ISSN 00201685, Inorganic Materials, 2011, Vol. 47, No. 9, pp. 952–956. © Pleiades Publishing, Ltd., 2011.
Original Russian Text © A.A. Ashcheulov, O.N. Manyk, T.O. Manyk, S.F. Marenkin, V.R. BilynskiySlotylo, 2011, published in Neorganicheskie Materialy, 2011, Vol. 47, No. 9,
pp. 1052–1055.
Chemical Bonding in Cadmium
A. A. Ashcheulova, O. N. Manyka, T. O. Manyka, S. F. Marenkinb, and V. R. BilynskiySlotyloa
a
b
Fed’kovich State University, ul. Kotsyubinskogo 2, Chernivtsi, 58012 Ukraine
Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences,
Leninskii pr. 31, Moscow, 119991 Russia
email: [email protected]
Received October 27, 2009
Abstract—The theory of elasticity and lattice dynamics are used to analyze the key features of chemical bond
formation in cadmium. The results increase the possibilities of finding appropriate technological solutions.
DOI: 10.1134/S0020168511090019
INTRODUCTION
Modern science and technology impose ever more
stringent requirements on cadmiumbased materials
for various electronic and optical applications [1–3],
which extend the application field of both cadmium
and cadmiumbased compounds.
Cadmium crystallizes with the hexagonal close
packed structure, which contains helical chains: each
atom has two nearest neighbors within its chain and
four more distant neighbors in adjacent chains. Within
the chains, the atoms are linked by covalent bonds.
The bonding between the chains is metallic [4].
This qualitative picture of chemical bonding in
cadmium accounts for some of its physical properties.
The ability to create novel materials is highly depen
dent on knowledge of technological parameters, with
allowance for the structure of chemical bonding.
The purpose of this work is to examine in detail the
chemical bonding in cadmium using mathematical
modeling.
2.99 Å. The coordination number is K = 12. Cadmium
has 8 stable and 12 radioactive isotopes.
The fact that there are different interatomic dis
tances determines the crystal structure, physicochem
ical properties, polymorphic transformations, and
phase transitions of cadmium and leads us to conclude
that the hexagonal closepacked structure of this ele
ment is due to the overlap of the outer, spherical
sshells, stabilized by d orbitals [4].
To adequately take into account all of the above
factors, we solved the inverse problem: using experi
mentally determined lattice parameters and inter
atomic distances [5, 6], we evaluated the atomic posi
tion coordinates in the unit cell of cadmium. The cal
culation results are presented in Table 1, and the atom
numbering scheme is shown in Fig. 1. The numerical
values correspond to the Cartesian coordinate system
with its origin at the center of the unit cell.
From these results, we inferred and assumed that
there are inequivalent interactions between the cad
mium atoms, which can be divided into five groups
according to the interatomic distance:
CHEMICAL BONDING IN CADMIUM
ϕ1 (R01 = 2.979 Å);
The lattice parameters of cadmium are а = 2.98 Å
and с = 5.62 Å, with с/а = 1.88 [5, 6]. The shortest
interatomic distances within the chains are 3.69 and
ϕ2 (R1''1 = 2.9845 Å);
ϕ3 (R01'' = 2.9899 Å);
Table 1. Atomic position coordinates in the unit cell of cadmium
Cd
X
Y
Z
Cd
x
y
z
1
2
3
4
5
6
0
2.5799
2.5799
0
–2.5799
–2.5799
2.979
1.4895
–1.4895
–2.979
–1.4895
1.4895
0
0
0
0
0
0
1'
2'
3'
1''
2''
3''
1.0656
1.0656
–2.1311
–0.8631
1.7262
–0.8631
1.8456
–1.8456
0
1.4949
0
–1.4949
3.0139
3.0139
3.0139
–2.4413
–2.4413
–2.4413
952
CHEMICAL BONDING IN CADMIUM
5
953
6
3'
1''
3''
4
0
Y
1
2'
1'
2''
3
2
X
Fig. 1. Unit cell projected onto the XY plane (viewed along the С6 axis): (0–6) atoms in the XY plane, spaced one lattice constant
(a = 2.979 Å) apart; (1 '−3 ') atoms above the XY plane (z = 3.0139 Å distance R ' = 3.6912 Å); (1 ''− 3 '') atoms below the XY plane
(z = –2.44125 Å, distance R '' = 2.9899 Å from the central atom).
ϕ4 (R1'1 = 3.3917 Å);
time t. The deformation of the continuum can then be
described by the symmetric strain tensor
ϕ5 (R01' = 3.6912 Å).
