Recent experimental reports bring out extreme size sensitivity in the heat
capacities of Gallium and Aluminum clusters. In the present work we report
results of our extensive {\it ab initio} molecular dynamical simulations on
Ga$_{30}$ and Ga$_{31}$, the pair which has shown rather dramatic size
sensitivity. We trace the origin of this size sensitive heat capacities to the
relative order in their respective ground state geometries. Such an effect of
nature of the ground state on the characteristics of heat capacities is also
seen in case of small Gallium and Sodium clusters indicating that the observed
size sensitivity is a generic feature of small clusters.Comment: 4 pages, 6 figure

D. G. Kanhere

D. G. Kanhere

Kavita Joshi

Sailaja Krishnamurty

English

arXiv

Recent experimental reports bring out extreme size sensitivity in the heat capacities of gallium and aluminum clusters. In the present work we report results of our extensive ab initio molecular dynamical simulations on Ga30 and Ga31, the pair which has shown rather dramatic size sensitivity. We trace the origin of this size sensitive heat capacities to the relative order in their respective ground state geometries. Such an effect of nature of the ground state on the characteristics of heat capacity is also seen in case of small gallium and sodium clusters, indicating that the observed size sensitivity is a generic feature of small clusters

Joshi, Kavita

Krishnamurty, Sailaja

Kanhere, D. G.

