The energetics
and structure
of nickel clusters:
Size dependence
Charles L. Cleveland and Uzi Landman
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 303324430
(Received4 September1990;accepted11 February 1991)
The energeticsof nickel clusters over a broad size range are explored within the context of the
many-body potentials obtained via the embeddedatom method. Unconstrained local minimum
energyconfigurations are found for single crystal clusters consisting of various truncations of
the cube or octahedron, with and without ( 110) faces,as well as some monotwinnings of these.
We also examine multitwinned structures such as icosahedraand various truncations of the
decahedron,such as those of Ino and Marks. Theseclusters range in size from 142 to over
5000 atoms. As in most such previous studies, such as those on Lennard-Jonessystems,we
find that icosahedralclusters are favored for the smallest cluster sizesand that Marks’
decahedraare favored for intermediate sizes (all our atomic systemslarger than about 2300
atoms). Of course very large clusters will be single crystal face-centered-cubic(fee) polyhedra:
the onset of optimally stable single-crystal nickel clusters is estimatedto occur at 17 000 atoms.
We find, via comparisonsto results obtained via atomistic calculations, that simple
macroscopicexpressionsusing accuratesurface,strain, and twinning energiescan usefully
predict energy differencesbetweendifferent structures even for clusters of much smaller size
than expected.These expressionscan be used to assessthe relative energeticmerits of various
structural motifs and their dependenceon cluster size.
I. INTRODUCTION
The structural, electronic, and chemical characteristics
of materialsdependon the state (phase) and degree(size) of
aggregation.Clusters, i.e., finite aggregatesof up to tens of
thousandsof atoms, exhibit unique physical and chemical
propertiesand a systematicstudy of the dependenceand evolution of theseproperties with size allows elucidation of the
transition from the molecular to condensed matter regimes.‘-4 Such investigationshave becomepossibleover the
last two decadesdue to the developmentand proliferation of
methods of preparation and characterization of clusters of
various elementalmaterials and their compounds,as well as
in differing charge states.
A particularly complex and intriguing issueis the structure of clustersand the dependenceof structure on size. Difficulties in investigating this question are found on both the
experimental and theoretical fronts. On the experimental
side, the problem is twofold. First, since cluster systemsare
larger than ordinary molecules,but too small to be dealt with
as ordinary bulk materials, they are not well suited to exploration of their structure using the traditional approachesof
spectroscopyand x-ray diffraction. As a result, structural
studiesusingelectron diffraction are the preferredmethod of
investigation and have beenapplied with significant success
in studies mainly of gas phaseclusters of rare gaseP and
molecular clusters.‘.” For metallic clusters, data obtained
through electron diffraction experimentsis scarce,but valuable information has beenobtainedvia high-resolution electron microscopy for metal clusters from a few hundred to
thousandsof atoms condensedon substrates.“-26 In addition, a scanning tunneling microscope probe has recently
beenusedto examinesilver and gold clusters supportedon a
graphite surface.27Furthermore, metallic clusters support7376
J. Chem. Phys. 94 (1 I). 1 June 1991
ed on graphite surfaces have also been investigated using
extendedx-ray-absorptionfine-structure (EXAFS) and related x-ray techniques.“-” Analysis of this data is complicated, however, by the interaction between the deposited
cluster and the substrateand the multitude of cluster sizes
presenton the surface.”
In the case of structural studies of gas-phaseclusters,
analysisof the electron diffraction data is complicated by the
clusters’finite size: they do not fill all spaceand so neednot
possesstranslational symmetry. Consequently,the analysis
reliesto a large extent on comparisonsbetweenthe measured
diffraction patterns (i.e., diffraction functions which give
the intensity of the scatteredelectron beam as a function of
momentum transfer) with the results of model calculations
which simulate the experimental data, using prescribedinteratomic interactions and ambient conditions. The analysis
of the gas phase data is further complicated by the dependence of the structural characteristics of the system on the
generation processand ambient conditions. Furthermore,
small clusters (below a few tens of atoms) may undergo a
radical structural changeby the addition of a singleatom (or
molecule), and for both small and large clusters, a distribution (equilibrium or nonequilibrium dependingon the preparation conditions) of structural isomersmay be found for a
given cluster size.
The aboveconsiderationsmotivate systematic theoretical studiesof the structure and energeticsof clusters as well
asof modesof cluster growth, phasechanges,and isomerization. A principal objective of such structural studies is the
determination of size boundaries of energetic stability for
various structural forms and the question of the transition
from behavior characteristic of a finite system to that of a
bulk system with increasingcluster size. Ideally, we would
seekanswersto the abovequestionsfor all cluster sizesand
0021-9606/91 /‘I 17376-21$03.00
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C. L. Cleveland and U. Landman: Structure of nickel clusters
for a range of temperatures. However, the plethora of possible structures and the isomerizations between them make
such studies rather prohibitive. Instead, as severalresearchers previously have done, we limit ourselvesin this study to
consideration of a number of structures whose selection is
motivated by energetic and morphological considerations,
as well as guided by accumulated past experience. As a
further simplification, the energetic optimization of the
structures is performed at zero temperature and thus neglects entropic effects. Nevertheless, such studies are relevant for developing understanding and establishing a conceptual framework which could guide a rational approach to
thesecomplex structural issues,as well as aiding in the analysis of experimental data. In addition, obtaining the structural (and underlying energetic) trends, via minimization of
the clusters’energiesusing a conjugate gradient technique,
permits us to assesscritically the validity of formulas which
we derive on the basis of macroscopic arguments for the
energies of various structural motifs. Our results demonstrate the remarkable adequacy for many cluster types of
such “macroscopic” expressions,provided the input energies (such as crystal plane dependentsurface energies,twinning energies,and strain energiesassociatedwith particular
deformations) are known.
The majority of experimental’-” and theoretica18*33-45
investigations to date on the structure of clusters have been
on rare-gas clusters produced in supersonic beams during
gas expansion. In such experiments, clusters can obtain stable structures (for eachcluster size) sincethey are formed at
high temperatures and thus may anneal effectively.46Using
electron diffraction data and analysis via molecular dynamics simulations employing Lennard-Jones(LJ) pair interactions, thesestudies have establishedthe presenceof icosahedral structures for small rare gas clusters (RGCs)
containing 40-800 atoms. Much bigger argon clusters
( 104-10’atoms) exhibit a crystalline face-centered-cubic
(fee) structure containing twin boundariesET4’
(which are
detected by extra reflections in the diffraction patterns).
However, the structural evolution of rare gas clusters in the
intermediate regime is somewhat elusive.
All theoretical studies of the energeticsand structure of
RGCs employed pair interactions (mostly LJ potentials).
The most complete study of these systemsis the recent one
by Raoult et al.* These authors conclude on the basis of a
relaxation method,48and using LJ potentials, that for clusters with less than 1600 atoms, the most stable structures
belong to the icosahedral sequence,while for larger clusters
the decahedra1sequenceis energetically more stable. Additionally, they found that although monotwinned fee models
have a slightly lower energy than crystalline fee polyhedra,
they are not expectedto becomethe structurally stable form
for RGCs containing lessthan - IO5atoms.
Theoretical investigations of the energetics and structure of metal clusters are complicated by the complex nature
of bonding and cohesionin metals. While for very small clusters (mainly of alkali metals and aluminum) rigorous quantum chemical calculations haveprovided valuable structural
information,49v5ofor larger clusters one must resort to approximate descriptions of the interatomic interactions. Such
7377
atomistic studiesemploying various levels of approximation
havebeenperformed5’-55for clusters containing up to - 100
atoms.
Another approach in the investigation of the structure
of metal particles, which has been used for a rather long
time ,22*5c6’
is founded on macroscopicconceptsrather than
atomistic structural optimization. This approach is basedon
the Wulff construction56’57which determines the equilibrium macroscopicmorphology of the crystallite (seeSec.III)
and on the theory of elasticity which allows estimatesof the
elastic strain energyfor a given structure.58The first systematic study employing the above principles was that of InoS8
who calculated the free energy of multiply twinned particles
and compared it to that of an fee crystal of the samevolume
using as input to the calculation macroscopic quantities
characteristic of the material, such as cohesive, surface,
twin-boundary, and elastic strain energies.In addition, the
adhesive energy of interaction between the metal particle
and the substrate support was included. The sequenceof
structures obtained from Ino’s calculations is as follows: for
metal particles smaller than a critical value ( - 100 w in diameter for gold), structures basedon the icosahedronare the
most stable. For larger clusters, the optimal structures are
truncated octahedra (these are semiregular polyhedra of 14
faces,sometimescalled tetrakaidecahedra,or TKDs). In addition, it was concluded that the decahedra, specifically
those called Ino’s decahedra (seeSec.IV), are not structurally preferred at any size.
A modified Wulff construction was introduced by
Marks” to include twin boundaries. Motivated by experimental observations,” this modified construction allows for
reentrant faces at the twin boundaries of the decahedron,
thus decreasing its surface energy. Consequently, Marks
predicts the evolution in cluster structure with increasing
cluster size to progress from icosahedral to decahedra1and
finally to truncated octahedral structures. The energetics
underlying the above progression results from a balancebetween the relatively lower surface energies of the fivefold
symmetric structures and the elastic strain energiesinherent
in them. Since the total surface energy of a cluster varies
roughly as its surfacearea while its total elastic strain energy
varies roughly as its volume, the latter must increase more
rapidly with the cluster’s size. At a critical size, the decahedral sequenceof clusters becomesenergetically more favorable than the icosahedral sequencesince it is characterized
by a smaller strain energy. It should be noted, however, that
numerical estimates” failed to show stability of the decahedral structures in the intermediate cluster sizeregime. In this
context, we remark that application of the modified Wulff
construction requires, as input, quantities which are often
difficult (or impossible) to obtain from experiments (such
as elastic strain energies,surface and twinning energies). In
this paper, we show how to calculate these quantities for a
given model of the interaction potentials.
Motivated by the above considerationsand the increasing experimental efforts aimed at determinations of the
structures of metal clusters and elucidation of correlations
betweenthe structure and physical and chemical properties,
we embarked on a systematic study of the dependenceof the
J. Chem. Phys.,subject
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June or
1991
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7378
C. L. Cleveland and U. Landman: Structure of nickel clusters
structure of metal clusters on cluster size. Our studies are
basedon unconstrainedminimizations (relaxations) of the
energiesof a multitude of structural motifs, with the interatomic potentials described via the many-body embedded
atom method (EAM)624 parametrizedfor nickel.65
In our studies,both the macroscopicand microscopic,
atomistic approachesare pursued, thus providing a test on
the adequacyof the macroscopicconcepts.We concludethat
the macroscopicexpressionswhich we develop,in conjunction with accurately calculated valuesfor the characteristic
energiesasinput, provide a reliableframework for structural
analysis.
We find that in the small cluster size regime (less than
2300 atoms), icosahedral structures are energetically preferred, while for somewhatlarger clusters,the decahedraare
more stable.Furthermore, the optimal singlecrystal polyhedron doesnot becomemore favorablethan the best Marks’
decahedronof similar size for clusterssmaller than - 17 000
atoms.
