P r o p e r t i e s o f t h e m a t t e r at h i g h p r e s s u r e s 

b y Y . P . T R U B I T S Y N 

Ricevuto il 7 dicembre I960 

The ionic crystal MgO is the most abundant crystal of the Earth's 
rocks. The state equation for MgO was obtained in this paper for pres-
sures between zero and million atmospheres. The state equations for 
metal at high pressures are significant for physics of the Earth's core. 
In this paper these equation were obtained for metal in the analytical 
form. 

Usual Tliomas-Fermi-Dirak theory of the electron gas is applicable 
only at extremelly high pressures p > 109 atm. Statistical Gombas-
Jensen theory of the ionic crystals and metals gives good results only 
at not high pressures. Therefore in this paper the modified statistical 
theory is used. By statistical theory the energy of some system of the 
electrons with density q is (Goinbas 1949) 

U(Q)=f\*k Q5'3 + (Vjc + V. ve) e - < Q1'31 OT 
xk = 2.871 xa' - 0.8349 

I t is the sum of kinetic (Fermi) and potential energy of the electrons 
in the field of the nuclear and of the exchange-correlation energy. If 
the electronic density q is known from the experiment and is substituted 
into the formula for the energy, then the problem leads only to integration. 

In this paper the energy of the ionic crystal MgO is calculated. 
The electron density in crystal is taken as the sum of the overlapping 
electron densities of the ions. The electron density of the oxygen defor-
mated in crystal was taken from Yamasliita's and Koyima's works 
(1952). That of magnesium was taken according to Hartrec-Fock of 
isolated ions. For pressures 0-106 atm. the calculated energy (with 
account for the quantum correction) can be approximated by the for-
mula (Trubitsyn 1958). 

E(v) = 93.7 exp (— 1.69 «Vs) — 3.495 trVa + 0.582 at. unity 
atom 



I \ 

1 0 4 V . P . T R U B I T S Y N 

The pressure is found by the formula 

P = 
dE 
dV 

(1 at. unity of pressure = 3 X 10' atm., 1 at unity of volume = 
0,0886 

cm , /i — is the atomic number of the matter), 
s 

Figure 1 plots the calculated energy of the crystal MgO at T = 0°K 
(a) and the experimental energy of Bridgman (1949) extrapolated by 
Davydov's method (1956) (b). 

F 
atun. 

a font 

0.6 

0.9 

0.2 

-0.2 

109 10* 10' 

atm 

0 6 0.8 7.0 V y 

K 

Pig. 1. - The energy of t h e ionic crystal MgO calcu-
lated by t h e author(curve a) and extrapolated 
from t h e experiment (curve b). 

For metals the electron density in crystal is taken as the sum of 
the density of the ionic cores and that of valence electrons. The first 
was taken according to Hartrec-Fock for isolated ions and was appro-
ximated by the formula o+ (r) = Aexp (—br). The second was appro-
ximated by the constant gv (r) = q/V (Theis 1955). q — is the number 
of valence electrons. The integrand was taken in form of power series 



P R O P E R T I E S OF T H E M A T T E R AT H I G H P R E S S S U R E S 1 0 5 

and integrated. The dependence of energy and pressure upon volume 
was obtained for metals in the analytical form (Trubitsyn 1960) 

E(v) = 2.871 v^—qll3 v^k[0.8349 + 1.4508 g2'3] 
+ g6'3 b~a v-*'* [784.12 — 525.06 x + 16.4934 x2 — 12.026a?] 
+ qll3 b~3 tr4/3 [720.26 + 244.81 x + 31.517 x2 + 3.4972 a;3] 
+ q Zr3 v-1 [405.88 A^ — 755.40 A1'* — 631.65 A 6"2] 

x = In (Aq'1 v) . 

The pressure is P = — dE/dV. Every metal was characterized by 
means of three parametres A, b and q. 

Por magnesium A — 45, b — 4,9 and q = 2. 
Then 

p(v) = 21.52 v-1 + 6.095 tr'/s — 2.6352 v*'* 
+ ir7'3 [15.3236 + 5.6403 x + 0.6753 x2 + 0.09985 a:3] 
+ ir8'3 [49.4291 — 24.5025 x + 1.7152 x2 — 0.51083 a;3] ; 

x = In (22.5 v) . 

(curve a) and extrapolated from t h e experiment (curve c), as well as 
calculated by Gombas (curve 6). 



1 0 6 V . P . T R U B I T S Y N 

Figure 2 plots the energy and pressure of the metallic magnesium 
a t T = 0° K, calculated by the author (a), this one calculated by Gombas 
(1949) (b) and the experimental energy and pressure of Walsh and others 
(1957), recounted for T = 0° K (c). 

One can account for the temperature by the Davydov's method 
(1956). 

I t is a pleasure to thank Professor B. I. Davydov for helpful dis-
cussions. 

SUMMARY 

The modified statistical method is proposed for calculation the equation 
of state for ionic crystal and metals. The equation of state for metals is 
obtained in analytical form. The method is illustrated on magnesium and 
magnesium oxide. 

RIASSUNTO 

Si propone il metodo statistico modificato per il calcolo delVequazione 
di stato per il cristallo ionico e relativi metalli. L'equazione di stato per 
i metalli si ottiene in forma analitica. II metodo e illustrato nel caso del 
magnesio e dell'ossido di magnesio. 

R E F E R E N C E S 

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