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pISSN 1229-3008 eISSN 2287-6251
Progress in Superconductivity and Cryogenics
Vol.17, No2, (2015), pp.36~40 http://dx.doi.org/10.9714/psac.2015.17.2.036
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1. INTRODUCTION
There are some demands for increased intensities of
highly charged heavy ions, which leads to higher
performance electron cyclotron resonance (ECR) ion
sources. Strong confinement by highly charged ions is
required for this system, and superconducting magnets are
needed in 28 GHz ECR ion source [6].
Plasma is produced in ECR ion source. In a high
magnetic field, outer electron of gas in plasma chamber
will conduct cyclotron motion (Larmor gyration). When
cyclotron angular frequency is equal to microwave angular
frequency, resonance happens and plasma is formed [7].
For an ECR source, it is convenient to describe the
magnetic field strength relative to BECR, which is the
magnetic field in tesla for electron cyclotron resonance. If
the frequency is 28 GHz, BECR becomes 1 T [6].
In order to produce more ion beams, plasma diffusing to
plasma chamber wall must be prevented. To avoid
diffusing, magnetic fields are used to provide a force in the
transverse direction to confine plasma. In this way,
charged ions becoming getting close to the center of the
axis. The magnetic fields are also used to reflect electrons
back into the plasma chamber.
Magnetic field mainly plays two roles in the discharge
process. One is to provide an electron cyclotron resonance
and the other is to confine electrons and ions. The
confinement is mainly generated from the appropriate
distribution of magnetic field in the plasma chamber.
Plasma is confined by making the magnetic field B
minimum at the center of the plasma chamber, and this
structure is called by minimum B structure. In the
minimum B structure, magnetic field from sextupole is
superimposed for radial confinement. For the axial
confinement, asymmetry displacement of solenoids can be
used to realize the confinement. Two groups of
optimization based on the 3, 4 and 6 solenoid system are
introduced in this paper.
2. ECR ZONE
2.1. Introduction of ECR Zone
The confinement fields in an ECR ion source provide
closed surface, and the magnetic field on this closed
surface is equivalent everywhere. On the closed surface
where the ECR condition is satisfied, the electrons can be
heated by rf power through electron cyclotron resonance.
The physical region over which the ECR condition is
satisfied is called as ECR zone [8]. By using Opera-3dTM
program, the ECR zone was visualized in Fig. 1. The
center part in the magnets represents the ECR zone.
The microwave power can be coupled to the plasma
only in ECR zones which occupy a small percentage of the
ionization chamber volume and leave the remainder of the
plasma chamber as “unheated” zones, as shown in Fig. 2.
Therefore, the absorptivity of the plasma region is not
Comparison analysis of superconducting solenoid magnet systems
for ECR ion source based on the evolution strategy optimization
Shaoqing Wei and Sangjin Lee*
Uiduk University, Gyeongju, Korea
(Received 23 February 2015; revised or reviewed 10 June 2015; accepted 11 June 2015)
Abstract
Electron cyclotron resonance (ECR) ion source is an essential component of heavy-ion accelerator. For a given design, the
intensities of the highly charged ion beams extracted from the source can be increased by enlarging the physical volume of ECR
zone [1]. Several models for ECR ion source were and will be constructed depending on their operating conditions [2-4]. In this
paper three simulation models with 3, 4 and 6 solenoid system were built, but it’s not considered anything else except the number of
coils. Two groups of optimization analysis are presented, and the evolution strategy (ES) is adopted as an optimization tool which
is a technique based on the ideas of mutation, adaptation and annealing [5]. In this research, the volume of ECR zone was calculated
approximately, and optimized designs for ECR solenoid magnet system were presented. Firstly it is better to make the volume of
ECR zone large to increase the intensity of ion beam under the specific confinement field conditions. At the same time the total
volume of superconducting solenoids must be decreased to save material. By considering the volume of ECR zone and the total
length of solenoids in each model with different number of coils, the 6 solenoid system represented the highest coil performance.
By the way, a certain case, ECR zone volume itself can be essential than the cost. So the maximum ECR zone volume for each
solenoid magnet system was calculated respectively with the same size of the plasma chamber and the total magnet space. By
comparing the volume of ECR zone, the 6 solenoid system can be also made with the maximum ECR zone volume.
Keywords: Coil performance, ECR ion source, evolution strategy, superconducting magnet, length of solenoids, volume of ECR zone
* Corresponding author: sjlee@uu.ac.kr
Shaoqing Wei and Sangjin Lee
(a) ECR zone in ECR magnet
.
(b) Enlarged view of ECR zone.
Fig. 1. ECR zone (for f=28 GHz). The isopotential lines of
1 T constitute the outline of ECR zone.
Fig. 2. A structure for ECR ion source.
controlled by the size of the plasma, but by the size of the
ECR zone. Hence, the ECR zone is one of the key elements
to increase the intensity of highly charged ion beams.
