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Electronic acoustic attenuation for [001] longitudinal wave propagation in copper
Volume 24A, number 4
PHYSICS LETTERS
ELECTRONIC
13 February 1967
ACOUSTIC ATTENUATION. FOR [001] WAVE PROPAGATION IN COPPER
LONGITUDINAL
R. E. M c F A R L A N E and J . A. RAYNE
Carnegie Institute of Technology, Pittsburgh, Pennsylvania and C. K. JONES
Westinghouse Research Laboratory, Pittsbu~'gh, Pennsylvania Received 13 January 1967
Electronic attenuation measurements for [001] longitudinal wave propagation in copper are reported in the range 30 to 450 Mc/s. A detailed comparison with theory is given using the known properties of the Fermi surface. No d e t a i l e d c o m p a r i s o n between e x p e r i m e n t and the t h e o r y of a c o u s t i c attenuation in r e a l m e t a l s [1] h as been m a d e . In this l e t t e r , we r e p o r t such a c o m p a r i s o n f o r c o p p e r , taking into account the d e t a i l e d g e o m e t r y of i ts F e r m i s u r f a c e and using a m o d e l for its d e f o r m a t i o n c o n s i s t e n t with o t h er e x p e r i m e n t s . F o r longitudinal w a v e s p r o p a g a t i n g along a p u r e mode d i r e c t i o n in a r e a l m e t a l , the a c o u s t i c attenuation i s given by [1]
2.5
l
=
t
A
A
z.o
n
E =
1.5
=
t~-
tif 21t2pv 2
tf
~2qldS
~ 2
.S. 1 + (q/)2 cos2~
+
F.S. 1 + (ql)2 cos 2 g
f
I,
(1)
0.5
ql c o s 2 ~ dS
F.S. 1+ (ql)2 COS2 ( 0.0
w h e r e f i s the f r e q u e n c y , q is the a c o u s t i c wave v e c t o r , I is the e l e c t r o n m e a n f r e e path, and ~ is the angle between q and the n o r m a l to the F e r m i s u r f a c e . In the s p e c i a l c a s e of [001] propagation, the l o c a l d e f o r m a t i o n p a r a m e t e r ~ is g iv e n by ~) = K 3 3 + k 3 c o s ~ , w h e r e K33 i s one e l e m e n t of the static d e f o r m a tion t e n s o r , and b 3 i s the component of the e l e c t r o n wave v e c t o r [001]. A c o m p u t e r p r o g r a m has been d e v e l o p e d to p e r f o r m the i n t e g r a l s of eq. (1) by an equal i n t e r v a l m e t h o d using the r e -
0
, I00
= 200
;0
0
400
Frequency (Mc/se¢)
Fig. 1. Graph of ct/f versus f for copper showing experi. mental data and theoretical fits: curve A is for an isotropic deformation (K33 = -1.52), curve B is for a deformation consistent with other experiments (Ao = -1.52, A 4 = - 0 . 0 9 , A 6 =0, B2 =0.008, B4=0.326, andB 6 = = 0.40), and curve C shows the prediction of the freeelectron model (K33 = -1.64). Deformations are given in units of 1/a ~-1 where a is the lattice constant. p r e s e n t a t i o n of the F e r m i s u r f a c e of c o p p e r due to S h o e n b e r g and Roar [2]. T h e s i m p l e s t a s s u m p t i o n about K33 is that it i s i s o t r o p i c . C u r v e A of fig. 1 shows the r e s u l t 19'/
Volume 24A, number 4
PHYSICS LETTERS
ing a / f v e r s u s f plotted f o r l = 5.9 x 10 -3 cm, the m e a n f r e e path computed f r o m the e x p e r i m e n tal r e s i s t i v i t y r a t i o . It is s e e n that a / f is c l o s e to the e x p e r i m e n t a l data [3] at low f r e q u e n c i e s , but that it r ap i d l y b e c o m e s too l a r g e at h i g h e r f r e q u e n c i e s . T h i s b e h a v i o u r is to be c o n t r a s t e d to ~ / f f o r the f r e e - e l e c t r o n m o d e l (curve C) which a p p r o a c h e s its l i m i t i n g v a l u e v e r y quickly. B a s i c s y m m e t r y a r g u m e n t s show that the d e f o r m a t i o n can be expanded as a s e r i e s of s y m m e t r i z e d h a r m o n i c s with the f o r m
K33 =A oXo +A4X4 +A6X6 + . . . + B2 Y2 + B4 Y4 +. • •,
(2) w h e r e the X I have point group s y m m e t r y Oh and the YI have s y m m e t r y D4h. F u r t h e r i n f o r m a t i o n on the f o r m of K33 can be obtained f r o m de H a a s van Alphen e x p e r i m e n t s which have been p e r f o r m ed on c o p p e r u n d er h y d r o s t a t i c p r e s s u r e [4] and static uniaxial tension [5]. By fitting eq. (2) to the four a r e a changes o b s e r v e d in t h e s e e x p e r i m e n t s and r e q u i r i n g c h a r g e c o n s e r v a t i o n , five equations r e l a t i n g the expansion c o e f f i c i e n t s a r e o b t a i n e d . If a ll t e r m s of o r d e r g r e a t e r than s ix a r e a s s u m ed to v an i s h , t h e r e is one d i s p o s a b l e p a r a m e t e r which can be a d j u s t e d to give the b e s t fit to the a c o u s t i c attenuation data. C u r v e B of fig. 1 is the r e s u l t of following this p r o c e d u r e f o r copper using I = 5.9 x 10 -3 c m ; c l e a r l y the a g r e e m e n t with e x p e r i m e n t is quite good.
