Defect melting models for cubic lattices and universal laws for melting temperatures

Abstract

We set up simple harmonic lattice models for elastic fluctuations in bcc and fcc lattices and the excitation of dislocations and disclinations. From these we derive, in the lowest approximation, universal formulas which predict melting temperatures in good agreement with the experiments. This new theory is more precise than Lindemann's rule by factor 2, and more predictive, since the size of the Lindemann number has to be fixed by experiments. In addition, our theory allows for systematic improvements.

Full text

Page 1

arXiv:cond-mat/0301453v1 23 Jan 2003
Defect Melting Models for Cubic Lattices and
Universal Laws for Melting Temperatures
Hagen Kleinert1, ∗and Ying Jiang1, †
1Institut f¨ ur Theoretische Physik, Freie Universit¨ at Berlin, Arnimallee 14, D-14195 Berlin, Germany
(Dated: Received February 6, 2008)
We set up simple harmonic lattice models for elastic fluctuations in bcc and fcc lattices and
the excitation of dislocations and disclinations. From these we derive, in a lowest approximation,
universal formulas which predict melting temperatures in good agreement with the experiments.
This new theory is more precise than Lindemann’s rule by factor 2, and more predictive, since the
size of the Lindemann number has to be fixed by experiments. In addition, our theory allows for
systematic improvements.
PACS numbers: PACS Numbers: 61.72.Bb, 64.70.Dv, 64.90.+b
Melting transitions [1, 2, 3, 4] are important phenom-
ena of both technical and theoretical interest [5, 6, 7,
8, 9, 10, 11, 12, 13]. A hundred years ago, Sutherland
[14] found an empirical rule, that the product bTmM
nearly constant for metals, where Tmis the melting tem-
perature, b the mean coefficient of expansion, and M the
atomic mass. In the following year, he advanced a kinetic
theory of solids [15], in which he postulated that melting
would occur when the space between atoms reaches a cer-
tain value relative to the atomic diameter, so that atoms
are able to escape from the imprisonment by their neigh-
bors. He carried out a hard-sphere simulation experiment
by shaking a box containing one layer of marbles, and
noting by removing marbles one at a time that the ap-
parent solid-liquid transition occurs when a free volume
of 25-33% is reached. From the vibration amplitude and
the thermal kinetic energy, together with the above em-
pirical melting rule, Sutherland [15] calculated the period
of vibrations of metal atoms at the melting temperature,
and found that the ratio of vibration amplitude to atomic
spacing is nearly the same for all elements at melting.
In 1910, Lindemann [16] combined Sutherland’s ideas
with the Debye theory of specific heat in solids, and de-
rived his famous rule according to which a quasi-universal
parameter, the Lindemann number, should be the same
for all melting transitions. The number is usually stated
in the form
1
6 is
L = θv1/3(M/Tm)1/2
where θ is the Debye temperature and v the volume per
atom. Subsequent extensive tests of this rule were car-
ried out by Gschneidner [17], Ross [18], and Crawford
[19] who found, however, that Lindemann’s rule is not
very reliable. First, the Debye temperature cannot be
precisely specified since the Debye theory of specific heat
is itself approximate. Second, extracting θ from the data
by a best fit of the Debye theory, the Lindemann number
varies considerably from material to material, as empha-
sized by Wallace [20].
The most unsatisfactory feature of the Lindemann rule
is that its size is unknown and has to be extracted
from an average of many melting transitions.
the Sutherland-Lindemann approach to melting has re-
mained a purely phenomenological rough description of
the transition. In spite of its roughness, it is often used
even today in many works to estimate melting tempera-
ture, this being due to a lack of a more precise melting
theory.
During the past two decades, several other melting the-
ories have been developed. One is based on density func-
tionals [21, 22], and has provided us with an alterna-
tive insight into the melting transition. This work, how-
ever, describes satisfactorily only the long-wave length
limit of fluctuations, and neglects the important role of
the lattice structure, thus requiring essential corrections,
which sometimes have been inserted these heuristically
[23, 24, 25]. Another theory of melting is based on a Lan-
dau expansion in a symmetric tensor order parameter of
rank four [26]. This theory is purely phenomenological
and thus only descriptive, not predictive.
For a proper description for the melting transition, it
seems necessary to incorporate information of the lat-
tice structure and the associated crystal defects into the
theory. About 15 years ago, one of the authors (HK)
did this successfully by constructing lattice models of
the melting transition [3, 4].
of these models [27] was that they started out from the
lattice version of the elastic lattice energy which allow for
the thermal creation and annihilation of dislocations and
disclinations and keeps track of their lattice elastic en-
ergy [27]. They are included by means of discrete-valued
defect gauge fields, the disclinations being essential for
explaining the first-order nature of transition [28], which
otherwise would be of second order, as in the vortex-line
induced λ-transition of superfluid helium.
In this respect, the older models went beyond those of
more recent authors [24, 29] who considered only the pro-
liferation of dislocation lines which, if properly treated,
would have produced only second-order phase transitions
[30], thus being unable to describe the melting transition.
A simple universal melting formula was obtained from
Thus,
The important progress

