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Oscillatory magnetoacoustic attenuation
Volume 22, n u m b e r 5
PHYSICS
LETTERS
15 S e p t e m b e r 1966
(1965) 1685. 12. J . M . Ziman, Advances in P h y s i c s 13 (1964) 89. 13. C . P . Flynn, J.Appl. Phys. 35 (1964) 1641.
10. E . F . W . Seymour and G.A. Styles, P h y s i c s L e t t e r s 10 (1964) 269. 11. R. L. Odle and C. P. Flynn, J. Phys. Chem. Solids 26 $$$$$
OSCILLATORY
MAGNETOACOUSTIC
ATTENUATION
L. F L A X Lewis Research Center, National Aeronautics and Space Administration Cleveland, Ohio and J. TRIVISONNO * John Carroll University, Cleveland, Ohio Received 1 August 1966
E x p r e s s i o n s for attenuation coefficients, suitable for n u m e r i c a l evaluation and valid for all ql, have been developed f r o m the t h e o r y of Cohen, H a r r i s o n and H a r r i s o n . A considerable reduction in the complexity of the equations m a k e s it f e a s i b l e to r e t a i n all o r d e r s in the s e r i e s . Shifts in the e x t r e m a for the m a g n e toacoustic oscillations a r e obtained.
S e v e r a l t h e o r i e s of u l t r a s o n i c a t t e n u a t i o n in m e t a l s h a v e b e e n g i v e n f o r t h e c a s e of f r e e e l e c t r o n s in a m a g n e t i c f i e l d a n d i n z e r o m a g n e t i c field. These theories have demonstrated that the a t t e n u a t i o n of u l t r a s o n i c w a v e s d e p e n d s o n t h e p r o d u c t of t h e w a v e n u m b e r q of t h e u l t r a s o n i c w a v e a n d t h e e l e c t r o n ' s m e a n f r e e p a t h 1. F o r t h e z e r o f i e l d c a s e t h e t h e o r i e s b y P i p p a r d [1] a n d M o r s e [2] a r e in a g r e e m e n t . C o h e n e t a l . [3] d e v e l o p e d a g e n e r a l t h e o r y of t h e m a g n e t i c f i e l d d e p e n d e n c e of u l t r a s o n i c a t t e n u a t i o n f o r ql >> 1, a n d f o u n d a n o s c i l l a t o r y d e p e n d e n c e on f i e l d strength. Ref.4 gave theoretical results for int e r m e d i a t e v a l u e s of ql. T h e p r e s e n t n o t e p r e s e n t s a n a p p l i c a t i o n of t h e t h e o r y of C o h e n e t a l . f o r a r b i t r a r y ql v a l u e s f o r t h e c a s e of t h e m a g n e t i c f i e l d (H) p e r p e n d i c u l a r to t h e s o u n d p r o p a g a t i o n v e c t o r a n d i s v a l i d i n b o t h t h e h i g h a n d low f i e l d l i m i t s . T h e p r e s e n t c a l c u l a t i o n p r e d i c t s in t h e v a l u e of m a g n e t i c f i e l d f o r w h i c h t h e a t t e n u a t i o n i s at m i n i m u m f o r i n t e r m e d i a t e ql v a l u e s , a result not mentioned explicitly by either ref. 3 o r 4. The solutions obtained by Cohen et al. for the relative attenuation contained the following rest r i c t i o n s : (1) T h e p r o d u c t of t h e u l t r a s o n i c w a v e * Supported by the A i r F o r c e , AFOSR 862-65
Table 1 R e p r e s e n t a t i v e values of e x t r e m a in the r e l a t i v e a t t e nuation coefficient S l l V a l u e s of X V a l u e s of X Relative Relative ql f o r m a x i m u m for minimum attenuation attenuation attenuation attenuation 50
