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1/96 animus-p.14

Rigorous Field Analysis of Superconducting Magnets and the Influence on
Nonlinear Dynamics in Particle Accelerators

By

Michael Lindemann

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy

1998

ABSTRACT

Rigorous Field Analysis of Superconducting Magnets and the Influence on
Nonlinear Dynamics in Particle Accelerators

By

Michael Lindemann

The nonlinear dynamics of particles in modern accelerators are governed by the
fields of the guiding and focusing superconducting magnets. While for the linear de-
sign of the lattice it is suflicient to treat the fields by a rather coarse approximation,
a thorough analysis of the magnetic fields is necessary in order to study the nonlinear
effects. As an example, the Large Hadron Collider (LHC) is studied, using detailed
field data for the High Gradient Quadrupoles in the interaction region. The influence
of the resulting nonlinearities on the dynamics is analysed via high-order maps deter-
mined with Differential Algebraic (DA) techniques and the code COSY INFINITY.
Normal form methods are utilised to determine amplitude dependent tune shifts as
well as resonance strengths. It is shown that the end effects change the nonlinear

characteristics of the lattice significantly.

This made a rigorous treatment of the fields desirable, which is presented in the second
part. Using DA-techniques the Taylor expansion of the field is calculated. Based on
this expansion two analytical algorithms to determine the multipole content of the

field are developed.

ACKNOWLEDGEMENTS

First of all I would like to thank my wife Wencke and my parents for their continued

support during my studies.

I am grateful to Prof. Martin Berz for giving me the opportunity to come to Michigan
State University and work in his outstanding research group. This group consisted of
Béla Erdélyi, Jens Hoefkens, Khodr Shamseddine, Kyoko Makino, Jens von Bergmann
and Lars Diening, who have all in various ways supported me. Especially I would
like to express my gratitude to Béla Erdélyi, who was my tutor in the beginning and
became my friend and partner in research later. I also appreciate the mighty help in

all questions related to computers by my friend Jens Hoefkens.

Outside the group I got the chance to work with several scientists from Fermi National
Accelerator Laboratory, namely Dr. Jim Holt, Dr. Carol Johnstone, Dr. Weishi Wan
and Dr. Gianlucca Sabbi. I am grateful to them for many stimulating discussions.
Specifically I would like to thank Gianlucca Sabbi for his help with the magnetic field

calculation.

I appreciated the financial support from the German National Scholarship Founda-

tion.

iii

Contents

LIST OF TABLES
LIST OF FIGURES

1 Introduction

1.1 The Large Hadron Collider ........................
1.2 Methods and Tools ............................
1.2.1 Coordinates and Maps ......................
1.2.2 Differential Algebras .......................
1.2.3 Computation of Transfer Maps .................
1.2.4 Tune Shifts and Resonance Strengths ..............
1.2.5 Fields and Potentials ......................

2 The AT to COSY Converter

3 The SXF to COSY Converter

3.1 Introduction ................................
3.2 The Converter ...............................
3.2.1 General Features .........................
3.2.2 Description ............................

4 Beam Dynamics Studies

4.1 Introduction ................................
4.2 Magnetic Field Data ...........................
4.2.1 Introduction ............................
4.2.2 Fitting ...............................
4.3 Lattice Description ............................
4.4 Results ...................................
4.4.1 Tune Shifts ............................

iv

vi

vii

mwwHI-I

18
20
20

23

25
25
25
25
26

28
28
29
29
31
32
35
35

4.4.2 Resonance Analysis ........................ 39

5 Differential Algebraic Field Calculation 43
5.1 Introduction ................................ 43
5.2 Magnetic Field Calculation by ROXIE ................. 43

5.2.1 Introduction ............................ 43
5.2.2 Image Currents .......................... 44

5.3 Magnetic Field Calculation in COSY .................. 44
5.3.1 Introduction ............................ 44
5.3.2 Line Currents ........................... 45
5.3.3 Implementation in COSY .................... 46

5.4 Extraction of the Multipole Content ................... 49
5.4.1 Analytical Fourier Transform .................. 50
5.4.2 Example .............................. 51
5.4.3 Differential Algebraic Multipole Extraction ........... 53
5.4.4 Example .............................. 58

A SXF to COSY Converter 62

List of Tables

4.1
4.2
4.3

4.4

4.5
4.6

5.1

5.2

5.3

Time Shifts in the a: -— a plane with and without fringe fields.
Tune shifts in the y - b plane with and without fringe fields.

Tune shifts in the a: — a plane with fringe fields in IR 5 and in both
IR1 and IR5 ...............................

Tune shifts in the y-b-plane with fringe fields in IR 5 and in both IR1
and IR5 ..................................

Resonance strengths with and without fringe fields. ..........

Resonance strengths with and without refitting the linear tune.

Skew multipoles as obtained by the DA-based extraction scheme in
comparison with the results of the analytical Fourier transformation
at s=0. ..................................

S-Derivatives of the solenoid component of the field as obtained by the
DA—based extraction scheme in comparison with the results of the an-

alytical Fourier transformation at 3:0. n is the order of the derivative.

Comparison of the allowed multipoles in the lead end at z = 0cm in
the coordinate system defined in [23] calculated by the DA-Extraction
and the analytical Fourier transformation. ...............

vi

36
36

37

39
40
41

58

59

List of Figures

2.1

4.1

4.2

4.3
4.4
4.5

5.1
5.2
5.3

5.4

5.5

5.6

5.7

The AT to COSY converter ....................... 24

Comparison of the quadrupole component of the field in the return end
as computed by COSY and ROXIE at different radii. From top left to
bottom right: 1‘ = 5 mm, r = 10 mm, 1' = 20 mm and r = 30 mm. . 33

Comparison of the quadrupole component of the field in the lead end
as computed by COSY and ROXIE at different radii. From top left to
bottom right: 1‘ = 5 mm, 1‘ = 10 mm, 1' = 20 mm and r = 30 mm. . 34

Layout of the left low-,6 triplet. ..................... 35
Tune footprint without fringe fields (top) and with fringe fields (bottom). 38

Average logarithmic resonance strength for orders 3 to 7. The values
are given by squares for the case without fringe fields and circles for

the case with fringe fields. ........................ 42
Demonstration of the general element GE in COSY .......... 48
Model for the High Gradient Quadrupole’s lead end [23]. ....... 52
Model for the High Gradient Quadrupole’s return end [23]. ..... 52

Comparison of 52 in the lead end as computed by the differential alge-
braic field calculation in COSY and as given in [23] for different radii.
From top left to bottom right : r = 5 mm, 1' = 10 mm, r = 20 mm
and r = 30 mm. ............................. 53

Comparison of 52 in the return end as computed by the differential
algebraic field calculation in COSY and as given in [23] for different
radii. From top left to bottom right : r = 5 mm, 1' = 10 mm, 1' = 20
mm and r = 30 mm. ........................... 54

Comparison of the normal multipoles b2; (top) and b6,6 (bottom) with

52 respectively 56 in the lead end for r =2 5 mm (left) and r = 20 mm
(right). .................................. 60

Comparison of the skew multipoles (12,2 (top) and (16,6 (bottom) with
62 respectively 66 in the lead end for r = 5 mm (left) and r = 20 mm
(right). .................................. 61

vii

Chapter 1

Introduction

1.1 The Large Hadron Collider

The Large Hadron Collider (LHC) will be the next accelerator to be built at CERN
in Geneva. One of the the main goals of LHC is to study the actual mechanism
for symmetry breaking in the electroweak sector of the standard model. This phe-
nomenon is associated with the nature of the Higgs mechanism, the existence of the

Higgs particle and the origin of mass.

The LHC poses unprecedented challenges in terms of accelerator physics :
High Luminosity

In the LHC the energy available in the collisions between the constituents of the
protons, the quarks and gluons, will reach the 8 TeV range, which is about 10 times
that of the Fermilab Tevatron. In order to maintain an equally effective physics
program at a higher energy E the luminosity of a collider which is proportional to
the number of collisions per second, should increase in proportion to E2. This is
because the cross section of the particle decreases like 1/E2. Whereas in past and
present colliders the luminosity culminates around L = 1032cm"2s‘1, in the LHC it

2

is expected to reach L = 1034cm‘ 8—1. The luminosity is given by the formula [27]

1

L 1 7 [£][Nkf]F (1.1)

= EB: 45,,

where 'y is the energy of the protons divided by their rest energy, ,8“ is the value of
the betatron function, corresponding to the width of the beam at the collision point,
N is the number of protons in the k bunches, 6,, is the invariant transverse emittance,
f is the revolution frequency and F is a reduction factor due to the finite crossing angle
which is 0.9 for the LHC. In formula (1.1) 7 is limited by the bending magnet field
and 6* is similarly largely determined by the available technology of high gradient
quadrupole lenses. The first bracket is proportional to the beam-beam parameter
which is limited by the electromagnetic interaction of colliding bunches. The second
bracket is proportional to the beam current, which has to be chosen very large. This
will be achieved by filling each of the two rings, in which the particles are rotating
in opposite direction, with 2835 bunches of 1011 particles each. The resulting large
beam current 1;, = 0.53A is particularly challenging in a machine made of delicate

superconducting magnets operating at cryogenic temperatures.

Furthermore it is necessary to decrease B“ in order to reach the luminosity goal.
This requires the use of High-Gradient Quadrupoles in the interaction region. These
quadrupoles combine relatively short length, large aperture, and short focal length
with a rather peculiar configuration of the return coils, all of which enhances the
relevance of their fringe field effects. This is why a substantial part of this work will

deal with the influence of the fringe fields on the nonlinear dynamics of the particles.

Longterm Stability

The nonlinear components of the guiding and focusing magnetic fields of the machine
are also very important for the question of long term stability. The beams will

be stored at high energy for about 10 hours. During this time the particles make

four hundred million revolutions around the 26.7 km circumference of the machine.
Meanwhile the amplitude of their oscillations around the central orbit should not

increase significantly, because this would dilute the beams and degrade luminosity.

1.2 Methods and Tools

In this section we present the main methods used in this work. The goal is to provide
the reader with a basic intuitive understanding of these tools. We follow closely and

summarise [1], [2] and [3]. The methods are discussed in a broader context in [4].

1.2.1 Coordinates and Maps

Usually when studying dynamics, the time t plays the role of the independent variable,
and we study the motion of positions 57' and velocities 27 or momenta 15' as coordinates.
Using the Lagrange mechanism, it is easy to transfer to new coordinates, in particular
the coordinates that describe the relative motion around the reference orbit. Further-
more, instead of using If , we usually use the arc length 3 along the reference orbit as

independent variable.

For the understanding of the motion in relative coordinates, let us assume we have
studied and understood the motion of the reference orbit. In the case when there is no
field at all, this reference orbit will merely follow a straight line. Furthermore, there
are a host of devices used in accelerators that have fields, but along one given straight
line, all the fields vanish, and the device is lined up in such a way that the reference
particle follows this line. Another important device uses magnetic fields, and along
the reference orbit one tries to hold them constant, in which case the reference orbit
is circular, at least within the element. In all other cases, it is usually necessary to

numerically integrate the reference orbit.

We assume the position and momenta of the reference particle are 17,8 f(s), fire f(3)
are known. As a technical detail, let us also assume that for no point 3, we have
13'"; f(3) II é; , i.e. the motion is never pointing straight up,which for most real acceler-
ators is no limitation whatsoever. Let furthermore pt be smaller than the minimum
radius of curvature that the reference orbit experiences in the section of the machine
that we want to study. We now consider a ”flexible tube” of radius pt centred around
the reference orbit, and restrict the particles that we want to describe to only those
within the tube. Again, for practical devices this hardly represents a limitation; in
the LHC, for example, the ”tube” would be more than 4 km wide, much larger than

the region required by the beam particles.

