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NONLINEAR EFFECTS IN THE VERTICAL
MOTION OF IONS IN A
SUPERCON DUCTIN G CYCLOTRON

By

Dong-o J eon

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requireme nts
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1995

ABSTRACT

NONLINEAR EFFECTS IN THE VERTICAL MOTION OF IONS IN A
SUPERCONDUCTING CYCLOTRON

By

Dong—o J eon

Coupling effects near the 1/2 = 3/ 4 nonlinear resonance were studied both numer-
ically using the new Z W Orbit Code and theoretically utilizing an expansion of the
Hamiltonian to the fourth order in the vertical motion. A simple one-dimensional
model provided only a qualitative agreement, however close the operating point came
to the resonance. It turned out that inclusion of third-order coupling terms in the
Hamiltonian that are associated with certain nonlinear coupling resonances relatively
far away from the operating point provides an excellent explanation of phase space

diagrams both quantitatively and qualitatively.

The effects of higher order terms in the magnetic field components were studied
numerically using the new Z2N Orbit Code. The following cases were used for this
study: the change brought about to phase space diagrams by higher order terms near
the 1/2 = 3/4 resonance, the evolution of an eigenellipse during acceleration from 10
MeV/u to the final energy, 40 MeV/u, and the higher order effects on the sixth-order

V; = 3/ 6 resonance.

The new Z 2N Orbit Code uses “improved differentiators” based on a finite differ-
ence technique, which overcome the difficulties presented by the “old differentiators”

of the original Z4 Orbit Code. A comparison for the individual terms in the field

components shows that the “old differentiators” are less effective in suppressing high
frequency components and also tend to wash out physically important slowly varying
components of data. The orbit computation results from the original Z4 Orbit Code
using the “old differentiators” show a highly chaotic behavior that appears unrealis-
tic. A detailed description of the improved differentiators is given in the appendix
together with the applications to the magnetic field produced by two saturated iron

bars where the exact analytical expressions for the field components are available.

To Father and my family.

iii

ACKNOWLEDGEMENTS

I am deeply grateful to God for having guided me during the course of my life
and for having blessed me so much. I owe what I am today totally to God. I’d like
to express my deep gratitude to Prof. M. M. Gordon for his years of guidance and
kindness. I can’t help mentioning that his advice and criticism have been enlightening
and helpful. I’d also like to thank Prof. H. Blosser, Dr. F. Marti, and Mr. D. Johnson
for their advice and guidance. Also I’d like to thank all the guidance committee
members and fellow graduate students. Many thanks goto the National Science

Foundation that has made my study possible.

iv

Contents

LIST OF TABLES vii
LIST OF FIGURES viii
1 Introduction and summary 1
2 Hamiltonian theory 9
2.1 Expansion of Hamiltonian ........................ 10
2.2 Canonical transformation using angle and action variables ...... 14
3 Comparison of theory and orbit computations 19
3.1 Study of 1/2 = 3/4 resonance at 1/2 = 0.740 ............... 19
3.2 Study of 1/2 = 3/4 resonance at 1/2 = 0.749 ............... 27
3.3 Conclusions ................................ 33
4 Effects of higher order terms 35
4.1 The Z2N Orbit Code ........................... 36
4.2 Effects of higher order terms on certain orbits ............. 38

4.3 Effects of higher order terms on the 1/, = 3/ 6 resonance at 1/2 = 0.497 44

5 Comparison of old and new methods for computing field derivatives 49
5.1 Application to K1200 cyclotron field .................. 50

5.2 Application to orbits near the 1/2 = 3/ 4 resonance ........... 52

A Finite difference method for calculating magnetic field components

off the median plane using median plane data 61
A.1 Design of the first and second order differentiators in one dimension . 63
A2 Design of the first and second order differentiators in two dimensions 73
A3 Application to the field produced by magnetized iron bars ...... 82
A4 Application to data with noise ...................... 90

LIST OF REFERENCES

vi

96

List of Tables

4.1 rms differences in orbit computations obtained from the Z2N Orbit

Code withN=2andN=3, and withN=2andN=4 ....... 44
A.l Resolving Efficiency 61 (e) of the First-Order Derivative Schemes . . . 66
A2 Resolving Efficiency 62 (e) of the Second-Order Derivative Schemes . . 73
A.3 Comparison of the improved and old differentiators using data without

noise .................................... 84

A4 rms differences between the exact derivatives for the field without the
noise and those for the field with the noise for the two different numer-
ical differentiators ............................. 91

A.5 rms differences between the values of numerically differentiated deriva-
tives for the field with the noise and those for the field without the
noise for the two numerical differentiators ............... 91

A.6 Effects of the secondary filter on data with noise ............ 95

vii

List of Figures

1.1

1.2

1.3

3.1

3.2

3.3

3.4

3.5

3.6

4.1

4.2

Tune diagram for a K1200 superconducting cyclotron field with q/A =
0.25 and a nominal final energy of 40 MeV/ u ..............

Initial and final (2, 19,.) phase plots for orbits starting on two different
eigenellipses through the traversal of V; = 3/ 4 resonance .......

Initial and final (:15, 1),.) phase plots for the two sets of orbits through
the traversal of V2 : 3/ 4 resonance ...................

Two z-space diagrams at V2 = 0.740 (E = 34.8 MeV 11) for two sets of
five orbits at (/2 = 0.740, one obtained from the Z2 Orbit Code with
N = 2 and the other from the simple Hamiltonian ...........

Two z-space diagrams at uz = 0.740 (E = 34.8 MeV/u) for two sets of
three orbits at V; = 0.740, one obtained from the Z 2N Orbit Code with
N = 2 and the other from the Hamiltonian that includes the coupling
terms ....................................

Six x-space phase plots for the same two sets of three orbits at V; =
0.740. One set is obtained using the Z2N Orbit Code with N = 2,
while the other derived from the Hamiltonian including the coupling
terms ....................................

Two z-space diagrams at V; = 0.749 (E = 35.5 MeV/u) for two sets
of two orbits, one obtained from the Z 2” Orbit Code with N = 2 and
the other from the simple Hamiltonian .................

Two z-space diagrams at V2 2 0.749 (E = 35.5 MeV/u) for two sets of
three orbits, one obtained from the Z 2N Orbit Code with N = 2 and
the other from the Hamiltonian including the coupling terms .....

Six :r-space phase plots for the same two sets of three orbits at 12,, =
0.749. One set is obtained using the Z2N Orbit Code with N = 2,
while the other derived from the Hamiltonian including the coupling
terms ....................................

Three maps of orbits quite close to the separatrices at V; = 0.740
obtained by using the three different options of the Z21" Orbit Code
withN=2, N=3,andN=4 .....................

Three maps of an orbit quite close to the stable fixed point at 112 :2
0.740 obtained by using the three different options of the ZZN Orbit
Code withN=2,N:3,andN=4 ..................

viii

24

25

26

30

31

32

40

41

4.3

4.4

4.5

4.6

5.1

5.2

5.3

5.4

5.5

5.6

5.7

A.1

A.2

A.3

A4
A5

Evolution of an eigenellipse in z-phase space during it is accelerated
from 10 MeV/ u up to E f = 40 MeV/u obtained by using the Z21"
Orbit Code with N = 2 .......................... 42

Evolution in m-phase space for the orbits in the previous figure during
they are accelerated from 10 MeV/ u up to E f = 40 MeV/u obtained

by using the Z2N Orbit Code with N = 2 ................ 43
Two z-space diagrams near the V; = 3/6 resonance at V2 = 0.497
obtained from two different Z2N Orbit Codes .............. 47

m-space diagram near the V; = 3/ 6 resonance at 11,, = 0.497 showing
coupled motion in a: space obtained from the Z2N Orbit Code with
N = 3 ................................... 48

Map of a magnetic field of the K1200 superconducting cyclotron with
q/A = 0.25 and a nominal final energy E] = 40 MeV/u ........ 53

A map of V33 x 0.52/2! of the magnetic field of the K1200 cyclotron
computed by using the improved differentiators and a map of difference
between this and the map computed by using the “old differentiator” 54

A map of V38 X 0.54/4! of the magnetic field of the K1200 cyclotron
computed by using the improved differentiators and a map of difference
between this and the map computed by using the “old differentiator” 55

A map of $23 of the magnetic field of the K1200 cyclotron com-

puted at z = 0.5 (in) by using the improved differentiators and a map
of difference between this and the map computed by using the “old
differentiator” ............................... 56

Two maps of an orbit close to the separatrix for V; = 0.740, one at
the top is obtained from the Z4 Orbit Code which utilizes the “old
differentiators” and one at the bottom obtained from the Z2N Orbit
Code with N = 2 that uses the improved differentiators ........ 58

Two maps of an orbit in the z-dimensional phase space just outside the
separatrices of V; = 0.740 computed using the two different Z4 Orbit
Codes ................................... 59

Two maps of an orbit in the z-dimensional phase space just inside the
inner separatrix of V, = 0.740 computed using the two different Z4

Orbit Codes ................................ 60
Plots of (11(1) (w) — iw) /'iw for several first—order differentiators . . . 67
Plots of H (1) (w) /i for several first-order differentiators ........ 68

Plots of (Hm (w) + w?) /(—w2) for several second-order differentiators 71

Plots of [1(2) (w) for several second-order differentiators ........ 72

Plot of the frequency response H(F) (w) of the filter with Q = 1.01 . . 75

ix

A.6 Plot of the frequency response of the improved first-order partial dif-

ferentiator with respect to :c divided by 2', H£1)(wx,wy) /z' ....... 79
A.7 Plot of the frequency response of the improved second-order partial

differentiator with respect to x multiplied by (—1), -H_,(,2) (wmwy) . . 80
A8 Map of the magnetic field produced by two saturated iron bars . . . . 83
A9 Map of V38 x 0.52/2! obtained from the analytical expression . . . . 86
A.10 Two maps of the difference between exact values and those from the

two differentiators for V38 X 0.52/2! term ............... 87
A.11 Map of V38 x 0.54/4! obtained from the analytical expression . . . . 88
A.12 Two maps of the difference between exact values and those from the

two differentiators for V38 X 0.54 / 4! term ............... 89

A13 Two maps of the difference in V38 x 0.52/2! between the exact deriva-
tives for the field without the noise and those obtained from the two
numerical differentiators applied to the field with the random noise . 93

A.14 Two maps of the difference in V38 x 0.54 / 4! between the exact deriva-
tives for the field without the noise and those obtained from the two
numerical differentiators applied to the field with the random noise . 94

Chapter 1

Introduction and summary

The Z 4 Orbit Code [1] was originally developed for evaluating non-linear effects that
can seriously affect the beam quality during the process of extraction from supercon-
ducting cyclotrons. In addition, this orbit code can be used for semi-empirical studies
of third and fourth-order resonances that can occur in these cyclotrons]. One such

study, that of the 21/2 2 11,. resonance, has already been reported on [2].

The Z 4 Orbit Code uses exact equations of motion with magnetic field components
(8,, 89, 82) that are evaluated up to fourth-order in z. This evaluation requires up
to four successive derivatives of the measured median plane field data. Because of the
noise inherent in such data, the calculation of successive derivatives can sometimes
lead to spurious results. To overcome this difficulty, a new method of calculating
field derivatives has recently been implemented [3]. This method makes use of finite
difference techniques like those used in digital signal processing to suppress the noise

and produce smoother derivatives.

The main part of this thesis deals with the results obtained from a study of a
fourth-order resonance, (/2 = 3 / 4, that usually occurs in three-sector superconducting
cyclotrons when the field level is relatively low. For this purpose, we chose a K1200

superconducting cyclotron [4, 5, 6] field used for ions with q/A = 0.25 and a nominal

final energy of 40 MeV/ u. Figure 1.1 shows a plot of 11,, vs. 11,. for energies between
10 and 40 MeV/ u. The relevant third-order and fourth-order resonances are shown

by various straight lines, and we note that uz = 3/ 4 occurs at 35.6 MeV/u.

Even though the V; = 3/ 4 resonance does not lead to an actual instability, accel-
eration of the beam through this resonance can produce a noticeable deformation of
the vertical phase space. This is shown by the phase plots in Fig. 1.2 for two sets of
orbits starting on eigenellipses at 34 MeV/ u (V; = 0.721) with vertical widths A2 = 5
mm and A2 = 10 mm. These orbits are run for 99 turns out through the resonance
to 39 MeV/u (V1 2 0.782). The resultant two phase plots shown at the final energy

indicate how the deformation depends on amplitude.

All of these orbits start with identical values of (r, p.) on the same (accelerated)
equilibrium orbit at 34 MeV/u. Figure 1.3 shows the corresponding pair of radial
phase space areas at the final energy that are produced by certain nonlinear coupling
effects. The radial widths here, Ax = 0.08 mm and A.7: = 0.28 mm, are considerably
smaller than the corresponding A2 values given above, and moreover, are roughly
proportional to (Az)2. Such coupling effects are clearly undesirable, and indicate once

more the importance of controlling the range of vertical (and radial) displacements.

The 11, = 3/ 4 and 11,, = 3/ 4 resonances are obviously different since the former in-
volves motion entirely in the median plane and is therefore a simple one-dimensional
resonance [7, 8, 9]. For the u, = 3/4 case, however, coupling between the z and a:
motion is inevitable as shown in Figs. 1.2 and 1.3. Nevertheless, one would expect
that for small vertical amplitudes close to the resonance, the vertical phase space
properties, and especially the location of fixed points, could be described by the sim-
ple one-dimensional analysis. Although our results Show that this is qualitatively
true, they also show that the simple theory does not predict any of the fine struc-

ture apparent in the phase plots, no matter how close the operating point is to the

 

 

 

 

 

 

Figure 1.1: Tune diagram for a K1200 superconducting cyclotron field with
q/A : 0.25 and a nominal final energy of 40 MeV/ u. The diagram covers an en-
ergy range from 10 MeV/ u (marked with “a”) to 40 MeV/ u (marked with “b”) with
points shown as small circles having integer energy values. Also shown are three solid
lines for the third-order resonances (311,. = 3, 21/2 == 11,-, ur+2uz = 3) and four dot-dash
lines for the fourth-order resonances (41/2 = 3, 411,. = 3, 211,. = 21/2, 21/2 +214. = 3). The
V; = 3/4 resonance occurs at 35.6 MeV/u, and this point (11, = 1.136, V,, = 0.750) is
apparently not very close to either of the third-order coupling resonances. Neverthe-
less, these two coupling resonances do affect appreciably the vertical motion near the
11,, := 3/ 4 resonance.

 

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at uz=0.721

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— 1 0 -—5 O 5 10
z(mm)

Figure 1.2: Initial and final (2, 1),) phase plots for orbits starting on two different
eigenellipses at E = 34 MeV/u where V; = 0.721 (at top) and accelerating through
the resonance to 39 MeV/ u where V, = 0.782 (at bottom). The vertical widths of the
eigenellipses are A2 = 5 mm and A2 = 10 mm. These orbits were run for 99 turns
using the Z W Orbit Code with N = 2. Clearly, acceleration of the beam through the
V2 2 3/ 4 resonance can produce significant deformation of the vertical phase space.
(We should note that in all our orbit codes, momenta are expressed in length units by
setting if —> 17/qu where 80 is a given central field value. Here 80 = 35.1 ICC and
our lengths are expressed in mm. p = 988.7 mm is the total momentum at E = 35.6

MeV/u.)

 

 

 

 

 

I I I I I I I I I I I I I I Fl I I I I I II I—I
Eh at 112:0.721 —-_:
T— —J
C— —_.
;l I LII I I I I I I I I I I I I I l l l I l;
:1 I I I I I I I I I I I I I I I I I I I I I I I t
0-4 T at u,=0.782 ‘3
0.2 :— ——E
A _ _.
I3 : :
g 0.0 —— —1
- -1
>4 _ _
0* : :
—0.2 '_—' -—_‘
_0.4 :— .._:
— I I4 I LLI l l l I I I 11 l I l I ll JJ V

—0.4 —0.2 0.0 0.2 0.4

x(mm)

Figure 1.3: Initial and final (2:, 17,) phase plots for the two sets of orbits depicted
in Fig. 1.2, which demonstrate the result of coupling from the 2-motion into the
x-motion. Because initially all of the orbits share identical values of (r, p,) at 34
MeV/u, they are depicted as a point (shown at top). Finally reaching 39 MeV/u
after the acceleration (shown at bottom), the radial widths, A2: = 0.08 mm and
A3: = 0.28 mm, are much smaller than the corresponding A2 values in Fig. 1.2 and
roughly proportional to (A2)2.

resonance. We have, moreover, also found that this fine structure can be very well
reproduced when the effects of third-order coupling terms are included in the analysis.
One usually associates these terms with certain coupling resonances (12,. — 21/, = 0
and u, + 21/, = 3), and although these resonances are not very close to our operating

points, neither are they very far away (see Fig. 1.1).

