The new approach for definition of volume confined by ECR surface and its area in ECR ion source
The volume confined by resonance surface and its area are one of the important parameters of the balance equations model for calculation of ion charge–state distribution (CSD) in electron–cyclotron resonance (ECR) ion source. The new approach for definition of these quantities is given. This approa...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1114292017-01-10T03:04:37Z The new approach for definition of volume confined by ECR surface and its area in ECR ion source Philippov, A.V. Новые методы ускорения, сильноточные пучки The volume confined by resonance surface and its area are one of the important parameters of the balance equations model for calculation of ion charge–state distribution (CSD) in electron–cyclotron resonance (ECR) ion source. The new approach for definition of these quantities is given. This approach allows to reduce the number of parameters of balance equations model. У моделі рівнянь балансу для розрахунку зарядових розподілів іонів (ЗРІ) в іонному джерелі, заснованому на електронно-циклотронному резонансі (ЕЦР), площа резонансної поверхні і об'єм, обмежений резонансною поверхнею, є важливими параметрами моделі. У даній роботі запропоновано новий підхід за визначенням даних величин, що дозволяє зменшити число параметрів моделі. В модели уравнений баланса для расчёта зарядовых распределений ионов (ЗРИ) в ионном источнике, основанном на электронно-циклотронном резонансе (ЭЦР), площадь резонансной поверхности и объём, ограниченный резонансной поверхностью, являются важными параметрами модели. В данной работе предложен новый подход по определению данных величин, позволяющий уменьшить число параметров модели. 2008 Article The new approach for definition of volume confined by ECR surface and its area in ECR ion source / A.V. Philippov // Вопросы атомной науки и техники. — 2008. — № 3. — С. 163-165. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.25.Xz, 52.25.Ya, 52.50.Sw, 52.55.Dy, 52.55.Jd http://dspace.nbuv.gov.ua/handle/123456789/111429 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Новые методы ускорения, сильноточные пучки Новые методы ускорения, сильноточные пучки |
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Новые методы ускорения, сильноточные пучки Новые методы ускорения, сильноточные пучки Philippov, A.V. The new approach for definition of volume confined by ECR surface and its area in ECR ion source Вопросы атомной науки и техники |
| description |
The volume confined by resonance surface and its area are one of the important parameters of the balance equations
model for calculation of ion charge–state distribution (CSD) in electron–cyclotron resonance (ECR) ion source. The new approach for definition of these quantities is given. This approach allows to reduce the number of parameters of balance equations model. |
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Article |
| author |
Philippov, A.V. |
| author_facet |
Philippov, A.V. |
| author_sort |
Philippov, A.V. |
| title |
The new approach for definition of volume confined by ECR surface and its area in ECR ion source |
| title_short |
The new approach for definition of volume confined by ECR surface and its area in ECR ion source |
| title_full |
The new approach for definition of volume confined by ECR surface and its area in ECR ion source |
| title_fullStr |
The new approach for definition of volume confined by ECR surface and its area in ECR ion source |
| title_full_unstemmed |
The new approach for definition of volume confined by ECR surface and its area in ECR ion source |
| title_sort |
new approach for definition of volume confined by ecr surface and its area in ecr ion source |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2008 |
| topic_facet |
Новые методы ускорения, сильноточные пучки |
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http://dspace.nbuv.gov.ua/handle/123456789/111429 |
| citation_txt |
The new approach for definition of volume confined by ECR surface and its area in ECR ion source / A.V. Philippov // Вопросы атомной науки и техники. — 2008. — № 3. — С. 163-165. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT philippovav thenewapproachfordefinitionofvolumeconfinedbyecrsurfaceanditsareainecrionsource AT philippovav newapproachfordefinitionofvolumeconfinedbyecrsurfaceanditsareainecrionsource |
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2025-07-08T02:10:12Z |
| last_indexed |
2025-07-08T02:10:12Z |
| _version_ |
1837042881157660672 |
| fulltext |
THE NEW APPROACH FOR DEFINITION OF VOLUME CONFINED
BY ECR SURFACE AND ITS AREA IN ECR ION SOURCE
A.V. Philippov
Joint Institute for Nuclear Research, Laboratory of Particle Physics, Dubna, Russia
E-mail: philippov@jinr.ru
The volume confined by resonance surface and its area are one of the important parameters of the balance equa-
tions model for calculation of ion charge–state distribution (CSD) in electron–cyclotron resonance (ECR) ion
source. The new approach for definition of these quantities is given. This approach allows to reduce the number of
parameters of balance equations model.