For any particle within the tube, there is now a closest point on the reference
orbit; because only particles within the tube are allowed, this point is indeed unique.
Let s be the arc length at this point, and Fref(s) the position of the reference particle
on the reference orbit. Then the relative coordinates of the point F are obviously
F — Fref(s).

Let now 6', be a unit vector in the direction of 118;. Consider now the plane
perpendicular to 6,. Of all the unit vectors in this plane, let é'y be the one with the
largest ”upward” component; because fire, and hence (2', are not allowed to go straight
up, this vector is well defined. Finally choose a third vector é} as e; = 6,, x é}.
Because 6,, has maximum ”upward” component, 6,, has vanishing upward component

and hence lies in the horizontal plane.

Denote now by x the component of F— Fref(s) in the direction of (3'3, and by y the
component of 1" — 77,8):(3) in the direction of 63,. Similarly, define p3 and py to be the

momentum components in the directions é’x and é’y.

Furthermore, denote by 6 the relative difference between the total (kinetic plus

potential) energy E of the particle under consideration and the reference energy E0,
i.e. 6 = (E - E0)/E0. Finally, introduce a space-like variable I to be the time of
flight t minus the time of flight to of the reference particle, multiplied by a constant
k of dimension ”velocity”, i.e. l = k (t — to) . Then we form the vector 2 of particle

optical coordinates as

f 1" )

y
'* [2 [C(t—to)
Z = 1.2
0’ =px/p0 ( )
b=py/po

 

 

\5=(E-Eo)/Eo)

where p0 is some previously chosen scaling momentum; a natural choice may be to

select the momentum of the reference particle at the beginning.

Note that due to the definition of Z, the reference particle itself corresponds to

Z = 0, and hence the vector 2 does indeed describe the relative motion.

The entire action of a beam physics device can now be expressed by how it ma-
nipulates the coordinates in the vector Z. In fact, usually a set of initial conditions
20 at position so uniquely determines the future evolution and hence 2 at any later
position 3, so we can define a function relating the initial conditions at so to the

conditions at 3 via

2(3) 2 M (30, s) (Z(so))

The function M (.90, s), which formally summarises the entire action of the system,
is of great importance for the description and analysis of beam physics systems. It is
often called the transfer function, the transfer map, or simply the map of the system.

Note that the transfer functions satisfy the relationship

M (81,32) 0M (30,31) 2 M (50:32):

which merely says that transfer maps of systems can be built up from the transfer

maps of the pieces.

Since M describes the motion in relative coordinates, we always have

M(6)=6

Furthermore, since by the very definition of a beam, the coordinates of Z are
”small”, M is usually only weakly nonlinear; because of this, it is often represented
by its Taylor series expansion. In the following section we will present the differential
algebraic methods which allows for the computation of the Taylor series expansion

very efficiently.

1.2.2 Differential Algebras

In this section we will provide the mathematical background of the theory of differ-
ential algebras required for the study of nonlinear particle dynamics via the Taylor

series representation of maps.

Historically, the treatment of functions in numerics has been done based on the
treatment of numbers; and as a result, virtually all classical numerical algorithms
are based on the mere evaluation of functions at specific points. As a consequence,
numerical methods for differentiation, which are so relevant for the computation of
Taylor representations of the map, are very cumbersome and prone to inaccuracies

because of cancellation of digits, and not useful in practice for our purposes.

The method of differential algebra is based on the observation that it is possible
to extract more information about a function than its mere values, specifically its
Taylor expansion. We define the operation T to be the extraction of the Taylor

coefficients of a pre—specified order n of the function. In mathematical terms, T is an

equivalence relation, and the application of T corresponds to the transition from the
function to the equivalence class comprising all those functions with identical Taylor
expansion to order n. Obviously Taylor coefficients of order n for sums and products
of functions as well as scalar products with reals can be computed from those of the
summands and factors. This means that the set of equivalence classes of functions can
be endowed with well-defined operations, leading to the so—called Truncated Power
Series Algebra [12],[13].This led to a method to extract maps to any desired order
from a computer algorithm that integrates orbits numerically. Similar to the need
for algorithms within floating point arithmetic, the development of algorithms for
functions followed, including methods to perform composition of functions, to invert
them, to solve nonlinear systems explicitly, and to introduce the treatment of common

elementary functions [6], [3].

However, very soon afterwards it became apparent [1] that this only represents
a half-way point, and one should proceed beyond mere arithmetic operations on
function spaces of addition and multiplication and consider their analytic operations
of differentiation and integration. This resulted in the recognition of the underlying

differential algebraic structure and its practical exploitation.

In passing we note that in order to avoid loss of order, in practice the derivations
have the form 8 2: h-d/dzri, where his a function with h(0) = 0. As a first consequence,
it allowed to construct integration techniques to any order that for a given accuracy
demand are substantially faster than conventional methods [3]. Subsequently, it was
realized that the differential algebraic operations are useful for a whole variety of
other questions connected to the analytic properties of the transfer map [6]. It was
possible to determine arbitrary order generating function representations of maps [7],
[3] and normal form methods [8], [9] could be performed to arbitrary order. [3] On

the practical end, based on the latter concept, there are also several improvements

regarding methods of computational differentiation [10],[11].
In order to show how this method works in practice , we first address the simplest

case of differential algebras, the structure 1D1.

The Structure 1D1

Consider the vector space R2 of ordered pairs (a0, a1), a0, a1 6 R in which an addition

and a scalar multiplication are defined in the usual way:

(00,611) + (130,61) 2 (Clo + b0,a1+ b1) (1.3)

t-(ao,a1) = (t - a0,t - a1) (1.4)

for a0, a1, b0, b1 6 R. Besides the above addition and scalar multiplication a multipli-

cation between vectors is introduced in the following way:

(ao,a1)-(b0,b1) = (0.0 ° ()0, (10 ° b1+ 01' ()0) (1.5)
for a0, (11, b0, b1 6 R. With this definition of a vector multiplication the set of ordered
pairs becomes an algebra, denoted by 1D1.

Note that the multiplication is the same one would obtain by multiplying (a0 +

a1 - 11:) and (be + b1 - :13) and keeping terms linear in x.

In the same way than in the case of complex numbers, one can identify (a0, 0) as
the real number do. Where in the complex numbers, (0,1) was a root of -1, here it

has another interesting property:

(0,1) - (0,1) = (0,0) (1.6)

which follows directly from equation (1.5). So (0,1) is a root of 0. Such a property
suggests thinking of d = (0,1) as something infinitely small, small enough that its
square vanishes. Because of this we call d = (0,1) the differential unit. The first
component of the pair (a0, a1) is called the real part, and the second component is

called the differential part.

It is easy to verify that (1,0) is a neutral element of multiplication, because

according to equation (1.5)

(1,0) - (a0,a1) = (a0,a1) - (1,0) = (ao,a1) (1.7)

It turns out that (a0, a1) has a multiplicative inverse if and only if no is nonzero;

so 1D1 is not a field. In case (10 ¢ 0 the inverse is

(a0,a1)"1=(aiO,—:—é) (1-8)

Using equation (1.5) it is easy to check that in fact ((10, a1)“1 - (amal) = (1,0).

The space 1D1 is a subspace of the field R“ introduced in Nonstandard Analysis
[8],[9]. Besides the usual real numbers, R“ contains a variety of infinitely small and
infinitely large quantities. The outstanding result of the theory of Nonstandard Anal-
ysis is that differentiation becomes an algebraic problem: a function f is differentiable

if and only if for any arbitrarily small quantity 6, the real part of the quotient

f(z+6)—f(x)
a

 

(1.9)

is independent of the choice of the specific 6. Thus, given any differentiable function
f, we can compute its derivatives by just evaluating the formula for a special choice

of 6. We choose 6 = d = (0, 1) and thus obtain

10

fl$+®-f@)
d

D[f(x + d) — f(x)] = D[f(:c + d)] (1.10)

WW): Rl
we)

 

]or

where R denotes the real part and D denotes the differential part. In the last step
use has been made of the fact that f (:17) has no differential part. Hence differential
algebras are useful to compute derivatives directly, without requiring an analytic

formula for the derivative and without the inaccuracies of numerical techniques.

The computation of derivatives shall be illustrated in an example using the fol-

lowing function:

 

 

1
flfl=$+l (LU)
The derivative of the function is
_1,_ — 1

Suppose we are interested in the value of the function and its derivative at x=2.

We obtain

flfl=§.flm=—— am)

Now take the definition of the function f in equation (1.11) and evaluate it at

2 + d = (2,1). One obtains:

 

11

 

 

_ 1

— (211)+(%1_%)

_ 1

_ (31%)

= «2%

= (E,-%) (1.14)

As we can see, after the evaluation of the function the real part of the result is
just the value of the function at :1: = 2, whereas the differential part is the derivative

of the function at a: = 2.

This is exactly what was to be expected from the theory of Nonstandard Analysis.
However, for the sake of not relying on the quite advanced techniques of this relatively
new field of mathematics, we also present an elementary but less elegant proof of the

result.

By our choice of the starting vector (2,1), initially the vector contains the value
I (2) of the identity function I : a: ——> :1: in the first component and the derivative of

I '(2) = 1 in the second component.

Now assume that in an intermediate step two vectors of value and derivative
(g(2), g’(2)) and (h(2), h’(2)) have to be added. According to (1.3) one obtains (g(2)+
h(2), g’ (2) + h’ (2)) But according to the rule for the differentiation of sums, this is

just the value and derivative of the sum function (g + h) at :1: = 2.

The same holds for the multiplication: Suppose that two vectors of value and
derivatives (9(2), 9’ (2)) and (h(2),h’ (2)) have to be multiplied. Then according to
(1.5) one obtains (g(2) - 11(2), 9(2) - h’(2) + g’(2) - 11(2)). But according to the product

rule, this is just the value and derivative of the product function (g - h) at :r = 2.

The evaluation of the function f at (2,1) can now be viewed as successively

12

combining two intermediate functions g and h, starting with the identity function
and finally arriving at f. At each intermediate step the derivative of the intermediate
function is automatically obtained as the differential part according to the above

reasoning.

An interesting side aspect is that with the search for a multiplicative inverse in
equation (1.8) one has derived a rule to differentiate the function f (:r) = 1/:r without

explicitly using calculus rules.

After discussing the algebra 1D1 and its virtues for the computation of derivatives,
we now address the most general differential algebra, the structure “D”. It will
eventually allow us to arithmetically compute partial derivatives of functions of 12

variables through order n.

The Structure an

We define N (71,12) to be the number of monomials in 12 variables through order n.
We will show that N(n, v) = %)—' = C(n + 11,12) where C(z',j) is the familiar bino-
mial coefficient. First note that the number of monomials with exact order n equals
N (71,1) — 1). This is true because each monomial of exact order n can be written
as a monomial with one variable less times the last variable to such a power that
the total power equals 71. Thus we have N(n,v) = N(n — 1,12) + N(n,v — 1): the
number of monomials in 12 variables through order n equals the number of monomials
of one order less plus the ones of exact order n. This recursive relation is satisfied by
C(n + 11,12). Since also obviously C(l + 1,1) = 2 = N(1,1), the formula follows by

induction.

Now assume that all these N monomials are arranged in a certain manner order by
order. For each monomial M we call 1M the position of M according to the ordering.

Conversely, with M1 we denote the I th monomial of the ordering. Finally, for an I

13

with M1 = 11:? ~... 45:," we define F, 2 2'1! - . M.

