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Electromagnetic excitation of elastic modes in aluminium
Volume26A. n u m b e r 7
PHYSICS
F o r t h e c o n d i t i o n s of t h i s e x p e r i m e n t
LETTERS
e n t i n t h e d i r e c t i o n of i t s m o t i o n . F o r a s u d d e n d i s p l a c e m e n t A X of t h e n o r m a l r e g i o n t h e r e i s a relaxation time T associated with restoring thermodynamic equilibrium. For our particular case w e e s t i m a t e f r o m t h e h e a t c a p a c i t y [3] a n d t h e t h e r m a l c o n d u c t i v i t y [1] t h a t AX/'r ~ 120 m / s e c in f a i r a g r e e m e n t w i t h t h e v a l u e of Vo.
T = 6.5 °,
d H c / d t = - 2 0 0 g a u s s / d e g a n d H c = 140 g a u s s [1], w e o b t a i n f r o m eq. (2) t h a t h Q o = 7 × 10 -6 j o u l e . If w e f i t o u r d a t a to a n e x p r e s s i o n of t h e f o r m A Q = AQ1 exp ( - V / V o) ,
(3)
we f i n d t h a t AQ 1 = 8 × 10 -6 j o u l e i n g o o d a g r e e m e n t w i t h t h e c a l c u l a t e d Qo" Vo = 60 c m / s e c . In t h e l i m i t of low v e l o c i t i e s a l l of t h e h e a t i n the normal region is carried along. This is a c o n s e q u e n c e of t h e p r i n c i p l e of a d i a b a t i c i n v a r i ance. Here the adiabatic invariant state is the s t a t e of u n i f o r m t e m p e r a t u r e t h r o u g h o u t t h e s a m p l e . A s t h e v e l o c i t y of t h e n o r m a l r e g i o n i n creases the system is no longer in equilibrium and the normal region develops a thermal gradi-
ELECTROMAGNETIC
EXCITATION
26 F e b r u a r y 1968
References 1. D.Shoenberg. Superconductivity (Cambridge U n i v e r sity P r e s s , London. 1960). 2. P . R . S o l o m o n . Phys. L e t t e r s 26A (1968) 293. 3. F . E . Hoare. L. C. Jackson and N. Kurti. E x p e r i m e n tal cryophysics (Butterworths. London. 1961).
OF
ELASTIC
MODES
IN
ALUMINIUM
P . K. L A R S E N a n d K. S A E R M A R K
Fysisk Laboratorium I, The Technical Uni~,ersity of Denmark, Lyngby, Denmark Received 18 J a n u a r y 1968
E x p e r i m e n t s on e l e c t r o m a g n e t i c excitation of elastic modes in A l - d i s k s at liquid-helium and r o o m - t e m peratures are described.
