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Evaluation of force constant ratio in the AlCu system
Solid State Communications,Vol. 15~,pp.1401—1402, 1974.
Pergamon Press.
Printed in Great Britain
EVALUATION OF FORCE CONSTANT RATIO IN THE Al—Cu SYSTEM R. Munjal, D. Raj and S.P. Pun Punjab Agricultural University, Department of Physics, 141004 Ludhiana, India (Received 22 July 1974 by A.A. Maradudin)
Band model calculations, using Mannheim’s theory of nearest neighbour central force approximation in a cubic crystal, have been carried out to estimate the impurity—host to host—host coupling ratio X’/X in the Al—Cu system. The obtained value of 1.22 for X’/X suggests about 20 per cent stiffening of the impurity to host binding.
f
1 + (M?~~M)2P ~m g(w)d w~2~0 w ~
INTRODUCTION of an isolated substitutional impurity in an otherwise perfect lattice site can produce two types of vibrational modes, the local mode1 and the resonant mode.2 The local mode has frequency higher than the maximum frequency of the perfect host crystal and its amplitude dies out very rapidly with distance from the defect site. A resonant (or band) mode, on the other hand, has a frequency in the allowed band of frequencies to the perfect crystal and its amplitude though peaked, does not fall off as rapidly as that of the local mode. The existence of a particular type of mode depends upon the characteristics of the impurity—host system and is the result of one or a combination of possible changes in mass, force constant and charge at the defect site.
(1)
where M and M’ are the masses of the host and impurity atoms, g(w) is the density of pure crystal states with Wm as the maximum frequency and the principal value of the integral is required. Mannheim5 and Takeno6 in addition to the mass defect have also considered the force constant changes. In the nearest neighbour central force approximation their expression for the impurity mode in case of b.c.c. and f.c.c. crystal is given by5 1 + p(w)s(w) = 0 (2) where M 2w2 1 (3)
~ ( ~) —
Various experimental techniques,3 most promi-
‘2
nent among them being Infrared absorption, Raman spectroscopy, Neutron inelastic scattering, etc., have been instrumental in finding out their existence. Model calculations based upon different assumptions and approximations in certain type of crystal symmetries have also been carried out to predict the impurity mode positions. The most simple and common among these is the mass defect approximation (MDA), where only change in mass at the defect site (i.e. isotropic impurity) is taken into account. In this approximation, the vibrational frequencies of the impurity modes are given by4 the solution of
s(w)
=
I
~ Jr wwg(w ~ )dw w2 —
~
(4)
A is the host—host and A’ is the impurity—host coupling constant respectively. For A’ = A equation (2) reduces to equation (1) of Dawber and Elliott.4 The equation (2) can also be used to estimate the X’/A ratio if g(w) of the host and the position of the impurity mode is known by any of the mentioned techniques. Recently local modes have been observed7 at about 9.2 X 1012 Hz by inelastic neutron scattering at 1401
1402
FORCE CONSTANT RATIO IN THE Al—Cu SYSTEM
modest concentrationof light mass impurities in Al—Cu alloys. Kaplan and Mostaller8 using Coherent Potential approximation (CPA) and MDA and employing the density of state functiong(w) of Nicklow et al.9 have determined the position of local mode peak at 8.9 X 1012 Hz. Though the agreement with the experimentally reported value is not very unsatisfactory the difficulties encountered in matching both the in band and local mode frequencies, the change of volume on alloying and the different outer electronic structure of Cu and Al are indicative of that the forces do vary significantly around the different constituents. They in fact feel the necessity of incorporation of force constant changes in the CPA. In view of this discrepancy and the recently reported new experimentally observeU g(w) of Cu by Svensson et al.1°we have carried out calculations based on equation (2) to estimate the impurity—host to host— host force constant ratio in Al 0~1Cu0~9 system. The frequency wave vector dispersion relations in copper at room temperature has been determined by inelastic neutron scattering by ~venssonet al)°The frequency distribution function has been obtained using M2 (fourth neighbour), M3 (sixth neighbour) and M4 (eighth neighbour) general force constant model. Using these g(w) the value of force constant ratio A’/A has been calculated using equation (2) and the results are given in Table 1.
1. 2. 3. 4~ 5. 6. 7. 8. 9. 10.
