Lattice specific heat and elastic constants of hcp sodium

Abstract

Sodium undergoes a martensitic transformation passing from a bcc structure to an hcp structure, below 35 K. Using a first principles pseudopotential, we calculated the constant volume lattice specific heat and elastic constants of both hcp and bcc sodium for temperatures below 35 K, for comparison.

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Volume 143, number 3 PHYSICS LETTERS A 8 January 1990
LATTICE SPECIFIC HEAT AND ELASTIC CONSTANTS OF hcp SODIUM
G.J. VAZQUEZ and L.F. MAGAI~IA
lnstituto de Fisica, Universidad Nacional Aut6noma de Mbxico, apartado postal 20-364, Mexico DF 01000, Mexico
Received 22 September 1989; revised manuscript received 27 October 1989; accepted for publication 6 November 1989
Communicated by A.A. Maradudin
Sodium undergoes a martensitic transformation passing from a bcc structure to an hcp structure, below 35 K. Using a first
principles pseudopotential, we calculated the constant volume lattice specific heat and elastic constants of both hcp and bcc
sodium for temperatures below 35 K, for comparison.
For simple metals the interionic potential can be
constructed from first principles using pseudopoten-
tial theory. In this work we constructed a first prin-
ciples pseudopotential following a method proposed
by Manninen et al. [ 1 ] who had followed the spirit
of the work of Rasolt and Taylor [2], with some
differences.
In previous work we have employed this kind of
pseudopotential (and, as in this work, within the
model of the nucleus embedded into a jellium va-
cancy [ 1 ] ) with success in the calculation of the lat-
tice specific heat of lithium [3], and aluminum [4],
and of the pressure dependence of the lattice specific
heat of lithium, and aluminum [ 5], and also in the
calculation of the pressure dependence of the elastic
constants of aluminum and lithium [6]. More re-
cently we also explored, with good results, its appli-
cation in the calculation of the phonon limited re-
sistivity of aluminum [7] and of sodium and
potassium [8 ].
We started by calculating the displaced electron
densities around a nucleus in an electron gas for so-
dium (Na). This was done using the density func-
tional formalism [9,10], and the model of the nu-
cleus embedded into a jellium vacancy. Taking into
account that in the pseudopotential formulation the
pseudodensity must not contain wiggles near the ion,
these wiggles in the calculated density had to be re-
moved, as we explain below.
We have used atomic units (i.e., magnitude of the
electron charge = electron mass = h = 1 ). The energy
is given in double Rydbergs.
From pseudopotential theory and linear response
theory [ 1 1 ], the interionic potential is given by
~(r)= Z( 1+-2
~Z 2
× i dqsin(qr)e(q)[Sn(q) ] 2)
q[1-e(q)] ' (1)
0
where r is the separation between the two ions, Z is
the charge of the metal ion, ~ (q) is the dielectric re-
sponse function of the electron gas and 8n (q) is the
Fourier transform of the induced charge pseudo-
density.
For the model of the nucleus embedded in a jel-
lium vacancy, the induced electronic density is cal-
culated by taking the difference [ 1 ]:
5n(r)=n(r)-nv(r)-2 ~
I~'b(r) [ 2 , (2)
b
where n (r) is calculated with the total charge density
corresponding to a nucleus located at the center of
a vacancy in jellium, and
nv(r)
is the electron den-
sity around a jellium vacancy alone. Charge neu-
trality of the metal is a necessary condition. The
bound states are represented by q/b-
We calculated 5n (q), the Fourier transform of the
displaced electron pseudodensity, taking the Fourier
transform of the density given by eq. (2), after
smoothing. In this smoothing, the conditions that the
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Volume 143, number 3 PHYSICS LETTERS A 8 January 1990
electronic charge is conserved and that
8n(r)
and
(O/0r)&t(r) are continuous are imposed [ 1 ]. It is
convenient to mention that in the pseudopotential
formulation, the pseudodensity must not contain
wiggles near the ion, and the induced density cal-
culated from density functional theory contains those
wiggles in that region due to the orthogonalization of
conduction states to core orbitals.
The unscreened pesudopotential form factor, u(q),
is related to 5n(q) by
4~ 8n(q) e(q)
u(q)- q2[l_{(q)] (3)
Eq. (3) is used to obtain an effective local pseu-
dopotential, which in linear response will give the
exact induced displaced electronic density outside
the region of smoothing. In this way some of the non-
linear screening effects are included into the pair po-
tential calculated from this pseudopotential.
The dielectric function we used satisfies by con-
struction the compressibility theorem which is im-
portant in connection with the interionic potential
[1,12]. lt is given by [1,13]
4rt
~(q)=l+ ~sG(q ) ,
(4)
where
Go(q)
G(q)=
l_(4~/k2v)Go(q)(l_L),
(5)
Go(q)
is the usual Lindhard polarizability,
kvv
is the
Fermi-Thomas screening constant, and L is the ratio
OltlOr,
L- (6)
OEF/Or s "
In eq. (6) II is the chemical potential,
EF
is the
Fermi energy and
l~(r~) =Ev(r~)+ lZ~c(r~)
, (7)
where/Zx~ (r~) is the exchange-correlation contribu-
tion to the chemical potential.
Using the expression of Gunnarson and Lundquist
[ 14] for exchange-correlation (which is the one we
used in the calculation of the induced electronic den-
sity), the corresponding value of L is
(4)'/3(
)
L=I- 9~ r~ l+ 0.6213
r~ + 1 l.~ r~ . (8)
With the induced charge pseudodensity and the
dielectric function already given, we used eq. ( 1 ) to
calculate the interionic potential.
From the interionic potential we calculated the
phonons and associated force constants, using the
harmonic approximation.
From the tensor force model and using the nota-
tion of ref. [ 15 ], the force matrix, q~j, is defined as
the force on the origin atom in the i direction when
atom S moves one unit distance in the j direction.
This force matrix is symmetric and it is denoted by
,:iOus')-/fls, j
-
o~
fl'?]
. (9)
The point S is one of a set of points according to
the symmetry of the lattice. This set of points is de-
noted by S, where S= I, 2, 3, etc., corresponding to
every shell of neighbors. The force matrices of the
other members of the set consist of rearrangements
of the same set of force constants.
The elastic constants, C~,
C44 ,
~'12 are given by
[15]
n S
aC44=42~ ~ j~ [(h)S+l)2+(h)~2)2]°t) ';"
HA"
a(CII-}-C44)=16T~-~ h)S+lh~jS+2~,
(10)
where ce is the lattice constant, n s is the number of
lattice points for neighbour shell S; h s corresponds
to three non-negative integers such that ht >h2>h3
and the coordinates of a point in shell S are
hja/2,
h2a/2, h3a/2.
For fcc ~= 1 and for bcc r= 1/2.
The relations between the force constants of the
tensor force model and the axially symmetric model
are [16]
c~'~ " = CB( S) + ( h~/ h2)k~ ( S) ,
as = CB( S) + ( h~/ h 2)k, ( S) ,
as = CB(S) + ( h2/h2)k, (S) ,
fit'= ( h2h3/ h2 )kl ( S) ,
fl~'= (h3hl/h2)kl(S) ,
156