0
166.6
2.92
4.04
149.9
7.24
11
111.8
6..01
60.19
77.15
9.12
43.49
0
8.06
2.59
6.12
4.00
8.30
5.78
5.06
7.24
5.85
8.93
5.10
n u m b e r a n d t h e g y r o m a g n e t i c r a d i u s of t h e e l e c t r o n d e n o t e d by X = qR, w a s of o r d e r u n i t y [2]; ~Vcv/(1 - iwT) I z >> 1 w h e r e w c a n d w a r e t h e c y c l o t r o n f r e q u e n c y a n d t h e f r e q u e n c y of t h e w a v e , r e s p e c t i v e l y , a n d T t h e r e l a x a t i o n t i m e ; a n d [3] t e r m s of o r d e r 1/(ql) 2 w e r e n e g l e c t e d . B e c a u s e of t h e s e r e s t r i c t i o n s t h e t h e o r y of r e f . 3 i s v a l i d o n l y f o r l a r g e ql. S i n c e m a n y e x p e r i m e n t s do n o t satisfy the above restrictions, the conductivity t e n s o r w a s r e f o r m u l a t e d to i n c l u d e a l l o r d e r s of n and ql ff WT<< 1. The expressions developed by Cohen et al. for t h e c o n d u c t i v i t y t e n s o r a r e a s s u m e d to b e v a l i d 569
Volume 22, number :5 :
P HYSICS LE T T EHS
f o r all ql and qR. v a l u e s if a l l o r d e r s of n : a r e inc l u d e d . T h i s i nfini-t e s e r i e s c o n t a i n s an i n t e g r a l of B e s s e i f u n c t i o n s in e a c h t e r m ; h o w e v e r , the e x p r e s s i o n s : h a v e b e e n found to i n f i n i t e s e r i e s c o n t a i n i n g no i n t e g r a l s , and h e n c e n u m e r i c a l c a l c u l a t i o n s of the q u a n t i t i e s a r e f e a s i b l e f o r i n t e r m e d i a t e ql a s w e l l a s the l i m i t i n g c a s e s ~ The r e s u l t ing r e l a t i v e a t t e n u a t i o n c o e f f i c i e n t s can be w r i t t e n
E x t r e m a in the a t t e n u a t i o n f o r v a r i o u s ql v a l u e s , e v a l u a t e d by c o m p u t e r , a r e g i v e n in t a b l e 1 f o r S l l . The t a b l e s h o w s that f o r i n c r e a s i n g ql v a l u e s , the v a l u e s of x f o r m i n i m u m a t t e n u a t i o n s h i f t h i g h e r . F o r the c a s e of $22 is s h o w n in r e f . 5 to shift in the v a l u e s of x f o r m a x i m u m a t t e n u a tion but t h e s e s h i f t s a r e e x t r e m e l y s m a l l c o m p a r e d to the s h i f t s in the m i n i m u m p o s i t i o n of S 11. The s m a l l e r v a l u e s of ql g i v e n in the t a b l e a r e in the r a n g e a c c e s s i b l e to e x p e r i m e n t s . The v a l u e s f o r ql = 50 a r e in a g r e e m e n t with r e s u l t s of r e f . 3, w h i c h s h o w s that the p r e s e n t r e s u l t s a g r e e with r e f . 3 f o r l a r g e ql. R e c e n t e x p e r i m e n t a l i n v e s t i g a t i o n s by T r i v i sonno et al. [6] on p o t a s s i u m h a v e v e r i f i e d that the s h i f t s in t a b l e 1 do e x i s t . C a l c u l a t i o n s b a s e d on e q s . (4) to (7) a r e in good a g r e e m e n t with the m a g n i t u d e s of the s h i f t s in the m i n i m u m r e p o r t e d by T r i v i s o n n o . The m a g n i t u d e p r e d i c t e d f o r the r e l a t i v e a t t e n u a t i o n a l s o a g r e e s with the e x p e r i m e n t a l r e s u l t s [6].
