JOURNAL OF SCIENCE Ol- MNUE
Natural Sci., 2011 Vol. 56, No. 7, pp. 58-64
THERMODYNAMIC PROPERTIES OF SOME RARE-EARTH METALS
Vti Van riimg(*)
//(/;//,(;/ National University of Education
Dang Tlianh Hai
Vietnam Education Publi.shing House
^*^ Ivinail: bangvu57'" vaho(j.com
Abstract. Thermodynainic proptMties of rare-earth metals have been studied using st;iislical moment method. The analytic expressions of tlK> llehnholtz hvv. (Mieigy and thermodynamic quantities were obtained.
Present SMM r(^snlts of nc^arest neighbor distance, liiu^ar thermal expaiLsion- coefficient and specific heats at constant pressure for Th and Ce metals are compared with the experimental results.
Keywords: Thermodynamic, rare-earth metals, staistical moment method.
1. Introduction
Recently, there has been a great interest in the stud}' of rare-earth metals since they provide us a wide vari(>ty of academic problems as well as the technological ap- plications. Various theoretical studies of simple, noble and transition metals [1-3]
have been made so far based on pseudopotential theory but relat ivel>- few attempts have been made on rare-(>arth and actinide elements such as La. ^'b, Ce and Th [ 1-6] by the same method. Rosc^ngren et al [7] have investigated the lattice dynam- ics of thorium using pseudopotential due to Krasko and Gurskii [8]. N.Singh and S.P.Singh [9] have calculatcHl the phonon dispersion of La, \h, Ce and Th. Recently, Pandya c4 al [10], have in\-estigated the phonon dispersion cur\es, phonon density of state's. Del)ye-Walkr factor, mean sciuare displacements and equation of state for thorium. J.K.Baria and A.R.,lani [11] also have calculated the phonon dispersion cuiA-es. phonon density of state's, Debye temperature, Griineisen parameters and dynamic elastic constants for La, \h. Ce and Th.
.Most of the previous theoretical studies, however, are concerned with the ma- terials properties of rare-earth metals at absolute zero temperature and temperature dependence of the thermodynamic quantities has not been studied in detail. The purpose of the present article is to investigate the temperature dependence of the thermodynamic properties of some rare-earth metals using the analytic statistical moment method (SMM) [12-15]. The thermodynamic quantities are derived from the Helmholtz free energy.
Thermodynamic properties of som.c rare-earth meta,ls
2. C o n t e n t 2.1. T h e o r y
To derive the temperature dependence of the thermodynamic properties of rare-earth metals, we use the statistical moment method. This mc^tJiod allows us to take into account the anharinoiiicit>' effects of thermal lattice vibrations on the thermodynainic ciuantities in the analytic formulations.
The essence of the SM.M schc-^me can be summarized as folkjws: for simplicity, we derive the thermodynamic quantities of cr^•stalline materials witJi cubic .symme- try, taking into account the higher (fourth) order anharmonic contributions in the thermal lattice vibrations going be>-ond the ciuasi-Hamonic (QH) approximation.
The extentions for the SMM formalism to non-cubic .systems is straightforward.
The basic equations for obtaining thermodynamic ciuantities of the given crystals are derived in a following manner: the eciuilibrium thermal lattice^ expansions are calculated by the force balance criterion and then the thermodynamic quantities are determinded for the equilibrium lattice spacings. The anharmonic contributions of the thermodynamic quantities are given explicitly- in terms of the power moments of the thrnial atomic displacements.
Let us first define the lattice displacements. We denote Uu the vector defining the displacement of the ith atom in the 1th unit cell, from its equilibrium position.
The potential energy of the whole crystal U{uii) is expressed in terms of the positions of all the atoms from the sites of the eciuilibrium lattice. We use the theory of small atomic vibrations, and expand the potential energy f/ as a powder series in the cartesian components, IL{^ of the displacement vector uu around this point.
For the evaluation of the anharmonic contributions to the free energy (/', we consider a quantum s>-stem, whicli is influenced by supplemental forces a, in the space of the generalized coordinates g,. For simplicity, we only discuss monatomic metallic systems and hereafter omit the indices I on the sublattices. Then, the Hamiltonian of the crystalline system is given by
H = HQ-Y,a,qi. (2.1)
i
where HQ denotes the crystalline Hamiltonian without the supplementary forces tti and upper huts A represent operrators. The supplementary forces a, are acted in the direction of the generalized coordinates g,;. The thermodynamic quantities of the anharmonic crystal (harmonic Hamiltonian) wiU be treated in the Einstein approximation.