Volume 143, number 3 PHYSICS LETTERS A 8 January 1990
fls= (h,h2/h 2)k,(S) , (11)
where
h2=h 2 2 2
+h2+h3, and
k,(S)
and
CB(S ) are
the two force constants of the axially symmetric
model for the Sth shell of neighbors [ 16]. We had
for bcc a good convergence with 12 neighbor shells.
We could relate easily the force constants kl (S)
and
C,(S)
to the derivatives of the interionic po-
tential and we obtained
_Fd=V(r)
1 dV(r)] (12)
k'(S)- L
dr 2
r d--~ J(s)
and
c,o, F1
dV(r)]
J(,,
(13)
In this way, from the interionic potential
V(r),
we
could find k, (S) and
CB(S)
and using eqs. (10) and
( 11 ) we could calculate the elastic constants.
For the hcp structure we had a good convergence
with four neighbor shells. For this case the elastic
constants are [ 17 ]
C1, =V#3 (3c~--Al
-L)/2c,
(14)
L= [3(B, +B2 +G, q-G2) ] -I
X [ (2B2 +G2 + 3GI) (3Bt +B2 +8G2)
+2G2(3B, +B2)] , (15)
c
C33- x/-~a 2 [-3(B3+G3)+46 ] , (16)
2
C44 --~
~CC (3A2+B3+4G3) ,
(17)
1
C12 - - xfjc [3 (or- 3AI)- 3B1 -B2 - 12Gl
-4G2+P] +C~l , (18)
p= (Bt -B2--2Gl +2G2) 2
Bi +B2 +G,
+G2 '
(19)
Ci3= _2 (2G4-B4)-C44. (20)
a
In the case of an ideal hcp structure,
c/a= 8v/~.
The relationships of the force constants with the
derivatives of the interionic potential are
a=- [kt(l)+Cn(1)]
,
A,=A2=CB(1),
A3=0,
B, =C,(2),
B2=][k,(2)+3CB(2)],
B3 =2Bz-BI, B4 =N/2 (B2 -B! ) ,
G~=CB(3),
G2=~[2kl(3)+3CB(3)],
G2 - GI
G3=½(G, +G2), G4- ~ ,
6=-[k,(4)+C,(4)].
For this case, again, from the interionic potential
we can find kl (S) and
CB(S)
and, from these, the
corresponding elastic constants for hcp.
To calculate the phonon frequency distribution
6
v
1
-2 ~ ) l
r (O,u.)
I I I
25 30 35
Fig. 1. Calculated interionic potential for sodium. We use
rs= 3.93ao, where a0 is the Bohr radius.
3o
5-
N 2,5
I
l
2.0
©
-~ to
LI_
0,5
0 0
A
II
f k
I
~\v l/
/
I II
///
LO 2.0 3,0
I/( 10 lz Hz )
Fig. 2. Phonon spectra for sodium. (
(---) results for hcp.
~,A
i
4,0 5.0
) Results for bcc;
157

