Dynamics of Liquid Lutetium

Dynamics of liquid lutetium

R V Gopala Rao and R Vcnkaiesh

Division of Physical Chcmisiry, Jadavpur University, Calcuita-700 032, India R eceived 5 M arch 19QI, accepted 10 September 1991

Abstract : The potential function of liquid lutetium has been calculated using Ashcroft’s pseudo potential in ihc harmonic approximation due to Class and Rice [11 By this method wc have successfully calculated the p< wer spectrum as well as the longitudinal and transverse phonon frequencies I'he characteristic frequency calculated from the potential function and power spectrum, are found to be in agreement with that obtained from sjiecific heal. The elastic constants of the liquid have been calculated from phonon frequencies and have also been computed from Schofield’s /j and 12 integrals. The values so calculated are found to be in good agreement with those calculated from phonon frequencies Also from the compressibility an average frequency has been calculated using Einstein’s equation In analogy with average velocity, an average frequency is defined and calculated. The value so calculated is found to be close to that obtained from Einstein’s equation Isothermal compressibility has been calculated from the long wave limit of the stmciurc factor, from the potential function and from C |i. The values calculated arc found to be self-const stent It may be noted that the theory along with the Ashcroft potential predicts the physical properties reasonably well

Keywords : Phonon, structure factor, elastic constants, pseudopotenlial, autocorrelation function, dispersion

PACSNo. : 61.25 Mv

1. Introduction

Dynamics of a system o f interacting particles are usually described by the time correlation lunctions [2, 3]. The molecular dynamics experiments have renewed the interest in understanding the basic nature of the dynamical correlations in liquids. Both computer experiments [4, 5] and scattering techniques have [6] helped in revealing the nature of the lime correlation functions. The velocity correlation function the power spectrum g(o}) and the effective pair potential 0 (r) are some of the important properties that give information regarding the dynamical properties of liquids. The derivatives of the effective potential can be used to compute longitudinal and transverse phonon frequencies as well as the elastic constants of liquid lutetium. The power spectrum which shows the dispersion in frequency can be used to calculate the characteristic frequency of the liquid.

It may be pointed out that in recent years several attempLs have been made to utilize the simple but extremely useful Ashcroft’s pscudopotential which has been applied to a variety o f metals treating the Ashcroft core radius and Z the valency as parameters [7-lOJ.

Irt this paper, we describe the dynamical correlations in liquid lutetium using Ashcroft’s

© 1992 lACS

pseudopotential formalism. This potential has been successfully used in deriving the structure factors of several rare earth metals [9] including lutetium. At this juncture it may be pointed out that Badiali and Regnaut [8] applied Ashcroft’s potential to metallic gallium to obtain thermodynamic properties while Rao and Bandyopadhyay [9] used the same potential to derive the structure factors of alkaline earth metals including both IIA and IIB group metals. Rao and Venkatesh [10] applied this potential to obtain the structure factors of vansition metals. According to this theory the metal is considered as a sea of valence electrons in which the massive metallic ions are embeded.

600 R V Gopala Rao and R Venkatesh

2. Theory

The model which we use is that of a diffusing particle in a harmonic well with a characteristic frequency d>o. Glass and Rice [1] obtained the following differential equation for \p{t)

^ ^ ^ = 0

Here v^O is the velocity autocorrelation function defined as

(1)

<v > ⁽2⁾

<v> is the average thermal velocity. Along with others [1, 11], the following physically acceptable boundary conditions are used.

lim iKO = 1 (3) r^-»0 ' f->0

d V(0

lim — = -<»„■ =

i^{-* 6} dt"' 3m

From these equations the is obtained as [IJ V(0 = [cos (^0 + (j8/2§) sin (§f)]

Here ^

ft m_£ „ 2 P " kaT

(4) (5)

(6) (2) (8⁾ Here D is the self diffusion coefficient while the rest of the symbols have their usual connotation. We may point that mDcaJkaT = 0.75, which is therefore accq>table as it gives a real value of ^ (vide eq. (7)). As is well known statistical mechanically, the thermal average potential in terms of the radial disu'ibution function is given by

< > = 4;rp \dr r" g(r) W ' (r) + 2 ^ ' (r)/r] (9) Here p is the average number density, 0'(r) and 0"(r) are the first and second derivatives of the potential function respectively and we write (ji (r) as

( 10)