After the action of the supplementary forces ai the system passes into a new equihbrium state. If the 0th atom in the lattice is affected by a supplementary force ap, then the total force acting on it must be zero, and one gets the force balance
Vu Viiu Hung and Dang Thanh Hai relation as
2 - ^ \ ()ti,n(>ii,.i / 4 -"^-^ \ (hi,„()n,,^()Ui^ I
l.n \ I ,n . , " . 7 \ • / Cf/
E
(IV/O,o / , •) •) •) "3 ('"'n'/.„fa,„) - r^.i - 0 12 '^^-^ \ (hi.,„(hi.,,i()n,^(hr,,,
I .l\ 'i.l]
(2.2)
The thermal a\'(>ra}',es of th(> atomic displaceiiKnits {u^a'n.,:)) and {u.nii,:ra-ni) (called as s(>(()n(l and third order monunits) at givcm site R, can be expressed in terms of Ihe hist moment (//,„) with the aid of the leciirence formula [12. 14].
Jlicni (Miuation (2.2) is trarisfornuxl into the new differential equation:
•.()''''•' I li-^Oir^ I Vl' I ^'/ I -"^f (-^coth-r 1)// o - 0. (2.3)
do' d(\ '• k
where
r;V»o'
niu;
1 12 - ^
3 " • < /
+ 6
O^^iOdii^fidu^. 3^"-i' eq.
(2.4) hw
^' -" " . ^ : '/ = ^'">)-
9 = krjT:
with A;^ is the Boltzmann constant and u; is the atomic frequence.
Then, the solutions of the non-linear differential equation (2.-3) can be expanded in the power series of the supplemental force o as
// A/- -|- liO -f 4.JO" (2.5)
Here. A/ is the atomic displacement in the limit of zero of supplemental force a.
After a f)it of algebra, it can be slitjwn that the atomic displacement Ar in cubic .systems is given by [12]
ir --
'276/^
3A^ A (2.6)
Once the thermal expansion A r in the lattice is found, one can get t h e Helmholtz free energy of the system in the following form
•(/.' = Ho -\- i'o 4- il'i (2.7)
Thermodynamic properties of some raxe-earth m,etals
where ^^o denotes the free energy in the harmonic approximation and V'l the anhar- monic contribution to the free energy. We calculate the anharmonic contribution to the free energy t/'i by applying the general integral formula [12]
ill = UQ -f- t/i '0 + / (K);
./O
dX (2.8)
where AU represents the Hamiltonian corresponding to the anharmonicity contribu- tion. Then the free energy of ther system is given by
V = UQ + ?>N9 26^
k^ L3 " -7.?.rcothx( 1 -I
.T -I- ln(l - p"^"^) .rcothx \
4- 3A 9 2 r
A;2L 72a"'^coth .r - ; 7i( 1 o .Tcothx \
/ , .TCOth.T \
('-'—2-)l
2 - ) - 2 7 i ( 7 i + 2 7 2 ) ( l + ^ ^ ^ ^ ^ j ( l + x c o t h . T ) | W2.9) where the second term denotes the harmonic contribution to the free energy.
With the aid of the "real space" free energy formula •;/' = E — TS, one can find the thermodynainic quantities of given systems. The thermodynamic quantities such as specific heats and elastic modul at temperature T are directly derived from the free energy xl' of the system. For instance, the isothermal compressibihty XT is given bv
\ao/
XT =
1 V2/dH'
op -j- —-1 3A a Kdr'^Jr
(2.10)
where
Qj.2
^^Ald'^UQ ^r.rcothxS^A: 1 fOkx"^/ , ;r^
) | | , (2.11) here P is the pressure and a is the nearest neighbor distance at temperature T
a = ao + Ar, (2.12) where ag is the nearest neighbor distance at zero temperature.