In the next section, we briefly review the embedded
atom method. In Sec. III, we describethe developmentof
macroscopicexpressionsfor the energeticsof various structural forms, aswell as provide detailsof methodsfor calculations of surface,strain, and twinning energies.Results obtained via atomistic energy minimization are presentedin
Sec. IV and compared to those derived from macroscopic
estimates.Finally, we summarize our results in Sec.V.
c
I( #i)
p;(R,)
++I
,x,
q$j(R,)],
(1)
J(#l)
where@ is the spherically averagedatomic electron density,
the function ~+4~
dependsonly on the speciesof atoms i andj,
and R, is the distancebetweenatoms i andj locatedat Ri and
Ri. Thus the background density for each atom i is determined as the superpositionof electronic densitiesfrom other
atoms, evaluatedat the nucleusof the ith atom. As seenfrom
Eq. ( 1)) the total cohesiveenergyof the material is expressed
asa sum over all atoms in the systemof atomic energycontributions. In the EAM, the functions F and 4 are determined
by choosing for them functional forms which meet certain
generalrequirementsand fitting parameters in these functions to a number of bulk equilibrium propertiesof the solid
such as lattice constant, heat of sublimation, elastic constants, vacancyformation energy,etc. From the severalparametrization procedures which have been suggested,we
have chosenthe one describedrecently by Adams et a1.65
III. MACROSCOPIC CALCULATIONS
A. Single crystals
For a macroscopicsingle crystal cluster of a given size
and shapewhich only has flat facets,it is simple to calculate
the energyif we know the energyper atom in the bulk crystal
and the surface energy (surface energy per unit area) for
each type of crystal face that is exposed.The macroscopic
character is invoked in the neglect of corrections to bulk
II. INTERACTION POTENTIALS
energyother than surfaceenergies,such as “edge”and “verAs is well known, pair potentials are generally not ade- tex” energies.
Alternatively, if we are given surfaceenergiesfor all the
quate for a description of the energeticsin metals due to the
(possibly) relevant types of facets, we can ask what shape
density dependenceof the interactions which underlie the
cohesionof thesematerials. Nevertheless,for simple metals will minimize the energy under the constraint that its volume remain constant.Wulff 56s7has provided a simple con(e.g., free-electronmetals), formulations basedon densitystruction that gives the answer. Proceedfrom the origin in
dependentexpressionsfor the total energy6”69(employing
effective Hamiltonians within the context of pseudopoten- any direction. This direction definesthe normal to a crystal
tial theory) have been used with some success.However, surface. At a distance from the origin numerically equal to
since the effectiveHamiltonian formulations, basedon sec- the surfaceenergyof that surface,erect a normal plane. Go
ond-order perturbation theory with respectto the electron back to the origin and repeatthe processfor all other direcion interaction, are limited to free-electron (sp-bonded) tions which may point to facetson the cluster. The smallest
volume containing the origin insideall erectedplanesis simimetals, we have representedthe energeticsof the material
under study (nickel) using the embeddedatom method lar (in the geometricalsense)to the minimum energyshape
for a cluster of given size. The construction is perfectly gen(EAM)62-64 which has been applied recently with significant success in studies of various transition-metal sys- eral and handlesevenunfacetedsurfaces.
We will call the minimum energyshapefor a material its
tems.62-65,7G72
Wulff polyhedron. Someresearchersuse this term to denote
The EAM is a semiempirical method which provides a
an Archimedian solid, the truncated octahedron,whosesurconvenientframework for atomistic calculations for metalfaceholds six square( 100) facesand eight equilateral hexaglic systems.In this method, the dominant contribution to the
onal ( 111) faces.While this shapeis indeed the minimum
energyof the metal is viewed as the energyof embeddingan
energyfee polyhedron for surfaceenergiesgiven by a simple
atom into the local electron density provided by the other
atoms of the system, representedby an embedding-energy bond-breakingcalculation,76it at best only resemblesthe
minimum energy polyhedra for any realistic materials exfunction F. This dominant contribution is supplementedby
cept by coincidence,so such a usageseemssomewhat misshort-range,two-body interactions due to a core-core repulleading.
sion 4. The basic idea underlying this method is thus the
In order to apply such macroscopic methods, we need
sameas that which motivated the developmentof the effecsurfaceenergies,and possiblybulk energies,for the material
and both find their roots in the dentive-medium theory,73,74
from which the cluster is made. These may be difficult to
sity-functional theory.75
acquireexperimentally.Often the bestinformation available
The cohesiveenergyEcoh of the metal is given in EAM
for surfacesis only an averagesurfaceenergy,undifferentiatby the ansatz
J. Chem. Phys., Vol. 94, No. 11,i June 1991
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C. L. Cleveland and U. Landman: Structure of nickel clusters
ed by crystal face. One of the problemsin applying the macroscopicapproachto small clustersis being unsurewhether
any deficienciesthat may arisedo so from the method’sfundamental approximationsor from poor values for material
properties. Thus such methods are frequently applied only
qualitatively. However, in cases such as ours where the
atomic interactions are specified,the required energiesare
not difficult to obtain. This will permit us to test, in at least
one case,how well the macroscopicapproach works when
appliedto small clusterswhen the input energiesare precisely known.
The crystaline bulk systemmay be describedas consisting of a basisof N atoms (where N is on the order of several
hundred) within a parallelapipedcomputationalcell defined
by three fundamentaltranslation vectors which describethe
periodicity of the cell (i.e., the periodic boundary conditions). Furthermore, these vectors are treated as dynamic
variableson the samelevel as atomic coordinates.77In fact,
the atomic coordinates are only describedin terms of the
prevailing periodicity of the bulk system.Consequently,if so
desired,a uniform strain could be applied to the systemby
imposing suitable constraints on the three translation vectors that representthe correspondingperiodicity of the system under the specifiedstrain.
To determinethe bulk energyper atom and the density
for our material (at 0 K), it is sufficient to m inimize the
energyof the systemby varying the vectors defining the unit
ce11,77
which we accomplishby using the conjugategradient
method.” This providesus with the bulk (per atom) energy
for our material and also determinesthe bulk density. Using
the EAM potentials for nickel, the calculated bulk energy
(per atom) is eg = - 4.450 008 08 eV and the volume per
atom is 10.9163A3.79
To determinethe surfaceenergyat 0 K for an unstrained
surface,we removea periodic boundary condition, creating
two free surfaces,” while holding the computational cell
fixed sinceit is determinedby the bulk crystal. Subsequently,
we relax the energyof the systemby varying atomic coordinates using the conjugategradient procedure. The surface
energyis then simply the differencebetweenour relaxedsystem’s energyand that of the relaxedbulk systemdivided by
twice the area of one of the newly exposedsurfaces.
We have performedsuch calculationsfor the low M iller
index surfaces[ ( 100)) ( 1lo), and ( 111)] which contribute
to the Wulff polyhedron. In addition, we can also insert a
single ( 111) twinning fault into a bulk system, and relax
both the computational cell and atomic coordinatesto obtain the twinning energy. The results of these calculations
are shown in Table I. We remark that the atomic energies
near the centersof the slabswere very closeto the bulk ener-
7379
gy given above,which servesas evidencethat theseslabsare
bulklike in their m iddles. We also note that relaxation has
very little effecton the surfaceenergiesin nickel, all of which
turned out to be somewhatsmaller than the experimental,
facially averagedvalue” of 0.1485 eV/A2 (2.380 J mT2)
(however, the experimental value may be questioned;see
Ref. 62 for a discussion).
Note that the twinning energyis negative.This suggests
that the m inimum energycrystal structure for theseinteractions m ight be hexagonalclosepacked (hcp) insteadof fee.
If we construct an hcp bulk systemand m inimize its energy,
we find that the energyper atom is lower than that of the fee
systemby only 4.6 x 10 - 4 eV. This is of little if any consequencein the study presentedhere, sinceother energiesare
much more important for our clusters.
With theseenergies,we are in a position to construct the
Wulff polyhedron for nickel. Using the zero temperature
surface energies” for Lennard-Jones6-12 interactions, we
can also do the samefor that material (seealso Ref. 5 1 for a
more inclusiveexploration of facetingfor this and other M ie
potentials, and Morse potentials, neglecting relaxation).
Theseresults are comparedin Fig. 1.
The Wulff polyhedronfor nickel is markedly more similar to the truncated octahedron than the one for LennardJonesium.We will seethat the differencesbetweenthesepolyhedra for nickel and Lennard-Jonesiumexactly parallel
differenceswe will observein comparing our results for m icroscopicnickel clusterswith thoseof Raoult et al8 for LennardJonesium.
More quantitatively, Table II compares the ratios
among the ( 111), ( lOO), and ( 110) surface energiesfor
nickel, Lennard-Jonesium,and an ideal material whose
Wulff polyhedron is exactly the truncated octahedron.Both
nickel and Lennard-Jonesiumcrystallites prefer to have
their ( 100) facessomewhatnearerto their centersthan does
the truncated octahedron (i.e., the ratio yloo/y,, , for these
materials is smaller than 2/d), increasingthe relative area
of those faces.Both also have ( 110) facets which the truncated octahedronlacks, but in nickel that facet is very narrow (less than one nearest-neighbordistance in width for
clustersup to about 15 000 atoms) and, unlike the LennardJonescase,’thesewill never end up being advantageousfor
clusters as small as the ones we will examine later on an
atomistic scale.
Let us now calculatethe energyE,(N) of a Wulff polyhedron containing Natoms. Since( 110) facetsdo not occur
on clusters as small as the atomistic oneswe will treat later,
we will omit thesefacesin this calculation. Let us usea unit
of length uN for our cluster such that the distancefrom the
center of given Wulff cluster of N particles to a face with
TABLE I. Interfacial energiesfor unstrained nickel slabs.
Interface
Unrelaxed surfaceor twinning energy
Relaxed surface or twinning energy
(1ll)surface
( 100) surface
(110)surface
( 111) twinning plane
0.094 14 eV A - 2
0.103 41 eV A-*
0.11244eVA-2
0.094 675 8 eV A-’
O.l034068eV&’
0.112341 2eV&*
-O.O000883eVA-*
J. Chem. Phys., Vol. 94, No. 11,l June 1991
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C. L. Cleveland and U. Landman: Structure of nickel clusters
FIG. 1. The Wulff polyhedra for (a) nickel and (b) LennardJonesium.
Also shown for comparison is (c) the truncated octahedron.
surfaceenergyy is numerically equalto yuN. For a given N,
the surface energiesthus determine the distancesfrom the
center of the cluster to its facesand specify the geometryof
the polyhedronso that we may calculatethe areasof its faces
AllI and Al, and its volume V. For nickel, we find (with
energyin units of eV and lengths in units of A) that
E,(N)
= Eg(N)
+ 8y,,,A,,,
+ 6y,ooA,,
E,N + Cu”,,
(2)
where C = 0.014970 96. Sincethe volume is given as
=
F’= 8Am Y,IIU,/~ + ~mo~m~d3
= cl&./3
(3)
and the volume per atom V/N = 10.9163,we can conclude
that uN = 12.9812N 1’3and obtain for the Wulff polyhedron
energy
Ew(N) = EON+ 2.522 79N2’3.
(4)
6. Multitwinned
particles
1. lcosahedra
An icosahedron(seeFig. 2) may be thought of as being
composedof 20 distorted tetrahedral units all meetingat the
center and each sharing three of its faces with neighboring
units, leaving the last face to serve as one of the external
icosahedralsurfacefacets.Crystallographically, eachfaceof
the tetrahedronis an fee ( 111) face and the plane of atoms
which two neighboringtetrahedrashareis a twinning plane,
or put in other words, the stacking sequencein the direction
normal to this plane is mirrored in it, ignoring the distortion
of the tetrahedron.
Twenty perfect tetrahedral units, however, cannot be
put together to make an icosahedron.If surface vertices of
neighboringunits are aligned,the units themselveswill overlap at the icosahedron’scenter. To make a proper icosahedron, the tetrahedral units must be strained to make three of
their sides, those meeting at the central vertex, about 5%
shorter than the three which lay on the surface.This sort of
strain is easy to accommodatewithin the Parrinello-Rahman description” employedfor extendedsystems,in order
to obtain the bulk and surfaceenergiesrequired for calculations of the cluster energiesas describedabove.
For an unstrainedsystem,eachfundamentaltranslation
vector defining the periodically repeatedcell could be taken
as an integral multiple of one of the conventional primitive
translation vectors for an fee material [each pair of which
definesa ( 111) plane1. This situation is representedin Fig.
3. In this figure, all the distancesOA, OB, OC, AB, BC, and
AC are equal. In order to apply icosahedralstrain, we need
only changethe translation vector c so that the distancesOC,
AC, and BC become4.894...% smaller than they were originally. [For further discussionof this sort of description of
icosahedral(and decahedral)materials, seeRef. 59.1When
minimizing the energy of a system with such a computational cell, the only parameter which we may vary is the
volume. Doing so, and comparing with the corresponding
result for the unstrained bulk, gives a strain energy (per
atom) for uniform icosahedralstrain of E’, = 0.032 921 95
eV and a volume per atom of 11.0124A3, slightly greater
than that for an unstrainedsystem.