The phase space properties near the V; = 3 / 4 resonance were explored by compar-
ing orbit code results with those obtained from a standard theoretical analysis based
on an expansion of the Hamiltonian. In chapter 2, we present a detailed description on
the expansion procedure of the Hamiltonian and on the subsequent canonical trans-
formation using angle and action variables. The Hamiltonian was expanded around
the equilibrium orbit order by order. In addition to the linear part, the part per-
taining to the V; = 3/ 4 resonance was kept. Concerning coupling terms, we retained
only the most slowly varying components of the lowest order coupling terms in the
Hamiltonian for the sake of simplicity. A canonical transformation using angle and
action variables was carried out assuming that the terms relevant to the (/2 = 3/4

resonance and the coupling terms are perturbations on the linear motion.

In chapter 3, comparisons between orbit computations and the theory are pre-
sented at two different values of Vz, 0.740 and 0.749. We considered two different
theoretical models, the simple one-dimensional Hamiltonian model by itself and the
same simple model together with the coupling terms. These studies show, for example,
that inclusion of the coupling terms reduced the error in the location of the unstable
fixed points from about 40 % to about 10 % for both V; = 0.740 and V; = 0.749. In
addition to providing a good explanation of the fine structure in the 2-space phase
plots, the simple theory including the coupling terms also provides a remarkably good
representation of the :c-space phase plots resulting from the coupling of the vertical

motion into the radial motion.

Chapter 4 starts with a brief description of the Z 2” Orbit Code that enables one
to use magnetic field components containing all terms up to 221" with N =1, 2, 3, or 4,
as desired. The orbit computations described in chapter 3 were carried out with this
code using N = 2. To see how important higher order terms might be, we repeated
some of these computations using the N :: 3 and N = 4 options, which corresponds

6 and 28 respectively. The results of this

to including magnetic field terms up to 2
study, which are presented in section 4.2, show that for the 11,, = 3/4 resonance, the
higher order terms are not significant for the range of 2 values of interest in the K1200

superconducting cyclotron.

To investigate further the higher order effects, we examined briefly the sixth-order
resonance V; = 3/6 using the Z2N Orbit Code with N = 2 as well as N = 3. Both
versions of the code produced phase space patterns with fixed points that are typical
of a sixth-order resonance, as shown in section 4.3. Moreover, the locations of the fixed
points in the two cases differ by only about 14 % which seems somewhat surprising

since the N = 2 option has no field components with terms higher than fourth-order.

Data are presented in chapter 5 showing the difference in results obtained by using
the “old differentiators” and the “improved differentiators” to calculate magnetic field
components off the median plane. First a comparison is given for the individual terms
in these field components evaluated at 2 = 0.5 in. The results show that the “old
differentiators” are less effective in suppressing high frequencies associated with noise
and at the same time, they also tend to wash out physically important slowly varying
parts of the data. In addition, a comparison is made of some orbits near the uz = 3 / 4
resonance using the old Z4 Orbit Code and the new ZQN Orbit Code with N = 2,
which utilized the old and improved differentiators, respectively. These orbits lie in a
highly sensitive region of phase space close to the separatrices, and the results from

the old Z4 Orbit Code show a highly chaotic behavior that appears unrealistic.

Finally in the appendix, a detailed description of the improved differentiators is
presented. As shown by frequency response curves, the improved method for eval-
uating derivatives suppresses high frequency signals effectively, while maintaining
reasonable accuracy over a sufficiently wide range of low frequencies. The improved
first-order and second-order differentiators for a uniform mesh in both one and two
dimensions are presented. As a test, these differentiators were applied to the magnetic
field produced by two magnetized iron bars where an exact analytical expression for
the magnetic field is known, and also applied to the same field when a small amount
of noise is superimposed. These results clearly demonstrate the superiority of the

improved differentiators.

The work described in this thesis has produced two papers. One dealing with the
coupling effects at the 12,, = 3 / 4 resonance has been published in Nuclear Instruments
and Methods A [10]. The other deals with the new method for calculating field
components off the median plane and has been accepted for publication in the Journal

of Computational Physics [3].

Chapter 2

Hamiltonian theory

The K1200 and K500 superconducting cyclotrons at this laboratory [11, 12] both have
three magnet sectors with three dees in the intervening valleys. For this study, we have
assumed that the field has perfect three-sector symmetry, and have, for simplicity,

ignored all imperfections.

In both the computations and the analysis, the magnetic field components in

cylindrical polar coordinates up to 24 are calculated using:

2 4

Z . Z
Bz = — B (1“,0) —- yViB (7‘,0) + 17V38 (7',6) , (2.1)
B — —30( a ) (2 2)
r — 05 7‘, a2 a '
BO : —WC (T',0,Z), (23)

where 8 (1',0) is the measured median plane field, and

 

82 3 (92
2 ___ _ __
V2 _ 87“? + 707' + r2002, (2'4)
is the two-dimensional Laplacian, and
23
C (r,0, 2) E 28 (730) — 5V38 (730). (2.5)

10

The above field components satisfy V-8 = 0 and can be derived from the following

fourth-order vector potential components:

1 a 22 24 2
A9 — —;/r8 (130) dr + a (58 (r,0) — EV2B (730)) , (2.6)
8 22 24 2
A, — T—a—é (—2TB (7‘, 0) + III-V23 (736)) , (2.7)
A, = 0. (2.8)

In the analysis of large synchrotrons and storage rings where Serret-Frenet coordi-
nates are used, it is usually assumed that the component of A along the reference
orbit is dominant and that the other components can be neglected. But for sectored

cyclotrons, one obviously needs both A, and A9.

With the expressions for the vector potential components on hand, we can now
write down the Hamiltonian required for the analysis. With 0 as the independent

variable, this Hamiltonian is given by:

He = —r (p2 — 103- 193% — qer,

=—r 2—II—A'2—H2
(p (. q.) .)

NIH

— qTAg, (2.9)

where p is the total momentum, and where p, and p, are components of the mechanical
momentum, while II, and II, are the corresponding canonical momenta. Note that
H, = p, since A, = 0 and (II,)Z=0 = (10,)1=0 since (A,)2=O = 0. We should also note
that the equations of motion used in the Z4 Orbit Code [1] can be derived from this

Hamiltonian.

2.1 Expansion of Hamiltonian

The theoretical analysis proceeds by expanding the above H9 about a given reference

orbit, namely the equilibrium orbit (EO). For a given energy (and hence p) value, the

11

coordinates of this orbit (r0, pm) can be determined as a function of 0 using the E0
Code [13]. Because midplane symmetry is assumed for magnetic field and the E0 is
on the median plane with z = P2 (01‘ Hz) = 0, it is natural to expand the Hamiltonian
Hg in Eq. 2.9 with respect to 2 and II, (or p,) in the first place. And subsequent

expansion of the Hamiltonian up to fourth-order in 2 and IIz is presented as follows:

= Iq/rBdr—rPeI+I—Hi+§ (W 2).},

 

 

2 pg 89 81‘
qII, 88
11422112
[8p9 2+” 4p 3 80 +
2 2 2 2 2
q 1+H—2’ 8—8 _,qII _0__V28+qr8V38 24 +...,
87‘pg pg 89 —4ng 09 4I 01‘

where Ho is the zero-order Hamiltonian in 2 and I12, Hg is the second-order Hamil-

tonian, and H4 18 the fourth- order Hamiltonian and where pg: (1)2 — IV)?

After the completion of expansion with respect to 2 and Hz, the above Hamiltonian
is again expanded around the E0 with respect to :L' and II, for the purpose of taking
account of the coupling effects and the subsequent deviations from the E0. Deviations

from the E0 are specified as :1: and p, defined by:

:rzr—ro, p,=p,—p,0. (2.11)

Since II, = p, + qA,, the corresponding canonical deviation is

Ha: 2 Hr — Pro : pa: + qAT'! (212)

and, of course, II, 2 pz as noted above.

For the sake of simplicity, we restrict ourselves to the lowest order coupling terms

in the Hamiltonian which come from the expansion of H2 with respect to r and II,.

12

After being expanded around the E0, H2 can be expressed as follows:

.. _"_ 2 2 E10_B_ (“LB 2
H2 _ [(2P9)0Hz+2(p9 60 Tar 02 +

1 m W
I(—) $113+ (r 3) 11,113+ (3— (1+—,1) Q3.) II,22+
2P9 0 2p.) 0 2109 P9 30 0

 

q n. 023 as 023 ,
2 (P6 0081‘ (97‘ _ r 81:2 02:2 + ' ' ., (2'13)
2 H§°)+H§”+..., (2.14)

where H30) is the zero—order part of H2 in :c and II, and H31) is the first-order part,
and where ()0 means that the quantity in the bracket is evaluated on the E0. It
should be noted that 830) contains all the terms pertaining to linear oscillations in

2-space.

Most of the terms in 831) are neglected for the sake of simplicity that are propor-
tional to 2H3, lelIZ, and II,,22 because they are small compared with the dominant

term by an order of magnitude. The dominant term to be kept is

Hcoupl E (2.15)

q (n. 323 BB 523) ,
"‘ 1'2
2 0

It should be noted that Hcoupl consists of the important third-order coupling terms;

that is, those proportional to 2:22 with the coefficients evaluated on the BC.

In a similar manner, H; can also be expanded around the E0 with respect to 2:

and II, as follows:
H, = Hp) + HI” + . . ., (2.16)

where HID) is the zero-order part of H4 with respect to a: and II, and HI” the first-
order part and so forth. Of course, there are higher order coupling terms coming from

H2 and H4 and so forth, but the results presented here are obtained by neglecting

13

all of them. Because all the higher order coupling terms are omitted for the sake of

simplicity, only H50) is retained for the analysis presented here:

II 63 q2 II2 BB 2
: (L) awe—s —(I+—r><—>
4 8p? 0 2 4p? 80 0 7' 8rpg pg 30

qn. 8V§B + 2139:7319) 24.
0

—4lp9 60 4! 07‘ (2'17)

 

 

 

Most of the terms in Him are again ruled out for the sake of simplicity that are
proportional to II: and 2211: because they are small compared with the dominant
term that is defined to be H,,_.,:

2 2 2 2 2
_ q H, BB qH, asz qr BVZB 4
HT” _ [81‘pg (1+ pg) (00) 4lp9 00 + 4! 07‘ z 0 (2.18)

 

 

 

o
It should be pointed out that H,“ contains the important terms relevant to the
V2 = 3/4 resonance, that is, the fourth-order terms proportional to 24 with their

coefficients evaluated on the E0.

Finally concerning H0 in Eq. 2.10, only the terms relevant to linear oscillations in
x—space are kept and the other higher order terms are neglected because the amplitude

of the induced motion in :r-space is small compared with that in z-space.

The expansion of the Hamiltonian, as a result, is divided into three parts as

follows:
H0 : Hlin + IIres + Hcoupla (2°19)

where Hun contains all of the terms pertaining to the linear oscillations in the z-space
and :r-space, and Hm and Hump; consist of the dominant fourth-order resonant terms
and the dominant third-order coupling terms respectively. As was noted previously, in
the expansion process we have for simplicity omitted many small terms that appeared

insignificant even though they might affect the results.

14

2.2 Canonical transformation using angle and ac-
tion variables

We next introduce angle and action variables, (452, J.) and ((251,, Jr). If y stands for

either 2 or :r, then the required canonical transformation is

y = (2J,fl,)%cos(¢,+5,), (2.20)

Hy : " (2Jy/flyfi [sin (iby + 6y) 'l' 03/ C05 (96y + 624)] a (2-21)

where the periodic Courant-Snyder parameters (1,, = ay(0), fly 2 fly (0), and the
phases 6,, = 6,, (0) are determined as a function of 0 from the transfer matrix elements
generated by the E0 Code [13]. This transformation reduces the linear part of the

Hamiltonian to:
Hun =1 l/sz + Ver, (2.22)

so that for the linear motion, Jy =const. and 455, = uy0+const., as expected.

Following the customary procedure, the nonlinear parts of the transformed Hamil-
tonian are treated as perturbations, and for this purpose, only the terms varying
slowly with respect to 9 are retained. Detailed description of the procedure is pre-

sented below for Hm, and Hemp).

Using the angle and action variables defined in Eq. 2.20 and 2.21, Hm, can be

written as follows:

H,” = Jff (0) [c054 (Q52 + 52) + 4cos 2 (452 + 6;) + 3], (2.23)

with

_flZ q2 1_I_3_ a_B 2 g_n_.avgB flax-233
f(0)_—_ ( Hp; 80 —4lpg 00 +4! 07' 0’ (2'24)

 

 

 

15

where ()0 means that the quantity in the parentheses is evaluated on the E0. Separat-
ing the functions dependent on (152 from those dependent on 0 which is an independent
variable, and Fourier transforming the functions with respect to 0 using the fact that

they are periodic, we can rewrite Hm, as follows:

H”,s = Jff (0) [cos 46,, cos 4% — sin 46z sin 4¢z+
4 cos 26,, cos 2gb, — 4 sin 2(5Z sin 2¢z + 3], (2.25)

= J3 [(A0 + 2: A... cos (M + o,)) cos4¢z
— (Bo + E 8,, sin (M + on) sin 445,
+ (Co + g 0,, cos (729 + c,,)) cos 2%
— (Do + 2 D, sin (no + cm) sin 2s,

+ G0 + Z 0,, cos (720 + g,,)] , (2.26)

where an, b,,,..., and gn are the phase of the nth harmonic and An, BMW, and 0,, are
the amplitude of the nth harmonic under Fourier transformation with respect to 0.

Reshuffling the terms using the properties of trigonometric functions, we get:

Hm 2 J3 {Ascos4o, — B0 sin 4% + Co cos 2o, — D0 sin 245. + 00
+ % 2713M” cos (4(15z + 126 + a,,) + An cos (4% — n0 — an)
+ 3,. cos (4s. + 720 + on) — 3,. cos (4o. — n0 — on)
+ 0.. cos (2%. + me + c.) + 0.. cos (2% — n0 — cs)
+ 1),. cos (2s. + no + d.) — 0,, cos (2o. — n0 — or.)

+ 20,, cos (720 + gn)]}- (2.27)

16

Retaining only the most slowly varying components and neglecting all the other
rapidly varying terms in the vicinity of the V; = 3/4 resonance, we obtain as a

final form of the resonant Hamiltonian to be used for analysis:

Hrs, % J: [00 + 1‘23- cos (4422 — 36 — a3) — % cos (4452 — 30 — b3) , (2.28)
= J; [G0 + Q cos (4o, —— 30 + 120)], (2.29)
where
_ 1 2 2 i
Q = -2- (A3 + B3 — 224383 cos (a3 — 123)) ,
B3 sin ()3 — A3 sin (13
tan Ibo E

 

A3 cos (1;; -—- 8;, cos ()3.

When we follow again a similar procedure for Hump, just as for Hr”, using angle

and action variables defined by

a: : (2Jrfir)%COS(¢x+6r)v

Z = (2Jzflz)% COS ((152: + 62) 2
Hemp) can be rewritten as follows:

1 _ _
Hes... = 1.31.9 (o) (swan + («so»)

(e2i(¢>z+6z) + 6-2i(¢z+6z) + 2) , (2.30)

wit—2
Alr—

with

__2 E323 213 33.13 s

It should be noted that g (0) is a real function of 9.