In r e c e n t c o m m u n i c a t i o n s [1,2] e l e c t r o m a g n e t i c e x c i t a t i o n of a c o u s t i c w a v e s of b o t h s h e a r - a n d compressional types in metals have been reported. T h e p u r p o s e of t h e p r e s e n t n o t e i s t o d e s c r i b e e x p e r i m e n t s in w h i c h a v e r y l a r g e n u m b e r of r e s o n a n c e s b e l o n g i n g to a t l e a s t 3 d i f f e r e n t t y p e s of e l a s t i c m o d e s h a v e b e e n o b s e r v e d i n t r a n s m i s s i o n e x p e r i m e n t s on A l - s a m p l e s in t h e f r e q u e n c y r a n g e 0.1 - 5.0 M H z . The Al-samples - mono- as well as polycryst a l l i n e - w e r e d i s k - s h a p e d w i t h a d i a m e t e r of 1 0 - 13 m m a n d a t h i c k n e s s of 0.5 - 1.5 m m . T h e m o n o c r y s t a l l i n e m a t e r i a l h a d a R R R of 14.000, while the polycrystalline material was quite s i m i l a r t o t h e a l l o y A1 6151. A c o n v e n t i o n a l t w o coil, orthogonal-geometry set-up was used and the d.c. magnetic field was perpendicular to the sample faces. The transmitted r.f. signal was m e a s u r e d by a H P 8 4 0 5 A V e c t o r - V o l t m e t e r (amplitude and phase) or by a HP-310 WaveAnalyzer (amplitude only). At 4.2°K a superconducting solenoid was used, at room-temperature 296
a V a r i a n 6 - i n c h m a g n e t w a s u s e d . T h i s t y p e of s e t - u p g i v e s m o r e d e t a i l e d i n f o r m a t i o n on t h e e x c i t e d e l a s t i c m o d e s t h a n m a y b e o b t a i n e d by a p u l s e e c h o - m e t h o d [2]. For a fixed d.c. magnetic field recorder t r a c e s of t r a n s m i t t e d r . f . s i g n a l v e r s u s f r e q u e n cy s h o w a r e s o n a n t b e h a v i o u r . An e x t r e m e l y l a r g e n u m b e r of r e s o n a n c e s w e r e o b s e r v e d in t h e r a n g e 0.1 - 5.0 MHz i n p o l y - a n d m o n o c r y s t a l l i n e s a m p l e s a t 4.2 a n d 3 0 0 ° K . S o m e of t h e s e m a y b e i d e n t i f i e d w i t h s t a n d i n g s h e a r w a v e s [1], s o m e w i t h e x t e n s i o n a l w a v e s [3], w h i l e t h e r e s t remain as yet unidentified. Fig. 1 shows a recorder trace for a monocrystalline sample at 4 . 2 ° K , w h i l e t a b l e 1 g i v e s a s u m m a r y of o b served resonances for polycrystalline material i n t h e r a n g e 0.1 - 1.2 MHz a l l of w h i c h c a n b e i d e n t i f i e d a s e x t e n s i o n a l w a v e s . By m e a n s of t h e f r e q u e n c y r e l a t i o n f o r e x t e n s i o n a l w a v e s [3] o n e m a y c a l c u l a t e t h e m o d e f r e q u e n c i e s vnrn . A s suming that the observed pronounced resonance a t 341.05 k H z c o r r e s p o n d s t o v01 o n e m a y c a l -
Volume 26A, n u m b e r 7
PHYSICS
LETTERS
26 F e b r u a r y 1968
Table 1 O b s e r v e d r e s o n a n c e f r e q u e n c i e s for polycrystalline Al-disk, diam. 10.97 m r n t h i c k n e s s 0.909 ram. The r e s o n a n c e s shown have been o b s e r v e d on 5 s a m p l e s with different d i m e n s i o n s . The calculated values a r e based on the o b s e r v e d r e s o n a n c e at 341.05 as explained in the text.
1
t--
n,m
Vn m (calc.) ' (kHz)
2,1
223.79
1,1 0.1 3,1
262.83 341.05 343.82
2,2 3,2
408.41 560.99
1,2
580.48
1,3
655.63
2,3
735.87
2,4 3,3 1,4
850.52 868.18 949.44
_J Q_
200
400
600 kHz
BOO
1000
Fig. 1. r.f. -amplitude in r e c e i v e r coil v e r s u s frequency for a monocrystalline Al-disk, B ( N 3 0 kG) is n o r m a l to the disk and oriented in a (1,1,0)-direction. The e l a s t i c r e s o n a n c e s a p p e a r on a background of helicons. c u l a t e a l l of t h e m o d e s f r o m a k n o w l e d g e of P o i s s o n ' s r a t i o ~, w h i c h w a s t a k e n t o b e 0 . 3 3 1 . The results are shown in table 1 together with (n,m). It is seen that the agreement is excellent. The following observations may be noted: 1) Tl~e p o s i t i o n of t h e r e s o n a n c e s d e p e n d s l i g h t l y o n H~; 2) m e a m p l i t u d e s of t h e p e a k s a r e p r o p o r t i o n a l t o / / 2 ; 3) t h e Q - v a l u e s a r e v e r y h i g h a n d a l m o s t t e m p e r a t u r e i n d e p e n d e n t ; 4) t h e r e s o n a n ces may be seen for as low fields as ~ 0.6 kG; 5) t h e d e t a i l e d l i n e - s h a p e a p p e a r s t o b e s o m e what dependent on the position the sample relative to the coils and in some cases pronounced d e p e n d e n t o n t h e d . c . m a g n e t i c f i e l d ; 6) f o r t h e m o r e c o m p l e x e l a s t i c m o d e s t h e o b s e r v a t i o n of a particular resonance is dependent on the precise p o s i t i o n of t h e s a m p l e r e l a t i v e t o t h e c o i l s ; 7) f o r a f i x e d f r e q u e n c y r e c o r d e r t r a c e s of t r a n s m i t t e d r.f. signal versus m~etic field show de Haasvan Alphen oscillations. In this respect the ~fferent elastic modes show a different behavionr. Financial support from Statens Almindelige Videnskabsfond is gratefully acknowledged.