It is instructive to use Ino’s5’expressionfor the energy
due to a uniform icosahedralstrain, with the elastic constants C,, , C,, , and C, for our model of nickel, which are
given in Ref. 65. The resulting value for the strain energyis
0.0349eV. The agreementbetweenthis value and that obtained hereby direct minimization indicates that for our interactions, nickel under uniform icosahedralstrain is within
the region of linear elastic behavior.
If we removethe periodic boundary condition associated with translation vector c, we create two surfaceslike the
OADB plane in Fig. 3, which under the type of strain described abovecorrespondsto the exposedface of a tetrahedral unit in an icosahedron.Fixing the shapeof the computational cell as beforeand relaxing atomic coordinatesyields a
surfaceenergyof 0.0795eV/A’, nearly 16% lower than the
unstrained value. (Without relaxation, the icosahedralsurface energy would have been 0.0894 eV/A’.) Notice that
unlike the caseof the unstrainedcrystal (cf. Table I), surface
relaxation has a significant effect here.
We may perform for the icosahedronan analogouscalculation to that which gave us an expression [see Eq. (4) ]
for E,(N) for the Wulff polyhedron. If we ignore the twinning energy,which will be very small, we obtain (in units of
eV)
TABLE II. Surfaceenergyratios for nickel and Lennard-Jonesium,along with thoseneededto produce a Wulff
polyhedron that is exactly the truncated octahedron.
Nickel
Truncated octahedron
Lennard-Jonesium
1.092
1.187
2/g/3(1.154...)
>m(
> 1.224...)
1.054
1.118
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C. L. Cleveland and U. Landman: Structure of nickel clusters
7381
for decahedraand single crystals. In the icosahedron,strain
can be relieved by letting the surface facets bow out in a
nonplanarand nonuniform fashion at the expenseof surface
energy (see the discussion on icosahedrain Sec. IV). The
two corrections nearly cancel for the range of cluster sizes
where the energiesof the icosahedracross thoseof the Wulff
polyhedra, so that the prediction made on the basis of the
macroscopiccalculation is deceptively good.
However, we will seethat in systemsunder lower strain,
such as the single crystals and the decahedra(whose uniform strain energy is 30 times smaller than the icosahedron’s) , lhe macroscopiccalculations can indeedbe usedto
produce quantitatively informative energy differences
betweencluster forms.
2. Decahedra
FIG. 2. The icosahedron and its relationship to the tetrahedral units of
which it is composed.
E,(N) = (eB +&IV+
2.02709N2'3.
(5)
Comparing Eqs. (4) and (5), we can determine at what
valueof N the Wulff polyhedronbecomesmore energetically
favorable than the icosahedron.We find the energiesequal
when N = 3413. In our microscopic atomistic calculations
(see Sec. IV), we will find that the two energy curves for
thesetwo structural forms cross when N = 3540.
However, the remarkable agreementbetweenthe macroscopic estimateand that obtainedfrom microscopic calculations is fortuitous in this case.If we comparethe macroscopically calculated icosahedralenergieswith those obtained
by atomistic minimizations of icosahedralclusters’energies,
we find that the “real” clusters (i.e., those obtained via
atomistic minimization) act as if their strain energy is substantially smaller than the value we have usedhere. In fact,
relaxation of an icosahedralcluster from the “ideal” size and
shapegiven by the macroscopicapproachreducesthe energy
by a fraction almost an order of magnitude larger than that
Figure 4 illustrates the kinds of “decahedra”we will be
concernedwith here. Of these,only the classic decahedron
or pentagonaldipyramid actually has ten faces, the others
being so called only becauseof their derivation from it.
The classic decahedronis, like the icosahedron,composedof slightly distorted tetrahedral units, five of which are
joined at a common edgeand eachof which sharestwo of its
( 111) faces as twinning planes with neighboring units and
contributes two other ( 111) facesto the surfaceof the decahedron. The faceson the classic decahedronare equilateral
triangles. The principal advantageof the decahedronover
the icosahedronlies in its lower strain energy.Part of this is
due to the lower strain in the tetrahedral units of the classic
decahedronand part is becausein addition to being able to
minimize strain energyby varying the volume of the units,
we can also relax an additional degreeof freedom corresponding to the length of the decahedron’scentral axis,
shown as a dashedline in Fig. 4(a).
The relaxedclassic decahedronshareswith the icosahedron the advantageof only having ( 111) faces,but because
of its much lower sphericity, its surfaceareais comparatively
larger and energetically it is a poor competitor. Ino5* suggested truncating the tetrahedral units as shown in Fig.
(a)
FIG. 3. An unstrainedcomputational cell for fee crystals. The three vectors
a, b, and c are translations under whose repeatedapplication the computational cell periodically fills all spAce.
FIG. 4. (a) The classic decahedron, (b) an example of Ino’s decahedron,
and Cc) an example of Marks’ decahedron.
J. Chem. Phys., Vol. 94, No. Ii,1 June 1991
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7382
C. L. Cleveland and U. Landman: Structure of nickel clusters
4(b), exposing( 100) facesand making the structure more
spherical.While this improvesmatters and alsoprovidesanother parameterwe may vary to m inimize the energyof the
polyhedron,Ino’s decahedronis not the lowestenergystructure for any known material, sinceas the polyhedrabecome
larger and larger, the single crystals becomemore favorable
than either the icosahedronor Ino’s decahedronbefore the
decahedronsurpassesthe icosahedron.
In an attempt to m inimize the surfaceenergyof the decahedron,however, we may give each of its single crystal
tetrahedral units a surface like that of the single crystal
Wulff polyhedron. This motivates Marks’ modified Wulff
construction*’for decahedra,resultingin a nonconvexpolyhedron like that in Fig. 4(c). In this construction, the distancesfrom the origin to eachsurfaceplane are againtaken
to be simply proportional to that crystal plane’ssurfaceenergy, and in addition, new “faces” are addedin the directions
corresponding to the decahedra1unit’s ( 111) twinning
planes,ignoring strain. Sincethe interfacial energiesof the
twinning planes are very nearly zero, these planes almost
passthrough the origin and excisea wedgefrom the Wulff
polyhedron that would otherwise have resulted. Thus the
exposedsurfaceof a unit in the decahedron,apart from being
slightly stretched to fit, looksjust like part of the surfaceof
the Wulff polyhedron if the necessarystrain doesnot significantly effect the surfaceenergies.Figure 5 illustrates the relationshipbetweenthe units of which Marks’decahedronis
composedand the single crystal Wulff polyhedron.
For uniform decahedra1strain, we may calculate the
strain energy, and surface energiesfor the ( 100) face, the
( 111) “capping” face which touchesthe axis of the decahedron, and the (111) “notch” (or reentrant) face which
makesMarks’ decahedronnonconvex,as well as the twinning planeenergy,by m inimizing the energyof systemswith
two- or three-dimensionalperiodic boundary conditions, in
a mannersimilar to our treatment of icosahedralstrain. The
detailsof such calculationsare much more involved than the
icosahedralcaseand are given in Appendix A. The resulting
uniform strain energy per atom is e = 1.124426X 10- 3
eV. The volume per atom is 10.9186A3, only 0.02% more
than that of the perfect singlecrystal. Table III displaysthe
resultsfor the surfaceand twinning energies,with the ( 110)
surface omitted for simplicity [for nickel clusters in the
rangeof sizeswe concentrateon in this paper, ( 110) facets
FIG. 5. The “tetrahedral” units of Marks’decahedron(left) and their relation to the Wulff polyhedron (right).
do not occur]. Notice that in contrast to the icosahedral
case,relaxation effectsfor surfaceenergiesare small. Comparisonwith Table I showsthat the changefrom the surface
energiesof an unstrainedcrystal are also small. The percentagechangefor the capping ( 111) face is - 0.26%, that for
the reentrant ( 111) faceis - 0.5 l%, and that for the ( 100)
face is - 0.48%. The twinning plane energy, although
slightly larger and with the oppositesign than the unstrained
case,is still negligible.
Given this information, we may seekan equationfor the
energyof the best Marks’decahedron,for a given volumeor
number of particles,just as we did for the Wulff polyhedra.
This evaluationis describedin Appendix B where we obtain
for the optimal Marks’decahedronlacking ( 110) facets,the
expressionfor the cluster’senergyas a function of the numberofatomsinit,E,(N);seeEq.
(Bl).
If we compareED (N) with the Wulff polyhedron’senergy E,(N) from Eq. (4), we can estimatethat the single
crystal will becomemore favorable than a Marks’decahedron whenNZ 17 000.Clearly, clustersof this sizemay have
( 110) facets-in fact their width would be about one nearest-neighbordistance.Our neglectof them, however,occurs
on both the Wulff polyhedronand the decahedron.They will
effect the energiesof both kinds of shapessimilarly and that
effect will be small since they are much smaller than the
other faces.For nickel, at least,we do not feel that our estimation would be improvedenoughto warrant the increasein
the complexity of the derivation given in Appendix B. As we
will seefrom comparing energydifferencesbetweenm icroscopicatomistic clustersof lessthan about 5000atoms with
corresponding macroscopic ones, the macroscopic treatment of clusters can be expectedto be quantitatively quite
good for clustersmuch smaller than 17 000 atoms.
IV. MICROSCOPIC CALCULATIONS
A. Single crystals and icosahedra
Prior to describing our m icroscopic, atomistic treatment, it is appropriateto discussthe nomenclaturethat we
will usefor different structural modelsof singlecrystal clusters. In general,it is not possibleto fit an arbitrary numberof
atoms together in the shapeof any given polyhedron, especially one like the Wulff polyhedronfor nickel, the lengthsof
whoseedgeshavenonsimpleratios. In addition, we examine
clusterswhich depart from the ideal shape,not only because
no clusters of m icroscopicdimensionsexist for that precise
shape,but alsosincewe do not know apriori how reliablethe
macroscopiccalculationsare for such small clusters. Some
of the cluster shapeswhich we examinealreadyhaveconventional names,suchasthe cuboctahedron,the truncated octahedron, and the icosahedron.Others have no namesoutside
the community of scientists studying small clusters and
therefore needto be introduced for clarity.
The most extensivem icroscopicstudy to date is that of
Raoult et aZ.* who studied the structure of Lennard-Jones
m icroclustersand who havenamedmany cluster modelsfor
the first time. Sincethere are no precedents,we will adopt
much of their terminology in this paper.
All the fee singlecrystal shapeswhich we will consider
may be thought of as originating from a cubewhosecenter is
J. Chem. Phys., Vol. 94, No. 11.1 June 1991
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C. L. Cleveland and U. Landman: Structure of nickel clusters
7383
TABLE III. Interfacial energiesfor nickel decahedra.
Interface
Unrelaxed surfaceor twinning energy
( 111) surface (capping)
( ill) surface (notch)
( 100) surface
Twinning plane
0.094 60eV A;-’
0.094 80 eVA-’
O.l0306eVA-’
Relaxed surfaceor twinning energy
the origin and whoseeight verticespoint in the ( 111) directions. A cube,along with someother Platonic and Archimedean solids, is shown in Fig. 6.
There are two fundamental families of such cubesthose with an atom at the origin and those with an fee octahedral site there.*’ We will generallydiscussonly atom-centered clusters in this section, becausethe octahedral-sitecentered clusters have little new to offer after the
atom-centeredoneshavebeenexamined.Apart from the decahedra,if the number of atoms in a single-crystalcluster is
odd, the cluster is atom centered; if even, octahedral site
centered.
Other polyhedracan be producedby “shaving off layers of atoms correspondingto crystal planeswe want to exposeon the surface.When shavingoff any face of atoms, we
also removeall other similar ones,sincewe want to keepthe
shapeswe consider as symmetric, and therefore as small in
number, as possible.Repeatedlyshaving ( 111) facesfrom a
cubeallows us to successivelyproduce,at leastapproximately, a sequenceof shapeslike the Archimedeansolids in Fig.
