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On phenomenological models of lattice dynamics
SolidState Communications,
Vol. 14, pp. 307—308, 1974.
Pergamon Press.
Printed in Great Britain
ON PHENOMENOLOGICAL MODELS OF LATTICE DYNAMICS Manjeet Kaur and SS. Kushwaha Department of Physics, Banaras Hindu University, Varanasi—221005, India (Received 20 June 1973 by 0. V. Lounasmaa)
The metallic interaction is split into (i) closed-shell part and (ii) ion— electron—ion part. The former is evaluated using non-central forces upto the third neighbour and the latter is obtained from the concept of bulk modulus of the electron gas. Phonon dispersion relations of h.c.p. scandium have been calculated and compared with the experimental results.
1. INTRODUCTION 13 authors used electron—gas models t 0 SEVERAL study the lattice dynamics of h.c.p. metals. These authors have neglected the ion—electron—ion contri-
I
B
11,101
~
II
IOtI0~
00011
TO~
-
types of atoms in the basis of h.c.p. structure. This bution co~dbe corresponding possible in a to naive themodel interaction wherebetween electronstwo do not contribute to the interaction between atoms
TO,, 2
As constituting a matter of thefact basis. electrons Physically should thiscontribute is not tenable. to this interaction and this should be stronger than the contribution for farther neighbours.4 The aim of this short communication is therefore two-fold: (a) to obtain the correct expressions for the ion—electron—
r
$~
r
N
A
FIG. 1. Phonon dispersion relations for scandium have been shown by solid lines. Experimental points X (LA/LO), O(TA/TO) along ~A ,v (TA 11), A (TA1), • (LA), 0 (TO11 ITO1). X (LO) along F’M and 0 (TA1/T01) along l’K have been shown for compari.
ion interaction in h.c.p. metals and (b) to study the phonon dispersion relations of scandium, which so far did not receive proper attention from the theoretical view point. Scandium is the lightest element having electronic configuration similar to that of rare earth metals.
D(q)
IA (q) ~*(q)
B(q)\ A(q))
(2)
A(q) and B(q) are 3 X 3 submatrices and B* is the complex conjugate of B. In the present approach the
2. THEORETICAL MODEL The secular determinant for phonon frequencies is usually written as 2II = 0 (1) ID(q) —mw where m is hte mass of the atom,! is the unitary matrix. The dynamical matrix D(q) is expressed by
matrix elements are expressed as (Afl)
=
(A1j)c + (A1.)e
(3) 11) = (B11)c + (B1~)e where superscripts c and e stand for closed-shell part and ion—electron—ion part respectively. (B
307
308
MODELS OF LATTICE DYNAMICS
Six force constants (three central a,(3,7 and three angulara’,~’,y’)appear in the matrix elements for the closed-shell part. The contribution for ion—electron— ion part has been obtained following Sharma and Joshi5 method for cubic metals. it is assumed thai the bulk modulus of the electron gas in the basal plane ~ is different from that in the c direction kez. The new expressions for (Au)e and (Ba) are (A,1)e = qgqjke 2 (x)1Z?(x) exp {—iq-r (B11)e = q, q1 ke flg 12
}
(4) (5)
where k4 isisobtained the bulk using modulus of the electron gas and its value expression for the bulk modulus of Bhatia.6 Symbols in expressions (4) and (5) are the same as given in reference 5. TI2 is the vector distance between the two atoms in the unit cell. The secular determinant (1), when expanded in long wavelength limit (q 0), gives five elastic constants in terms of eight parameters. The Born—Huang condition for the lattice equilibrium gives a relation —‘
a +
— ~,,
iP
~
,
3 c2 ~ -~(o+’y) a ,
=
,
(6)
VoL 14, No. 4
determined using experimental elastic constants expression (6) for lattice equilibrium, and two frequencies from the neutron scattering data of Sinha et aL8 The input data for fixing the force constants is as follows: C 11 = 0.993, C12 = 0.268, ~ = 0.336, C33 = 1.069, C13 = 0.294 (all in units of 1012 dyn/cm), vro1(M) = 623, PLO(r) = 6.91 (in units of THZ). The lattice parameters are: a = 3.3 A and c = 5.268 A. The value of CIA has been arbitrarily increased from 0.277 X 1012 dyn/cm to 0.336 X 1012 dyn/cm to get an overall good fit. The force constants are found to 4dyn/cm2), bea= 1.804l,~=1.7971,7= —0.772,a= 0.2771, = 0.458 and ~y= —0.1692 (all in units of lO ke~= —0.004 and ke~= 0.074 (in units of lO’2dyn/cm). These force constants are used to calculate the phonon dispersion relations of scandium. The calculated frequencies are not periodic in the reciprocal space. This shortcoming may, however, be removed by using a screened coulomb interaction between the ions.9 The expressions (4) and (5) will therefore involve the summation over reciprocal lattice vectors. The calculated results have been plotted along with the neutron scattering data of Sinha et aL in Fig. 1. There is good agreement between the theoretical and experimental results except in the case of TO
In this model the angular forces also contribute to the Cauchy-discrepancy through the following relation C13
—
CIA
=
2ke2
2..J3c —
~
(~.V+ ~‘)
(7)
11 mode along [O1TO] direction, where the discrepancy between the calculated and experimental results is quite significant. It has been noted by the present authors that there is significant contribution to CIA as well as to TO11 from fourth and fifth neighbours. But we have restricted the number of parameters for obvious reasons.