Indian Academy of Sciences

PRL 96, 135703 (2006) P H Y S I C A L R E V I E W L E T T E R S
week ending
7 APRIL 2006‘‘Magic Melters’’ Have Geometrical Origin
Kavita Joshi, Sailaja Krishnamurty, and D. G. Kanhere
Department of Physics and Center for Modeling and Simulation, University of Pune, Ganeshkhind, Pune-411 007, India
(Received 30 December 2005; published 7 April 2006)0031-9007=Recent experimental reports bring out extreme size sensitivity in the heat capacities of gallium and
aluminum clusters. In the present work we report results of our extensive ab initio molecular dynamical
simulations on Ga30 and Ga31, the pair which has shown rather dramatic size sensitivity. We trace the
origin of this size sensitive heat capacities to the relative order in their respective ground state geometries.
Such an effect of nature of the ground state on the characteristics of heat capacity is also seen in case of
small gallium and sodium clusters, indicating that the observed size sensitivity is a generic feature of small
clusters.
DOI: 10.1103/PhysRevLett.96.135703 PACS numbers: 64.70.Nd, 36.40.Ei, 65.40.Ba, 82.20.Wt 1
 1.3
 1.6
 1.9
 2.2
 2.5
 0  300  600  900  1200
H
ea
t C
ap
ac
ity
 (
C
v/
C
0)
Temperature (K)
Ga30
Ga31
FIG. 1 (color online). The heat capacity of Ga31 and Ga30
computed over 90 ps.The finite temperature behavior of clusters has shown
many interesting and intriguing properties [1–5]. Recently,
the calorimetric measurements reported by Jarrold and co-
workers found that small clusters of tin and gallium in the
size range of 17–55 atoms have higher than bulk melting
temperatures (Tmbulk) [2,3]. A striking experimental result
from the same group showed extreme size sensitivity in the
nature of the heat capacity for Ga clusters in the size range
of 30–55 atoms [4]. It turns out that the addition of even
one atom changes the heat capacity dramatically. For ex-
ample, Ga30
 has a rather flat specific heat curve, whereas
the heat capacity of Ga31
 has a well-defined peak and has
been termed ‘‘magic melter.’’ A similar size sensitive
feature has also been observed in the case of Al clusters [5].
The explanation and understanding of various experi-
mental observations have come from the first-principles
density functional (DF) simulations [6–10]. For example,
the higher than bulk melting temperature for Sn and Ga
clusters is understood as being due to the difference in the
nature of bonding between the cluster and the bulk
[6,8,10]. However, the extreme size sensitivity displayed
in gallium and aluminum clusters is still an unexplained
phenomena. The present work addresses this issue by
employing first-principles DF methods. In this Letter we
report our results of ab initio molecular dynamical (MD)
simulations carried out on Ga30 and Ga31. It is of some
interest to note that similar size sensitive heat capacities
have been observed in case of Gan (n  17; 20) [9] and
Nan (n  40; 50; 55) [11] clusters. In both these cases the
addition of a few atoms changes the nature of heat capaci-
ties significantly. By analyzing the geometry of the ground
state, we establish a definitive correlation between the
nature of the ground state and the observed heat capacity.
Our detailed calculations show that an ‘‘ordered’’ ground
state leads to a heat capacity with a well-defined peak,
while a cluster with ‘‘disordered’’ ground state leads to a
flat heat capacity with no distinct melting transition. In
what follows we will make the meaning of ‘‘order’’ and
‘‘disorder’’ precise and provide an explanation for the size
sensitive heat capacities.06=96(13)=135703(4)$23.00 13570We have carried out constant temperature Born-
Oppenheimer MD simulations using ultrasoft pseudopo-
tentials within the generalized gradient approximation
[12]. For all the clusters reported here we have obtained
at least 200 equilibrium structures. For computing heat
capacities of Ga30 and Ga31 the MD calculations were
carried out for 16 different temperatures, each with the
duration of 150 ps or more, in the range of 100  T 
1100 K, which results in a total simulation time of 2.4 ns.
In order to get converged heat capacity curves especially in
the region of coexistence, more temperatures were required
with longer simulation time. The resulting trajectory data
have been used to compute the ionic specific heat by
employing the multiple histogram method [13,14].
Figure 1 shows the calculated heat capacity of Ga30 and
Ga31. Evidently the dramatic difference in the heat capaci-
ties of Ga30 and Ga31 observed in the experiments is well
reproduced in our simulations. Thus Ga31 has a well-
defined peak in the heat capacity, whereas the heat capacity
for Ga30 is rather flat. We also note that both Ga30 and Ga313-1 © 2006 The American Physical Society
PRL 96, 135703 (2006) P H Y S I C A L R E V I E W L E T T E R S
