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Magnetoacoustic attenuation in liquid mercury
Volume 20, number 6
PHYSICS LETTERS
1 April 1966
M A G N E T O A C O U S T I C A T T E N U A T I O N IN L I Q U I D M E R C U R Y Y. S H A P I R A
National Magnet Laboratory *, Massachusetts Institute of Technology, Cambridge, Massachusetts Received 2 M a r c h 1966
The attenuation of 6 and 18 M e / s e c sound waves propagating in liquid m e r c u r y was m e a s u r e d in d.c. magnetic fields up to 145 kG. The r e s u l t s a r e in v e r y good a g r e e m e n t with A n d e r s o n ' s theory.
W e r e p o r t t h e f i r s t o b s e r v a t i o n of a c h a n g e i n the ultrasonic attenuation in conducting liquids d u e to a m a g n e t i c f i e l d . A m a g n e t o - h y d r o d y n a m i c m o d e l f o r u l t r a s o n i c p r o p a g a t i o n in c o n d u c t i n g l i q u i d s i n t h e p r e s e n c e of a m a g n e t i c f i e l d w a s d e v e l o p e d b y A n d e r s o n [1] a n d b y A l p h e r a n d R u b i n [2]. T h e b a s i c a s s u m p t i o n s of t h i s m o d e l a r e : (a) T h e e q u a t i o n of m o t i o n of a c o n d u c t i n g l i q u i d i n the p r e s e n c e of a m a g n e t i c f i e l d i s p~
(J X B ) ,
=-VP+
(1)
w h e r e p i s t h e d e n s i t y , v i s t h e v e l o c i t y of t h e liquid, P is the pressure, Jis the current dens i t y , a n d B i s t h e m a g n e t i c i n d u c t i o n . (b) T h e current density J is given by
J = (~[E + 1 (v × B)]
(2)
where E is the electric field associated with the sound wave, and G is the electric conductivity of the liquid. While the original treatments [1, 2] specialized to the case of a transverse magnetic field, it is easy to generalize the results to the case in which the applied magnetic field /-/makes an arbitrary angle 0 with the propagation vector q of the sound wave. Using eqs. (1) and (2) together with Maxwell's equations, one obtains the following relation between the frequency w and the propagation constant q of a longitudinal sound wave ** w2
= ysq22
q2~H2 s i n 2 0 ie2q2/4~w) '
4~p(1 -
(3)
w h e r e Vs is the s o u n d v e l o c i t y at z e r o f i e l d a n d p i s t h e p e r m e a b i l i t y of t h e l i q u i d . In d e r i v i n g eq. (3) we h a v e r e t a i n e d o n l y t h o s e t e r m s w h i c h are linear in the disturbance associated with the s o u n d w a v e . C o n s e q u e n t l y , eq. (3) a p p l i e s o n l y to a s m a l l - a m p l i t u d e s o u n d w a v e .
604
T h e c h a n g e s i n t h e a t t e n u a t i o n a n d v e l o c i t y of t h e s o u n d w a v e c a n b e c a l c u l a t e d f r o m eq. (3). Under the present experimental conditions these c h a n g e s c a n b e o b t a i n e d to a h i g h d e g r e e of a c curacy by making the following approximations: (a) T h e t e r m q2 w h i c h o c c u r s i n t h e d e n o m i n a t o r o n t h e r i g h t s i d e of eq. (3) i s r e p l a c e d b y i t s z e r o f i e l d v a l u e (w/Vs)2. (b) In s o l v i n g f o r q, t e r m s w h i c h a r e p r o p o r t i o n a l to t h e f o u r t h p o w e r of H, or higher, are neglected. Using these approximat i o n s o n e c a n s h o w t h a t t h e a m p l i t u d e of t h e s o u n d w a v e d e c a y s w i t h t h e d i s t a n c e x a s e - ~ x w h e r e *** = (r~2H2/32 s i n 2 ~
2pV.c2(l +f12) cm-1
'
= c2~/4~v2.
