the beam matrix is printed after every physical element, by

the insertion of a 13. 3. ; element, a heading will be printed to identify the numbers in the beam matrix. The beam matrix will be printed in

a single line. If the beam matrix is printed only after selected elements via a 13. 1. ; type code entry, no heading is printed. The quantities printed are

Cl.y ~x' 

- 4 -

The parameters tx and ty are the phase advances in the horizontal and

vertical planes. The phase advance is given by

ds

BCs)

where s is the distance along the reference trajectory. The normal units

for t in TRANSPORT are degrees. However, the units for t may be changed

via a 15. 12 .... element. for example if tis desired in "tune" units

then a 15 12 'TUNE' 360 ; element should be inserted before the type code

1. 0 beam entry.

CENTROID 1338Al

A TWO-DIMENSIONAL BEAM PHASE ELLIPSE

The area of the ellipse is given by:

The equation of the ellipse is:

~x 2 + 2axe + BS 2 = e 

- 5 -

where

[ [ -: ] .,, ] a,, B

a = = «:

az1 azz -a.

and

a. a.

rz1 = r12 = =

J1+tt 2 &

B'Y

- a.2 = 

- 6 -

rRINGING rIELDS AND POLE-rACE ROTATIONS roR BENDING MAGNETS:

Type Code 2.0

If the bending magnets to be specified are rectangular as seen from

the top, then the sum of the input and output pole face rotation angles

will equal the bend angle. Several options exist for automatically setting the pole face rotation angles in terms of the bend angle. Such

options are given by entries of 13. 40. ; through 13. 43. ; inclusive.

Even when one of these options is selected, the 2. element must be

included in the data before and after the type code 4. entry if fringingfield effects are to be calculated. Details of the various options are

found in the section describing type code 13. 

- 7 -

WEDGE BENDING MAGNET: Type Code 4.0

Alternatively, a wedge bending magnet may be specified in terms of its

length and bend angle. If a 13. 48. ; entry is inserted in the data, all

subsequent bend magnet entries are taken to be expressed in terms of L,

a, and n. The four parameters to be specified are then

Type code 4.0 (specifying a wedge bending magnet).

2 The (effective> length L of the central trajectory in meters.

3 The bend angle « in degrees.

4 The field gradient Cn-value).

The quantities L, a, and n may be varied. The units for bend angle

may be changed via a 15. 7. element preceding the beam card.

At a later point one can revert to the normal description in terms of

L, B, and n by the insertion of a 13. 47. ; element. 

- 8 -

QUADRUPOLE: Type Code 5.0

A device which is focusing in both transverse planes or defocusing in

both transverse planes may also be represented via a 5. type code. The

description is the same as for an ordinary quadrupole, except that the

sign of B now applies to both planes. A 13. 98. ; element inserted

before a 5. type code will cause it to be taken as such a device. A subsequent 13. 97. ; entry causes the 5. type code to be taken as representing a conventional quadrupole. 

- 9 -

SHlfT IH THE BEAM CENTROID: Type Code 7.0

If accelerator notation is selected for the beam matrix via a 13. 7.

element, then the 7. type code is used to initiate the ~ function. The

coordinates specified are those of the initial ~ function. The ~ function is conventionally taken to represent a trajectory whose momentum

differs by one unit from the design momentum of the beam line. In this

case the 6 parameter on the shift element would be 1.0. The units for 6

may be changed via a 15. 6. type code entry. A complete set of unit

changes appropriate for the use of accelerator notation is given under

type code 1. Typically an initial ~function would be specified via a

type code 7.0 as follows:

7. 0 

- 10 -

VARY CODES ANO FITTING CONSTRAINTS: Type Code 10.0

First-Order Vary Codes

In a first-order run, the additional quantities marked v may be

varied:

BEAM (rotated ellipse) ... 12.vvvvvvvvvvvvvvv - All elements of

the correlation matrix may be varied.

Second-Order Vary Codes

In a second-order run the following parameters may be varied:

ROTAJ ...•.....•...•....•. 2.v - The pole face angle of a bending

magnet may be varied.

DRIFT .................•.. 3.v - The drift length may be varied.

BEND .......•......••..... 4.vvv - The length, the field, and/or the

n-value may be varied. Alternatively,

the length, the bend angle, and/or the

n-value may be varied.

QUAD .•...•...••....•••... 5.vvO - The length may be varied; the

field may be; the aperture may not be.