Vol. 15, No.8
Table 1. Force constant ratio A’/A due to Al impurity ifl Qs wlocal mode = 9.2 X 1012Hz g(w) model M2 M3 M4
Wm
X 10’2Hz 7.495 7.395 7 295
A/A 1.233 1 222 1219
The value of local mode has also been calculated in the MDA equation (1) and it comes out to 8.45 X 1 012 Hz. This value as compared to the experimentally reported value of 9.2 X 1012 Hz clearly indicates that MDA is inadequate for treating the Al impurity in copper lattice and suggests the necessity to incorporate the change of force constant in the vicinity of the impurity. Furthermore, the lower value of the localised mode frequency as compared to the experimental one also suggests an increase in stiffening of approximately 20 per cent and this is in reasonably good agreement with the calculated value of 1.22 (Table I) from the equation (2) which incorporate both the mass change as well as force constant change. The results of the table also show that the value of X’IA is independent of the models used to calculate the g(w) from the dispersion relation. Acknowledgements Two of the authors R.M. and D.R. are thankful to Council of Scientific and Industrial Research, India for the award of Junior Research Fellowship and the award of Poolship respectively. —
REFERENCES LIFSHITZ I.M.,Nuovo cim. 3 Suppl., 716 (1957). BROUT R. and VISSCHERW.M.,Phys. Rev. Lett. 9,54(1972). MARADUDIN A.A., Elementary Excitation in Solids (edited by MARADUDIN A.A. and NARDELLI G.F.) p. 35, Plenum Press, New York (1969). ~ D.G. and ELLIOTF R.J.,F~oc.R. Soc. (London) A273, 222 (1963). MANNHEIMP.D.,Phys.Rev. 165, 1011 (1968). TAKENO S., Localised Excitation in Solids (edited by WALLIS R.F.) p. 85, Plenum Press, New York (1968); SIEVERS A.J. and TAKENO S.,Phys. Rev. 140A, 1033 (1965). (The same equation (2) used in the text had been obtained in a mathematically different form). NICKLOW R.M., VIJAYARAGHAVAN P.R., SMITH H.G., DOLLING G. and WILKINSON M.K.,Neutron Inelastic Scattering (IAEA, Vienna) 1, 47 (1968). KAPLAN T. and MOSTOLLER M.,Phys. Rev. B9, 353 (1974). NICKLOW R.M., GILAT G., SMITH H.G., RAUBENHEIMER L.J. and WILKINSON M.K.,Phys. Rev. 164, 922 (1967). SVENSSON E.C., BROCKHOUSE B.N. and ROWE J.M.,Phys. Rev. 155, 619 (1967).
Pergamon Press.
Printed in Great Britain
EVALUATION OF FORCE CONSTANT RATIO IN THE Al—Cu SYSTEM R. Munjal, D. Raj and S.P. Pun Punjab Agricultural University, Department of Physics, 141004 Ludhiana, India (Received 22 July 1974 by A.A. Maradudin)
Band model calculations, using Mannheim’s theory of nearest neighbour central force approximation in a cubic crystal, have been carried out to estimate the impurity—host to host—host coupling ratio X’/X in the Al—Cu system. The obtained value of 1.22 for X’/X suggests about 20 per cent stiffening of the impurity to host binding.
f
1 + (M?~~M)2P ~m g(w)d w~2~0 w ~
INTRODUCTION of an isolated substitutional impurity in an otherwise perfect lattice site can produce two types of vibrational modes, the local mode1 and the resonant mode.2 The local mode has frequency higher than the maximum frequency of the perfect host crystal and its amplitude dies out very rapidly with distance from the defect site. A resonant (or band) mode, on the other hand, has a frequency in the allowed band of frequencies to the perfect crystal and its amplitude though peaked, does not fall off as rapidly as that of the local mode. The existence of a particular type of mode depends upon the characteristics of the impurity—host system and is the result of one or a combination of possible changes in mass, force constant and charge at the defect site.