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the above lattice defect models in a lowest-order ap-
proximation which combines the properties of the high-
temperature expansion of the defect contributions with
the low-temperature expansion of the free energy density.
From the intersection of these two curves, one can obtain
a formula for the melting temperature in accordance with
Lindemann’s rule, but with a prediction of the absolute
size of the Lindemann number [3].
This successful theory has, however, one important
drawback: For simplicity, it was constructed only for
the physically somewhat exotic case of simple cubic lat-
tices. This makes it strictly applicable only to very few
materials—there are almost no simple cubic lattices in
nature. If the derived formulas are applied to other cu-
bic crystal structures such as body center cubic (bcc)
lattices and face center cubic (fcc) lattices, the accuracy
must necessarily be bad. Hence there is an obvious need
to generalize the theory to physically more prolific crystal
structures.
The purpose of this letter is to present such a gener-
alization of the melting model to fcc and bcc lattices.
When generalizing the previous melting model to other
cubic lattices, We shall see in the high-temperature limit,
that for all cubic crystals the free energy density is only
related to the elastic constants of the crystals and does
not depend on the lattice structure of the system. This
is in contrast to the low-temperature limit, where the
free energy density involves integrals over the lattice mo-
menta Kiand Kiwhich strongly depend on the detailed
lattice structure of the crystals. It is therefore important
to give a proper description of the low-temperature limit
of the free energy density for different lattice structures.
The lattice model for the cubic crystals will be based
on the elastic energy
E =
a3
nA
?
x



1
4
?
l?=m
?∇(l)u(m)+ ∇(m)u(l)
?2+B
1
2ui(x)
?2
+ C
?
l
?∇(l)u(l)
2A
??
l
∇(l)u(l)
?
? x−? e(l)a
2
??2


= −a3
nA
?
x,i,j,l,n
?
ei
(l)ej
(l)
?
m
∇(m)∇(m)
+ 2(C − 1)ei
?B
(l)ej
(l)∇(l)∇(l)
+
A+ 1
?
ei
(l)∇(l)∇(n)ej
(n)
?
uj(x), (1)
where n is the number of atoms per unit cell, ? e(l)are
oriented link vectors of cubic lattice pointing from any
lattice site to its nearest neighbors along the positive di-
rection, and uidenote the Cartesian displacement field,
u(l)= uiei
(l)their components along the link directions.
For fcc lattices, there are six link vectors:
? e(1)= (1,1,0),
? e(4)= (1,−1,0), ? e(5)= (0,1,−1), ? e(6)= (−1,0,1), (2)
? e(2)= (1,0,1),? e(3)= (0,1,1)
each lattice site x having twelve nearest neighbors at x±
? e(l)a/2. For bcc lattices, there are four link vectors
? e(1)= (1,1,1),
? e(3)= (1,−1,1), ? e(4)= (−1,1,1),
? e(2)= (1,1,−1),
(3)
and each lattice site possesses eight nearest neighbors.
The symbols ∇(l) and ∇(l) denote lattice derivative,
defined by ∇(l)f(? x) =
∇(l)f(? x) =?f(? x) − f?? x −? e(l)a/2??/b, where b is the dis-
calculation, the prefactors A, B, and C are related to the
usual elastic constant of the crystal µ, λ and ξ defined
by the continuum elastic energy
?f?? x +? e(l)a/2?− f(? x)?/b and
tance between the nearest neighbors. After some algebra
E = µ
?
d3x