MATRIX ..............•.... 14.vvvvvvO - Any of the first order matrix

elements may be varied.

INITIAL COORDINATES ...... 16.0v - Any of the three initial position

floor coordinates or two angle coordinates

may be varied. 

- 11 -

E (1) .•.••....•.••.•.•... 16.0v 1. - The normalized quadratic

term (sextupole component) in the midplane

expansion for the field of a bending magnet

may be varied.

1/R1 ................••... 16.0v 12. - The pole face curvature of a

bending magnet entrance may be varied.

1/R2 ......•....•....•..•. 16.0v 13. - The pole face curvature of a

bending magnet exit may be varied.

SEXTUPOLE ....•...•....••• 18.0v - The field strength may be varied.

SOLENOID ....•••....••...• 19.vv - The length and/or field may be

varied.

BEAM·ROTATIOH .•..••..•••• 20.v - The angle of rotation may be varied.

Variation of any parameter which may also be varied in first order

should be done with caution as it may affect the first-order properties

of the beam line. But, inversely coupled drift spaces straddling a sextupole will, for example, show only second-order effects.

The special parameter cards (type code 16.0) once introduced apply to

all subsequent ~agoets in a beam line until another type code 16.0 specifying the same parameter is encountered. Thus, if such a parameter is

varied, the variation will apply simultaneously to all subsequent magnets

to which it pertains. The variation will persist until the parameter or

vary code attached to the parameter is changed by the introduction of

another type code 16.0 element specifying the same parameter. 

- 12 -

The first-order parameters of a pole face rotation, bend, or solenoid

may not be varied to satisfy a second-order constraint. They are variable in a second-order run only to permit first- and second-order constraints to be imposed simultaneously.

Possible Fitting Constraints

Any elements of the first-order transfer matrix, the floor coordinates, and the system length may now also be fit in a second-order run. 

- 13 -

Beam Matrix fitting Constraints Using Accelerator Notation

The correspondence between the notation for a beam constraint and the

convention for beam parameter input is the same as when accelerator notation is not used. There are five parameters to be specified.

Type code 10.

2 Code digit Ci).

3 code digit Cj).

4 Desired value of parameter to be fit.

5 Desired accuracy of fit (standard deviation).

If i = j, the constraint applies to the i-th physical parameter on the

beam· Ctype code 1.) element. For example, if Bx is to be constrained,

then i = j = 1. If i = O, the constraint applies to the j-th physical

parameter on the shift (type code 7.) element. Thus, to fit 7)x, one

would set j equal to 1.

Some typical accelerator notation beam constraints are as follows:

Desired Optical Condition Typical Fitting Constraint

Bx = . 5 10. 1. 1. . 5 0.001 "Fl"

0. x = 0 10. 2. 2. 0. 0.001 "F2"

13 y = .3 10. 3. 3. .3 0.001 "F3"

0. y = -.2 10. 4. 4. -.2 0.001 "F4"

7) x = . 5 10. 0. 1 . . 5 0.001 "FS"

7)y , = -.2 10. 0. 4. -.2 0.001 "F6" 

- 14 -

Algebraic Combination of Matrix Elements Constraint

Any quantity which can be constrained may also be used to form an

algebraic expression which may itself be constrained. The algebraic combinations are formed using the 22. and 23. type code elements. The

results are placed in numbered registers. To constrain the contents of a

numbered register, five parameters are required:

Type code 10.

2 Code digit 9.

3 Register number Cj).

4 Desired value of register contents.

5 Desired accuracy of fit.

Complete descriptions on the formation of algebraic combinations of

matrix elements and their placement in storage registers are given in the

sections on type codes 22. and 23.

The following example shows the use of the 22, 23, and 10 type code

elements. It shows a fit of R11 + Rzz to a value of .2.

22. -1.

22. -2.

23. 1.

10. 9.

1.

2.

2.

1.

1.

2.

1.

.2

1.

0. 001 .

"Store R11 in register 1"

"Store Rzz in register 2"

"Add contents of register 1 to

the contents of register 2 and

store in register 1"

"Fit contents of register 1 to

a value 0.2" 

- 15 -

INPUT-OUTPUT OPTIONS: Type Code 13.

Several new options for input-output control have been implemented.

The format for type code 13 is:

Type code 13.0.