as [5]: (~)2 [ ~ _ 1 $1I = " ~ Lb + ~ / w
~ -1
(1)
j
1
$22 -
3[w+¼u2/b]-I
(2)
.1
(3)
$33 = (~-~ -
1)
w h e r e S 11 r e p r e s e n t s the r e l a t i v e a t t e n u a t i o n c o efficient for a longitudinal wave moving perpendic u l a r to an a p p l i e d m a g n e t i c f i e l d ; $22 and 833 c o r r e s p o n d to a t r a n s v e r s e w a v e m o v i n g p e r p e n d i c u l a r and p a r a l l e l to an a p p l i e d f i e l d , r e s p e c t i v e l y . E x p r e s s i o n s f o r b, u, v and w, d e r i v e d in r e f . 5, a r e n=
c~
b=-~ n =1
15 September 1966
(-1)nx2n
(4)
(2n +1)[(12 +X2/q212) "'" (n 2 +X2/q212)]
n =°0
(-1 )n2nx2n- 1
(5)
u = ~ (2n + 1) [(12 + X2/q212) (n 2 +X2/q212)] n=l " ""
n=~ w
(-1)n2nx 2n
_ 1 ~-~
= - n=l ~ (2n + l ) ( 2 n +3)[(12 +X2/q212). . .(n 2 +Z2/q212)]
(-1)nx2n
(q/)2n~== 1_ (2n +1)[(12 +Z2/q212)... (n 2 +X2/q212)]
(6) n =~
(- 1 )nx2n
Y = ~ n__~l (2n + 3) (2n + 1)[(12 +X2/q212)... (n 2 +X2/q212)] . W h e n the m a g n e t i c f i e l d is e x t r e m e l y l a r g e the a t t e n u a t i o n t e n d s to a l i m i t in a d i f f e r e n t way f o r e a c h of the Sij c o e f f i c i e n t s . T h i s l i m i t is found s i m p l y f r o m e q s . (4) to (7) by l e t t i n g x to go z e r o as Happroaches infinity. Hence L i m S l l - (ql)2 15
L i m $22 = L i m $33 = 0 .
In the low f i e l d l i m i t the a t t e n u a t i o n c o e f f i c i e n t s a p p r o a c h P i p p a r d ' s r e s u l t [1] w h e n x ~ ~ .
570
(7)
References 1. A. B. Pippard, Phil. Mag. 46 (1955) 1104. 2. R. W. Morse, Phys. Rev. 97 (1955) 1716. 3. M. H. Cohen, J. J. Harrison and W. A. Harrison, Phys. Rev. 117 (1960) 937. 4. T. Kjeldaas Jr. and T. Holstein, Phys. Rev. Letters 2 (1959) 349. 5. L. Flax, Unpublished thesis, John Carroll University, proposed NASA TN. 6. J. Trivisonno, S. Said and V. Pouer, Phys. Rev., to 13e"published.
PHYSICS
LETTERS
15 S e p t e m b e r 1966
(1965) 1685. 12. J . M . Ziman, Advances in P h y s i c s 13 (1964) 89. 13. C . P . Flynn, J.Appl. Phys. 35 (1964) 1641.
10. E . F . W . Seymour and G.A. Styles, P h y s i c s L e t t e r s 10 (1964) 269. 11. R. L. Odle and C. P. Flynn, J. Phys. Chem. Solids 26 $$$$$
OSCILLATORY
MAGNETOACOUSTIC
ATTENUATION
L. F L A X Lewis Research Center, National Aeronautics and Space Administration Cleveland, Ohio and J. TRIVISONNO * John Carroll University, Cleveland, Ohio Received 1 August 1966
E x p r e s s i o n s for attenuation coefficients, suitable for n u m e r i c a l evaluation and valid for all ql, have been developed f r o m the t h e o r y of Cohen, H a r r i s o n and H a r r i s o n . A considerable reduction in the complexity of the equations m a k e s it f e a s i b l e to r e t a i n all o r d e r s in the s e r i e s . Shifts in the e x t r e m a for the m a g n e toacoustic oscillations a r e obtained.
