ANOMALOUS SKIN EFFECT IN CYLINDERS

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ANOMALOUS SKIN EFFECT IN CYLINDERS

L.J.M. van de Klundert, H.P. van de Braak

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C6, supplPmenr au no 8, Tome 39, aolit 1978, page

_{~ 6 -}

_{1 133}

kNOMALOUS S K I N E F F E C T I N C Y L I N D E R S

L.J.M. Van de Klundert and H.P. Van d e Braak

ZZjente University

of Technology, P.O.Box

217,

Enschede, The NetherZands

Rdsurn6.- La rdponse d'un conducteur c y l i n d r i q u e B un champ magndtique s i n u s o i d a l p a r a l l s l e

B l ' a x e peut s ' o b t e n i r 1 p a r t i r des composantes azimuthales d e s champs Q l e c t r i q u e s ou d e s c o u r a n t s i n d u i t s

.

@and il d e v i e n t n d c e s s a i r e d e t e n i r compte d e s e f f e t s de l i b r e p a r c o u r s moyen d e s d l e c t r o n s , l e s v a I e u r s de c e s champs ou c o u r a n t s s o n t ddterminQs par d e s dquations i n t d g r a l e s . On prG- s e n t e une mdthode de s o l u t i o n numdrique pour c e s 6 q u a t i o n s i n t C g r a l e s .

A b s t r a c t . - The induced response of a c y l i n d r i c a l conductor due t o a s i n u s o i d a l magnetic f i e l d p a r a l l e l t o t h e a x i s of t h e c y l i n d e r can b e obtained from t h e azimuthal components of t h e induced e l e c t r i c a l f i e l d s o r c u r r e n t s .

I n case mean f r e e p a t h e f f e c t s of t h e e l e c t r o n s have t o b e considered t h e v a l u e s of t h e s e e l e c t r i c a l f i e l d s o r c u r r e n t s a r e governed by i n t e g r a l e q u a t i o n s . A numerical s o l u t i o n method f o r t h e s e i n t e g r a l e q u a t i o n s i s p r e s e n t e d .

The s u s c e p t i b i l i t y of an i n f i n i t e rod of a conductor i n p a r a l l e l s i n u s o i d a l magnetic f i e l d

i s given by

X

= Jz(ka)/Jo(Ka),

J o and J 2 a r e Bessel f u n c t i o n s and k 2

=

iwpoo

,

o i s t h e c o n d u c t i v i t y , a i s t h e sample r a d i u s . This r e l a t i o n i s only v a l i d i f a l o c a l c o n s t i t u t i v e e q u a t i o n l i k e

j

= oE e x i s t s , t h a t i s when mean f r e e p a t h e f f e c t s can be n e g l e c t e d .

The i n f l u e n c e of t h e mean f r e e p a t h h beco- mes important whenever A

2

lk-I

1

o r A _> a.

The non-local c o n s t i t u t i v e e q u a t i o n t h e n r e a d s _a

j ( r ) = o

6

F ( r , r l , k , p ) E ( r 1 ) d r 7 ,

F i s a d i s t r i b u t i o n f u n c t i o n t o be determined from t h e Boltzmann e q u a t i o n and p accounts f o r t h e mode of r e f l e c t i o n s of t h e e l e c t r o n s a t t h e s u r f a c e of t h e c y l i n d e r ; p = 1 means s p e c u l a r r e f l e c t i o n , p = 0 d i f f u s e r e f l e c t i o n . Independently of t h e n a t u r e of t h e c o n s t i t u - t i v e e q u a t i o n j = j (E) t h e apparent s u s c e p t i b i l i t y

X

can b e w r i t t e n i n terms of ,T o r E a s f o l l o w s ( 1 ) :

X

= {E(a)

-

E T ( a ) ] / { E ( a ) + ~ ' ( a ) ) w i t h E ' ( a ) = = a-'

b2

J ( r ) d r , w i t h J ( r ) = p O j ( r ) / B a

,

'0 Ba i s t h e amplitude of t h e a p p l i e d f i e l d .