(4) (5)
T h e c h a n g e AV s i n t h e s o u n d v e l o c i t y i s g i v e n by
aVs/gs
=
~H 2 sin20 . 8~pV2(1 +/32)
(6)
* Supported by the U . S . A i r Office of Scientific Research. ** It can be shown that in addition to the longitudinal mode t h e r e also exist two t r a n s v e r s e modes. In liquid m e r c u r y , at frequencies above 5 M c / s e c and at fields below 2 × 105 gauss, the t r a n s v e r s e waves decay exponentially over a c h a r a c t e r i s t i c distance which is c o m p a r a b l e to the c l a s s i c a l skin depth. In g e n e r a l , the longitudinal mode and one of the s h e a r modes a r e coupled to each other. Howe v e r , the effects of this mode coupling on the d i s p e r s i o n relation, eq, (3), for the (nearly) longitudinal sound wave a r e negligibly s m a l l under the p r e s e n t e x p e r i m e n t a l conditions. *** A n d e r s o n ' s theory neglects the attenuation at zero magnetic field. The coefficient (x r e p r e s e n t s t h e r e fore the change in the attenuation caused by the magnetic field.
Volume 20, number 6
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P H Y S I C S L E TT E RS
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Fig. 1. Change in the ultrasonic attenuation of 6 and 18 Mc/sec longitudinal waves in a transverse magnetic field. The solid line has been calculated on the basis of eq. (4). F o r liquid m e r c u r y , at f r e q u e n c i e s above 5 M c / s e c , ~2 > 100. Consequently, the attenuation c o e f f i c i e n t ~ should be v i r t u a l l y f r e q u e n c y independent in this r a n g e . The u l t r a s o n i c attenuation of longitudinal sound w a v e s p r o p a g a t i n g in t r i p l e - d i s t i l l e d liquid m e r c u r y was m e a s u r e d in d.c. m a g n e t i c f i e l d s up to 145 kG. The m e r c u r y was contained in a c y l i n d r i c a l cavity, ~ 0.5 in. long and ~ 0.5 in. in d i a m e t e r , whose w a l l s w e r e made of p l e x i g l a s s . The p l e x i g l a s s s a m p l e h o l d e r was p l a c e d inside the t a i l of a m e t a l D e w a r * which w a s i n s e r t e d into the b o r e of w a t e r - c o o l e d s o l e n o i d c a p a b le of g e n e r a t i n g a m a g n e t i c f i e l d of 150 kG. The t e m p e r a t u r e of the m e r c u r y w a s ~ 10°C. The a t t e n uation m e a s u r e m e n t s w e r e p e r f o r m e d by conv e n t i o n a l p u l s e t ech n i q u e s . The data w e r e taken point by point, i n c r e a s i n g the m a g u e U c f i e l d f r o m z e r o to a p r e d e t e r m i n e d value, m e a s u r i n g the change in the attenuation, and then d e c r e a s i n g the f i e l d back to z e r o . This m e t h o d r e d u c e s subs t a n U a l l y the e r r o r s c a u s e d by the d r i f t of the i n s t r u m e n t s . The a c c u r a c y of the a t te n u a t io n m e a s u r e m e n t s is * 5%. The attenuation of 6 and 18 M c / s e c longitudinal w a v e s was m e a s u r e d in a t r a n s v e r s e m a g n e t i c * The purpose of the Dewar is to prevent the heating of the mercury when the magnet is turned on.
