INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf ·...
Transcript of INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf ·...
![Page 1: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/1.jpg)
IC/94/247
INTERNATIONAL CENTRE FOR
THEORETICAL PHYSICS
^-DYNAMICS IN SIMPLE LIQUID METAL
INTERNATIONALATOMIC ENERGY
AGENCY
UNITED NATIONSEDUCATIONAL,
SCIENTIFICAND CULTURALORGANIZATION
S.K. Lai
and
H.C. Chen
MIRAMARE-TRIESTE
![Page 2: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/2.jpg)
![Page 3: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/3.jpg)
IC/94/247
International Atomic Energy Agencyand
United Nations Educational Scientific and Cultural Organization
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
/J-DYNAMICS IN SIMPLE LIQUID METAL
S.K. Lai 'International Centre for Theoretical Physics, Trieste, Italy
and
H.C. ChenDepartment of Physics, National Central University,
Chung-li 32054, Taiwan, Republic of China.
M1RAMARE- TRIESTE
August 1994
'Permanent address: Department of Physics, National Central University, Chung-li32054, Taiwan, Republic of China.
Abstract
The modified hypernetted chain theory is applied to calculate the static liquid structure
factor for simple liquid metals at rapid cooling. With a proper choice of bridge function,
it is found that this integral equation theory is capable of yielding reasonably accurate
liquid structure factor in the supercooled liquid region. To exploit the usefulness of the
calculated liquid structure, we combine these structure data with the mode-coupling theory
to investigate the dynamics of the /3-process. It is demonstrated in this work that the
critical temperature for an ideal glass transition predicted here for liquid metals Na and
K are slightly lower than those determined previously [S.K. Lai and H.C. Chen, J. Phys.
Condens. Matter 5, 4325 (1993)] using computer simulated structure factors. In particular
we show that the material-dependent exponent parameter A which is used widely in the
literature as a fitting parameter in the analysis of light scattering experiments can be given
more physical significance if one correlates the change of A with the microscopic interaction
of particles specifically for the liquid metal, Lennard-Jones fluid and hard-sphere system.
The implication of the present results and the possibility to extract useful information for
more complicated systems will be discussed in the text.
![Page 4: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/4.jpg)
I. Introduction
The .^-relaxation for a glass-forming supercooled liquid has been a subject of current in-
terest. Empirically this phenomenon was studied using different experimental probes. The
dielectric loss spectroscopy is perhaps one of the earlier experiments that provides evidence
for the existence of /3-process. Such an experimental technique was applied extensively by
Goldstein and .Johari [1,2] to understand the /^-resonances for a wide class of rigid molecular
substances and amorphous polymers. Specifically they observed in their measured dielectric
loss spectra a secondary absorption band. To explain its origin they [3,4] proposed a potential
energy barrier picture which is a concept comprising the cooporative and the hindered types
of molecular rearrangements. In an entirely different approach Knaak et al [5] carried out
an inelastic neutron scattering experiment for an ionic glass. This experiment focused on the
scaling behavior of & resonances. The dynamics of/^-relaxation was investigated also by Cum-
mins and coworkers [610] for the ionic and molecular glasses using the Brillouin-scattering
spectroscopy. This study was made feasible through an improved technique on the Sander-
cock tandem multipass Fabry-Perot interferometer which is an experimental setup specifically
designed to circumvent the overlapping order problem. Their Brillouin spectra measurements
for the dynamical structure factor exhibit a fractal law which varies as i j"1 ' "" ' , where a is an
exponent parameter to be defined below. In an attempt to look into the dynamical behavior
of a supercooled liquid, Pusey and Megen [11] prepared suspensions of small, equal size, and
long-hours metastable spherical colloidal particles and used the dynamical light scattering tool
to investigate the structural and dynamical properties of the system. Because of the unique
means the colloidal system was prepared, the interactions of particles resemble that of a fluid
of hard spheres thus permitting a tractable theoretical analysis of the /^-process.
Theoretically Goldstein [3] and Johari [4] attempted to interpret the /^-process as origi-
nated from the mobile clusters of atoms which, due to stringent atomic rearrangements, suffer
the hindered type of motion for encaged atoms. Such an idea has been questioned, however,
by Gotze and Sjogren [12] to be physically unsound since this latter feature has not been
observed generally. In a scries of works [12-19] Gotze, Sjogren and coworkers have studied the
/3-relaxation within the recently developed mode-coupling theory (MCT) of glass transition.