Volume 143, number 3 PHYSICS LETTERS A 8 January 1990
F(v) from the force constants obtained in the phonon
dispersion curve, we followed the method of Gilat
and Raubenheimer [ 18 ].
From
F(v)
the specific heat is calculated numer-
ically by the integral
O<E>
C,- aT
F(u)
=kB dv (½flhV)esinh(flhv/2) ,
0
(21)
where (E> is the average of the internal energy, T
is the temperature and vm is the maximum phonon
frequency.
For high temperature (for example, larger than the
corresponding Debye temperatures) the anhar-
monic effects become more important, as it happens
with the specific heat [ 19-21 ].
In fig. 1 we show the resulting interionic potential.
From this interionic potential we calculated the de-
rivatives to be used in order to obtain the elastic con-
stants. Also, from the interionic potential we ob-
tained the phonon spectra and the force constants,
by the harmonic approximation and the method of
Gilat and Raubenheimer [ 18 ]. The resulting phonon
spectra for bcc and hcp sodium are shown in fig. 2,
for a temperature of 5 K. We could not find the cor-
responding experimental results for this tempera-
Table 1
Predicted elastic constants of bcc and hcp sodium, Sodium
undergoes a martensitic transformation below 35 K to an hcp
structure. The Debye temperature for sodium bcc is 156 K, and
for hcp sodium it is 160 K [22 ]. This calculation is for 5 K. The
units are 10 ~ dynes/era 2. Notice that C33 and C~ 3 are for the hcp
structure only.
bcc hcp
C44 0.654 0.028
('~ 0.616 0.168
CI2
0.474 0.059
C33 0.144
('~3 0.018
ture. However, for larger temperatures (above the
temperature of the martensitic transformation) there
is a good agreement between our predicted phonon
dispersion curves [21] and experimental results.
From here we should expect a good prediction for
the elastic constants and the lattice specific heat.
The results for the lattice specific heats for sodium
bcc and sodium hcp are shown in fig. 3. We could
not find experimental results for the specific heat of
hcp sodium alone, nor for bcc sodium alone below
35 K.
Our calculated elastic constants, for hcp and bcc
sodium for temperatures below 35 K are shown in
table 1.
~> 2 I /
/
/~//// _1 1 .... 1
o2s oI~ o75 ~,o
T/~ D
Fig. 3. Lattice specific heat for sodium. ( ) Results for bcc;
(---) results for hcp. (-..) Experimental results for tempera-
tures above 35 K. The Debye temperatures are taken as 153 K
for bcc and 160 K for hcp [ 22 ].
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