I , I , I 0.2 0.4 0.6 sin2e
,
I L0.8 I.O
Fig. 2. Angular variation of the attenuation of 6 Mc/sec longitudinal sound waves caused by a magnetic field of 140.8 kG. The solid line has been calculated on the basis of eq. (4). field. The r e s u l t s of t h r e e e x p e r i m e n t a l r u n s a r e r e p r e s e n t e d in fig. 1 by t r i a n g l e s , s q u a r e s and c i r c l e s . Also shown in this f i g u r e a r e theor e t i c a l c u r v e s which w e r e c a l c u l a t e d using eq. (4). In t h ese c a l c u l a t i o n s we used the m e a s u r e d sound v e l o c i t y Vs = 1~45 × 105 c m / s e c , and the v a l u e s [3] 13.57 g / c m ~ and 94.9 ~ 2 c m for the density and e l e c t r i c a l r e s i s t i v i t y r e s p e c t i v e l y . The p e r m e a b i l i t y ~ was s e t equal to unity [3]. As can be s e e n f r o m fig. 1 the a g r e e m e n t b et w een e x p e r i m e n t and t h e o r y is v e r y good. It is noteworthy that the t h e o r e t i c a l c u r v e s do not i n v o l v e any adjustable parameters. The a n g u l a r v a r i a t i o n of the attenuation c a u s e d by a m a g n e t i c f i el d of 140.8 kG was m e a s u r e d u si n g a 6 Mc longitudinal sound wave. The r e s u l t s a r e shown in fig. 2 t o g e t h e r with the p r e d i c t i o n s of eq. (4). It is c l e a r that the t h e o r y g i v e s a good acco u n t of the an g u l ar dependence of the attenuation. The data of fig. 2 a l s o indicate that the r e s u l t s shown in fig. 1 a r e not due to heating of the liquid m e r c u r y which m i g h t have b e e n caused, f o r e x a m p l e , by eddy c u r r e n t s induced by the changing m a g n e t i c field. Such heating e f f e c t s should not be s e n s i t i v e to the angle ~ b e t w e e n / - / and q. The p r e s e n t r e s u l t s , t o g e t h e r with r e c e n t m e a s u r e m e n t s [4, 5] of the u l t r a s o n i c attenuation and v e l o c i t y in s e v e r a l i m p u r e solid m e t a l s and in the m i x e d st at e of Nb-25%Zr, s u g g e s t that the m a g n e t o - h y d r o d y n a m i c m o d e l [1, 21 a p p l i e s to 605
Volume 20, number 6
PHYSICS LETTERS
s e v e r a l b r o a d c l a s s e s of m a t e r i a l s viz. conducting liquids, s o l i d m e t a l s with s h o r t e l e c t r o n i c m e a n f r e e path, and s o m e h i g h - f i e l d s u p e r c o n d u c t o r s .
References 1. N.S. Anderson, J. Acoust. Soc. Am. 25 (1953) 529. Some of the numerical results in this reference are in e r r o r [2]. $~$*$
MAGNETOACOUSTIC
DENSITY
1 April 1966
2. R.A. Alpher and R. J. Rubin, J. Acoust. Soc. Am. 26 (1954) 452. 3. Handbook of chemistry and physics (Chemical Rubber Co., Cleveland, Ohio, 44th edition, 1962). 4. Y. Shapira and L. J. Neuringer, Phys. Rev. Letters 15 (1965) 724; erratum 15 (1965) 873; Y. Shapira and L. J. Neuringer, Physics Letters 20 (1966) 148. 5. G.A. Alers and P. A. Fleury, Phys. Rev. 129 (1963) 2425.
OF
STATES
RESONANCE
Y . ECKSTEIN
Argonne National Laboratory, Argonne, Illinois Received 5 March 1966
A magnetoacoustic resonance in antimony is described. It is due to a singularity in density of states. This resonance suggest that the F e r m i surface of antimony deviates from an ellipsoid.
In a p r e v i o u s l e t t e r [1] we d e s c r i b e d a m a g n e t o a c o u s t i c effect in the oblique g e o m e t r y , i.e. the m a g n e t i c f i el d not p e r p e n d i c u l a r to the wave v e c t o r of the sound. It w a s shown t h e r e that b e s i d e s the u s u al g e o m e t r i c r e s o n a n c e o s c i l l a t i o n s it i s a l s o p o s s i b l e to o b s e r v e a b s o r p t i o n e d g e s f r o m which v a l u a b l e i n f o r m a t i o n about the l i m i t ing points of the F e r m i s u r f a c e can be obtained. In this l e t t e r we d e s c r i b e y e t a n o t h e r effect which was o b s e r v e d in a n ti m o n y . T h i s i s a r e s o nance which cannot o c c u r f o r an e l l i p s o i d a l F e r m i s u r f a c e , and i s t h e r e f o r e additional e v i d e n c e that the F e r m i s u r f a c e of antimony i s not e l l i p s o i d a l [2,3]. In this e x p e r i m e n t 467 m c / s e c longitudinal sound wav es w e r e p r o p a g a t e d along the t r i g o n a l a x i s of a single c r y s t a l of antimony. T h e m a g n e t i c fi el d was r o t a t e d in the b i s e c t r i x - t r i g o n a l plane. At a r a n g e of a n g l e s between 21 ° and 27 ° f r o m t r i g o n a l a x i s , a v e r y s h a r p r e s o n a n c e and it s s u b h a r m o n i c s w e r e o b s e r v e d . T h e e x p e r i m e n t a l t r a c e s a r e shown in fig. 1. At 23 °, w h e r e the r e s o n a n c e i s s h a r p e s t , the line i s split into two peaks. B e c a u s e of the shape of the peaks, and e s p e c i a l l y b e c a u s e the p e a k s a p p e a r only f o r a v e r y n a r r o w r a n g e of a n g l e s , we b e l i e v e that * Based on work performed under the auspices of the U.S. Atomic Energy Commission. 606