(1)
where M and M’ are the masses of the host and impurity atoms, g(w) is the density of pure crystal states with Wm as the maximum frequency and the principal value of the integral is required. Mannheim5 and Takeno6 in addition to the mass defect have also considered the force constant changes. In the nearest neighbour central force approximation their expression for the impurity mode in case of b.c.c. and f.c.c. crystal is given by5 1 + p(w)s(w) = 0 (2) where M 2w2 1 (3)
~ ( ~) —
Various experimental techniques,3 most promi-
‘2
nent among them being Infrared absorption, Raman spectroscopy, Neutron inelastic scattering, etc., have been instrumental in finding out their existence. Model calculations based upon different assumptions and approximations in certain type of crystal symmetries have also been carried out to predict the impurity mode positions. The most simple and common among these is the mass defect approximation (MDA), where only change in mass at the defect site (i.e. isotropic impurity) is taken into account. In this approximation, the vibrational frequencies of the impurity modes are given by4 the solution of
s(w)
=
I
~ Jr wwg(w ~ )dw w2 —
~
(4)
A is the host—host and A’ is the impurity—host coupling constant respectively. For A’ = A equation (2) reduces to equation (1) of Dawber and Elliott.4 The equation (2) can also be used to estimate the X’/A ratio if g(w) of the host and the position of the impurity mode is known by any of the mentioned techniques. Recently local modes have been observed7 at about 9.2 X 1012 Hz by inelastic neutron scattering at 1401
1402
FORCE CONSTANT RATIO IN THE Al—Cu SYSTEM
modest concentrationof light mass impurities in Al—Cu alloys. Kaplan and Mostaller8 using Coherent Potential approximation (CPA) and MDA and employing the density of state functiong(w) of Nicklow et al.9 have determined the position of local mode peak at 8.9 X 1012 Hz. Though the agreement with the experimentally reported value is not very unsatisfactory the difficulties encountered in matching both the in band and local mode frequencies, the change of volume on alloying and the different outer electronic structure of Cu and Al are indicative of that the forces do vary significantly around the different constituents. They in fact feel the necessity of incorporation of force constant changes in the CPA. In view of this discrepancy and the recently reported new experimentally observeU g(w) of Cu by Svensson et al.1°we have carried out calculations based on equation (2) to estimate the impurity—host to host— host force constant ratio in Al 0~1Cu0~9 system. The frequency wave vector dispersion relations in copper at room temperature has been determined by inelastic neutron scattering by ~venssonet al)°The frequency distribution function has been obtained using M2 (fourth neighbour), M3 (sixth neighbour) and M4 (eighth neighbour) general force constant model. Using these g(w) the value of force constant ratio A’/A has been calculated using equation (2) and the results are given in Table 1.
1. 2. 3. 4~ 5. 6. 7. 8. 9. 10.
Vol. 15, No.8
Table 1. Force constant ratio A’/A due to Al impurity ifl Qs wlocal mode = 9.2 X 1012Hz g(w) model M2 M3 M4
Wm
X 10’2Hz 7.495 7.395 7 295
A/A 1.233 1 222 1219
The value of local mode has also been calculated in the MDA equation (1) and it comes out to 8.45 X 1 012 Hz. This value as compared to the experimentally reported value of 9.2 X 1012 Hz clearly indicates that MDA is inadequate for treating the Al impurity in copper lattice and suggests the necessity to incorporate the change of force constant in the vicinity of the impurity. Furthermore, the lower value of the localised mode frequency as compared to the experimental one also suggests an increase in stiffening of approximately 20 per cent and this is in reasonably good agreement with the calculated value of 1.22 (Table I) from the equation (2) which incorporate both the mass change as well as force constant change. The results of the table also show that the value of X’IA is independent of the models used to calculate the g(w) from the dispersion relation. Acknowledgements Two of the authors R.M. and D.R. are thankful to Council of Scientific and Industrial Research, India for the award of Junior Research Fellowship and the award of Poolship respectively. —
REFERENCES LIFSHITZ I.M.,Nuovo cim. 3 Suppl., 716 (1957). BROUT R. and VISSCHERW.M.,Phys. Rev. Lett. 9,54(1972). MARADUDIN A.A., Elementary Excitation in Solids (edited by MARADUDIN A.A. and NARDELLI G.F.) p. 35, Plenum Press, New York (1969). ~ D.G. and ELLIOTF R.J.,F~oc.R. Soc. (London) A273, 222 (1963). MANNHEIMP.D.,Phys.Rev. 165, 1011 (1968). TAKENO S., Localised Excitation in Solids (edited by WALLIS R.F.) p. 85, Plenum Press, New York (1968); SIEVERS A.J. and TAKENO S.,Phys. Rev. 140A, 1033 (1965). (The same equation (2) used in the text had been obtained in a mathematically different form). NICKLOW R.M., VIJAYARAGHAVAN P.R., SMITH H.G., DOLLING G. and WILKINSON M.K.,Neutron Inelastic Scattering (IAEA, Vienna) 1, 47 (1968). KAPLAN T. and MOSTOLLER M.,Phys. Rev. B9, 353 (1974). NICKLOW R.M., GILAT G., SMITH H.G., RAUBENHEIMER L.J. and WILKINSON M.K.,Phys. Rev. 164, 922 (1967). SVENSSON E.C., BROCKHOUSE B.N. and ROWE J.M.,Phys. Rev. 155, 619 (1967).