After separating functions dependent on 43’s from those depending on 0 and reshuf-

fling the terms, we get:

H Pl : .12.] g_(_0_2. {26161614}: + 26—i6Ie—i¢x + ef(26z+61)ei(2¢z+¢1)+
cou 1: z 8

e_i(252+61)e—i(2¢z+¢x) + 65(251—6I)ei(2¢z—¢1) + 6—i(261-6I)e-i(2¢z—¢1)] ,‘(2.32)

17

E JEJZ [S (0) eiafis + S‘- (0) e—t'tb: + 11(0) ei(2¢z+¢s) + Tit (0) e—i(2¢>z+¢s)

+ (1(9) ei('2¢>z—¢x) + U’ (0) e-i(2¢>z-¢I)] , (2.33)

where * stands for complex conjugate and where

5(0) 5 g—flcw‘r,

71(0) E 9(9)€r(26.+6s)
8 ?

U(0) : Mei(252-51).

8

Fourier transforming complex functions 5 (0), T(0), and U (0) with respect to 0
exploiting the fact that they are periodic functions of 0, we can rewrite Hcoupl as

follows:

Hcoupl = JéJz [Z (Snei(¢1+n9) + 5:6-i(¢>x+n0))

n

+ Z (Tnei(2¢s+¢s+n0) + Tge-i(2¢s+¢s+n9))

+ Z (Unei(2¢z—¢x+n9) + U;€_i(2¢z_¢r+n0))] , (234)

where Sn, Tn, and Un are the complex amplitudes of eino components of S (9), T(0),

and U (0) respectively.

Just as in the case of Hres, we again assume that the most slowly varying compo-
nents of each part contribute most. Keeping only the most slowly varying components

while taking into account the operating points near V; = 3/4, we get:

1 . . . , .
[Icoupl z J1? J2 [5064” + SEC—“bx + T—3ez(2¢z+¢1_36) + T:36—1(2¢z+¢r—30)

wows—M +Ugc-‘Wz—q’rl]. (2.35)

Defining 5'0 5 Clei‘f’1/2, T.;; E Cgei'l’2/2, and U0 E C3ei‘f3/2 where (0,, 2%) for

j = 1,2, 3 are real constants, we then obtain, as a final form of Hcoupl to be used for

18

analysis:

Hcoupl z Jlg—Jz [Cl COS ($3 + z()1) + 02 COS (2¢z + 451‘ — 30 + 2("2)

+ C3 COS (29b: _ ¢x + 1(5)] ' (2'36)

When this procedure is completed, as a summary the results are:

Hm z 60.]: + QJf cos (4s, — 30 + zbo), (2.37)
l.
Hcoupl z J1? J2 [Cl COS (¢.r + $1) + C2 C03 (29252 + 451: _ 30 + tp2)

+ C3 COS (2¢z _ <st + $3”, (2'38)

where Go, Q, db, and the set (Cj, 2/21') forj = 1, 2, 3 are all constants which are defined

above and numerical values are given in the following sections.

To see how slowly these terms vary, consider the tune values for a case that will
be discussed later in considerable detail: 11,, = 0.740 and V, = 1.129, which occur at
34.8 MeV/ u. In this case, the cosine term in Eq. 2.37 has the (zero—order) frequency:
(3 — 41/2) = 0.040. The corresponding frequencies of the three terms in Eq. 2.38 are:
11,. = 1.129, (3 - 21/2 — 14.) 2: 0.391, and (21/2 — 11,.) = 0.351. These frequencies are
obviously much greater than the 0.040 in the resonant term, but in spite of this, the
coupling terms play an important role in the uz = 3/4 resonance (as will be seen

below), and this is presumably because they are of lower order.

Chapter 3

Comparison of theory and orbit
computations

In this chapter, we compare the results obtained from the Z2N Orbit Code with
N = 2 and those obtained from the Hamiltonian derived in the previous chapter.
To emphasize the difference between the new and old orbit codes, we called the new
one the Z2N Orbit Code with N = 2 to distinguish it from the old Z4 Orbit Code.

Further information about the new code can be found in Chapter 4.

3.1 Study of V2 2 3/4 resonance at V2 2 0.740

Consider first the results obtained from the simple theory of the V, = 3/4 resonance,
assuming it is “isolated”. In this case, the Hamiltonian is obtained by combining

Eq. 2.22 and Eq. 2.37, that is,
H9 R1 11sz + Gon2 + QJ: cos (4(152 — 30 + 1110), (3.1)

where the values of the constants depend on the energy chosen.

We find that the “tune-shifting” term 00.122 is positive and, moreover, Go > Q > 0,
so that the resonance produces no instability. More specifically, the phase space

diagram shows four stable as well as four unstable fixed points for 11,, < 3/4, but none

19

20

at all for V; > 3/4. In addition, the theory shows that these fixed points have the

following angle and action values:

cos (44L — 30 + 2&0)“, = i1, (3.2)
and
Us)” = (3/4 - V2) /2(Go i Q), (3-3)

where the +(—) Sign applies to the unstable (stable) fixed points. These conclusions

are in good qualitative agreement with the computational results given below.

Although the theory is expressed in terms of angle and action variables, the in—
put/output of the Z2N Orbit Code are in terms of (1', pr) or (2', pm). In order to make
direct comparisons, the orbit code results are translated into 45’s and J’s by using the
inverse of the transformation Eq. 2.20 together with the definitions of Hz and II;c in
Eq. 2.21. We should also note that for linear motion, 27rJ is the invariant area of an

eigenellipse, and we will therefore use

NIH

p = (ZJ) (3.4)

as a “radius” in phase space. In the phase space diagrams presented here, we plot
p sin 45 vs. p cos (25 since such plots provide the symmetric diagrams shown in theoretical

papers [7, 8, 9]. Moreover, for linear motion, such a plot yields a circle of radius p.

As an example, take the case where (3/4 — V2) = 0.010 occurring at 34.8 MeV/u.

Here, numerical analysis yields the following values of the constants:
(GO,Q) = (975,342) x 10—5 (mm)_2, (3.5)
and

20 = —44.3°. (3.6)

21

All of the orbit computations reported on here start with various initial values
of (2, p2), but with initial (r, p,) values always on the E0 in order to avoid added
complications. The two z-space diagrams shown in Fig. 3.1 were obtained by plotting
points once per sector at 6 = 27m / 3. The diagram at the top came from the output
of the Z2N Orbit Code with N = 2 while the diagram at the bottom was derived
from the simple Hamiltonian in Eq. 3.1. The two diagrams are plotted to the same
scale and as can be seen, the fixed points in the top diagram are closer to the origin
than those in the bottom one. The (p2, qfiz) coordinates of the fixed points in the first

quadrant have the following values:

_ (6.13 mm, 9.7 deg)

(pz’¢2)“ _ { (8.72 mm, 11.1 deg) (3'7)
_ (10.6 mm, 54.1 deg)

(pz’¢z)’ — { (12.6 mm, 56.1 deg) (38)

where the subscripts “u” and “3” refer to the unstable and stable fixed points, and
where the top and the bottom lines on the right give the values obtained from the top
and the bottom diagrams in Fig. 3.1. The fractional difference in the pz for stable
(unstable) fixed point is 19 % (42 %).(Actually, the bottom values are derived from

the formulas in Eq. 3.2 and 3.3 above with the given V2 value.)

In addition to the significant differences in the fixed point locations, the plots
obtained from the orbit code (especially those near the stable fixed points) show a

complicated fine structure that is completely missing from the theoretical plots.

To see if the differences between the theoretical results and those from the Z”,
Orbit Code with N = 2 can be traced to the third-order coupling terms, consider
next the complete theoretical Hamiltonian obtained by combining Eq. 2.22, 2.37, 2.38.

Again using as an example the numerical constants determined analytically for 34.8

22

MeV/u where V2 2 0.740, this Hamiltonian becomes:

H9 = usz + VJ, + J3 [9.75 + 3.42 cos (4gb, — 30 — 44.3°)] x 10-5
.1,
+ J} J; [5.65 cos (45,, —— 77.4°) + 7.19 cos (2qu + (bx —— 30 —— 146.2°)

+ 4.30 cos (245. -— as, +104.4°)] x 10-3. (3.9)

Evidently, when J, = 0, this Hg reduces to that in Eq. 3.1 with the constants given

in Eq. 3.5 and 3.6.

The equations of motion derived from He in Eq. 3.9 can be integrated and the
results compared to those obtained from the Z” Orbit Code with N = 2. Such
a comparison is shown in the phase space diagrams presented in Fig. 3.2 with the
orbit code results given at the top and the theoretical results given at the bottom,
just as in Fig. 3.1. But here in Fig. 3.2, we restrict ourselves to phase plots for three
particular orbits that have special interest. Plot # 1 circulates around the stable
fixed points coming close to the separatrices and the unstable fixed points; this plot
closes in about 200 turns. Plot # 2 resembles a flower with many petals that encloses
each of the stable fixed points; this plot requires about 180 turns. (These two plots
are also shown at the top of Fig. 3.1.) Finally, plot # 3 (which runs about 100 turns)
is a small blur of points that are as close as one can get to the stable fixed points.
Comparison of the diagrams in Figs. 3.2 and 3.1 clearly shows that the qualitative and
quantitative agreement between the theoretical and orbit code results is significantly

improved by inclusion of the coupling terms in the Hamiltonian.

This conclusion is reinforced by the evidence presented in Fig. 3.3 which show
the :r-space phase plots associated with the three orbits whose z-space plots are
given in Fig. 3.2. Here, the three plots on the left come from the Z2” Orbit Code
with N = 2 while those on the right come from the theoretical results. The good

agreement between these results is quite impressive. One should keep in mind that all

23

of these plots are simply twosdimensional projections of the four-dimensional phase

space trajectories traced out by the three orbits.

As a test of the quantitative agreement between the two sets of results, we again
compare the locations of the fixed points shown in the two diagrams of Fig. 3.2. Using

the same notation as in Eq. 3.7 and 3.8 above for the data in Fig. 3.1, we now find:

_ (6.13 mm, 9.7 deg)

(pg, “52)“ _ { (6.85 mm, 10.0 deg) (3°10)
_ (10.6 mm, 54.1 deg)

(p2, Q52)’ _ { (12.7 mm, 56.4 deg) (3'11)

where the top and bottom numbers represent the orbit code and theoretical results,
respectively. The fractional difference in the p, for the stable (unstable) fixed point
is 20 ‘70 (12 %). Comparison with the data in Eq. 3.7 and Eq. 3.8 above shows that
including the coupling effects provides a very significant improvement in the location
of the unstable fixed points, but practically no change at all in the location of the
stable fixed points. The accuracy of the theory is, of course, limited by its reliance

on an expansion of the Hamiltonian in Eq. 2.9 with many “small” terms omitted.

It should also be pointed out that in producing the diagrams shown in Figs. 3.2
and 3.3, we used slightly different initial conditions for the two sets of orbits. As
noted in the beginning of section 3.1, all of the Z2” Orbit Code runs were started
with (1', p,) values on the E0 so that initially, :r = p, = 0. But since IL, 2 pa, + qAr,
and since A, 75 0 for z 75 O, the initial values of Hg, and hence J1. and pr differ from
zero. In all cases reported here, computer runs based on the theory were started with
the same values of (px, (151,) as those from the Z2N Orbit Code with N = 2. However,
the initial values of (pz, (92) for the theory runs were changed slightly so as to match

as well as possible the resultant phase plots with those obtained from the Z 2N Orbit

24

 

I- 1. .1
.d“ .. o
P ‘ d
. j . . I. .00.
L . 0‘ no“. . .f “L _
8 . .f 9 3
)- f ,p ‘ '. ,o g -(
I : ‘ I. ” '
p-
. : fl
1- ‘ a... : J
I
'- \

 

o. o.
i "
4 f -
~
I : " \J '. .5 . s
I- ! . “w ' -l
o a g .
(- U ' t a d
I ' f 1' .r
5‘: . I a I
- 2 ‘0' ..-’ '.,""“” -4
l ‘ ’. a I ’
— ' o . _
’ ‘- .Iv
' I - ° , "H ”"o ~
b O 0 c d
.e '
I O
b d
D d
L L 1 l l l l l I L 1 mL L L l l l
I I I I I I l I I l I r 1 I
I- a
'I .. .0
h . o I ' . ' . . d
. 0" . . . . ---- o '
:- ... ..- on u. c. 00- . 0 .. n. . .. .4
c' o . n. ' o .
1 0 b- o. o . . . . ' I . .. i I c z —(
o a u. . a
. ‘ . . n o a g... a .. . ' .
I- z . ‘ .' .5. .°. I . q
: I 9 ......... #- . :
h I. . I. ..... ..'. .I d
,. . . . “I I... 'l/ o. '. n . .. '0
.- .. . u .do". .. ... I .-

Plen¢z (mm)
0

. . I l '

1 I 4 4 1 1 l

I
I
1 L l I

 

 

 

Figure 3.1: Two z-space diagrams at V, = 0.740 (E = 34.8 MeV/u) obtained by
plotting pz sin (9, vs. pz cos 45,, once per sector for two sets of five orbits. One set (shown
at top) is obtained from the Z W Orbit Code with N = 2 and the other set (shown at
bottom) is derived from the simple Hamiltonian in Eq. 3.1 with its constants evaluated
at the given energy. The simple theory obviously fails to reproduce quantitatively
the results from the orbit code, including the location of the fixed points. It also
fails to account for the fine structure in the phase plots from the orbit code which, as
shown below, are produced by the coupling effects. Note that (45, J) are angle-action
variables and that p = (2J)1/2 corresponds to a “radius” in phase space.

 

-
p

 

).

t

L.

t

‘F

F
+-

I
l

\
s

Plen¢z (mm)
o
l .

‘ 1
v.1".
Lim

10s a...

. ’ F-‘H
. .'
CD ’ .

I
l

- X

 

 

 

.J

.. ‘ .(

f >( ~"r'.

\wa

—10 _ (Of ' . —
- 1s. 4

i- 4 l I I l l l l l l l l l I l 1 LA

—10 0 10

P2003952 (mm)

Figure 3.2: Two z—space diagrams (just like those in Fig. 3.1) for two sets of three
orbits obtained by plotting once per sector, again at 34.8 MeV/ u where V; = 0.740.
The set at the top was obtained using the Z” Orbit Code with N22, while the set
at the bottom was derived from the Hamiltonian in Eq. 3.9 that includes the third-
order coupling terms. In contrast to the comparable situation in Fig. 3.1, the results
presented here show that this Hamiltonian provides a relatively good explanation of
the orbit code results, including the location of the fixed points and the fine structure
in the phase plots. Note that all the orbits from the Z21” Orbit Code with N = 2
reflect coupling effects. Unlike the situation predicted by the simple theory in Fig. 3.1,
the stable and unstable fixed points are not sharply defined due to the coupling effects.