week ending
7 APRIL 2006become ‘‘liquidlike’’ at temperatures much higher than
Tmbulk (303 K), i.e., above 500 K, consistent with the
experiments. In order to gain insight into these observa-
tions, we analyze the ground state of Ga30 and Ga31. In
Fig. 2 we show the ground state geometries of Ga30 and
Ga31 with two different perspectives. A cursory analysis of
Fig. 2(a) may lead to the conclusion that the only differ-
ence between Ga30 and Ga31 ground states is the presence
of the capped atom in Ga31. However, a different view
obtained by rotating the cluster by 90 brings out the
significant differences in Ga30 and Ga31, clearly indicating
that Ga31 is more ordered. A careful examination of
Fig. 2(b) shows the presence of well-ordered planes in
Ga31. Such planes are only in a formative stage and con-
siderably deformed in Ga30. In fact, an addition of just one
atom in Ga30 displaces all the atoms by a significant
amount which makes Ga31 more ordered. That a single
atom makes a substantial rearrangement is also seen by the
fact that there is a noticeable difference in the coordination
number of atoms in these two clusters. In Fig. 3 we show
the number of atoms as a function of the coordination
number. It can be noted that in Ga30, 5 atoms have 4 or
more coordination number, whereas in Ga31, 14 atoms
have fourfold or higher coordination. Therefore we term
Ga30 as a disordered structure relative to Ga31.
Thus when the system is disordered, each atom (possibly
a group of atoms) is likely to have different local environ-
ment. That means different atoms are bonded with the rest
of the system with varying strength. Consequently, their
dynamical behavior as a response to temperature will
differ. Some of the atoms may pick up kinetic energy atFIG. 2 (color online). The ground state geometry of Ga30 and
Ga31 with two different perspectives. Perspective (b) is rotated
by 90 with respect to perspective (a) about an axis shown in the
figure.
13570low temperatures, while the others may do so at higher
temperatures. In a given structure, if a large group of atoms
are bonded together with a similar strength forming an
island of local order, it is reasonable to expect that they
will ‘‘melt’’ together. In this case the cluster can be con-
sidered as (at least partially) ordered and will show a well-
defined peak in the heat capacity. However, if the system is
disordered in the sense that there are no such islands of
significant sizes having local order, then we expect a very
broad continuous phase transformation. Indeed, our analy-
sis of mean square displacement (MSD) for individual
atoms brings out this fact clearly. The MSD for individual
atoms is defined as
hr2I ti 
1
M
XM
m1
RIt0m  t 	RIt0m2; (1)
where RI is the position of the Ith atom and we average
over M different time origins t0m spanning the entire
trajectory. In Fig. 4 we show MSDs of individual atoms
for Ga30 and Ga31 at 250 K. The contrast between the
kinetic response of individual atoms in Ga30 and Ga31 is
very clear. For Ga30, the MSDs of individual atoms show
that at least 10 atoms have picked up more kinetic energy
compared to other atoms and hence have significantly
higher displacements (9:0 A2 as compared to 0:45 A2),
whereas in Ga31 all atoms are oscillating about their
mean positions and exhibit small values of MSDs
(0:45 A2). Thus MSDs clearly indicate that in Ga30 differ-
ent atoms have different mobilities at low temperatures.
This wide distribution of MSDs in Ga30 indicates that the
cluster is in coexistence phase around 250 K and is con-
tinuously evolving. This is precisely what is expected if the
cluster is disordered in the sense described above. This 0
 2
 4
 6
 8
 10
 12
 14
 0  1  2  3  4  5  6  7
N
um
be
r 
of
 A
to
m
s
Coordination Number
’Ga30’
’Ga31’
FIG. 3 (color online). Coordination numbers for Ga30 and
Ga31. We note that in Ga30 most of the atoms have either 2 or
3 as a coordinations number, whereas in Ga31 the coordination
number is either 3 or 4.
3-2
 0
 2
 4
 6
 8
 10
18 36 54 72 90
M
ea
n 
Sq
ua
re
 D
is
pl
ac
em
en
ts
 (
A
0 )
2
Time (ps)
Ga30 250K
 0
 2
 4
 6
 8
 10
18 36 54 72 90M
ea
n 
Sq
ua
re
 D
is
pl
ac
em
en
ts
 (
A
0
)2
Time (ps)
Ga31 250K
FIG. 4 (color online). The MSD for individual atoms of Ga30
and Ga31 computed over 90 ps.
PRL 96, 135703 (2006) P H Y S I C A L R E V I E W L E T T E R S
week ending
7 APRIL 2006phenomenon has also been observed in the extended sys-
tems and a similar analysis has been used to characterize
the nature of spatial inhomogeneities with considerable
success [15].
The difference in the mobilities of individual atoms in
these two clusters is also reflected in the root-mean square
bond-length fluctuations (rms) shown in Fig. 5. rms shows
a clear signal for the beginning of the change of phase