MAGNETIC
STRUCTURE
13 February 1967
The equal i n t e r v a l method b e c o m e s i n a c c u r a t e f o r l a r g e ql; h o w e v e r , as shown by P i p p a r d , the s u r f a c e i n t e g r a l s in eq. (1) can then be r e p l a c e d by line i n t e g r a l s along the e f f e c t i v e zone. When such a c a l c u l a t i o n is p e r f o r m e d f o r this d e f o r m a tion, it is found that ~ / f will continue to i n c r e a s e to a v a l u e at l e a s t 3.7 t i m e s the f r e e e l e c t r o n l i m i t by the t i m e ql r e a c h e s 100. It is i m p o s s i b l e to obtain the f a m i l i a r l i m i t i n g b e h a v i o u r with low o r d e r h a r m o n i c ex p an si o n s, so that e i t h e r K33 is highly a n i s o t r o p i c or a is not a l i n e a r function of f at high f r e q u e n c i e s . A s o u r c e of u n c e r t a i n t y in our a n a l y s i s is the effect of a n i s o t r o p y in l; s e r i o u s e r r o r could r e s u l t if this p a r a m e t e r w e r e to v a r y s t r o n g l y around the belly. T h i s work was s u p p o r t e d by a g r a n t f r o m the National S c i e n c e Foundation.
References 1. A.B.Pippard, Proc. Roy. Soe. (London)A257(1960) 165. 2. D. Shoenberg and D.J.Roaf, Phil. Trans. Roy. Soc. London A255 (1962) 85. 3. R.E.McFarlane, J.A.Rayneand C.K.Jones. Phys. Letters 19 (1965) 87. 4. I.M.Templeton, Proc. Roy. Soc. (London)A292 (1966) 413. 5. D.Shoenberg, private communication.
TRANSFORMATION
E. KRI~N, M. CSELIK, G. K / ~ R
IN M n P t
and L. P/kL
Central Research Institute for Physics, Budapest. Hungary Received 13 January 1967
A magnetic structure transformation has been observed in MnPt. On increasing the temperature the magnetic moments turn from the tetragonal axis to the basal plane. In Mn0.9 Ptl.1 between 5°K and T N the magnetic moments lie in the basal plane.