According to the latter works the /3-phenomenon can be understood microscopically by ex-
amining the temporal and spatial changes of the normalized intermediate scattering function
R(r,t). Specifically the MCT predicts the presence of three relaxation regimes for R(r,t): a
microscopic relaxation time to ~ 10 Ti|s corresponding to uncorrelated binary collisions such
as those happened in an equilibrium liquid state, an intermediate mesoscopic time- tg typically
on the order of 10~8-10~ns and a long-lasting time scale t,, ~ l-103s (so-called a-relaxation).
The /3-relaxation belongs to the mesoscopic time scales. It can be shown in the MCT that
near a critical temperature Tc the normalized density correlation function R.(r,l) exhibits a
factorization property which expresses R(r,t) in terms of an universal temporal scaling func-
tion uncorrelated with two material-dependent spatial functions. This factorization property
has been investigated first for a Lennard-Jones system [20], subsequently for a hard-sphere
system [17,18,21], recently for a lattice gas system [22] and Cummins and coworkers [6-10]
have quantitatively examined this property for the ionic and molecular glasses. In this paper
we intend to supplement these works for a metallic system. Our purpose for carrying out this
calculation is threefold.
First of all we plan to perform a realistic calculation for the supercooled liquid metals
Na and K by applying the modified hypernetted-chain integral equation theory to the liquid
structure. This recently proposed approach has been evaluated critically [23] by us to be
highly reliable and accurate. As the static liquid structure factor S[q) is a sole input to MCT
this application is important not only because it offers an alternative means to calculate Tc its
comparison with that Tc determined previously [24] using the molecular dynamics simulated
S(q) further exploits the potentiality of the method to an undercooled liquid. Second, we
intend to compare the scaling properties of the /^-relaxation for the liquid metal, Lennard-
Jones and hard-sphere systems. This comparison is instructive since it enables us to give more
physical significance to the MCT exponent parameter A (to be defined below) which has been
widely used as a fitting parameter in experiments on glass transition. In particular one can
![Page 5: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/5.jpg)
infer from the variation of A the microscopic interactions of more complex systems. Finally,
we shall calculate the lagged particle distribution function P*(r,t) and see if the Ps(r,t) result
for a metallic system can give any hint to the large discrepancies in this quantity [18] between
the hard-sphere and binary soft-sphere systems.
The format of this paper is as follows. In the next section we briefly summarize the for-
mulas used in the modified hypernetted-chain theory and draw attention to a bridge function
that we found l.o yield reasonable S(<;). The interatomic potential used in the calculation is
also outlined briefly. Section Ifl is devoted to an introduction of the mode-coupling theory.
Here the nonlinear integral equation which R(r,t) satisfies will be introduced. The explicit
expressions for the scaling formula R(r,t) and for the distribution function P'(r,t) will be given
in this same section. We discuss our numerical results in section IV and make a concluding
remarks in section V.
II. Static structure factor
In this section we summarize essential formulas that appear in the modified hypernetted-
chain theory and outline briefly the evaluation of the S(g).
The modified hypernetted-chain integral equation begins with the Ornstein-Zernike rela-
tion
-r'ijc(r') (1)
where /> is the number density; c(r) is the direct correlation function; and d(r) = g(r) - 1 is
the total correlation function, g(r) being the pair correlation function. To solve this equation
one must supplement it with a closure. The most frequently used form is
g{r)=exp[d(r)-c(r)-/3<4(r)-B(r)], (2)
where /? = (1<BT) ' is the inverse temperature; <f>(r) is the interatomic potential, and B(r)
is the bridge function. Following our previous applications to liquid metals we construct the
pair potential at each T using the modified generalised non-local model pseudopotential of Li
A
et al [25] and Wang and Lai [26]. According to the latter works <4(r) can be written
• (
where G^c(q) is the normalized energy-wave-number characteristics with the Singwi et al [27]
exchange-correlation factor included and 'L\g = 7? - p\,l and pd being the nominal valence and
the depletion charge, respectively. It should be emphasized that in Eq. (3) proper attention
has been given to the one-electron energy and pseudo-wave-function by carefully incorporating
higher-order perturbative corrections through a parameter in the bare-ion pseudopotential (see
[25] for details). For the bridge function B(r) we have chosen the empirical B(r) of Malijevsky
and Labile [28] which reads
fi + ^(r/ff - 1 )][r/(T - 1 - f3][r/<r - I - v^/iWi) r < I^/T
Aiexp[—(^s(r/c — 1 - f4)]sin[A2{r/CT — 1 - f4)]/r r > i^cr
(r being the hard-sphere diameter and the parameters A, and v, are determined, respectively,
by continuity conditions and by fitting to all known structural and thermodynamic computer
simulation data of hard spheres over the entire fluid range up to the density of freezing.
