- Email: [email protected]
Anomaly in the phonon dispersion of a disordered alloy, Cu0.715Pd0.285
ELSEVIER
Physica B 219&220 (1996) 490-492
Anomaly in the phonon dispersion of a disordered alloy, Cuo.715Pdo.285 Y. N o d a a'*, K. Ohshima b, Y. Endoh c aSendai National College of Technology, 1 Kitahara, Kamiayashi, Aoba-ku, Sendai 989-31, Japan blnstitute of Applied Physics, University of Tsukuba, Tsukuba 305, Japan CDepartment of Physics, Tohoku University, Sendai 980, Japan
Abstract The phonon frequencies of a disordered Cu0.715Pd0.2 s 5 alloy were measured using neutron experiments to investigate the relationship between Kohn anomaly and the maxima of the diffuse scattering due to the atomic short-range order, in which the form of the Fermi surface is reflected. Diffuse satellites have been observed clearly at the position expected from the form of Fermi surface. The weak Kohn anomaly of [1 10]L branch has been observed at the position expected from the diffuse satellites.
1. Introduction By considering the screening effect of a free-electron gas on a fluctuating charge distribution such as a lattice vibration, Kohn predicted that there should be lines in reciprocal space through which the dispersion surfaces of the lattice vibration will have infinite slopes [1]. This effect has become known as the Kohn anomaly and, in pure metals, has been observed in Pb by Brockhouse et al. [2], in Cu by Nilson and Rolandson [3], and in Nb by Sharp [4]. The Kohn anomaly corresponds to the logarithmic singularity of the static dielectric function e(q) at q = 2kv, for a spherical Fermi surface of radius kv. Flat parallel sections of the Fermi surface enhance the singularity in the direction perpendicular to the surface; thus the strength of the Kohn anomaly depends on the geometry of the Fermi surface and the details of the electron-phonon interaction. In the case of disordered alloys, Krivoglaz has pointed out that the form of Fermi surface is reflected in the distribution of atomic short-range order diffuse scattering *Corresponding author.
of X-rays, electrons and neutrons via the singularity in the static dielectric function at q = 2kF [5]. The phonon frequencies at room temperature along high-symmetry directions of disordered Cuo.saml0.16 and Cu0.715Pd0.285 alloys have been measured by Chou et al. [6] and Noda et al. [7], respectively, to investigate the relationship between the diffuse maxima due to the atomic short-range order and the Kohn anomalies in phonon dispersion curves. In both cases, no strong anomaly was observed either in the phonon dispersion itself or in its derivative near the expected wave vector. In the CuPd alloy, fourfold split diffuse maxima have been observed sharply at (3, 0.865, 0) and three equivalent positions around the (3, 1, 0) reciprocal lattice point [7], as expected from the geometry of Fermi surface [8]. We have performed phonon measurements under high resolution condition at low temperature to observe weak anomalies of dispersion curve of CuPd alloy. 2. Experimental results Phonon frequencies were measured on the triple axis spectrometer of Tohoku University (TOPAN) installed
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 7 8 8 - 1
491
Y. Noda et al. /Physica B 219&220 (1996) 490 492 100
(a)
=
/~, Eq=21.905
.
~
50
.
.
.
I~t), "
o
31)
i
q)2
~
~.
Eq =20.708
0
-'--..,
°i
*
- - . * = 0 9,
(~0)
(oo~)
c2-~, 2-~, o>
•
. . . . . . . .
",
oe~ 2 0
L
r..
O 2O
i0
)
30
E (meV)
10
I00
(b)
=
--@
z
(2-c, 2-~, o)
E,,=20.708
.... o.o o
3
,,~0
,~ = 0 , 9 4
50
~/~
,
• ./
"%
20 )
'
'
'
'
E (raeV)
0.5
~.,
.0
Fig. 2. Phonon dispersion curves for [~ (0] and [00 ~] directions at 25 K. Curves are results for the second-neighbor BvK model. The arrow indicates the position of 2kv as derived from the position of diffuse satellites.
'k,
oOj
. . . . .
