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Phonon dispersion of alkali metals and the LA23(1, 1, 1) anomaly in bcc structures
15 September
1997
PHYSICS ELSEVIER
LETTERS
A
Physics Letters A 234 (1997) 134-140
Phonon dispersion of alkali metals and the LA$( 1, 1, 1) anomaly in bee structures M. Li, N.X. Chen CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China Institute of Applied Physics, Beijing University of Science and Technology, Beijing 100083, China Received 31 January
1997; revised manuscript received 27 May 1997; accepted for publication Communicated by A.R. Bishop
’
12 June 1997
Abstract The phonon dispersion of body-centered cubic metals Li, Na, K and Rb is calculated based on the Mijbius inversion-converted pair potentials and a Slater-Kirkwood-type three-body interaction. The calculated results are in agreement with the experimental data from the inelastic neutron scattering. Then the LA*(l) 1, 1) anomaly is discussed. The results indicate that the LA$(l, 1, 1) anomaly can also be explained by use of the three-body interaction in the Born-von Karman model
framework. 0 1997 Elsevier Science B.V. PACS: 63.2O.Dj; 63.20. - e; 02.30.Em
1. Introduction
The phonon dispersion in alkali metals Li, Na, K and Rb has been measured using the inelastic neutron scattering method [l-6]. Various theoretical methods have been proposed to explain the experimental data. Some ab initio calculations give good results on phonon dispersion in alkali metals [7-141, especially, the calculations by Frank et al. [ 151 based on the local density approximation and norm-conserving pseudopotentials give some nice results, in which the crossing of the longitudinal and transverse branches for Li along the [loo] direction is explained. Some classical calculations of the phonon dispersion in the Born-von Karman model have been carried out [ 161. In these studies it is found that there is a large dip in the phonon frequency near q = f along the [ 1111 direction for the longitudinal branch. This interesting phenomenon attracted attention and has been studied carefully [17-231. Falter et al. pointed out that this was caused by an intrinsic geometrical effect of the bee lattice and this so-called LA$l, 1, 11 anomaly is characteristic of bee metals [20]. In the present paper the phonon dispersion in alkali metals Li, Na, K and Rb has been calculated using the method developed by the authors [24] based on the MGbius theorem in number theory [25], by which the pair potential can be obtained from the cohesive energy and then the force constant and the phonon dispersion can be
’ Mailing address 0375-9601/97/$17.00 0 1997 Else&r PII SO375-9601(97)00497-O
Science B.V. All rights reserved.
M. Li. N.X. Chen/Physics
Letters A 234 (1997) 134-140
135
Table 1 The values of B. r,\. U and I) [261
Materials
B (10” dyne/cm)
r(, (IO-” cm)
U (eV/atom)
p (108/cm)
Li Na K Rb
0.116 0.068 0.032 0.031
3.491 4.225 5.225 5.585
1.630 1.113 0.934 0.852
0.5911 0.6025 0.5017 0.5346
calculated. The results are in agreement with the neutron scattering data and indicate that the LA$[l, 1, l] anomaly can be explained by means of a three-body interaction with a suitable choice of adjustable parameters. In Section 2 the cohesive energy is given as a function of some properties of the metals in order to convert their pair potentials. In Section 3 the pair potentials are converted by use of the Mobius theorem and the calculated cohesive energies. A three-body interaction is used and the phonon dispersion is calculated in Section 4. In Section 5 the results are discussed.
2. The cohesive
energies
of Li, Na, K and Rb
The cohesive energy curve for bee structure can be obtained by ab initio calculations. paper it is assumed that the cohesive energy can be expressed as a Morse potential E(r)
= U(ep2P(‘-‘o)
But in the present
- 2e-P(‘-‘0)),
(1)
where r is the lattice constant of the metal, U the cohesive energy for each atom at equilibrium, r. the equilibrium lattice constant and p a parameter related to the bulk modulus B. For alkali metals, the values of U, B and r,, are taken from experimental data [26] and the modulus B can be written as B = 4Up2/9ro.
(2)
Therefore, p=;JBr,/u.