2.2. Calculation of ECR Zone Volume
The volume of ECR zone VECR can be calculated
approximately by using the magnetic field. On the surface
of ECR zone, the magnetic field B is equal to BECR, while
inside of ECR zone B value is less than BECR. We can
calculate VECR with MatlabTM and Opera-3dTM program in
rectangular coordinate system or cylindrical coordinate
system. In the rectangular coordinate system, the volume
of each unit Vr is calculated by
stepstepsteprzyxV ⋅⋅=
(1)
where xstep , ystep and zstep indicate the unit distance along x, y
and z axis directions. In the cylindrical coordinate system,
per unit cylindrical arc is shown in Fig. 3 and its volume Vc
is calculated by
i
zV
ρφρ
⋅⋅⋅=
stepstepstepc
(2)
where ρi is the ith radius of per unit cylindrical arc; and
step
ρ
, ρi ·
step
φ
and
step
z
indicate the unit distances along ρ,
φ
and z axis directions.
Fig. 3. Per unit cylindrical arc.
Fig. 4. Comparison of VECR with different steps between
two methods.
TABLE I
COMPARISON OF vECR AND CALCULATION TIME WITH DIFFERENT STEPS.
ρstep
(mm)
Cylindrical coordinate
system
Rectangular coordinate
system
V
ECR
Time(m)
V
ECR
Time(m)
5 702352 15.60 673000 17.51
8 717670 5.09 679936 6.32
10 717932 3.08 667000 4.97
12 642755 2.20 698112 3.36
15 747502 1.62 681750 2.50
20 780372 1.18 704000 2.02
The calculation time of VECR with two kinds of
coordinate systems was also checked. The results and
diagram about different steps in two coordinate systems
are respectively shown in Fig. 4 and table I.
In considering of the accuracy of VECR and calculation
time, the cylindrical coordinate system is used to get the
ECR zone volume. VECR volume is more accurate with
smaller ρstep ,
step
φ
and zstep . In this paper, ρstep=zstep=8 mm
and
step
φ
=0.05π are used.
3. COMPARISON ANALYSIS
Evolution Strategy (ES) optimization method was
adopted for the 2 groups of comparison analysis. All the
current density in the optimization was set as 100 A/mm2
in solenoids and 270 A/mm2 in sextupole. In addition, the
size of each sextupole for each case is equivalent. The
confinement conditions is given as
plasma
solenoid
sextupole
plasma chamber
ECR zone
BECR=1T
ρ
i
+
ρ
step
/2
ρi – ρstep/2
z
step
step
φ
37
Comparison analysis of superconducting solenoid magnet systems for ECR ion source based on the evolution strategy optimization
2
0.8)~(0.4
2.3)~(1.8
3.5
ECRr
ECRmin
ECRext
ECRinj
BB
BB
BB
BB
≈
≈
≈
≥
(3)
where Binj is the axial magnetic field at injection part, Bext
is the axial magnetic field at extraction part, Br is the radial
magnetic field at chamber wall, and Bmin is the minimum
axial magnetic field [4] .
The radius of solenoids can be defined as matrix A, and
the depth of solenoids can be defined as matrix B.
=
24232221
14131211
aaaa
aaaa
A
=
24232221
14131211
bbbb
bbbb
B
(4)
where the general components of matrix A and B are
defined as aij and bij respectively. For aij, i means the inner
and outer radii. j means the jth solenoid. In matrix A, the
first row indicates the inner radius of solenoid and the
second row indicates the outer radius of solenoid. In matrix
B, the first row indicates the lower position of solenoid and
the second row indicates the larger position of solenoid in
the z axis. These parameters are shown in Fig. 5. The total
solenoid number is defined as N, which is the maximum
value of j. And N is equal to 3, 4 and 6 respectively for the
3, 4 and 6 solenoid system.
The solenoids can be designed in the area which is
shown by the dashed line in Fig. 5. We assumed L=1000
mm and W=600 mm and used them to determine the range
of variables. So the range of parameters are
Waa
jj
<<
21
,
2/
1
Lb
j
≤
,
2/
21
Lbb
ji
≤<
. (5)
To make the size of plasma chamber constant, a same
constant was used for the first row in matrix A.
3.1. Coil Performance
To increase intensity of ion beams, we had to enlarge the
ECR zone under the confinement field. Meanwhile, the
total volume of solenoids must be reduced to save material.
Based on this idea, the object function 1 was defined as (6).
Fig. 5. Parameters for solenoid systems.
TABLE II
CONFINEMENT FIELDS FOR COIL PERFORMANCE.
N Binj (T) Bm in (T) Bext (T)
3 3.5002 0.4231 1.8371
4 3.5041 0.4139 1.8141
6 3.5128 0.4020 1.8006
TABLE III
OPTIMIZATION RESULTS FOR COIL PERFORMANCE.
