Parametric instability influence on isotope separation by ion-cyclotron resonance method
Solution of the dispersion equation for linear potential ion cyclotron oscillations with characteristic parameters used for selective ion cyclotron resonance separation of gadolinium isotopes has revealed the presence of the shortwavelength unstable oscillation in the ion cyclotron frequency range...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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| Цитувати: | Parametric instability influence on isotope separation by ion-cyclotron resonance method / V.V. Olshansky, K.N. Stepanov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 204-206. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-823042015-05-28T03:02:16Z Parametric instability influence on isotope separation by ion-cyclotron resonance method Olshansky, V.V. Stepanov, K.N. Low temperature plasma and plasma technologies Solution of the dispersion equation for linear potential ion cyclotron oscillations with characteristic parameters used for selective ion cyclotron resonance separation of gadolinium isotopes has revealed the presence of the shortwavelength unstable oscillation in the ion cyclotron frequency range. Computer simulation of this instability by means of macro particle technique has shown that this instability may be responsible for turbulent heating of both resonant and nonresonant ions obtaining large transversal energy. It can decrease the efficiency of this method. 2006 Article Parametric instability influence on isotope separation by ion-cyclotron resonance method / V.V. Olshansky, K.N. Stepanov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 204-206. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.35.-g; 52.35.Mw; 28.60.+s http://dspace.nbuv.gov.ua/handle/123456789/82304 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Low temperature plasma and plasma technologies Low temperature plasma and plasma technologies |
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Low temperature plasma and plasma technologies Low temperature plasma and plasma technologies Olshansky, V.V. Stepanov, K.N. Parametric instability influence on isotope separation by ion-cyclotron resonance method Вопросы атомной науки и техники |
| description |
Solution of the dispersion equation for linear potential ion cyclotron oscillations with characteristic parameters used
for selective ion cyclotron resonance separation of gadolinium isotopes has revealed the presence of the shortwavelength
unstable oscillation in the ion cyclotron frequency range. Computer simulation of this instability by means
of macro particle technique has shown that this instability may be responsible for turbulent heating of both resonant and
nonresonant ions obtaining large transversal energy. It can decrease the efficiency of this method. |
| format |
Article |
| author |
Olshansky, V.V. Stepanov, K.N. |
| author_facet |
Olshansky, V.V. Stepanov, K.N. |
| author_sort |
Olshansky, V.V. |
| title |
Parametric instability influence on isotope separation by ion-cyclotron resonance method |
| title_short |
Parametric instability influence on isotope separation by ion-cyclotron resonance method |
| title_full |
Parametric instability influence on isotope separation by ion-cyclotron resonance method |
| title_fullStr |
Parametric instability influence on isotope separation by ion-cyclotron resonance method |
| title_full_unstemmed |
Parametric instability influence on isotope separation by ion-cyclotron resonance method |
| title_sort |
parametric instability influence on isotope separation by ion-cyclotron resonance method |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2006 |
| topic_facet |
Low temperature plasma and plasma technologies |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/82304 |
| citation_txt |
Parametric instability influence on isotope separation by ion-cyclotron resonance method / V.V. Olshansky, K.N. Stepanov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 204-206. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT olshanskyvv parametricinstabilityinfluenceonisotopeseparationbyioncyclotronresonancemethod AT stepanovkn parametricinstabilityinfluenceonisotopeseparationbyioncyclotronresonancemethod |
| first_indexed |
2025-07-06T08:48:49Z |
| last_indexed |
2025-07-06T08:48:49Z |
| _version_ |
1836886765223280640 |
| fulltext |
204 Problems of Atomic Science and Technology. 2006, 6. Series: Plasma Physics (12), p. 204-206
PARAMETRIC INSTABILITY INFLUENCE ON ISOTOPE SEPARATION
BY ION-CYCLOTRON RESONANCE METHOD
V.V. Olshansky, K.N. Stepanov
National Science Center “Kharkov Institute of Physics and Technology”,
Akademicheskaya Str. 1, 61108, Kharkov, Ukraine
Solution of the dispersion equation for linear potential ion cyclotron oscillations with characteristic parameters used
for selective ion cyclotron resonance separation of gadolinium isotopes has revealed the presence of the short-
wavelength unstable oscillation in the ion cyclotron frequency range. Computer simulation of this instability by means
of macro particle technique has shown that this instability may be responsible for turbulent heating of both resonant and
nonresonant ions obtaining large transversal energy. It can decrease the efficiency of this method.