Using the expression of the free energy ih from (2.9), after a bit of algebra, the specific heat at constant volume Cy is derived as
u^ 3A^^" B
u
sinh^i /,-;2 X+
29(2.. I )
-^K-
sinh ,T X ) - - ( / X'7i \ x^cotha:
sinh^x 2x'*coth^x
+
sinh X
+
sinh X)i)
(2.13)Vu Vail Hung and Dang Thanh Hai
Jlieii tlu^ sp(M ific heat at constant pressure Cp is given from the thermodynamic relation as:
WViy'j.
r;,-c„ +
\r
(2.14)
(2.15) wluMe th(> linear 1 hernial expansion coefficient rvj- is given by
''' ""''" :\ii^'~:\NiWifn
for simplicity, we lake the effective pair interaction energ>' in rare-earth metals as the power law, similar to the Lciimard Jones
D
; • ' • )
[n -in) 711
CfT-'^-r
(2.16)where D,/o are determiiKHl to fit to the (experimental data (e.g.. cohesixe energy and elastic modulus). Using Ihe (4fectiv(> pair potentials of Ec[uation (2.16). it is straightforward to g(>t the interaction energy ii.o and the i)araineter k. ^. in the crystal as
1 ^r- , . D
" " •"" 2 / .^"'"(^'')
(•// in) .
rnA„(-^] -nA„J-)
(2.17)1 ^-^ \ (hi-^ / 2a^{n in) L 1 V a / Dnrn.
2(1-[n - in)
•J
• r o \ " '
(.nf2)4;,-.l„„,](^)
(2.18)
1
•E
; l l .,CJ
cl(/;^.c)^/;^^
eg.
= 4(-,,+-.2)
Dnni
12c/,'(// //?') (" + 2)(77 4 4 ) ( n 4 - 6 ) ( i : ^ , + 6 < : : - ) - 18(/; I 2)(-» I l).4;H'„-f 9 ( , M - 2 ) A , ^ , ] ( ' ^ ) ' ^
(777 4 2)(7/7-f i)(m4 6 ) ( ^ : ; ; ; , + 6 < J : ; J ) -
- 18(m -f 2)(m 4- 4)^;^, + 9{m + 2)A.n,+,\ ( ^ ) ' (2.19) where rrio is the mass of particle, UIQ is the frequency of lattice vibration and An.A„,.... are the structural sums for the given crystal.
Thermodynamic properties of some rare-earth metals
2.2. Results and discussions
Using the moment method in statistical dynamics, we calculated the thermo- dynamic properties of rare-earth metals Ce and Th. The; potential parameters are fisted in Table 1.
Table 1. Parameter D and ro detemiined by the experimental data [18]
M e t a l Th Ce
n 4.0""'
17.0"
m 3.5 I2.O'
D/kniK 4458.6
1966"
' • o ( A )
3.5898 _ 3.6496
Table 2. Temperature dependence of thermodynamic quantities of Th metal T{K)
"(A) _^
cAr(10-^A'-^)
£:.rp[17]
Cp{cal/mol.K) Ex2)[17]
300 3.1310
12.58 11.1 6.08 6.53
400 3.1349
12.61 11.9 6.16 7.00
500 3.1388
12.66 12.5 6.22 7.45
600 3.1428
12.72 13.1 6.29 7.90
700 3.1468
12,79 13.7"' 6.35 8.36
800 3.1508
12.86
"14.2 6.42 8.81 Table 3. Temperature dependence of thermodynamic quantities of Ce metal
T{K) a(A)
QT(IO-^A:-^)
Exp[n]
Cp{cal/mol.K) Exp[n]
300 3.4496
8.28
-
5.94 6.80
400 3.5525
8.37 6.0 6.15 7.30
500 3.5555
8.50 6.1 6.30 7.70
600 3.5585
8.66 6.3 6.43 8.10
700 3.5615
8.84 6.8 6.55 8.50
800 3.5647
9.05 7.6 6.66 8.90
We present in Tables 2 and 3 the hnear thermal expansion coefficient Qy, nearest neighbor distance a and specific heats at constant pressure Cp of Th and Ce metals calculated by the present SMM, together with those of the experimental results [17]. The calculated thermal expansion coefficients of Th and Ce metals are in good agreement with the experimental results. The thermal expansion coefficient of Th and Ce metals are also calculated as a function of the temperature T
The calculated specific heat at constant pressure Cp of Th and Ce metals are presented in Tables 2 and 3. As shown in these Tables, the specific heat Cp depends strongly on the temperature. The caculated lattice specific heat Cp at constant pressure are in good agreement with the experimental results.
Vu Van I lung and Dang Thanh Hai
3. Conclusion
The SMM calculations of tlK^rmodyiiamic quantities of Th and Ce metals were IxM-formed. Pres(>nt SMM results of the linear thermal (expansion coefficient QT, specific- hci\\s al constanI pressure^ C,, inv. in good agreement with the experimental data.
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