26

 

 

 

 

 

. I I l I I I I I I I I I I j I l l I I ‘ P I I I I I I I I I I I I I I T I I I I -
u- o 4 )- 0 d
. Orbit 1 .. Orbit 1 .
u- 1 b .4
-— —| — —
I- 4 h . : o. c-
- 1 D- .. 5 . -(
)- '.:..' ‘ ' a
'- n’ ” ..’ 'o‘t . '1
I— —- — ‘. $0. -—(
I. D . ‘: .l ' .4
h I- . .0 q
r- b ' +
n- r- 2...:‘ a; q
—- — — 0.; -—(
I- . ‘ H . a
I- - p . . u
h A A I A A A A l A A A A I A A A L. P A A A A I A A A A l A A A A l A A A A .
- V I 1 V ' l V V V T I Tr ' I ' ‘ p I V V ' I I ' '—' r v r v v I r t r i q
o u I- 0 -l
t Orbit. 2 .. Orbit 2 ,
p r- . d
— ‘ '.' .g \ $ ' _ I— .- ’..:: z.- —I
b. ‘0: 0.? o g0'fo1..: ‘ ‘ l' ”c .. .: . . . :':. ‘
~ -- .- -:.-.-: «.x .. rise. ..= . .
'- .. 0’...~:o .‘ ‘OIJ.".10.’}‘ ”p . .0 51‘; .00.“... '. .3; : ' T
r- o‘.' 30...: ' ’. :" ”.I‘. 1|- '0 .‘t..’|....:' 50.: ”:33 o -l
h—C‘ "’do~.:. ~| fi.‘ k _ _ 0‘ ...:.: ' ‘o:l: '.’o: ... :z. a .—
p . .2 I n‘. ' ’0‘ 1p . o In... . .0..I .IK’. : .1
' “r“ . - .-. . .-i“". t. 1‘» 4
" ::~~' 7: -' . ~ . .':' --.-'--:' . .
h I 3 l,- :1 '0‘. .. .‘ I: ‘I' : 0‘0... ~ .:.0 '
I- I. 3; :-¢ 2.... . .4. u )- g’;. a \fiof. 31:...
—- . ' a o‘ 3' . ' . _ _ . °. . . . \ ‘ 0°: .;.
'- l a. '0‘“. .. .!.¢' ¢ P ' .0 . 0 o ' o ' I. o q
p .u . . .. ‘2'! 1b “a : a . 2.. d
'- o ’ O 4 ’ c o .. . . ’ J
z p l 4 I L l A A A A Jb L L I I A .A I I A ‘
h r I I ' I l I ' T. h V I t r I ' I t I f V q
)- 0 1 b -(
. 011311: 3 .. Orblt 3 .
u- 1- d
”s 1- 2- ‘
h 1 I- -I
u- (p I d
E i- d v- “! ”.Q -4
v b d . {In \‘ d
.9’.‘ O . .7' .~' 2, ’o, 3 "j
q n 3' . ‘ 3‘ d
O“ I- n 0‘ “ v "} d
m ' 1"”
" L i 2‘ 3" d
_ u— v—— '. ‘ ' —-1
Q 1 h- ‘ b *"a" ‘
u- 4l- 4
:- lb -(
~ -1
_2 J 1 I l l 1 l l l l l l l l I I l l l h I l I I l I I l l I l I l l I l l l l

 

 

 

 

-2 -1 O 1 2
PxC°8¢x (mm)

Figure 3.3: Six x-space phase plots obtained by plotting p, sin 45,, vs. px cos 45,. once
per sector for the same two sets of three orbits whose z-space phase plots are shown
in Fig. 3.2. The three plots on the left were obtained using the Z2” Orbit Code with
N = 2, while the three on the right were derived from the Hamiltonian in Eq. 3.9
that includes the third-order coupling terms. The orbits labeled 1, 2, and 3 in Fig. 3.2
appear here at the top, center, and bottom, respectively. Since all of the orbits start
with (1', p,) values on the equilibrium orbit, the phase plots here demonstrate the
coupling of the z—motion into the x-motion. Moreover, the values of px here are an
order of magnitude smaller than the values of pz in Fig. 3.2 so that the coupling is
not very strong. However, this coupling is quite significant and it is clear from a
comparison of the two sets of phase plots that the theoretical Hamiltonian in Eq. 3.9
provides a good representation of the coupling effects found with the Z21” Orbit Code
with N = 2. Note that all six plots have exactly the same scales although only the
one at the bottom left is labeled.

27

Code with N = 2. The size of this change for the orbit # 3 representing the stable
fixed points can be gauged from the numbers given in Eq. 3.11; the changes for the

other two orbits are about the same.

3.2 Study of V2 2 3/4 resonance at V2 2 0.749

Similar calculations were carried out for (11,. = 1.136, V2 2 0.749) at 35.5 MeV/ u to
see if the third-order coupling terms still play an essential role just as in the case for
11,, = 0.740. One difference from the Vz = 0.740 case is that the scale of the major
features of z-space diagrams are reduced by about three times compared with those for
112 = 0.740. Because 3/4 — V; = 0.001 is ten times smaller than 3/4 — 0.740 = 0.01 for
V2 2 0.740, the subsequent values of (Jz)fp is reduced by a factor of ten approximately,
assuming the other constants remain about the same (refer to Eq. 3.3). So (pz)fp is

reduced by about three times.

Assuming again the V, = 3/4 resonance is “isolated”, the Hamiltonian without

the coupling terms is:
He = 44sz + .13 [11.0 + 3.98 cos (44>. — 30 — 45.5°)] x 10-5, (3.12)

and the phase space diagram and its features can therefore be determined. Remem-
bering that (Go, Q)=(9.75,3.42) x 10’5 and who = -—44.3° for V, = 0.740, the values
for Go and Q for V; = 0.749 are larger than those for V, = 0.740 by 13 ‘70 and 16 %

respectively and the value of 7,120 differs by 1.2°.

Just as in the case for V2 2 0.740, all of the orbit computations reported on here
start with various initial values of (2, 1),), but with initial (7', 1),) values always on
the E0 in order to avoid added complications. Figure 3.4 corresponds to Fig. 3.1 for

V2 2 0.740 and shows two different z-dimension phase space diagrams. The diagram

28

at the top is obtained from the numerical orbit integration using the Z 2N Orbit Code
with N = 2 and the bottom diagram is obtained by integrating the equations of
motion derived from the Hamiltonian in Eq. 3.12. It should be noted that Fig. 3.4
uses two orbits while Fig. 3.1 uses five orbits. The diagrams in this figure are very

similar to those in Fig. 3.1 except for the difference in the scale.

Again as a test of the quantitative agreement, the (p2, (25g) coordinates of the fixed

points in the first quadrant are given here:

__ (1.87 mm, 9.9 deg)

(p4, $2)“ _ { (2.59 mm, 11.4 deg) (3'13)
_ (3.24 mm, 54.8 deg)

(94,94). ‘ i (3.79 mm, 56.4 deg) (3'14)

“.3” refer to the unstable and stable fixed points, and

where the subscripts “u” and
where the top and the bottom lines on the right give the values obtained from the
top and the bottom diagrams in Fig. 3.4. For the stable (unstable) fixed point, the
values of p2 differ by 17 % (39 %) while they differ by 19 % (42 %) for the V2 = 0.740

case. (The bottom values are derived from the formulas in Eqs. 3.2 and 3.3 above

with the given 11; value just as in the case for 112 = 0.740.)

The fractional differences in the fixed point locations are almost equal. In addition
to that, the plots obtained from the orbit code show a complicated fine structure

because of the coupling between the z and :1: dimensions when the plots are magnified.

Now consider the complete theoretical Hamiltonian including the third-order cou-
pling terms just as in the case of V2 2 0.740. Using the numerical constants deter-

mined analytically for 35.5 MeV/ u where V, = 0.749, this Hamiltonian becomes:

H9 2 44...]. + 11er + J3 [11.0 + 3.98 cos (445,. — 30 —— 45.5°)] x 10-5

1
+ .1ng [5.78 cos ((15,, — 79.0°) + 7.37 cos (2% + 45x — 30 — 154.9°)

29

+ 4.48 cos (2% — (fix + 107.8°)] X 10-3. (3.15)

Compared with the corresponding values of C,- and z/Jj for V; = 0.740, the values of
C j differ by at most 4 % and the values of 1b,- vary by 1.6°, 8.7°, and 3.4° respectively

for j = 1,2,3.

Similar comparisons between theory and orbit computations are shown in the
phase space diagrams presented in Fig. 3.5, which corresponds to Fig. 3.2 with the
orbit code results given at the top and the theoretical results given at the bottom.
We restrict ourselves to three particular orbits, just as in Fig. 3.2. Plot # 1 closes in
about 1800 turns. Plot # 2 requires about 400 turns. Finally, plot # 3 runs about
400 turns. It should be noted that in Fig. 3.2 for uz = 0.740, it takes about 200 turns
for plot # 1, about 180 turns for plot # 2, and about 100 turns for plot # 3. Because
:12 = 0.749 is closer to the 112 = 3/ 4 resonance, it naturally takes more turns to close
in. When considering only linear motion, it should take ten times more turns to close
in than the case for 112 = 0.740. Comparison of the diagrams in Fig. 3.4 and Fig. 3.5
clearly shows that the qualitative and quantitative agreement between the theoretical
and orbit code results is significantly improved by inclusion of the coupling terms in

the Hamiltonian for V2 = 0.749, just as for uz : 0.740.

F igure.3.6 which corresponds to Fig. 3.3 shows the two sets of three x-space
phase plots associated with the corresponding orbits whose z-space plots are given
in Fig. 3.5. The three plots on the left come from the Z2” Orbit Code with N = 2
while those on the right from the theoretical results. Again the Hamiltonian with the
coupling terms provides good agreement between these results. Except for the scale

of the plots and the turn numbers, these plots are very similar to those in Fig. 3.3.

As a test of the quantitative agreement between the two sets of results, we again

compare the locations of the fixed points shown in the two diagrams of Fig. 3.5. Using

30

 

TFI’YTIYIYIITTTTTI‘ITTijlr

f

qu-

 

ITTrFITIlIlUIIIIIIIUII'1-_FYIYTIIlil‘TI'YYIYIIIII‘I

 

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e." 0
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m
N
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-4
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-4 —2 0 2 4
P2003952 (mm)

Figure 3.4: Two z-space diagrams at V2 2 0.749 (E = 35.5 MeV/u) for two sets of
two orbits obtained by plotting p2 sin (:52 vs. pz cos 432 once per turn. These diagrams
were obtained by plotting once per turn instead of once per sector as in Fig. 3.1, and
the scale is 3.6 times smaller. One set (at the top) was obtained from the Z21v Orbit
Code with N = 2 and the other set (at the bottom) was derived from the simple
Hamiltonian in Eq. 3.12. Unlike the plots in Fig. 3.1, the fine structure is not clearly
observable to the eye. Otherwise, the plots are very similar except for the scale and
the turn numbers of the orbits when compared with those in Fig. 3.1.

 

PI I I I I I I I I I I I I I I I I I I I I I I I

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ch°S¢z (mm)

 

Figure 3.5: Two z-space diagrams (like those in Fig. 3.2) for two sets of three orbits
obtained by plotting p2 sin qfiz vs. pz cos 452 once per turn at 35.5 MeV/u where V2 =
0.749. But Fig. 3.2 was plotted once per sector and its scale is 3.6 times larger. The
set at the top was obtained using the Z 2N Orbit Code with N = 2, while the set at the
bottom was derived from the Hamiltonian in Eq. 3.15 that includes the third-order
coupling terms. The plots are very similar except for the scale and the turn numbers
of the orbits compared with those in Fig. 3.2. Even though the fine structure may not
be observable for the given scale, it becomes apparent when the plots are magnified.

32

 

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-0.1 0.0 0.1
Px°°S¢x (mm)

Figure 3.6: Six x-space phase plots (like those in Fig. 3.3) obtained by plotting
p,c sin 45$ vs. pa, cos ¢x once per turn for the same two sets of three orbits whose 2-
space phase plots are shown in Fig. 3.5. But Fig. 3.3 was plotted once per sector
and its scale is 12.5 times larger. Because px is proportional to p: and because p2
is reduced by about 3.2 times for uz = 0.749 compared with that for uz = 0.740,
p3,. is reduced by about ten times. The three plots on the left were obtained using
the Z2N Orbit Code with N = 2, while the three on the right were derived from
the Hamiltonian in Eq. 3.15 that includes the third-order coupling terms. The orbits
labeled 1, 2, and 3 in Fig. 3.5 appear here at the top, center, and bottom, respectively.
The plots are very similar except for the scale and the turn numbers of the orbits
compared with those in Fig. 3.3 for V2 = 0.740. Note that all six plots have exactly
the same scales although only the one at the bottom left is labeled.

33

the same notation as in Eq. 3.13 and 3.14 above for the data in Fig. 3.4, we now find:

_ (1.87 mm, 9.9 deg)

(pg, (#2)” _ { (2.05 mm, 11.6 deg) (3'16)
_ (3.24 mm, 54.8 deg)

(”2’ 452). — { (3.85 mm, 56.6 deg) (3'17)

where the top and bottom numbers represent the orbit code and theoretical results,
respectively. The fractional difference in the p2 for the stable (unstable) fixed point is
19 % (10 %) while the corresponding value for the stable (unstable) fixed point is 20 %
(12 %) for V2 2 0.740. A comparison with the data in Eq. 3.13 and Eq. 3.14 above
shows that including the coupling effects provides a very significant improvement in
the location of the unstable fixed points, but little change in the location of the stable

fixed points.

It should also be pointed out that in producing the diagrams shown in Fig. 3.5
and Fig. 3.6, we used again slightly different initial conditions for the two sets of
orbits as in the case for V; = 0.740. In all cases reported here, computer runs based
on the theory were started with the same values of (p35, (153) as those from the Z2N
Orbit Code with N = 2. However, the initial values of (pz, 96;) for the theory runs
were changed slightly so as to match as well as possible the resultant phase plots with

those obtained from the Z” Orbit Code with N = 2.

3.3 Conclusions

From the results of the two different cases for V; = 0.740 and V2 = 0.749, we can
conclude that the uz : 3/4 resonance is not “isolated” and that the coupling terms
have significant effects on the orbits. Moreover, the coupling effects between the z

and a: motion can be reproduced, within a reasonable accuracy, using only the most

34

slowly varying components of the lowest order coupling terms in the Hamiltonian.
By inclusion of the coupling terms, both the qualitative and quantitative agreement
between the Z2N Orbit Code results and the theoretical results is significantly im-
proved. In addition, the success of the theory here also provides evidence that the

Z 21" Orbit Code is functioning properly.

Chapter 4

Effects of higher order terms

Up to this point, we have ruled out the effects of the terms of the magnetic field
components which are of order higher than four in 2. It’s very important to decide
whether it is worth going to the trouble of including higher order terms in orbit
computations and hence at what order one can safely truncate the series. For the

field components given in Eqs. 2.1- 2.5, the higher order increments up to 28 are:

26 6 28 8
AB; = — —-6—!V2B+§V23 , (4.1)
0
AB... = —5;AC, (4.2)
A33 2 _7'8WAC’ (4.3)
with
25 27

Before setting out on actual comparisons based on orbit computations, there are

several points we should note.

First of all, the representation for the magnetic field components assumes there
are no sources (no singularities). As the value of 2 increases, one approaches various
sources such as trim coils and pole face, and the representation becomes less appro-

priate. However, in actual operation of the K1200 superconducting cyclotron, the

35

36

maximum vertical displacement of beam, 2m, is usually limited to be zm < 5 mm,
while the height of the liner is about 19 mm. Because the effects of higher order
terms of the magnetic field components become significant only for large 2 ampli-
tudes, higher order terms beyond a certain order can be neglected without making

much difference if one is interested in motion within a zm much less than the magnet
gap.
Secondly, including higher order terms requires more orbit computation time and

more memory capacity of a computer. Determining the lowest possible order of

truncation is therefore crucial in the light of cost—effectiveness.

Effects of the terms of the magnetic field components that are of order higher
than four were studied by comparing the results of orbit computations obtained by
using the magnetic field components up to 26 or 28 terms with those derived by using
the magnetic field up to 24. Three different calculations were carried out. One was
carried out using an orbit with the same initial conditions as those of the orbit # 1
in the top diagram of Fig. 3.2. Another was done using an eigenellipse in 2 phase
space at E = 10 MeV/u. This eigenellipse was accelerated until the energy of the
central ray reached the final energy E f = 40 MeV/u to see how much difference the
higher order terms make. For both cases, the Z2” Orbit Code was used with N = 2,
N = 3, and N = 4 for comparison. And thirdly, a brief study of one of the sixth-order
resonances, the V2 2 3/ 6 resonance, was carried out using the Z 2N Orbit Code with

N = 2 and N = 3 to show the effects of higher order terms.

4.1 The ZZN Orbit Code

Before the results are presented, a brief description of the orbit code used for the

orbit computations is given here. The ZgN Orbit Code is a modified version of the

37

Z 4 Orbit Code based on the development of new differentiation schemes using a finite
difference technique as was pointed out in the beginning of Chapter 1 (for details,

refer to the appendix). There are two differences between the two orbit codes.

One difference is that the ZzN Orbit Code has an option specified by N while the
Z4 Orbit Code uses the magnetic field components only up to 2.4 terms. For each
choice of N = 1,2, 3, or 4, the magnetic field components up to 221V terms are used.
(For example, with N = 2, the magnetic field components up to 24 terms are used,
which corresponds to the Z 4 Orbit Code.) This option enables one to study nonlinear

effects order by order.