.0
F /
0
0
~_
~
30
Fig. 1. Example of the phonon peaks in Cuo,715Pd0.285 disordered alloy at 25 K. Curves are least-squares fit to the data with the calculation of convolution. Eq is the phonon frequency which is a fitting parameter.
at JRR-3M reactor of JAERI. The sample used was the same single crystal as used in the previous measurements of p h o n o n and diffuse scattering [7]. The sample has an FCC structure with lattice parameter a = 3.719 A and a mosaic spread of 44 min. Pyrolytic graphite was used as both a m o n o c h r o m a t o r and an analyzer. A pyrolytic graphite filter was used before the analyzer to eliminate the higher-order contamination in the beam. Constant-Q scans were made with fixed final energy Ef = 13.8 meV with a set of collimators of 15' 30' 30'-30' or 3 0 ' - 3 0 ' 30' 30'. Measurements were performed at a low temperature of 25 K with an energy resolution of about 0.2 meV. The resolution along the direction parallel to the scattering vector is about 0.02 in reduced wave n u m b e r ( = q/(2rt/a). Additionally, to investigate the temperature dependence of the anomaly, the p h o n o n dispersion of [1 1 0]L mode was measured at 13 K and at room temperature (295 K), with Ef = 13.7 meV and a set of collimators of 3 0 ' - 6 0 ' - 60'-60'. P h o n o n frequencies were measured along the [ ( ~ 0 ] direction in the range of ~ = 0.5-1.0, and along the [( 0 0] direction in the range ( = 0.3 0.7. Anomaly of p h o n o n dispersion along the [ ( ~ 0 ] is expected near ( = 0.935
from the positions of the observed satellites (q = 2kv). Fig. 1 shows examples of observed p h o n o n peaks together with fits using a damped harmonic oscillator function convoluted with the instrumental resolution function. The peak positions, Eq (meV), can be determined with a standard deviation better than 0.1 meV in worse cases of [1 1 0]L mode near the zone boundary. All p h o n o n peaks were fitted with the convolution procedure. Intrinsic p h o n o n line width Y (meV), which is a fitting parameter, at ( = 0.88, 0.92 and 0.94 is 0.83 _+ 0.58, 1.10 ___0.86 and 1.34 + 0.98, respectively. The p h o n o n energy decreases more and more as ( increases from 0.86 to 0.94, as shown in Fig. l(a), while it hardly changes for ( > 0.94, as shown in Fig. l(b). Fig. 2 is a plot of the p h o n o n frequencies together with the curve for a second nearest-neighbor B o r n - v o n K a r m a n (BvK) fit to the data of three observed branches. The force constants of fitting parameters are listed in Table 1. The arrow in Fig. 2 indicates the position of 2kF as derived from the positions of the diffuse satellites. The small anomalous change in the slope of the dispersion curve is detected just at the position of arrow. The p h o n o n frequencies and their derivatives of the [ ( ( 0 ] L branch in the region ( = 0.5-1.0 are plotted in Fig. 3. The open circle and triangle in Fig. 3(a) and (b) mean the different conditions of measurements, where the first collimation is 15' and 30% respectively. Solid line in Fig. 3(b) is guide to the eye. Error bars of p h o n o n frequencies are less than 0.1 meV, then the derivatives (dE/d0 have errors about 10 meV. The dotted line is obtained by the BvK model using force constants shown
492
Y. Noda et al. / Physica B 219&220 (1996) 490-492
Table 1 Force constants of disordered Cu0.715Pdo.2s5 alloy at 25 K Force constants (dyn cm-i) 17470 ± -- 2090 ± 17090 ± -- 720 ± -- 2700 ±
lxx lzz lxy 2xx 2yy
200 250 300 350 410
28
~D 2~
E
........°O""O"-...~ .......
[ ~ ff 0 ] L
o o~ 22
(a)
.....o '~O
....
I ,o .
~
[
t
3. Conclusion The anomaly of the [1 1 0]L p h o n o n dispersion in Cuo.vlsPdo.28s disordered alloy was observed at the position (q = 2kv) expected from the diffuse satellite positions, as reflected by the form of Fermi surface [7]. The dispersion relation of the [1 ( 0 ] branch was measured in the region around the (1,0.865,0) reciprocal lattice point, where the diffuse maximum is observed in the elastic scattering. No anomalous change with the wave vector was observed in the frequency and the intensity of phonon. It is difficult to consider that the anomaly of the derivative of [1 1 0]L p h o n o n dispersion is caused by an incommensurate short-range atomic order. The p h o n o n dispersion and its derivative in the [1 1 0]L mode at 13 and 295 K were also measured. There is not much change of the derivative with temperature. This means that the measurements at a higher temperature, for instance at 850 K, are required to observe the temperature dependence of the derivative, as observed in Pd [9]. The composition dependence of the anomaly position would also need to be studied, changing the size of the Fermi surface in CuPd disordered alloy system.