?
i?=j
u2
ij+ ξ
?
i
u2
ii+
λ
2µ
??
i
uii
?2
, (4)
and can be expressed for fcc as follows:
A=µ
8(2ξ − 1), B=λ+2µ(ξ − 1)
8
, C=1−8ξ − 1
2ξ − 1, (5)
and for bcc as
A =
3
16µξ,B =
3
16(λ + µξ − µ),C =2 − ξ
ξ
,(6)
For very low temperatures, the atomic positions devi-
ate very little from those of an ideal crystal. It is there-
fore suggestive to use these small deviations for defining
the displacement field. But this definition can be con-
sistent only for a small time span. Due to fluctuations,
thermals as well as quantum, the atoms are capable of
exchanging positions with their neighbors and migrate,
after a sufficiently long time, through the entire crystal.
This process of self-diffusion makes it impossible to spec-
ify the displacement field uniquely. Thus, as a matter of
principle, the displacement field is undetermined up to an
arbitrary lattice vector, it is impossible to say whether
an atom is displaced by u(l)(x) or by
u(l)(? x) + bN(l)(? x)
where N(l)(? x) is the jumping field with integer value.
Because of this, we should introduce an extra sum over
an integer-valued field n(lm)(x) with a gauge fixing term
Φ[n(lm)] in the corresponding partition function, thus the

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3
partition function of the defect melting model reads
Z =
?
n(lm)
?
?
x,i
??
dui(x)
a
?
Φ[n(lm)]
×exp
−
a3A
nkBT
?
?2
x
?
1
2
?
?
2)
l<m
?
?∇(l)u(l)− bn(ll)
∇(l)u(m)+ ∇(m)u(l)
−b(n(lm)+ n(ml))+ C
l
?2
+B
2A
??
l
?∇(l)u(l)(? x−? e(l)a
2)??2??
−bn(ll)(? x−? e(l)a
(7)
The partition function is invariant under the following
defect gauge transformations
u(l)→ u(l)(x) + bN(l)(x), n(lm)→ n(lm)+ ∇(l)N(m)(x).
This partition function can be expressed in terms of the
stress field σijby
Z =
?
8ξ3
?
1+3λ
2µξ
??−N/2?
1
2πβ
?3N?
?
x,i≤j
?
??
dσij
?
×
?
1
2ξ
n(lm)
Φ[n(lm)]
?
x,i
??
dui(x)
a
exp
−1
2β
?
x
??
i<j
σ2
ij
+
?
?
i
σ2
ii−
λ
4µξ2+ 6λξ
??
i
σii
?2??
× expiπ
?
x,l,m
σ(lm)
?
∇(l)u(m)+∇(m)u(l)− 2bn(lm)
??
,(8)
where σ(lm) = (√3/16)ei
σ(lm) = (√2/16)ei
(a3µ)/(nkBT(2π)2). In the calculation process from (7)
to (8), the corresponding expressions of A, B, C and ei
are used. In spite of the tedious intermediate calculation,
the prefactors in the partition functions (8) are the same
for all cubic lattices.
For the-lowest order approximation, we need only to
investigate the system in low-temperature and high-
temperature limits. It has been shown in the textbook
[3] (see Fig. 12.1 on p. 1084) that the intersection of
the free energies of the two limiting curves yields good
estimates for the melting temperatures in simple-cubic
lattices.
For T → 0, the defect configuration is completely
frozen out, and the partition function has a classical
limit. From the lattice energy (1) we obtain the clas-
sical partition function of cubic lattice
(l)ej
(m)σij for bcc lattice and
(m)σij for fcc lattice, and β =
(l)ej
(l)
Zcl=
?
x,i
??
dui(x)
a
?
e−E/kBT=
?
2πnkBT
Aa3
3N
e−N1
2ℓ,
where the dimensionless parameter ℓ is defined by the
trace log
ℓ =
2π/a
?
−2π/a
d3ka3
(4π)3trlogM,(9)
and M denotes the matrix
Mij=4δija2?
+
m
K(m)K(m)+2(C−1)a2?
A+1
l,n
l
ei
(l)K(l)K(l)ej
(l)
?B
?
a2?
ei
(l)K(l)K(n)ej
(n)
(10)
and K(l)and K(l)are lattice momenta, for fcc
K(l)=
√2
ai(ei?k·? e(l)a
2− 1), K(l)=
√2
ai(1 − e−i?k·? e(l)a
2), (11)
and for bcc
K(l)=
2i
√3a(1−ei?k·? e(l)a
2),K(l)=
2i
√3a(e−i?k·? e(l)
a
2−1).(12)
Thus we find the free energy density of the fcc and bcc
lattices in the low-temperature limit
−fT→0
kBT
=3
2log
?2πnkBT
µa3
?
+3
2logµ
A−1
2ℓ.(13)
In the opposite limit of high temperature, the defects
are prolific and the sum over n(lm)can be approximated
by integral enforcing σij≡ 0. Then the partition function
(8) yields the free energy density
−fT→∞
kBT
=3log
?2πnkBT
µa3
?
−1
2log
?
8ξ3
?
1+3λ
2µξ
??
. (14)
From the intersection of this with the low-temperature
approximation (13), we obtain the melting relation for
the fcc and bcc lattices
a3µξ
2πnkBTmelt
=A
µ
1
81/3
?
1 +
3λ
2ξµ
?−1/3
eℓ/3
.(15)
The formula is structurally very similar to the Linde-
mann rule L = const., but predicts, in addition, the size
of L to be averagely of the order of 126. Moreover, from
our expression, one can recognize clearly that the melting
temperature will be zero when the anisotropic parameter
ξ equals to zero, this is in contrast to recent phenomeno-
logical theories of melting by other authors [24]. The
novelty of this formula with respect to a similar formula
in Ref. [3] [see Eq. (12.13) on p. 1079] lies in a determi-
nation of trace log term ℓ for bcc and fcc lattices. The
resulting melting temperatures from Eq. (15) are shown
in Table 1, where they are compared with experimental
numbers and with the results found from Lindemann’s