The other difference is that the Z 2N Orbit Code uses the improved differentiators
for evaluating various derivatives while the Z4 Orbit Code utilizes the simple “old
differentiators” (see Eqs. A.1 and A2 in the appendix). In order to maintain consis-
tency in differentiation, the Z2N Orbit Code needs to store 3N maps of derivatives
and a map of the midplane field data before orbit computations. For example, for
N = 1, the ZzN Orbit Code needs the maps of 5%? (related with B,) and {Ba—I: (related
with Bg), in addition to the map of V38 (related with B2). But the maps of 83? and
123% are automatically computed and stored by a subroutine in the Z2” Orbit Code
when the map of B is given as an input. The Z 2N Orbit Code with N = 2 requires six
maps of derivatives besides a map of the midplane field data, while the old Z4 Orbit
Code needs only two maps of derivatives plus the map of the midplane field data.

Thus, improved accuracy in the evaluation of derivatives is achieved at the expense

of computation time and memory space.

Except for the two differences mentioned above, the two nonlinear orbit codes are
the same, including the use of exact equations of motion that can be derived from a

Hamiltonian given in Eq. 2.9.

38

4.2 Effects of higher order terms on certain orbits

The orbit computation results presented in section 3.1 using the Z M Orbit Code with
N = 2 reflect the effects of the magnetic field components only up to 24. To see the
significance of the higher order terms in the magnetic field components, we chose the
orbits # l and # 3 in the top diagram of Fig. 3.2 and repeated similar calculations
using the ZZN Orbit Code with N = 2, N = 3, and N = 4. The orbit # 1 was
chosen again because it defines the separatrices for the resonance and because of the
existence of a chaotic layer surrounding the separatrices which greatly amplifies small

differences in the orbits.

Figure 4.1 shows three z-space maps of orbits for which one can get as close as
possible to the separatrices at 112 = 0.740 without jumping out of the island. These
three maps were obtained by plotting p2 sin ¢z vs. p2 cos ¢z once per sector for 190
turns using the Z W Orbit Code with N = 2 (top diagram), with N = 3 (bottom left
diagram), and with N = 4 (bottom right diagram). The orbit in the top diagram has
the same initial conditions as those of the orbit # 1 in Fig. 3.2, which are (20 = 3.9192,
1920 = 0). But the corresponding initial conditions for the bottom left (right) orbit
are (20 = 3.9243 (3.9522), on = 0) for N = 3 (N = 4), which differ by less than 1
‘70. All these initial conditions are in units of mm. It should be noted that the basic
features of the diagrams, such as the location of the unstable fixed points, remain
almost unchanged even with inclusion of the terms of order higher than four in z to
the magnetic field components. In these diagrams, the maximum p, is 12 mm which
corresponds to a maximum 2 value of 15 mm while, as noted above, the usual beam

height 2m 2 5 mm and the height of the liner is about 19 mm.

Figure 4.2 shows three maps of an orbit that has the same initial conditions as

those of the orbit # 3 in the top diagram of Fig. 3.2. These were obtained by plotting

39

p2 sin 9152 vs. p2 cos 452 at every sector 9 = 2n7r/ 3 for 100 turns using three different
options of the Z2” Orbit Code with N = 2 (top diagram), N = 3 (bottom left
diagram), and N = 4 (bottom right diagram). The length and width of the small
blur of points expand by less than 1 % compared with the average pz values of the
points. It is clear from this figure that inclusion of higher order terms does not make

a significant difference in the location of the “stable fixed points”.

As a second investigation, we chose an eigenellipse in 2 phase space at E = 10
MeV/u at 0 2 0° with emittance 27rJz/p = 7.64 wmm-mrad (assuming p = 1.0 m)
and accelerated the orbits until the energy of the central ray reached E f = 40 MeV/ u
in order to determine the integrated effects of higher order terms. Figure 4.3 portrays
the resultant “ellipse” in z-phase space at every 10 MeV/ u during the acceleration,
which ran for 619 turns from 10 MeV/ u to 40 MeV/ u. Initially, all the points on the
eigenellipse share the same values of (To, pro) which are on the accelerated equilibrium
orbit. Figure 4.4 shows the corresponding zit-phase space diagrams depicted at every
10 MeV/ u. During the course of acceleration, the eigenellipse passes through various
resonances depicted in Fig. 1.1 and some of these resonances couple the z—motion
into the m-motion as shown in Fig. 4.4. (A similar calculation was performed over a

limited range of energy across the V, = 3/ 4, resonance and is presented in Figs. 1.2

and 1.3.)

Table 4.1 shows the rms differences in the final values of (2, pg), (3', p3), and (E, <I>)
for the orbits at the final energy E f = 40 MeV/ u. The left and right columns show
the rms differences in the results derived from the Z2N Orbit Code with N = 2 and
N = 3, and with N = 2 and N = 4, respectively. The final ellipse has A2 = 8 mm
at 0 2 0° and for all 0 values, A2 = 12 mm, which is only 20 ‘70 greater than the
assumed beam height 22m = 10 mm. The rms differences listed in the table are

negligible compared with the corresponding values.

40

 

 

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—1o —5 o 5 10
pzcosqbz (mm)

Figure 4.1: Three maps of orbits for which one can get as close as possible to the
separatrices at Vz = 0.740 without jumping out of the island. These were obtained by
plotting pz sin (15,, vs. ,0; cos 452 at every sector 0 = 2n7r/ 3 for 190 turns. The orbit at
the top (with N = 2) has the same initial conditions as those of the orbit # 1 in the
top diagram of Fig. 3.2, which are (20 = 3.9192, p20 = 0). The orbit at the bottom left
(right) has the initial conditions (20 = 3.9243 (3.9522), p20 2 0) for N = 3 (N = 4).
Here all the initial values are in units of mm. The variation in the initial conditions is
less than 1 % and the basic features, such as the location of the unstable fixed points
remain almost unchanged even with inclusion of higher order terms proportional up
to 26 and 28.

41

 

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IIII

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Illlllll

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”IIIIIIIILLLUIIIIILIILILbllllllllIlLIIllllLlllll d

 

 

8.00
5.50 5.75 6.00 6.25 6.50 6.75
P2603952 (mm)

Figure 4.2: Three maps of an orbit that has the same initial conditions as those of the
orbit # 3 in the top diagram of Fig. 3.2. These were obtained by plotting p2 sin (15,,
vs. p2 cos QSz at every sector 0 = 2n7r / 3 for 100 turns using the three different options
of the ZzN Orbit Code with N = 2 (top diagram), N = 3 (bottom left diagram),
and N = 4 (bottom right diagram). The length and width of the small blur of points
expand by less than 1 % compared with the average pz values of the points. It is clear
from this figure that inclusion of higher order terms does not significantly affect the
location of the “stable fixed points”.

42

 

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Figure 4.3: Evolution of an eigenellipse in z-phase space starting at 10 MeV/ u (top
left diagram) depicted at every 10 MeV/ u during its acceleration for 619 turns after
which the central ray reaches the final energy E f = 40 MeV / u (bottom right diagram).
These plots obtained by using the Z W Orbit Code with N = 2 show the evolution of
the ellipse at 20 MeV/u (top right diagram), at 30 MeV/ u (bottom left diagram) and
at 40 MeV/u (bottom right diagram). It should be noted that the initial emittance
is 27rJz/p = 7.64 7rmm-mrad (assuming p = 1.0 m) for the initial eigenellipse at
10 MeV/u. (These diagrams should be compared with the those in Fig. 1.2 which
are the results of a similar calculation.) Initially, all the points on the eigenellipse
share the same values of (To, Pro) which is on the accelerated equilibrium orbit at 10
MeV/ u. The bottom right diagram is deformed appreciably due to the traversal of
the 1/Z = 3/ 4 resonance between 30 MeV/ u and 40 MeV/ u. Almost exactly the same
results were obtained with the N = 3, and N = 4 options for the Z2N Orbit Code.
(The same is true for the results in Fig. 4.4 below.)

1.5

1.0

0.5

0.0

43

 

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PX (mm)

Figure 4.4: The evolution in x-phase space of the orbits in Fig. 4.3 obtained by using
the Z2” Orbit Code with N = 2. These plots show the evolution from 10 MeV/u
(top left) to 20 MeV/u (top right) to 30 MeV/u (bottom left) to 40 MeV/u (bottom
right). Initially, all the points share the same values of (To, Pro) which are on the
accelerated equilibrium orbit at E = 10 MeV/ u (thus depicted as a point in top left
diagram). Due to the traversal of the coupling resonances 21/2 = 11,. and 21/2 + 21/, = 3
depicted in Fig. 1.1 between 10 MeV/ u and 20 MeV/ u, relatively large x-px spreads
of Arc ~ 1 mm and Apr ~ 2 mm develop between the two top diagrams. Afterwards,
the orbits do not cross any coupling resonances up to fourth order, and the area of
the :r-phase space diagram shrinks. These diagrams should be compared with those

-0.5

—1.0

—15

'—1.5—1.o-o.5 0.0 0.5 1.0

in Fig. 1.3.

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1.

5

Table 4.1: rms differences in orbit computations obtained from the Z2” Orbit Code

44

withN=2andN=3,andwithN=2andN=4

 

 

 

 

 

 

 

 

N=2andN=3 N=2andN=4
A2 6.74 x 10‘3 mm 6.81 x 10‘3 mm
Apz 5.77 x 10‘3 mm 5.80 x 10"3 mm
Ax 6.38 x 10‘4 mm 6.53 x 10’4 mm
Apr 3.85 x 10"4 mm 4.01 x 10‘4 mm
AB 1.72 X 10‘2 keV 1.83 X 10-2 keV
A4) 1.08 x 10"3 deg 1.14 x 10‘3 deg

 

 

 

From the results shown in Figs. 4.1—4.4 together with Table 4.1, it can be concluded
that the Z 2” Orbit Code with N = 2 can be used for all orbit computations relevant to
the K1200 cyclotron. This restriction makes orbit computations relatively inexpensive

as far as computing time and memory are concerned.

4.3 Effects of higher order terms on the V2 2 3/ 6
resonance at V2 2 0.497

The effects of higher order terms were studied near the V2 2 3/ 6 resonance at V; =
0.497 with E = 16.63 MeV/u using the Z21" Orbit Code with N = 2 and N = 3.
The Vz = 3/6 resonance is at E = 16.73 MeV/u where V, = 1.070. At V; = 0.495,
we could not find clear separatrices, and for 1/2 values even closer to the resonance, it
takes a large number of turns to get the results. Because of these reasons, 11,, = 0.497
was chosen as a point in between. It should be noted that at 11,, = 0.50, there are
many overlapping resonances such as V; = 1/2, V2 2 2/4, V2 2 3/6, and so forth.
The V; = 3/ 6 resonance is excited by the third harmonic component of the magnetic
field which is one of the main harmonics of the K1200 superconducting cyclotron
with three-fold symmetry in the magnet. On the other hand, the V, = 1/2 resonance

and the Vz = 2/4 resonance are generated by imperfection field components which

45

were excluded for this study by making the magnetic field symmetric for the sake of

simplicity.

Figure 4.5 shows two z-space diagrams of an orbit obtained by plotting pz sin (15,,
vs. p, cos 45,, once per sector. The top diagram was obtained by using the ZzN Orbit
Code with N = 3 for 1000 turns at (V, = 1.070, 112 = 0.497) with E = 16.63 MeV/u.
The initial conditions (.20 = 7.11 mm, H20 2 0 mm) of this orbit were chosen in such
a way that one can get as close as possible to the separatrices using the ZZN Orbit
Code with N = 3. The initial values (r0, pro) are again on the E0 to avoid added

complexity.

The bottom diagram in Fig. 4.5 showing maps of two different orbits was derived
from the Z2” Orbit Code with N = 2. The inner orbit in bottom diagram obtained
by running for 250 turns has the same initial conditions as those of the orbit in the
top diagram. The outer orbit was derived by running 1300 turns in such a way that
one can again get as close as possible to the separatrices. But it has different initial

conditions, (20 = 8.13 mm, H20 2 0 mm).

The distinct features of a sixth-order resonance are observed for the top diagram
which are composed of six islands and a central stable region. But the inner orbit,
which has the same initial conditions as those of the orbit at the top, is still inside
the central stable region. On the other hand, as is also depicted in Fig. 4.5, when the
initial conditions are changed into (zo = 8.13 mm, I120 2: 0 mm), the characteristic
structure pertaining to the sixth-order resonance emerges here also. (See the outer
orbit in the bottom diagram.) This originates from the 26 terms coming from the
terms proportional to A? or A? that are obtained when the square root term of
the Hamiltonian in Eq. 2.9 is expanded, even though the magnetic field components
contain terms only up to fourth-order in 2. But the width of the six islands of the

bottom diagram is less than that of the top diagram, which suggests a weaker driving

46

force for the V; = 3/ 6 resonance in the case of the orbit at the bottom.

Due to coupling effects, the maps of the orbit in z-space have finite thickness and
this thickness grows as the value of J, grows. Figure 4.6 shows a diagram of the
coupled motion in :c-space for the orbit at the top in Fig. 4.5. The maximum value
of p3 of the coupled motion is about 20 ‘70 of the maximum value of p2, and this
percentage is about two times larger than the ratio of the maximum value of pm to
that of pz for the orbit # 1 in Fig. 3.3. Besides, all the unstable fixed points are
fuzzier than for 1/3 = 3 / 4. All these results indicate that coupling effects are stronger

here than for the V; = 3/ 4 resonance.

From Figs. 4.5 and 4.6, it is interesting to find that even though all the sixth-order
terms in the magnetic field components are omitted, the typical phase space structure
for a sixth-order resonance emerges and that the value of p2 for the unstable fixed
points differs only by 14% compared with the results obtained by using the magnetic
field components up to 26 terms. However, it is clear that one can not obtain accurate
results for the V2 2 3/6 resonance without using the ZzN Orbit Code with N = 3, at

least.

47

 

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Figure 4.5: Two z-space diagrams for V; = 0.497 at E = 16.63 MeV/u obtained by
plotting p2 sin 43,, vs. ,0, cos 45,, once per sector. The top diagram was obtained by using
the ZZN Orbit Code with N = 3 for 1000 turns with the initial conditions (20 = 7.11
mm, H20 = 0 mm). These are chosen in such a way that one can get as close as possible
to the separatrices. The bottom diagram showing maps of two different orbits was
derived from the Z 2N Orbit Code with N = 2. The inner orbit in the bottom diagram
obtained by running for 250 turns has the same initial conditions as the top diagram.
The outer orbit was derived by running 1300 turns with (20 = 8.13 mm, H20 2 0
mm). These turn numbers are required to make approximately one revolution in
the diagram. Without the higher order terms in the magnetic field components, the
inner orbit is still inside the central stable region without showing any sign of the
sixth-order resonance. But as shown for the outer orbit, even with the magnetic field
components containing terms only up to 24, the typical phase space structure of a
sixth-order resonance still emerges. Besides, all the unstable fixed points are fuzzier
than those for the V2 = 3/ 4 resonance due to stronger coupling effects here.

ilLLLLIIIIIIIIIIIILLIII

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48

 

 

 

 

 

 

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PxC°S¢x (mm)

Figure 4.6: The x-space diagram showing the result of coupled motion in a: space near
the V2 = 3/ 6 resonance at V2 = 0.497 for the same orbit shown in the top diagram of
Fig. 4.5. This was obtained by plotting p1, sin (15,, vs. p3 cos 43,, once per sector using
the Z2” Orbit Code with N = 3 for 1000 turns. The maximum value of p, for this
orbit is about 20 % of the maximum value of p2, which is about two times larger than
the ratio of the maximum value of pi. to that of p, for the orbit # 1 in Fig. 3.3. This
reflects stronger coupling effects here than for the V; = 3/ 4 resonance.

Chapter 5

Comparison of old and new
methods for computing field
derivatives

Recently a new finite difference technique for evaluating the off median plane magnetic
field components has been developed [3]. The main motivation was based on the
observation that maps of the higher order terms in the magnetic field components
obtained by using the “old differentiators” in the original Z4 Orbit Code become
increasingly more noisy. The objective was to design new differentiators to suppress
noise effectively without destroying important components by maintaining reasonable
accuracy over a sufficiently wide range of low frequencies. For a detailed discussion,
refer to the appendix. In this chapter, some results on the comparison between the

improved differentiators and the “old differentiators” are given.