I
Acknowledgements ................~ ...............................................................................i ~ .
",
Q)
-10
I
A
We would like to thank Dr. K. Yamada of Tohoku University for much helpful advice for neutron scattering experiments.
...
-20
t
References
(b) -30 i
0.5
i
i
i
R e d u c e d wave v e c t o r
1.0
Fig. 3. (a) Dispersion curve and (b) its derivative of the [-~(0]L branch at 25 K. Dotted lines are calculated by BvK model and solid line is a guide to the eye. Open circle and triangle mean that the first collimator is 15' and 30', respectively. in Table 1. The m i n i m u m of the derivative of the [ 1 1 0] L p h o n o n dispersion curve, which is anomalous and sharp, was found at the position indicated by the arrow in Fig. 3(b).
[1] W. Kohn, Phys. Rev. Lett. 2 (1959) 393. [-2] B.N. Brockhouse, T. Arase, G. Caglioti, K.R. Rao and A.D.B. Woods, Phys. Rev. 128 (1962) 1099. [3] G. Nilsson and S. Rolandson, Phys. Rev. B 9 (1974) 3278. [4] R.I. Sharp, J. Phys. C 2 (1969) 432. [5] M.A. Krivoglaz, Theory of X-ray and Thermal Neutron Scattering by Real Crystals (Plenum Press, New York, 1969). [-6] H. Chou, S.M. Shapiro, S.C. Moss and M. Mostoller, Phys. Rev. B 42 (1990) 500. [7] Y. Noda, D.K. Saha and K. Ohshima, J. Phys.: Condens. Matter 5 (1993) 1655. [-8] B.L. Gyorffy and G.M. Stocks, Phys. Rev. Lett. 50 (1983) 374. [9] A.P. Miiller, Can. J. Phys. 53 (1975) 2491.
Physica B 219&220 (1996) 490-492
Anomaly in the phonon dispersion of a disordered alloy, Cuo.715Pdo.285 Y. N o d a a'*, K. Ohshima b, Y. Endoh c aSendai National College of Technology, 1 Kitahara, Kamiayashi, Aoba-ku, Sendai 989-31, Japan blnstitute of Applied Physics, University of Tsukuba, Tsukuba 305, Japan CDepartment of Physics, Tohoku University, Sendai 980, Japan
Abstract The phonon frequencies of a disordered Cu0.715Pd0.2 s 5 alloy were measured using neutron experiments to investigate the relationship between Kohn anomaly and the maxima of the diffuse scattering due to the atomic short-range order, in which the form of the Fermi surface is reflected. Diffuse satellites have been observed clearly at the position expected from the form of Fermi surface. The weak Kohn anomaly of [1 10]L branch has been observed at the position expected from the diffuse satellites.
1. Introduction By considering the screening effect of a free-electron gas on a fluctuating charge distribution such as a lattice vibration, Kohn predicted that there should be lines in reciprocal space through which the dispersion surfaces of the lattice vibration will have infinite slopes [1]. This effect has become known as the Kohn anomaly and, in pure metals, has been observed in Pb by Brockhouse et al. [2], in Cu by Nilson and Rolandson [3], and in Nb by Sharp [4]. The Kohn anomaly corresponds to the logarithmic singularity of the static dielectric function e(q) at q = 2kv, for a spherical Fermi surface of radius kv. Flat parallel sections of the Fermi surface enhance the singularity in the direction perpendicular to the surface; thus the strength of the Kohn anomaly depends on the geometry of the Fermi surface and the details of the electron-phonon interaction. In the case of disordered alloys, Krivoglaz has pointed out that the form of Fermi surface is reflected in the distribution of atomic short-range order diffuse scattering *Corresponding author.