We now define an addition, a scalar multiplication and a vector multiplication on

R” in the following way:

(0.1,...,aN)+(b1,....bN) = (0.1+ b1, ..., (IN + bN) (1.15)
t' (0.1,...,0.N) = (t-a1,...,t-aN) (1.16)
((1.1, ...,CLN) ' (b1, ....bN) '2 (61,...,CN) (1.17)

where the coefficients 0,- are defined as follows:

 

(1.18)

To help clarify these definitions, let us look at the case of two variables and second
order. In this case, we have n = 2 and v 2: 2. There are N 2 C(2 + 2,2) = 6

monomials in two variables, namely

1, :12, y, 23:6, my, 111/ (1-19)

As an example, using the ordering in (1.19), we have Izy = 5 and M3 = y. Using

the ordering in (1.19), we obtain for CI through as in equation (1.18):

C1 = al-bl

C2 = al-b2+a2-b1

14

al-b3+a3-bl

C3
C4 = 2-(a1~b4/2+a2-b2+a4-b1/2)
C5 = a1°b5+a2-b3+a3-b2+a5-b1

66 = 2'(0.1°b6/2+a3°b3+0.6'b1/2) (1.20)

On "D” we introduce a third operation 8,:

8,,(a1, ...,aN) = (01,...,cN) (1.21)
with
Ci: { 0 if M,- has ordern (122)
WWW) else

So 6,, moves the derivatives around in the vector. Suppose a vector contains the
derivatives of the function f, then applying 8,, to it one obtains the derivatives of
5655 through one order less. With this third operation, "DU becomes a Differential
Algebra as defined in [12].

While in 1D1, d = (0,1) was an infinitely small quantity, here we have a whole
variety of infinitely small quantities that have the property that high enough powers
of them vanish. We give special names to the ones in components I belonging to
first order monomials, denoting them by dMI. In the example of 2D2, we have
dx = (0, 1,0, 0,0,0) and dy = (0, 0,1,0, 0, 0). It then follows from the theory of

Nonstandard Analysis that instead of equation (1.10) we obtain

f(:1:+d:1:,y+dy) =

(191 5’1 62f 52f 32% )
’6x’6y’632’6z6y’0y2 ,y

 

(1.23)

15

In the general case of v variables and order n, after evaluating f in the differential

algebra one obtains:

ai1+i2+...+ivf

 

. . . = CI 1' 1’ (1.24)
8:12:11 83:32 . . .8333! (.11......,,v)
where I x11. -:1:‘”) is the index of the monomial (£13211 - - 1:2”), as defined in the
1 ... v

beginning of the section.

Important Functions on Differential Algebras

In this section we will generalise standard functions like exponentials, logarithmic
and trigonometric function to differential algebra. As we will see below, virtually all

functions existing on a computer can be generalised in a straightforward way.

We start our discussion by noting that for any differential algebra vector of the
form (0, a1, ..., (IN) E nDv, i.e. with a zero in the component belonging to the zeroth

order monomial, we have the following property:

(0,111, ...,aN)‘ = (0,0, ....,0) for z‘ > n (1.25)

which follows directly from the definition of the multiplication in “D” defined in

equation (1.17).

Let us begin our discussion of special functions with the exponential function
exp(:r). Assume we have to compute the exponential of a differential algebra vector
that has already been created by previous operations. First we note that the func-
tional equation exp(:1: + y) = exp(a:) . exp(y) also holds in Nonstandard Analysis. As

we will see, this facilitates the computation of the exponential considerably.

exp[(a0,a1,a2,...,aN)] = exp(a0)~exp[(0,a1,a2,...,aN)]

 

 

i=0 2'
n 01 a 1'“, i

: exp(a0)-Z( “1 a; a”) (1.26)
i=0 '

In the last step use has been made of equation (1.25) which entails that the sum
has to be taken only through order n and thus allows the computation of the root
in finitely many steps. Hence the evaluation of the real number exponential exp(a0)
which internally on a computer requires a power series summation and hence cannot
be done accurately, is more subtle then the rest of the operations in differential

algebra.

A logarithm of a differential algebra vector exists if and only if an > 0. In this

case one obtains

a a
log[(a0,a1,a2,...,aN)] = log[a0- (1+(0,a—-1,—2-,...—N
0.0 0.0 (1.0

00.1 (12 (IN -
= (log(a0)0 .0 +1)"+11.—,—,...,—z
) :(—ano 00 00)

n

- 1 0,1 02 (IN -
= 1 ,0,...,0 —1'+1— 0,—,——,...,—'
(og<a.> Hg ) z.( a. a0 a0)

(1.27)

Again use has been made of the fundamental property of the logarithm log(m-y) =
log(:1:) + log(y) which transforms directly into Nonstandard Analysis and leads to

simplifications by virtue of equation (1.25).

As the last example, we will derive a formula for the root function. Even though
there is a direct method to compute roots by solving a set of linear equations for the

coefficients of the root, we present here a technique based on power series following

17

an approach similar to the exponential and logarithm. The root has the following

power series expansion:

 

V 1+ ”3 : :81)” '23. 1711.? (Jag) f “'28)

Using this formula and the definitions of addition and multiplication (1.15), (1.17),

one directly obtains for the square root of a differential algebra vector:

 

\/( (C110, 01, 0.2, (IN)

= +(0_1_2_Ha_N
r-1+(\/ ,)

€10

 

= fig.:(_::1°23o-uu(2i—j3) .(0__ a1 11—2 ....fli)‘

 

 

4...(2Z) 0.0 (1.0 0.0
--3 ...-(21—3) a1 a2 aN -
= . — _ 000 1 1'2
\F 2(— 2 4 (2,.) (.a0.a0, .ao) ( 9)

Using the addition theorems for sine and cosine, one obtains formulas with finite

sums in a quite similar way; in general, suppose a function f has an addition theorem

of the form

f(a + b) = 90(0) (1'30)

and ga(b) can be written in a power series, then by the same reasoning its differential
algebraic extension is computable exactly in only finitely many steps. In practice it
turns out that this can be done for all commonly supported functions in a FORTRAN

computer environment.

18

1.2.3 Computation of Transfer Maps

Differential Algebras can be used very efliciently to compute the transfer map of

particle optical systems in its Taylor series representation.

To illustrate this, let us start the discussion with a very simple example, the
midplane motion in a 90° homogenous bending magnet. Let :17,- and a,- = sin(a,~)
denote the initial distance and scaled transverse momentum relative to the reference
trajectory. Then we are interested in the values :17, and a, = sin(ozf). Since the

trajectories in the magnet are circles, we can readily infer:

A = R sin(oz,-) = R a,-

B = R(1—cos(a.-))+:1:,°=R(l—y/l—a?)+:1:,-

. B
of = s1n(af)= -E
:r, = A—R (1—cos(af))=A—R (1— (/1—a}) (1.31)

These equations allow the computation of the final coordinates :1: ha, in terms
of the initial coordinates 12,, a,. However, taking these equations and performing all
operations in differential algebra allows us to obtain all derivatives of 51:}, a, with
respect to 33,-, a,. These so obtained derivatives, evaluated at x, = 0, a,- = 0, are then
the expansion coefficients of the map. For the sake of clarity, let us explicitly show

how :13, and of are computed.

Using the ordering in (1.19) and identifying the variable a with y, we obtain using

the arithmetic defined in equations (1.15), (1.16) and (1.17)

:r,- = (0,1,0,0,0,0)

19
a,- = (0,0,1,0,0,0)
A = (0,0,R,0,0,0)

B = (0,1,0,0,0,R)

1
(If 2 (0,—§,0,0,0,—1)
1
z, = (0,0,R,——I§,0,0) (1.32)

As a quick check, the fact that the second component of :13, is zero implies that

g3. : 0 and hence (x,x) = 0 which is a well known property of 90° bends.

In case an additional particle optical element is to follow this bending magnet,
one does not have to start all over evaluating this new element at :12,- = (0, 1, 0, 0, 0,0),
a,- = (0,0, 1,0, 0,0), but one can start already with 51:, and af of equation (1.32).
This way one can save the usually quite involved concatenation process and increase

performance significantly.

In the general case in which no closed solution of the problem exists, there is
still a way to computationally relate the final to the initial coordinates, by numer-
ical integration of the equations of motion. In this case, the final coordinates are
still computed from the initial coordinates using standard arithmetic and functions,

however the relations are more complex than in the case of the homogeneous sector.

Now performing all these operations in differential algebra automatically gives all
desired derivatives of the transfer function, regardless of the form of the equations
of motion. Still there are more elegant DA-based schemes for numerical integration,

which are described in [3].

20

1.2.4 Time Shifts and Resonance Strengths

We study the Poincare section of an accelerator lattice. In the linear approximation
the particles move on ellipses, called invariant ellipses in this section. This is shown
for example in [5]. The average angles in the :1: —- a and y — b plane by which the
particles rotate during one revolution around the accelerator ring are called the tunes
of the system. They usually depend on the distance of the particle from the reference
particle in :1: — a respectively y — b plane, which is called the amplitude, as well as on
parameters like the chromaticity. The tunes are of prime importance for the stability
analysis of a lattice, as a low order resonance between them entails instability of the
particles. The tune shifts with amplitude and parameters are an outcome of the DA
Normal Form algorithm described in [3], as well as the resonance strengths, which
tell how sensitive the system is to a certain resonance. Given the emittance of the
beam we can use the nonlinear tune shifts to calculate the tune foot print. This is
the region in the two dimensional tune space which is occupied by the beam. There

should be no low order resonances with high resonance strength in the tune foot print.

1.2.5 Fields and Potentials

As a significant part of this thesis will be devoted to the calculation of magnetic fields
an introduction to this topic, especially with respect to the terms used later on , is
in order. We will restrict ourself to systems with straight reference orbit, because the
quadrupoles with which we will deal are falling in this class. Many elements with a
straight reference orbit possess a certain rotational symmetry around the axis of the
reference orbit, and it is most advantageous to describe the potential in cylindrical
coordinates with a z-axis that coincides with the reference orbit. We first begin by

expanding the r and 0‘) components of the potential in Taylor and Fourier series,

21

respectively; the dependence on the cylindrical z coordinate, which here coincides

with the particle optical coordinate s, is not expanded. So we have

hi) f: M,” )COS ([05 + 0!: [)7‘k (1.33)

(:0

In cylindrical coordinates, the Laplacian has the form

AV:

2 2
13(7‘3V) 18V 8__8V:0; (1.34)

1' Br 81“ 1‘2 8052
By inserting the Fourier-Taylor expansion of the potential, we obtain the following

recursion relation
Ml(2n)(8 )
Ml+2n,z(8) = n (1.35)
H (l2 — (1+ 211)?)
v=l

 

The Mu(s) are free parameters as are the 911,0 The M1,) that cannot be obtained
by the above recursion are zero. The number I is called the multipole order, as it
describes how many oscillations the field will experience in one 27r sweep of 05. The free
term M1,)(s) is called the multipole strength, and the term 01,, is called the multipole
phase. Apparently, frequency 1 and radial power k are coupled : The lowest order in
r that appears is l, and if the multipole strength is s-dependent, the powers I + 2,
l + 4, will also appear. These terms that are induced by the s—dependence of the
multipole strength M1,,(s) via equation (1.35) are called pseudo-multipoles. For a
multipole of order I, the potential has a total of 21 maxima and minima, and is called
a 2l-pole. Often Latin names are used for the 21 poles, and we have the following
table: For a quadrupole the potential is quadratic, so the resulting field B is linear.
Indeed, the quadrupole is the only s-independent element that leads to linear motion
similar to that in glass optics, and thus has great importance. In practice, of course,
s-dependence is unavoidable: the field of any particle optical element has to begin and
end somewhere, and it usually does this by rising and falling gently with s, entailing

s-dependence. This actually entails a crux of particle optics: even the quadrupoles,

22

 

Leading Term in V Name
M0,0(3) COS (00,0)
M1,1(s) cos (05 + 01,1)1' Dipole
M2,2(s) cos (20‘) + 02,2) 7'2 Quadrupole
M3,3(s) cos (3(1) + 03,3)1‘3 Sextupole
( ) ( )1”4
( ) ( )7‘5

 

MM 3 cos 4¢+04,4 Octupole
M5,5 3 cos 505 + 05,5 Decapole

mthI—‘ON

 

 

 

 

 

the ”linear” elements, have nonlinear effects at their edges, requiring higher order
correction. The region in which the field falls off from its value in the magnet body
to zero is called the fringe field region. Chapter 4 will be devoted to studying the

eflect of fringe fields on the beam dynamics.