As a direct test, we present maps of the second—order and fourth-order terms of
B, in Eq. 2.1 and the first-order term of B, in Eq. 2.2 which are evaluated at z = 0.5
(in) by using the improved differentiators. We also present maps of the differences
between these maps and those computed using the “old differentiators”. In addition,
as an indirect test, the results of orbit computations for three different orbits with

distinct physical characteristics obtained from the Z2” Orbit Code with N = 2 using

49

50

the improved differentiators are compared with those for the same three orbits derived
from the Z4 Orbit Code utilizing the “old differentiators”. This is to show explicitly
how a small amount of noise in the off median plane magnetic field components affects
orbits when it is not suppressed effectively. It should also be pointed out that all the

orbit computation results presented in the previous chapters were obtained by using

the new Z”, Orbit Code.

5.1 Application to K1200 cyclotron field

The magnetic field here is exactly the same as that used in the preceding chapters.
The median plane field map was measured on a polar mesh with A0 2 1° and Ar = 0.1
(in) while the pole radius is 42 (in). But for the orbit studies in Chapter 3, we
used a field map restricted to the range from r = 32 (in) to r = 40 (in) and from
0 2 0° to 9 2 119°. As shown in the appendix, the improved differentiators have a
cutoff frequency Lawton % 1.05 and strongly suppress higher frequency signals. This
cutoff frequency corresponds to the 20th harmonic obtained from a Fourier analysis
of the field with perfect three-fold azimuthal symmetry. Moreover, the improved
differentiators accurately evaluate the derivatives over a much broader range of low
frequencies than the “old differentiators”, while suppressing more effectively the high

frequency signals associated with noise.

We have applied the improved differentiators and the “old differentiators” to eval-
uate the coefficients of the magnetic field components in Eq. 2.1 and Eq. 2.2 using
the measured median plane magnetic field data shown in Fig. 5.1. The maximum

and minimum field values here are 45 [CG and 30 kG, respectively.

As an illustration, Fig. 5.2 shows at the top a map of the second—order term, V§B X

0.52/21, evaluated at z = 0.5 (in) obtained by using the improved differentiators, and

51

the map of the difference at the bottom between this and the map for the same term
obtained by utilizing the “old differentiators”. The maximum and minimum of the
map at the top for V§B X 0.52 / 2! are (248 G, —352 G) respectively, and those of the
difference map at the bottom are (13.0 G, —11.9 G). The map of the difference at
the bottom shows the relative defects of the old differentiators in suppressing high

frequency signals.

Similarly Fig. 5.3 shows, at the top, a map of the fourth-order term, V§B X 0.54/41,
evaluated at z = 0.5 (in) obtained by using the improved differentiators and at the
bottom, a map of the difference between this and the map of the same term obtained
by utilizing the “old differentiators”. The maximum and minimum of the map at the
top are (82.3 G, -—104 G) respectively, and those of the difference map at the bottom
are (33.9 G, —28.5 G). Because useful data which are slowly varying are washed out
to an appreciable extent by the “old differentiators” when they are applied twice to
get the fourth-order term, the values of the bottom map depicting the difference are
significant compared with those of the fourth-order term itself. The amount of slowly
varying components washed out by the old differentiators is so large compared with
noise that it is difficult to observe rapidly varying noise compared with the case for
the map portraying the difference in the second-order term shown in Fig. 5.2. Refer to
Figs. A.10 and A.12 in the appendix for the wash out of physically important slowly
varying components by the “old differentiators”. For the case dealing with data with
noise, refer to Figs. A.13 and A.14 in the appendix. As is discussed in detail in the
appendix, the “old differentiators” are accurately evaluating derivatives only over a

very limited range of low frequency.

Figure 5.4 shows the map of 63—128 evaluated at z = 0.5 (in) at the top which
is the leading term of B, in Eq. 2.2 obtained by using the improved differentiators,

and at the bottom, a map of the difference between this and the map of the same

52

term obtained by using the old differentiators. The maximum and minimum values
of £28 evaluated at z = 0.5 (in) are 1.96 kG and —1.92 kG respectively. This is the
leading term of B,, and B, plays a dominant role compared with B9 in the z focusing.
Because of these reasons, 52B is a physically important term. The maximum and
minimum values of the difference at the bottom are 14.7 G and —16.5 G respectively,
and these values are less than 1 % compared with 1.96 [CG and -1.92 kG. Just as
in Fig. 5.2, the difference map shows the relative defects of the old differentiators in

suppressing rapidly varying components of data.

5.2 Application to orbits near the u, = 3/4 reso-
nance

As an indirect test, comparison of the results of orbit computations was conducted
near the V2 2 3/ 4 resonance at u, = 0.740. Three different orbits were chosen. One
set of orbit computations was carried out for the three orbits using the Z2N Orbit
Code with N = 2 and the other set of orbit computations was performed for the same

three orbits using the old Z4 Orbit Code.

One orbit exactly the same as # 1 in Fig. 3.2 was chosen again. This orbit
is close to the separatrices and due to the chaotic layer, any kind of error in the
computation can easily be amplified. Figure 5.5 shows clearly the difference in the
orbit computation results which portray two maps of this orbit obtained by plotting
,02 sin g6, vs. p, cos <13, once per sector for 400 turns. The map at the bottom obtained
from the Z4 Orbit Code spirals inward and is asymmetric, while that at the top
obtained by using the Z2N Orbit Code with N = 2 exhibits a more physical behavior.
It is remarkable that small differences in the various order terms of the magnetic field

components can produce such a noticeable difference.

53

        
 
  

     
 
     
   
 
 
   
  

 
     
          

     

  
 
      
   
  

    
 

  

 

 

      
 

  
        

 

    

    
 
  

 
 
  

  

    

    
                        

 

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Figure 5.1: Map of B(r,0) = —B,(r,0,z = 0) of a magnetic field of the K1200

superconducting cyclotron with q/A = 0.25 and a nominal final energy E; = 40

MeV/u over the range 32 (in) S r S 40 (in) and 0° S 9 S 119° used for orbit
computations in the preceding chapters. This field has a perfect three-fold azimuthal

symmetry. The maximum field value is 45 [CG and the minimum is 30 kG. The

magnetic field within this radius range is very nonlinear.

54

   
   

    

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Figure 5.2: A map of VgB X 0.52/2! evaluated at z = 0.5 (in) by using the improved
differentiators (at the top) and a map of difference between this and the map for the
same term computed using the “old differentiator” (at the bottom) for the magnetic
field shown in Fig. 5.1. The maximum and minimum values of the map at the top
are (248 G, —352 G) respectively while those of the difference map at the bottom
are (13 0 G, —11.9 G). Note that the two maps are not plotted to the same scale
The map at the bottom depicting the difference shows the relative defects of the old

differentiators in suppressing noise.

55

   
     
      

   

        
    

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Figure 5.3: A map of VgB x 0.54/4! evaluated at z = 0.5 (in) by using the improved
differentiators (at the top) and a map of difference between this and the map for the
same term computed using the “old differentiator” (at the bottom) for the magnetic
field shown in Fig. 5.1. The maximum and minimum values of the map at the top
are (82.3 G, —104 G) respectively while those of the difference map at the bottom
are (33.9 G, —28.5 G). Note that the two maps are not plotted to the same scale.
Because useful slowly varying components of data are washed out to an appreciable
extent by the “old differentiators” when they are applied twice to get the fourth-order
term, the values of the bottom map depicting the difference are significant compared
with those of the fourth-order term itself at the top. (See Figs. A.10 and A.12 in
the appendix for the wash—out of physically important slowly varying components of
data.) Moreover it is not easy to observe the rapidly varying small noise because it
is buried in the washed out data.

56

       

     

                 

 

   
 

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Figure 5.4: A map of £23 evaluated at z = 0.5 (in) by using the improved dif—
ferentiators (at the top) and a map of difference between this and the map for the
same term computed using the “old differentiator” (at the bottom) for the magnetic
field shown in Fig. 5.1. This is the leading term of B, and B, plays a dominant
role in z-focusing. The maximum and minimum values of the map at the top are
(1.96 160, —1.92 kG’) respectively while those of the difference map at the bottom
are (14.7 G, —16.5 G). Note that the two maps are not plotted to the same scale.
The map at the bottom depicting the difference shows the relative defects of the old

differentiators in suppressing noise.

      
  
      
   
    

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57

Figure 5.6 shows two z-space diagrams of an orbit just outside of the separatrices
obtained by plotting pz sin (I); vs. p2 cos (#2 once per sector for 500 turns. The diagram
at the top was obtained by using the Z2” Orbit Code with N = 2 and that at
the bottom by using the old Z4 Orbit Code for the same orbit. Highly chaotic
characteristics are observed for the map derived from the Z4 Orbit Code, which
reflects the effects of remaining rapidly varying components of field data that were

not effectively suppressed by the old differentiators.

Figure 5.7 shows two z-space diagrams of an orbit just inside the inner separatrix
obtained by plotting p2 sin 45; vs. p2 cos Q32 once per turn (instead of once per sector) for
1000 turns. The map at the top was obtained from the Z 2” Orbit Code with N = 2
and that at the bottom was derived from the Z4 Orbit Code. The corresponding
values of zmax and 2min are 5.3 mm and —5.3 mm. The bottom map of this orbit slowly
spirals outward as the turn number increases, while the map at the top remains almost
unchanged and shows more physical behavior. This results from a rather inaccurate
evaluation of field derivatives and poor suppression of rapidly varying components of

data by the old differentiators.

It turns out that noise in the calculated magnetic field components makes numer-
ically computed orbits highly chaotic especially near the chaotic region. Without a
proper handling of the noise inherent in the measured median plane magnetic field

map, one can apparently obtain orbit computation results that appear unrealistic.

 

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Figure 5.5: Two maps of an orbit close to the separatrix for 11,, = 0.740 which is
near the 11, = 3/4 resonance. This orbit has the same initial conditions as those of
the orbit # 1 in top diagram of Fig. 3.2. These were obtained by plotting once per
sector for 400 turns. Due to the chaotic layer on the separatrix, any kind of errors
in orbit computations can be visualized with ease. The bottom diagram of this orbit
obtained from the Z4 Orbit Code using the “old differentiators” spirals inward and
is asymmetric, while that obtained from the ZZN Orbit Code with N = 2 exhibits
more physical behavior (see the top diagram).

59

 

 

 

 

 

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Figure 5.6: Two maps of an orbit in the z-dimensional phase space just outside the
separatrices for V2 = 0.740 which is near the 11, = 3 / 4 resonance. The map at the top
was obtained from the Z 2N Orbit Code with N = 2 and that at the bottom by using
the Z4 Orbit Code for the same orbit. These diagrams were obtained by plotting
once per sector for 500 turns. The map at the bottom shows highly chaotic behavior,
which is a result of poor suppression of noise in numerical computation of magnetic
field components by the “old differentiators”.

60

 

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Figure 5.7: Two maps of an orbit in the z-dimensional phase space just inside the
inner separatrix for 11,, = 0.740 which is near the V2 = 3/ 4 resonance. The map at the
bottom was obtained from the Z4 Orbit Code and that at the top was derived from
the Z 2N Orbit Code with N = 2 for the same orbit. These were obtained by plotting
once per turn (instead of once per sector) for 1000 turns. The map at the bottom
slowly spirals outward while that at the top remains almost the same.

Appendix A

Finite difference method for
calculating magnetic field
components off the median plane
using median plane data

Gordon and Taivassalo [1, 14] evaluated the coefficients in Eqs. 2.1- 2.3 using the

second-order central difference scheme as follows:

f.’ = (f.'+1-f.-—1)/2A (A-I)

f." = (fi+2 + fi-Z - Zfi) /4A2 (A2)

where f; = f (mi) and A is the step size of the mesh. These are what we call the old
differentiators which are used by the Z 4 Orbit Code. This scheme presents difficulties
both in accuracy and maximum order of expansion due to amplified noise produced
by taking derivatives. By “noise”, we mean any high frequency components like those

usually associated with noise.

In an effort to overcome the difficulties, new differentiators using a finite difference
technique were developed and we call them the “improved differentiators”. Here, a

detailed description is presented.

The finite difference scheme, which is a special case of the compact finite difference

61

62

scheme, is used to avoid additional complexity imposed by compact schemes. If one
wants to enhance accuracy, a plausible way would be to use compact finite difference
schemes. A lot of work has been done for compact finite difference schemes [15,
16, 17, 18], and work for non-uniform mesh has also been done [19, 20]. Especially
Lele [18] did recent work on compact finite difference schemes. Even though the
operators designed by Lele have a frequency response reasonably close to that of
mathematical differentiation over the complete frequency range, they are not adequate
for processing measured data containing noise because sufficient suppression of high
frequency signals is not provided.

The frequency w which is going to be used extensively later is defined as follows.

:17

Let’s suppose there is a uniform mesh and a sinusoidal signal 6”“ is applied to it
where k is the wavenumber. When the distance between two consecutive mesh points
is A, the subsequent phase advance is 1913. Thus we define w E kA. So the largest w

without the problems of “aliasing” is 7r.

There are two possible ways to compute derivatives of a magnetic field. One way
to do so is to process a magnetic field with a certain filter to remove high frequency
components before taking any derivatives. If the undesirable high frequency com-
ponents are suppressed to a satisfactory extent, comparatively simple and standard
differentiators can be used for computing derivatives. The other way is to mix pro-
cesses of computing derivatives and filtering. We decided to take the latter approach
for the following reasons. First of all, there are many orbit codes used in this labora-
tory (and many other laboratories) to calculate various orbit properties which do not
require any of the off median plane magnetic field components, but only the median
plane field map itself, and all the researchers in this laboratory prefer having the field
data unaltered for orbit computations. Secondly, it is vital to preserve the consistency

between the results obtained by the orbit codes using only the unprocessed median

63

plane field map and those obtained by the nonlinear orbit codes utilizing both the

median plane field map and the off median plane field components.

We will use a composite operator of more than two finite difference operators that
use only three nodes. In this case, the use of separate algorithms to evaluate deriva-
tives at nodes near the boundaries can be avoided by imposing simple conditions for
each 3-node operator. The following property is useful to get the frequency response

of composite operator obtained in this way.

Let’s consider a composite operator of two linear operators L1 = Z: hl (n) fn and
L2 = 2h; (m)gm with the corresponding frequency response H1 and H2. Then
the frequency response of the composite operator of these two is H1H2, which is
independent of the ordering of application of the linear operators. If one considers
the properties of Z transformations [21, 22], it is straightforward to verify the previous

statements. We are going to use this for the design of the operators.

A.1 Design of the first and second order differ-
entiators in one dimension

The standard second-order central difference scheme composed with a filter is used
to improve the frequency response of differentiators. The use of such a filter is to
provide a strong suppression of high frequency signals and by adjusting a parameter

Q of the filter, to improve the low frequency response at the same time.

First of all, the filter F is introduced that transforms {fn} to {fa} suppressing
high frequency signals. For the first-order and the second—order differentiator, the
parameter Q is adjusted respectively to produce a reasonable result. It is given

below:

fn, forn=0,...,N+1

64

U

9n = (fn—l + 2fn + fn+1) /4

90 = f0, 9N+1 = fN+1

U

hn = 9n — %(gn—l + gn+1 — 29-71)

ho = 90, hN-H = 9N+1

U

kn =(hn_1+ 2h.” + hn+1) /4

[Co = ’10, kN+1 = hN+1

U

. 1

fn = kn _ :4— (kn—l "l’ kn+1— 2kn)

f0 = k... f~+1 = kN+1-

The frequency response HlF) of the filter is as follows:

HlF) (w) = [cos2 (cu/2) + 9 sin2 (52)] [cos2 (cu/2) + Ell-sin2 (w)[ .