3,4 2,5
1043.3 1090.1
1,5
1139.8
v (observed) (kHz) 4.2°K room-temp. 262.84 263.35 341.05
249.32 249.76 325.94
408.29 409.07 561.30 578.45 578.81 655.10 655.50 656.51 732.47 733.94 734.30 846.20 847.17 866.58 ? 1034.23 1035.08 ? 1122.22 1123.13
387.26 388.04 550.40 550.73 621.04 695.90 696.30 803.77 804.55 821.07
1068
References
1. K . S a e r m a r k and P . K . L a r s e n , Phys. L e t t e r s 24A (1967) 668. 2. A . G , B e t j e m a n n et al., Phys. L e t t e r s 25A (1967) 753. 3. A. E. H, Love, A t r e a t i s e on m a t h e m a t i c a l theory of e l a s t i c i t y (4th ed., Dover, 1944).
297
PHYSICS
F o r t h e c o n d i t i o n s of t h i s e x p e r i m e n t
LETTERS
e n t i n t h e d i r e c t i o n of i t s m o t i o n . F o r a s u d d e n d i s p l a c e m e n t A X of t h e n o r m a l r e g i o n t h e r e i s a relaxation time T associated with restoring thermodynamic equilibrium. For our particular case w e e s t i m a t e f r o m t h e h e a t c a p a c i t y [3] a n d t h e t h e r m a l c o n d u c t i v i t y [1] t h a t AX/'r ~ 120 m / s e c in f a i r a g r e e m e n t w i t h t h e v a l u e of Vo.
T = 6.5 °,
d H c / d t = - 2 0 0 g a u s s / d e g a n d H c = 140 g a u s s [1], w e o b t a i n f r o m eq. (2) t h a t h Q o = 7 × 10 -6 j o u l e . If w e f i t o u r d a t a to a n e x p r e s s i o n of t h e f o r m A Q = AQ1 exp ( - V / V o) ,
(3)
we f i n d t h a t AQ 1 = 8 × 10 -6 j o u l e i n g o o d a g r e e m e n t w i t h t h e c a l c u l a t e d Qo" Vo = 60 c m / s e c . In t h e l i m i t of low v e l o c i t i e s a l l of t h e h e a t i n the normal region is carried along. This is a c o n s e q u e n c e of t h e p r i n c i p l e of a d i a b a t i c i n v a r i ance. Here the adiabatic invariant state is the s t a t e of u n i f o r m t e m p e r a t u r e t h r o u g h o u t t h e s a m p l e . A s t h e v e l o c i t y of t h e n o r m a l r e g i o n i n creases the system is no longer in equilibrium and the normal region develops a thermal gradi-
ELECTROMAGNETIC
EXCITATION
26 F e b r u a r y 1968
References 1. D.Shoenberg. Superconductivity (Cambridge U n i v e r sity P r e s s , London. 1960). 2. P . R . S o l o m o n . Phys. L e t t e r s 26A (1968) 293. 3. F . E . Hoare. L. C. Jackson and N. Kurti. E x p e r i m e n tal cryophysics (Butterworths. London. 1961).
