A Quantum Statistical Calculation of Electric Field

International Journal of Applied Physics.
ISSN 2249-3174 Volume 4, Number 1 (2014), pp. 21-28
© Research India Publications
http://www.ripublication.com
A Quantum Statistical Calculation of Electric Field
Gradient in Indium
Ashok Kumar and B. C. Rai
PG Centre, Department of Physics,
College of Commerce, Patna-20
(Magadh University)
Abstract
Quantum statistics of phase space is applied to the available space for
conduction electron in Indium metal, and Electric Field Gradient is calculated
at nucleus in crystalline asymmetric environment. The charge distribution is
obtained using essentially the Thomas Fermi Statistical model as initiated for
hcp metal by E Bodenstedt, and developed by us for tcp metals. The ionic
charge is calculated using Roothaan-Hartree-Fock tabulation of wave
functions, andconduction electronic charge distribution is exclusively
determined by the available volume in phase space. Calculations are reported
at room temperature using a FORTRAN program developed by us and
Mathematica, and compare well with recent value.
Key words: Quantum statistics, Indium, Electric Field Gradient.
1. Introduction
Electric Field Gradient (EFG) to which a nucleus can respond is not yet available in
laboratories to be applied to it externally. Crystalline asymmetric environment is,
however, able to do so, and gives rise to nuclear quadrupole interaction with the field
gradient. The determination of EFG involves quantum mechanical methods of atomic,
molecular and solid states, microwave spectra, and others. Accurate calculation of
EFG requires large basis set to encompass core and valence polarization, and often
face slow convergence with standard basis sets. On the other hand, qualitatively good
basis set for heavy elements are rare. Thus, the situation has given rise to several
empirical and semi-classical models of metals. . A precise knowledge of ionic and
conduction electronic wavefunctions is, however, not essential for distant particles.
Conduction electrons are lumped into small pockets in semi-classical models (of point
lattices) to required degree of accuracy. Charge shift model of Bodenstedtet al [1],
Paper Code:26589 IJAP
22
Ashok Kumar and B. C. Rai
Thomas –Fermi statistical model of atom when applied to hcp metals like zinc [2],
and others take help of such an approach. Similar models have successfully computed
electric field gradients and nuclear quadrupole moments [3, 4, and 5], and cohesive
energies [6, 7, and 8].
The conduction electrons in Indium, as we assume, fill up the space between
spherically symmetric metal ions according to quantum statistics as assumed for free
atoms by Thomas and Fermi in statistical atomic model [9-12]. The assumption does
not exhibit the electronic shell effects of interference pattern but is a good
approximation for electrostatic potential of atoms used till date. We assume the shell
effects in our three dimensional Thomas-Fermi lattice calculations as insignificant.
Here we find a charge distribution exclusively determined by the available volume in
phase space of Indium metal. Once the charge distribution is at hand we can calculate
EFG.
2. Formulations
As conduction electronscan have momentum from zero to Fermi momentum, pf, their
charge density may be written, using quantum statistics, as
4
×2
3ℎ
The factor 2 cares for two spin states of electrons, and a phase cell is of volume ℎ .
As discussed in the previous work [13],the potential (r) at position r,is of
electrostatic origin only, and the kinetic energy of electron corresponds to Fermi
momentum, as
( )
= (− )
= (− ) ( ) +
2
The conduction electronic charge distribution may now be written, using above two
equations, as
8
( )=−
(2 ) [ +
( )] … (1)
3ℎ
As the unit cell is neutral, the volume integral of the charge density over the unit cell
must be zero. The Fermi energy is fixed by this secondary condition.
The total charge density ( ) and potential ( ′) are related in the Poisson equation
formed as
( )
( ′) =
… (2)
4
Here the integration runs over entire volume.
To know density, we need potentials, and to know potentials we need density!
In cases of such transcendental equations, numerical or graphical methods are used
for solutions, andonly an approximate solution can be obtained. As described in the
previous work [13], instead of dividing the crystal volume into infinitesimal volume,
N small Wigner-Seitz unit cells are devised filling up the Bravais unit cell. Their
centres fulfil the symmetry of the Bravais lattice. Using crystal symmetry, nonequivalent sites are decided [fig 1]. Charges in such sites, each of volume ,are
A Quantum Statistical Calculation of Electric Field Gradient in Indium
23
related to their potentials as
8
(2 ) [ +
]
=−
… (3)
3ℎ
Let there be
sites per unit cell, for i-th non-equivalent type. The cell neutrality
condition is expressed as
ℵ
= 0
… (4)
The Fermi energy
is calculated from eq. (4) with eq. (3) inserted in it. Iterations
give us self-consistent charges as discussed in the next section.