(3)
The experimental values of U, B, r,, and the calculated resultant cohesive energies are
parameter
p
for alkali metals are listed in Table 1. The
Li:
E(r)
= 1.630(61 .9954e-‘~‘822’ - 15.7474~-~~~~“‘),
(4)
Na:
E(r)
= 1.113( 196.6155e-‘~2050’ - 28.0439e-“~6025r)
(5)
K:
E(r)
= 0.934( 189.1925eC’ .0034r- 27.5095e-0.50’7’),
(6)
Rb:
E(r)
= 0.852(392.0863e-‘.0692r
(7)
3. Pair potentials
- 39.6025e-0.5346r),
of Li, Na, K and Rb
If many-body
interactions
E(r)=3
5 d)(IRI), IRl#O
are ignored,
the cohesive energy per atom can be approximated
as [27]
(8)
136
M. Li, N.X. Chen/
Physics Letters A 234 (1997) 134-140
where C#Iis the pair potential and /R /the distance structure Eq. (8) can be written as [28] E(r)
=3
+(nr)
t
+6
f
n=l
P-4’
q!~(/m
of an atom from the reference
one at the origin. For bee
r)
I
+4Pi=,[+(G777r)+O(/(~-f)2+(~-f)2+(u-i)2 7I
+4
r)]
[+(@T&7r)+~({(p-f)2+(q-f)2+(u-f)2r
e
r)],
(9)
p#q#u=l
where r is the lattice constant. E(r)
We can divide the cohesive energy
E into two parts,
+E2(r),
(10)
[4(~&nr)+~q5(nr)]
(11)
=E,(r)
where E,(r)=4C
n=l
and E2( r) = 6
(P(h-7 r)
i
f4
e
[+(@T&7r)+4(jl(p-f)2+(g-f)2+(u-i)2
We introduce
r)].
I
p#q#u=
two operators
(12)
B, and B, such that
E,(r)
= &4(r)
(13)
E,(r)
= B,+(r).
(14)
and
So we have E(r)=(B,+B,)+(r)=B,(l+B,‘B:,)4(r).
Applying
an inverse operation
(15)
the pair potential
c#dr) can be obtained
immediately,
+(r)=(l+B;‘B,)-‘B-‘E(r).
(16)
By using the Mobius theorem in number theory it has been proved that [28]
B;‘f(r)=am~l(-l)“‘(~)m-‘~(n)~[(z/a)”,,], where fir> is a real function = 1, /..L(“) =(+, = 0,
and /.~(a> is the Mobius function
if n=
(17)
[25]
1,
if n includes
s distinct primes,
if n includes repeated factors.
(‘8)
M. Li, N.X. Chen/Physics
Letters A 234 (1997) 134-140
137
Table 2 The values of &,. n, and q Materials
&, (ev)
a,, (1O-8 cm)
q (108/cm)
Li Na K Rb
0.07833 0.07480 0.06215 0.06745
4.78 5.08 6.08 6.04
0.6356 0.6407 0.5271 0.5660
The cohesive energy E(r) decreases rapidly when r is increased. Therefore, when Eq. (16) is expanded, just a few terms need to be taken into account and then the pair potential can be calculated using Eq. (17). The calculations have been carried out for alkali metals Li, Na, K and Rb. It is interesting that the calculated results can also be fitted as Morse potentials with the parameters in Table 2, +(r)
= +O(e-W-%)
Li:
I
= 0.07833(e-‘.2712(‘-4.78)
Na:
b(r)
= 0,07480(e-
K:
+(r)
= ()06215(~-1.054X-6.08)
_ 2e-O-527’(‘-6.08)),
Rb:
$(r)
= 0.-~6745(~-‘.t32O(r-6.04)
_ 2e-0.5660W6.04)).
Based on the calculated
4. Phonon dispersion
- 2e-4(r-%)),
(19) _ 2,-0.6356(r-4.78)),
(20)
1.2814(r-5.08)_ 2e-0.6407(‘-5.08)),
pair potential
(21) (22) (23)
we can give the vibrational
in Li, Na, K and Rb and a three-body
potential
interaction
Using the calculated pair potential we can give the total potential. potential @ can be expressed by the displacement of atoms [29], @= 3 C
C
@.