Equations (l)-(4) constitute a set of equations which are to be solved self-consistently for
g(r). The static structure factor S(</) is then obtained by Fourier-transformation.
III. Mode-coupling theory
Original derivation of the non-linear self-consistent equations has been well documented
in several papers by Gotze and Sjogren [29-32]. Here we review the essential ones sufficient
for our present purpose.
A. Basic Formulas
Central to the MCT is the density-density correlation function F(r,t) = < 6p(r,t) Sp(0,0)>
where 6p(r,t) is the density fluctuation for the position vector r at time t and < ., > denotes
the ensemble average. According to the MCT the normalized correlator R(q,z) = P(q, z)/S(q)
![Page 6: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/6.jpg)
satisfies
Rfa.z) = — —• (5)zM(q, z)
where m is the atomic mass and M(g, z) is the generalised fractional term or the so-called
memory function. Here the Laplace transform of a quantity O(g,t), denoted by a caret, is
defined as
O(q,z) = i / dte"'O(q,t) (6)Jo
Since we shall be concerned with the long-time behavior, we approximate M(q,t) RJ T(q,t) and
ignore contributions from the transient part of M(g,t). P((/,t) has been derived previously by
Sjogren [33] and Bengtzelius, Gotze and Sjolander [34] and is given by
-W) ~ <
r",t). (7)
On the basis of many-body methods and kinetic theory, Bengtzelius et al [34] derived suc-
cessfully n closed non-linear self-consistent dynamical equation for the solidification which
reads
= ;—r(^, t —* oo) = ^(/(fc)) (8)1 - JW r
In Eq. (8) the Debye-Waller form factor f(q) = R(<J,t—» oo) and J",(/(i)) is a functional of
}{k) which appears in Eq. (7). Equations (7) and (8) are basic formulas which have to be
solved iterativcly.
Similar equation for a tagged particle exists and it can be written [34]
where-
dq'q'
(10)
In solving for f'(q) one requires f(q) as input but again it has to be solved self-consistently.
We shall apply these equations in our calculation of the tagged particle motion.
B. 0— relaxation
Following the work of Gotze and Sjogren [30,32] we present in this subsection those equa-
tions in MCT which are essential for an analysis of /^-process. The readers arc referred to
original papers [24,29-32] for technical details.
According to Gotze and Sjogren [30,32] the correlator R(</,t) in the vicinity of Tr can be
expressed in a factorized form
R(</,t) = fc(q)+ hc(q)ii\/l^Ay'\e\g±{k/t,p}. (11)
Here /c(<?) is the Debye-Waller form factor which is the solution of Eqs. (7) and (8); hc(<;)
= [1 - IcMY^l is t l l e c r i t i c a l amplitude [24] where fj (fj) being the right-hand (left-hand)
eigenvector of the stability matrix
ji2 = [ £ ^Tc((),F,/flT)/(t)]/(l-A), A being a material-dependent exponent parameter which
can be calculated from Eq. (8) [24]
1(13)
£=(TC-T)/TC is the separation parameter, and fl±(t/ty3) is a scaling function which satisfies
Tl+C29±(C) + ACi / dr^glir) = 0, (14)
Jo
where f=wt/j and T = ijtp both being rescaled variables. The plus and minus in <7±(t/t,g)
refer to glassy £ >0 and liquid € <0 sides of the critical Tc respectively. Note that the rescaled
time ip = to|/j2(l — A)£|~'''(2'1> where the exponent parameter a can be determined from Eq.
(13)
= r2(l -«)/r(l-2a)
7
(15)
![Page 7: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/7.jpg)
in terms of the Gamma function T(x).
In the same vein the tagged particle correlator can be shown to be
(16)
where P(ij,t) = < <V(q, t)Sp'(O,Q) >, f'(q) is the Lamb-Mossbauer factor, being the solution
of Eqs. (9) and (10), and h'c(q) is given by [29]
KM = (17)k,k'
in which V = (1-C;)"1 is a.n inverse matrix and C'qk = [l - f'(k)?dT'!df'(k) is the tagged
particle stability matrix. The above equations constitute the starting point for our discussion
of the /^-relaxation.
