1. INTRODUCTION:
The understanding of the ionic solids at non-ambient conditions
received considerable attention in solid state physics which is routinely
confronted with problems involving ionic solids at high pressure and high
temperature. In such studies, the equation of state (EOS) of condensed matter
plays a vital role. A relation between pressure and volume, which is capable of
predicting 
behavior of a
substance at constant temperature, is usually known as an isothermal EOS. The
study of the forces between atoms is essential to explain an EOS for
thermodynamic as well as thermo-elastic properties of substance. The exact
evaluation of these forces from atomic theory is one of the most difficult
problems of the quantum theory and wave mechanics. Hence, due to the lack of a
precise knowledge of inter-atomic forces, a theoretical EOS cannot easily be
obtained. Therefore, different simplifying models and approximations are being
used to obtain an EOS and due to this reason semi-empirical EOS’s have been
developed. The commonly available EOS’s are those given by Birch-Murnaghan1,
Vinet-Rydberg2, Shanker3, Bardeen4, and Kumar5.
Among these EOS’s, one of the most successful EOS is that proposed by Vinet et
al.2 which is valid for all classes of solids in compression and in
the absence of phase transition. The basis of this EOS is a universal relation
for the binding energy in terms of the inter-ionic distance.
In the derivation of this EOS, contribution of the thermal pressure is neglected, and the volume derivative of the binding energy is used to approximate the internal energy. In the present paper, we have exploited an analysis of bulk modulus for solid Al under extreme compression by using isothermal, Vinet-Rydberg EOS. The variation of relative compression volume and isothermal bulk modulus with pressure is studied and compared with available experimental results. Moreover, the effect of high pressure on melting temperature of Al is studied using the Lindemann-Gilvarry law of melting.
2. ANALYSIS:
2.1 Vinet-Rydberg equation of state (V-R EOS)
The pressure-volume (
)
relationship for a solid at a given temperature can be expressed as
non-inverted type6
(1)
where 
is the
volume at pressure 
. 
, 
and 
are the values of
bulk modulus, first pressure derivatives and second pressure derivatives
respectively, all at 
.
The EOS for such non-inverted type of solids is derived from the relation
(2)
Where, 
is the
inter-ionic potential energy.
The bulk modulus 
for
non-inverted type can be derived from the following equation:
(3)
The Rydberg potential energy 
expressed as a
function of inter-ionic distance 
can
be written as:
(4)
where 
and 
are the potential
parameters.
The V-R EOS ased on can
be written as2
(5)
where ,
The isothermal bulk modulus corresponding to V-R EOS can be written as
(6)
2.2 Lindemann-Gilvarry law of melting:
One of the most widely used attempts to predict the melting curves of solids is the Lindemann-Gilvarry law of melting. Sunil et al.7 obtained an expression for melting temperature of solids in terms of volume dependence of the Gruneisen parameter as
(7)
3. RESULTS AND DISCUSSION:
The pressure has been calculated at different relative compression
volume ranging from 
to 
at room temperature
for solid Al using equations (5) and (6). The input parameters 
and 
are taken from
Hänström and Lazor8. The results are presented in Figure 1 and
compared with experimental data8,9. From this figure one can see
that pressure decreases continually with the increase in the relative
compression volume and agree with experimental data8,9.
Figure 1: Pressure dependence of relative compression volume at room temperature for solid Al. solid line indicates our result, symbols are for experimental data8,9.
Isothermal bulk modulus has
been calculated at same relative compression volumes from 
for solid Al and dependence
of with pressure 
is developed in
Figure 2. This figure indicates that bulk modulus increases continuously with
increase in pressure. One can see from this figure that pressure dependence of
bulk modulus based on V-R EOS agree with Singh10 based on
first-principles results up to 
.
However, there is deviation in bulk modulus above this pressure range with
experimental data.
Figure 2: Pressure dependence of bulk modulus at room temperature for solid Al. solid line indicates our result, symbols are for results of Singh10.
The melting temperature 
for
solid Al has been calculated by employing Lindemann-Gilvarry law of melting
obtained by Sunil el al.7 i.e equation (7) and V-R EOS. Figure 3
shows a comparison between the calculated and melting
pressure for Al with
experimental data from Hänström and Lazor8 and Boehler and Ross9.
The input parameters10 for Al are 
, 
and 
. From this figure
it is evident that the results obtained from Lindemann-Gilvarry law of melting
and V-R EOS are in agreement with experimental results8,9.
Figure 3: Pressure dependence of melting temperature for Al. solid line indicates our result, symbols are for experimental data8,9.
4. CONCLUSION:
The present analysis describes adequately some important features for solid Al as:
i)
Pressure decreases
exponentially with relative compression volume 
at room
temperature.
ii) Isothermal bulk modulus increases
continuously with increase in pressure . 
iii) Melting temperature increases
continuously with increase in melting pressure .
It should also be noted that the pressure dependence of bulk
modulus for solid Al suggests that for high pressure region (
), V-R EOS deviates
from experimental results. This is due to the disregard of higher order
pressure derivatives in formulating V-R EOS. Pressure dependent melting
temperature for solid Al using Lindemann-Gilvarry law of melting is in
reasonable agreement with experimental data for entire pressure range. We,
therefore, recommend this law at extreme compression of any type of solids.
5. REFERENCES:
1. Anderson OL. Equations of State of Solids for Geophysics and Ceramic Science. Oxford University Press, New York. 1995: 370.
2. Vinet P, Ferrente J, Smith JR and Rose JH. A universal equation of state foe solids. Journal of Physics C. 19; 1986: L467-L473.
3. Shanker J, Kushwah SS and Kumar P. Equation of state and pressure derivatives of bulk modulus for NaCl crystal. Physica B. 239; 1997: 337-344.
4. Butler R and Anderson DL. Equation of state fits to the lower mantle and outer core. Physics of the Earth and Planetary Interiors. 17; 1978: 147-162.
5. Kumar M. High pressure equation of state for solids. Physica B. 212; 1995: 391-394.
6. Digpratap S, Rakesh K and Arunesh K. Analysis of temperature dependence of thermal pressure of solids. Indian Journal of Pure and Applied Physics. 45; 2007: 654-657.
7. Sunil K, Anand K and Sharma BS. Pressure dependence of melting temperature for alkali halides. Indian Journal of Pure and Applied Physics. 51; 2013: 444-447.
8. Hänström A and Lazor P. High pressure melting and equation of state of aluminium. Journal of Alloys and Compounds. 305; 2000: 209-215.
9. Boehler R and Ross M. Melting curve of aluminium in a diamond cell to 0.8 Mbar: implications for iron. Earth Planetary Science Letters. 153; 1997: 223-227.
10. Singh PK. Pressure dependence of bulk modulus for solids based on the first-principles results. Indian Journal of Pure and Applied Physics. 49; 2011: 829-832.
Received on 10.10.2014 Modified on 20.10.2014
Accepted on 27.10.2014 © AJRC All right reserved
Asian J. Research Chem. 7(12): December, 2014; Page 1059-1061