PACS: 52.35.-g; 52.35.Mw; 28.60.+s
1. INTRODUCTION
According to this method [1] in the plasma coming
with the thermal velocity from an ion source into the
heating region under the antenna the resonant ions which
are to be separated obtain their transverse energy from the
RF field under the ion cyclotron resonance and are heated
up on the average. The nonresonant ions obtain lesser
energy due to Doppler effect and one can neglect their
heating if the generator frequency (more precisely the
magnetic field) is matched so that
Tici VkVk 0||||0||0 >>⋅−− ωω , where ||V is the particle
velocity along the magnetic field, TiV is its thermal
velocity, Lk /20|| π= , L is the antenna length.
Transverse RF field speeds up the resonant particles and
they are heated. For a nonresonant particle the cyclotron
frequency differs from one for a resonant particle by the
value ( )AA∆=∆ 0ωω , where A is the atomic number of
the isotope, and the condition of resonant absorption is
not met for them, if
( )( )AAVLL Ti ∆=>> 08 ωπ . (1)
In the RF field during the time TiVkt ||1~∆ the resonant
ions obtain the current velocity of the order of
( )Tiires VkmeEu 0||02π≈ ,
where 0E is the complex amplitude of the field rotating
in the direction of cyclotron gyration of ions. Current
velocity of nonresonant ions will be of the order of
( ) ( )AABeEunores ∆⋅00 /~ . The velocity resu can be, by
virtue of inequality (1), considerably greater than the
velocity of nonresonant ions.
If the relative velocity of resonant and nonresonant
ions will become of the order of initial thermal velocity of
ions, then in such plasma, due to the relative oscillations
of ions, parametric instabilities will arise associated with
potential ion cyclotron oscillations [2].
In the present work the linear growth rates of the
parametric ion cyclotron instability, conditioned by the
oscillations of different species ions, are found and
computer simulation, based on the macro particle model
(2D2V model [3]), of this instability is carried out for the
gadolinium isotopes.
2. LINEAR THEORY OF PARAMETRIC ION
CYCLOTRON INSTABILITIES
Let us consider the particle motion in the pumping
field [ ]( )tzki 00||0 (expRe ω−= EE that is switched on at
the moment 0=t . Because the particle displacement is
small in comparison with the antenna radius, then one can
consider the electric field to be uniform. Integrating the
equations of motion, neglecting small longitudinal electric
field ||E and carrying out the averaging over initial
Maxwell distribution, we obtain the expression for the
average transversal velocity of particles
( ) ( ) ( )[ ]{
( )
,
2
222
exp
exp
exp
2
0||
0
0||
0||
0
2
0||
0
0||
0
−
−
−
−
⋅
−×
×−−
−
=⊥
α
α
α
α
αα
α
α
α
α
ωω
ωω
ωω
ωπ
T
c
T
T
cT
c
T
caa
Vk
W
tVk
i
Vk
W
tVk
ti
Vk
tiEme
V
(2)
where yx iVVV ααα +=⊥ is the transversal velocity in the
complex form, αωc is the ion cyclotron frequency of the
ion of α -species, 0E is the complex amplitude of the
pumping field, 0ω is the frequency of this field, that is
equal to the cyclotron frequency of the ion species under
separation. In expression (2)
( ) ( ) ( )
+⋅−= ∫
z
dtti
k
k
zzW
0
2
0||
0||2 exp2
||
exp
π
.