6. It also enablesus to produce many other intermediate
shapesas well. However, we cannot produceclustersof arbitrary shape,as we could macroscopically,sinceatoms must
be removedas discrete units.
This aspectof atomicity is revealedexplicitly in the way
the choice of a particular geometric shapefor a cluster re-
FIG. 6. A sequenceof Platonic and Archimedian solids demonstrating the
duality of the octahedron and the cube. Shown are (a) the cube; (b) the
truncated cube; (c) the cuboctahedron; (d) the truncated octahedron; and
(e) the octahedron.
0.094423 3 eVA-’
O.O941898eVA--2
O.l029129eVA-*
0.0001663eVA-2
stricts the number of atoms which it can contain. If we want
to construct an icosahedronor cuboctahedron,we must use
a number of atoms in the set {13,55, 147,309,561, ...I. or to
construct a truncated octahedron,we must choosefrom {38,
201, 1289,4033, ...}. etc.
This posesthe questionof which modelswe should consider?The truncated octahedron [Fig. 6 (d) ] is an obvious
choicesinceof all the namedshapes,it is closestto the Wulff
polyhedron. We also consider local variations from this
shapesinceit occurs sparselyand we want to examinesimilar shapesto fill the gapsin cluster size,as well as explore its
local neighborhood in configuration space. Many of the
modelswhich we haveconsideredare lessthan encouraging.
In this paper, we concentrateon the more energeticallyfavorable models and a posteriori explain the principles that
m ight have led us to such models from the outset.
Let us construct our clusters from cuboctahedra[Fig.
6(c) ] which have the convenient property that removing
one atomic layer from each of its faces producesthe next
smallercuboctahedron.(A similar observationholds for the
icosahedra.) This causesthe difference in the number of
atoms betweena cuboctahedronand the next one in size to
be as small as possible.Taking the sequenceof cuboctahedra,84we find that only for every fourth one can a truncated
octahedron be produced by shaving off ( 111) planes of
atoms. Nevertheless,Raoult et al.’ have defineda sequence
of clusters (named“model C”) which closelyresembletruncatedoctahedraand which can be madefrom everycuboctahedron. Accordingly, a cluster belongs to model C if its
( 111) facesare just further from the center than would be
the casefor the truncated octahedronand consequentlyare
slightly smaller in comparisonwith the ( 100) faces.In this
context, we remark that the Wulff polyhedron also has its
( 111) facesslightly further from its center than the truncated octahedrondoes.For any cuboctahedron,we havemerely
to shave off ( 111) planes until we come to the last one
further from the origin than the ideal truncated octahedral
distance.Thus we have four model C clustersfor every truncated octahedron.85
Another sequenceof clustersmay be obtainedby exposing ( 110) facets.Given any model C cluster, we can simply
strip off the line of atoms at the edgewhere two ( 111) faces
meet. This produces a small (110) face consisting of two
parallel lines of atoms. Following Raoult et al., we call these
HKI clusters.86
The last sequenceof single-crystal clusters which we
considerare cuboctahedra.Many previousresearchershave
studiedthem, eventhough the shapeis not very optimal,
J. Chem.Phys.,Vol.94,No.11,1
June1991
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7384
C. L. Cleveland and U. Landman: Structure of nickel clusters
becauseof the special relationship, alluded to earlier, that
the cubocthedronhas with the icosahedron.Not only do the
two sequencescontain clusters with the same numbers of
atoms,but any cuboctahedralcluster can convert to an icosahedralone,and vice versa,by meansof a simple twinning
transformation.” Thus the competitionbetweenicosahedra
and cuboctahedrais more direct and physical than that involving other single-crystalclusters,which generallycannot
convert to other high symmetry clusters without gaining or
losing atoms or at least undergoingsubstantialatomic rearrangement.
Finally, we will also include in our m icroscopiccalculations the icosahedra,which are expectedto be energetically
optimal for the smallestclusters,and sincewe wish to investigate as a function of cluster size the transition from icosahedra to other structural forms, such as single crystals and
cuboctahedra.
In Table IV some of the geometric properties of these
clusters are collected. Here and elsewhere,m denotesthe
numberof atomson an edgejoining a ( 111) faceand a ( 100)
face; n denotesthe number of atoms on an edgebetweena
( 111) face and any non-( 100) face-either ( 111) or ( 110)
for our clusters.The number of atoms on an edgebetweena
( 100)faceand a ( 110) faceon an HKI cluster is alwaystwo.
First we will plot ,vsN “’ the energies(obtainedby energy m inimization using the EAM potentials) of theseclusters
less the bulk energy of the same number of atoms
EB (N) = l BN divided by LV”~. Using macroscopicconsiderationsas a guide, we expectthat the total energyof a given
cluster shape, with N atoms and surface area S, could be
expressedin the form5’
E(N) = E&v + E&v + ep,
(6)
whereeBis the bulk energyper atom, 6, is the uniform strain
energyper atom (zero for the single-crystalclusters), and cy
is an averagesurface energy per unit area for that cluster
shape.Denoting by a a geometric factor suitable for that
shape,we may write
[E(N)
-ESN]/N2'3~=E,N1'3+aEY.
(7)
Equation (7) is an approximateequality becausequite large
clustersmay be necessarybeforethe surfaceareacan be well
approximatedby aN 2’3for a given cluster shape.
Thus a simple macroscopicapproach would lead us to
expectthat plotting the energiesof m icroscopicsingle-crystal clusters in this way will producehorizontal lines ranked
according to how well they m inimize surfaceenergy,while
the icosahedralcurve will be a straight line with a positive
slopecausedby its internal strain.
In the upper panel of Fig. 7, we display the results of
energym inimization for m icroscopicsingle-crystalclusters
and the aboveexpectationsare born out. We also find that
icosahedralclustersare most favorablefor clusterscontaining lessthan 3540atoms,at which point the truncatedoctahedra and model C clustersare very competitive with each
other, as expected since their shapesbracket that of the
Wulff polyhedron.As expected,we alsofind the ( 110) faceting of HKI clusters to be energetically unfavorable for
smaller clusters and to becomemore attractive as cluster
sizes increase,although we do not explore clusters large
enough for it to becomethe optimal structural form. The
cuboctahedrondoesnot competeat all with the other clusters in this size range. However, extrapolation” of the cuboctahedraland icosahedralenergiesenablesus to estimate
that the cuboctahedralsare favoredover the icosahedralsfor
clusterslarger than about 14 000 atoms.
At this point we wish to comparethe resultsobtainedvia
the atomistic energym inimization with thoseof macroscopic calculations.To perform macroscopiccalculationsfor an
unrelaxedatomic cluster, we must be able to associatea polyhedron with it; i.e., we must take the cluster and find its set
of surfaceplanes.This definesthe polyhedron and allows us
to calculateits volume and various surfaceareas,which are
usedin a macroscopiccalculationof the cluster’senergy.We
note that sincewe are dealingwith rather small,atomic clus-
TABLE IV. Single crystal atom-centeredclusters and icosahedra:number of atoms N, number of atoms on an edgebetweena ( 111) and a ( 100) face m,
number ofatoms on an edgebetweena ( 111) faceand a non-( 100) facen. Each truncated octahedroncan be producedfrom the cuboctahedronin the same
row by removing one or more setsof eight ( 111) planes,and ifone fewer set of ( 111) planesis removedfrom eachtruncated octahedron,the model C cluster
in the samerow is produced.
Icosahedra/cuboctahedra”
Truncated octahedra
N
m
n
147
309
561
923
1415
2057
2869
3871
5083
6525b
o/4
O/5
O/6
O/7
O/8
o/9
o/10
o/11
o/12
o/13
4/l
5/l
6/l
7/l
8/l
9/l
10/l
1 l/l
12/l
13/l
Model C
N
m
”
201
3
3
1289
5
5
4033
7
7
HKI
N
m
n
N
m
n
405
711
1139
1709
1925
2735
3739
4957
5341
4
5
6
7
6
7
8
9
8
3
3
3
3
5
5
5
5
7
369
675
1103
1673
1865
2675
3679
4897
5257
2
3
4
5
4
5
6
7
6
4
4
4
4
6
6
6
6
8
“Form and n, the left-hand value is for icosahedra;the right is for cuboctahedra.
‘Not studied here.
J. Chem. Phys., Vol. 94, No. 11,l June 1991
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C. L. Cleveland and U. Landman: Structure of nickel clusters
5
53
cu’
z
$3
hy:
fci
“,
ui’s:
8,”
x
0-e.
o---E)---*---
Q---Q---Ct---@..--Q
X
-,
x HKI
*
4
F
-&
5
$3,
Y-s:
2d
%*
sd
7385
would initially be discouraged.All the singlecrystal clusters
would have scaled energy differencessubstantially lower
than any of the icosahedra,and for smaller clustersthe deficiency would be much worse. However, upon closer examination, we observedthat the relative ordering in energy of
the clusters of similar size had beenremarkably improved.
Results in which the energeticsof various structural
modelsare comparedmay be presentedin various ways. In
particular, we may use as our referencenot E&V, but the
energiesof the membersof a standard referencefamily of
clusters, such as the cuboctahedra, somewhat as van de
Waa140does.The advantageof this approachover using the
bulk energy as referenceis that it subtracts cluster surface
energiesas well. While this is unimportant when presenting
energiesobtained via atomistic m inimization, it is advantageouswhen displaying energiesfrom macroscopiccalculations for small clusters, sincesuch a subtraction can largely
correct for systematicerrors, such as may be connectedwith
the positioning of the surfaceplanes.We will usethe cuboctahedron as our referencecluster sinceit has no strain energy, it occurs at frequent intervals for better interpolation,
and it will not be greatly m issedfrom the figures since it is
such a poor competitor.
Resultsobtainedby atomistic energym inimization and
via macroscopiccalculations which take surface planes to
passthrough surfaceatoms are shown in Fig. 8. At the top,
the m icroscopic results, using an interpolation88 formula
based on m icroscopic cuboctahedraare displayed. At the
bottom, we show the results for macroscopicsingle-crystal
polyhedra, using an interpolation formula basedon macroscopic cuboctahedra.The icosahedralcurve in both figures
is the one obtained from our m icroscopiccalculationssince
the macroscopicapproachfails to provide an accurateestimate for this structure, as discussedin Sec.III. The overall
agreement between the results obtained by the two approachesis remarkable.Note, however, the overestimation
of the energiesof the HKI clustersby the macroscopiccalculations. Nevertheless,this discrepancydecreasesas clusters
become larger and the m icroscopic HKI clusters become
more competitive energetically.
The use of cuboctahedralreferenceenergiesinstead of
bulk ones does not affect the relative energeticordering of
clusters of similar size, regardlessof which convention is
employedfor positioning the surfaceplaneswith respectto
atomic ones. The advantageof using cuboctahedralreference energieslies in being able to changethe conventional
location of the surfaceplanesin order to improve the relative
energeticordering of clustersof similar size,sinceonly clusters of similar size can energeticallycompetewith one another.89
*sQ--.o
---Q.-~~---EP-.-c3---0
,.QQBL
Y HKI
d.0 0.0 IA.0 I;.0
IA.0
do
NV3
FIG. 7. In the upper panel of this figure, the minimized energiesof microscopic single clusters are representedrelative to the bulk, together with
those of the icosahedra. In the lower panel, also relative to the bulk, are
shown the energiesresulting from macroscopiccalculations for unrelaxed
versions of the same clusters, along with the same icosahedral curve. In
these macroscopiccalculations, surface planes were taken to fall between
atomic layers.
ters, it may make a substantial differencewhere we define
those surfaceplanesto fall. We m ight choosefor one to cut
through the atoms in the surfacelayer of its facet. Alternatively, we m ight choosefor the surface plane to fall outside
that plane by a distanceequal to one-half the layer spacing
for that crystal face.Each choiceproducesa different energy
for every cluster we examine.