R e c e n t l y a growing i n t e r e s t has a r i s e n in the m a g n e t i c s t r u c t u r e p r o p e r t i e s of the m a n g a n e s e a l l o y s having o r d e r e d t e t r a g o n a l CuAuI type s t r u c t u r e . The f i r s t n e u t r o n d i f f r a c t i o n i n v e s t i gation of K a s p e r and Kouvel on MnNi in 1959 [1] was followed by the d e t e r m i n a t i o n of the m a g n e tic s t r u c t u r e s in MnPt [2], MnPd [3] and Mn2Pd 3 198
[4]. The m a g n e t i c p r o p e r t i e s of MnPt [5], MnIr [6], MnSi [7] and MnRh [8] w e r e studied by s u s ceptibility m e a s u r e m e n t s . The basic a n t i f e r r o m a g n e t i c s t r u c t u r e , which was found to be the s a m e in all t h e s e s y s t e m s , is i l l u s t r a t e d as an i n s e r t in fig. 1. The angle qoc of the m a g n e t i c m o m e n t to the c r y s t a l l o g r a p h i c a x i s c is 0 U in MnPt
PHYSICS LETTERS
ELECTRONIC
13 February 1967
ACOUSTIC ATTENUATION. FOR [001] WAVE PROPAGATION IN COPPER
LONGITUDINAL
R. E. M c F A R L A N E and J . A. RAYNE
Carnegie Institute of Technology, Pittsburgh, Pennsylvania and C. K. JONES
Westinghouse Research Laboratory, Pittsbu~'gh, Pennsylvania Received 13 January 1967
Electronic attenuation measurements for [001] longitudinal wave propagation in copper are reported in the range 30 to 450 Mc/s. A detailed comparison with theory is given using the known properties of the Fermi surface. No d e t a i l e d c o m p a r i s o n between e x p e r i m e n t and the t h e o r y of a c o u s t i c attenuation in r e a l m e t a l s [1] h as been m a d e . In this l e t t e r , we r e p o r t such a c o m p a r i s o n f o r c o p p e r , taking into account the d e t a i l e d g e o m e t r y of i ts F e r m i s u r f a c e and using a m o d e l for its d e f o r m a t i o n c o n s i s t e n t with o t h er e x p e r i m e n t s . F o r longitudinal w a v e s p r o p a g a t i n g along a p u r e mode d i r e c t i o n in a r e a l m e t a l , the a c o u s t i c attenuation i s given by [1]
2.5
l
=
t
A
A
z.o
n
E =
1.5
=
t~-
tif 21t2pv 2
tf
~2qldS
~ 2
.S. 1 + (q/)2 cos2~
+
F.S. 1 + (ql)2 cos 2 g
f
I,
(1)
0.5
ql c o s 2 ~ dS
F.S. 1+ (ql)2 COS2 ( 0.0
w h e r e f i s the f r e q u e n c y , q is the a c o u s t i c wave v e c t o r , I is the e l e c t r o n m e a n f r e e path, and ~ is the angle between q and the n o r m a l to the F e r m i s u r f a c e . In the s p e c i a l c a s e of [001] propagation, the l o c a l d e f o r m a t i o n p a r a m e t e r ~ is g iv e n by ~) = K 3 3 + k 3 c o s ~ , w h e r e K33 i s one e l e m e n t of the static d e f o r m a tion t e n s o r , and b 3 i s the component of the e l e c t r o n wave v e c t o r [001]. A c o m p u t e r p r o g r a m has been d e v e l o p e d to p e r f o r m the i n t e g r a l s of eq. (1) by an equal i n t e r v a l m e t h o d using the r e -
0
, I00
= 200
;0
0
400
Frequency (Mc/se¢)
Fig. 1. Graph of ct/f versus f for copper showing experi. mental data and theoretical fits: curve A is for an isotropic deformation (K33 = -1.52), curve B is for a deformation consistent with other experiments (Ao = -1.52, A 4 = - 0 . 0 9 , A 6 =0, B2 =0.008, B4=0.326, andB 6 = = 0.40), and curve C shows the prediction of the freeelectron model (K33 = -1.64). Deformations are given in units of 1/a ~-1 where a is the lattice constant. p r e s e n t a t i o n of the F e r m i s u r f a c e of c o p p e r due to S h o e n b e r g and Roar [2]. T h e s i m p l e s t a s s u m p t i o n about K33 is that it i s i s o t r o p i c . C u r v e A of fig. 1 shows the r e s u l t 19'/
Volume 24A, number 4
PHYSICS LETTERS
ing a / f v e r s u s f plotted f o r l = 5.9 x 10 -3 cm, the m e a n f r e e path computed f r o m the e x p e r i m e n tal r e s i s t i v i t y r a t i o . It is s e e n that a / f is c l o s e to the e x p e r i m e n t a l data [3] at low f r e q u e n c i e s , but that it r ap i d l y b e c o m e s too l a r g e at h i g h e r f r e q u e n c i e s . T h i s b e h a v i o u r is to be c o n t r a s t e d to ~ / f f o r the f r e e - e l e c t r o n m o d e l (curve C) which a p p r o a c h e s its l i m i t i n g v a l u e v e r y quickly. B a s i c s y m m e t r y a r g u m e n t s show that the d e f o r m a t i o n can be expanded as a s e r i e s of s y m m e t r i z e d h a r m o n i c s with the f o r m
K33 =A oXo +A4X4 +A6X6 + . . . + B2 Y2 + B4 Y4 +. • •,
(2) w h e r e the X I have point group s y m m e t r y Oh and the YI have s y m m e t r y D4h. F u r t h e r i n f o r m a t i o n on the f o r m of K33 can be obtained f r o m de H a a s van Alphen e x p e r i m e n t s which have been p e r f o r m ed on c o p p e r u n d er h y d r o s t a t i c p r e s s u r e [4] and static uniaxial tension [5]. By fitting eq. (2) to the four a r e a changes o b s e r v e d in t h e s e e x p e r i m e n t s and r e q u i r i n g c h a r g e c o n s e r v a t i o n , five equations r e l a t i n g the expansion c o e f f i c i e n t s a r e o b t a i n e d . If a ll t e r m s of o r d e r g r e a t e r than s ix a r e a s s u m ed to v an i s h , t h e r e is one d i s p o s a b l e p a r a m e t e r which can be a d j u s t e d to give the b e s t fit to the a c o u s t i c attenuation data. C u r v e B of fig. 1 is the r e s u l t of following this p r o c e d u r e f o r copper using I = 5.9 x 10 -3 c m ; c l e a r l y the a g r e e m e n t with e x p e r i m e n t is quite good.