In [4] the device was proposed for isotope
gadolinium-157 separation with the parameters: 0B =3 T,
0n =1012 cm-3, initial temperature of ions is 10 eV,
L =2 m, the frequency of alternating field is
rescωω = =2⋅106 s-1, the plasma radius is 10 cm, the
electric field strength is ϕE =3 V/cm. In this case
resu ~3·106 cm/s, the nonresonant ions velocity is
approximately 2 times smaller; the initial thermal velocity
of ions is TiV ~2.5·105 cm/s.
In Fig. 1 the dependence against time t0ωτ = of the
velocities of the gadolinium isotopes G157 and G158 and
their relative velocity, divided by the initial thermal
velocity of ions, is shown for the conditions given above
[4]. Oscillations of the relative velocity of the isotopes
lead to the appearance of the amplitude beats with the pe-
riod ( ) 3102 ≈∆=′ AAπτ , that arise from “instantaneous”
switching on of the pumping field. At the initial stage the
velocity growth is almost the same for the isotopes of
different species. At the stage, when the relative velocity
205
of different isotopes is close to the maximum value,
oscillations of the components of the transversal velocity
take place in opposite phase. In this case the value of the
relative velocity achieves 20 TiV . If switching on of the
pumping field takes place at the moment −∞→t , then
the oscillations with the frequency ciω disappear. Only
the oscillations with the frequency 0ω remain. In this
case the relative amplitude u at 400≥τ is equal to 4 TiV
approximately.
tc1ω
1/ TVu
11 / TVV
12 / TVV
Fig. 1. Time dependence of the velocities of the gadoli-
nium isotopes G157 ( 1V ), G158 ( 2V ) and their relative
velocity u divided by the initial thermal velocity of Gd157
( 1TV )
0.5 1.0 1.5
kρ1
0.4
0.6
0.8
1.0
1.2
ω
/ω
c1
0.00
0.04
0.08
0.12
γ/
ω
c1
γ/ωc1
ω/ωc1
Fig. 2. Dependence of frequencies and growth rates of the
unstable oscillations against wave number ( 1|| ρk = 0.01).
For oscillations with larger ||k growth rate is 1.5÷3 times
less than maximal one
The dispersion equation for unstable oscillations has a
form of the infinite determinant that is equal to zero
0||det =mnA , (3)
( ) ( ) ( )
( )0
0,1
1
ωωδε
ωωδε
δ
α
α
p
aJaJ
m
A
p
anpamp
e
mnmn
+×
×
++
+= ∑ ∑
∞
−∞=
++k ,
where αa is the displacement of ions of α -species in the
pumping field, multiplied by the wave vector of the
unstable oscillations, ( )αaJ p is the Bessel functions of
the first kind, αδε is the contribution to the longitudinal
plasma permittivity of the particles of α -species in the
presence of the oscillations with the average velocity (2).
The solution of the dispersion equation (3) for the
parameters [4] in the case of the rest plasma, 0|| =V , is
presented in Fig. 2 (it is assumed in computations, that
nonresonant ions have a mass of the gadolinium-158 and
concentration 0.843, and concentration of gadolinium-157
is equal to its concentration in the natural mixture, 0.157).
In Fig. 2 there are two branches of the unstable
oscillations, the frequencies of which join the ion
cyclotron frequencies (because heavy ions are considered
the difference of their cyclotron frequencies is negligible).
One can see, that in the region 1≈ikρ ( iρ - is the
Larmor radius of the ions) the difference between
unstable oscillations frequencies and the frequency
ciωω =0 is ciωω 05.0≈∆ . The growth rates have a
maximum at 1~ikρ . The growth rate of the branch with
lesser frequency achieves the value ciω06.0 , and the
growth rate of the branch with larger frequency achieves
the value ciω025.0 . The longitudinal wave number is
small, ik ρ|| =0.01, but it exceeds the longitudinal wave
number of the pumping field 0||k =0.03 cm-1 three times,
that allows to neglect the pumping field nonuniformity .