Dynamics o f liquid lutetium 5qi

Here the first term on the right hand side is the coulomb repulsion between ions of valence Z and the second is the indirect interaction through conduction electrons which brings in attraction between the positive ions and is the normalised energy wave number characteristic function which can be written as

We use Ashcroft’s pseudopotential for (Oi,{k) which c«n be written as

a>b {k) CO& kr, (12)

where r<; is the Ashcroft core radius and £} is the v(4ume per electron, e (k) the dielectric screening constant and is given by

e ( k ) = 1 - (4ne^Jk^)X(k) --- :2//z V —

\ + ( 4 n e V t ) X { k ) m

X (k) takes into account the interaction between electrons and is given by I n . ^{^ k , - k}

| 2* < ± * n

I

IJ

(13)

(14)

2 " ⁸kjk ■■■ I²k f - k IJ ■ Hk + k

while y(k) takes into account the exchange and correlation energies among the elecu'ons 112]. We use Vashisia and Singwi’s formulation [13] which can be written as

Aifc) = A [ 1 ( 1 5 )

Here A and B have been computed as given by them and k f - (3«^2p)‘^ is the wellknown Fermi wave vector.

A detailed study of the microscopic motion of the atom is carried out through the computation of the so called power spectrum which gives the frequency dependence of the spectral density. The power spectrum is calculated through the following eqfuation,

f T*

g(a>) = \di iKO cos (fi)t) (16)

An importaint application of the potential function given by eq. (10) is in the evaluation of both longitudinal and transverse phonon frequencies with me use of radial distribution function derived earlier by Waseda and Miller [ 14].

The phonon frequencies are calculated through the equation derived by Takeno and Goda [15] which are given by

(k) s Idr g(r) [r0' (r) [1 - ] + [r^ 0" (r) - np\r)]

. 1 sin (kr) 2 cos ( M . 2 sin (kr) , I3 " kr " (kr) {krY ^ a 'rtt) - 1* s w w w I' -

(17)

1 cos (kr) sin (kr)

{kry (k r y (18)

Here g{r) is the radial distribution function, p is the number density and M is the atomic mass.

602 R V Gopala Rao and R Venkaiesh

Results and discussion

The effective pair potential obtained by using eq. (10) is given in Figure 1. The input parameters used are given in Table 1. As can be seen from Figure 1 the potential has a well defined characteristic depth. The minimum of the potential is seen to be occurring near the same position as the first peak of the radial distribution function. The oscillations as expected occur for long distances and the depth of the potential accounts for the softness of the repulsive core. The characteristic frequency cOo in the harmonic approximation as obtained from eq. (5) is in close agreement with that obtained through the use of Debye’s law of specific heats and the various values of (Oo are given in Table 1. The specific heats of lutetium are taken from literature 116]. It is important to note that all (o^ values are nearer to each other.

Table 1, Input parameters and characteristic frequency. Temperature = 1953°K

Packing _P D Z

fraction cm^/scc effective

charge

(A)

0.44 0.0316 3.01 X 10 • 2.7 0 830 83 2.32

co„xl0'¹³

7^ law spectrum

2.1 2.1

The velcx:ily autocorrelation function as given by eq. (6) is shown in Figure 2. The function shows a negative region and is due to back scattering of the atoms by short range core collisions, Wc also note that oscillations of V'CO approach zero as / becomes large while it shows a rapid initial decay starting with an initial value of unity. It may be mentioned that if r is the collision lime then v^(/) becomes zero as / » r i.e. in the hydrodynamic regime [15],

In Figure 3 we show the variation of g(o) with frequency. It is important to note that the power spectrum peaks at a value close to <0o = < >/3m, the characteristic

rigurc 2. Velocity auto-correlation function

Ircqucncy ol the liquid. The peak value of the power spectrum (Op^ is also given in Table 1.

The average value of ^(Op is 2.13 x l()'^ sec ' and is close to other values obtained by different methods.

t j K lo'^ s»c'

Figure 3. Vanaiion of ^{a>) with frequency

The phonon frequencies have been evaluated through eqs. (17) and (18) using Waseda-Millcr values of g i f ) [14] and arc given in Figure 4. We sec from Figure 4 that (0^\

and (⁰, are zero at it = 0. (⁰\ shows oscillations for large values of k whHe it attains a maximum at smaller value of k than o)i. Thus the oscillations at large k in fi>i(A:) indicate the existence of collective excitons at large momentum transfer while a>^k) attains a smooth

6(>4 R V Gopala Rao and R Venkaiesh

Figure 4. Ix>ngiiudmal and iransverse phonon frequencies in liquid luteiium.

maximum at larger k and remains constant. At this stage it is important to point out that in fi)i (k) the first minimum at A = 2.4A^’ is close to the values of the first peak of the structure factor S{k) (k = ^{2 . 2} A '). This was predicted to be nearly true by Bhatia and Singh (171.