MAGNETIC
STRUCTURE
13 February 1967
The equal i n t e r v a l method b e c o m e s i n a c c u r a t e f o r l a r g e ql; h o w e v e r , as shown by P i p p a r d , the s u r f a c e i n t e g r a l s in eq. (1) can then be r e p l a c e d by line i n t e g r a l s along the e f f e c t i v e zone. When such a c a l c u l a t i o n is p e r f o r m e d f o r this d e f o r m a tion, it is found that ~ / f will continue to i n c r e a s e to a v a l u e at l e a s t 3.7 t i m e s the f r e e e l e c t r o n l i m i t by the t i m e ql r e a c h e s 100. It is i m p o s s i b l e to obtain the f a m i l i a r l i m i t i n g b e h a v i o u r with low o r d e r h a r m o n i c ex p an si o n s, so that e i t h e r K33 is highly a n i s o t r o p i c or a is not a l i n e a r function of f at high f r e q u e n c i e s . A s o u r c e of u n c e r t a i n t y in our a n a l y s i s is the effect of a n i s o t r o p y in l; s e r i o u s e r r o r could r e s u l t if this p a r a m e t e r w e r e to v a r y s t r o n g l y around the belly. T h i s work was s u p p o r t e d by a g r a n t f r o m the National S c i e n c e Foundation.
References 1. A.B.Pippard, Proc. Roy. Soe. (London)A257(1960) 165. 2. D. Shoenberg and D.J.Roaf, Phil. Trans. Roy. Soc. London A255 (1962) 85. 3. R.E.McFarlane, J.A.Rayneand C.K.Jones. Phys. Letters 19 (1965) 87. 4. I.M.Templeton, Proc. Roy. Soc. (London)A292 (1966) 413. 5. D.Shoenberg, private communication.
TRANSFORMATION
E. KRI~N, M. CSELIK, G. K / ~ R
IN M n P t
and L. P/kL
Central Research Institute for Physics, Budapest. Hungary Received 13 January 1967
A magnetic structure transformation has been observed in MnPt. On increasing the temperature the magnetic moments turn from the tetragonal axis to the basal plane. In Mn0.9 Ptl.1 between 5°K and T N the magnetic moments lie in the basal plane.
R e c e n t l y a growing i n t e r e s t has a r i s e n in the m a g n e t i c s t r u c t u r e p r o p e r t i e s of the m a n g a n e s e a l l o y s having o r d e r e d t e t r a g o n a l CuAuI type s t r u c t u r e . The f i r s t n e u t r o n d i f f r a c t i o n i n v e s t i gation of K a s p e r and Kouvel on MnNi in 1959 [1] was followed by the d e t e r m i n a t i o n of the m a g n e tic s t r u c t u r e s in MnPt [2], MnPd [3] and Mn2Pd 3 198
[4]. The m a g n e t i c p r o p e r t i e s of MnPt [5], MnIr [6], MnSi [7] and MnRh [8] w e r e studied by s u s ceptibility m e a s u r e m e n t s . The basic a n t i f e r r o m a g n e t i c s t r u c t u r e , which was found to be the s a m e in all t h e s e s y s t e m s , is i l l u s t r a t e d as an i n s e r t in fig. 1. The angle qoc of the m a g n e t i c m o m e n t to the c r y s t a l l o g r a p h i c a x i s c is 0 U in MnPt