PACS: 52.25.Xz, 52.25.Ya, 52.50.Sw, 52.55.Dy, 52.55.Jd
1. INTRODUCTION
For some transport models [–] for calculation of ion
CSD in ECR ion sources [] the values of the volume pV
confined by resonance surface and its area pS are one
of model parameters. For example in model [] these val-
ues are using in balance equation for neutrals in ECR
plasma:
( )
, 0,
1 1 1
,
cx
,
1
,
ion
,0,
0,
0,0,
s
M
m
S
s
Z
mz
zsmzzs
m
K
k
kekms
m
ss
p
pss
nnn
nn
V
Sv
dt
dn
s
∑ ∑ ∑∑
= =′ +=
′−→′
=
→
+−
−−=
′
νν
(1)
here agreed notation are: s , s′ are ion species index; z
, z′ are ion charge state index; m is process multiplici-
ty; k , k ′ are electron component index; 0,sv is a neutral
velocity; ion
,, kzzs
m
′→ν , cx
, zzs
m
′→ν are ionization and charge
exchange rates from charge state z to z′ ; zsn , , ken , are
ions and electrons densities; 0,sn , sn are neutral density
inside and outside of the source chamber.
In this case the proper calculations of these ones im-
portant parameters are presented.
2. ECR ION SOURCE MAGNETIC MAP AP-
PROXIMATION
The approximation of the ECR ion source magnetic
map uses the next well known fact: that minimum–B
field configuration created by external magnetic system
of ion source segmented on two different parts. One of
these parts is solenoid magnet and the other one is mul-
tipole magnet.
2.1 EXTERNAL SOLENOID FIELD
In cylindrical coordinate system we describe the ex-
ternal magnetic field of solenoid by ( )zAA ,ρθθ = − az-
imuthal component of vector potential []:
( ) ( )
( ) .
,,
2
2
1
1
BzBz
z
dz
dJzA
+=Φ
Φ
= ρρθ (2)
Here
dz
dJ ρ1 is a Bessel function of first order; ( )zΦ
is a magnetic field at the axis; 1B and 2B are numerical
coefficients in Gs and Gs∙cm–2 units correspondingly.
When in use approximation only first order term was
taken in decomposition (2).
2.2. EXTERNAL FIELD OF MULTIPOLE LENS
The external multipole magnet of sextupole lens we
describe by ( )θρ ,zz AA = − longitudinal component of
vector potential []:
( ) .
3
3sin
2
0
0
3
R
θBρθρ,Az = (3)
Here 0B is a pole tip magnet field and 0R is a lens ra-
dius. The dimension of a quantity 2
00 RB is Gs∙cm–2.
2.3. FITTING OF TOTAL MAGNETIC FIELD
The total vector potential ( )z,, θρAA = of mini-
mum–B configuration is on the form:
( ) .
0
,,
=
zA
Az θθρA (4)
Here θA , zA defined in (2) and (3) correspondingly.
The total magnetic field ( )z,, θρBB = of ECR ion
source can be expressed as follow:
,AB ×∇= (5)
here ∇ is gradient operator.
Fitting of the numerical coefficients in formulas (2)
and (3) was performed separately for solenoid field and
for sextupole magnet. These coefficients was found for
magnetic field maps of the three different ECR ion
sources: for INFN, LNS, SERSE ion source of two dif-
ferent working frequencies 14, 18 GHz and for ECR ion
source with working frequency 14.4 GHz of the Frank-
furt University, IKF. The results of this calculation are
presented in the table 1 below.
Table 1. Numerical coefficients
ECR ion source 2
00 RB 1B 2B
INFN, LNS, SERSE
14 GHz 270 4813 27
INFN, LNS, SERSE
18 GHz 334 5374 42
____________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 3.
Series: Nuclear Physics Investigations (49), p.163-165.
163
Frankfurt UNI, IKF
14.4 GHz 251 4797 21
3. ECR RESONANCE SURFACE
The subsequent discussion we advance in Cartesian
coordinate system. The ECR resonance surface
( )zyxFF ,,= is determined by condition that absolute
value ( )zyxBB ,,= of total magnetic field (5) is equal
to the resonance value resB i.e.
.resBB = (6)
For absolute value of total magnetic field ( )zyxB ,, we
have:
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ).
,2,,
,,,
,,,,,,,
20
20
22
222
zzB
BzBxyzyxB
BzxByxzyxB
zBzyxBzyxBzyxB
z
y
x
zyx
Φ=
+−=
−−=
++=
(7)
The coefficient 2
00 RB here was redefined as 0B and
therefore expression for ( )zyxF ,, is given by:
( )
( ) .
2
62
,,
2
res
2
2
22
0
22
2
2
42
2
22
21
2
20
2
20
3
2
1
2
0
42
0
4
BBzByx
BzBzyBBz
BBzyxBBzx
BByBxzyxF
−++
++++
++−
−++=
(8)
The equality to zero of the expression (8) defines the
implicit equation of ECR surface.
On Fig.1 the ECR resonance surfaces for different
ion sources are shown. The fitted numerical coefficients
are taken from Table 1.
Fig.1. ECR resonance surfaces (8) for different ion
sources
4. DEFINITION OF VOLUME CONFINED
BY RESONANCE SURFACE AND ITS AREA
Now when the equation defines is known we can to
develop the method for calculation of volume confined
by resonance surface and its area. The volume can be
define as
( ) ( ){ },0:Ω
,,
Ω
<=
== ∫ ∫ ∫
zy,x,Fzy,x,
dzdydxdVdVVp
(9)
and for resonance surface area:
( ) ( ){ }.0,,:,,
,,
,
==
==
⋅= ∫ ∫
zyxFzyxS
dydxdSdSd
dS
S
p
nS
Sn
(10)
Using the Ostrogradsky-Gauss theorem we reduce the
last expression i.e.