Once the charges are known, EFG is calculated as follows. Using the well-known
notations of previous work [4], let us take a charge q at position ⃗relative to an ionic
site as origin of co-ordinates in the crystal. It produces electric field intensity at the
origin whose space rate of variation along c-axis as Z-axis (the electric field gradient
along c-axis) is given by
Vzz=
∊
… (5)
Taking in the tcp lattice a-axis, b-axis and c-axis as X-axis , Y-axis and Z-axis
respectively, the crystallo-physical co-ordinates of lattice sites are denoted by real
number triplets N1, N2 and N3 in units of lattice parameters.To find the EFG in
equation (5), we must take contributions from all the sites in the lattice. For this a
lattice summation is required. Using notation as in the previous work [4], the EFG
may be written as
ℵ
=
Here
are charges given in equation (3), and ionic charge. The lattice sum is
contained in
=
4
Here
Bj= ∑ ∑
∑ ∑
(
(
)
) /
… (6)
The net EFG at nuclear site due to ionic and electronic charges can, thus, be
calculated. However, the nucleus that is put now in the crystal environment at the
origin interacts with the field gradient splitting its energy levels that is strongly
dependent on the degree of deviation of wave function from free state. A correction to
account for this is to multiply the distant sum by Sternheimer factor (1- ) for ions
[14]. Thus, we write for EFG (eq) at nuclear site in crystalline environment as
(1 −
)
eq= ∑ℵ
…(7)
The ℵ dimensionless lattice sums Bj in the above equation are independent of the
lattice parameter a, and depend only on the axial ratio c/a of the crystals with
interfacial angles 90°. The charges Qjand anti-shielding factor are dependent on
element in question.
24
Ashok Kumar and B. C. Rai
3. Numerical computation for Indium
3.1 Electron lattices:
The parameters along a- and b- axes of Indium unit cell are divided into 4 equal parts
each and c-axis into 8 equal parts, i.e., N=128. In these, a volume of 128th part of unit
cell volume belongs to each site: ω=a2 c/128. Equivalent sphere replaces the sites for
computational ease, whose radii are quickly obtained. We have used here the realistic
ionic radius obtained by enlarging it to cover 12 sites [13]. These ions occupy the
corners and the body center of the tcp unit cell. Their multiplicity is Mi=2.
Electronicsites, taken in planes perpendicular to c-axis, are composed of four nonequivalent types, and arepositioned in nonequivalent planes normal to c-axis.
Tetragonal symmetry is maintained about
c-axis
b-axis
a-axis
Fig 1: Dividing unit cell in non-equivalent points
new ionic sites, giving three non-equivalent planes and
with certain multiplicities [fig 1, and Table 1].
non-equivalent sites
Table 1: Non-equivalent sites
Nonequivalent
type( )
1
2
3
4
5
Diagrammatic Multiplicity
Multiplicity after
symbol
before
ionic
ionic
enlargement[128] enlargement[104]
2
2
48
24
32
32
24
22
24
22
A Quantum Statistical Calculation of Electric Field Gradient in Indium
25
3.2 Ionic charge:
TheRoothaan-Hartree-Fock wave function of indium [15] is used to first find orbital
wise charge. Then ionic charge is calculated using the ionic radius as obtained in part
3.1. After calculation of energetically allowed ionic charge, the size of ion is enlarged
covering 12 nearby sites. Then only 104 sites per unit cell remain with us for further
calculations. Ionic volume is now 13 times the electronic volume given above. Putting
lattice parameters of indium, the ionic radius turns out to be 2.34 au, which compares
well with its known value.
Integrations of orbitals are obtained for this radius of ion and tabulated [Table 2].
We observe that contribution of 5S and 5P orbitals are small. The electrons belonging
to these orbitals have greater probability of being found outside the ionic radius. It
appears factual, as we need proper concentration of charge carriers for the element to
behave as metal. The charges that are more populating the space outside the ionic
radius must contribute to this concentration.