of X-rays, electrons and neutrons via the singularity in the static dielectric function at q = 2kF [5]. The phonon frequencies at room temperature along high-symmetry directions of disordered Cuo.saml0.16 and Cu0.715Pd0.285 alloys have been measured by Chou et al. [6] and Noda et al. [7], respectively, to investigate the relationship between the diffuse maxima due to the atomic short-range order and the Kohn anomalies in phonon dispersion curves. In both cases, no strong anomaly was observed either in the phonon dispersion itself or in its derivative near the expected wave vector. In the CuPd alloy, fourfold split diffuse maxima have been observed sharply at (3, 0.865, 0) and three equivalent positions around the (3, 1, 0) reciprocal lattice point [7], as expected from the geometry of Fermi surface [8]. We have performed phonon measurements under high resolution condition at low temperature to observe weak anomalies of dispersion curve of CuPd alloy. 2. Experimental results Phonon frequencies were measured on the triple axis spectrometer of Tohoku University (TOPAN) installed
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 7 8 8 - 1
491
Y. Noda et al. /Physica B 219&220 (1996) 490 492 100
(a)
=
/~, Eq=21.905
.
~
50
.
.
.
I~t), "
o
31)
i
q)2
~
~.
Eq =20.708
0
-'--..,
°i
*
- - . * = 0 9,
(~0)
(oo~)
c2-~, 2-~, o>
•
. . . . . . . .
",
oe~ 2 0
L
r..
O 2O
i0
)
30
E (meV)
10
I00
(b)
=
--@
z
(2-c, 2-~, o)
E,,=20.708
.... o.o o
3
,,~0
,~ = 0 , 9 4
50
~/~
,
• ./
"%
20 )
'
'
'
'
E (raeV)
0.5
~.,
.0
Fig. 2. Phonon dispersion curves for [~ (0] and [00 ~] directions at 25 K. Curves are results for the second-neighbor BvK model. The arrow indicates the position of 2kv as derived from the position of diffuse satellites.
'k,
oOj
. . . . .
.0
F /
0
0
~_
~
30
Fig. 1. Example of the phonon peaks in Cuo,715Pd0.285 disordered alloy at 25 K. Curves are least-squares fit to the data with the calculation of convolution. Eq is the phonon frequency which is a fitting parameter.
at JRR-3M reactor of JAERI. The sample used was the same single crystal as used in the previous measurements of p h o n o n and diffuse scattering [7]. The sample has an FCC structure with lattice parameter a = 3.719 A and a mosaic spread of 44 min. Pyrolytic graphite was used as both a m o n o c h r o m a t o r and an analyzer. A pyrolytic graphite filter was used before the analyzer to eliminate the higher-order contamination in the beam. Constant-Q scans were made with fixed final energy Ef = 13.8 meV with a set of collimators of 15' 30' 30'-30' or 3 0 ' - 3 0 ' 30' 30'. Measurements were performed at a low temperature of 25 K with an energy resolution of about 0.2 meV. The resolution along the direction parallel to the scattering vector is about 0.02 in reduced wave n u m b e r ( = q/(2rt/a). Additionally, to investigate the temperature dependence of the anomaly, the p h o n o n dispersion of [1 1 0]L mode was measured at 13 K and at room temperature (295 K), with Ef = 13.7 meV and a set of collimators of 3 0 ' - 6 0 ' - 60'-60'. P h o n o n frequencies were measured along the [ ( ~ 0 ] direction in the range of ~ = 0.5-1.0, and along the [( 0 0] direction in the range ( = 0.3 0.7. Anomaly of p h o n o n dispersion along the [ ( ~ 0 ] is expected near ( = 0.935
from the positions of the observed satellites (q = 2kv). Fig. 1 shows examples of observed p h o n o n peaks together with fits using a damped harmonic oscillator function convoluted with the instrumental resolution function. The peak positions, Eq (meV), can be determined with a standard deviation better than 0.1 meV in worse cases of [1 1 0]L mode near the zone boundary. All p h o n o n peaks were fitted with the convolution procedure. Intrinsic p h o n o n line width Y (meV), which is a fitting parameter, at ( = 0.88, 0.92 and 0.94 is 0.83 _+ 0.58, 1.10 ___0.86 and 1.34 + 0.98, respectively. The p h o n o n energy decreases more and more as ( increases from 0.86 to 