This information can be used to examine the force
constants and energy characteristics of cadmium and
estimate characteristic temperatures of the inequiva
lent chemical bonds corresponding to the different
interatomic distances.
FORCE AND ENERGY
PARAMETERS OF CADMIUM
In studies of solidstate transformations, structural
changes of elements heated to above their critical tem
perature or cooled to subcritical temperatures are of
special interest.
The various concepts used to analyze such pro
cesses are semiphenomenological [7]. The develop
ment of microscopic theory has highlighted the neces
sity to supplement available theoretical approaches by
quantitative calculations using mathematical models
that take into account the detailed structure and
chemical bonding of the materials of interest.
The first step in such an approach is to establish
relationships between the theory of elasticity [8] and
lattice theory [9] by comparing the corresponding
equations of motion. The theory of elasticity treats a
crystal as a continuum of constant density ρ in an
undistorted, equilibrium state. Its state can be
described by a vector field, S(k, t), which specifies the
displacement of a point (equilibrium coordinate R) at
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(1)
ε mn = 1 ( S m n + S n m ) ,
2
where Sm/n = ∂Sm/∂xn characterizes the change in the
displacement vector of component xn. According to
Hooke’s law, stress and strain are related by a fourth
order tensor:
σ ik =
∑C
ik,mnε mn,
(2)
m,n
where σik is the stress tensor and Сik, mn is the elasticity
tensor.
When the forces of inertia are equal to the forces
produced by stress, the equation of motion in the the
ory of elasticity has the form
ρ Si =
∑C
ik, mnS m nk .
(3)
This equation can be compared to the equation of
motion in lattice theory [9]:
∑Φ
MSim = −
mn n
ik S i ,
(4)
m,k
2
where Φ ikmn = ∂mΦ n
is the force acting on atom m
∂xi ∂x k R = R0
in direction i when atom n is displaced distance S in
direction k, whereas all of the other atoms are at equi
librium: R = R0.
954
ASHCHEULOV et al.
(a)
z
2 ⁄ 3 × 2D
R2
y
D
0
x
(c)
(b)
D
D⁄2 3
D⁄ 3
y
D⁄ 3
D⁄2 3
x
D⁄2 3
D⁄ 3
Fig. 2. Hexagonal close packing: (a) lattice structure; (b) coordination of an atom in sublattice 1 (the atoms of sublattice 2 reside
a distance 2 3D above and below); (c) coordination of an atom in sublattice 2 (the atoms of sublattice 1 reside a distance 2 3D
above and below).
Replacing Sim in (4) by a displacement field, Si (R,
t) [9],
(5)
S im = S Am, t
( A is a matrix that specifies the position of a nucleus in
the lattice), expanding S kn into a power series about
point R = Am ), and introducing density ρ = М/Vz (М
is the average mass and Vz is the unitcell volume), we
obtain
(
ρ Si =
∑C
)
(6)
ik,mnS k mn,
kmn
where
C ik,mn = − 1
2V z
∑ Φ ( Ah) ( Ah)
h
ik
m
(7)
.
n
)
(
Cik,mn =
C33 : C11 : C12 : C13 : C44 = 4 : 3.625 : 1.375 : 1 : 1. (10)
Real cadmium has a different ratio of these con
stants,
C33 : C11 : C12 : C13 : C44 = 2.52 : 5.67 : 1.96 : 1.99 : 1,(11)
which confirms that Cd has a double hexagonal close
packed crystal structure.
In describing the elastic properties of cadmium, we
employed the springbond approximation, which
means that vibrations along interatomic bonds are
independent and can be characterized by the elastic
constant f (ᐉ), where 1 ≤ ᐉ ≤ 5.
Relation (9) can be modified to the form
h
In the case of hexagonal packing and under
assumption that there are spring bonds only between
nearest neighbors, force matrices have the form
h
f Φ ikmν = − 2 Ah + Rμ − Rν Ah + Rμ − Rν , (8)
i
k
D
for Ah + Rμ − Rν = D , where D is the nearest neighbor
distance (Fig. 2). Therefore,
(
where хi are projections of interatomic distances onto
XYZ axes (1 ≤ i, k, m, n ≤ 3) [9]. According to Leibfried
[9], the anisotropy of elastic constants in this ideal
model is
)(
∑
f
[ xi x k x m x n ],
2V z D 2 h,μ,ν
)
(9)
Cik,mn = 1
2V z
∑
()
Φ ik ( Ahk ) m ( Ahk ) n,
(h)
(12)
where the superscript l refers to a particular type of
inequivalent bond, and hk characterizes the transfor
mation of atomic bonds within a given family by sym
metry elements.