S e v e r a l t h e o r i e s of u l t r a s o n i c a t t e n u a t i o n in m e t a l s h a v e b e e n g i v e n f o r t h e c a s e of f r e e e l e c t r o n s in a m a g n e t i c f i e l d a n d i n z e r o m a g n e t i c field. These theories have demonstrated that the a t t e n u a t i o n of u l t r a s o n i c w a v e s d e p e n d s o n t h e p r o d u c t of t h e w a v e n u m b e r q of t h e u l t r a s o n i c w a v e a n d t h e e l e c t r o n ' s m e a n f r e e p a t h 1. F o r t h e z e r o f i e l d c a s e t h e t h e o r i e s b y P i p p a r d [1] a n d M o r s e [2] a r e in a g r e e m e n t . C o h e n e t a l . [3] d e v e l o p e d a g e n e r a l t h e o r y of t h e m a g n e t i c f i e l d d e p e n d e n c e of u l t r a s o n i c a t t e n u a t i o n f o r ql >> 1, a n d f o u n d a n o s c i l l a t o r y d e p e n d e n c e on f i e l d strength. Ref.4 gave theoretical results for int e r m e d i a t e v a l u e s of ql. T h e p r e s e n t n o t e p r e s e n t s a n a p p l i c a t i o n of t h e t h e o r y of C o h e n e t a l . f o r a r b i t r a r y ql v a l u e s f o r t h e c a s e of t h e m a g n e t i c f i e l d (H) p e r p e n d i c u l a r to t h e s o u n d p r o p a g a t i o n v e c t o r a n d i s v a l i d i n b o t h t h e h i g h a n d low f i e l d l i m i t s . T h e p r e s e n t c a l c u l a t i o n p r e d i c t s in t h e v a l u e of m a g n e t i c f i e l d f o r w h i c h t h e a t t e n u a t i o n i s at m i n i m u m f o r i n t e r m e d i a t e ql v a l u e s , a result not mentioned explicitly by either ref. 3 o r 4. The solutions obtained by Cohen et al. for the relative attenuation contained the following rest r i c t i o n s : (1) T h e p r o d u c t of t h e u l t r a s o n i c w a v e * Supported by the A i r F o r c e , AFOSR 862-65
Table 1 R e p r e s e n t a t i v e values of e x t r e m a in the r e l a t i v e a t t e nuation coefficient S l l V a l u e s of X V a l u e s of X Relative Relative ql f o r m a x i m u m for minimum attenuation attenuation attenuation attenuation 50
0
166.6
2.92
4.04
149.9
7.24
11
111.8
6..01
60.19
77.15
9.12
43.49
0
8.06
2.59
6.12
4.00
8.30
5.78
5.06
7.24
5.85
8.93
5.10
n u m b e r a n d t h e g y r o m a g n e t i c r a d i u s of t h e e l e c t r o n d e n o t e d by X = qR, w a s of o r d e r u n i t y [2]; ~Vcv/(1 - iwT) I z >> 1 w h e r e w c a n d w a r e t h e c y c l o t r o n f r e q u e n c y a n d t h e f r e q u e n c y of t h e w a v e , r e s p e c t i v e l y , a n d T t h e r e l a x a t i o n t i m e ; a n d [3] t e r m s of o r d e r 1/(ql) 2 w e r e n e g l e c t e d . B e c a u s e of t h e s e r e s t r i c t i o n s t h e t h e o r y of r e f . 3 i s v a l i d o n l y f o r l a r g e ql. S i n c e m a n y e x p e r i m e n t s do n o t satisfy the above restrictions, the conductivity t e n s o r w a s r e f o r m u l a t e d to i n c l u d e a l l o r d e r s of n and ql ff WT<< 1. The expressions developed by Cohen et al. for t h e c o n d u c t i v i t y t e n s o r a r e a s s u m e d to b e v a l i d 569
Volume 22, number :5 :
P HYSICS LE T T EHS
f o r all ql and qR. v a l u e s if a l l o r d e r s of n : a r e inc l u d e d . T h i s i nfini-t e s e r i e s c o n t a i n s an i n t e g r a l of B e s s e i f u n c t i o n s in e a c h t e r m ; h o w e v e r , the e x p r e s s i o n s : h a v e b e e n found to i n f i n i t e s e r i e s c o n t a i n i n g no i n t e g r a l s , and h e n c e n u m e r i c a l c a l c u l a t i o n s of the q u a n t i t i e s a r e f e a s i b l e f o r i n t e r m e d i a t e ql a s w e l l a s the l i m i t i n g c a s e s ~ The r e s u l t ing r e l a t i v e a t t e n u a t i o n c o e f f i c i e n t s can be w r i t t e n