IV. Numerical results and discussion
We have applied Eqs. (l)-(4) to obtain the static liquid structure factor for the liquid
metals Na and K from near freezing point down to the supercooled liquid temperatures. For a
liquid metal these calculations pose a great difficulty since the mass densities of liquid metals
are a priori unknown. To circumvent this difficulty we have resorted to our recently proposed
novel method [24,35] which determined this physical quantity indirectly from the simulated
Wendt-AbraJiam parameter [36], We refer the interested reader to these works for further
details. Now, given an atomic volume at each lower T, it is then a straightforward matter to
evaluate S(</) [23,37,38]. The calculated S(q)s are displayed in Fig. 1 and they are compared
with those obtained from the molecular dynamics simulation. An immediate conclusion can be
drawn from this comparison: it justifies our present choice of the modified hypernetted-chain
theory to the calculation of S{q) for an undercooled liquid.
To proceed further we input the above S(<?) to Eqs. (7) and (8) and iteratively solve for
/(</). Subsequently, the obtained solution /(<?) is used in conjunction with Eqs. (9) and (10) to
solve iteratively for the non-ergodic form factor f'(q). The critical temperatures Tcs obtained
for the Na and K liquid metals are Tc = 211 K and 194.18 K respectively. These temperatures
g
are slightly lower than the ones determined previously using simulated S(q)s (for Na, T<; =
215.3 K; for K, Tc = 200.2 K). In the following we present and discuss some interesting results.
A. Exponent parameter A
Using Eq. (13) we have calculated the exponent parameter A and hence the parameter a
from Eq. (15). These exponent parameters are given in table 1 along with the ones corre-
sponding to the HS calculated using the empirical B(r) given by Eq.(4) and the Lennard-Jones
[20] system. A notable feature in this table is that A shows a systematic change; it decreases
from a large value of 0.772 for hard spheres (characterized by a purely infinite repulsive part),
to a value of 0.714 for a Lennard-Jones system (delineated by a still highly repulsive but a
weakly attractive parts), and to values of 0.712 and 0.710 for liquid metals (described by
a relatively softer repulsive followed by an oscillatorily and weakly attractive parts). This
variation of A is surely intimately related to the details of microscopic interactions. In fact,
it is possible to infer from this change of A that for a simple monatomic system such as the
one-component plasma (having a much softer repulsive and zero attractive parts), it will be
expected to have an exponent parameter that lies within the range 0.772 > A > 0.714. This
prediction is based on the assumption that the attractive part of pair potential does play some
role in the dynamics of liquid structure, and that the Lennard-Jones system has a sufficiently
weak attractive tail. It would be interesting to carry out such kind of calculation to check this
conjecture.
B, Master function for the ^-process
From the As given above, it is numerically straightforward to calculate the master functions
<7±(T). They are given in Fig. 2 for the hard-sphere (HS) and liquid metals Na and K. However,
the calculated results are equally instructive if they are expressed in their Laplace transform
g±{z). For real frequencies u, z —> u)+ iO, one obtains
, 0
= % /
JO
ro
/
JO
![Page 8: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/8.jpg)
and the corresponding susceptibility spectra X±{u) c a n De calculated from x±iljJ) = ^?±(^)-
The A'if'4') f°r lifJLiid metals Na and K are depicted in Fig. ,'} together with (.hat for the
HS system included for comparison. It is interesting to iiote from these figures the following
discernible differences.
(a) For given e, the master function <f±(r) for a metallic system varies more steeply with r
for T -Cl and r > 1 compared with that of hard-spheres.
(b) The susceptibility spectra \"(w) for the HS system has a broader dispersion relative to
those of liquid metals Na and K.