As a further lest and use of the derived (⁰\(k) and (⁰,(k), we calculate the elastic constants Cn and C44 from the maxima obtained in <w,(k) and (⁰,{k) versus k plot. Since liquids are isotropic we calculate C12 from the Cauchy relation

C „ - C i2 = 2C44 (^9)

The values so calculated are given in Table 2. At this juncture it may be mentioned that the elastic constants can also be calculated through the use of l\ and ¹² integrals defined by Schofield (18) as

h _ _ e _

■ 2M _V

d<l> j r )

d r d r

2 d 0 { r )

~ d P ~ ‘^^

(20)

(21⁾

These equations are related to elastic constants in liquids as (18]

(22⁾

(23) (24) we evaluated the /i and I² integrals using the radial distribution function obtained by

Cii =

2/, + ‘j )

=

(1 2fi h ^.

C

^T5 15^

4/1

/2 ,

Cj4

=

(1 ^15 15 ^

Fourier transform ation o f the structure factor. From these values o f / i and I2 we com pute the elastic co n stan ts an d the resu lts so obtained are given in T able 2 for com parison. The agreement obtained by the tw o different m ethods can be considered as good.

Table 2. Elastic constants from phonon frequencies and and / j integrals.

C u C44

in units of 10^^ dyncss/cm^

Cj2

From a'l and 2 92 _1.22 0.85

From /i and I2 2.75 _0.95 0 90

integrals

A n intersting application o f the derived potential from eq. (10) is the evaluation o f com pressibility in ar. approxim ate way. It can be shown that in condensed system s [19] that the isotherm al com pressibility is related to the potential function ^ a s

1 - - I - r

^ 9v-

L

dr^ ^2r r= r„ _{(2 5 )}

Here r„ is the nearest neighbour distance and Vp is the volume per particle. However can also be o b tained from 5(0) the structure factor in the long wave limit which is given by

(2 6 ) Wc use W ased a-M ilicr [14] value o f 7710 obtain S (0) and from the com puted phonon frequencies w e have already prescntetl C n from which we can obuiin since

C n = r /A s o (2 7 )

Assuming unity for / t h e specific heat ratio for liquid lutetium at 1953'^K we can obtain Aso from eq. (27). Further, we have the well known com pressibility summation rule as

5(0) - p kg I P^t

Table 3. Compressibilities from differcni methods

(28)

From from From

S{0) polenual function ^{cq (22)}

p x 10'‘*

cm /dyne2

3 1 3.7 3.42

H ence, from cqs. (26) and (28) wc obtain com pressibility. The values, calculated by various m ethods, are given in Table 3 and arc found to be self consistent. At this stage we make an im p o rta n t an d in te re stin g calc u la tio n w ith the th e o re tic a lly c o m p u ted com pressibility. E instein [19] derived an equation connecting com pressibility p and the average frequency (Og^ and is given by

V ^ )

6 ( ) 6 R V Ciopalu Rao and R Venkaiesh

(29)

Here X ^ slruclural consUinl and is nearly unity, r„ is the nearest neighbour distance which we obtain Irom the first peak of the radial distribution function (RDF), p. is effectively the mass of the single atom and j3r is the isothermal compressibility. We get from the RDF obtained be Fourier transformation of the structure factor for ^{r^,} = 3.3A. We calculate rOo„ using ^-/•= 3.1 x 10 cm^/dync and obtain a value of 1.5 x lO'^scc"'. At this stage we define in analogy with average velocity an average frequency as

^ 2 1

Using the pcitk values of (o, and « i, we obuiin an average value of as 1.9 x lO'^ .sec ' which IS close to the value ol I..S 10

given in cq. (29).

n see computed through Einstein’s equation as

Acknowledgments

RV IS thankfui to Department ol Science and Technology, New Delhi and RVGR is

thankful to Council of Scientific and Industrial Research, New Delhi for financial assistances.

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(191 H A Moclwyn Hughes 1961 P h y s ic a l C h e m is tr y (Ix^ndon Pergamon) p 101,322