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Particle diffusion along the direction of an electric field acting transverse to a non-uniform magnetic field
Volume 32A, number 2
PHYSICS LETTERS
PARTICLE DIFFUSION ALONG THE DIRECTION ACTING TRANSVERSE TO A NON-UNIFCRM
15 June 1970
OF AN E L E C T R I C FIELD MAGNETIC FIELD
S. PURI Institut fiir P l a s m a p h y s i k , 8046 Garching/M~inchen, Germany Received 12 May 1970
A uniform electric field transverse to a spatially non-uniform magnetic field produces particle diffusion even when the electric field spectrum contains no dc component. This diffusion is additional to the usual E x B diffusion and occurs isotropically in the plane perpendicular to the magnetic field direction.
It is generally accepted that only the dc component of the electric field spectrum in the p a r t i c l e ' s frame of motion could produce a diffusion t r a n s v e r s e to the magnetic field. This diffusion is due to the pole at o~ = 0 when the position of the particle in the E × B direction is expressed as a function of the electric field. In this note it is shown that the parametric resonances due to the longitudinal motion of the particle in a non-uniform magnetic field cause the appearance of the additional poles in the equation relating the particle position and the electric field. The resultant drift and diffusion occur isotropically in the plane perpendicular to the magnetic field direction. Seidl [I] has shown that (in the absence of an electric field) for paraxial particles in a parabolic
magnetic m i r r o r , ~c(t)
= Wc[1 + A c o s (pt+~)]
(1)
w h e r e O)c(t) is the i n s t a n t a n e o u s value of the p a r t i c l e g y r o f r e q u e n c y at the position of the p a r t i c l e , A = (R - 1)/(R + 1), R b e i n g the r a t i o of the m a x i m u m to the m i n i m u m magnetic field seen by the p a r t i cle, ½p is the f r e q u e n c y of the longitudinal motion along the m i r r o r axis and ~ specifies the i n i t i a l position. F r o m eq. (54) of ref. [2] we w r i t e down the equation of t r a n s v e r s e motion of a p a r a x i a l p a r t i c l e of charge q and m a s s m as Jr(t)
= - j Wc[1 + A cos ~ t + ~)] Vr(t) + ~TE(t)
(2)
w h e r e v r = v x + i v y , 71 = q / m and o)c is the a v e r a g e value of the cyclotron frequency seen by the p a r t i cle. The x - d i r e c t e d e l e c t r i c field E(t) is spatially uniform and i s a s s u m e d to be v a n i s h i n g s m a l l in m a g n i t u d e so that it l e a v e s eq. (1) unaffected. The solution of eq. (2) is given by eq. (55) of ref. [2]. On i n t e g r a t i n g v r ( t ) , we obtain the position, r = x + jy of the p a r t i c l e , r(t)
= ~7~Jm(~)Jm-s(B) S ~/ × r - e x r {j(0~ - sp) L W -sp
exp (-is ~) ~ /~(w) do) 0)C + rt2P -oo
t}
-
1
+ exp {-j (wc + rap) t} exp{j (w + w c + m p - sp) t} - 1~[ w + Wc + m p - s p J
(3)
exp {-j(~c + rap)t} - 1 + j vr(O) e x p (j~ sin ~ ) ~ J m ( ~ ) exp (-jm ~) + r(0) m o)c + m p w h e r e all s u m m a t i o n s extend from - ~ to +~o. Of the two sets of poles in eq. (3) the f i r s t set at o) = s p gives r i s e to guiding c e n t e r drift, as may be seen by the application of l'HSpital r u l e to one of these poles, r(t)
= 71~Jm(~)Jm-s~)
exp (-j s ~) ; / ~ ( o ) ) d o ) ( - j t )
(4a) 55
Volume 32A, number 2
PHYSICS LETTERS
15 June 1970
Since ~ can a s s u m e a r b i t r a r y v a l u e s between 0 and 2~, the d r i f t produced is i s o t r o p i c for the c a s e oJ = sp ¢ 0. F o r the c a s e w = sp = O, h o w e v e r , r(t) is p u r e l y i m a g i n a r y and the d r i f t o c c u r s in the y d i r e c t i o n only c o r r e s p o n d i n g to the E × B d r if t in a uniform m a g n e t i c field. The second set of p o l e s at co = Wc + np, w h e r e n = rn - s , r e p r e s e n t the effect of expanding p a r t i c l e o r b i t s due to the c o r r e s p o n d i n g p o l e s in the v e l o c i t y equation and do not c o n t r i b u t e to guiding c e n t e r motion. T h i s is r e a d i l y v e r i f i e d by applying once again l 'H 6 p i t al r u l e to one of t h e s e p o l e s , co