These two points, the validity of one implies the other, can be understood qualitatively by
examining the details of interactions between particles. For HS we note that the 'microscopic
interaction" is purely geometrical being characterized by an infinite repulsive potential. The
structural behavior is therefore determined solely by the excluded volume effect. On the
other hand, the microscopic interaction of a liquid metal is described by a relatively softer
finite repulsive force embellished by an oscillatorily and weakly attractive part. Thus the
structural behavior is determined both by the geometrical factor similar in nature to that
of a HS system, and by an additional Coulomb attraction due to the presence of valence
electrons. When the temperature (density) of an equilibrium liquid is decreased (increased)
rapidly it is anticipated that the non-linear coupling between particles is enhanced by the
attractive force that prevails in a metallic system. Since /?-relaxation is believed to be a
localized excitation describing particles moving dynamically with their surrounding cages,
such enhancement in the attractive interaction with decreasing temperature (or increasing
density) has the consequence of robusting the occurrence of /̂ -process as evident by a longer
characteristic time scale t̂ for the liquid metal than for the HS system (sec table 1). This
would imply that the atomic motion in its associated cage is relatively more stable for a
metallic atom than for a HS particles. We will see more clearly of such a picture below when
we discuss the tagged particle distribution function.
Before proceeding to the latter discussion, we should mention one interesting aspect of
Fig. 3. This figure reminds us of recent theoretical fittings of the MCT to the light scattering
10
experiments on Cao.jKo.etNOa)!^ [8-10]. There the fitted values X = 0.81-0.85 surely imply
that the details of inter-particle interactions should differ considerably from the presently con-
sidered monatomic systems. Indeed for this particular fragile glass which consists of argon-like
ions K+ and Ca+, and optically anisotropic ions (NO3)~, it is believed that the orientational
motion for the planar (NO3)" is probably the most relevant cause for the distinctly large value
of A.
C. Tagged particle distribution function
We turn now to a discussion of the tagged particledistribution function Ps(r,t) = 47rr2F'(r,t)
which can be obtained from Eq. (16) by a back Fourier transformation. This equation, which
(18)
describes the probability density of locating a tagged particle at time t and position r, if it
was found to appear at the origin r = 0 at t = 0. Figures 4 and 5 display the P^(r,r=t/t^) vs.
r/<7 for the liquid metal K and HS system calculated using the <?-(T) and jf+(r) respectively.
(The results for the liquid metal Na is virtually the same and is therefore not given here).
There are two interesting points that merit emphasis.
(a) For given e < 0, the HS tagged particle Ps(r,T-) at different T intercepts at r/V = 0.172
(for K, r/<r = 0.211). Since the probability of finding the HS lagged particle is higher
for increasing T for r/cr > 0.172 (for K, r/cx > 0.211), the HS tagged particle is seen to
push its way through the cage of neighboring particles. Such a picture for an atomic
motion in a cluster of particles appears to persist longer for a metallic atom than for a
HS particle.
(b) For given £ > 0, the tagged particle P*(r,r) at different r displays the characteristic
localized motion and shows a slightly more stable configuration for metallic particles
than HSs.
The first point can be explained by referring to the master function given in Fig. 2. There
we see that the metallic ff-(i") lies above that of HSs throughout the time window of the
11
![Page 9: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/9.jpg)
/3-process. Further evidences for this difference in the diffusive motion can be gleaned also
from table 2 where we give the P'(rmax,T) at the maximum position rmwt for increasing T. AS
regards the second point, it can be attributed physically to the attractive part of <£(r) being
more important when the system is at a lower (higher) temperature (density).
Finally, it is of great theoretical interest to compare our calculated monatomic metallic
H"(r) with that, of the computer simulated soft sphere mixture (see Fig. 1 in Ref. 18). Our
metallic H"(r) shows the same peak-valley behavior similar to that of HSs. Quantitatively,
however, the magnitude of the first peak is lower and the whole curve is shifted to larger
rvalues (see Fig. ft). This latter feature being qualitatively similar to the soft sphere mixture
may be attributed to the more realistic repulsivenessof <#>(r). Comparison between the present
result for K and the soft sphere mixture further implies that the single peak feature followed
by an extensive spatial H*(r) for the latter might be due to the size effect of different soft
spheres.
V. Conclusion
In this paper we first examined the appropriateness of the modified hypernetted-chain
theory applied to the evaluation of trie static liquid structure factor for an undercooled liquid.
We compared the calculated liquid structure factor with MD simulated S(q) and found that
they agree very well for the supercooled liquid region of interest in this work, This prompted
us to apply the calculated S(q) in conjunction with the mode-coupling theory to study the /?-
relaxation. We have chosen the liquid metal as a prototype system since the latter is described
by the simplest realistic interactions which, when compared with the US and Lennard-Jones
systems, permits extraction of fruitful information about the supercooled liquid dynamics.