First, let us take a surfaceplane to lie abovethe surface
atoms by one-half a layer spacing. Results of macroscopic
calculations with this convention are shown in the lower
panel of Fig. 7. The overall energiesare are in adequate
agreementwith the correspondingm icroscopicones,shown
in the upper panel of Fig. 7, with the scaledenergy differencesin the figure too low by 0.03 eV for large clusters and
worsefor smallerones.However, the relative ordering of the
clusters in energy, particularly the truncated octahedron,
model C, and HKI clusters, is poorly articulated and frequently wrong, evenfor larger ones.
Let us explorea different conventionfor positioning the
surface planes in the macroscopic calculation and move
theseplanesso that they passthrough the cluster’s surface
atoms. If we were to plot the results in the usual way, we
B. lcosahedra
There is substantialinterest in the characteristicsof relaxed icosahedralclusters. As we pointed out in Sec. III,
systemsunder uniform icosahedralstrain have a relatively
large strain energy. Nonetheless,m icroscopically relaxed
icosahedral clusters are energetically favored over other
typesevenfor clustersof severalthousandatoms.It is worthwhile to examinethe nature of the relaxation of icosahedral
J. Chem. Phys.,
Vol. 94,toNo.
Juneor1991
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AIP11,l
license
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7386
C. L. Cleveland and U. Landman: Structure of nickel clusters
N
‘i
X.
..
‘ic
A-
561
N= 55 147
1415
2869
5063
2I
9
&t
_ -.
-x- - -.-.*
..-I-‘F _ _ _ -x---_ _ .+-.*&-...c.-~
_ --Q- - - +.‘&*---
3
.-SC
Z+
‘%
%
f
/
)RA
70
t
iI
-?
T
I
0.0
I
2.0
I
4.0
I
6.0
I
8.0
I
10.0
12.0
Shell Number
x.
--_
FIG. 9. The energiesof vertex atoms on successiveshells of icosahedraof
various sizes. Lines connect energiescorresponding to atoms in different
shells in the same icosahedron.Results for icosahedracomprising three,
five, seven,nine, and elevenshellsare shown. The 55-atom, two shell icosahedron is also shown, but with a dashedline.
x- - _ _ x
* ____ +-_..*-A----&.-----A
--__
+X-.*
-‘x----x.-~-~+
+--A-+..*:
1
x
+
bdi
)RA
I
6.0
/
8.0
I
I
10.0
12.0
NV’
1
I
14.0
16.0
18.0
FIG. 8. In the top graph, the minimized energiesof microscopicclustersare
represented,relative to an interpolated cuboctahedralenergy.In the bottom
graph, equivalentmacroscopicquantities are displayed,with the exception
of the curve for icosahedra,which is the sameas in the upper graph.
clusterswhich enablesthem to competeenergeticallyas well
as they do.
As mentioned above, we may analyze the energy of a
cluster atom by atom. In Fig. 9 we display how the energies
of atoms on the fivefold axeschangeas we move from the
insideof a cluster to its outside.We observea great regularity
with overall cluster sizeif we comparethe outermostlayer of
each cluster and then successivelyproceed inwards. Our
overall observationis that the energiesof atoms in the outer
shellsof theseclustersare very similar, but that as the interior of the cluster is penetrated,energiesincrease.The further
one penetrates,the greater is the increasein atomic energy
with a particular concentrationof energyat the center of the
cluster. Sincea large part of the cohesiveenergy in metals
derivesfrom the embeddingof the metal ions in the electronic chargedensity of the medium, atoms will seekto occupy positionswhich optimize that embeddingenergy.Consequently,atoms on the outer shellsof the cluster attempt to
compensatefor the m issingof neighborswhich would have
beenpresentin the bulk by displacinginward toward regions
of higher electronic density. (The optimal structure is of
course a balance between the cohesiveembeddingenergy
and the core-core repulsions.) The atoms that experience
the greatestlossof electronicembeddingdensity are the ones
at the outside verticesof the cluster. Under the geometryof
the icosahedron,theseatoms can only move to a region of
higher electronicdensityby pressingdirectly on atomsat the
verticesof the shell below theirs. Becausethe vertex atoms
form a linear chain from the outside of the cluster to its
center, and since the stresson theseatoms is radial, stress
accumulatesalong the fivefold axes and especially at the
clusters’centers. It is not surprising that the energy of an
atom at a cluster’scenteris still increasingalmost as strongly
with the cluster’s radius when the cluster hasten or 11shells
as when it had two or three. (We suspectthis behavior to be
largely independentof the interatomic interaction for potentials that favor close-packedstructures, since the geometry
of the icosahedronalmost forces such a concentration of
stress.)
Similar analysisto that leading to the results shown in
Fig. 9 could also be made for atoms on the edgesadjoining
faceson shells within the icosahedraor atoms on the faces
themselves.By way of definition, let us say that asidefrom
the vertex atoms on an icosahedralshell, we also have edge
atoms, which adjoin two faces,but do not include the vertex
atoms, and face atoms, which include every atom that is
neither a vertex or an edgeatom. It is worth pointing out that
although unrelaxedclustersshowa considerablevariation in
atomic energiesfor the edgeatoms in a shell, or for the face
atoms in a shell, the edgeenergiesare remarkably uniform
within a given shell of a cluster after relaxation, as are those
of the face atoms. For example,in the 1l-shell, 5083 atom
icosahedron, the standard deviation of energies of edge
atoms is greatestfor the tenth shell, where it is 0.003 27 eV,
as comparedwith an edgeaveragefor that shell of - 4.4 183
eV. For the samecluster, the standard deviation of energies
of face atoms is greatest for the 11th shell, where it is
0.005 02 eV, as comparedto a facial averagefor that shell of
- 3.93929 eV. Within a given shell of an icosahedron,the
energydifferencebetweena pair of atoms chosenfrom two
different categories(vertex, edge,or face) is much greater
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C. L. Cleveland and U. Landman: Structure of nickel clusters
than the energydifferencebetweenany pair of atoms within
any single category.
Fargeset aL9 consider similar issuesfor clusters whose
atoms interact through a Lennard-Jones 6-12 potential.
They show how the spacingbetweenpairs of adjacentatoms
lying on one of the fivefold axes of an icosahedral cluster
changesin moving from the inside of a cluster out, and for a
given distance from the center, how it changeswith cluster
size. We call this spacing the radial nearest-neighbordistance. Furthermore, they examinethe nearest-neighbordistancebetweenatoms near the center of a face for eachof the
shells of which an icosahedralcluster is composed.We take
this to mean the distance betweenthose two atoms in the
shell nearestthe center of a face, whether one of the atoms is
exactly in the centeror not, and call it the tangential nearestneighbor distance for that shell. In Fig. 10, we show the
results for the icosahedral clusters that we have studied,
which are qualitatively similar to the results of Fargeset al.
although somewhatlarger in degree.We note that both the
c;i -Bulk
4,
Shell Pail
r
c-4
gp
L-4
-0E
an
as
fE
a
ci
Shell
kig2@
8
ci
-2
-a53
2C”
4
3
2
$J
e
0
3
z
c-4
-yt:
2.0
I
4.0
Total Number
6.0
a.0
10.0
12.0
of Shells in Icosahedron
FIG. 10. In the top graph, the distancesbetweenadjacent atoms on a fivefold axis ofan icoshedralcluster are shown. The curves connect distancesin
different clusters betweenatoms in the same two shells and are marked by
the pair of shell numbers involved. In the bottom graph, the distance between the two atoms in a shell closest to the center of one of its faces is
shown, marked by the number of the shell. Note that the numbering of
shells is such thatshell zerohasone atom,shellonehas12,etc.
7387
radial and tangentialdistancesbetweenatoms decreasenonuniformly as we move from the outside toward the inside of
the cluster. Furthermore, as we consider larger and larger
clusters, the distancesat the surfaceof the clusters increase
and the distancesnear its center decrease.For the icosahedral clusters we have examined,the radial nearest-neighbor
distanceis alwayslessthat the nearest-neighbordistancein a
perfect bulk crystal, while the tangential nearest-neighbor
distanceis larger than that distancefor a cluster’s outer layers and smaller for its inner ones.
Farges et a1.9 also provide information about the convexity of faces on the 923-atom, six-shell icosahedralLennardJones cluster and suggestthat the convexity is more or
less uniform from shell to shell for any given icosahedral
cluster. Following their suggestion,we take a radial unit vector through the center of a face and project onto it two vectors pointing from the center of the cluster, with one pointing to a vertex atom on that faceand the other pointing to the
atom closestto the face’scenter. We definethe layer position
of the faceto be the meanof theseprojecteddistancesand the
layer distortion to be their difference. We find for the sixshell icosahedralcluster that the ratio of the distortion for
the outermost shell to the differencebetweenthe layer position of the two outermost shells is about 13%. Fargeset al.,
obtain 9% for an apparently similar quantity for their LennardJones material. They observethat this ratio increasesin
a nearly linear mannerwith shell number if it is also calculated for the interior shells of the cluster, suggestinga uniformity of convexity throughout the shells of the cluster. We
also observea roughly linear relation betweenshell number
and this ratio. Moreover we note that this ratio, for a given
shell number within a containing icosahedralcluster, is remarkably independentof the size of the cluster containing it.
An alternative measure of the convexity of a surface
may be obtainedin the following manner: Let us definetwo
spheres.One is chosenso that its center is at the center of the
icosahedralcluster and its surfacepassesthrough the vertex
atoms on a given shell of the cluster. The other is chosenso
that its surfacepassesthrough the vertex atoms of a particular face and also through the atom closest to the center of
that face (or rather through the projection of that atom’s
position onto the radial unit vector through the center of the
face). As the mean curvature C of the face,we take the reciprocal of the radius of the secondsphere,although the face
will tend to be flatter in its center than at its edges.To normalize this curvature in a way that should be independentof
which of the shells we are examining if they are all similar,
we multiply it by the radius R, of the first sphere.Thus if the
faceswere completely Rat, the product would be zero, while
if the cluster had relaxed to a perfect sphere, it would be
unity. Results are shown in Fig. 11.We find that as expected,
this relative curvature uniformly decreasesas we move to
larger and larger clusters, since it ultimately must conform
to the macroscopiclimit.
C. Decahedra
The principle difficulty in studying the energies of
Marks’decahedrais the richnessof the parameterspace[see
Figs.4(c) and 51.As a simplification,we ignore( 110)faces
J. Chem. Phys., Vol. 94, No. 11,l June 1991
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7388
C. L. Cleveland and U. Landman: Structure of nickel clusters
8
d
8
d
$8
vo
-7
;:
6
8
6
-a
B---+-49
w
I
2.0
I
4.0
Total Number
I
6.0
I
a.0
I
10
$ II
10.0
FIG. 12. An Ino’s decahedronwith m = 7 and n = 5 is shown on the left.
On the right is shown the resulting Marks’ decahedronif theoutermost line
of atoms on eachtwinning plane is removed.This decahedronhass = 1 and
p = 2 as well as m = 5 and n = 5.
of Shells in lcosahedron
FIG. 11. The relative curvature of faceson shells within icosahedraof various sizes.
sincein the rangeof cluster sizeswhich we study, such faces
do not appearto be favorable. Even so, we have two inequivalent kinds of ( 111) faces,which we have called capping
facesand notch faces.Thus while we can specify one of our
single-crystalpolyhedra without ( 110) facesjust by listing
the number of atoms on the edgejoining ( 100) and ( 111)
faces m and the number of atoms on the edgejoining two
( 111) facesn, we will find an additional parameteris needed
for decahedra.For decahedra,we will use m to denote the
number of atoms on an edgejoining a ( 100) face and a capping ( 111) face, n to denote the number of atoms on the
other kind of edgeon a (100) face, which connectsit with
either a notching ( 111) face or if no notching has occurred
with a neighboring (100) face, andp to denotethe number
on an edgejoining the two kinds of ( 111) face.This notation
is the sameas that usedby Raoult et al* Instead ofp, however,we shall usually refer to s = p - 1, the number of ( 111)
notching planeswhich havebeenremovedfrom the sidesof
eachdecahedra1unit to producethe desiredMarks’decahedron. We chooseto do this since our schemefor organizing
the plethora of Marks’ decahedrarefers each back to the
Ino’s decahedronfrom which it could be madeby notching,
and which is simply a Marks’ decahedronwith s = 0 [see
Figs. 4(b) and 4(c) 1. An Ino’s decahedronand any Marks’
decahedronthat can be obtainedfrom it by notching all belong to the same “family.” Two Ino’s decahedrabelong to
the same family if for them m - n, which measuresthe
“squareness”of the ( 100) face, has the samevalue. Thus in
Fig. 12, both the Ino’s decahedronand the Marks’ decahedron belongto family m - n = 2. The Ino’s decahedronhas
m = 7, n = 5, s = 0 (andp = l), while the Marks’ decahedronhasm=S,n=5,s=l
(andp=2).