Page 4

4
rule where the Lindemann number is initially undeter-
mined and must be extracted from the average over all
materials. Our lowest-order theory has an average preci-
sion of about 12%, which is better by a factor 2 than the
22% of the numbers from derived Lindemann’s rule.
Element
Ag (fcc)
Au (fcc)
Ba (bcc)
Ca (fcc)
Co (fcc)
Cu (fcc)
Li (bcc)
Nb (bcc)
Ni (fcc)
Pb (fcc)
Pd (fcc)
Pt (fcc)
Sr (fcc)
Ta (bcc)
Th (fcc)
Tl (bcc)
V (bcc)
W (bcc) 163.10 204.90 1.005
µλξTm,theor. Tm,exp. κ(%) δ(%)
1246.51234.0
1201.41336.2
875.4998.0
889.01112.0
1892.71765.0
1299.81356.0
450.2454.0
2778.2 2741.0
2034.91726.0
507.4 600.6
1723.01825.0
2527.32042.0
868.0 1045.0
3879.03271.0
1882.82024.0
609.9576.0
2444.32178.0
3497.4 3653.0
51.10
45.40 169.70 0.351
9.508.00 0.242
16.3018.20 0.294
128.00 160.00 0.320
81.80 124.90 0.314
10.8012.50 0.110
28.70 134.00 1.951
124.70 147.30 0.398
14.4039.20 0.222
71.20 176.10 0.407
76.50 250.70 0.627
9.9010.30 0.253
81.80 157.40 0.627
47.80 48.90 0.278
11.0034.00 0.309
46.00 119.40 1.228
97.30 0.335124
6
22
-14
-2
16
-14
-39
16
26
-10
-12
-20
7
-4
-1
1
17
-15
-6
23
-17
18
-7
7
29
11
-32
43
58
12
-4
-15
3
Table 1. Theoretical results of melting temperatures
(units in K) derived from the elastic constants of cu-
bic crystals (units in GPa, gotten from [31]) compared
with the experimental data (units in K, gotten from
[17]). The second-last column shows the relative error
κ = (Tm,theor/Tm,exp− 1) · 100 of our theory, which
is by a factor 2 lower than the relative error δ =
(Tm,Lind/Tm,exp− 1) · 100 found from Lindemann’s rule.
The Lindemann melting temperatures Tm,Lindare from
Ref. [32].
There is no problem, in principle, to calculate system-
atic improvements to formula (15) by including defect
excitations into the low-temperature approximation (13)
and stress corrections into the high-temperature approxi-
mation (14). This was done in Ref. [3] for the unphysical
simple cubic lattices, and will be done for bcc and fcc
lattices in future work.
One of the author (Y.J.) gratefully acknowledges the
financial support from Alexander von Humboldt Foun-
dation.
∗Electronic address: kleinert@physik.fu-berlin.de
†Electronic address: jiang@physik.fu-berlin.de
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