In this approximation, vibrations along atomic
bonds are determined by the corresponding elastic
constant f (ᐉ). The values of () Φ (ikh) are related to f (ᐉ) by
()
Φ (ikh) = f ()
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()
α (ikh) ,
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2011
CHEMICAL BONDING IN CADMIUM
where () α (ikh) = cos ϕ hi cos ϕ hk is the product of the
direction cosines of bonds ᐉ with the xi and xk (1 ≤ i,
k ≤ 3) axes of atoms in the unit cell [10].
Setting up equations of the type (12), which char
acterize the relationship between the force matrices
and elastic constants of Cd, and solving this system of
equations for f (ᐉ), we found analytical expressions for
and numerical values of the force constants of micro
scopic theory, f (ᐉ). The elastic constants of Cd needed
for f (ᐉ) calculation can be found in Drits [6].
Analytical expressions for force constants have the
form
(1) −1
⎧ f a = −1.904C11 + 6.635C12 − 0.8097C13
⎪
⎪− 4.227C33 + 1.671C44,
⎪ f (2)a −1 = −0.1487C11 + 4.3599C12 − 0.5345C13
⎪
⎪− 0.3607C33 + 1.0704C44,
⎪⎪ f (3)a −1 = 2.142C − 6.3072C + 0.4506C
11
12
13
(14)
⎨
⎪+ 0.7378C33 − 9.0242C44,
⎪ f (4)a −1 = 0.4774C − 3.0151C + 1.9502C
11
12
13
⎪
⎪− 0.8417C33 − 3.8713C44,
⎪ (5) −1
⎪ f a = 7.0311C11 − 20.5037C12 − 4.102C13
⎪⎩+ 0.86283C33 + 8.21499C44,
where а is the lattice constant.
These results were used to evaluate energy parame
ters of Cd. In our case, these are the characteristic fre
quencies of atomic vibrations in individual chemical
bonds.
955
Table 2. Characteristic frequencies (ωᐉ) and temperatures
(Tᐉ)
(K
(K
ϕᐉ
ωᐉ × 1012, Hz
Tᐉ, K
ϕ1
ϕ2
ϕ3
ϕ4
ϕ5
42.26
34.5
23.19
18.99
16.43
594
521.8
416
378
353
)(
ω )( K
) (
ω ) ... ( K
11
− T11ω2 K 12 − T12ω2 ... K 1n − T1nω2
21
− T21
n1
− Tn1ω
2
22
− T22
2
2n
− T2nω2
........
(K
2
)( K
n2
) (
− Tn2ω2 ... K nn − Tnnω2
x j = x j0 e iω t ,
(15)
where ω is the circular frequency (conventional fre
quency, ν (the number of vibrations per unit time)
times 2π), we obtain
n
∑(Kᐉ
j =1
j
)
(16)
∑A T
i j
i
⎧1 for i = j
= δ ij ⎨
⎩0 for i ≠ j.
(18)
With
Dij =
∑A K
i
j .
(19)
i
system (16) can be written in diagonal form. Determi
nant (17) then takes the form
(20)
Thus, calculation of characteristic frequencies
reduces to finding the interaction coefficients of the
dynamic matrix Dij. It follows from (19) that, to this
end, one should find the kinematic (Aiᐉ) and dynamic
(Kᐉj) coefficients at all possible values of i, ᐉ, and j. The
calculation procedure is similar to that in the case of
orthorhombic crystals [10].
The force coefficients Кᐉj were also determined in
molecular models by solving the inverse problem of
the theory of elasticity [10]. According to (14), the
elastic constants f (ᐉ) of bonds of the corresponding
inequivalent orbitals of Cd are
f (1) = –5.1 × 104 d/сm;
f (3) = –3.9 × 104 d/сm;
f (4) =
Here, Тᐉj are constants dependent on the mass and
geometry of the equilibrium configuration, and Kᐉj are
potential energy constants, which characterize the
force structure of the system under consideration.
Equation (16) has a solution when
INORGANIC MATERIALS
)
f (2) = 4.1 × 104 d/сm;
− Tᐉ j ω = 0, (j = 1, 2 … n).
2
) = 0. (17)
Formula (17) can be reduced to a more convenient
form if ω2 appears only in the diagonal elements of the
determinant. To this end, we multiply (16) by coeffi
cients meeting the conditions
Dij − δ ij ω2 = 0.
