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PARAMETRIC EXCITATION OF ELECTRONS AT
THE LIQUID HELIUM SURFACE IN A MAGNETIC
FIELD
V. Shikin
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, suppl6ment au no 8 , Tome 39, aozit 1978, page C6-336
PARAMETRIC EXCITATION OF ELECTRONS AT THE L I Q U I D HELIUM SURFACE I N A MAGNETIC F I E L D
V.B. Shikin.
S o l i d S t a t e Physics I n s t i t u t e USSR Academy o f Sciences 142432, ChemogoZovka, Moscm d i s t r i c t USSR.
Rdsum6.- Dans c e t t e n o t e , on p r g s e n t e d e s r g s u l t a t s t h e o r i q u e s s u r l e s p a r t i c u l a r i t e s de l a resonance parametrique pour l e s B l e c t r o n s 5 l a s u r f a c e de 11h61ium 1iqui.de.
A b s t r a c t . - T h e o r e t i c a l r e s u l t s a r e p r e s e n t e d on i n v e s t i g a t i o n of p e c u l i a r f e a t u r e s of p a r a m e t r i c re- sonance f o r e l e c t r o n s a t t h e l i q u i d helium s u r f a c e .
A unique f e a t u r e of a system of e l e c t r o n s lo- reduces e q u a t i o n (3) t o t h e u s u a l form f o r t h e para- c a l i z e d a t t h e l i q u i d helium s u r f a c e i s t h e weakness m e t r i c resonance t h e o r y / I /
of i t s i n t e r a c t i o n w i t h t h e l i q u i d phase. Such a system i s a s u i t a b l e o b j e c t f o r o b s e r v a t i o n of d i f - f e r e n t s o p h i s t i c a t e d c l a s s i c a l and quantum e f f e c t s . One of t h e s e e f f e c t s is parametric e x c i t a t i o n of e l e c t r o n s i n a magnetic f i e l d . The q u e s t i o n t o be
t r e a t e d concerns t h e behaviour of a charged p a r t i - c l e i n a magnetic f i e l d H + h ( t ) where H i s a cons- t a n t f i e l d normal t o t h e helium s u r f a c e , h ( t ) i s an o s c i l l a t i n g a d d i t i o n a l f i e l d of low amplitude which i s c o l l i n e a r w i t h H. Under these c o n d i t i o n s t h e c y c l o t r o n frequency of s u r f a c e e l e c t r o n s i s a p e r i o d i c a l f u n c t i o n of time and t h e s i t u a t i o n i s s i - " Q2
5
+ - $ ( I + 2€i0 cos wt)5
= 0 ( 4 4 According t o (4a) p a r a m e t r i c e x c i t a t i o n of e l e c t r o n s t a k e s p l a c e f o r t h e f r e q u e n c i e s wn = Q o / n n = 1 , 2, 3,...
(5) Expressions (3)-(5) a r e w e l l known, s e e f o r i n s t a n c e book by Lenert / 2 / .2.- Let us change t h e e l e c t r i c f i e l d symmetry, name- l y , l e t t h e f i e l d be c r e a t e d i n a r e c t a n g u l a r s o l e - n o i d of a c r o s s e c t i o n a x b , a >> b. Under t h e s e c o n d i t i o n s a c u r l e l e c t r i c f i e l d can be t r e a t e d a s one-dimensional m i l a r t o t h a t a r i s i n g i n problems of p a r a m e t r i c ins- E
= - h ' x
-
-
b < X < - b t a b i l i t y / I / Y c 2 - - 2 eH I n t h i s c a s e e q u a t i o n s of e l e c t r o n motion a r e g i v e n a c t ) = n o ( ,+
61,no
=-
mc' 6= h
H << I , h = h o coswt bv ( 1 )Here e and m a r e t h e charge and t h e mass of a f r e e e l e c t r o n , c is t h e v e l o c i t y . o f l i g h t , w i s t h e os-
+
+
c i l l a t i o n frequency of h ( t ) , H =
I H I
and h = I h l .1 . For t h e following d i s c u s s i o n a c u r l e l e c t r i c f i e l d E i s e s s e n t i a l . We suppose t h a t t h i s f i e l d is e x c i t e d by the f i e l d h ( t ) i n t h e volume of a long c y l i n d r i c a l s o l e n o i d of r a d i u s R << -f-
.
I n t h i s c a s eW
t h e f i e l d E i n c a r t e s i a n c o o r d i n a t e s i n which t h e OZ a x i s c o i n c i d e s w i t h c y l i n d e r a x i s i s a s f o l l o w s
D e f i n i n g e q u a t i o n s f o r e l e c t r o n motion w i t h a l l o - wance f o r t h e f i e l d E and i n t r o d u c i n g t h e new v a r i a - b l e Z % X + i Y we can f i n d
t h e f o l l o w i n g c o o r d i n a t e t r a n s f o r m a t i o n
i !