Table 2: orbital wise charge and ionic charge
Orbital(occupancy)
1S(2)
2S(2)
3S(2)
4S(2)
5S(2)
2P(6)
3P(6)
4P(6)
5P(1)
3D(10)
4D(10)
Square integral of wave function
0.999994
1.00028
0.947083
0.551193
0.328504
0.999998
0.999995
0.997806
0.137715
1.000000
0.970358
Ionic charge=3.5 e
3.3 Conduction electron charges
With ionic charge with us, we distribute the double of ionic charge among 102 sites in
the unit cell equally, with negative signs. Potentials at non-equivalent sites are
calculated. These potentials are used in equation (4) with (3) to determine Fermi
energy numerically. With this Fermi energy and these potentials, charges for second
round of calculation are obtained using equation (3). The process is repeated until we
get convergent value of charges.
Thepotential at a site is supposed to arise due to own spherical mini-cell,
called local or self-potential, and due to point charges sitting at all the remaining sites
26
Ashok Kumar and B. C. Rai
in the lattice.
=
+
Lattice points are grouped into those within own unit cell and those outside it.
=
1
4
ℵ
[
−
ℵ
+
+
−
]
… (8)
If we select our field points nearest to the origin so that the source points at larger
distances, we can use the series for the sum. We have to be careful to avoid self-point
calculations also. With these points in mind, we have evaluated lattice sums required
to write potentials as
1
=
(
) + 1.2 /
… (9)
4
Here the matrix element
is the co-efficient of
in the equation (8), and is
evaluated in this work by using Legendre polynomial method [13]. The matrix
is
obtained by lattice sum of inverse of distances. It depends upon the axial ratio of tcp
lattice. We have obtained the matrix
for Indium at 300 K (Table 2). The ratio
(
/ ) is found to be constant and for the Table 2, its value for IDN=20 is about
3.41×103 (IDN is number of a, a and c units along the unit cell edges up to which
unit cells are taken in calculations). This observation means
=
That is the property obeyed by lattice summations in such cases [16].
Table 3: The
↓
matrix for axial ratio 1.52 of tcp lattice in unit of
1
2
3
4
5
→ 1
6823.48 81923.18 109267.32 81928.53 75098.83
2
6825.23 81915.48 109241.56 81921.99 75086.16
3
6826.25 81842.16 109249.66 81928.23 75089.73
4
6823.62 81924.42 109244.82 81920.53 75092.63
5
6828.62 81924.38 109268.42 81927.63 75090.53
⁄
A Quantum Statistical Calculation of Electric Field Gradient in Indium
27
The self-consistent electronic charges as obtained by us are given in the second
column of Table 4.
3.4 Electric Field Gradient in Indium
Lattice parameters of indium at 300 K are
= 3.253Å; and
= 1.52.
Sternheimer’s antishielding factor: 1 − is 25.9 for neutral atom. Using the charges
and another lattice sum for , given in the third column of Table 4, we have obtained
EFG of Indium in au (the last column of Table 4). Its value is in excellent agreement
with the recently reported one by Stralen and Visscher[17].
Table 4: Charges and EFG in Indium
Non-equivalent site in a a c Non-equivalent charges in e
EFG in au
Ionic (0 0 0)
+3.5
2.1724 -3.39
Elec. (0.5 0.25 0)
-0.068
64.546
Elec.(0.25 0.25 0)
-0.278
141.319
Elec.(0.5 0 0)
-0.079
-39.2953
Elec.(0 0 -0.125)
-1.627
617.382
4. Result and Discussion:
The value of 2.34 au for the ionic radius of indium, defined as described in the
protocol of the model, nicely compares with known value. The ionic charge is close to
tri-valence by about 16%. The square integral of 5S and 5P orbitals indicate that these
electrons have greater probability outside the ionic radius and appear responsible for
metallic character of Indium.
The electric field gradient is computed due to ions and effect of core polarization is
taken care of by Sternheimerantishielding factor. Conduction electronic EFG is also
computed and the value of Raghvan’s correlation is found about 5 and is comparable
[18]. The total electric field gradient along c-axis of lattice inatomic unit is -3.39
which compares well with the values obtained by a bit sophisticated work of quantum
physics [17].
The results of ionic radius, ionic charge and electric field gradient calculations give us
very closer values. This is encouraging for further work along the line or off the line
set by this work.
5. Conclusion
The charge distribution obtained in Indium using quantum statistics and a ThomasFermi type of model for tcp metals gives correct sign, order and order of magnitude of
28
Ashok Kumar and B. C. Rai
EFG in indium as compared by reference work of Joost N. P. van Stralen and Lucas
Visscher(2002). The model is believed working well for EFG in tcp metals as verified
by the work on Indium.
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[14] Sternheimer, R. M.: Phys. Rev. 159, 266(1967)
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