t h e s e p e a k s a r e not the a b s o r p t i o n ed g es d i s c u s s e d in r e f . 1, which a l w a y s a p p e a r e d o v e r a v e r y l a r g e r a n g e of a n g l e s . We s u g g e s t that t h e s e p e a k s a r e r e s o n a n c e s of the type d i s c u s s e d by K a n e r et al. [4] and S. G. E c k s t e i n [5]. As d i s c u s s e d in r e f . 1, the e l e c t r o n s which c o n t r i b u t e to the attenuation a r e t h o s e which fulfill the D o p p l e r - s h i f t e d c y c l o t r o n r e s o n a n c e condition: + n w c = w(1 +
(1)
w h e r e w c = eH/m*c, ¢0/2w i s the sound f r e quency, v s the v e l o c i t y of sound, and< VH> the a v e r a g e d r i f t v e l o c i t y in the m a g n e t i c field d i r e c t i o n . F o r s i m p l i c i t y n e g l e c t 1 in c o m p a r i s o n with VF/COS ely s. Equation (1) can then be w r i t t e n in the f o r m
lcw l c {aA~ H = n e v s rn* = n e q ~ H J , . (2) w h e r e q = W/Vs, A is the c r o s s s e c u o n a l a r e a of the F e r m i s u r f a c e p e r p e n d i c u l a r to the m a g n et i c field, and PH is the e l e c t r o n m o m e n t u m in the f i el d d i r e c t i o n . The well-known r e l a t i o n = - ~A/rn*~p H was used. The r e s o n a n c e d e s c r i b e d h e r e o c c u r s when ~A/aPH is not a monotonic function of PH, but r a t h e r has a r e l a t i v e m a x i m u m in the i n t e r v a l 0
PHYSICS LETTERS
1 April 1966
M A G N E T O A C O U S T I C A T T E N U A T I O N IN L I Q U I D M E R C U R Y Y. S H A P I R A
National Magnet Laboratory *, Massachusetts Institute of Technology, Cambridge, Massachusetts Received 2 M a r c h 1966
The attenuation of 6 and 18 M e / s e c sound waves propagating in liquid m e r c u r y was m e a s u r e d in d.c. magnetic fields up to 145 kG. The r e s u l t s a r e in v e r y good a g r e e m e n t with A n d e r s o n ' s theory.
W e r e p o r t t h e f i r s t o b s e r v a t i o n of a c h a n g e i n the ultrasonic attenuation in conducting liquids d u e to a m a g n e t i c f i e l d . A m a g n e t o - h y d r o d y n a m i c m o d e l f o r u l t r a s o n i c p r o p a g a t i o n in c o n d u c t i n g l i q u i d s i n t h e p r e s e n c e of a m a g n e t i c f i e l d w a s d e v e l o p e d b y A n d e r s o n [1] a n d b y A l p h e r a n d R u b i n [2]. T h e b a s i c a s s u m p t i o n s of t h i s m o d e l a r e : (a) T h e e q u a t i o n of m o t i o n of a c o n d u c t i n g l i q u i d i n the p r e s e n c e of a m a g n e t i c f i e l d i s p~
(J X B ) ,
=-VP+
(1)
w h e r e p i s t h e d e n s i t y , v i s t h e v e l o c i t y of t h e liquid, P is the pressure, Jis the current dens i t y , a n d B i s t h e m a g n e t i c i n d u c t i o n . (b) T h e current density J is given by
J = (~[E + 1 (v × B)]
(2)
where E is the electric field associated with the sound wave, and G is the electric conductivity of the liquid. While the original treatments [1, 2] specialized to the case of a transverse magnetic field, it is easy to generalize the results to the case in which the applied magnetic field /-/makes an arbitrary angle 0 with the propagation vector q of the sound wave. Using eqs. (1) and (2) together with Maxwell's equations, one obtains the following relation between the frequency w and the propagation constant q of a longitudinal sound wave ** w2
= ysq22
q2~H2 s i n 2 0 ie2q2/4~w) '
4~p(1 -
(3)
w h e r e Vs is the s o u n d v e l o c i t y at z e r o f i e l d a n d p i s t h e p e r m e a b i l i t y of t h e l i q u i d . In d e r i v i n g eq. (3) we h a v e r e t a i n e d o n l y t h o s e t e r m s w h i c h are linear in the disturbance associated with the s o u n d w a v e . C o n s e q u e n t l y , eq. (3) a p p l i e s o n l y to a s m a l l - a m p l i t u d e s o u n d w a v e .