In the harmonic
@ahp.tK.jvUa,ixUfi.jv)
approximation
the total
(24)
i~~r jv/3
where uOI,iKis the displacement of the K th atom in the ith unit cell along the a direction from the equilibrium position. - @abip,iK,jVis known as the atomic force constant. It is the a-component of the force exerted on the K th atom in the ith unit cell when the v th atom in the jth unit cell moves a unit displacement along the p direction. In the lattice dynamics, the atomic force constant - @a,8iK,jV can be obtained as
(25) where the subscript 0 means the derivatives Solving the secular equation det I w2&&
- Dcrp.J
we can obtain the dispersion
are taken at the equilibrium
k) I = 0, relations
D crp.K”(k) = wKK-“2
or frequency
(26) w as a function
,FR @c2p.itr,jvexP[-ik * (Ri 1
position.
of wave vector k. In Eq. (26), Dap,,,(k)
-Rj)])
is
(27)
I
which expresses the elements of the dynamic matrix D. M, is the mass of the K th atom and Ri is the position vector of the ith unit cell. In our calculation contributions up to fourth nearest neighbours are involved. By using Eqs. (20)-(27) the dispersion curves for Li, Na, K and Rb are calculated for three major directions [ IOO], [ 1 IO] and [ 1111. The calculated results qualitatively agree with the experiments. The discrepancy might be
M. Li, N.X. Chen/ Physics Letters A 234 (1997) 134-140
138 Table 3 The values of A, C and Q Materials
A (eV)
C (eV cm9)
a (108/cm)
Li Na K Rb
-4.5718 - 1.6513 - 2.0685 - 1.6728
5802.2 10439.0 128460.0 164140.0
0.17835 0.14915 0.11279 0.11095
caused by many-body as follows,
interactions.
If we consider
three-body
interactions
the cohesive energy must be rewritten
(28)
where I$(‘) and $J(~) are the two-body and three-body interactions respectively. In order to inverse the pair potential from Eq. (28) the Slater-Kirkwood-type
three-body
interaction
is used
[301, (p(3)
=
[ A
e-a(r~+r~+r,~)
+
~(r~rp-~~)-~] (I+ 3 cos
Oi
cos
Oj
cos
Oij),
where ri, rj and rij are the three sides of a triangle formed by three atoms and Oi, Oj and Oij are its interior angles. The factor A is the strength of the exchange three-body interaction which describes the alterations of the charge densities of two interacting atoms by the presence of a third one. The factor C is the strength of the
Fig. 1. Comparison of the calculated phonon dispersions theory and the full points for the experiment [I].
Fig. 2. Comparison of the calculated phonon dispersions the theory and the full points for the experiment [3].
with three-body
with three-body
interactions
interactions
and the experimental
and the experimental
ones for Li; the solid lines for the
ones for Na; the solid lines for
M. Li. N.X. Chen/
0. 0 0.0
0. d
0. I
0.8
Physics Letters A 234 (1997) 134-140
I.0
0.8
0. I
9
0.1 9
0.2
0.0
0. 1
Cl.2
0. 3 q
139
0. 5
Fig. 3. Comparison of the calculated phonon dispersions with three-body interactions and the experimental theory, the full points and the open circles are experimental data from Ref. [4] and Ref. [5], respectively.
Fig. 4. Comparison of the calculated phonon dispersions the theory and the full points for the experiments [6].
with three-body
interactions
and the experimental
ones for K; the solid lines for the
ones for Rb; the solid lines for
triple-dipole interaction which is obtained from perturbation theory and known as the Axilrod-Teller one. LYis a decay factor. By use of the obtained pair potentials, the Slater-Kirkwood three-body interaction and Pqs. (24)-(27) we recalculate the phonon dispersions in alkali metals. Properly adjusting the parameters A, C and a good results are obtained. The values of the parameters A, C and (Y used in the calculation are listed in Table 3 and calculated phonon dispersions are plotted in Figs. l-4.