E x t r e m a in the a t t e n u a t i o n f o r v a r i o u s ql v a l u e s , e v a l u a t e d by c o m p u t e r , a r e g i v e n in t a b l e 1 f o r S l l . The t a b l e s h o w s that f o r i n c r e a s i n g ql v a l u e s , the v a l u e s of x f o r m i n i m u m a t t e n u a t i o n s h i f t h i g h e r . F o r the c a s e of $22 is s h o w n in r e f . 5 to shift in the v a l u e s of x f o r m a x i m u m a t t e n u a tion but t h e s e s h i f t s a r e e x t r e m e l y s m a l l c o m p a r e d to the s h i f t s in the m i n i m u m p o s i t i o n of S 11. The s m a l l e r v a l u e s of ql g i v e n in the t a b l e a r e in the r a n g e a c c e s s i b l e to e x p e r i m e n t s . The v a l u e s f o r ql = 50 a r e in a g r e e m e n t with r e s u l t s of r e f . 3, w h i c h s h o w s that the p r e s e n t r e s u l t s a g r e e with r e f . 3 f o r l a r g e ql. R e c e n t e x p e r i m e n t a l i n v e s t i g a t i o n s by T r i v i sonno et al. [6] on p o t a s s i u m h a v e v e r i f i e d that the s h i f t s in t a b l e 1 do e x i s t . C a l c u l a t i o n s b a s e d on e q s . (4) to (7) a r e in good a g r e e m e n t with the m a g n i t u d e s of the s h i f t s in the m i n i m u m r e p o r t e d by T r i v i s o n n o . The m a g n i t u d e p r e d i c t e d f o r the r e l a t i v e a t t e n u a t i o n a l s o a g r e e s with the e x p e r i m e n t a l r e s u l t s [6].
as [5]: (~)2 [ ~ _ 1 $1I = " ~ Lb + ~ / w
~ -1
(1)
j
1
$22 -
3[w+¼u2/b]-I
(2)
.1
(3)
$33 = (~-~ -
1)
w h e r e S 11 r e p r e s e n t s the r e l a t i v e a t t e n u a t i o n c o efficient for a longitudinal wave moving perpendic u l a r to an a p p l i e d m a g n e t i c f i e l d ; $22 and 833 c o r r e s p o n d to a t r a n s v e r s e w a v e m o v i n g p e r p e n d i c u l a r and p a r a l l e l to an a p p l i e d f i e l d , r e s p e c t i v e l y . E x p r e s s i o n s f o r b, u, v and w, d e r i v e d in r e f . 5, a r e n=
c~
b=-~ n =1
15 September 1966
(-1)nx2n
(4)
(2n +1)[(12 +X2/q212) "'" (n 2 +X2/q212)]
n =°0
(-1 )n2nx2n- 1
(5)
u = ~ (2n + 1) [(12 + X2/q212) (n 2 +X2/q212)] n=l " ""
n=~ w
(-1)n2nx 2n
_ 1 ~-~
= - n=l ~ (2n + l ) ( 2 n +3)[(12 +X2/q212). . .(n 2 +Z2/q212)]
(-1)nx2n
(q/)2n~== 1_ (2n +1)[(12 +Z2/q212)... (n 2 +X2/q212)]
(6) n =~
(- 1 )nx2n
Y = ~ n__~l (2n + 3) (2n + 1)[(12 +X2/q212)... (n 2 +X2/q212)] . W h e n the m a g n e t i c f i e l d is e x t r e m e l y l a r g e the a t t e n u a t i o n t e n d s to a l i m i t in a d i f f e r e n t way f o r e a c h of the Sij c o e f f i c i e n t s . T h i s l i m i t is found s i m p l y f r o m e q s . (4) to (7) by l e t t i n g x to go z e r o as Happroaches infinity. Hence L i m S l l - (ql)2 15
L i m $22 = L i m $33 = 0 .
In the low f i e l d l i m i t the a t t e n u a t i o n c o e f f i c i e n t s a p p r o a c h P i p p a r d ' s r e s u l t [1] w h e n x ~ ~ .
570
(7)
References 1. A. B. Pippard, Phil. Mag. 46 (1955) 1104. 2. R. W. Morse, Phys. Rev. 97 (1955) 1716. 3. M. H. Cohen, J. J. Harrison and W. A. Harrison, Phys. Rev. 117 (1960) 937. 4. T. Kjeldaas Jr. and T. Holstein, Phys. Rev. Letters 2 (1959) 349. 5. L. Flax, Unpublished thesis, John Carroll University, proposed NASA TN. 6. J. Trivisonno, S. Said and V. Pouer, Phys. Rev., to 13e"published.