It was attempted further in this work to establish the connection between the details of
microscopic interactions and the parameters that appeared in the mode-coupling theory. For
simple monatomic systems, we found that the strength of the attractive part of pair potential
(a) varies inversely with the X parameter and (b) robusts the occurrence of /3-process and
hence has the tendency of stabilizing the motion of the tagged particle in its surrounding
neighbours.
Acknowledgments
This work has been supported iti part by the National Science Council, Taiwan, R.O.C. under
contract No.:NSC83-0208-M008-056. We thank the Computer Center of the Department of
Physics, National Central University (NSC82-0208-M008-094) for the support of computing
facilities. One of the authors (SKL) would like to thank Professor Abdus Salain, the Inter-
national Atomic Energy Agency and UNESCO for hospitality at the International Center for
Theoretical Physics, Trieste, Italy.
References
1. G.P. Johari and M. Goldstein, J. Chem. Phys. 74, 2034 (1970).
2. G.P. Johari and M. Goldstein, J. Chem. Phys. 53, 2372 (1970).
3. M. Goldstein, J. Chem. Phys. 51, 3728 (19G9).
4. G.P. Johari, J. Chem. Phys. 58, 17G6 (1973).
5. W. Knaak, F. Mezei and B. Farago, Europhys. Lett. 7, 529 (19S8).
6. N.J. Tao, G. Li and H.Z. Cummins, Phys. Rev, Lett. 66, 1334 (1991).
7. G. Li, M. Du, A. Sakai and H.Z. Cummins, Phys. Rev. A 46, 3343 (1992).
8. G. Li, M. Du, X.K. Chen and H.Z. Cummins, Phys. Rev. A 45, 3867 (1992)
9. G. Li, M. Du, J. Hernandez and H.Z. Cummins, Phys. Rev. E 48, 1192 (1993)
10. H.Z. Cummins, W.M. Du, M. Fuchs, W. Gotze, S. Hilderbrand, A. Latz, G. Li and N.J.
Tao, Phys. Rev. E 47, 4223 (1993).
11. P.N. Pusey and W. van Megen, Phys. Rev. A 59, 2083 (1987).
13
![Page 10: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/10.jpg)
12. W. Gotze and L. Sjogren, J. Non-cryst. Solids 131-133, 161 (1991).
13. W. Gotze and L. Sjogren, J. Phys. C: Solid State Phys. 21 , 3407 (1988).
14. G. Buchalla, U. Dersch, W. Gotze and L. Sjogren, J. Phys. CrSolid State Phys. 21 ,
4239 (1988).
15. VV. Gotze and L. Sjogren, J. Phys,:Condens. Matter 1, 4183 (1989).
16. VV. Gotze, J. Phys.:Condens. Matter 2, S485 (1990).
17. W. Gotze and L. Sj6grei>, Phys. Rev. A 43, 5442 (1991).
18. M. Fuc.hs, W. Gotze, S. Ililderbrand and A. Latz, Z. Phys. B - Condensed Matter 87,
43 (1992).
19. M. Fuchs, VV. Gotze, S. Hilderbrand and A Latz, J. Phys.:Condens. Matter 4, 7709
(1992).
20. U. Bengt.zelius, Phys. Rev. A 34, 5059 (1986).
21. J.L. Barrat, VV. G6t7.e and A. Latz, J. Phys.:Condens. Matter 1, 7163 (1989).
22. W. Knob and II.C. Andersen, l'hys. Rev. E 47, 3281 (1993).
2a. H.C. Chen and S.K. Lai, Phys. Rev. F,, 49, R982 (1994).
24. S.K. Lai and H.C. Chen, J. Phys.:Condens. Matter 5, 4325 (1993).
25. D.H. Li, X.R. Li and S. Wang, .1. Phys. F 16, 309 (1986).
26. S. Wang and S.K. Lai, J. Phys. F 10, 2717 (1980).
27. K.S. Singwi, A. Sjolander, M.P. Tosi and R.H. Land, Phys. Rev. B 1, 1044 (1970).
28. A. Malijevsky and S. Labi'k, Mol. Phys. 67, 663 (1987).
29. W. Gotze, Z. Phys. B: Condensed Matter 60, 195 (1985).
30. W. Gotze, in Liquids, Freezing and the Glass Transition, edited by J.P. Hansen, D.
Levesque, J. Zinn-Justin (North Holland, Amsterdam, 1991), p.287
31. W. Gotze, in Amorphous and Liquid Materials, edited by E. Luscher, G. Fritsch, and
G. Jacucci (Nijhoff, Dordrecht, 1987), p.34.
32. W. Gotze and L. Sjogren, Rep. Prog. Phys. 55, 241 (1992).
33. L. Sjogren, Phys. Rev. A 22, 2883 (1980).
34. U. Bengtzelius, W. Gotze and A. Sjolander 17, 5915 (1984).
35. S.K. Lai, M.H. Chou and H.C. Chen, Phys. Rev. E 48, 214 (1993).
36. H.R. Wendt and F.F. Abraham, Phys. Rev. Lett. 41 , 1244 (1978); F.F. Abraham, J.
Chem. Phys. 72,359 (1980).