5. Discussion
and conclusion
From Figs. l-4 it can be seen that all of the calculated phonon dispersions along the [ 1001, [ 1101 and [ 11 l] directions are in agreement with the experimental data for metals Li, Na, K and Rb. This indicates that by use of the above simple method we can obtain good results for phonon dispersion in alkali matals in the Born-Von Karman framework. The calculation is much simpler than that by use of the ab initio method. Of course the ab initio method can give deeper physical insight into the problem. A similar calculation for alkali metals was carried out by Ramji Rao et al. and. very good results were obtained [16]. But in that work 12 adjustable parameters were introduced. In the present work only three adjustable parameters for the three-body interaction are used. In the [1 1l] direction a pronounced dip in the frequency appears in the vicinity of q = f for the longitudinal branch. This anomaly has been studied carefully by Falter et al. [20-231 based on the density response function method. They indicate that in a diagonal approximation for the density response function .in the dynamical matrix the matrix elements can be expressed as a product of an intrinsic geometrical factor and the Fourier components of an effective ion interaction. When the wave vector q = 5 the geometrical factor will vanish and
M. Li. N.X. Chen/ Physics Letters A 234 (1997) 134-140
140
the Fourier components will be switched off. Therefore, this work can give a very good explanation for the position of the large dip in the frequency. In the above approximation which means central forces are dominant they give a calculation for the phonon dispersion in Na and obtain a good result. In the present work the phonon dispersion in alkali metals is calculated in the pair potential (central force) aproximation and the calculated phonon dispersion qualitatively agrees with the experiments. Some discrepancies between theory and experiments are obvious. Therefore, we consider the effect of a three-body interaction for the calculation of the phonon dispersion in Li, Na, K and Rb. Figs. l-4 demonstrate that not only the position of the dip, but also the calculated values of the frequencies are in agreement with the experiments. Based on the results we could conclude that the so-called LA$[l, 1, l] anomaly might be also explained by means of a three-body interaction in the Born-von Karman model. The experimental data of the T2 branch in K are not reported in Ref. [4] and are taken from Ref. [5].
Acknowledgement This work was supported by the National Foundation Materials Committee of China.
of Natural Science of China and the National Advanced
References [ 1] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
H.G. Smith et al., in: Neutron inelastic scatering, Vol. 1 (IAEA, Vienna, 1968) p. 149. M.M. Beg and M. Nielson, Phys. Rev. B 14 (1976) 4266. A.D.B. Woods et al., Phys. Rev. 128 (1962) 1112. R.A. Cowley et al., Phys. Rev. 150 (1966) 487. G. Dolling and J. Meyer, J. Phys. F 7 (1977) 775. J.R.D. Copley et al., in: Neutron inelastic scatering, Vol. 1 (IAEA, Vienna, 1968) p. 209. L.J. Sham, hoc. R. Sot. A 283 (1965) 33. N. Singh and S.K. Joshi, Physica 32 (1966) 127. D.C. Wallace, Phys. Rev. 176 (1968) 832. DC. Wallace, Phys. Rev. 178 (1968) 900. S. Prakash and S.J. Joshi, Phys. Rev. 187 (1969) 808. N.W. Ashcroft, J. Phys. C 1 (1968) 232. V. Bortolani and G. Pizzichini, Phys. Rev. Lett. 22 (1969) 845. D. Sen and S.K. Sarkar, Phys. Rev. B 22 (1980) 1856. W. Frank, C. Elslsser and M. Fahnle, Phys. Rev. Lett. 74 (1995) 1791. R. Ramji Rao et al., Phys. Stat. Sol. b 147 (1988) kll. A.D.B. Woods and S.H. Chen, Solid State Commun. 2 (1964) 233. S.H. Chen and B.N. Brockhouse, Solid State Commun. 2 (1964) 73. A.D.B. Woods, Phys. Rev. A 136 (1964) 781. C. Falter, W. Ludwig, M. Selmke and W. Zierau, Phys. Lett. A 90 (1982) 250. K.-M. Ho, C.-L. Fu and B.N. Harmon, Phys. Rev. B 28 (1983) 6687. C. Falter, W. Ludwig, W. Zierau and M. Selmke, Phys. Len. A 93 (1983) 298. C. Falter, Phys. Rep. 164 (1988). M. Li et al., in: Materials theory and modelling, eds. J. Broughton, P.D. Bristowe and J.M. Newsam, (Materials Research Society, Pittsburgh, 1993) p. 497. M.R. Schroeder, Number theory in science and communication, 2nd Ed. (Springer, Berlin, 1990). C. Kittel, Introduction to solid state physics, 5th Ed. (Wiley, New York, 1976). A.E. Carlsson et al., Philos. Mag. A 41 (1980) 241. N.X. Chen et al., Phys. Rev. B 45 (1992) 8177. J. Callaway, Quantum theory of the solid state (Academic Press, New York, 1976). L.W. Brunch and I.J. McGee, J. Chem. Phys. 59 (1973) 409.