The azimuthal components E of t h e e l e c t r i - c a l f i e l d s t r e n g t h and t h e reduced c u r r e n t d e n s i t y J obey t h e f o l l o w i n g i n t e g r a l e q u a t i o n s : dE E ) = r - d r r = o

-

k 2 dy E(y) G(r,y)

6'

I"

w i t h G(r,y) : d x ( r 2

-

X ~ ) F ( X , ~ ) / Z ~ , 1 J ( r ) = - i . k 2 H ( r , a ) + i k 2

I'

J ( x ) H ( r , x ) d x , According t o t h e non l i n e a r c o n s t i t u t i v e e q u a t i o n above, F ( r , r T ) h a s t o be determined from Boltz- mann's t r a n s p o r t e q u a t i o n . This p a r t of t h e problem has a l r e a d y been solved a n a l y t i c a l l y i n a p r e v i o u s paper / l / . A n a l y t i c s o l u t i o n s of t h e i n t e g r a l equa- t i o n s have n o t been o b t a i n e d so f a r . It may be noted t h a t by p u t t i n g F ( r , r f ) = 6 ( r

-

r ' ) t h e nor- mal s k i n e f f e c t e x p r e s s i o n f o r

X

i s r e a d i l y o b t a i - ned from t h e . s o l u t i o n s of t h e i n t e g r a l equations.

Since b o t h t h e r e a l and t h e imaginary p a r t s of E and J have t o be determined and k 2 i s a p u r e l y imaginary number i t i s convenient t o s u b s t i t u t e E(y) i t s v a l u e given by t h e i n t e g r a l e q u a t i o n . For t h e r e a l and imaginary p a r t s of E we then o b t a i n :

fa fa

E ( r ) =

r

+

k 4 d. E=(x)

j

dy ~ ( r , y ) ~ ( y , x )

,

E i ( r ) = k 2

1

y G(r,y)dy + k v

i"

dx E ~ ( x )

I'

dy P u t t i n g (dE/dr)r=o= 1 does n o t a f f e c t t h e r e s u l t s . The i n t e g r a l s a r e now approximated by a 40 p o i n t s Gauss Legendre sum formula and Er and Ei c a n be found by s t a n d a r d s o l u t i o n methods. Analogous re- s u l t s hold f o r t h e i n t e g r a l e q u a t i o n f o r t h e reduced c u r r e n t J ( r )

.

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F i n a l l y t h e r e s u l t s f o r

X

d e r i v e d e i t h e r way a g r e e d f a i r l y w e l l i n a wide range of k and

R e f e r e n c e s

h / a v a l u e s . Only f o r k v a l u e s w i t h k > l00 toge- _{/ l /} _{Van d e Braak, H.P., and Van d e Klundert,L.J.M.,}

t h e r w i t h small X/a v a l u e s , X/a c 0.1 rounding P h y s i c a

77

(1974) 5 3 2 .

e r r o r s a f f e c t t h e r e s u l t s . / 2 / D i n g l e , R.B., P h y s i c a

19

(1953) 3 1 1 .

Good agreement was a l s o found w i t h Dingle's / 3 / L y a l l , K.X. and Cochran, J . F . , Phys. Rev.

159

r e s u l t s f o r t h e anomalous s k i n e f f e c t a t f l a t bouG (1967) 5 1 7 .

/ 4 / Van d e K l u n d e r t , L.J.M., G i j s b e r t s e , E.A. and d a r i e s a l t h o u g h r e c a l c u l a t i o n o f h i s r e s u l t s seems _{Van d e r Marel, L.C. Phys. L e t t e r s}

_g

₍₁₉₇₂₎ n e c e s s a r y because o f i n t e r n a l i n c o n s i s t a n c i e s / 2 / . 373.

The comparison between e x p e r i m e n t s / 3 , 4 /

and t h e p r e s e n t r e s u l t s w i l l a l s o have t o b e r e i n - t e r p r e t a t e d .