37. S.K. Lai, Wang Li and Tosi, M.P., Phys. Rev. A 42, 7289 (1990).
38. H.C. Chen and S.K. Lai, Phys. Rev. A 45, 3831 (1992).
15
![Page 11: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/11.jpg)
Table 1: Parameters A and a calculated from the present work, characteristic time t/s scaledby microscopic time t0 for the liquid metals Na and K compared with those of the hard-sphere(HS) calculated at the critical packing ratio 0.5324 and the Lennard-Jones (LJ) [20] systems.
System a t,?/toxl0
IIS 0.772 0.294 6.1fi
LJ 0.711 0.321
Na 0.712 0.322
K 0.710 0.323 20.71
16
Table 2: Tagged particle distribution function Ps(rmax,r,) evaluated at the maximum positionrmax f° r T% — W t f = 5xlO"3+> where i = 0,..4 for the HS and liquid metal K in the e < 0liquid region. In the e > 0 glassy region T ( + 4 = t(+ 4 / t^ = 10"3+* where I = 1,..3.
T2 T5
HS system
rm.x/ir 0.120 0.126 0.132 0.144 0.184 0.120 0.124 0.125
P"(rmM,Tv) 6.876 6.181 5.751 5.309 5.044 6.804 6.379 6.289
K system
r™*/ff 0.152 0.157 0.159 0.164 0.187 0.152 0.157 0.159
P*(rmax,r,) 5.582 5.252 5.066 4.867 4.409 5.507 5.289 5.229
17
![Page 12: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/12.jpg)
FIGURE CAPTIONS
Fig.l - Static liquid structure factor S(q) calculated by the modified hypernetted-chain
theory (dashed curve) with the Malijevsky and Labik empirical bridge function (28)
compared with the Fourier-transforming molecular dynamics g(r) (full curve) for (a)
sodium and (b) potassium liquid metals.
Fig.2 - Master function g±(i") versus T—tji^ for the hard-sphere system (full curve) and
liquid mt'tals Na (light dashed curve) and K (heavy dashed curve).
Fig.3 - Susceptibility spetra x"(^) versus fi = ui/wp (where up = l/t^) for the hard-sphere
system (full curve) and liquid metals Na (light dashed curve) and K (heavy dashed
curve),
Fig.4 - Tagged particle distribution function PI(r) for the (a) hard spheres and (b) potassium
element evaluated at r , /^ = 5xlO~3 (full curve), 5xlO~2 (dotted curve), 5x10"' (short
dashed curve), 5x10° (long dashed curve), 5x10' (chain curve) for e < 0 liquid region.
Fig.5 - Tagged particle distribution function P+(r) for the (a) hard spheres and (b) potassium
element evaluated at r./t/j = xlO"2 (full curve), xlO"1 (dotted curve), xlO° (dashed
curve) for £ > 0 glassy region.
Fig.6 - Spatial functions HJ(r) and p*(r) for the tagged particic distribution for the hard-
sphere system (full curve) and liquid metal K (dashed curve).
18
I 11 I I I I I I I I I I I I I I I I I I I M I I I I I I I I I I I I I I
2 3 4 5q (A"1)
F i g . l
19
2 3 4 5 6q (A"')
![Page 13: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/13.jpg)
15
10
0 r—
-5
-10
-15
' 4 nl ,J>10° 10' 102
F i g . 2
2021
![Page 14: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/14.jpg)
P-(r)
o
(a)
0.50
0
Fig.
22
Fig.523
![Page 15: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/15.jpg)
05 O
24
![Page 16: INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/94/247.pdf · 2005-02-27 · by Gotze and Sjogren [12] to be physically unsound since this latter feature](https://reader034.fdocuments.in/reader034/viewer/2022042111/5e8d1043e41d5551900d3508/html5/thumbnails/16.jpg)