Chapter 2

The AT to COSY Converter

In order to make use of the capabilities of COSY INFINITY for dynamics studies in
the LHC we first had to produce the lattice description in the COSY language. The
LHC lattice is available from [19] in the optics output format of MAD version 8 [20].
This output is created by MAD using the OPTICS command giving a flat sequence
of elements that can be translated to COSY easily. For this purpose we wrote the
program given in figure 2. The output created by the OPTICS command is also
called AT output, because it explicitly contains the position of each element. We will
therefor refer to the conversion tool as the AT to COSY Converter . It is important
to note the limitations of the program. First it is only designed to translate particle
optical elements. This means that it is not possible to process a MAD-program that
performs an analysis task, like the computation of tunes, using AT to COSY and
obtain a program that does the same in COSY. Secondly we restricted the program
to those elements actually occuring in the LHC lattice version 5.0. These are Drifts,
Quadrupoles, Bends, Sextupoles. RF Cavities, Octupoles, Markers and Multipoles
are converted to drifts of respective length. This limitation will be remedied by the

SXF to COSY Converter described in the next chapter.

23

24

 

 

ipoles are
t

t

l

S

* Michael Lindemann

' 11/24/97

w

* The Program takes the optics output cre
ated by MAD

* and converts it to COSY Input.

t

' Translated are drifts, quadrupoles, ben
ds, sextupoles.

i

RfCavities, Octupoles, Markers and MUlt

converted to drifts of according length

Program nacoqy

character text(7)‘24
double precision zahll?)
double precision angle,Pi
character zeile'200
integer a,b,n,status,max

Pi=3.141592653589793
max=10000

open (UNI T=10 , FILE: ' double2.da1' ,
IOSTATtstatus,STATUS='OLD‘,ERR=1000

open (UNIT=11 , file: 'double2_f77.fox ' , for

m: ' fonnatted' ,

I

t

10

20

$

STATUS: ' UNKNOWN ' )

wri te (11 , ’(A)') 'PROCEDURE RING; '
READI10,'(A)’) zeile
if Izeile(1:1).NE.")
n=0

do while (n<max)
n=n+1
a=l

do 60 j=1.7.1

DO While (zeile(a:a).EQ.' '

goto 10

.AND. IC

HAR(zeile(b:b)) .NE. 13)
40 a=a+1
if (zeile(a:a).EQ." .AND. ICHAR(ze
ilela:a)) .NE. 13) goto 40
b=a
' DO While (zeile(b:b) .NE. ' ' .AND.
ICHAR(zeile(b:b)) .NE. 13)
50 b=b+1
if (zeile(b:b) .NE. " .AND. ICHAR(

zeile(b:bll

.NE. 13) goto 50
text(j)=zeile(a:b)

a=b
60 continue
! Conversion to float
do 70 j=3,7,1
read (text(j).'(62¢l6)') zah1(j)
70 continue
if (textll) . EQ. '"MARKER"') then
write (11, '(ALAIO,A3.A4.¢24.16,A)') ' {' ,
text(2),’ ]', 'dl',0,’$
else
if (text(1) .EQ. '"QUADRUPOLE"') then
write (11, '(A2,AIO.A3.A4.A8.¢24.l6.A)') ' ['
,textlZ),'}', 'nqk'
,text(4),zah1(6)/zah1(4).’005$
else
if (textll) .EQ. '"SEXTUPOLE”') then
write (11, '(A2.AIO.A3,A4.A8.e24.l6,A)' ) ' l ’
,text(2),' )', 'nmk'
$ ,textl4),zah1(7)/zah1(4)/2,’0057

else
if (text(1).EQ.'”OCTUPOLE"') then
write (11,'(A40.AIO,A4)') ' [OCTUPOLEsct

Iodfifl',text(2),')'

wri ca (11 , ’(A2,AIO.A3.A4.c24.l6.A) ' l ' l ' .

dil'
$

 

 

text(2).'l‘.
$

text(2).'l'.
S

(2). zahllS). 'l
1

ngle/2,‘
$

2),zahl(5),’)'
i

Id] 1

.zahl(4).':’

else

if (text(1) .EQ. '"MULTIPOLE"') then

write (11, '(A40.AIO.A4)') '{MULTIPOLEse
itodrifr '.text(2) .' 1'
write (11, '(AZAIO.A3,A4.¢24.I6.A)') ' l',
textIZ).'l‘. ’dl'
$ .zahl(4).':'
else

if (text(1) .EQ. '"RFCAVITY")
write (11, '(A40.Al0.A4)')

then
' l RFCAVITY sell

write (11, '(A2.AIO.A3.A4.624.16.A)') ' l' .
dl '

odrift’,text(2),' } '

textIZ),'l‘ '
$

, zahl (4) . ';'
else
if (text(1) .EQ. '"DRIFT"') then
wri ted1 (11 , ' (A2.A10.A3.A4,e24.16.A) ' )
. 23111 (4) . ':'
else
if (textll) .EQ. '“RBEND”') then
if (zah1(5).GT.O) then
angle=zahl(5)/Pi*180
write (11 , ’(A2.Al0,e24.l6.A3)') ' l' , text

.{1'

write (11, ’(e24.16)’) Pi
write (11,'(A4.e24.l6.e24.16.A6)') '

,zahl(4),angle,'005'
write (11, ’(e24.l6,A4.e24.l6,A5)') a
o l
,angle/2,'0;'
else
angle=-zah1(5)/Pi*180
write (11, '(A)') ' lnegativebend I]
write (11. ’(A2.A10.e24.l6.A3)') ' [' ,textl
write (11,’(924.16)’) Pi
write (11,’(A)') 'cbr
wri te (11 , ’(A4,e24.16.e24.l6.A6) ' )

1d“ I
,zahl(4),angle,'005'
wri te (1 1 , ’ (e24.16.A4.c24.l6,A5) ' )
angle/2
$ .‘0',angle/2,'O;'
write (11, '(A)') 'cb;'
endif
else
write (11, '(A30,A10.Al)') ' (warning
:unknown element ' ,
$ textIl),'}'
print " . ’ warning : unknown clement!‘ .
text(1)
max=max+1
end if
end if
end if
end if
end if
end if
end if
end if
read (10,'(A)', IOSTAT=status,end=1000
) zeile
if (n.LT.max) goto 20
1000 print '
if (status.EQ.-1) then
print " , ' complete file converted ‘
endif

print *, max
write(11,'(A)’)
close (10)
close (11)

end

' cndprocedure; ‘

 

 

Figure 2.1: The AT to COSY converter

 

Chapter 3

The SXF to COSY Converter

3. 1 Introduction

At the “Berkeley National Laboratory Workshop on the Unified Accelerator Libraries
and US-LHC software” a new machine file format was proposed [30]. This format is
meant to be a general lattice description language and is intended to facilitate the
c00peration between different groups and the comparison of results obtained with
different codes. The language was named Standard Exchange Format and abbreviated
as SXF. The language specifications were developed by F.Pilat et al.. The SXF to
COSY converter can be used online at our web page [18] where the current SXF

specifications are also available.

3.2 The Converter

3.2.1 General Features

The SXF to COSY Converter is written in PERL. The source code is given in Ap-
pendix A. As the SXF language is still evolving, it is very likely that the converter
will have to be adapted to future changes of the language standard. This made it

very important to optimise the code for ease of reading and modifying.

25

26

3.2.2 Description

In order to facilitate the understanding and changing of the program in the future we
give a brief description of how the translation is done. We will especially stress the
points where differences between COSY and SXF make the translation less transpar-

cut.

The AT attribute

The AT attribute in SXF gives the position of the middle of the element. As COSY
does not allow an AT command this is emulated by drifts, keeping in mind that the

drift should only extend to the beginning of the element but not to the middle of it.

The bends

There are two bends allowed in SXF, the rectangular bend RBEND and the sector
bend SBEND. Because they are in most cases equivalent the use of rectangular bends
is deprecated. As described in the MAD-Manual [20] and in the MAD Physics Manual
[21] these two elements only differ by the local reference system in which the entrance
and exit angles are measured. This results in the addition of half the bending angle
to entrance and exit angles for the rectangular bend. By doing this both rectangular
and sector bends can be translated to the COSY command DIL. The direction of
the bend is determined by the sign of the bending angles. For positive angles the

direction is clockwise. The bending direction is changed for negative angles using

COSY’s CB command.

Another important point concerning the rectangular bends is that there are two
ways of specifying their length. One can either use I, the end to end length of the

element, or arc, the length along the circular orbit. arc and l are redundant since

27

they satisfy
kl [0] / 2

sin(kl[0]/2) (3'1)

arc=l

where kl[0] is the bending angle. If I is given it is converted to arc using this formula.

If I and arc are given, I is ignored.
Multipoles

SXF allows dipoles, quadrupoles, sextupoles and octupoles as well as a general mul-
tipole. Each of these elements can contain a list of multipole strengths with not only
the strength corresponding to the name of the element nonzero. This means that
one might even specify a quadrupole by calling the element dipole in SXF with zero
dipole strength and nonzero quadrupole strength. In this sense the element specifier
is redundant (and the SXF to COSY converter translates all multipole elements to
a general multipole in COSY. The information given in the element specifier is pre-
served as a comment. Following the convention that dipoles are usually described as
bends, a dipole component in multipoles is not supported at this point. If a dipole is

specified by using a multipole an error message is issued.

Chapter 4

Beam Dynamics Studies

4. 1 Introduction

Accelerator lattices are usually described by the position, length and field strength
in the main body of their elements. The field of the magnets is considered to change
from zero to the value in the main field at the magnet entrance and drop again to
zero at the magnet exit. Although this approximation is widely used in beam physics
it is very unrealistic. Using COSY INFINITY it is possible to take into account the
effect of the exact shape of the magnetic field at the ends of the magnet. To do this
the magnet is split into a main section in which the field is independent of the particle
optical coordinate s, and an s-dependent element representing the fringe field. The
fringe field map, which has finite length, is composed with two negative drifts, to

produce a zero-length insertion [14].

28

29
4.2 Magnetic Field Data

4.2.1 Introduction

We have seen in section 1.2.5 that in the curl and divergence free region the most

general form of the potential in cylindrical coordinates can be written as

 

V = Z Z Mk,((8) (208(105 + 0k,()1'k (4.1)
k=0 [=0
with
Ma") S

M..2.,1(s) = , ‘1‘ ( ) (4.2)

H (l2 — (l + 211)?)

11:1

The Ml,1(s) are free parameters as are the 01,). In the magnetic case 91.1 = —7r / 2 is

chosen by convention. The Mk) that cannot be obtained by the above recursion are

zero. For ease of notation we define
Bz(8) = Ml,z(S) COS(61’()

141(8) 1‘ M1’1(8)Sln(01,1) (4.3)

Using these equations we can rewrite 4.1 as

B" 8(4)
V = sin(2¢)r2(Bg(s) + #653273 + %r4 + . . .)

A3(8) 2 Ag”

 

 

 

— cos(2¢)r2(A2(s) + 7' + 77A + . . .)
311(3) 3(4)

+ sin(6¢)'r6(BG(s) + 6 r2 + jS—r" + . . .)
A"(S) 24(4)

— cos(6¢)r6(A6(s) + 6 r2 + i1‘4 + . . .)

c of

BM 8(4)
+ sin(10¢)r1°(B10(s) + —10—('flr2 + —leO—T4 + . . .)