OF
ELASTIC
MODES
IN
ALUMINIUM
P . K. L A R S E N a n d K. S A E R M A R K
Fysisk Laboratorium I, The Technical Uni~,ersity of Denmark, Lyngby, Denmark Received 18 J a n u a r y 1968
E x p e r i m e n t s on e l e c t r o m a g n e t i c excitation of elastic modes in A l - d i s k s at liquid-helium and r o o m - t e m peratures are described.
In r e c e n t c o m m u n i c a t i o n s [1,2] e l e c t r o m a g n e t i c e x c i t a t i o n of a c o u s t i c w a v e s of b o t h s h e a r - a n d compressional types in metals have been reported. T h e p u r p o s e of t h e p r e s e n t n o t e i s t o d e s c r i b e e x p e r i m e n t s in w h i c h a v e r y l a r g e n u m b e r of r e s o n a n c e s b e l o n g i n g to a t l e a s t 3 d i f f e r e n t t y p e s of e l a s t i c m o d e s h a v e b e e n o b s e r v e d i n t r a n s m i s s i o n e x p e r i m e n t s on A l - s a m p l e s in t h e f r e q u e n c y r a n g e 0.1 - 5.0 M H z . The Al-samples - mono- as well as polycryst a l l i n e - w e r e d i s k - s h a p e d w i t h a d i a m e t e r of 1 0 - 13 m m a n d a t h i c k n e s s of 0.5 - 1.5 m m . T h e m o n o c r y s t a l l i n e m a t e r i a l h a d a R R R of 14.000, while the polycrystalline material was quite s i m i l a r t o t h e a l l o y A1 6151. A c o n v e n t i o n a l t w o coil, orthogonal-geometry set-up was used and the d.c. magnetic field was perpendicular to the sample faces. The transmitted r.f. signal was m e a s u r e d by a H P 8 4 0 5 A V e c t o r - V o l t m e t e r (amplitude and phase) or by a HP-310 WaveAnalyzer (amplitude only). At 4.2°K a superconducting solenoid was used, at room-temperature 296
a V a r i a n 6 - i n c h m a g n e t w a s u s e d . T h i s t y p e of s e t - u p g i v e s m o r e d e t a i l e d i n f o r m a t i o n on t h e e x c i t e d e l a s t i c m o d e s t h a n m a y b e o b t a i n e d by a p u l s e e c h o - m e t h o d [2]. For a fixed d.c. magnetic field recorder t r a c e s of t r a n s m i t t e d r . f . s i g n a l v e r s u s f r e q u e n cy s h o w a r e s o n a n t b e h a v i o u r . An e x t r e m e l y l a r g e n u m b e r of r e s o n a n c e s w e r e o b s e r v e d in t h e r a n g e 0.1 - 5.0 MHz i n p o l y - a n d m o n o c r y s t a l l i n e s a m p l e s a t 4.2 a n d 3 0 0 ° K . S o m e of t h e s e m a y b e i d e n t i f i e d w i t h s t a n d i n g s h e a r w a v e s [1], s o m e w i t h e x t e n s i o n a l w a v e s [3], w h i l e t h e r e s t remain as yet unidentified. Fig. 1 shows a recorder trace for a monocrystalline sample at 4 . 2 ° K , w h i l e t a b l e 1 g i v e s a s u m m a r y of o b served resonances for polycrystalline material i n t h e r a n g e 0.1 - 1.2 MHz a l l of w h i c h c a n b e i d e n t i f i e d a s e x t e n s i o n a l w a v e s . By m e a n s of t h e f r e q u e n c y r e l a t i o n f o r e x t e n s i o n a l w a v e s [3] o n e m a y c a l c u l a t e t h e m o d e f r e q u e n c i e s vnrn . A s suming that the observed pronounced resonance a t 341.05 k H z c o r r e s p o n d s t o v01 o n e m a y c a l -
Volume 26A, n u m b e r 7
PHYSICS
LETTERS
26 F e b r u a r y 1968
Table 1 O b s e r v e d r e s o n a n c e f r e q u e n c i e s for polycrystalline Al-disk, diam. 10.97 m r n t h i c k n e s s 0.909 ram. The r e s o n a n c e s shown have been o b s e r v e d on 5 s a m p l e s with different d i m e n s i o n s . The calculated values a r e based on the o b s e r v e d r e s o n a n c e at 341.05 as explained in the text.