4

The improved first-order differentiator is a combination of the central difference
scheme (refer to Eq. A.1) and a filter, which ends up with being an eleven-point
formula. With the choice of Q = 1.70, reasonable low frequency response was obtained
for the first-order differentiator. More detailed discussion about the choice Q = 1.70
is given at the section for resolving efficiency. The operator D“) which approximates

first-order derivative {if (.rn)} for n = 1,...,N from a given set of data {fn

f(a:,,)} for n = 0, . . . ,N +1 is as follows:

fn, forn=0,...,N+1

U

65
(1,, =(fn+1—fn_1)/2A, forn =1,...,N
U
3n = (drH-l + 2d,. + dn-l) /4
31: (11, SN = dN
U
9n = 3n — 1174311“ + Sn—i — 2371)
91: 31, 9N = 8N
U
hn = (9n+1 + 29n + Sin—1V4
hl =91, hN =9N
U
£115.): 1..—

d
fiffih) = (11,

(hn+l + hn-l — th)

grlkrbH—t

f(1'N) = Im-

Complete expression of the operator D“) is applied for the nodes with n = 5 to
n = N — 4 but we can still get good evaluation of derivatives for the rest of nodes
at the same time. It is convenient to express an operator as a composition of several
3—node linear operators with simple treatment of nodes at end. This keeps us from
the trouble of using a separate algorithm to evaluate derivatives at each of the nodes

withn=1,...,4andn=N—-3,...,N+1.

The frequency response H (1) of the first-order differential operator, D“), is:
(1) - - 2 1'7 . 2 2 1 - 2
H (w) = zsm (w) cos (cu/2) + T 8111 (w) cos (ca/2) + Z sm (w) . (A.4)

Notice that this is pure imaginary, so it doesn’t have any phase shift. Additionally

it has a frequency response reasonably close to that of mathematical differentiation

 

 

ord

66

Table A.1: Resolving Efficiency 61 (e) of the F irst-Order Derivative Schemes

 

 

 

 

 

 

 

Scheme 6 = 0.1 e = 0.01 e = 0.001
Improved first-order differentiator 0.29 0.16 0.10
Old first-order differentiator 0.25 0.08 0.02

 

 

for low frequency, and suppresses high frequency signals sufficiently. The fractional
differences of frequency response for the old and improved differentiators are depicted
in Fig. A.1 to show the low frequency characteristics. By composing the central
difference scheme with the filter, low frequency response was improved considerably

compared with that of the old differentiator (refer to Fig. A.1).

Before making comparison, let’s define resolving efficiency of approximate first-
order differentiators, 61 (c) E (pf/71' [18]. The value to} is the maximum wave-number
of well-resolved wave satisfying the error tolerance relation I [1(1) (w) —— iw I /w S e for
any given positive value of e. The resolution characteristics of the old and improved
differentiators are tabulated in Table A.1. In this paper, Q = 1.70 was chosen to
maximize the resolving efficiency for e = 1.0 x 10’4 (refer to Fig. A.1). If one wants
to maximize the resolving efficiency for c = 5.2 x 10”, the natural choice would be
Q = 1.75 (refer to Fig. A.1). According to Fig. A.1 and Table A.1, the improved
first-order differentiation scheme is better than the old differentiator. At the same
time, it is superior to the old differentiator in suppressing high frequency signals (refer
to Fig. A.2). It should be pointed out that the cutoff frequency wcutoff is about 1.05

for this improved first-order differentiator judging from the frequency response.

A second-order differentiator is designed separately because when we apply the
first-order differentiator twice (which is acceptable in the light of mathematics), it
effectively is a 22 point formula and is computationally inadequate if we can get

results with comparable accuracy with small number of points to be used. The

67

 

_

0.0010 TTTT]IITT[IIII[IIIIIIIIIIIIII

— .

0.0005— —

0.0000

(H(1)(co)—ico)/ico

F
—0.0005

  

—I1

 

 

 

.\
_0.0010111lL111LL1\IIllli'lilllllllll

0.0 0.1 0.2 0.3 0.4 0.5 0.6
cu (Wavenumber)

 

Figure A.1: Plot of (H(1)(w) - iw)/iw over the range 0 S (.0 S 0.6 for the old
differentiator (real line “b”), and that for the improved differentiator (solid line “a”)
with Q = 1.70. The dotdash line corresponds to Q = 1.65 and the dotted line to
Q = 1.75. Composition of the central difference scheme “b” with the filter improves

the low frequency behavior significantly, which results in “a” with Q = 1.70.

68

 

 

 

 

3 T 1 r 1 1 1 r 1 1 1 1 1
’0? _ ..
ID
Cl _ _
O
as . —
>‘ 2 L..__ ._
0 _ _
I:
Q) _ ...
:3
C" — ..
Q)
a ._ _
V __ b __
.-1 1
\ - ..
A
3 - a a
X
'11: 1- -
1 1 1 1 l 1 1 1 1 l 1 1
0
0 1 2 3

w (Wavenumb er)

Figure A.2: Plots of H (1) (w) / i, as a function of the wavenumber to over the range
0 S to S 7r. Curve “a” is for our improved differentiator, curve “b” for the old
differentiator, and curve “c” for the mathematical first-order differentiator. “a” is
superior to “b” in suppressing high frequency signals and shows a frequency response
reasonably close to that of the mathematical differentiator for low frequency as well.

It should be pointed out that the cutoff frequency wcumff is about 1.05.

69

improved second-order differentiator again is a combination of the central difference
scheme (refer to Eq. A5) and a filter, which ends up with being an eleven-point
formula, just like the improved first-order differentiator presented previously. The

second-order central difference scheme is defined as follows:

f1” = (fi+1 + fi—I — inl /A2 (1615)

where f, = f(:1:,-) and A is the step size of mesh. With the choice of Q = 1.36,
reasonable low frequency response was obtained for the second-order differentiator.
More detailed discussion about the choice Q = 1.36 is given at the section for resolving
efficiency. The operator 0(2) that approximates second-order derivatives {% f (1%)}

for n = 1,...,N from agiven set of data {fn} forn = 0,...,N+1 is given below:
fn, forn=0,...,N+1

U

dn =(fn+1+fn_1—2f,,)/A2, for n =1,...,N
U

3n = (dn+1 +2dn+dn-1)/4

31 =d1, sdeN

U

1.36
_ — (Sn-H + 371—1 _ 2571)

gnzsn 4

91 = 31, 9N = 3N

U

hn = (9n+1 + 291: + gn—1)/4
hl = 91, (IN = m

U

d2

1
Elihu) = hn — Z(hn+1 + hn-l _ 2h")

70

d2 d2
'55 ($1)=h1, @f($N)=hN-

The frequency response H (2) of the second—order differential operator, D”), is:

[1(2) (w) = —4 sin2 (ta/2) [cos2 (ca/2) + 14g sin2 (w)[

x [cos2 (LU/2) + Ill—sin2 (1.0)] . (A.6)

Notice that this is purely real, so it doesn’t have any phase shift. The cutoff frequency
wcuton is about 1.05 for this improved second-order differentiator judging from this
frequency response. The fractional differences of frequency response for the old and
improved differentiators are depicted in Fig. A.3. By composing the standard central
difference scheme with the filter, the low frequency response was improved consider-

ably compared with that of the old differentiator (refer to Fig. A.3).

In a similar way, let’s define a resolving efficiency of approximate second-order
differentiators, 82 E wf/7r where the value (.12; is the maximum wave-number of well-
resolved wave satisfying the tolerance relation I [1(2) + w2 I /<.122 S 6 for any given value
of e. In this paper, Q = 1.36 was chosen to maximize the resolving efficiency for e =
1.0 x 10’4 (refer to Fig. A.3). If one wants to maximize the resolution characteristics
for e = 4.3 X 10‘“, the natural choice would be Q = 1.40 (refer to Fig. A.3). The
resolving efficiency for the old and improved differentiators are tabulated in Table A2.
The improved second-order differentiation scheme is better than the old differentiator
for low frequency signals (refer to Fig. A.3 and Table A2). At the same time, it

is superior to the old differentiator in suppressing high frequency signals (refer to
Fig. A.4).
The reason why Gordon and Taivassalo [1, 14] used Eq. A.2 instead of Eq. A5 is

that the frequency response of Eq. A2 at w = 7r is equal to 0 while that of Eq. A5

is not equal to 0. Due to this, Eq. A.5 does not properly suppresses high frequency

71

 

0.001011111111 llllllfllllllfl
t l l' l l | _

0.0005 —— ——

0.0000

(H‘2)(w)+wz)/-mz

—0.0005

 

E .
_0.00101111l1111i1\11l111l1111lL11
0.0 0.1 0.2 0.3 0.4 0.5 0.6

cu (Wavenumber)

 

 

 

Figure A.3: Plot of (11(2) (02) + w2)/ — 1.02 over the range 0 S 0) S 0.6 for the old
differentiator (solid line “b”), and that for the improved differentiator (solid line “a”)
with Q = 1.36. The dotdash line corresponds to Q = 1.30 and the dotted line to
Q = 1.40. Composition of the central difference scheme with the filter improves the
low frequency behavior significantly, which results in “a” with Q = 1.36.

72

 

 

 

 

0 11'111111
s b _
A
3,) _. _
1:: “ a ‘
o-ZL— —
Q -1
m P— -1
0) _ _.
05 _. _
:>,_ __ _
o 4- .1
c: _ _
CD _ _
5 t _
CD— _.
1. 65— _
it.
v "' _.
A " "l
\3/ 8_ C F
e”: _ — 1
CE: 5 _
— -1
b1111l1111l1111l
0 1 2 3

w (wavenumb er)

Figure A.4: Plot of the frequency response [1(2) (02) as a function of the wavenumberw
over the range 0 S 02 S 1r. Curve “a” is for our improved second-order differentiator,
curve “b” for the old differentiator, and curve “c” for the mathematical second-order
differentiator. “a” is superior to “b” in suppressing high frequency signals and shows
a low frequency response reasonably close to that of the mathematical differentiator

as well. It should be noted that the cutoff frequency wcutoff is about 1.05.

73

Table A.2: Resolving Efficiency 62 (c) of the Second-Order Derivative Schemes

 

 

 

Scheme 6 = 0.1 e = 0.01 c = 0.001
Improved second-order differentiator 0.30 0.17 0.10
Old second-order differentiator 0.18 0.06 0.02

 

 

 

 

 

 

signals, which makes it difficult to apply these schemes successively to get higher

order derivatives of data with noise.

A.2 Design of the first and second order differ-
entiators in two dimensions

Mathematically one can evaluate a partial derivative in two-dimension only by ap-
plying the differentiator designed for one-dimensional data to two-dimensional data.
It is true for data which do not contain any noise at all. As is presented at the later
section of this appendix (see section AA), it turns out to be very useful to add an
additional filtering in y when a partial derivative with respect to :1: is taken of data

which contain noise from any source including truncation errors.

For the partial differentiators in two dimension, the differentiators for one dimen-
sional data presented in the previous section composed with a filter in y are used,
when partial derivative is taken with respect to 3:. For the sake of convenience, a:
filter in the differentiators in one-dimension will be called “primary filter” and y fil-
ter will be called “secondary filter” when a partial derivatives with respect to :1: is
taken. In other words, the difference between the differentiators in one-dimension
and two-dimension is the addition of “secondary filter” such that when partial dif-
ferentiation with respect to a: (y) is performed, suppression of high frequency signals

with respect to y (as) should be performed at the same time which passes through low

74

frequency signals and provides a strong suppression of high frequency signals. This
ensures the suppression of noise signals when partial differential operators are applied
successively to get higher order derivatives. The importance of this filter is very well

demonstrated in Table A.6.

The filter Fy that transforms {fmm} to {13mm} suppressing high frequency signals

with respect to y with a choice of Q = 1.01 is given below:

fmm, forn=0,...,N+1, andm=0,...,M+1
U
9mm 2 (fn,m—l + 2fn,m + fn,1n+l) /4

911,0 = fn,01 gn1M+1 = fn.M+1

U

1.01

hmm : gmm — T (gnaw—1 + gn,m+1 — 29mm)

hn,0 = 971,01 hn,M+1 : 911,114+]

U
kmm : (hn,m-1 + 2h'n,m + hn,m+l) /4

1511.0 = [171,01 kn,M+1 = thW-i-l

U

. 1
fn,m : kmm _ Z (kmm—l + kmm-l-l — 2kmm)

fn,0 : 1911,01 fn,M+1 : kn,M+l-

The frequency response Hf) of the filter, Fy, with Q = 1.01 is as follows:

Hf) (wmwy) = [cos2 (coy/2) + jig—1 sin2 (wy)] [cos2 (Lay/2) + isin2 (wy)[ , (A.7)

and Fig. A.5 shows the plot of this frequency response.

75

 

1.50

—
_
—
—1
——1
—.
—4
.4
-
—
-
-
-
-—1

l

1.25

r
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__
._
._
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[—

 

1.00‘

0.75

0.50

illlliLLJLilllliillliIJl

 

0.25

H(F)(w) (Frequency Response)

llllllllllllllllTTf

llll

 

 

p.—

 

0.00

l l l l 1 l 1 l L l
1 2
co (Wavenumber)

O
m...—

Figure A.5: Plot of the frequency response, H (F l (02) over the range 0 S to S 1r, of
the filter used for the improved partial differentiators with Q = 1.01 as a function
of wavenumber, w. The nice low frequency response and the suppression of high
frequency signals are shown.

 

76

Let’s define a resolving efficiency of the filter, 81? E wf/7r where the value w;
is the maximum wave-number of well-resolved wave satisfying the tolerance relation

I H”) —-1 IS 6. ep(e = 0.1) = 0.32, 61: (c = 0.01) = 0.17, and ep(e = 0.001) = 0.10.

A first—order partial differentiator with respect to :1: is considered. A combination
of the standard central difference scheme (refer to Eq. A.1) and a a: filter (with
Q = 1.70) and a y filter (with Q = 1.01) is used. In this case, the :r filter will be
called “primary filter” and the y filter “secondary filter”. The differential operator
D9) that approximates first-order partial derivatives with respect to :r, {%f (mm ym)}
for n =1,...,N and m = 0,...,M+1 from the data {fmm = f(:rn,ym)} for
n =0,...,N+1 and m =0,...,NI+1 is:

fwn, forn=0,...,N+l, andm=0,...,M+1
U

dn,m = (fn+1,m - fn—l,m) /2Ar

forn=1,...,N, and m=0,...,M+1

U

533mm = (dn+1.m + 2dn,m + dn—1,m) /4

33:1,", 2 d1,ma 333M", 2 de

U

gmn,m : 333mm _ 4 (ser-Lm + 3xn—l,m — 23$n,m)
gml,m = 31:1,m1 ng,m = 317N,m
(“5mm : (g$n+1,m + 2937mm + 937n—1.m)/4

ham. = gum, hrvN,m = gmzvm

U

77

1
kxnmz : hmn,m —' Z (h$n+1,m + h3n—l,m _ 2ha7n,m)
kxlnn : h$1,m7 kxN,m = th,m

U

symm = (kxn,m+l + 2‘33an + kmn,m—1)/4

33/n,0 : kxnfla Syn,M+1 : kxn,M+1

U
1.01
gymm : 331mm _ T (Syn,m+l + Syn,m-1 — 2symm)
93171.0 : 3yn,07 gyn,M+l = 3yn,M+l
U

hym = (gyn,m+1 + 293mm + gyn,m_1)/4
hymo : 9.7/n.0, hyn,M+1 = gyn,M+l
U

a 1
ng (3371: gm) : hymm _ 21- (hyn:m+1 + hyn’m_l _ 2hyn’m)

a 8
53f (33113 y0) = hynfla $f(xn’ylw+l) : hyn'M+1’

where Ax is the step size of memesh.

Evaluation of derivatives at nodes near boundaries is handled properly and with
simplicity without introducing additional algorithms to handle them. In a similar
way, the approximate first-order partial differentiator D51) with respect to y can be

obtained. The corresponding frequency response of 05)) is:

Hy) (W...) = (at) [cos (Wm/2) + 1—43 sin? on]
x [cos2 (cum/2) + isin2 (wx)] [cos2 (coy/2) + 1—2-1- sin2 (0%)]

X [cos2 (Lay/2) + isin2 (wy)] , (A.8)

and Fig. A.6 shows the plot of the frequency response in frequency domain. It should

78

be noted that because this is pure imaginary there isn’t any phase shift.