3. RESULTS AND DISCUSSION
Six parameters besides kex and kez have been REFERENCES 1.
GUPTA R.P. and DAYALB.,Phys. Status. Solidi. 8, 115 (1965).
2. 3.
SHARAN B. and BAJPAI R.P.,Phys. Lett. 31A, 120(1970). RAJPUT J.S. and KUSHWAHA S.S.,Phys. Lert. 38A, 497 (1972).
4.
JOSH! S.K., Private communication.
5. 6. 7.
SHARMA P.K. and JOSHI S.K.,J. Chem. Phys. 39, 2633 (1963). BHATIA A.B.,Phys. Rev. 92, 363 (1955). FISHER E.S. and DEVER D., Proc. 7th Rare Earth Research Conf 1, 237 (1968).
8.
WAKABAYASHI N., SINHA S.K. and SPEDDING F.H.,Phys. Rev. B4, 2398 (1971).
9.
KR.ESK.,Phys. Rev, 138,Al43(1965). L’interaction métallique est divisée en deux parties: (i) la contribution des couches completes, Ct (ii) la contribution ion—electron—ion’ La premiere est Cvaluée en considCrant des forces non-centrales jusqu’au troisiIme voisin, et la seconde est dCduite du concept de compressibilite du gas d’électrons. Las relations de dispersion des phonons dans le scandium h.c. ont Cté calculCes et comparCes aux rCsultats expCrimentaux.
Vol. 14, pp. 307—308, 1974.
Pergamon Press.
Printed in Great Britain
ON PHENOMENOLOGICAL MODELS OF LATTICE DYNAMICS Manjeet Kaur and SS. Kushwaha Department of Physics, Banaras Hindu University, Varanasi—221005, India (Received 20 June 1973 by 0. V. Lounasmaa)
The metallic interaction is split into (i) closed-shell part and (ii) ion— electron—ion part. The former is evaluated using non-central forces upto the third neighbour and the latter is obtained from the concept of bulk modulus of the electron gas. Phonon dispersion relations of h.c.p. scandium have been calculated and compared with the experimental results.
1. INTRODUCTION 13 authors used electron—gas models t 0 SEVERAL study the lattice dynamics of h.c.p. metals. These authors have neglected the ion—electron—ion contri-
I
B
11,101
~
II
IOtI0~
00011
TO~
-
types of atoms in the basis of h.c.p. structure. This bution co~dbe corresponding possible in a to naive themodel interaction wherebetween electronstwo do not contribute to the interaction between atoms
TO,, 2
As constituting a matter of thefact basis. electrons Physically should thiscontribute is not tenable. to this interaction and this should be stronger than the contribution for farther neighbours.4 The aim of this short communication is therefore two-fold: (a) to obtain the correct expressions for the ion—electron—
r
$~
r
N
A
FIG. 1. Phonon dispersion relations for scandium have been shown by solid lines. Experimental points X (LA/LO), O(TA/TO) along ~A ,v (TA 11), A (TA1), • (LA), 0 (TO11 ITO1). X (LO) along F’M and 0 (TA1/T01) along l’K have been shown for compari.
ion interaction in h.c.p. metals and (b) to study the phonon dispersion relations of scandium, which so far did not receive proper attention from the theoretical view point. Scandium is the lightest element having electronic configuration similar to that of rare earth metals.