2 Code number.

The new options with their code numbers are as follows:

Accelerator Notation for Beam Matrix 7.

Both input and output for the beam matrix are in accelerator notation.

The parameters B, a, and n, of accelerator theory are used. Constraints

on the beam are also taken-to be in terms of B, a, and n. The forms of

input and output for the beam matrix are explained under type codes 1 and

7. The form of constraints on the beam are explained under type code 10.

To specify that accelerator notation is to be used in a run, the 13. 7. ;

element must be before the beam element (type code 1.).

Positions of Waists 14.

The longitud-ina~ positions and transverse dimensions of Creal or virtual) waists in both transverse planes are printed at the point of insertion of this element. The waist location and characteristics are computed from the dimensions of the current location assuming infinitely

long drift spaces both upstream and downstream. 

- 16 -

Inverse Transverse Matrix Print Control -4.

The inverse of the current transformation matrix R1 will be printed by

this code. If the program is computing a second-order matrix. the inverse

of the second-order transformation matrix will be included in the print

out.

Refer Transfer Matrix to Original Coordinate System 13.

If the transverse coordinates are rotated through an angle via a 20.

type code entry, the rotation is taken only as specifying the orientation

of the rotated magnet. The transfer matrix, whenever printed, is given

in the unrotated coordinate system. The 13. 13. ; element must precede

any 20. type code elements to be used. When using a 13. 13. ; element,

all coordinate rotations must be (positive or negative) multiples of 90°.

Shift in Reference Trajectory 9.

At the location of this element, the reference trajectory is shifted

to line up with the first-order image of the original beam centroid.

Thus, the reference trajectory of the beam line, as followed by the program will not b~ coutinuous. Quadrupoles will also show dispersive

effects. All the fitting options connected with this element operate as

expected. Thus the original beam centroid displacement parameters can be

varied to fit values of floor coordinates specified after the shift.

Precise Values of Varied Parameters 16.

The values of any varied parameters will be printed in F18.10 format.

This option is useful primarily for investigating the mathematical characteristics of a solution. 

- 17 -

Pole Face Rotation Angle Specification 40., 41., 42., 43.

The pole face rotation angle normally specified with type code 2. may

alternatively be calculated from the bend angle of the associated bend

magnet. The pole face rotation element must still be present. The

options described here merely cause the value of B Cthe pole face rotation angle) to be filled in automatically by the program.

The element specifying the means of determination of the pole face

rotation angle must precede the bending magnet specification, including

pole face rotation angle elements, to which it applies. It will remain

in effect until the option is respecified via another 13. type code element~ Some caution must be taken in the use of these options. If a bend

magnet is segmented, then the pole face rotation angle will be calculated

from the bend angle of the adjacent segment.

C13. 41. ;) Both input and output pole face rotation angles are equal

to half the bend angle.

(13. 42. ;) The entry pole face rotation angle will be zero. The

exit pole face rotation angle will equal the bend angle.

(13. 43. ;) The entry pole face rotation angle will equal the bend

angle. The exit pole face rotation angle will be zero.

C13. 40. ;) The normal option is restored. The pole face rotation

angles will be read from the data. 

- 18 -

Bend Magnet Input Specifications 47., 48.

C13. 48. ;) Bending magnets encountered subsequent to the insertion

of this element are to be specified by the length, bend angle, and

n-value. thus a bend magnet element will take the form

4. L n

C13. 47. ;) The normal option for specification of a bending magnet

is restored by this element. A bend magnet element now takes the form

4. L B n

Lithium or Plasma Lens 97 .• 98.

C13. 98. ;) All quadrupoles subsequent to this element will be taken

as having the same effect in both transverse planes. Thus, if the pole

tip field is positive, both planes will be focusing. If the pole tip

field is negative, both planes will be defocusing.

C13. 97. ;) The normal option is restored for a quadrupole element.

It is now focusing in one plane and defocusing in the other. 

- 19 -

SPECIAL INPUT PARAMETERS: Type Code 16.0

Code Digit for New Special Parameter

2. Ko An integral related to the transverse displacement of the reference trajectory ar;sing from passage through the fringe

field of a bending magnet. If the C16. 5. g/2 ;) element has

been inserted, the program assumes a default value of Ko = .5.

Insertion of the (16. 2. Ko ;) element supplants this default

value. The transverse displacement is given by Ax = K0 g2 /p.