A” s A“)
——10()r2+——10r4+...)+...

— cos(10¢)r1°(A10(s) + f

30

where the coefficients (1,0,. . . , f can be calculated from 4.2. So we can calculate the

components of the magnetic field

av
31" ‘5;
II , (4)
282(5)? + 3—B; r4 + . . .)

 

- —2 sin(2¢))r(Bg(s) +

n A(4)
Mr2+3—;—r4+...)+...

+2 cos(2¢)r(A2(s) +

1 6V
B¢'- ‘rst
———B§I(S)r2 + gig?“4 + . . .)
a

- 2cos(2¢)r(Bo(s) + b

(4)

£213)-152+Air“+...)+...

-—2 sin(2¢)r(A2(s) + 0

G. Sabbi defines the quantities 6,, in [23]. In the notation introduced above 62 and 66

are given by

 

 

__ B” 8(4)(S) 1
b = B 2—2— 2 —2—— 4
2(3) ( 2(8) + a 1" +3 b 'r + )B2(So)
— " B(4)(s) 1 1‘
_ _6 2 __6_ 4 __ 4
b0(8) — (380(8) + 4 C 7' + 5 d T . . . 82(30) 1‘0) (4.4)

where so is chosen sufficiently inside the magnet, such that the derivatives of Bn(s)

with respect to s vanish. Taking a look at the definition of B,, in (4.3) reveals that

62 can be written as

 

II M(4)(5) 1
22 2 22 4
’r +3—’—r +...

b )M2,2(S0)

 

52(8) = (M2,2(3)+2

31

 

 

M" M“)
6’6 r2 + 5———6’6 (3)1'4 + 1 (1)4 (4'5)

6(8) ( 6,6(5) C d M23090) 70

4.2.2 Fitting

We model M2,2(s) such that equation (4.5) is approximately satisfied for the given
radii r = 5, 10,20, 30 mm. In order to calculate 62 from M23 via (4.5) we calculate
the out-of—plane expansion of the field by the recursion relation (4.2). This is done
in COSY INFINITY using the differential algebraic approach. As the model class we

choose Enge functions with 6 parameters namely

1
1+ exp (a1 + 112(5) + 113(3)2 + + a6(§)5)

 

M2,2(S) = (4.6)

where s is the Cartesian distance to the field boundary. The quantity d is the full
aperture, respectively twice the radius, of the quadrupole. For the fitting procedure
we use the nonlinear optimisers implemented in COSY INFINITY. It has proven
useful to obtain the initial values for the Enge coefficients by calculating coefficients
such that the resulting Enge function passes exactly through six data points. To do

this we calculate the solution of a system of six linear equations.

In the optimisation process we shift M2,2(s) such that the effective field boundary

coincides with the origin. This means we have to satisfy the equation

0 s
f 1— M230) ds = f ’ Mme) 113 (4.7)
3,- 0
where s,- is sufficiently inside the magnet, such that M2,2(s,-) % 1, and s, is suffi-
ciently outside the magnet, such that M2,2(sf) a: 0 .

In order to access the quality of the fit we compare in figure 4.1 the B, component

of the field in the return end of the magnet as calculated by COSY with the 62 as

32

calculated by ROXIE for different radii. The same comparison is given for the lead
end in figure 4.2. This comparison is valid , because the field generated by the M2,2(s)
via equation (4.2) is a pure quadrupole field. In the figures 4.1 and 4.2 the ROXIE
values are given by the line while the result of the COSY calculation is given by dots.
In each case the origin of the z-axis is chosen as defined in [29]. The z-axis is directed
outwards from the magnet body. The iron yoke starts at z = -—15 cm for the lead
end, at z = —5 cm for the return end. The vertical axis gives the magnetic field in
units of 104/62, where 62 is the quadrupole strength in the main body at the given
radius. The plots show clearly that for the region up to r = 20 mm the fitting agrees
very well with the simulated measurements. This is reassuring because the beam
stays within approximately 1' = 17 mm . On the other hand it is obvious from the
plot for r = 30 mm that this method does not work reliably any more for larger radii,
especially because the rather peculiar shape of the fringe field at r = 30 can not be

modelled accurately by an Enge function.

Nevertheless there is a way of calculating the real multipole content of the field
more accurately using the differential algebraic approach which will be described in

chapter 5.

4.3 Lattice Description

In our analysis we use the LHC lattice model Version 5.0, which is available from [19].
We used the AT to COSY converter to translate the lattice description given in the

@-output format of MAD 8.0 to COSY language.

In the present study the ring is subdivided in three regions, the two inner triplets
left and right of the interaction point 5, for which the detailed field data described

above is available, and the rest of the ring. The layout of the triplets, which are mirror-

2&3

 

 

 

 

 

 

-1r—————o————.._______.3£E:¥1> :——=> :: ;,i_____‘£§§¥¥Lr
8000» 8000»
6000» 6000»
4000» 4000»
2000 2000»
.280 -200 Affso -100 -50

 

 

 

 

 

 

-—1r————«o————1,._____q.g55¥¥1> e: : ;_______.3£§¥¥1
3000- 8000»
6000» 6000»
4000» 4000»
2000 2000»
A, .230 -200 ~1So -100 ~50

 

 

~2so -200 -130 -100 -50

Figure 4.1: Comparison of the quadrupole component of the field in the return end
as computed by COSY and ROXIE at different radii. From top left to bottom right:
r=5mm,r=10mm,r=20mmandr=30mm.

34

 

.__————Ch—_4L__j"~.._:3:::J :-—————.—__.___1"‘~._:3:::;
8000' 8000
6000> 5000
4000’ 4000'
2000' 2000*
-300 -200 '100 100 200 '300 '200 -I00

 

 

   

 

10000r

8000-
6000-
4000’

2000'

 

 

 

 

 

-3oo ~2oo -1oo -300 200 100 100 200

Figure 4.2: Comparison of the quadrupole component of the field in the lead end as
computed by COSY and ROXIE at different radii. From top left to bottom right:
=5mm,r=10mm,r=20mmandr=30mm.

35

symmetric with respect to the interaction point, is shown in figure 4.3. The High
Gradient Quadrupoles are denoted by Q. The elements between them are multipole

correctors. For details about the LHC naming convention refer to [22].

We study the seventh order map for the triplets, with and without fringe fields,
while the rest of the ring is treated in its linear approximation. Using this approach
we can study the nonlinear effects of the fringe fields in the triplets, which is the
most critical part of the lattice, as explained earlier. There are no main field errors

considered.

<—————— towards the IP

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q1 Q2A Q2B Q3 DI
/
- :fll I I I-
'\
BPM beadend MCBX bl/nl MCQS 213 MCBX IIl/hl /
Lead end

MCDD hh MCDDS .10 MCDD bl»

Figure 4.3: Layout of the left low-[3 triplet.

4.4 Results

4.4.1 Tune Shifts

We computed the amplitude dependent tune shifts with and without fringe fields.

The result is given in table 4.1 for the :1: — a plane and in table 4.2 for the y — b plane.

36

The tables show two effects. First, the nonlinear tune shifts increase considerably.
Secondly, the center tune decreases by 6 . 10'3. This can be corrected using the trim
quadrupoles in the LHC lattice. But even after refitting the center tune to its original
value, the nonlinear tune shifts are considerably bigger with fringe fields than without

fringe fields. In fact they are still of the same order of magnitude as without refitting

 

 

 

 

 

 

 

the tune.
without fringe fields with fringe fields order exponents
original center tune refitted

.310000 .303921 .310000 0 0 0 0 0
58.5018 592.429 610.796 2 2 0 0 0
20.45303 873.057 902.598 2 0 0 0 2
8352.403 1889188 2191916 4 4 0 0 0
-3360.344 -13279249 —13536156 4 2 0 2 0
9216.0682 11570624 12740611 4 0 0 4 0
13692934 —399181479286 —422995493944 6 6 0 0 0
—51336671 88048290859 96863534946 6 4 0 2 0
124626604 -1035625378625 -1159771282208 6 2 0 4 0
-15542851 1394274127638 1515548018372 6 0 0 6 0

 

Table 4.1: Tune Shifts in the :1: -— a plane with and without fringe fields.

 

 

 

 

 

 

without fringe fields with fringe fields order exponents
original center tune refitted

.320000 .313929 .320000 0 0 0 0 0
20.4538 873.058 902.600 2 2 0 0 0
58.5018 589.143 610.797 2 0 0 0 2
-1680.1 -7496999 —7667610 4 4 0 0 0
18432.1 24842663 272712816 4 2 0 2 0
11342.9 2181737 2516249 4 0 0 4 0
-17112223 1289593405561 1373994850952 6 6 0 0 0
124626604 -11277831999539 4259605227516 6 4 0 2 0
-46628554 4712527272105 5749819744288 6 2 0 4 0
13545598 —399572595609 -427111891097 6 0 0 6 0

 

 

Table 4.2: Tune shifts in the y — b plane with and without fringe fields.

 

 

37

The tune footprints shown in figure 4.4 clearly demonstrate the significance of the

fringe field effects.

Compared to the previous study done by F. Méot et. al. [26], the center tune
change shown here is about 20 times larger. Yet we notice the fact that the two
studies use different sets of Enge coeflicients. Specifically, in the Méot study the
detailed shape of the fringe field was not considered but assumed to be similar to
that of other standard quadrupoles. In reality, however, the end fields used here look

much worse.

As the interaction region 1 has the same design as the interaction region 5, one
can apply the fringe fields to the triplets in the interaction region 1 as well. The
result is that as expected the center tune shift grows by a factor of two, while the
nonlinear tune shifts are still of the same order of magnitude. In the tables 4.3 and

4.4 we compare the tune shifts for the :1: —- a and y — b plane respectively.

 

 

 

 

 

 

 

without fringe fields with fringe fields order exponents
in IR 5 in IR5 and IR1

.310000 .303921 .297997 0 0 0 0 0
58.5018 592.429 1127.460 2 2 0 0 0
20.45303 873.057 1721.358 2 0 0 0 2
8352.403 1889188 6319767 4 4 0 0 0
-3360.344 -13279249 -48466669 4 2 0 2 0
9216.0682 11570624 32720546 4 0 0 4 0
13692934 -399181479286 -624020378914 6 6 0 0 0
-51336671 88048290859 4464097212016 6 4 0 2 0
124626604 -1035625378625 -8236728752799 6 2 0 4 0
-15542851 1394274127638 3983026534205 6 0 0 6 0

 

Table 4.3: Tune shifts in the :1: — a plane with fringe fields in IR 5 and in both IR1

and IR5

 

38

 

 

 

 

 

 

 

 

LHC
I r fir f l V Y I 7—7
3.2oosx1o“1 — —
L .1
>\ _1 '- F
g 3.2000x10 — e —«
3.1995x1o“— _
3.1990x10—1 . . . 1 1 . . . . 1 . . . . 1 . . . .
3.0990x10“ 3.0995x10‘1 3.1000x10’1 3.1005xio'1 3.1010x1o‘1
mUX
LHC
V Y I f V I V I I V
[ 1
3.2005x10‘1 — —
r- q
)— -I
>\ __1 L d
g 3.2000x10 — —
3.1995x10’1 1— _,
1 _
3,1990x10—1 A 1 l J l 1 1 L J l l l 1 l l I 1 l 1
3.0990xio"1 3.0995x10‘1 3.1000x10’1 3.1005x10‘1 3.1010x1o“
mUX

Figure 4.4: Tune footprint without fringe fields (top) and with fringe fields (bottom).