3. COMPUTER SIMULATION RESULTS
Two-dimensional numerical simulation of nonlinear
evolution of the parametric ion cyclotron instability by
macro particle technique was carried out for the ions of
gadolinium isotopes with the parameters [4]. As distinct
from the previous section we will assume, that the
oscillations of the ions of different species are defined by
formulae (2) with 00|| →k . In this case the velocity of
the resonant isotopes, for which ωω =resc , will grow
linearly with time, ( ) tBEcV ⋅= 001 / , and the simulation
results obtained can be used up to the time lesser than
( ) 1
0||
−=∆ TiVkt , i.e. 300≈∆⋅ trescω .
In Fig. 3 the time dependence of the averaged energy
density of unstable oscillations of the electric field is
shown with the initial value ( ) 3
00 100 −⋅= TnWE .
0 50 100 150 200
ωc1t
0.00
0.01
0.02
0.03
0.04
W
E/
n 0
T
Fig. 3. Time dependence of energy density of the self
consistent electric field π82EWE = divided by the
initial thermal energy density
206
The frequencies of these oscillations are of the order of
cinω . The characteristic wave numbers at 20≤tciω are
large, 1>ikρ . At the large time the oscillations are
excited with 1~<ikρ . Note, the dependence of the value
00TnWE against time is nonlinear. It relates to the
variation of the current velocities of the particles in time
and nonlinear effects in the particles motion in the electric
field of unstable oscillations.
0 50 100 150 200
ωp1t
0
1
2
3
4
5
6
7
8
T/
T 1
0
Fig. 4. Time dependence of the averaged chaotic energy
of ions of the isotope Gd157 divided by its initial value
Development of the parametric instability is
accompanied by heating the particles. It is illustrated by
Fig. 4. In Fig. 4 the time dependence of the relation of
transversal chaotic energy of the resonant isotope G157 to
the initial temperature is shown. At 180=tciω this value
exceeds the initial value seven times. The heating is
conditioned by nonlinearity of the equations of the
particles motion and by the dynamic chaos that can arise.
In Fig. 4 at 50>tciω one can see the oscillations with the
frequency ciω2 and with the amplitude growing in time.
It is related to high energy ions generation. Transversal
chaotic energy of the nonresonant isotope G158 grows in
time too and it has the value of the order of that is
obtained for the G157isotope.
4. CONCLUSIONS
On the ground of computer simulation by means of
macro particle technique it is shown, that for the typical
parameter values of plasma, magnetic field and
alternating electric filed under condition of the ion
cyclotron resonance for the gadolinium isotopes mixture,
the small-scale potential ion cyclotron oscillations with
transversal wave number of the order of the inverse
Larmor radius of ions arise in plasma as a result of the
parametric instability after the relative velocity of the
different isotopes achieves the value of the order of their
thermal velocity. This instability results in the turbulent
heating of both resonant and nonresonant ions and it can
reduce the efficiency of the isotope separation method
given.
REFERENCES
1. Yu.A. Muromkin. Razdeleniye izotopov v plazme s
pomoschyu ionno-tsiklotonnogo nagreva // Itogi nauki i
tekhniki. Series: “Fizika Plazmy”. M: VINITI, 1991,
N 12, p. 83.
2. K. N. Stepanov. Non-Linear Parametric Phenomena
during Radio Frequency Heating in the Ion Cyclotron
Frequency Range // Plasma Physics and Controlled
Fusion. 1996, v. 38, N 12, p. A13.
3. V.V. Ol’shansky. KC-A Kinetic Computer Code for
Investigation of Parametric Plasma Instabilities //
Seibersdorf Report, Juli 1995, OEFZS-4752, Seibersdorf,
Austria.
4. V. I. Volosov, I. A. Kotel’nikov, S. G. Kuz’min. O
razdelenii izotopov tyazhelyh elementov metodom
izotopicheski selektivnogo ICR-nagreva // Fizika Plazmy.
1998, v. 24, N 6, p. 517.
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