604
T h e c h a n g e s i n t h e a t t e n u a t i o n a n d v e l o c i t y of t h e s o u n d w a v e c a n b e c a l c u l a t e d f r o m eq. (3). Under the present experimental conditions these c h a n g e s c a n b e o b t a i n e d to a h i g h d e g r e e of a c curacy by making the following approximations: (a) T h e t e r m q2 w h i c h o c c u r s i n t h e d e n o m i n a t o r o n t h e r i g h t s i d e of eq. (3) i s r e p l a c e d b y i t s z e r o f i e l d v a l u e (w/Vs)2. (b) In s o l v i n g f o r q, t e r m s w h i c h a r e p r o p o r t i o n a l to t h e f o u r t h p o w e r of H, or higher, are neglected. Using these approximat i o n s o n e c a n s h o w t h a t t h e a m p l i t u d e of t h e s o u n d w a v e d e c a y s w i t h t h e d i s t a n c e x a s e - ~ x w h e r e *** = (r~2H2/32 s i n 2 ~
2pV.c2(l +f12) cm-1
'
= c2~/4~v2.
(4) (5)
T h e c h a n g e AV s i n t h e s o u n d v e l o c i t y i s g i v e n by
aVs/gs
=
~H 2 sin20 . 8~pV2(1 +/32)
(6)
* Supported by the U . S . A i r Office of Scientific Research. ** It can be shown that in addition to the longitudinal mode t h e r e also exist two t r a n s v e r s e modes. In liquid m e r c u r y , at frequencies above 5 M c / s e c and at fields below 2 × 105 gauss, the t r a n s v e r s e waves decay exponentially over a c h a r a c t e r i s t i c distance which is c o m p a r a b l e to the c l a s s i c a l skin depth. In g e n e r a l , the longitudinal mode and one of the s h e a r modes a r e coupled to each other. Howe v e r , the effects of this mode coupling on the d i s p e r s i o n relation, eq, (3), for the (nearly) longitudinal sound wave a r e negligibly s m a l l under the p r e s e n t e x p e r i m e n t a l conditions. *** A n d e r s o n ' s theory neglects the attenuation at zero magnetic field. The coefficient (x r e p r e s e n t s t h e r e fore the change in the attenuation caused by the magnetic field.