A second-order partial differentiator with respect to a: is considered. The standard
central difference scheme (refer to Eq. A.5) composed with a a: filter (with Q = 1.36)
and a y filter (with Q = 1.01) is used. In this case, a: filter is called “primary filter” and
3] filter is called “secondary filter”. The differential operator D?) that approximates
second-order partial differentiation {53%, f (3", ym)} with respect to :c for n = 1,. . . , N
and m = 0,...,M+1 from data {fmm = f(a:n,ym)} for n = 0,...,N+1 and
m =O,...,M+ 1 is as follows:

fmm, forn=0,...,N+1, andm=0,...,M+1
U

dn,m = (fn+1,m + fn—1,m — 2fn,m) /A2
forn=1,...,N, andm=0,...,M+1

U

3.1:",m = (dn+1,m + 2dmm + dn_1,m) /4

3x1,m = d1,m, st,m = dN,m

U
1.36

gxn,m = S$n,m “ (3$n+l,m + 33n-1,m _ 23$",m)

4

g$1,m : 3x1,m7 ng,m = 337N,m
U
hxn,m : (gxn+l,m + 293mm + gxn—1,m) /4
hxl,m : gxlmn th,m : ng,m
U
1
kxn,m 2 [213mm — 1(hxn+l,m + hxn-1,m _ 2hxn,m)

kxl,m = h$1,ma kxN,m : th,m

79

 

Figure A.6: Plot of the frequency response of the improved first-order partial differen-
tiator with respect to :3 divided by 1', H9) (wmwy) /i, for 0 S a), 5 1r and 0 S (0,, S 7r.
The suppression of high frequency signals both in a: and 31 should be noted. It also
shows the correct linear behavior for low frequencies.

80

 

Figure A.7: Plot of the frequency response of the improved second-order partial
differentiator with respect to :r multiplied by (—1), —H£2) (wmwy), for 0 S w, S 7r
and 0 S 1.0,, S 1r. The suppression of high frequency signals both in a: and 31 should
be noted. It shows the correct quadratic behavior for low frequency components as
well.

81

U

symm : (kxn,m+l + 21611371,," + kxn,m—l) /4

3yn,0 = (Winn, syn,M+1 = kxn,M+1

U

1.01

gymm = Symm _ T (Syn,m+l + symm—l _ 23.9mm)

93111.0 = 331mm gyn,M+l 1‘ syn,M+1
U n
hymm : (gymm-H + 2gyn,m + gyn,m—1)/4

hymo = gymo, hyn,M+1 = gyn,M+1

U

82 1

% (xnaym) : hymm _ 1(hyn,m+l + hyn,m-—l _ 2hyn,m)
32 82

fifwmyo) = hymo: 55f (“Mai/M“) : hynvMH’

where AI is the step size of :r-mesh.

The approximate second-order partial differentiation D?) with respect to y can

be obtained in a similar way. The corresponding frequency response of D9) is:

H532) (wmwy) = —4 sin2 (wt/2) [cos2 (wt/2) + %§ sin2 (wx)]
x [cos2 (wt/2) + isin2 (w3)] [cos2 (Lay/2) + 1—2—1 sin2 (wy)]

x [cos2 (Lay/2) + ism“ (wy)] (A.9)

and Fig. A.7 shows the plot of the frequency response in frequency domain.

82

A.3 Application to the field produced by magne-
tized iron bars

Two long iron bars were considered with the geometry ——2 S :1: S 2, —2 S y S 2 and
z 2 1 for one iron bar and —2 S :L‘ S 2, —2 S y S 2 and 2 S —1 for the other. These
bars are uniformly magnetized in the +2 direction with a resultant internal field 8,.
Let’s define 3:1 = —2, x2 = 2, 311 = —2 and y2 = 2. The magnetic field due to the two

sheets of surface charge [1, 23] is given by:

. _ B, Xm- XM-
82 (.r,y, z) — 41r 21—1) [arctan (Z+R+) + arctan (Z_R_ (A.10)

iJ

 

 

where i,j = 1,2, and B, is taken to be 21.4 kG [1] and where:

X,- = :r 17,,
Y: = 31-3/1,
Z+ 2 1+2,
Z- = 1—2,

R1 = (X,2 +YJ-2+Zi)l/2.

The map of the magnetic field Bz(:1:,y,z = 0) in [CC is shown in Fig. A.8 over the

range—4SatS4and —4SyS4withAx=Ay=0.1.

B; is Taylor expanded around 2 = 0 just as in Eq. 2.1. For the sake of convenience,

let’s define:
B(:1:,y) = 3,, (a:,y,z = 0) (A.11)

The program “Mathematica” was used to obtain the analytical expressions of various

- ~ 2 6 - 2 _ a2 a2
derivatives such as V28, . . . , V28 WIth V2 — 23-7 + 532-.

We made comparisons between the results obtained from the analytically differ-

entiated formulas using the program “Mathematica” and the results from the two

83

 
  

    
 
 

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Figure A.8: Map of Bz(:z:, y, z = 0) produced by two saturated semi-infinite iron bars
which are magnetized in the +z direction whose geometry is specified in the text.
This field is plotted over the range —4 S m S 4 and —4 S y S 4 with Ax = Ay = 0.1.
Notice how fast the field values decrease as the distance from the origin increases.
The maximum field value is 12.6 kG.

84

Table A.3: Comparison of the improved and old differentiators using data without
noise

 

 

 

 

 

 

old differentiators improved differentiators
term for comparison rms(D) rms(D)
V38 x 0.52/2! 5.34 G 0.360 G
V38 x 0.54/4! 3.79 G 0.598 G
V38 x 0.56/6! 1.59 G 0.454 G

 

 

 

different approximate second-order differential operators, one is the old differentiator
used by Gordon and Taivassalo [1, 14] and the other is the improved differentiators

described in section A.2.

The numerical calculations start with values of B (:r, y) stored in a uniform square
mesh with A1: = Ay = 0.1. Data from —4 S :c S 4 and —4 S y S 4 were chosen for
comparison because this is the region where drastic changes occur. Exact evaluation
of the derivatives of various order was done by analytically differentiating Eq. A.10
evaluated on the median plane where z = 0 using the program “Mathematica”.
Numerical evaluation of the derivatives of various order was done by applying the
approximate differentiators to the field data of B. In obtaining numerical evaluation
of the derivatives, the field data expressed in kG up to the 11th decimal place obtained

from Eq. A.11 were used to keep truncation error as small as possible.

In each case, we calculated values of a particular term in the expansion of B, for
z = 0.5, which is halfway from the median plane to the poleface. The rms difference
in G between the values obtained from the two different approximate methods and
the exact values from the analytically differentiated expressions are given in Table A.3
where rms(D) is the rms difference between the numerical and analytical results. The
largest magnitudes of the second-order, the fourth-order, and the sixth-order terms

of the expansion are 948 G, 140 G, and 25.2 G, respectively.

85

Figure A.9 shows the map of the second-order term in 82, V38 x 0.52/21, derived
from the analytical expression whose maximum and minimum are 447 G and —948 G.
Figure A.10 shows two maps of the difference in the second-order term in 2 between
the analytical results and those obtained from the two different differentiators, the
improved and old differentiators. The top map represents the difference between
theory and the improved differentiators, while the bottom map depicts the difference
between theory and the old differentiators. These two maps are plotted to the same

scale.

In a similar manner, Fig. A.11 depicts the map of the fourth-order term in 83,
V38 X 0.54 /4l, obtained from the analytical expression whose maximum and minimum
are 140 G and —86.5 G, respectively. Figure A.12 shows two maps of the difference in
the fourth-order term in 2 between the analytical results and those obtained from the
two different differentiators. The top map represents the difference between theory
and the improved differentiators, while the bottom map depicts the difference between

theory and the old differentiators. These two maps are plotted to the same scale.

From these figures and Table A.3, it is clear that the improved differentiators are
an order of magnitude more accurate than the old differentiators for the nonlinear
terms in 8,, at z = 0.5. This is because the old differentiators wash out physically

important low-frequency components of the data.

86

       
    

         

 

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Figure A.9: Map of V38 X 0.52/2! obtained from the analytical expression. This is
plotted over —4 S a: S 4 and —4 S y S 4 with Ax = Ay = 0.1. The maximum and

the minimum are 446 G and —948 G, respectively.

87

 

         

   

        
    

 
   
 
 
 
 

      
  

   

    

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Figure A.10: Two maps of the difference between exact values and those from the two
differentiators for V33 x 0.52/2! term over the range —4 S a: S 4 and —4 S y S 4 with
Ax = Ay = 0.1. The top map depicts the difference between theory and the improved
differentiators, while the bottom map portrays the difference between theory and the
old differentiators. Both of these are plotted to the same scale. The maximum and
the minimum of the top map are 1.1 G and —1.5 G respectively, while those of the
bottom map are 13.2 G and —19.4 G. Clearly, the improved differentiators are about
twelve times more accurate than the old differentiators.

88

      
     

 
 

  

   

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Figure A 11 Map of VgB x 0.54/4! obtained from the analytical expresmon This Is

plotted over 4 < a: S 4 and —4 S y S 4 with Ax = Ag 0 1 The max1mum and
the minimum are 140 G and —86.5 G, respectlvely

89

   
       

 
         
  

    

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Figure A.12: Two maps of the difference between exact values and those from the two
differentiators for VgB x0.54/4! term over the range —4 S 1' S 4 and —4 S y S 4 with
A1: = Ay = 0.1. The top map depicts the difference between theory and the improved
differentiators, while the bottom map portrays the difference between theory and the
old differentiators. Both of these are plotted to the same scale. The maximum and
the minimum of the top map are 2.9 G and —2.0 G respectively, while those of the
bottom map are 16.3 G and —10.9 G. Clearly, the improved differentiators are about

five times more accurate than the old differentiators for the fourth—order term in B2
at z = 0.5.

90

AA Application to data With noise

We assumed that noise in the data could be simulated by generating random numbers
and adding them to the values of B stored in the square mesh described above.
Comparison was then made between the results obtained from the improved and old
differentiators to determine their characteristics when applied to data with noise.
In order to compare the results with those in Table A.3 above, we used the same
mesh spacing, Ax 2 Ag = 0.1. The random numbers added to the stored field data
are within i0.1 G, which is around the limit of measurement accuracy. The rms
difference between the field data with and without this noise is 5.75 X 10‘2 G (which

agrees quite well with the expected values, (IO/J3) x 10"2 G).

Table A.4 shows the values of rms differences between the exact derivatives of
the field without noise and the results derived from the two numerical differentiators
for the field with the noise. This table shows the results for the second, fourth, and
sixth-order terms in units of G. The rms values in Table A.4 are larger than those in

Table A.3 due to the noise.

It should be noted that the magnitude of random noise is very small compared
with the maximum field value of 12.6 kG (see Fig. A.8) for the field described in
section A.3, and yet its effect on the derivatives is not negligible at all. The magnitude
of the noise is also small compared with the largest magnitudes of the second, fourth,

and sixth-order terms which are 948 G, 140 G, and 25.2 G evaluated at z = 0.5.

For a comparison, we took rms values of the differences between the values of
numerically differentiated derivatives for the field data without the noise and those
for the field data with the noise using the two numerical differentiators up to the
sixth order. Table A.5 shows the values of rms differences for the two numerical

differentiators. The results for the second, fourth, and sixth-order terms are presented

91

Table A.4: rms differences between the exact derivatives for the field without the noise
and those for the field with the noise for the two different numerical differentiators

 

 

 

 

old differentiators improved differentiators
term for comparison rm3(D) rms(D)
VgB x 0.52/2! 5.40 G 0.682 G
VgB x 0.54/4! 4.48 G 1.74 G
VSB x 0.56/6! 3.60 G 2.08 G

 

 

 

 

 

Table A.5: rms differences between the values of numerically differentiated derivatives
for the field with the noise and those for the field without the noise for the two
numerical differentiators

 

 

 

 

 

old differentiators improved differentiators
term for comparison rms(D) rms(D)
V38 x 0.52/2! 0.798 G 0.580 G
V33 X 0.54/4! 2.40 G 1.64 G
VgB x 0.56/6! 3.23 G 2.04 G

 

 

 

 

in units of G. Addition of noise within :720.1 G produced rms differences of 0.580 G
(0.798 G) for the second-order term, 1.64 G (2.40 G) for the fourth-order term, and
2.04 G (3.23 G) for the sixth—order term when the improved (old) differentiators were
used. It should be noted that these rms differences are large in comparison with the
magnitude of the noise. From the rms differences given in Table A.5, it is clear that
the improved differentiators are progressively superior to the old differentiators for
higher order terms by from 37 ‘70 (the second-order term) to 58 % (the sixth-order
term) in suppressing high frequency components. Besides, the differences between

the rms values in Table A.4 and those in Table A.3 are smaller than the rms values

in Table A5.

Figure A.13 corresponds to Fig. A.10 depicting the map of the difference in V33 x

0.52/27. The only difference is that in this case, median plane magnetic field data

 

92

contain random noise within i0.1 G. Figure A.13 shows the combined effects of the
errors produced by the two numerical differentiators for the field without the noise
and the additional errors due to the added noise. The surface of the top map derived
from the improved differentiators is smoother than that at the bottom from the old

differentiators.

Figure A.14 also corresponds to Fig. A.12 portraying the map of the difference in
V38 X 0.54/41. The only difference lies again in the inclusion of random noise within
:7:0.1 G. Figure A.14 shows the combined effects of the errors produced by the two
numerical differentiators for the field without the noise and the additional errors due
to the added noise. The surface of the bottom map derived from the old differentiators
becomes even sharper than that at the top from the improved differentiators. Besides,

the height of individual spikes is larger than that in the top diagram.

As shown in section A.2, our approximate partial differentiators are accompanied
by the “secondary filter” which filters data in y (x) direction when partial derivatives
are taken with respect to a: (3;). If the data contain no noise at all and are perfectly
analytical, the “secondary filter” makes little difference. But it becomes indispensable
when data with noise must be dealt with. Table A.6 shows the values of the rms
differences in units of G between results obtained from the improved differentiators
with and without the “secondary filter”. For the sake of convenience, let’s call the
former “with secondary filter” and the latter “without secondary filter”. As can
be seen from this table, the “secondary filter” becomes progressively more important
with each succeeding term. Thus the “secondary filter” is indispensable for evaluating

higher order derivatives of data with noise.

93

    
      
   
 
     

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Figure A.13: Two maps of the difference in V38 X 0.52 / 2! between the exact deriva-
tives for the field without the noise and those obtained from the two numerical differ-
entiators applied to the field with the random noise within 27:0.1 G. The magnitude
of the noise is very small compared with the maximum field value of 12 6 [CG of 8.
These are plotted to the same scale over the range —4 S a: S 4 and —4 S y S 4 with
A.7: 2 Ag 2 0.1. This figure should be compared with Fig. A.10. The top (bottom)
map depicts the difference for the improved (old) differentiators. The maximum and
the minimum field values of the top map are 2.4 G and —2.4 G respectively, while
those of the bottom map are 14.8 G and —20.5 G. These figures show the combined
effects of the errors produced by the two numerical differentiators for the field without
the noise and the additional errors due to the noise. The surface of the top map is

smoother than that at the bottom.

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Figure A.14: Two maps of the difference in V33 x 0.54/4! between the exact deriva-
tives for the field without the noise and those obtained from the two numerical differ-
entiators applied to the field with the random noise within $0.1 G. These are plotted
to the same scale over the range —-4 S :1: S 4 and —4 S y S 4 with A1: 2 Ag 2 0.1.
This figure should be compared with Fig. A.12. The top (bottom) map depicts the
difference for the improved (old) differentiators. The maximum and the minimum
of the top map are 6.1 G and —5.8 G respectively, while those of the bottom map
are 20.3 G and —15.7 G. These figures again show the combined effects of the errors
produced by the two numerical differentiators for the field without the noise and the
additional errors due to the noise. The surface of the bottom map is even sharper
than that at the top, and the height of individual spikes is larger.

Table A.6: Effects of the secondary filter on data with noise

 

 

With Secondary Filter

Without Secondary Filter

 

 

 

 

 

term for comparison rms(D) rm3(D)
V38 x 0.52/2! 0.682 G 0.962 G
V38 X 0.54/4! 1.74 G 3.33 G
V33 x 0.56/6! 2.08 G 5.53 G

 

 

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