( ) ( ) ( )
( ) .
,,
,,,,,,
,
,
Ω
Ω
zyxB
zyxBzyx,zyx
dVS
dVd
p
S
∇
∇==
⋅∇=
⋅∇=⋅
∫ ∫ ∫
∫ ∫ ∫∫ ∫
nnn
n
nSn
(11)
We use the formulas (9), (11) for calculation of pV
and pS . This calculation was produced with using
Monte-Carlo method and tested for surfaces with ana-
lytical expression for volume and area i.e. sphere with
given radius and ellipsoid with given semiaxis.
Table 2. Result of calculation of pV and pS
ECR ion source Volume, cm3 Area, cm2
INFN, LNS, SERSE
14 GHz 64 79
INFN, LNS, SERSE
18 GHz 262 207
Frankfurt UNI, IKF
14.4 GHz 148 144
The numerical results of pV and pS values was
found for set of numerical parameters of approximated
magnetic field 0B , 1B , 2B of the three different ECR
ion sources The results of this calculation are presented
in the table 2 above.
CONCLUSION
From the point of view of author the calculation
problem of the volume confined by resonance surface
and its area is important.
There are some works when the assumption about
the ellipsoidal shape of resonance surface is given and
for this case these parameters was calculated. Also in
some works thus [] the numerical estimation is given:
,792
,150 2
dL.S
dL.V
p
p
=
=
(12)
here L the mirror–to–mirror distance and d is a work-
ing chamber diameter and two numerical factors in (12)
are for a very particular geometry of magnetic system of
ion source. But all of these examples have special case.
Therefore the presented above technique of calculation
of volume confined by resonance surface and its area
without any assumption about ECR surface shape in
general allows defining these parameters using only an
ion source magnetic fields map.
REFERENCES
1. H.I. West, Jr. UCRL–53391. LLNL. University of
California. 1982.
2. V.D. Dougar-Jabon, A.M. Umnov, V.B. Kutner.
Calculating method for confinement time and
164
charge distribution of ions in electron cyclotron res-
onance sources. // Review of Scientific Instruments.
1996. v.67, Iss. 3, p.1152-1154.
3. A.V. Philippov, G.D. Shirkov, F. Consoli, et al.
Development of the balance equations model for
calculation of ion charge–state distribution in ECR
ion sources: JINR Preprint P9-2007-75. Dubna,
2007, 18 p.
4. R. Geller. Electron Cyclotron Resonance Ion
Sources and ECR Plasmas. Institute of Physics
Publishing. Bristol and Philadelphia, 1996.
5. A.I. Morozov, L.S. Soloviev. Geometry of the mag-
netic fields // Voprosi teorii plazmi. 1965, №2, p.3-
176.
НОВЫЙ ПОДХОД К ОПРЕДЕЛЕНИЮ ОБЪЁМА ОБЛАСТИ, ОГРАНИЧЕННОЙ ЭЦР-ПОВЕРХНО-
СТЬЮ, И ЕЁ ПЛОЩАДИ В ИОННОМ ИСТОЧНИКЕ ЭЦР-ТИПА
А.В. Филиппов
В модели уравнений баланса для расчёта зарядовых распределений ионов (ЗРИ) в ионном источнике,
основанном на электронно-циклотронном резонансе (ЭЦР), площадь резонансной поверхности и объём,
ограниченный резонансной поверхностью, являются важными параметрами модели. В данной работе
предложен новый подход по определению данных величин, позволяющий уменьшить число параметров мо-
дели.
НОВИЙ ПІДХІД ДО ВИЗНАЧЕННЯ ОБ'ЄМУ ОБЛАСТІ, ОБМЕЖЕНОЮ ЭЦР-ПОВЕРХНЕЮ,
І ЇЇ ПЛОЩІ В ІОННОМУ ДЖЕРЕЛІ ЭЦР-ТИПУ
А.В. Філіппов
У моделі рівнянь балансу для розрахунку зарядових розподілів іонів (ЗРІ) в іонному джерелі,
заснованому на електронно-циклотронному резонансі (ЕЦР), площа резонансної поверхні і об'єм,
обмежений резонансною поверхнею, є важливими параметрами моделі. У даній роботі запропоновано новий
підхід за визначенням даних величин, що дозволяє зменшити число параметрів моделі.
____________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 3.
Series: Nuclear Physics Investigations (49), p.163-165.
165
Новый подход к определению объёма области, ограниченной ЭЦР-поверхностью, и её площади в ионном источнике ЭЦР-типа
Новий підхід до визначення об'єму області, обмеженою ЭЦР-поверхнею, і її площі в іонному джерелі ЭЦР-типу
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