CALCULATION OF CHARACTERISTIC
FREQUENCIES
Atomic vibration frequencies in cadmium crystals
can be found using Lagrange equations [10]. Repre
senting the potential and kinetic energies in the gener
alized coordinates xi (i = 1, 2 … n), which mean a devi
ation of the system of n particles under consideration
from equilibrium, and substituting a vibrational solu
tion in the form
)
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–3.2 ×
104
(21)
d/сm;
f (5) = 1.2 × 104 d/сm.
The results were used to determine the characteris
tic frequencies ω and the temperatures Тᐉ correspond
ing to the inequivalent interatomic distances (Table 2).
956
ASHCHEULOV et al.
The above approach allows one to solve a number
of technological problems pertaining to polymorphic
transformations and the “fine” structure of Cd melt
ing and crystallization.
Using a model of lattice vibrations with a single
characteristic frequency, νE, Lindemann obtained the
following expression for the melting temperature:
Tvibr = cν E2V n2 3M ,
(22)
which relates the characteristic vibration frequency of
a crystal to its vibrational melting temperature (Tvibr),
molar volume (Vn), and atomic mass (M), employed
by us to find “effective” characteristic temperatures of
inequivalent chemical bonds.
Given that cadmium embrittles at 353 K, that is, its
properties change, the constant с was determined with
allowance for the polymorphic transformations of Cd
in the temperature range 353–594 K. The numerical
values of the characteristic temperatures of inequiva
lent chemical bonds are listed in Table 2.
Analysis of these data indicates that cadmium
undergoes polymorphic transformations not only at
Т5 = 353 K but also at higher characteristic tempera
tures (Т1–Т4), with accompanying chemical changes
in the interatomic interaction of cadmium, which
determines both the physical properties of cadmium
and the properties of cadmium compounds.
CONCLUSIONS
We have carried out a detailed study of the structure
of chemical bonding in cadmium.
Five distinct interatomic distances have been iden
tified in the unit cell of cadmium, which determine
inequivalent interactions between cadmium atoms.
The proposed mathematical model made it possi
ble to determine the main characteristics of cadmium:
force constants of microscopic theory, characteristic
frequencies, and the corresponding characteristic
temperatures. This points to a complex structure of
chemical bonding, whose features should be taken
into account upon melting and crystallization and also
in determining the boundaries of the polymorphic
transformations of cadmium, leading to a real possi
bility of finding necessary technological solutions for
creating advanced materials with tailored properties.
REFERENCES
1. Ashcheulov, A.A., Gutsul, I.V., Manyk, O.N., et al.,
CdSb, ZnSb, and CdхZn1 – хSb LowSymmetry Crys
tals: Chemical Bonding and Technological Aspects,
Inorg. Mater., 2010, vol. 46, no. 6, pp. 574–580.
2. Ashcheulov, A.A., Voronka, N.K., Marenkin, S.F., et al.,
Preparation and Analysis of HighPurity Cadmium, in
Materialovedenie soedinenii gruppy A2B5 (II–V Materials
Research), Chernovtsy: ChNU, 1990.
3. Baranskii, P.I., Klochkov, V.P., and Potykevich, I.V.,
Poluprovodnikovaya elektronika: Spravochnik (Semi
conductor Electronics: A Handbook), Kiev: Naukova
Dumka, 1975.
4. Grigorovich, V.K., Metallicheskaya svyaz' i struktura
metallov (Metallic Bonding and Structure of Metals),
Moscow: Nauka, 1988.
5. Gorelik, S.S. and Dashevskii, M.Ya., Materialovedenie
poluprovodnikov i dielektrikov (Semiconducting and
Dielectric Materials Research), Moscow: MISIS,
2003.
6. Drits, N.E., Svoistva elementov (Properties of Ele
ments), Moscow: Metallurgiya, 1985.
7. Ubbelohde, A.R., The Molten State of Matter, Chiches
ter: Wiley, 1978.
8. Landau, L.D. and Lifshits, E.M., Teoriya uprugosti
(Theory of Elasticity), Moscow: Nauka, 1965.
9. Leibfried, G., in Handbuch der Physik, Flugge, S., Ed.,
Berlin: Springer, 1955, vol. 7, part 1, p. 104. 10. Manyk,
O.M., Bagatofaktornyi pidkhid v teoretychnomu materi
aloznavstvi (Multifactorial Approach in Materials The
ory), Chernivtsi: Prut, 1999.
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