39

 

 

without fringe fields with fringe fields order exponents
in IR 5 in IR5 and IR1

.320000 .313929 .307799 0 0 0 0 0
20.4538 873.058 1721.360 2 2 0 0 0
58.5018 589.143 1203.404 2 0 0 0 2
—1680.1 -7496999 -25882333 4 4 0 0 0
18432.1 24842663 68831273 4 2 0 2 0
11342.9 2181737 1386369 4 0 0 4 0
-17112223 1289593405561 3872032968262 6 6 0 0 0
124626604 -11277831999539 -8509355582662 6 4 0 2 0
46628554 4712527272105 4516085446489 6 2 0 4 0
13545598 -399572595609 -791311504674 6 0 0 6 0

 

 

 

 

 

 

 

Table 4.4: Tune shifts in the y — b plane with fringe fields in IR 5 and in both IR1
and IR5

4.4.2 Resonance Analysis

In order to further investigate the changes in the dynamical behavior of the system we
study the resonance strength. The detailed result is given in the tables 4.5 and 4.6.
Horn table 4.5 one can see the significant increase in the resonance strengths when we
take into account the effect of fringe fields. From table 4.6 one can see that refitting
the tune to its nominal value does not affect the resonances very much. To compare
the resonance strengths with and without fringe field, we calculate the logarithm of
the average absolute value of the resonance strengths for every order. The result is
given in figure 4.5. It shows clearly that the resonance strength increases by at least

one order of magnitude on average, and even more for higher orders.

40

 

Resonance Strength
without fringe fields

with fringe fields

order

exponents

 

 

-3.802e+01
9.937e+01
3.312e+01
-3.135e-09
1.472e+00
3.245e+01

-1.243e+00
-4.143e-01
3.245e+01
4.906e-01
3.709e+01

-9.893e+01

-3.298e+01

5.926e-07

-1.214e—05
-7.746e—07
-1.245e-06
-2.812e—06
4.018e-07
-9.641e—06
-3.814e—07
-3.374e-07
-5.320e-06
-5.293e-07
-2.796e-07
-1.407e-06
-3.686e—07
-4.420e-07
-2.267e—06

7 .545e—08

-4.701e-07
-9.907e—06
-2.446e—07
-9.698e-07
-3.134e-07
-9.280e-08
-2.626e—06
-9.609e—08

1 .268e+02
1.508e+02
-5.028e+01
1.163e+02
9.386e+01
1.020e+03
7.948e+01
-2.649e+01
-1.020e+03
-3.129e+01
9.772e+01
1.783e+02
-5.942e+01
-1 . 175e-07
-1.125e-05
-1.868e—06
-5.164e—07
-3.036e—06
2.360e-07
-9.068e—06
-6.630e-07
-2.458e—07
-5.630e—06
-4.194e—07
-3.529e—07
-1.330e-06
-5.061e-07
-2.543e-07
-2.410e-06
4.084e-07
-3.844e-07
-1.050e—05
-1.392e—07
-4.701e-07
-6.466e-07
-1.631e—07
-2.477e-06
-1.175e-07

 

-3.289e+02
8.412e+02
2.804e+02
1.054e-01
2.448e+01
5.842e+02
-2.041e+01
-6.785e+00
5.842e+02
8.141e+00
3.175e+02
-8.327e+02
-2.776e+02
4.011e-05
-9.245e—04
-9.106e—05
-1.239e—04
-2.106e-04
1.534e-04
-4.818e-03
-2.193e-04
-9.460e—05
-2.368e—03
-5.395e—04
-3.347e—04
-5.903e-04
-1.128e-04
-2.125e—04
-1.198e-03
4.945e-05
-1.049e—03
-3.498e—03
-1.858e-04
-4.024e—04
-1.809e—04
-9.970e-05
-9.322e-04
-2.364e—04

1.484e+03
1.395e+03
-4.650e+02
5.760e+03
3.672e+03
4.355e+04
3.0768+03
-1.025e+03
-4.355e+04
-1.224e+03
1.133e+03
1.655e+03
-5.518e+02
-1.689e-05
-8.424e-04
-1.858e—04
-6.071e—05
-2.311e-04
1.032e-04
-4.673e-03
-3.042e—04
-6.842e-05
-2.453e-03
-5.105e—04
-3.536e-04
-5.702e-04
-1.519e—04
-1.569e-04
-1.233e—03
1.482e-04
-1.000e—03
-3.690e—03
-1.567e-04
-2.629e-04
-2.745e-04
-1.183e—04
-8.844e-04
-2.479e-04

 

cncncncncncncncncncncncncncncncncncncncncnmmmmbpehhhaphase-11>A

HOHOI—‘OHOI—II—‘MQOOI—INWOHNWHNWAU‘OOOHMONOHNHWA

OI—‘OI—‘OHOHONHOWMHOWMHOACANHOOOOHONONHOWHO

OHI—IMNWWAADOOI—‘HI—II—IMMMNOOOOOHWVBCOHHMNNOOO

fiWCDleDI—II—IOONMNI—II—lI—IHOOOOOOOOCWHONNHHOOOOOO

 

Table 4.5: Resonance strengths with and without fringe fields.

 

41

 

Resonance Strength
without refitting

with refitting

order

exponents

 

 

-3.289e+02
8.412e+02
2.804e+02
1.054e-01
2.448e+01
5.842e+02
-2.041e+01
-6.785e+00
5.842e+02
8.141e+00
3.175e+02
-8.327e+02
-2.776e+02
4.01 1e-05
-9.245e-04
-9.106e—05
-1.239e-04
-2.106e—04
1.534e—04
-4.818e-03
-2.193e—04
-9.460e—05
-2.368e-03
-5.395e-04
-3.347e-04
-5.903e-04
-1.128e-04
-2.125e-04
-1.198e-03
4.945e-05
-1.049e-03
-3.498e—03
-1.858e-04
-4.024e-04
-1.809e—04
-9.970e-05
-9.322e-04
-2.364e-04

1.484e+03
1.395e+03
-4.650e+02
5.760e+03
3.672e-I-03
4.355e+04
3.076e+03
-1.025e+03
-4.355e+04
-1.224e+03
1.133e+03
1.655e+03
-5.518e+02
-1.689e-05
-8.424e—04
-1.858e—04
-6.071e-05
-2.31 1e-04
1.032e—04
-4.673e-03
-3.042e—04
-6.842e-05
-2.453e-03
-5.105e-04
-3.536e-04
-5.702e-04
-1.519e-04
-1.5699—04
-1.233e-03
1.482e-04
-l.000e—03
-3.690e-03
-1.567e—04
-2.629e-04
-2.745e-04
-1.183e-04
-8.844e-04
-2.479e—04

 

-3.320e+02
8.648e+02
2.883e+02
-1.223e-05
2.742e+01
5.947e+02

-2.323e+01

-7.722e+00
5.948e+02
9.118e+00
3.236e+02

-8.608e+02

-2.869e+02
6.353e-05
-1.488e—03
-1.179e-04
-1.643e-04
-3.488e-04
3.123e-04
-9.290e—03
-3.226e—04
-1.281e—04
-4.090e-03
-7.859e—04
-5.010e—04
-1.026e-03
-1.498e—04
-3.001e-04
-2.311e-03

1.301e-04
-2.432e—03
-6.874e-03

-2.805e-04
-5.897e—04
-2.634e—04
-1.467e—04
-1.794e-03
-5.623e—04

1.304e+03
1.604e+03
-5.347e+02
5.080e+03
4.182e-I-03
4.506e+04
3.547e+03
-1.182e+03
-4.506e+04
-1.394e+03
1.000e+03
1.891e+03
-6.302e+02
-5.748e-06
-1.395e—03
-2.465e-04
-7.860e-05
-3.721e-04
2.462e-04
-9.078e-03
-4.362e-04
-9.324e—05
-4.222e—03
-7.599e—04
-5.180e-04
-9.945e-04
-2.018e-04
-2.255e—04
-2.363e-03
2.621e-04
-2.339e-03
-7.122e-03
-2.271e-04
-3.869e—-04
-3.992e-04
-1.816e-04
-1.732e-03
-5.844e—04

 

010101010101mmmmmmmmmmmmmmmmmmmpefih11:11:11:-AAAAAA

HOHOHOHOHHNwOHMWOI—‘NWHMCDAUIOOOHNOMOI—‘MI—IwA
OI-‘OHOHOHONHOMNHOWMHOAWNHOOOOHOMOMHOOOr-‘O
CHI—tmmwwAAOOOHHHHNNNNOOOOOHQ‘DAOOHI—tNNMOOO
150:0»:MMHI—soowwtov-dI—II-IHOOOOOOOOOwHONMHI—IOOOOOO

 

Table 4.6: Resonance strengths with and without refitting the linear tune.

 

42

 

 

 

 

Lad-I

Figure 4.5: Average logarithmic resonance strength for orders 3 to 7. The values are

given by squares for the case without fringe fields and circles for the case with fringe
fields.

Chapter 5

Differential Algebraic Field
Calculation

5. 1 Introduction

So far the method of treating the fringe fields has one important shortcoming. The
real multipoles, which we have to know for for beam dynamics studies as they permit
the calculation of the field at any given point, have to be extracted from the available
data, which is in fact a sum of real and pseudo multipoles. This extraction can not
be achieved analytically but has to be done by fitting. This problem is remedied
by calculating the magnetic field in COSY using the DA method, which allows an

analytical scheme for the extraction of multipoles that is exact to machine precision.

5.2 Magnetic Field Calculation by ROXIE

5.2.1 Introduction

The magnetic field calculations for the LHC High Gradient Quadrupoles presented
in [23] and [24] are performed by ROXIE [25]. ROXIE uses a set of line currents
from the magnet model, specified by their starting and ending point in 3 dimensions

and the current they carry. The magnet is subdivided into two sections, the straight

43

44

section and the end field. The end of the straight section is given by the end of the
iron yoke. This is also the origin of the coordinate system used in ROXIE. The end

region extends from the origin to positive 2 values.

5.2.2 Image Currents

The high gradient quadrupole magnets are surrounded by an iron yoke with a cylin-
drical inner bore. It serves two purposes: the fringe field outside the coil is greatly

reduced and the field on the beam axis is enhanced by 10-20 percent.

The influence of the iron yoke is analysed using the method of image currents
assuming that the iron is not saturated and the permeability 11 is uniform. Then for
a current I inside a hollow iron cylinder of radius R the effect of the iron on the inner

field is equivalent to that of an image current I I, located at the radius r, :

I [1. —' 1
. R2

The current I ' is parallel to I. It thus increases the inner field.

It is important to notice that ROXIE calculates image currents only for the

straight section.

5.3 Magnetic Field Calculation in COSY

5.3.1 Introduction

G. Sabbi has implemented an output format in ROXIE that writes the positions and
strength of all line currents, including the straight, image and end currents to a file.

We use this information to calculate the magnetic field in COSY.

45

5.3.2 Line Currents

From the Biot-Savart law we can calculate the magnetic field contribution of an
infinitesimally short line current

6le

dB: 1
#0 47rr3

 

(5.3)

In order to calculate the magnetic field of a line current of finite length we have to

integrate along the line, which is for that purpose parametrised by
r6) = r. + 30?. — F.) (5.4)

where r, and r,- are giving start and end of the line current in three dimensions. Then

the magnetic field of a line current is given by

B = kl / 111(3) X,F(S)ds (5.5)

 

using the definition k = [lo/4W 2 10‘7. As we are integrating along a straight line,
Kl 2 00(3) is constant. It is Kl = Fe — 7",. The remaining integral is easily solved by

substituting (5.4) into (5.5).