MRS Symp. Proc. No. 291
1997
PHYSICS ELSEVIER
LETTERS
A
Physics Letters A 234 (1997) 134-140
Phonon dispersion of alkali metals and the LA$( 1, 1, 1) anomaly in bee structures M. Li, N.X. Chen CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China Institute of Applied Physics, Beijing University of Science and Technology, Beijing 100083, China Received 31 January
1997; revised manuscript received 27 May 1997; accepted for publication Communicated by A.R. Bishop
’
12 June 1997
Abstract The phonon dispersion of body-centered cubic metals Li, Na, K and Rb is calculated based on the Mijbius inversion-converted pair potentials and a Slater-Kirkwood-type three-body interaction. The calculated results are in agreement with the experimental data from the inelastic neutron scattering. Then the LA*(l) 1, 1) anomaly is discussed. The results indicate that the LA$(l, 1, 1) anomaly can also be explained by use of the three-body interaction in the Born-von Karman model
framework. 0 1997 Elsevier Science B.V. PACS: 63.2O.Dj; 63.20. - e; 02.30.Em
1. Introduction
The phonon dispersion in alkali metals Li, Na, K and Rb has been measured using the inelastic neutron scattering method [l-6]. Various theoretical methods have been proposed to explain the experimental data. Some ab initio calculations give good results on phonon dispersion in alkali metals [7-141, especially, the calculations by Frank et al. [ 151 based on the local density approximation and norm-conserving pseudopotentials give some nice results, in which the crossing of the longitudinal and transverse branches for Li along the [loo] direction is explained. Some classical calculations of the phonon dispersion in the Born-von Karman model have been carried out [ 161. In these studies it is found that there is a large dip in the phonon frequency near q = f along the [ 1111 direction for the longitudinal branch. This interesting phenomenon attracted attention and has been studied carefully [17-231. Falter et al. pointed out that this was caused by an intrinsic geometrical effect of the bee lattice and this so-called LA$l, 1, 11 anomaly is characteristic of bee metals [20]. In the present paper the phonon dispersion in alkali metals Li, Na, K and Rb has been calculated using the method developed by the authors [24] based on the MGbius theorem in number theory [25], by which the pair potential can be obtained from the cohesive energy and then the force constant and the phonon dispersion can be
’ Mailing address 0375-9601/97/$17.00 0 1997 Else&r PII SO375-9601(97)00497-O
Science B.V. All rights reserved.
M. Li. N.X. Chen/Physics
Letters A 234 (1997) 134-140
135
Table 1 The values of B. r,\. U and I) [261
Materials
B (10” dyne/cm)
r(, (IO-” cm)
U (eV/atom)
p (108/cm)
Li Na K Rb
0.116 0.068 0.032 0.031
3.491 4.225 5.225 5.585
1.630 1.113 0.934 0.852
0.5911 0.6025 0.5017 0.5346
calculated. The results are in agreement with the neutron scattering data and indicate that the LA$[l, 1, l] anomaly can be explained by means of a three-body interaction with a suitable choice of adjustable parameters. In Section 2 the cohesive energy is given as a function of some properties of the metals in order to convert their pair potentials. In Section 3 the pair potentials are converted by use of the Mobius theorem and the calculated cohesive energies. A three-body interaction is used and the phonon dispersion is calculated in Section 4. In Section 5 the results are discussed.
2. The cohesive
energies
of Li, Na, K and Rb
The cohesive energy curve for bee structure can be obtained by ab initio calculations. paper it is assumed that the cohesive energy can be expressed as a Morse potential E(r)
= U(ep2P(‘-‘o)
But in the present
- 2e-P(‘-‘0)),
(1)
where r is the lattice constant of the metal, U the cohesive energy for each atom at equilibrium, r. the equilibrium lattice constant and p a parameter related to the bulk modulus B. For alkali metals, the values of U, B and r,, are taken from experimental data [26] and the modulus B can be written as B = 4Up2/9ro.
(2)
Therefore, p=;JBr,/u.