. =
5
exp[-
4
I
n(t)dt] ( 4 )It i s easy t o s e e t h a t t h e second of t h e equations (7) can be i n t e g r a t e d by time
mjr =
-
x
(H + h)+
c o n s t (8)S u b s t i t u t i n g eq. (8) i n t h d f i r s t of t h e e q u a t i o n s (7) and r e t a i n i n g t h e homogeneous p a r t of t h e equa- t i o n a r i s i n g we o b t a i n
x +
n;
(1+
6 ) "x =
0 (9)This e q u a t i o n i s u n s t a b l e f o r f r e q u e n c i e s
w = 2Qo/n n = 1 , 2, 3 , . . .
n (1 0)
The d i f f e r e n c e between e x p r e s s i o n s (5) and (10) i s q u i t e e x p l i c i t . 3.- When a f r i c t i o n t a k e s p l a c e , p a r a m e t r i c e x c i t a - t i o n of a system u s u a l l y a c q u i r e s t h r e s h o l d proper- t i e s . I n t h e c a s e of e l e c t r o n s l o c a l i z e d a t t h e he- lium s u r f a c e t h e e x i s t e n c e of t h i s t h r e s h o l d i s ma- n i f e s t e d i n a v e r y p e c u l i a r way. Let us t r e a t t h e c a s e of low c o n c e n t r a t i o n of s u r f a c e e l e c t r o n s when
'Fne e l e c t r o n - e l e c t r o n i n t e r a c t i o n i s n e g l i g i b l e . I n t h i s case t h e odd p a r t of t h e e l e c t r o n d i s t r i b u t i o n f u n c t i o n i s found t o be e q u a l t o Here v i s t h e e l e c t r o n random v e l o c i t y a l o n g t h e helium s u r f a c e , f i s t h e s p h e r i c a l p a r t of t h e d i s - 0 t r i b u t i o n f u n c t i o n , u i s t h e e l e c t r o n d r i f t v e l o c i t y which can be found o u t from t h e s o l u t i o n of t h e c l a - s s i c a l e q u a t i o n of e l e c t r o n motion i n a magnetic f i e l d w i t h allowance f o r e l e c t r o n - r i p p l o n i n t e r a c - t i o n by t h e term v ( v ) u where v i s t h e e f f e c t i v e e l e c - t r o n - r i p p l o n c o l l i s i o n frequency depending on v. The s o l u t i o n of t h i s e q u a t i o n f o r an e l e c t r o n moving i n a c y l i n d r i c a l l y synrmetric f i e l d i n t h e v i c i n i t y of one of t h e resonance f r e q u e n c i e s ( f o r i n s t a n c e
w l =
R o ) l e a d s t o t h e following c h a r a c t e r i s t i c time de- pendence f o r u ( t l v ) e exp
[+
-
v ( v i l t I n s t a b i l i t y a r i s e s i n (12) under c o n d i t i o n 6 0 3 0 / 4 > v ( v ) (13) It should be n o t e d t h a t t h e f u n c t i o n v ( v ) f o r s u r - f a c e e l e c t r o n s i s l i m i t e d i n a r e g i o n of low v a l u e s by v = v min. P o s i t i o n and numerical v a l u e of vmin can be changed by v a r y i n g t h e p r e s s i n g e l e c t r i c f i e l d E It i s e v i d e n t t h a t i n e q u a l i t y (13) holds
1'
f i r s t of a l l f o r v = vmin. Hence, measurements of t h e p a r a m e t r i c i n s t a b i l i t y t h r e s h o l d f o r t h e s u r f a c e e l e c t r o n s g i v e s us some i n f o r m a t i o n about the nume- r i c a l v a l u e of v min' E x p l i c i t e x p r e s s i o n f o r v(v) was o b t a i n e d i n t h e l i m i t i n g case w-r >> 1 i n r e f .
131.
Here a and E a r e t h e s u r f a c e t e n s i o n c o e f f i c i e n t and t h e d i e l e c t r i c c o n s t a n t of l i q u i d helium. The mini- mum v a l u e of pin eq. ( 1 4) f o r weak p r e s s i n g f i e l d s i s ound t o be i n t h e p o i n t
c o n d i t i o n 6 >> 1 . The v a l u e of H i s t o be of t h e
0
o r d e r of 10' Oe.
References
/ I / Landau, L.D., Lif s h i t z , E.M., Mechanics, (Perga- mon) 1969, p. 80 1958, c t p . 103.
121 L e n e r t , B., Dynamics of charged p a r t i c l e s , Ato- mizdat, Moscow 1967. c t p . 51
/3/ Platzman, P.M. and Beni, G., Phys. Rev. L e t t . 36 (1976), 626.
-
For t h e f i e l d s t r e n g t h El = 1-lOv/cm and T