around 450 K for Ga31. However, in case of Ga30 the
transition is spread over a much broader range of tempera-
tures. In fact, the coexistence region for Ga31 is over 175 K
(from 425 to 600 K), and for Ga30 it extends over 425 K. It
is interesting to note that in Ga30 the isomerization begins
around 175 K and continues till 600 K.
The nature of the order can also be brought out by
examining the electron localization function (ELF). The 0
 0.08
 0.16
 0.24
 0.32
 0.4
 0  300  600  900  1200
δ r
m
s
Temperature (K)
Ga30
Ga31
FIG. 5 (color online). The rms for Ga30 and Ga31 computed
over last 90 ps.
13570ELF has been found to be extremely useful for elucidating
the bonding characteristics of a variety of systems [16]. For
a single determinantal wave function  i, the ELF is defined
as [17],
ELF  1 D=Dh2	1; (2)
where
Dh  3=10325=35=3; (3)
D  1=2
X
i
jr ij
2 	 1=8jrj2=; (4)
with  
 r is the valence-electron density. The ELF is
defined in such a way that its value is unity for completely
localized systems and 0.5 for homogeneous electron gas. It
is more convenient to analyze the topology of the ELF
surface by using concepts of attractors and their basins
[18]. The locations of the maxima of ELF are called
attractors. A set of all points in space which can be con-
nected to these attractors by maximum gradient paths is
called their basins. In the present case the attractors are
located at the ionic sites, the ELF being maximum there.
For large enough values of ELF, there are as many basins as
number of ions. As the value of the ELF is decreased, the
basins get connected, and finally at some low value we get
a single basin. The value of the ELF at which the basins get
connected is a measure of the strength of interaction be-
tween the different atoms. This means that the number of
atoms contained in the single basin are bonded to each
other with similar strength depending upon the value of the
ELF. Figure 6 shows the isosurface of ELF taken at ELF 
0:68, for Ga30 and Ga31. We note that for Ga30, 26 atoms
are connected via a single basin, whereas for Ga30, the
largest basin contains 12 atoms with other ‘‘fragmented’’
basins. This supports our earlier observation that Ga31 has
significantly more similarly bonded atoms than Ga30.
Further evidence for the amorphous nature of Ga30 comes
from the comparison of the entropies of these systems
(figure not shown). As expected the entropy of amorphous
structure (Ga30) rises rather sharply as compared to Ga31
(which is more ordered). Quite clearly, the amorphousFIG. 6. The ELF of (a) Ga30 and (b) Ga31 at the value of
ELF  0:68. The black lines show connected basins.
3-3
 1
 1.5
 2
 2.5
 3
 0  150  300  450  600
H
ea
t C
ap
ac
ity
 (
C
v/
C
0)
Temperature (K)
Na55
Na40
Na50
FIG. 7 (color online). The heat capacity of Na40, Na50, and
Na55 computed over 90 ps data.
PRL 96, 135703 (2006) P H Y S I C A L R E V I E W L E T T E R S
week ending
7 APRIL 2006nature leads to substantially large number of accessible
states in case of Ga30 and is more by a factor of 10 as
compared to Ga31 in the low energy region.
As mentioned earlier this size sensitive behavior is not
unique to the gallium clusters reported here and has been
observed in small Ga [9], Al [5], and Na [11] clusters. As
an example we show the heat capacities of Nan (n 
40; 50; 55) clusters in Fig. 7. The change in the nature of
the heat capacity as the cluster grows from Na40 to Na55 is
quiet evident from Fig. 7. Our detailed analysis of the
ground state geometries shows a direct correlation between
the nature of the ground state and calculated heat capaci-
ties. It may be noted that Na55 is highly symmetric and very
well ordered. Na40 is also ordered and has a basin contain-
ing a substantially large number of atoms, but Na50 is
relatively disordered, which is clearly reflected in their
heat capacities [19].
The main contribution of the present work is to bring out
a definitive relationship between the local order in the
cluster and its finite temperature behavior. As the cluster
grows in size, it is very likely that it will evolve through a
succession of such ordered and disordered geometries. In
such cases the addition of one or a few atoms is likely to
change (as demonstrated in this work) the nature of the
ground state abruptly. Thus, the size sensitive nature of
heat capacities is generic to small clusters and related to the
evolutionary pattern seen in their ground state geometries.
The evidence for this comes not only from gallium clusters
but also from clusters of sodium and aluminum having very
different nature of bonding.13570Partial support from IFCPAR-CEFIPRA (Project
No. 3104-2) New Delhi and the super computing facility
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Magic Melters' Have Geometrical Origin

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