 

 

 

46

For ease of notation we define

 

 

a. = |7=',|2, b = 2 (r, . 51), c = [51V (5.6)
Then the magnetic field is given by
- 2b 2(b + 20)
B = kIAl x 7, — 5.7
T (\f6(b2—4ac) \/a+b+c(b2-4ac)) ( )

The actual implementation of this formula is subtle. We observe for example, that in
the case F, H Al both the denominator and the numerator in (5.7) vanish. Calculating
the limit 0 -—> 0, where 0 denotes the angle between F, and A7, analytically, shows
that in fact for this case the constant part of the field vanishes. Nevertheless this
is problematic for high order calculations, as equation (5.7) becomes numerically
unstable for small 0. More numerical issues related to the application of the Biot-

Savart law are discussed in [31] and [32].

5.3.3 Implementation in COSY

Using the formula derived in the previous section and evaluating it in DA using COSY
we are able to calculate the Taylor expansion of the magnetic field produced by a line
current distribution. We implemented the formula in a new COSY function LIN EFLD
which calculates the y-component of the magnetic field in the midplane. LINEFLD
has to be called with two parameters, the x and z coordinate of the position in the
midplane where the field should be calculated. It reads the line current distribution
from the file LCD.DAT in the COSY directory. The format for LCD.DAT is very
simple. It contains only numbers, one number per line. The first line specifies the
total number of currents in this file. After that 7 lines always specify one line current
in the format x,,y,,z,,Al,,Aly,Alz,I, where I is the current . It is important to

notice that the units meter and ampere have to be used.

47

The results of the function LINEFLD have been tested against ROXIE and were
found to be correct. We checked both the the absolute field value at a given point,
including the x- and z-component that are not returned by LINEFLD, and the field
gradient. The gradient is a natural result of the computation of the field using
differential algebra, namely the term in the expansion of the By that is linear in
x. During the comparison a typo was discovered in [23]. The correct value for the
gradient in the magnet body is 18.22 T/m/kA and not 0.1822 T/m/kA as given in

the paper. This has been acknowledged by G. Sabbi.

In future the detailed field information shall be used for beam dynamics simula-

tions. I will outline in the following section how this can be done in COSY.

Creating a new Element

The first way is to create a new partical optical element. An element can be specified
by giving an explicit formula. In out case the formula is in fact a subroutine that
reads the positions and strengths of the line currents from a file and calculates the
resulting field. As the implementation of new elements in COSY is not documented

in the manual, I will give a brief prescription here.

All of the following changes are to be made in COSY.FOX. First you should create
a new procedure with the name of your new elements. In this case we call it LCD for

Line Current Distribution.

PROCEDURE LCD L D ; { Line Current Distribution }
NSDP := -6 ; LOFF :2 2 ; CE 2: ’LCD’ ;
SDELE 0 L L/500 L/ 10 L D ; ENDPROCEDURE ;

The new element has two parameters, length L and aperture D. The variable

N SDP determines which field formula is taken to calculate the matrix of the S-

48

Dependent ELEment SDELE. LOF F = 2 indicates that the formula for the field
in the midplane is given, while LOF F = 1 means that we supply the field on the
Optical axis. These formulas are specified in the procedure POTXZ. We create a new

entry for N SDP corresponding to the value we chose in LCD.

ELSEIF NSDP=-6 ; { LINE CURRENT DISTRIBUTION }
BY:=LINEFLD(X,Z) ;

General Element

The second way of making use of our additional fringe field information is to employ

COSY’S general element GE. We want to demonstrate this with a simple example.

The program given in figure 5.3.3 compares a quadrupole implemented with GE

to the standard quadrupole element in COSY.

include ’COSY' ;
procedure run ;
variable 3 1000 100 ;
variable h 1000 100 ;
variable v 1000 100 ;
variable w 1000 100 ;
variable i 1 ;

0v 2 2 O ;
rpp 1000 ;
um 3

loop i 1 6 ;
s(i):=i-1 ;

h(i):=0 ;

v(i):=0 ;
w(i):=DA(1)*DA(3) ;
endloop ;

GE 6 3 s h v w ;
pm 5 :

um 1'
mg 5 1/100 .01 ;
pm 6 I

endprocedure ; { run }

run; end ;

Figure 5.1: Demonstration of the general element GE in COSY

49

For a basic introduction to GE refer to [17]. The DA-vectors W are giving the
magnetic field at the respective point specified in S. For a quadrupole the potential is
proportional to :1: y. The coordinates are sorted :13, a, y, b, s, 6, so this proportionality
is specified by setting W(1') :2 DA(1) =1: DA(3), where DA(1) and DA(3) are the first

and third DA-base vector respectively.

From the definition of W it follows that the magnetic field at the pole tip of a
quadrupole of aperture D is equal to D. So the specified general element and the

following standard quadrupole are identical.

It is important to notice that for elements with curved optical axis H (1) contains
the DA vector describing the local curvature of the optical axis. This DA vector can
be obtained in the following way. Locally the magnetic field can be assumed to be
constant. Then we can use the well known equation for the radius of the circular

orbit of a particle carrying the charge q in a magnetic field B

_ p
or using the definition Xm = p/q
B
H = _ 509
Xm ( )

The value of Xm is available as a global variable in COSY which is initialised as

soon as the energy, mass and charge of the particle under study are specified.

5.4 Extraction of the Multipole Content

For beam dynamics studies it is essential to extract the multipole components from

the field calculation.

50

We use the expression for the scalar magnet potential in the curl- and divergence-
free region given in equation (1.33). By expanding the cosine and noting that a
”normal” quadrupole magnet has 92,2 = —7r / 2 , we can define the normal and skew
components of the potential by the fact that they reach the maximum, respectively

vanish at —71/2

bk,[(8) = —Mk,1(8)SlIl0k,1

ak,1(s) = Mk,((S)COS6k’l

After the expansion in normals and skews the potential becomes
VB 2 — Z (bk,¢(s) sinlqb + ak,1(s) cos lo) 1:" (5.10)
lc,l=0
Using equation (1.35) it is easy to see that the functions bk,z(s) and ak,¢(s) are related
by recurrence relations
2 1.5311(3)
Hfizl (l2 — (l + 2u)2)

 

bl+2n,l(3) (5.11)

all others being zero, and the same for the skews. We recall that the terms that contain
s—derivatives are called pseudo-multipoles while the others are called multipoles. To
analyse the multipole content of the field usually B, is evaluated at a certain radius
and different 43 values and then a numerical Fourier analysis is performed. Obviously
this approach does not allow to separate real and pseudo-multipoles as for this task
you need both the 1‘ and the (15 dependence of B,. This information is available in the

Taylor expansion of 8,.

5.4.1 Analytical Fourier Transform

The DA-based field calculation yields the Taylor expansion of the Cartesian com-

ponents of the field depending on :1:, y and s. We can now use the transformation

51

formulas from Cartesian to cylindrical coordinates to get B, and B, Obviously it is

sufficient to study one of them, so we focus on B,. We recall that

B, = B, cos 05 + 3, sin (b (5.12)

Inserting in the Taylor expansions for B, and By and using :1: = r cos (1), y = 1' sin 05

gives us B, as a function of r and (b in the form of

B, = Z Ti+j sini ¢cosl 43 lCiJ cos 05 + dio’ sin 05] (5.13)

131'
where 13,-,3- and did are the coefficients of the term proportional to ziyj in the Taylor
expansion of B, and B, respectively. We then use the representation of powers of
trigonometric functions in terms of functions of multiples of the argument [33] and
the formulas for products of trigonometric functions to perform an analytic Fourier
transformation of equation (5.13) with respect to the angle, while at the same time

we also preserve the Taylor expansion with respect to the radius that was already

achieved before.

5.4.2 Example

We demonstrate this method for the LHC High Gradient Quadrupoles. The field
calculations for these magnets presented in [23] and [24] are performed by ROXIE
[25]. The calculation is based on a set of line currents which are generated from the
magnet model shown in figure 5.2 and 5.3. Only the part of the coil extending beyond
the edge of the iron yoke is shown. G. Sabbi has implemented an output format in
ROXIE that writes the positions and strengths of all line currents, including the

straight, image and end currents to a file. We use this information to calculate the

magnetic field in COSY.

In [23] G. Sabbi defines E and calculates it for the lead and return end of the

magnet. For the definition of H see equation (4.4). Unlike the computation of M”

52

 

Figure 5.2: Model for the High Gradient Quadrupole’s lead end [23].

 

Figure 5.3: Model for the High Gradient Quadrupole’s return end [23].

53

which requires only a DA-computation of order l—l to be exact to machine precision,
it requires an infinite series to calculate 6,. Nevertheless, as you can see from the
figures 5.4 and 5.5 the agreement of 62(3) given in [23] and the DA-based computation

is already extremely good in order 9.

 

 

 

 

 

 

 

 

 

Figure 5.4: Comparison of b; in the lead end as computed by the differential algebraic
field calculation in COSY and as given in [23] for different radii. From top left to
bottom right : r = 5 mm, 1' = 10 mm, r = 20 mm and r = 30 mm.

5.4.3 Differential Algebraic Multipole Extraction

It is possible to extract the multipole content of magnetic fields by Differential Alge-

braic methods directly in a very elegant way that is arbitrary in order. We define

oo 0(2n)(8)
b n
=;‘+2’(8)2n=,§,n3=0212—(1+2u)2)’"

2n

 

54

10000-

8000*

10000*

8000*

6000* 6000*

4000* 4000*

2000* 2000*

 

 

 

 

 

-200 -100

 

 

-200 -100
10000 » $ : : - -10000»

8000

6000*

4000*

2000*

 

 

 

   

-2Aso -200 $50 ~1Aoo -50 -2‘50 -200 -1A50

Figure 5.5: Comparison of (32 in the return end as computed by the differential alge-
braic field calculation in COSY and as given in [23] for different radii. From top left
to bottom right : r = 5 mm, r = 10 mm, 1' = 20 mm and r = 30 mm.

 

oo oo (2")
(law (3) 2n
gazwmdsfiz n=znn III/:1 —(l +2102 )
Inserting this into equation (5.10) yields
VB 2 — Z (f,(r, s) sin l¢ + 91(7', 3) cos M9) 7" (5.14)

[:0
The magnetic field components in cylindrical coordinates can be calculated using the

well known formulas

_ 0V3
3' _ "55

__ 1 V3
Bl» — r 645

_ _8__VB
B3 79—3

resulting in the expressions

B,(r,¢,s) 2: §0(r,s)+

M8

(150", s) sinlcf) + §,(1~,3) coslqs) 7.1—1

~
ll

1

B¢(r,¢,s) = :[H [l(f(,—(rs)cosl¢ g,(r,s)sinl¢)]rll

Bs(r,q§,s) = Z(f,( 7“, s )sinl¢+g,(r, s)cosl¢)r
:0

where prime denotes derivative with respect to s, and

 

 

* °° n °° (1+ 2n)b‘§"’(s) n
W = 2H0 2;. was.”

.. (7‘ 3) — i“ + 2700. (8)72"... __ f: (l + 2n)a§2n)(s) 7‘2"
91 , — "=0 l+2n,l n=oHE—1(l'“’- (l +2102 )

It can be seen that every multipole strength, except for l = 0, is multiplied by

7“”. For the special case I = 0, we get

00 k _
B,(r,s) = —Z(—1)_k+1-2————2k lk!k!a((,2§)(s)r2k 1 (5.15)
B¢(7‘,S) = 0
°° 1
33(738) = -I§)(-1)““ma$€+”(s)r2k

In the DA picture, the field calculations are done locally, as a Taylor expansion of
the field with respect to Cartesian coordinates x, y, 3. Hence, we need the equations

relating the cylindrical and Cartesian components of the magnetic field.