The actual value of Ko is given by a double integral

Ko = I~ ds I~ CB0 - BCs)) ds

g2 Bocos 2B -s, -s,

where s is measured perpendicular to the pole face. The

parameter B is the pole face rotation angle and Bo is the

field interior to the magnet. The value of s 1 is chosen so as

to be well within the interior of the magnet. 

- 20 -

COORDINATE ROTATION: Type Code 20.0

Normally, the transformation matrix is expressed in the rotated system

of coordinates. If a 13. 13. element is included in the data before the

beam Ctype code 1.) element, the transformation matrix will be expressed

in the original (unrotated) coordinate system. In this latter case, all

coordinate rotations must be (positive or negative) multiples of 90°. 

- 21 -

ALGEBRAIC COMBINATIONS OF MATRIX ELEMENTS - DEFINING REGISTER CONTENTS:

Type Code 22.

Any of the quantities which can be constrained can also be used to

form algebraic combinations which in turn may be constrained. A set of

ten storage registers is available for formation of such algebraic combinations. The 22. type code element is used for storage into these registers. Four parameters are required.

Type code 22. (indicating a storage operation).

2 Code digit i.

3 Code digit j.

4 Register number.

The indices i and j have the same meaning as for the fitting constraint (10.) type code. Thus, for example, if one wished to place the

~ matrix element into register l, one would use the following element:

22. -3. 4. 1.

Numerical constants may also be placed in registers for use in forming

algebraic expressions. If i is equal to 100, then the third parameter is

taken to be the numerical constant. For example, the element:

22. 100. 3.14159 7.

causes the number 3.14159 to be placed in register 7.

The contents of a register may be constrained by a fitting constraint

element Ctype code 10.). Details are given in the section describing

fitting constraints. 

- 22 -

ALGEBRAIC COMBINATIONS Of MATRIX ELEMENTS - FORMING COMBINATIONS:

Type Code 23.

Algebraic combinations of elements in storage registers may be formed

and themselves placed in storage registers. The combinations formed may

be constrained or used to form further algebraic combinations.

five parameters are required:

- Type code 23. (signifying an algebraic combination).

2 First input storage register.

3 Second input storage register.

4 Algebraic operation to be performed.

· 5 Output storage register.

The algebraic operations are referred to by number according to the

following list:

addition (+)

2 subtraction (-)

3 multiplication (X)

4 division C+>

5 squ~re ~oot c,[">

If the square root is selected, only the first input register is used.

A dummy index should be inserted for the second input register.

As an example of the use of the algebraic combination element we

illustrate the process of taking the quotient of the contents of registers 2 and 7 and placing the result in register 2:

23. 2. 7. 4. 2. 

- 23 -

DEFINED SECTION: Type Code 24.

A system may contain a section which is repeated at some later point.

The repeated section may not occur immediately after the original section, so that the repeat element Ctype code 9.) is not appropriate. It

may also be useful to repeat a section, but with the elements listed in

the reverse order.

The 24. type code element allows definition of the section to be

repeated. It also indicates the locations at which the section is to be

repeated and whether the repetition is to be forwards or backwards.

Three parameters are required:

- Type code 24.

2 Code digit.

3 Section name. The section name is a maximum of four

characters long and is enclosed in single quotes.

The meaning of the code digit is as follows:

Indicates the beginning of the section to be defined.

2 Indi_cates the end of the section to be defined.

3 The section named is to be repeated at the present

location in the forwards direction.

4 The section named is to be repeated at the present

location with the elements in reverse order.

A number of rules apply to the use of the defined section element.

Each defined section must have both its beginning and its end indicated,

and the beginning must precede the end. The definition of a section must 

- 24 -

completely precede its use. A given name can be used only once to define

a section, although it can be used many times to indicate a repeat of a

section. A defined section may not refer to itself, either explicitly or

implicitly. The number of defined sections is limited to 10. The total

z rotation Ctype code 20.) within a defined section must sum to zero.

Finally, defined sections must nest properly with the repeat code Ctype

code 9.). Defined sections, however, need not nest properly with each

other.

Example of a

24. 1.

5. 10.

3. 8.

5. 10.

24. 2.

24. 3.

24. 4.

Defined

'DOUB'

5. 2.

-5. 2.

'DOUB'

'DOUB'

'DOUB'

Section

Begin defined section

End defined section

Repeat defined section

Repeat defined section

in reverse order