1
t--
n,m
Vn m (calc.) ' (kHz)
2,1
223.79
1,1 0.1 3,1
262.83 341.05 343.82
2,2 3,2
408.41 560.99
1,2
580.48
1,3
655.63
2,3
735.87
2,4 3,3 1,4
850.52 868.18 949.44
_J Q_
200
400
600 kHz
BOO
1000
Fig. 1. r.f. -amplitude in r e c e i v e r coil v e r s u s frequency for a monocrystalline Al-disk, B ( N 3 0 kG) is n o r m a l to the disk and oriented in a (1,1,0)-direction. The e l a s t i c r e s o n a n c e s a p p e a r on a background of helicons. c u l a t e a l l of t h e m o d e s f r o m a k n o w l e d g e of P o i s s o n ' s r a t i o ~, w h i c h w a s t a k e n t o b e 0 . 3 3 1 . The results are shown in table 1 together with (n,m). It is seen that the agreement is excellent. The following observations may be noted: 1) Tl~e p o s i t i o n of t h e r e s o n a n c e s d e p e n d s l i g h t l y o n H~; 2) m e a m p l i t u d e s of t h e p e a k s a r e p r o p o r t i o n a l t o / / 2 ; 3) t h e Q - v a l u e s a r e v e r y h i g h a n d a l m o s t t e m p e r a t u r e i n d e p e n d e n t ; 4) t h e r e s o n a n ces may be seen for as low fields as ~ 0.6 kG; 5) t h e d e t a i l e d l i n e - s h a p e a p p e a r s t o b e s o m e what dependent on the position the sample relative to the coils and in some cases pronounced d e p e n d e n t o n t h e d . c . m a g n e t i c f i e l d ; 6) f o r t h e m o r e c o m p l e x e l a s t i c m o d e s t h e o b s e r v a t i o n of a particular resonance is dependent on the precise p o s i t i o n of t h e s a m p l e r e l a t i v e t o t h e c o i l s ; 7) f o r a f i x e d f r e q u e n c y r e c o r d e r t r a c e s of t r a n s m i t t e d r.f. signal versus m~etic field show de Haasvan Alphen oscillations. In this respect the ~fferent elastic modes show a different behavionr. Financial support from Statens Almindelige Videnskabsfond is gratefully acknowledged.
3,4 2,5
1043.3 1090.1
1,5
1139.8
v (observed) (kHz) 4.2°K room-temp. 262.84 263.35 341.05
249.32 249.76 325.94
408.29 409.07 561.30 578.45 578.81 655.10 655.50 656.51 732.47 733.94 734.30 846.20 847.17 866.58 ? 1034.23 1035.08 ? 1122.22 1123.13
387.26 388.04 550.40 550.73 621.04 695.90 696.30 803.77 804.55 821.07
1068
References
1. K . S a e r m a r k and P . K . L a r s e n , Phys. L e t t e r s 24A (1967) 668. 2. A . G , B e t j e m a n n et al., Phys. L e t t e r s 25A (1967) 753. 3. A. E. H, Love, A t r e a t i s e on m a t h e m a t i c a l theory of e l a s t i c i t y (4th ed., Dover, 1944).
297