x p (~- j s~PP J / ) -f E(w)dw e x p [ - j ( w c + m P ) r(t) = ~ J m ( ~ ) Jm-s( ) e ~c
t ](jr)
(4b)
The o s c i l l a t o r y f a c t o r e x p [ - j (wc + rap)t] g i v e s r i s e to a s p i r a l i n g motion to Ir(t) l which i n c r e a s e s l i n e a r l y in t i m e . T h e r e is one exception, h o w e v e r , when COc/p has the i n t e g e r value - m , in which c a s e eqs. (4a) and (4b) c o m b i n e to give on applying l ' H 6 p i t a l r u l e
r(l)
= - ~ J _ c o c / p (fl) J_(cOc/P)_ s (fl) exp ( - i s ~) f~/~(o~) dw(t 2)
(4c)
-co
F o r a p r a c t i c a l p l a s m a c o n f in e m e n t s y s t e m c o u p >> fl so that J_coc/p(~)J_(cOc/P)_s(fl)
~ 0, and the q u a -
d r a t i c diffusion t e r m in eq. (4c) is negligibly s m a l l . In this p a p e r we shall c i r c u m v e n t this s i n g u l a r i t y by the p u r e l y m a t h e m a t i c a l a r t i f i c e of r e q u i r i n g ¢Oc/p to be a n o n - i n t e g e r . Applying F e r m i ' s golden r u l e to eq. (3) we obtain
:
~c ~
; ~(sp)
(5)
w h e r e the s u b s c r i p t ' g ' d e n o te s guiding c e n t e r diffusion and @(w) is the p o w er s p e c t r u m of the s t a t i o n a r y r a n d o m e l e c t r i c field E(t). It is to be r e m e m b e r e d that the diffusion is i s o t r o p i c f o r s ¢ 0 and in the E × B d i r e c t i o n f o r s = 0. Using the identity [3] ~ m J m ( f l ) J m _ s ( f l ) = Js(O), it is s e e n f r o m eq. (5) that f o r co = sp ¢ 0 the m a g n i tude of the diffusion t e r m is quite s m a l l c o m p a r e d to the c a s e co = sp = 0 f o r c o m p a r a b l e v a l u e s of the e l e c t r i c field. H o w e v e r , the contribution to diffusion p a r a l l e l to the e l e c t r i c f i el d may still be i m p o r tant due to the p r e s e n c e of r a t h e r l a r g e a m b i p o l a r f i e l d s in the r a d i a l d i r e c t i o n . T h i s work has been p e r f o r m e d as p a r t of the joint r e s e a r c h p r o g r a m m e of the Institut f o r P l a s m a physik and E u r a t o m .
References [1] M. Seidl, J. Nucl. Energy Pt. 66 (1964) 597. [2] S. Purl, Phys. Fluids 8 (1968) 1745. [3] G. N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, 1958) p. 360.
56
PHYSICS LETTERS
PARTICLE DIFFUSION ALONG THE DIRECTION ACTING TRANSVERSE TO A NON-UNIFCRM
15 June 1970
OF AN E L E C T R I C FIELD MAGNETIC FIELD
S. PURI Institut fiir P l a s m a p h y s i k , 8046 Garching/M~inchen, Germany Received 12 May 1970
A uniform electric field transverse to a spatially non-uniform magnetic field produces particle diffusion even when the electric field spectrum contains no dc component. This diffusion is additional to the usual E x B diffusion and occurs isotropically in the plane perpendicular to the magnetic field direction.
It is generally accepted that only the dc component of the electric field spectrum in the p a r t i c l e ' s frame of motion could produce a diffusion t r a n s v e r s e to the magnetic field. This diffusion is due to the pole at o~ = 0 when the position of the particle in the E × B direction is expressed as a function of the electric field. In this note it is shown that the parametric resonances due to the longitudinal motion of the particle in a non-uniform magnetic field cause the appearance of the additional poles in the equation relating the particle position and the electric field. The resultant drift and diffusion occur isotropically in the plane perpendicular to the magnetic field direction. Seidl [I] has shown that (in the absence of an electric field) for paraxial particles in a parabolic
magnetic m i r r o r , ~c(t)
= Wc[1 + A c o s (pt+~)]
(1)
w h e r e O)c(t) is the i n s t a n t a n e o u s value of the p a r t i c l e g y r o f r e q u e n c y at the position of the p a r t i c l e , A = (R - 1)/(R + 1), R b e i n g the r a t i o of the m a x i m u m to the m i n i m u m magnetic field seen by the p a r t i cle, ½p is the f r e q u e n c y of the longitudinal motion along the m i r r o r axis and ~ specifies the i n i t i a l position. F r o m eq. (54) of ref. [2] we w r i t e down the equation of t r a n s v e r s e motion of a p a r a x i a l p a r t i c l e of charge q and m a s s m as Jr(t)