(3)
The experimental values of U, B, r,, and the calculated resultant cohesive energies are
parameter
p
for alkali metals are listed in Table 1. The
Li:
E(r)
= 1.630(61 .9954e-‘~‘822’ - 15.7474~-~~~~“‘),
(4)
Na:
E(r)
= 1.113( 196.6155e-‘~2050’ - 28.0439e-“~6025r)
(5)
K:
E(r)
= 0.934( 189.1925eC’ .0034r- 27.5095e-0.50’7’),
(6)
Rb:
E(r)
= 0.852(392.0863e-‘.0692r
(7)
3. Pair potentials
- 39.6025e-0.5346r),
of Li, Na, K and Rb
If many-body
interactions
E(r)=3
5 d)(IRI), IRl#O
are ignored,
the cohesive energy per atom can be approximated
as [27]
(8)
136
M. Li, N.X. Chen/
Physics Letters A 234 (1997) 134-140
where C#Iis the pair potential and /R /the distance structure Eq. (8) can be written as [28] E(r)
=3
+(nr)
t
+6
f
n=l
P-4’
q!~(/m
of an atom from the reference
one at the origin. For bee
r)
I
+4Pi=,[+(G777r)+O(/(~-f)2+(~-f)2+(u-i)2 7I
+4
r)]
[+(@T&7r)+~({(p-f)2+(q-f)2+(u-f)2r
e
r)],
(9)
p#q#u=l
where r is the lattice constant. E(r)
We can divide the cohesive energy
E into two parts,
+E2(r),
(10)
[4(~&nr)+~q5(nr)]
(11)
=E,(r)
where E,(r)=4C
n=l
and E2( r) = 6
(P(h-7 r)
i
f4
e
[+(@T&7r)+4(jl(p-f)2+(g-f)2+(u-i)2
We introduce
r)].
I
p#q#u=
two operators
(12)
B, and B, such that
E,(r)
= &4(r)
(13)
E,(r)
= B,+(r).
(14)
and
So we have E(r)=(B,+B,)+(r)=B,(l+B,‘B:,)4(r).
Applying
an inverse operation
(15)
the pair potential
c#dr) can be obtained
immediately,
+(r)=(l+B;‘B,)-‘B-‘E(r).
(16)
By using the Mobius theorem in number theory it has been proved that [28]
B;‘f(r)=am~l(-l)“‘(~)m-‘~(n)~[(z/a)”,,], where fir> is a real function = 1, /..L(“) =(+, = 0,
and /.~(a> is the Mobius function
if n=
(17)
[25]
1,
if n includes
s distinct primes,
if n includes repeated factors.
(‘8)
M. Li, N.X. Chen/Physics
Letters A 234 (1997) 134-140
137
Table 2 The values of &,. n, and q Materials
&, (ev)
a,, (1O-8 cm)
q (108/cm)
Li Na K Rb
0.07833 0.07480 0.06215 0.06745
4.78 5.08 6.08 6.04
0.6356 0.6407 0.5271 0.5660
The cohesive energy E(r) decreases rapidly when r is increased. Therefore, when Eq. (16) is expanded, just a few terms need to be taken into account and then the pair potential can be calculated using Eq. (17). The calculations have been carried out for alkali metals Li, Na, K and Rb. It is interesting that the calculated results can also be fitted as Morse potentials with the parameters in Table 2, +(r)
= +O(e-W-%)
Li:
I
= 0.07833(e-‘.2712(‘-4.78)
Na:
b(r)
= 0,07480(e-
K:
+(r)
= ()06215(~-1.054X-6.08)
_ 2e-O-527’(‘-6.08)),
Rb:
$(r)
= 0.-~6745(~-‘.t32O(r-6.04)
_ 2e-0.5660W6.04)).
Based on the calculated
4. Phonon dispersion
- 2e-4(r-%)),
(19) _ 2,-0.6356(r-4.78)),
(20)
1.2814(r-5.08)_ 2e-0.6407(‘-5.08)),
pair potential
(21) (22) (23)
we can give the vibrational
in Li, Na, K and Rb and a three-body
potential
interaction
Using the calculated pair potential we can give the total potential. potential @ can be expressed by the displacement of atoms [29], @= 3 C
C
@.