0.94, as shown in Fig. l(a), while it hardly changes for ( > 0.94, as shown in Fig. l(b). Fig. 2 is a plot of the p h o n o n frequencies together with the curve for a second nearest-neighbor B o r n - v o n K a r m a n (BvK) fit to the data of three observed branches. The force constants of fitting parameters are listed in Table 1. The arrow in Fig. 2 indicates the position of 2kF as derived from the positions of the diffuse satellites. The small anomalous change in the slope of the dispersion curve is detected just at the position of arrow. The p h o n o n frequencies and their derivatives of the [ ( ( 0 ] L branch in the region ( = 0.5-1.0 are plotted in Fig. 3. The open circle and triangle in Fig. 3(a) and (b) mean the different conditions of measurements, where the first collimation is 15' and 30% respectively. Solid line in Fig. 3(b) is guide to the eye. Error bars of p h o n o n frequencies are less than 0.1 meV, then the derivatives (dE/d0 have errors about 10 meV. The dotted line is obtained by the BvK model using force constants shown
492
Y. Noda et al. / Physica B 219&220 (1996) 490-492
Table 1 Force constants of disordered Cu0.715Pdo.2s5 alloy at 25 K Force constants (dyn cm-i) 17470 ± -- 2090 ± 17090 ± -- 720 ± -- 2700 ±
lxx lzz lxy 2xx 2yy
200 250 300 350 410
28
~D 2~
E
........°O""O"-...~ .......
[ ~ ff 0 ] L
o o~ 22
(a)
.....o '~O
....
I ,o .
~
[
t
3. Conclusion The anomaly of the [1 1 0]L p h o n o n dispersion in Cuo.vlsPdo.28s disordered alloy was observed at the position (q = 2kv) expected from the diffuse satellite positions, as reflected by the form of Fermi surface [7]. The dispersion relation of the [1 ( 0 ] branch was measured in the region around the (1,0.865,0) reciprocal lattice point, where the diffuse maximum is observed in the elastic scattering. No anomalous change with the wave vector was observed in the frequency and the intensity of phonon. It is difficult to consider that the anomaly of the derivative of [1 1 0]L p h o n o n dispersion is caused by an incommensurate short-range atomic order. The p h o n o n dispersion and its derivative in the [1 1 0]L mode at 13 and 295 K were also measured. There is not much change of the derivative with temperature. This means that the measurements at a higher temperature, for instance at 850 K, are required to observe the temperature dependence of the derivative, as observed in Pd [9]. The composition dependence of the anomaly position would also need to be studied, changing the size of the Fermi surface in CuPd disordered alloy system.
I
Acknowledgements ................~ ...............................................................................i ~ .
",
Q)
-10
I
A
We would like to thank Dr. K. Yamada of Tohoku University for much helpful advice for neutron scattering experiments.
...
-20
t
References
(b) -30 i
0.5
i
i
i
R e d u c e d wave v e c t o r
1.0
Fig. 3. (a) Dispersion curve and (b) its derivative of the [-~(0]L branch at 25 K. Dotted lines are calculated by BvK model and solid line is a guide to the eye. Open circle and triangle mean that the first collimator is 15' and 30', respectively. in Table 1. The m i n i m u m of the derivative of the [ 1 1 0] L p h o n o n dispersion curve, which is anomalous and sharp, was found at the position indicated by the arrow in Fig. 3(b).
[1] W. Kohn, Phys. Rev. Lett. 2 (1959) 393. [-2] B.N. Brockhouse, T. Arase, G. Caglioti, K.R. Rao and A.D.B. Woods, Phys. Rev. 128 (1962) 1099. [3] G. Nilsson and S. Rolandson, Phys. Rev. B 9 (1974) 3278. [4] R.I. Sharp, J. Phys. C 2 (1969) 432. [5] M.A. Krivoglaz, Theory of X-ray and Thermal Neutron Scattering by Real Crystals (Plenum Press, New York, 1969). [-6] H. Chou, S.M. Shapiro, S.C. Moss and M. Mostoller, Phys. Rev. B 42 (1990) 500. [7] Y. Noda, D.K. Saha and K. Ohshima, J. Phys.: Condens. Matter 5 (1993) 1655. [-8] B.L. Gyorffy and G.M. Stocks, Phys. Rev. Lett. 50 (1983) 374. [9] A.P. Miiller, Can. J. Phys. 53 (1975) 2491.