= - j Wc[1 + A cos ~ t + ~)] Vr(t) + ~TE(t)
(2)
w h e r e v r = v x + i v y , 71 = q / m and o)c is the a v e r a g e value of the cyclotron frequency seen by the p a r t i cle. The x - d i r e c t e d e l e c t r i c field E(t) is spatially uniform and i s a s s u m e d to be v a n i s h i n g s m a l l in m a g n i t u d e so that it l e a v e s eq. (1) unaffected. The solution of eq. (2) is given by eq. (55) of ref. [2]. On i n t e g r a t i n g v r ( t ) , we obtain the position, r = x + jy of the p a r t i c l e , r(t)
= ~7~Jm(~)Jm-s(B) S ~/ × r - e x r {j(0~ - sp) L W -sp
exp (-is ~) ~ /~(w) do) 0)C + rt2P -oo
t}
-
1
+ exp {-j (wc + rap) t} exp{j (w + w c + m p - sp) t} - 1~[ w + Wc + m p - s p J
(3)
exp {-j(~c + rap)t} - 1 + j vr(O) e x p (j~ sin ~ ) ~ J m ( ~ ) exp (-jm ~) + r(0) m o)c + m p w h e r e all s u m m a t i o n s extend from - ~ to +~o. Of the two sets of poles in eq. (3) the f i r s t set at o) = s p gives r i s e to guiding c e n t e r drift, as may be seen by the application of l'HSpital r u l e to one of these poles, r(t)
= 71~Jm(~)Jm-s~)
exp (-j s ~) ; / ~ ( o ) ) d o ) ( - j t )
(4a) 55
Volume 32A, number 2
PHYSICS LETTERS
15 June 1970
Since ~ can a s s u m e a r b i t r a r y v a l u e s between 0 and 2~, the d r i f t produced is i s o t r o p i c for the c a s e oJ = sp ¢ 0. F o r the c a s e w = sp = O, h o w e v e r , r(t) is p u r e l y i m a g i n a r y and the d r i f t o c c u r s in the y d i r e c t i o n only c o r r e s p o n d i n g to the E × B d r if t in a uniform m a g n e t i c field. The second set of p o l e s at co = Wc + np, w h e r e n = rn - s , r e p r e s e n t the effect of expanding p a r t i c l e o r b i t s due to the c o r r e s p o n d i n g p o l e s in the v e l o c i t y equation and do not c o n t r i b u t e to guiding c e n t e r motion. T h i s is r e a d i l y v e r i f i e d by applying once again l 'H 6 p i t al r u l e to one of t h e s e p o l e s , co
x p (~- j s~PP J / ) -f E(w)dw e x p [ - j ( w c + m P ) r(t) = ~ J m ( ~ ) Jm-s( ) e ~c
t ](jr)
(4b)
The o s c i l l a t o r y f a c t o r e x p [ - j (wc + rap)t] g i v e s r i s e to a s p i r a l i n g motion to Ir(t) l which i n c r e a s e s l i n e a r l y in t i m e . T h e r e is one exception, h o w e v e r , when COc/p has the i n t e g e r value - m , in which c a s e eqs. (4a) and (4b) c o m b i n e to give on applying l ' H 6 p i t a l r u l e
r(l)
= - ~ J _ c o c / p (fl) J_(cOc/P)_ s (fl) exp ( - i s ~) f~/~(o~) dw(t 2)
(4c)
-co
F o r a p r a c t i c a l p l a s m a c o n f in e m e n t s y s t e m c o u p >> fl so that J_coc/p(~)J_(cOc/P)_s(fl)
~ 0, and the q u a -
d r a t i c diffusion t e r m in eq. (4c) is negligibly s m a l l . In this p a p e r we shall c i r c u m v e n t this s i n g u l a r i t y by the p u r e l y m a t h e m a t i c a l a r t i f i c e of r e q u i r i n g ¢Oc/p to be a n o n - i n t e g e r . Applying F e r m i ' s golden r u l e to eq. (3) we obtain
:
~c ~
; ~(sp)
(5)
w h e r e the s u b s c r i p t ' g ' d e n o te s guiding c e n t e r diffusion and @(w) is the p o w er s p e c t r u m of the s t a t i o n a r y r a n d o m e l e c t r i c field E(t). It is to be r e m e m b e r e d that the diffusion is i s o t r o p i c f o r s ¢ 0 and in the E × B d i r e c t i o n f o r s = 0. Using the identity [3] ~ m J m ( f l ) J m _ s ( f l ) = Js(O), it is s e e n f r o m eq. (5) that f o r co = sp ¢ 0 the m a g n i tude of the diffusion t e r m is quite s m a l l c o m p a r e d to the c a s e co = sp = 0 f o r c o m p a r a b l e v a l u e s of the e l e c t r i c field. H o w e v e r , the contribution to diffusion p a r a l l e l to the e l e c t r i c f i el d may still be i m p o r tant due to the p r e s e n c e of r a t h e r l a r g e a m b i p o l a r f i e l d s in the r a d i a l d i r e c t i o n . T h i s work has been p e r f o r m e d as p a r t of the joint r e s e a r c h p r o g r a m m e of the Institut f o r P l a s m a physik and E u r a t o m .
References [1] M. Seidl, J. Nucl. Energy Pt. 66 (1964) 597. [2] S. Purl, Phys. Fluids 8 (1968) 1745. [3] G. N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, 1958) p. 360.
56