If we take any Ino’s decahedronand beginnotching it at
the twinning planes,we find that, relative either to bulk energiesor cuboctahedralenergies,a sharp minimum exists. In
Fig. 13 we show the results for family four, i.e., the set of
decahedraarising from notching an Ino’s decahedronwith
m---=4.
Similar calculations were performed for members of
families two, six, and eight. The geometriesof the decahedra1
clusters which we investigatedare given in Table V. We observed that as one moves to larger and larger clusters, the
number of the energeticallyoptimal family increasesand the
degreeof notching of the optimal member of that family
increasesas well. The results of thesecalculations are summarized in Fig. 14, where we display the energiesfor those
decahedra1
clusters with energieslower than any other decahedron of similar size, along with the curve of microscopic
icosahedralenergieswhich we have seenin so many of the
other figures.We concludethat icosahedrabecomeunstable
with respect to Marks’ decahedrafor clusters larger than
about 2300atoms.
In examining Fig. 14, we note that minimum energy
decahedratend to havesquare ( 100) facesand that the size
of those faces and the depth of the notch embracing them
increaseswith increasingcluster sizejust aswe would expect
from macroscopicconsiderations.To make the comparison
with macroscopicexpectationsmore quantitative, we show
ii.0
A.0
Ii.0
Nm
Ii.0
Ii.0
‘LO
FIG. 13. The energiesof the decahedra1family four relative to the bulk
energyof the samenumber of atoms. The solid lines connect memberswith
the same degreeof notching, while the dashed lines connect each cluster
with that one producedfrom it by further notching and with its parent cluster from which it was similarly produced.Thes value for eachsolid curve is
usedto denote it, while each cluster is marked by its m and n values.
J. Chem. Phys., Vol. 94, No. ii,1
June 1991
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C. L. Cleveland and U. Landman: Structure of nickel clusters
7389
TABLE V. Decahedra1clusters: numbers ofatoms N, number ofatoms on an edgebetweenacapping ( 111) and a ( 100) face m, number ofatoms on the other
type of edgeon a ( 100) face n, and the depth of the notch s, for the decahedra1families 2,4,6, and 8, corresponding to decahedrathat may be produced by
various degreesof notching from Ino’s decahedrawith m - n = 2,4, 6, and 8, respectively.
Family 2
N
m
207
409
711
1133
1695
2417
3319
4421
192
389
686
1103
1660
2377
3274
4371
142
324
606
1008
1550
2252
3134
4216
461
838
1355
2032
2889
3946
6
7
8
9
10
11
12
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
1
2
3
4
5
6
5
Family 4
Family 6
Family 8
n
s
N
m
n
s
N
m
n
s
N
m
n
s
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
5
6
7
8
9
10
0
0
0
0
0
0
0
0
1
1
1
1
499
851
1333
1965
2767
3759
4961
484
831
1308
1935
2732
3719
434
766
1228
1840
2622
3594
339
646
1083
1670
2427
3314
7
8
9
10
11
12
13
5
6
7
8
9
10
3
4
5
6
7
8
1
2
3
4
5
6
3
4
5
6
7
8
9
3
4
5
6
7
8
3
4
5
6
7
8
3
4
5
6
7
8
0
0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
971
1513
2215
3097
4179
5481
1493
2190
3067
4144
1428
2110
2972
4034
1308
1965
2802
3839
1745
2547
3549
9
10
11
12
13
14
8
9
10
11
6
7
8
9
4
5
6
7
3
4
5
3
4
5
6
7
8
4
5
6
7
4
5
6
7
4
5
6
7
5
6
7
0
0
0
0
0
0
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
3397
4569
5911
13
14
15
5
6
7
0
0
0
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
results for family four in Fig. 15, where in the macroscopic
calculations the surfaceplanesare taken to cut through surface atoms. In the upper panel energiesobtained via microscopic, atomistic energyminimizations are shown relative to
27
c.4
. %I0
cuboctahedral energies,and in the lower panel, the corresponding macroscopic results are displayed. For reference,
the microscopic icosahedralenergiesare shown in the upper
graph. We find excellent agreementbetween the ordering
provided by the macroscopic calculations and the microscopic results. In evaluating the quality of the decahedra1
results, it should be borne in mind that the decahedrausedin
the macroscopiccalculation are made from unrelaxed,classic decahedraand not macroscopicdecahedrawhosegeometries are consistent with the bulk system whose energy we
minimized under decahedra1strain. Using the latter geometries insteadresults in an overall lowering of the macroscopic
energies,reducing the overall energy shift betweenthe two
panelsof Fig. 15.
D. Other clusters
I
d.0
El0
6.0
I;.0
Ii.0
1Lo
Ii.0
NV3
FIG. 14. Those decahedra1sequenceswe have studied which have somewhere an energy lower than any other decahedra1sequenceof corresponding size. Each curve is marked with its degreeof notching s and eachdecahedron’s point is marked with its m and n values. The icosahedral curve,
obtained from microscopic energy minimization, is included for comparison.
In addition to the abovestructures, other model structures of atomic clusters have been examined in order to
check that we haveindeedconsideredmost of the structural
motifs relevant to our study of the energeticsof nickel clusters. Thus, e.g.,we havestudied “model B” of Raoult et a&*
which is a shift from the truncated octahedronin the opposite direction to that taken by model C, so that it has smaller
( 100) faces.As expected,this is energeticallyquite unfavorable. Furthermore, unlike the Lennard-Jonesiumclusters of
Raoulteta1.,8
removing
the12vertexatomsontheicosahe-
J. Chem. Phys.,subject
Vol. 94,toNo.
11,l
June or
1991
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7390
C. L. Cleveland and U. Landman: Structure of nickel clusters
6
‘?
8
9& :’
I$
2 1.
Z
“0’00
u”$
A
$a
I
8
FQ
&
s* 9
$P
z-co
8s
yP
2
gc2
9
r
P
8.0
10.0
12.0
,,,I,3
14.0
16.0
18.0
FIG. 15. In the upper graph the minimized microscopicenergiesfor family
four are representedrelative to the cuboctahedralenergy. In the lower figure, the correspondingmacroscopicresultsare shown, where we have taken
eachsurface’splane to passthrough its atoms. The microscopic results for
the icosahedron(squares) are shown in the upper graph as well.
dral clusters makes them worse competitors energetically.
This type of variation never seemsto be advantageousfor
any cluster types and we havenot systematicallyexploredit.
We have also created monotwins of someof the singlecrystals, which invariably lowered the cluster’s energy,but to a
degreeso small comparedto other energydifferencesasto be
uninteresting.For example,the untwinnedHKI cluster with
3679atoms has a total energyof - 15 766.341eV. The correspondingtwinned cluster hasa total energyonly 0.086eV,
or 0.28 K lower.
There is another largegroup of clustersthat we havenot
extensivelystudied simply becausethey are too numerous:
those with broken symmetry, i.e., clusters with incomplete
shells,or those hybridized from piecesof severalclustersof
the more symmetric type which we have considered.Such
clusters may be abundant under experimentalconditions.
For example,for clusters producedvia a laser-vaporization
cluster source, where at low helium pressuresthe principal
mechanismof cluster growth is by monomer addition, the
oddsthat the number of atomsin a cluster turns out to be one
of the valuesfor which an energeticallyoptimum, symmetric
shapecan be made are small. Similarly, for clusters which
evolve from molten droplets, evaporation of excessatoms
will essentiallyceaseprior to any solidification becauseof
the large binding energyfor surfaceatoms ( - 3.5 eV).
The systematic trends revealedfrom our study of the
more symmetric structural forms guide us as to which structures may develop when the number of atoms in a cluster
does not correspond to one of the symmetric structures
which we have considered.The easiestway to interpolate
numbersof atoms is by breaking the symmetry of the clusters. One could usemodel C type faceson part of a cluster’s
surfaceand thosefrom a truncated octahedronelsewhere.If
somewhat fewer atoms are required, some vertex atoms
m ight be removed,or if the cluster is big enough,some( 110)
faces applied. In the decahedra,there may be variation
among the five tetrahedral units. In general,the surfaceof
each single crystal piece will attempt to resemblea piece
from the Wulff polyhedron even for clusters that are quite
small. Exceptionsare the icosahedra,where we expectthat
the best clustersof lower symmetry will have an icosahedral
core with a partial outer shell whoseatoms are as compactly
arrangedas possible.To avoid energeticallyexpensivesharp
atomic stepson a cluster’ssurface,the simplestprocedureis
to take a larger cluster in the family in questionand remove
whole atomic planesat a time. It is worth pointing out that
the truncated octahedra,model C’s, and HKI are all singlecrystal clusters related to eachother by just such operative
transformations and, except for the HKI clusters of small
sizes,all havesimilar energies.This indicatesthat interpolation between similar high-symmetry configurations may
provide someinformation about energiesof clustersof intermediatesizes.
In order to illustrate these ideas, we have considered
clusters with 201 atoms. Of the high-symmetry structures
which we have studied so far, only a truncated octahedron
can haveexactly this many atoms.On the other hand, as we
seefrom the upper half of Fig. 7, the icosahedrahavea much
lower scaledenergythan the truncated octahedrain this size
range (N “3 -6). The question naturally arises as to
whether some less symmetric icosahedrally based cluster
with 201 atoms m ight be lower in energy than the highly
symmetric truncated octahedron.
A cluster with the desirednumber of atoms may be constructed by addingextra atomsto a 147-atomicosahedronor
removing excessatoms from a 309-atom icosahedron.For
the purposeof our illustration, we have chosenthe second
construction,attempting to removewhole planesofatoms at
a time. Accordingly, starting from a 309-atomicosahedron,
we pick a vertex and removeall five surfaceplanesthat meet
at it, exposingpart of the 147-atomcore. We continue this
processby removing planesof atoms on facets adjacentto
the area we have already cleared until we have 201 or less
atoms left. In fact, following this procedure results in our
casein a cluster containing 200 atoms to which one atom is
replacedat the edgeof the partial shell which capsthe 147atom icosahedralcore. Direct m inimization of the energyof
that cluster yields a total energy E(N) = - 807.076 eV
which is slightly below that of the corresponding201-atom
truncated octahedron ( - 807.003eV). The resulting cluster is shown,front and back, in Fig. 16,where atoms belonging to the 147-atomicosahedralcore are light gray and those
J. Chem. Phys., Vol. 94, No. 11,l June 1991
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C. L. Cleveland and U. Landman: Structure of nickel clusters
I
FIG. 16. Front and back views ofa low symmetry 201-atom cluster with an
icosahedralcore of 147atoms (light gray) and a capping partial shell of 54
atoms (dark gray) constructed as detailed in the text.
of the capping partial shell are dark gray (the extra 201st
atom is the dark gray one nearestthe center of the left image). It is of interest to note that the effect of this extra atom
is rather large. The energy of the corresponding200-atom
icosahedrally basedcluster is - 803.353eV, while that of
the 201-atom truncated octahedron with one of its vertex
atoms removed is - 802.589eV, markedly increasing the
energetic advantageof the icosahedrally based cluster as
comparedwith the 201-atomcase.