D(q)
IA (q) ~*(q)
B(q)\ A(q))
(2)
A(q) and B(q) are 3 X 3 submatrices and B* is the complex conjugate of B. In the present approach the
2. THEORETICAL MODEL The secular determinant for phonon frequencies is usually written as 2II = 0 (1) ID(q) —mw where m is hte mass of the atom,! is the unitary matrix. The dynamical matrix D(q) is expressed by
matrix elements are expressed as (Afl)
=
(A1j)c + (A1.)e
(3) 11) = (B11)c + (B1~)e where superscripts c and e stand for closed-shell part and ion—electron—ion part respectively. (B
307
308
MODELS OF LATTICE DYNAMICS
Six force constants (three central a,(3,7 and three angulara’,~’,y’)appear in the matrix elements for the closed-shell part. The contribution for ion—electron— ion part has been obtained following Sharma and Joshi5 method for cubic metals. it is assumed thai the bulk modulus of the electron gas in the basal plane ~ is different from that in the c direction kez. The new expressions for (Au)e and (Ba) are (A,1)e = qgqjke 2 (x)1Z?(x) exp {—iq-r (B11)e = q, q1 ke flg 12
}
(4) (5)
where k4 isisobtained the bulk using modulus of the electron gas and its value expression for the bulk modulus of Bhatia.6 Symbols in expressions (4) and (5) are the same as given in reference 5. TI2 is the vector distance between the two atoms in the unit cell. The secular determinant (1), when expanded in long wavelength limit (q 0), gives five elastic constants in terms of eight parameters. The Born—Huang condition for the lattice equilibrium gives a relation —‘
a +
— ~,,
iP
~
,
3 c2 ~ -~(o+’y) a ,
=
,
(6)
VoL 14, No. 4
determined using experimental elastic constants expression (6) for lattice equilibrium, and two frequencies from the neutron scattering data of Sinha et aL8 The input data for fixing the force constants is as follows: C 11 = 0.993, C12 = 0.268, ~ = 0.336, C33 = 1.069, C13 = 0.294 (all in units of 1012 dyn/cm), vro1(M) = 623, PLO(r) = 6.91 (in units of THZ). The lattice parameters are: a = 3.3 A and c = 5.268 A. The value of CIA has been arbitrarily increased from 0.277 X 1012 dyn/cm to 0.336 X 1012 dyn/cm to get an overall good fit. The force constants are found to 4dyn/cm2), bea= 1.804l,~=1.7971,7= —0.772,a= 0.2771, = 0.458 and ~y= —0.1692 (all in units of lO ke~= —0.004 and ke~= 0.074 (in units of lO’2dyn/cm). These force constants are used to calculate the phonon dispersion relations of scandium. The calculated frequencies are not periodic in the reciprocal space. This shortcoming may, however, be removed by using a screened coulomb interaction between the ions.9 The expressions (4) and (5) will therefore involve the summation over reciprocal lattice vectors. The calculated results have been plotted along with the neutron scattering data of Sinha et aL in Fig. 1. There is good agreement between the theoretical and experimental results except in the case of TO
In this model the angular forces also contribute to the Cauchy-discrepancy through the following relation C13
—
CIA
=
2ke2
2..J3c —
~
(~.V+ ~‘)
(7)
11 mode along [O1TO] direction, where the discrepancy between the calculated and experimental results is quite significant. It has been noted by the present authors that there is significant contribution to CIA as well as to TO11 from fourth and fifth neighbours. But we have restricted the number of parameters for obvious reasons.
3. RESULTS AND DISCUSSION
Six parameters besides kex and kez have been REFERENCES 1.
GUPTA R.P. and DAYALB.,Phys. Status. Solidi. 8, 115 (1965).
2. 3.
SHARAN B. and BAJPAI R.P.,Phys. Lett. 31A, 120(1970). RAJPUT J.S. and KUSHWAHA S.S.,Phys. Lert. 38A, 497 (1972).
4.
JOSH! S.K., Private communication.
5. 6. 7.
SHARMA P.K. and JOSHI S.K.,J. Chem. Phys. 39, 2633 (1963). BHATIA A.B.,Phys. Rev. 92, 363 (1955). FISHER E.S. and DEVER D., Proc. 7th Rare Earth Research Conf 1, 237 (1968).
8.
WAKABAYASHI N., SINHA S.K. and SPEDDING F.H.,Phys. Rev. B4, 2398 (1971).
9.
KR.ESK.,Phys. Rev, 138,Al43(1965). L’interaction métallique est divisée en deux parties: (i) la contribution des couches completes, Ct (ii) la contribution ion—electron—ion’ La premiere est Cvaluée en considCrant des forces non-centrales jusqu’au troisiIme voisin, et la seconde est dCduite du concept de compressibilite du gas d’électrons. Las relations de dispersion des phonons dans le scandium h.c. ont Cté calculCes et comparCes aux rCsultats expCrimentaux.