Volume 20, number 6
I
P H Y S I C S L E TT E RS
I
0.5-
I
I
I
I
LIQUID MERCURY
0 . 4 --
18 Me, qJ-H =0o E X a E R I M E N T --THEORY
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oZ=J ~,~ " ./,~p/
~0.22 ~- 0 . 1 -
,,, I-
0.5
0.5
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--
1 April 1966
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' I ' MERCURY
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120
140
Fig. 1. Change in the ultrasonic attenuation of 6 and 18 Mc/sec longitudinal waves in a transverse magnetic field. The solid line has been calculated on the basis of eq. (4). F o r liquid m e r c u r y , at f r e q u e n c i e s above 5 M c / s e c , ~2 > 100. Consequently, the attenuation c o e f f i c i e n t ~ should be v i r t u a l l y f r e q u e n c y independent in this r a n g e . The u l t r a s o n i c attenuation of longitudinal sound w a v e s p r o p a g a t i n g in t r i p l e - d i s t i l l e d liquid m e r c u r y was m e a s u r e d in d.c. m a g n e t i c f i e l d s up to 145 kG. The m e r c u r y was contained in a c y l i n d r i c a l cavity, ~ 0.5 in. long and ~ 0.5 in. in d i a m e t e r , whose w a l l s w e r e made of p l e x i g l a s s . The p l e x i g l a s s s a m p l e h o l d e r was p l a c e d inside the t a i l of a m e t a l D e w a r * which w a s i n s e r t e d into the b o r e of w a t e r - c o o l e d s o l e n o i d c a p a b le of g e n e r a t i n g a m a g n e t i c f i e l d of 150 kG. The t e m p e r a t u r e of the m e r c u r y w a s ~ 10°C. The a t t e n uation m e a s u r e m e n t s w e r e p e r f o r m e d by conv e n t i o n a l p u l s e t ech n i q u e s . The data w e r e taken point by point, i n c r e a s i n g the m a g u e U c f i e l d f r o m z e r o to a p r e d e t e r m i n e d value, m e a s u r i n g the change in the attenuation, and then d e c r e a s i n g the f i e l d back to z e r o . This m e t h o d r e d u c e s subs t a n U a l l y the e r r o r s c a u s e d by the d r i f t of the i n s t r u m e n t s . The a c c u r a c y of the a t te n u a t io n m e a s u r e m e n t s is * 5%. The attenuation of 6 and 18 M c / s e c longitudinal w a v e s was m e a s u r e d in a t r a n s v e r s e m a g n e t i c * The purpose of the Dewar is to prevent the heating of the mercury when the magnet is turned on.
I , I , I 0.2 0.4 0.6 sin2e
,
I L0.8 I.O
Fig. 2. Angular variation of the attenuation of 6 Mc/sec longitudinal sound waves caused by a magnetic field of 140.8 kG. The solid line has been calculated on the basis of eq. (4). field. The r e s u l t s of t h r e e e x p e r i m e n t a l r u n s a r e r e p r e s e n t e d in fig. 1 by t r i a n g l e s , s q u a r e s and c i r c l e s . Also shown in this f i g u r e a r e theor e t i c a l c u r v e s which w e r e c a l c u l a t e d using eq. (4). In t h ese c a l c u l a t i o n s we used the m e a s u r e d sound v e l o c i t y Vs = 1~45 × 105 c m / s e c , and the v a l u e s [3] 13.57 g / c m ~ and 94.9 ~ 2 c m for the density and e l e c t r i c a l r e s i s t i v i t y r e s p e c t i v e l y . The p e r m e a b i l i t y ~ was s e t equal to unity [3]. As can be s e e n f r o m fig. 1 the a g r e e m e n t b et w een e x p e r i m e n t and t h e o r y is v e r y good. It is noteworthy that the t h e o r e t i c a l c u r v e s do not i n v o l v e any adjustable parameters. The a n g u l a r v a r i a t i o n of the attenuation c a u s e d by a m a g n e t i c f i el d of 140.8 kG was m e a s u r e d u si n g a 6 Mc longitudinal sound wave. The r e s u l t s a r e shown in fig. 2 t o g e t h e r with the p r e d i c t i o n s of eq. (4). It is c l e a r that the t h e o r y g i v e s a good acco u n t of the an g u l ar dependence of the attenuation. The data of fig. 2 a l s o indicate that the r e s u l t s shown in fig. 1 a r e not due to heating of the liquid m e r c u r y which m i g h t have b e e n caused, f o r e x a m p l e , by eddy c u r r e n t s induced by the changing m a g n e t i c field. Such heating e f f e c t s should not be s e n s i t i v e to the angle ~ b e t w e e n / - / and q. The p r e s e n t r e s u l t s , t o g e t h e r with r e c e n t m e a s u r e m e n t s [4, 5] of the u l t r a s o n i c attenuation and v e l o c i t y in s e v e r a l i m p u r e solid m e t a l s and in the m i x e d st at e of Nb-25%Zr, s u g g e s t that the m a g n e t o - h y d r o d y n a m i c m o d e l [1, 21 a p p l i e s to 605
Volume 20, number 6
PHYSICS LETTERS
s e v e r a l b r o a d c l a s s e s of m a t e r i a l s viz. conducting liquids, s o l i d m e t a l s with s h o r t e l e c t r o n i c m e a n f r e e path, and s o m e h i g h - f i e l d s u p e r c o n d u c t o r s .