We would like to note that other construction methods
that we haveattempted,including somewhich involved molecular dynamics simulated annealing,resultedin clusters of
substantially higher energythan the onewhich we havedemonstrated here. While we have undoubtedly not exhausted
all the possible construction schemes,our result indicates
that it is possibleto achieve energetically competitive lowsymmetry clusters by starting from judiciously chosenhighsymmetry structures and removing whole atomic planesat a
time, as discussedabove.Our generalconclusion is that in
attempting to build low-symmetry clusters whosenumber of
atoms is incompatible with high-symmetry structures, the
energy differencesbetweenthe latter (such as displayed in
Fig. 8) provide guidance in choosing the high-symmetry
structural motif on which to base the construction of the
low-symmetry clusters.
7391
energies and the modified Wulff construction due to
Marks” (seeSec.III) 1. Basedon comparisonsbetweenthe
results of atomistic energyminimization and the macroscopic formulas which we developed [see Eqs. (4), (5)) and
(Bl ) 1, we concludethat theseformulas, in conjunction with
accurately calculated, or experimentally measured, cohesive, surface,and strain energies[which in our studieswere
obtainedusing the EAM potentials (seeSets. II and III, and
the Appendices] allow a rather reliable framework for structural analysis. The macroscopic approach is particularly
useful in assessingthe relative energetic merits of various
structural motifs (seethe comparisonsbetweenthe results of
microscopic and macroscopic calculations in Sec. IV) and
thus aids in the analysisof experimentaldata.
The evolution with size of the optimal structures of
nickel, as well as Lennard-Jonesium(after the results of Ref.
8) clusters is summarized in Fig. 17. As seen from Fig.
17(a), our microscopic, atomistic calculations basedon the
EAM many-body interactions predict that the optimal
structures of nickel clusters containing less than -2300
atoms belong to the icosahedral sequence.Larger clusters
are predicted to optimally assumeMarks’ decahedra1geometries, while for clusters containing more than - 17 000
atoms, single crystal structures and their monotwins are energetically favored. Observe that we predict, on a macroscopic basis, that the evolution of optimal structures from
the decahedra1sequenceto the single crystal sequencehappens for much smaller cluster sizesfor nickel that has been
predicted* for Lennard-Jonesium,where on the order of 10’
atoms are expectedto be necessary.The transition for ener-
SINGLE
CRYSTALS
AND
MONOTWINS
MARKS
DECAHEDRA
V. SUMMARY
In this paper we have investigatedusing both a macroscopic approach and atomistic energy minimizations, the
sizedependenceof the evolution of structures in metal (nickel) crystallites ranging in size from small clusters containing
tens of atoms to rather large ones made of thousands of
atoms. In our studies we have used many-body interaction
potentials obtainedvia the embeddedatom method (seeSec.
II), which include as a dominant contribution to the cohesive energy of the metal the embeddingenergy of the metal
ions in the local electron density provided by the other atoms
in the system. In addition, we compare our results to those
obtained previously, employing simple pair-potential models.
One of the aims of our study is to test and assessthe
adequacyof macroscopicconceptsbasedon the Wulff construction and variants thereof [i.e., considerationsof elastic
ICOSAHEDRA
L-J
(a)
(W
FIG. 17. Diagrams, summarizing the evolution of optimal structural motifs, vs the number of atoms in nickel clusters (obtained in this study) and in
Lennard-Jonesium (after the results of Ref. 8). The results in (a) correspond to calculations including in the set of possible structures those of
Marks’ decahedra.The results in (b) were obtained without the inclusion of
Marks’ decahedrain the set of possiblestructures. Solid lines divide structures which belong to different internal structural forms (icosahedra, denoted by IC; Mark’s decahedra; and the single crystals and their monotwins, denoted by TO for truncated octahedra, model C, HKI, or W for
other Wulff-type structures). Dashedlines divide structures which have the
same internal form, but differ in their surface configuration (these are all
single crystals and their monotwins).
J. Chem. Phys., Vol. 94, No. 11,l June 1991
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7392
C. L. Cleveland and U. Landman: Structure of nickel clusters
getically favored structures between the icosahedral sequenceand the Marks’ decahedra1sequenceis predicted to
occur for somewhatlarger clusters in nickel than for LennardJonesium (2300 vs 1600atomsac’
). In this context, we
remind the reader that the HKI, model C, and truncated
octahedronare all singlecrystal clusters (seeSec.IV A) and
they all serveas approximations, forced by the discrete nature of matter, of the Wulff shape.As clustersbecomelarger
and larger, it becomesless and lessuseful to distinguish between clusters that only differ betweenone another by the
addition or removal of a singleplaneor row of atoms,and we
may just as well speakof the Wulff sequence,or that set of
cluster structures that are good approximationsto the Wulff
polyhedron. On the other hand, Marks’ decahedra,which
are basedon fivefold symmetrical structures, do not belong
either to the single crystal structures, or the icosahedral
structures.
The diagrams for nickel and Lennard-Jonesiumin Fig.
17(b) summarize the structural evolution vs size when
Marks’decahedraare not considered.As seen,such calculations predict the emergenceof single crystal structural motifs (i.e., truncated octahedra,model C, and HKI) as optimal clusters for much smaller sizes than when Marks’
decahedraare included, while the cluster size for the icosahedral to single crystal transition in nickel (3600 atoms) is
substantially greaterthan that for Lennard-Jonesium(2 100
atoms” ) .
For the caseof icosahedralstructures, we find that the
effect of relaxation (obtainedby atomistic minimization) of
theseclusters from the “ideal” structure given by the macroscopic approach is much larger than is the casefor decahedral and single crystal structures.Theserelaxationsof icosahedral clusters lead to relief of strain by nonuniform
relaxationsof atoms in the cluster, resulting in curved (i.e.,
nonflat) facets. Furthermore, the convexity of the structure
propagatesinto the interior of the cluster (seeSec.IV B).
While in this study we focus on structural motifs of high
symmetry, we also demonstratethat the results which we
obtained can guide the construction of energetically competitive low-symmetry clusters.
In closing, we reiterate that the results presentedin this
paper do not include thermal (and entropic) effects,which
can lead to isomerizations between accessiblestructural
forms and thus to a distribution of structures for a given size
cluster. While a more comprehensiveaccount of the behavior of nickel clusters at finite temperatureswill be given elsewhere, it is instructive, within the context of this study, to
comment on the stability of theseclusters at finite temperatures. As seenby molecular dynamics simulations which we
have performed, almost all the types of clusters which we
have considered here are stable up to the temperature at
which their surface atoms begin to diffuse and the clusters
can start to isomerize, generally - 1000 K dependingon
cluster size.
The exceptions are clusters which can convert to another of lower energy via a cooperative,nondiffusive transformation, i.e., the one which connects cuboctahedraand
icosahedra.We haveobservedthat on heating,cuboctahedra
undergo the multitwinning transformation well before any
atomic diffusion occurs, resulting in icosahedra.
As the temperature is successivelyincreasedfurther,
more outer layers melt and wet a structurally unchanged
solid core (i.e., statesof coexistencebetweenliquid, or quasiliquid, and solid phasesoccur). Finally, if enoughenergyis
added,the cluster will melt completely.
ACKNOWLEDGMENTS
This researchis supported by the U.S. Department of
Energy (DOE) under Grant No. FGO5-86ER-45234.Portions of the computations were carried out at the Florida
StateUniversity Computing Centerthrough a DOE grant of
computer time.
APPENDIX A: UNIFORM DECAHEDRAL STRAIN
In this appendix, we will show how the surface, twinning, and strain energiesneededfor a macroscopic calculation of the energyof oneof Marks’decahedrawere obtained.
In Fig. 18, we display an unstrainedunit cell with the tetrahedral unit embeddedin it. The periodic repetition of this
cell under the three translation vectors a, b, and c will fill all
space.Each of these three vectors points in a (100) direction. In order to changethis unit cell into one containing a
tetrahedral unit that will fit into a decahedron,we must
changethesethree vectors, so that they satisfy certain constraints.
We will choose to proceed so that the (100) surface
createdby removing the periodic boundary condition in the
c direction correspondsto the ( 100) surfacesthat occur on
Ino’s or Marks’decahedra.Then the line AB on the tetrahedron will correspondto the decahedra1axis. Our first constraints come from the requirementthat the lengths of AC,
BC, AD, and BD must all be the same.In terms of the transla-
D
C
I
[
- - -
c,’ .
’ .
* .
,
.
*
l
,
’
.
,
,
,
7
-iT
/
’’ . .’
.. .
.
..
,.
.
.
,.
,
.
’.
.. I
. . ’’
. .
,
,
,
,
L
,
,
FIG. 18. An unstrained computational cell for fee crystals. The three vectors a, b, and c are translations under whose repeatedapplication the computational cell periodically fills all space.They are currently pointing in
( 100) directions. We will apply decahedra1strain by changing thesethree
vectors to onesconsistent with decahedra1symmetry.
J. Chem. Phys., Vol. 94, No. 11,l June 1991
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C. L. Cleveland and U. Landman: Structure of nickel clusters
vectors, this implies that
tion
a=b
t-41)
c-a = cob = 0.
t-42)
and
We concludethat without loss of generality we can choosec
to lie in the + z direction and a and b to line in the x-y plane.
Further, since we can rotate the resulting setup about the z
axis without changing the system’s energy, we can further
take one of either a,, or b, zero. If, as is usual, we write the
three translation vectors as column vectors and take them as
the columns of an overall translation matrix H,
I-I = M-w),
we could so far chooseto write H as
(A3)
where (a,/Z,y,...)+ is the supremum of a, fi, y ,... . This
expressioncould be simplified by simply assertingthat a lies
along the x axis so that av = 0, correspondingto the case
where a,>b,,, but the conjugate gradient procedure will
treat a, and b,, as the two independentquantities in H, and
we cannot be sure that it may not try to make b, larger than
a, during minimization and must allow for that possibility.
The last constraint is of coursethe requirementthat the
dihedral anglebetweenthe ABC and the ABD planesbe 72”.
In terms of the translation vectors, this requirementmay be
written as
[c - (a + b)/2l*[c + (a + b)/2]
I[c--
(a+b)/2]jI[c+
0
(A4)
c,
=
Y
a,+
(O,az--bz)+]*+
by+
(a+b)/2]1
=cos(720)*
(A5)
d(O,a:-b:) +
a,
I-I== ,/(O,b:-a:) +
bY
0
7393
Equations (A4) and (A5) allow us to determinec, in terms
of a, and 6,. We find
(Ah)
(O,b:-aaL:)+]*,
where
Y = 0.5,/ [ 1 + cos(72”)]/[ 1 - cos(72”)j = 0.6882....
(A7)
That we haveonly two degreesof freedomleft is exactly
as expected. We can volumetrically expand or contract a
decahedronor changethe length of its axis and we will still
havea decahedron.Other kinds of changesto the tetrahedral
units will either prevent them from joining togetheror create
mismatchesbetweentheir edgesand vertices.
We can now fill this computational cell with an “fee”
lattice of 500 nickel atoms arranged so that there are five
conventional unit cells in the a direction, five in the b direction, and five in the c direction, and then minimize the energy by allowing a, and b,, to vary with the other components
of H then beingdeterminedas in Eqs. (A4) and (A6). Comparing the resulting energywith the bulk energyof the same
number of atoms in a perfect, unstrainedfee crystal gives us
the strain energy.
The Parrinello-Rahman method” returns the derivative of the potential energy with respectto each of the nine
componentsof H disregardingany constraints on them. It is
a small detail that we must use the chain rule to determine
from thesederivatives the derivatives with respectto a, and
b,, with constraints in place. For a, we have
(A81
Parrinello-Rahman methods give us the first partial derivative in the sum and Eqs. (A4) and (A6) give us the second.
by is of course treated in the sameway.
We may remove the periodic boundary condition in the
c direction and minimize the energyof the resultingslabby
varying atomic positions within a fixed computational cell,
obtaining the ( 100) surfaceenergyunder decahedra1strain.