In the harmonic
@ahp.tK.jvUa,ixUfi.jv)
approximation
the total
(24)
i~~r jv/3
where uOI,iKis the displacement of the K th atom in the ith unit cell along the a direction from the equilibrium position. - @abip,iK,jVis known as the atomic force constant. It is the a-component of the force exerted on the K th atom in the ith unit cell when the v th atom in the jth unit cell moves a unit displacement along the p direction. In the lattice dynamics, the atomic force constant - @a,8iK,jV can be obtained as
(25) where the subscript 0 means the derivatives Solving the secular equation det I w2&&
- Dcrp.J
we can obtain the dispersion
are taken at the equilibrium
k) I = 0, relations
D crp.K”(k) = wKK-“2
or frequency
(26) w as a function
,FR @c2p.itr,jvexP[-ik * (Ri 1
position.
of wave vector k. In Eq. (26), Dap,,,(k)
-Rj)])
is
(27)
I
which expresses the elements of the dynamic matrix D. M, is the mass of the K th atom and Ri is the position vector of the ith unit cell. In our calculation contributions up to fourth nearest neighbours are involved. By using Eqs. (20)-(27) the dispersion curves for Li, Na, K and Rb are calculated for three major directions [ IOO], [ 1 IO] and [ 1111. The calculated results qualitatively agree with the experiments. The discrepancy might be
M. Li, N.X. Chen/ Physics Letters A 234 (1997) 134-140
138 Table 3 The values of A, C and Q Materials
A (eV)
C (eV cm9)
a (108/cm)
Li Na K Rb
-4.5718 - 1.6513 - 2.0685 - 1.6728
5802.2 10439.0 128460.0 164140.0
0.17835 0.14915 0.11279 0.11095
caused by many-body as follows,
interactions.
If we consider
three-body
interactions
the cohesive energy must be rewritten
(28)
where I$(‘) and $J(~) are the two-body and three-body interactions respectively. In order to inverse the pair potential from Eq. (28) the Slater-Kirkwood-type
three-body
interaction
is used
[301, (p(3)
=
[ A
e-a(r~+r~+r,~)
+
~(r~rp-~~)-~] (I+ 3 cos
Oi
cos
Oj
cos
Oij),
where ri, rj and rij are the three sides of a triangle formed by three atoms and Oi, Oj and Oij are its interior angles. The factor A is the strength of the exchange three-body interaction which describes the alterations of the charge densities of two interacting atoms by the presence of a third one. The factor C is the strength of the
Fig. 1. Comparison of the calculated phonon dispersions theory and the full points for the experiment [I].
Fig. 2. Comparison of the calculated phonon dispersions the theory and the full points for the experiment [3].
with three-body
with three-body
interactions
interactions
and the experimental
and the experimental
ones for Li; the solid lines for the
ones for Na; the solid lines for
M. Li. N.X. Chen/
0. 0 0.0
0. d
0. I
0.8
Physics Letters A 234 (1997) 134-140
I.0
0.8
0. I
9
0.1 9
0.2
0.0
0. 1
Cl.2
0. 3 q
139
0. 5
Fig. 3. Comparison of the calculated phonon dispersions with three-body interactions and the experimental theory, the full points and the open circles are experimental data from Ref. [4] and Ref. [5], respectively.
Fig. 4. Comparison of the calculated phonon dispersions the theory and the full points for the experiments [6].
with three-body
interactions
and the experimental
ones for K; the solid lines for the
ones for Rb; the solid lines for
triple-dipole interaction which is obtained from perturbation theory and known as the Axilrod-Teller one. LYis a decay factor. By use of the obtained pair potentials, the Slater-Kirkwood three-body interaction and Pqs. (24)-(27) we recalculate the phonon dispersions in alkali metals. Properly adjusting the parameters A, C and a good results are obtained. The values of the parameters A, C and (Y used in the calculation are listed in Table 3 and calculated phonon dispersions are plotted in Figs. l-4.