Bx(r, 05, s) = B, (r, <15, 3) cos 43 — B¢(r, 05, s) sin 05
By(r, 03, s) : B,(r, <0, 3) sin 03 + B¢(r, 05, 3) cos (b

and Bs(r,¢, s) is unchanged. Obviously, if we evaluate the above equations in the

midplane (y = 05 = 0), then
Br(7', ¢ = 0’ 3) erx: Bx(xa y = 0a 3)
B¢(r, 45 = 0, s) In—n: By(x, y = 0, s)

We know that the Cartesian components of the fields from the DA calculation in the

midplane

oo
Bx(:c, y = 0, s) = §0(:r, s) + 2.6105, s) - 331—1
(:1

By(:r,y=0,s) = Elf1(:c,s)

This is all the information we need to extract the multipole strengths up to the order
of calculation, because, as previously mentioned, any multipole strength of order I is
multiplied by 23‘“. Starting at l = 1, a1,1(s) is extracted as the x-independent part of
Bx, and analogously b1,1(s) from By. Evaluating a1,1(s) and b1,1(s) at s = 0 yields the

skew and normal dipole component. From a1,1(s) and b1,1(s) the functions f1 (1:, s),

57

g1 (:13, 3) are generated up to the order of calculation and subtracted from B$(x, y =
0, s), respectively By(:r, y = 0, s). This cancels the pseudo-multipoles generated by
the s-dependence of a1,1(s) and b1,1(s), which otherwise would make the distinction
between sextupole terms and pseudo-dipole terms impossible. The procedure can be
iterated for the higher order multipoles, up to the order of calculation. After the k—th
step, the remainder of the field components should contain just (k +1)-th and higher

order multipoles.

However, there is an additional problem in the case of solenoidal fields (case I = 0).
In this case we have an (10,0(3) in the potential, but its contribution vanishes from
the field components BI and By, so the function §o(x, 3) cannot be generated from
the information available in B;c and By. Here comes to the rescue the B, component,
which evaluated at a: = y = 0 yields (133(3). From this function we can calculate
ao,o(s) up to a constant and generate the l = 0 contribution to Bx, §o(z, 3). Once
this is subtracted from B1,, the method works as previously described, starting with

1:1.

Finally, two notes: the method relies on the fact that the magnetic field can be
generated by a magnetic scalar potential that satisfies the Laplace equation. Hence,
it is important that the field is really curl-free. If the fields are calculated from
line currents by the Biot-Savart law, that means that the model should consist only
of closed circuits to ensure the curl is vanishes. Secondly, in the region where the
magnetic field is not s-dependent, the functions f; and g; are the real multipoles
and it is If, = f; and lg, = 57,, an assumption that is sometimes made even for the

s-dependent region.

The method has been implemented in the code COSY Infinity, and in the following

we will present results for two examples : a toy-model consisting of a single square

58

current-loop, and a realistic model of the LHC interaction region quadrupole’s fringe

region.

5.4.4 Example

First we demonstrate the algorithm using an easy example. We study the magnetic
field of a square current 100p of side length 20 cm, orientated parallel to the x, y plane
centered around the s-axis at s = 10 cm. The current is 10 kA clockwise, looking
along the s—direction. We calculate the field expansion around 1" = 0. Due to the
symmetry of this arrangement, only the 4k multipoles are allowed, where k E N. In
table 5.1 we show the skew multipoles and compare the result of the DA-Extraction
with the analytical Fourier Transform. The normal multipoles vanish. Table 5.2 gives
the derivatives of the solenoidal component (10,0 of the field. It should be noticed that
according to equation (5.16) from BT only the even order derivatives of 0,0,0 can be

obtained via Fourier transformation.

 

 

l DA-Extraction Analytical Fourier Transformation
1 0 0

2 0 0

3 0 0

4 04677606347601 04677606347601

5 0 0

6 0.8052817671947803E—13 0

7 0 0

8 -94.0084410633 -94.0084410633

 

 

 

 

 

Table 5.1: Skew multipoles as obtained by the DA-based extraction scheme in com-
parison with the results of the analytical Fourier transformation at s = 0.

For the second example we look at the end fields of the LHC High Gradient
Quadrupoles again. In order to apply the DA-Extraction in this case we have to

make sure that the line currents used in the magnet model form closed loops as

59

 

 

n DA-Extraction Analytical Fourier Transformation
1 0.2309401076758502E—01

2 -0307920143566 -0.307920143566

3 3.849001794594

4 -12.83000598197 - 1283000598198

5 -2035.694282476

6 11632538757 11632538757

7 -3789128.433348

8 1297256160401 - 129725378827

 

 

 

 

 

Table 5.2: S-Derivatives of the solenoid component of the field as obtained by the
DA-based extraction scheme in comparison with the results of the analytical Fourier
transformation at s = 0. n denotes the order of the derivative.

explained earlier. In table 5.3 we show the resulting normal and skew multipoles in

the lead for z = 0. The coordinate system is as defined in [23].

 

 

multipole strength DA-Extraction Analytical Fourier Transformation
b2 8270972386 8270972386
()6 28371.07332 28371.07796
a2 01043500051 01043500051
a6 -11935.21351 -11935.42465

 

 

 

 

 

Table 5.3: Comparison of the allowed multipoles in the lead end at z = 0cm in the
coordinate system defined in [23] calculated by the DA-Extraction and the analytical
Fourier transformation.

Finally, we show in the figures 5.6 and 5.7 a comparison of 62, 66, 52, 56 given in [23]
with the respective real multipoles as computed by our algorithm for the radii r = 5
mm and r = 20 mm. As before the values from [23] are given by the line and our
values are given by dots. One can see that while there is very good agreement for r = 5
mm there are substantial differences especially for the skew components at r = 20
mm. We chose 1‘ = 20 mm because this is close to the new LHC reference radius

r = 17 mm. Presumably this is the only radius for which magnet measurements will

60

become available in the future. The plots 5.6 and 5.7 show, that one can in general

not assume that the 6,, and 5,, measured at the reference radius are equal to the real

multipoles.

10000'

8000’

6000’

4000*

2000>

 

10000-

8000*

6000'

4000’

2000’

 

 

-2oo ~1Aso -

-20

Figure 5.6: Comparison of the normal multipoles bu (top) and b5,6 (bottom) with 52
respectively be in the lead end for r = 5 mm (left) and r = 20 mm (right).

-50

 

 

-SO

 

 

 

(51

 

 

 

 

 

-200 -150 -100 -‘

 

 

 

0.02
-200 -100 so 150 -200 50 1&0
0.01

 

 

 

Figure 5.7: Comparison of the skew multipoles 02,2 (top) and (16,6 (bottom) with 62
respectively 66 in the lead end for r = 5 mm (left) and r = 20 mm (right).

Appendix A

SXF to COSY Converter

62

63

 

 

 

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Bibliography

72

Bibliography

[1] M. Berz, Differential Algebraic Description of Beam Dynamics to Very High
Orders, Published in “Particle Accelerators”, Volume 24, 1989.

[2] Lecture Notes for the Michigan State University Course Physics 861, “Introduc-
tory Beam Physics” by Martin Berz , available upon request.

[3] M. Berz, High- Order Computation and Normal Form Analysis of Repetitive Sys-
tems, in: M. Month (Ed), Physics of Particle Accelerators, vol. AIP 249, p. 456.
American Institute of Physics, 1991.

[4] M. Berz, K. Makino, K.Shamseddine and W.Wan Applications of Modern Map
Methods in Particle Beam Physics, Academic Press, Orlando, Florida, in print
1998

[5] H.Wiedemann, Particle Accelerator Physics, Springer Verlag 1993.

[6] M. Berz, Truncated Power Series Techniques, Entry in ’Handbook of Accelerator
Physics and Engineering’, M. Tigner and A. Chao (Eds )Particle Accelerator
Physics, World Scientific, New York, in preparation, 1997.

[7] M. Berz, Symplectic Tracking in Circular Accelerators with High Order Maps,
Nonlinear Problems in Future Particle Accelerators World Scientific,1991.

[8] A. J. Dragt and J. M. Finn, Normal Form for Mirror Machine Hamiltonians,
Journal of Mathematical Physics, 20(12), 1979 .

[9] A. Bazzani, Normal forms for symplectic maps on R2", Celestial Mechanics, 42,
1988 .

[10] K. Shamseddine and M. Berz, Exception Handling in Derivative Computation
with Nonarchimedean Calculus, in: Computational Differentiation: Techniques,
Applications, and Tools, M. Berz, C. Bischof, G. Corliss, A. Griewank (Eds.),
SIAM, Philadelphia, 1996 .

[11] K. Shamseddine and M. Berz, Nonarchimedean Structures as Differentiation

Tools, Proceedings, Beirut International Conference on Numerical Analysis,
1997 .

[12] M. Berz, The new method of TPSA Algebra for the description of beam dynamics
to high orders, , Los Alamos National Laboratory, AT-6:ATN-86-16, 1986 .

73

74

[13] M. Berz, The method of power series tracking for the mathematical description
of beam dynamics, , Nuclear Instruments and Methods, A258, 1987, 431.

[14] G. H. Hoffstaetter and M. Berz, Symplectic Scaling of transfer maps including
fringe fields, Phys. Rev. E 54 (1996) 5664.

[15] M. Berz, Computational aspects of design and simulation: COS Y INFINITY,
NIM, A298 (1990) 473.

[16] K. Makino, M. Berz, COS Y INFINITY Version 8, submitted to NIM

[17] COSY INFINITY Version 8 User’s Guide and Reference Manual, MSUCL-1088,
1997.

[18] COSY INFINITY Homepage. [Online] Available
http: / / www.bt.nscl.msu.edu / cosy

[19] FNAL LHC Homepage. [Online] Available
http: //www-ap.fnal.gov/LHC/

[20] F.Christoph Iselin, The MAD Program - User’s Reference Manual,
CERN/SL/90-13 (AP)

[21] F .Christoph Iselin, The MAD Program - Physics Methods Manual, CERN / SL/ 92
(AP)

[22] The Large Hadron Collider - Conceptual Design, CERN/AC/95-05 (LHC), Oc-
tober 1995

[23] G. Sabbi, Magnetic Field Analysis of HGQ Coil Ends, Fermilab TD-97—040 ,
September 1997.

[24] G. Sabbi, H GQ 9711 Field Quality Table, Fermilab TD-97-050 , November 1997.

[25] S.Russenschuck, C.Paul (CERN), K.Preis(TU Graz), ROXIE : A Feature Based
Design and Optimisation Program for Superconducting Accelerator Magnets,
CERN LHC Project Report 046 , September 1996.

[26] F. Méot and A. Paris, Concerning Effects of Fringe Fields and Longitudinal
Distribution of blo in Low-fl Regions on Dynamics in LHC, FERMILAB-TM-
2017, 1997

[27] Jacques Gareyte, Accelerator Physics Issues of the LHC, CERN LHC Project
Report 10, Geneva, 24 June 1996

[28] M.Giovannozzi, W.Scandale, E.Todesco Inverse Logarithmic Decay of Long Term
Dynamic Aperture in Hadron Colliders, CERN LHC Project Report 118, Jun
1997

[29] J .Brandt, A.Simmons, Coil End Design for the LHC IR Quadrupole Magnet,
Fermilab TS-96—013 , November 1996.

[30] F.Pilat et al, RHIC/AP/(1998).

75

[31] S.Caspi, M.Helm, L.J.Laslett and V.O.Brady, An Approach to 3D Magnetic Field
Calculation Using Numerical and Differential Algebraic Methods, SC-MAG-395,
LBL-326240, Berkeley, July 17, 1992.

[32] S.Caspi, M.Helm and L.J.Laslett, 3D Field Harmonics, SC-MAG-328, LBL-
30313, Berkeley, March 30, 1991.

[33] I.S.Gradshteyn, I.M.Ryzhik , Table of Integrals, Series and Products, 1979

[34] W.Wan, C.Johnstone, J .Holt, M.Berz, K.Makino, M.Lindemann, B.Erdelyi, The
Influence of Fringe Fields on the Particle Dynamics in the Large Hadron Collider,
conference contribution to CPO5, April 1998, submitted to Nuclear Instruments

and Methods

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