Unfortunately, the systemas describedis not well suited
to determining the ( 111) surfaceenergiesor twinning energies,sinceit is not easyto createthe appropriateinterfaces.It
might appearthat we needto go to a different kind of computational cell for which ( 111) surfaces can be easily made,
such as the one in Fig. 3, reexaminethe consequences
of the
constraints neededfor decahedra1symmetry, and reminimize the bulk energyunder thoseconstraints, but we do not.
Any translationally periodic system may have its periodicity representedthrough any three noncollinear periodic
translation vectors. Imagine that the computational cell in
Fig. 18 correspondsto the one that was produced by the
minimization procedur~the changesare very small and do
not justify a secondfigure. Instead of describingthe periodicity of this bulk systemby using a, b, and c, we could just as
well have used
a’ = c - a,
b’ = c + b,
(A9)
c’ = b - a.
In terms of Fig. 18,a’points from A to C, b’ points from A to
D, and c’points from A to B. This is a right-hand coordinate
system analogousto the one in Fig. 3 and is suited for a
calculation of the (111) surface energy. Just as there the
three vectors define a tetrahedron, here they defineABCD.
The new parallelepipedunit cell has a volume that is twice
that of the old one,so it is unsurprising that when we periodically replicate the atomic positions in the old one to see
which copiesfall in the new one we find that 1000copiesdo.
At this point on it is probably better to refer to Fig. 3, where
a, b, and c can play the role of a’, b’, and c’.
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7394
C. L. Cleveland and U. Landman: Structure of nickel clusters
These1000atoms are arrangedas ten ( 111) layers parallel to the sidesof the computational cell, whether we scan
through the system along the a’, the b’, or the c’direction.
Unfortunately, the stacking sequencefor ( 111) layers in an
fee crystal only repeatsevery third layer, which meansthe
new systemwill contain seriousstacking faults if periodically repeated.Fortunately this is easily repaired. We simply
reduce the three new translation vectors by 10% and only
keep the copiesof atoms which fall in this newer, smaller
cell. We find that 93 do. Replicating these 729 remaining
atoms in this newest unit cell with its periodic translation
vectorsproducesexactly the sameatomic coordinatesas replicating the original 500 atoms with the original, relaxed
translation vectors: they are equivalent periodic systems.
Thus we have constructed a system with decahedra1strain
with periodic translation vectors suitable for determining
energiesinvolving ( 111) planes.
In terms of Fig. 3, the two planesOACand OBCmeet at
the line OC with a dihedral angle of 72”. Thus these two
planes have the proper orientation for either a twinning
plane or a notching ( 111) face. The other two faces of the
tetrahedronOAB and ABC have the orientation of capping
( 111) faces.Thus by removing the periodic boundarycondition in the c’direction, we can determinethe surfaceenergy
of the capping ( 111) facesby minimizing the energyof the
resulting slabwhile holding the computational cell fixed and
comparing the resulting energy with that of the bulk slab
under decahedralstrain.
Similarly, by removing the periodic
boundarycondition in the b’direction, we can determinethe
surfaceenergyof the notching ( 111) face. Leaving both periodic boundary conditions in place, we can introduce two
twinning planesin the original bulk system. Look at the system as a set of nine atomic ( 111) planesparallel to the a’-c’
plane. Thesehave a stacking sequenceabcabcabc, counting
away from the al-c’ plane. Let us take the last four of these
and rotate them 180”about their normals and put them back
so that stackingsequenceis abcabcbacabc..., wherewe have
periodically extendedthe sequenceand indicated the twinning planeswith capital letters for clarity. Thesetwo twinning planesare quite close together,but the twinning energy
will turn out to be so small that there is little point in being
more careful. We can now reminimize the bulk energyin the
presenceof these twinning planes, and by comparing the
resulting minimum energy with the energyof the bulk system under decahedra1strain we can determinethe twinning
energyappropriate to the decahedron.
The numerical results for nickel, obtainedby the above
method, may be found in Table III.
APPENDIX B: ENERGY OF IDEAL DECAHEDRA
In this section,we will walk through a calculation using
the modified Wulff construction of the energy of a Marks’
decahedrongiven the energy results of the previous appendix. However, insteadof carrying out the construction for an
unstrainedsinglecrystal, it is appropriateto carry it out for a
crystal under uniform decahedra1strain. Thus the normals
to the surfaceswill not quite point in the samedirections as
they would were no strain applied, and the faceswill be of
different sizes.Becausethe bulk strain energyis so small, this
can make a very large differencein our estimate of the size
theseclusters must be before single crystal clusters or monotwins of the samesize are energeticallypreferable.
We will neglect ( 110) faceting, for reasonsdiscussedin
the sectionon macroscopiccalculations.All distancesin this
sectionwill be in units of UN,introduced in that samesection.
From the computationalcell which gavethe lowest bulk
energyfor the given constraints, we know the normals to the
( 100) plane, as well as the ( 111) capping plane and the
( 111) twinning and notching planes.From the surfaceenergieswe found, we know how far eachsurfaceplaneshould be
from the origin. Thus if n is the unit vector normal to a
surface whose surface energy is y, a point r can be on the
planeif it satisfiesn-r = y. If we haveanother plane adjoined
to this one at an edge,the equationsof the edgefollow from
the requirement that the equationsof both planes must be
simultaneouslysatisfied,and a vertex where two edgesmeet
is at a point where the two sets of equationsdescribing the
edgeshold simultaneously.Thus the determination of all the
coordinatesof all the vertices is a straightforward exercise.
In the caseof the decahedron,we only have three different
vertices to find: a vertex of the ( 100) face; the vertex where
the capping facesmeet; and where a twinning plane, notching face, and capping face all meet.
Let us take a coordinate system with the origin at the
center of the polyhedron and where the z axis runs along
what will be the axis of the decahedron(we simply take the
twinning energy to be zero for the purposesof performing
the modified Wulff construction). We take the x axis to be in
the ( 100) direction and thus passingthrough the center of a
( 100) face. We place the y axis so that a right-handed orthogonal coordinate system is created.
Looking at the (100) face whose outward normal is in
the +x direction, we find that one of its vertices is at
(0.1029, 0.0417, 0.0421), the other four being given by
changingthe sign on one or both of they and z components.
Notice that the height of the (100) face exceedsits width.
This is not much affectedby the very small energy differencesbetweenthe capping and notching ( 111) faces, but
arisesbecausethe notching planesare more parallel to the
( 100) planeunder decahedra1
strain than the cappingplanes
are and thereforecut more deeplyinto it. It is interesting that
stretching the crystal out laterally to make it fit the decahedron makes its (100) face narrower rather than wider.
The vertex where the (upper) capping faces meet is
found at (0, 0, 0.1165). The lower capping faces meet at a
point just like the upperonewith the sign on the z coordinate
changed.
A vertex at the top of the notch, where capping, twinning, and notching ( 111) planesall intersect, is at (0.0801,
0.5821,0.0586). Others of its kind may be generatedby reversing one or both of the signs on they and z coordinates.
And of course, for each vertex on the decahedron,there is
another that can be found by rotating the decahedronabout
the z axis by a 72”angle.
With explicit coordinatesfor all the vertices, we can certainly calculate the areasof each type of face with a little
planegeometry,as well as the volume of the decahedron.On
J. Chem. Phys., Vol. 94, No. 11,l June 1991
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C. L. Cleveland and U. Landman: Structure of nickel clusters
the way we discover that the width and height of the ( 100)
face are 0.0833 and 0.0842, respectively,while the height of
the notch at its center is 0.1172 and the width of one of the
notch’s faces is 0.0282. The width of the capping face at its
widest point is 0.1164, slightly smaller that the height of the
notch as we should expect by now, and the length of the
decahedron’saxis is 0.2330. While the notching plane normal makes an angle with the ( 100) plane normal of 54”as it
must, the capping plane normal makes an angle of 54.13”
with it. For reference,in the unstrained crystal, both of these
angleswould be 54.74”,and in an unrelaxed classic decahedron, the capping plane’s angle with the (100) direction
would be 52.62”.
Equating the known bulk density under strain and
equating it to A’, the number of particles that may be in the
decahedron,divided by the decahedron’svolume in units of
u,,, cubed, allows us once more to find a connection between
uN and N, specifically uN = 13.1607NI” A.
The area in units of U; of a single notch face is
0.002 837, so in all, the notch surface area is 0.028 37. The
area of a single capping face is 0.008 564, so all the capping
faceshave a combined area of 0.085 64. The area of a ( 100)
face is 0.007 0 18 for a combined total ( 100) area of 0.035 09
and the area of a twinning plane is 0.017 34 for a total twinning plane surface area of 0.1734.92
Combining the bulk energy for N particles, including
strain energy, together with the surface areasand energiesof
the ten capping faces, ten notching faces, ten twinning
planes, and five ( 100) faces, we finally find the energy of a
decahedronof N particles in electron volts.
E,(N)
= (cB + t$‘)N
+ 2.493 87N2”,
(Bl)
where the uniform strain energy per atom of the decahedron
<= 1.124426x10-‘eV.
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AIP
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copyright, see http://jcp.aip.org/jcp/copyright.jsp
7396
C. L. Cleveland and U. Landman: Structure of nickel clusters
‘a Routine ZXCGR in the IMSL Fortran Subroutines for Mathematics and
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79Whenever we report properties for extendedsystems whose energieswe
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a3The octahedralsite is at the geometric canter of the conventional fee unit
cell. It is so called becauseits six nearestatomic neighborssit at the vertices of an octahedron.
*‘All perfect cuboctahedraare atom centered.
*‘These model C clusters are the sameas the onesusedby Raoult et al. (Ref.
8). Unfortunately, their description of model C doesnot produce this set
of clusters, but a similar set missing every fourth member of model C.
86HKI standsfor hexakaiicosahedrons,since they have 26 sides.
*‘A. Mackay, Acta Crystallogr. 15, 916 (1962).
a81nterpolationof cuboctahedral or icosahedral energiesproceedsin the
following way: for a given number of atoms, the cubic formula
N= 1 + (lln + 15n* + 10n3)/3forthenumberofatomsintheclusteris
inverted to obtain a nonintegral shell number. The energiesof the four
nearestclusters of that shape, two smaller and two larger, are used to
determine the four coefficientsof a cubic equation for energy usedin the
region betweenthe two nearestsuch clusters. If we want to extrapolate
energies,we simply usethe cubic equation in the closestregion for which
one could be obtained.This method can be testedby omitting a data point,
eitherinsideourdatarangeorat itsends,and thencheckinghow wellinterpolation or extrapolation can reproduceit. Accuracy is better than 1 meV.
The use of a cubic equation for interpolation is deliberate. If the surface
planesare taken through surfaceatoms, the shell number of a cuboctahedron or icosahedronis exactly proportional to any ofits linear dimensions,
includingits”radius”r. Itisveryreasonabletoexpecttheenergyofacuboctahedron or icosahedronto have terms varying as r ‘, r ‘, and r.
a91tremains unclear why taking surface planes betweenatomic layers insteadof through them should do better when estimating the overall energies of the clusters examined in this study, and worse when examining
relative energeticordering betweenclusters. If we focus on relative energy
differencesbetweenclusters of similar size and for that reasontake our
energiesrelative to cuboctahedra,it would appear for the clusters of this
study that the best location for a surface plane is just a little outside the
plane that passesthrough its surface’satoms.
gOThevalue of 1600atoms is given in Ref. 8 for the transition to decahedral
structures from icosahedrawith all vertex atoms removed. No transition
size is given for full icosahedra.
9’The value of 2 100atoms is given in Ref. 8 for the transition, with decahedra disallowed, to single crystal structures and their monotwins from full
icosahedra.For icosahedramissing their vertex atoms, which we havenot
consideredhere, the value reported in that referenceis 2400 atoms.
92In the modified Wulff construction, the twinning energy of a twinning
plane is divided betweenthe two decahedrathat meet at it. Thus the decahedron is treated as containing ten twinning planes, two per tetrahedral
unit.
J. Chem. Phys., Vol. 94, No. 11,l June 1991
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