5. Discussion
and conclusion
From Figs. l-4 it can be seen that all of the calculated phonon dispersions along the [ 1001, [ 1101 and [ 11 l] directions are in agreement with the experimental data for metals Li, Na, K and Rb. This indicates that by use of the above simple method we can obtain good results for phonon dispersion in alkali matals in the Born-Von Karman framework. The calculation is much simpler than that by use of the ab initio method. Of course the ab initio method can give deeper physical insight into the problem. A similar calculation for alkali metals was carried out by Ramji Rao et al. and. very good results were obtained [16]. But in that work 12 adjustable parameters were introduced. In the present work only three adjustable parameters for the three-body interaction are used. In the [1 1l] direction a pronounced dip in the frequency appears in the vicinity of q = f for the longitudinal branch. This anomaly has been studied carefully by Falter et al. [20-231 based on the density response function method. They indicate that in a diagonal approximation for the density response function .in the dynamical matrix the matrix elements can be expressed as a product of an intrinsic geometrical factor and the Fourier components of an effective ion interaction. When the wave vector q = 5 the geometrical factor will vanish and
M. Li. N.X. Chen/ Physics Letters A 234 (1997) 134-140
140
the Fourier components will be switched off. Therefore, this work can give a very good explanation for the position of the large dip in the frequency. In the above approximation which means central forces are dominant they give a calculation for the phonon dispersion in Na and obtain a good result. In the present work the phonon dispersion in alkali metals is calculated in the pair potential (central force) aproximation and the calculated phonon dispersion qualitatively agrees with the experiments. Some discrepancies between theory and experiments are obvious. Therefore, we consider the effect of a three-body interaction for the calculation of the phonon dispersion in Li, Na, K and Rb. Figs. l-4 demonstrate that not only the position of the dip, but also the calculated values of the frequencies are in agreement with the experiments. Based on the results we could conclude that the so-called LA$[l, 1, l] anomaly might be also explained by means of a three-body interaction in the Born-von Karman model. The experimental data of the T2 branch in K are not reported in Ref. [4] and are taken from Ref. [5].
Acknowledgement This work was supported by the National Foundation Materials Committee of China.
of Natural Science of China and the National Advanced
References [ 1] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
H.G. Smith et al., in: Neutron inelastic scatering, Vol. 1 (IAEA, Vienna, 1968) p. 149. M.M. Beg and M. Nielson, Phys. Rev. B 14 (1976) 4266. A.D.B. Woods et al., Phys. Rev. 128 (1962) 1112. R.A. Cowley et al., Phys. Rev. 150 (1966) 487. G. Dolling and J. Meyer, J. Phys. F 7 (1977) 775. J.R.D. Copley et al., in: Neutron inelastic scatering, Vol. 1 (IAEA, Vienna, 1968) p. 209. L.J. Sham, hoc. R. Sot. A 283 (1965) 33. N. Singh and S.K. Joshi, Physica 32 (1966) 127. D.C. Wallace, Phys. Rev. 176 (1968) 832. DC. Wallace, Phys. Rev. 178 (1968) 900. S. Prakash and S.J. Joshi, Phys. Rev. 187 (1969) 808. N.W. Ashcroft, J. Phys. C 1 (1968) 232. V. Bortolani and G. Pizzichini, Phys. Rev. Lett. 22 (1969) 845. D. Sen and S.K. Sarkar, Phys. Rev. B 22 (1980) 1856. W. Frank, C. Elslsser and M. Fahnle, Phys. Rev. Lett. 74 (1995) 1791. R. Ramji Rao et al., Phys. Stat. Sol. b 147 (1988) kll. A.D.B. Woods and S.H. Chen, Solid State Commun. 2 (1964) 233. S.H. Chen and B.N. Brockhouse, Solid State Commun. 2 (1964) 73. A.D.B. Woods, Phys. Rev. A 136 (1964) 781. C. Falter, W. Ludwig, M. Selmke and W. Zierau, Phys. Lett. A 90 (1982) 250. K.-M. Ho, C.-L. Fu and B.N. Harmon, Phys. Rev. B 28 (1983) 6687. C. Falter, W. Ludwig, W. Zierau and M. Selmke, Phys. Len. A 93 (1983) 298. C. Falter, Phys. Rep. 164 (1988). M. Li et al., in: Materials theory and modelling, eds. J. Broughton, P.D. Bristowe and J.M. Newsam, (Materials Research Society, Pittsburgh, 1993) p. 497. M.R. Schroeder, Number theory in science and communication, 2nd Ed. (Springer, Berlin, 1990). C. Kittel, Introduction to solid state physics, 5th Ed. (Wiley, New York, 1976). A.E. Carlsson et al., Philos. Mag. A 41 (1980) 241. N.X. Chen et al., Phys. Rev. B 45 (1992) 8177. J. Callaway, Quantum theory of the solid state (Academic Press, New York, 1976). L.W. Brunch and I.J. McGee, J. Chem. Phys. 59 (1973) 409.
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