ECR
21
ECR
2coil1
1
1
1function object
V
cVc
V
cVc
N
j
j+=
⋅+⋅=
∑
=1
(6)
where Vj is the jth solenoid volume, c1 is the weight value
for Vcoil, and c2 is the weight value for VECR. c1 and c2 can be
decided by the importance of two factors. In this paper,
c1=10-6, c2=107 was used to make Vcoil and 1/VECR same
order of magnitude. Hence, we can consider the values of
Vcoil and VECR equally. If VECR is more important, the weight
value of VECR could be enlarged, and vice versa. In the
optimization process for coil performance, (6) was used as
object function, (3) is some constraints for design variables
(4) and equation (5) is upper/ lower boundaries of design
variables (4).
After the optimization for coil performance, we got
three optimized designs for ECR magnet for the 3, 4 and 6
solenoid system respectively. Since the weight values in
objection function 1 can be decided for design purpose, the
ratio of VECR/Vcoil was used as an intrinsic parameter for
coil performance that means we used the object function
1as a tool for its process only to make the problem
converge.
The decided confinement fields of each optimized
model are shown in table II. And the final values results of
VECR, Vcoil and objection function 1are shown in table III
and Fig. 6. By comparing the ratio of VECR/Vcoil in table III,
the 6 solenoid system presents the highest coil
performance.
3.2. ECR Zone Volume
The second analysis aims to get the maximum ECR zone
volume without considering the coil volume within the
given working space in Fig. 5. Based on this idea the
objection function 2 was defined as (7).
ECR
1
2 functionobject V
=
(7)
The same constraints and boundaries for coil
performance was used in this optimization. After the
optimization for ECR zone volume, we got three optimized
N
V
ECR
(mm
3
)
V
coil
(mm
3
)
VECR/Vcoil
object
function 1
3
818563
34433278
0.02377
46.65
4 858977 36070784 0.02381 47.71
6 1138372 38304303 0.02972 47.09
38
Shaoqing Wei and Sangjin Lee
050 100 150 200 250
46
48
50
52
54
56
58
iter ation
Value
Ob ject F uction 1
3 solenoids
4 solenoids
6 solenoids
(a) Object function 1
050 100 150 200 250
7
8
9
10
11
12 x 105
iter ation
V
ECR
( mm
3
)
ECR Zo ne Volume
3 solenoids
4 solenoids
6 solenoids
(b) ECR zone volume
050 100 150 200 250
3.4
3.6
3.8
4
4.2
4.4 x 10
7
iter ation
V
coil
( mm
3
)
To tal Co il Volume
3 solenoids
4 solenoids
6 solenoids
(c) Total coil volume
Fig. 6. Process of ES optimization for coil performance.
designs for the 3, 4 and 6 solenoid system respectively.
The decided confinement fields for each model are shown
in table IV. And the results of VECR, Vcoil and objection
function 2 for each model are shown in table V and Fig. 7.
According to the result, the 6 solenoid system shows the
greatest ECR zone volume.
TABLE IV
CONFINEMENT FIELDS FOR ECR ZONE VOLUME.
N Binj (T) Bmin (T) Bext (T)
3
3.5454
0.4015
1.8028
4
3.6014
0.4614
1.9244
6 3.5329 0.4046 1.8497
TABLE V
OPTIMIZATION RESULTS FOR ECR ZONE VOLUME.
N
V
ECR
(mm
3
)
V
coil
(mm
3
)
VECR/Vcoil
Object
function 2
3 812733 36174552 0.0225 1.2304e-06
4
820212 41860113 0.0196 1.2192e-06
6
1298900 51674617 0.0251 7.6988e-07
010 20 30 40 50 60
0.5
1
1.5
2
2.5
3
3.5 x 10
-6
iter ation
Value
Ob ject F uction 2
3 solenoids
4 solenoids
6 solenoids
(a) Object function 2
010 20 30 40 50 60
2
4
6
8
10
12
14 x 10
5
iter ation
V
ECR
( mm
3
)
ECR Zo ne Volume
3 solenoids
4 solenoids
6 solenoids
(b) ECR zone volume
010 20 30 40 50 60
3.5
4
4.5
5
5.5 x 10
7
iter ation
Vcoil ( mm3 )
To tal Co il Volume
3 solenoids
4 solenoids
6 solenoids
(c) Total coil volume
Fig. 7. Process of ES optimization for ECR zone volume.
4. CONCLUSION
In this paper, we have presented two kinds of
optimization for ECR ion source. One is about the coil
performance of solenoid system and the other is about
ECR zone volume itself. Through these optimizations, we
got the ECR ion source with 6 solenoid system has the
highest coil performance and the largest volume of ECR
zone at the same working space.
This result seems to be caused that the 6 solenoid system
has more parameters to be controlled than those of other
solenoid system. But practically, if the number of solenoid
is increased, the things to be considered will be increased
such as the number of power supply, etc. In addition, the
sextupole in ECR ion source was not considered i.e. the
value of parameters for the sextupole magnet is fixed in
this paper. So the research about that must be the next.
39
Comparison analysis of superconducting solenoid magnet systems for ECR ion source based on the evolution strategy optimization
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