References 1. N.S. Anderson, J. Acoust. Soc. Am. 25 (1953) 529. Some of the numerical results in this reference are in e r r o r [2]. $~$*$
MAGNETOACOUSTIC
DENSITY
1 April 1966
2. R.A. Alpher and R. J. Rubin, J. Acoust. Soc. Am. 26 (1954) 452. 3. Handbook of chemistry and physics (Chemical Rubber Co., Cleveland, Ohio, 44th edition, 1962). 4. Y. Shapira and L. J. Neuringer, Phys. Rev. Letters 15 (1965) 724; erratum 15 (1965) 873; Y. Shapira and L. J. Neuringer, Physics Letters 20 (1966) 148. 5. G.A. Alers and P. A. Fleury, Phys. Rev. 129 (1963) 2425.
OF
STATES
RESONANCE
Y . ECKSTEIN
Argonne National Laboratory, Argonne, Illinois Received 5 March 1966
A magnetoacoustic resonance in antimony is described. It is due to a singularity in density of states. This resonance suggest that the F e r m i surface of antimony deviates from an ellipsoid.
In a p r e v i o u s l e t t e r [1] we d e s c r i b e d a m a g n e t o a c o u s t i c effect in the oblique g e o m e t r y , i.e. the m a g n e t i c f i el d not p e r p e n d i c u l a r to the wave v e c t o r of the sound. It w a s shown t h e r e that b e s i d e s the u s u al g e o m e t r i c r e s o n a n c e o s c i l l a t i o n s it i s a l s o p o s s i b l e to o b s e r v e a b s o r p t i o n e d g e s f r o m which v a l u a b l e i n f o r m a t i o n about the l i m i t ing points of the F e r m i s u r f a c e can be obtained. In this l e t t e r we d e s c r i b e y e t a n o t h e r effect which was o b s e r v e d in a n ti m o n y . T h i s i s a r e s o nance which cannot o c c u r f o r an e l l i p s o i d a l F e r m i s u r f a c e , and i s t h e r e f o r e additional e v i d e n c e that the F e r m i s u r f a c e of antimony i s not e l l i p s o i d a l [2,3]. In this e x p e r i m e n t 467 m c / s e c longitudinal sound wav es w e r e p r o p a g a t e d along the t r i g o n a l a x i s of a single c r y s t a l of antimony. T h e m a g n e t i c fi el d was r o t a t e d in the b i s e c t r i x - t r i g o n a l plane. At a r a n g e of a n g l e s between 21 ° and 27 ° f r o m t r i g o n a l a x i s , a v e r y s h a r p r e s o n a n c e and it s s u b h a r m o n i c s w e r e o b s e r v e d . T h e e x p e r i m e n t a l t r a c e s a r e shown in fig. 1. At 23 °, w h e r e the r e s o n a n c e i s s h a r p e s t , the line i s split into two peaks. B e c a u s e of the shape of the peaks, and e s p e c i a l l y b e c a u s e the p e a k s a p p e a r only f o r a v e r y n a r r o w r a n g e of a n g l e s , we b e l i e v e that * Based on work performed under the auspices of the U.S. Atomic Energy Commission. 606
t h e s e p e a k s a r e not the a b s o r p t i o n ed g es d i s c u s s e d in r e f . 1, which a l w a y s a p p e a r e d o v e r a v e r y l a r g e r a n g e of a n g l e s . We s u g g e s t that t h e s e p e a k s a r e r e s o n a n c e s of the type d i s c u s s e d by K a n e r et al. [4] and S. G. E c k s t e i n [5]. As d i s c u s s e d in r e f . 1, the e l e c t r o n s which c o n t r i b u t e to the attenuation a r e t h o s e which fulfill the D o p p l e r - s h i f t e d c y c l o t r o n r e s o n a n c e condition: + n w c = w(1 +
(1)
w h e r e w c = eH/m*c, ¢0/2w i s the sound f r e quency, v s the v e l o c i t y of sound, and< VH> the a v e r a g e d r i f t v e l o c i t y in the m a g n e t i c field d i r e c t i o n . F o r s i m p l i c i t y n e g l e c t 1 in c o m p a r i s o n with VF/COS ely s. Equation (1) can then be w r i t t e n in the f o r m
lcw l c {aA~ H = n e v s rn*