Vibration modes of a cluster
Consider a cluster or a molecule formed of an assembly of atoms bound due
to a specific potential. First, the structure must be relaxed to its ground
state, or at least a local minimum. This can be done numerically by several
local optimization methods. The most widely used methods are steepest
descent (SD), conjugate gradients (CG) and quasi-Newton (QN). Once the
local minimum is reached, one can expand the potential energy about this
minimum in terms of the powers of atomic displacements. Since all forces
on all atoms are zero, the Taylor expansion does not have linear terms. The
Harmonic approximation (HA) consists in neglecting all the powers of displacements
larger or equal to 3. Being at a local minimum, the matrix of
second derivatives must be positive definite, and thus will have only positive
eigenvalues. The potential energy thus becomes:
where u is the "small" displacement vector from the equilibrium position:
ri(t) = R0i+ u (t), and the label t= (i, α) refers to an atom i and the
cartesian component _ of its displacement. For an N atom cluster in 3D,
_ varies from 1 to 3N, and _ is a 3N×3N matrix. The latter is the second
derivative of the potential energy evaluated at the equilibrium position and
is called the force constants matrix.
Symmetries of the force constants
Apart from the invariance under permutation of atoms expressed above,
which comes from the fact that the total energy is an analytic function of
the atomic coordinates, there are other relations between different elements
of this tensor due to symmetries of the system. An important relation satisfied
by the force constants comes from the translational invariance of the
potential energy: under any arbitrary translation, the potential energy and
the forces should remain the same: E(u + c) = E(u); F(u + c) = F(u).
Substituting for the forces its harmonic expression, we find:
implying, since the displacements are arbitrary:
This relation defines the diagonal element of the force constants matrix as
a function of its non-diagonal elements meaning that,
effectively, the atom _ is bound by a spring to its equilibrium position.
Other relations come from symmetry operations, such a rotations or mirror
symmetries, elements of its point group, which leave the molecule invariant.
If such a symmetry operation is denoted by S, we must have
As is a second rank tensor, we have by definition:
where S_,_0 are the 3x3 matrix elements of the operation S. The above
relation implies that for any symmetry operation, the matrix of the force
constants must commute with that of S.
Classical theory of vibrations
Given the expression for the potential energy as a function of displacements,
it is an easy task to derive the Newtonian equations of motion:
This formula is linear in the atomic coordinates, and can be interpreted,
within the harmonic approximation, as the particles being connected by
"springs". A (harmonic) solution of the form can
be substituted in (1.3). The resulting set of linear equations in the amplitudes
e_ and frequencies ! define the vibrational modes of the system. First
to make the system of equations symmetric, one needs to make a change of
variable by setting . The resulting equations on e_ become:
This linear system has a nonzero solution for e if the determinant of the
matrix is equal to zero. Consequently, the square of
the vibrational frequencies of the cluster are the eigenvalues of the matrix
The system is 3N × 3N, in three dimensions, and has 3N
eigenvalues, six of which will be due to pure translations and rotations ,
and therefore equal to zero. All the rest are positive, reflecting the fact that
the total energy was a minimum and any deviation of the atoms from their
equilibrium position results in an increase in E For each eigenvalue there is an eigenvector of 3N components,defined by which is also called the normal mode. For this mode , thepolarization vector can be represented by N three-dimensional vectors
associated with the N atoms in the cluster, and showing the amplitude and
the direction along which the atom oscillates in that mode
Furthermore the eigenvectors e being a complete orthonormal set satisfy two
relations of orthonormality and completeness:
A general displacement of the atoms defined by the two initial conditions
on positions and velocities and the equations of motion, can be expanded
on the set of eigenvectors which form a complete orthonormal basis of the
3N-dimensional space:
where the 6N coefficients must be determined from the 6N initial
conditions on positions and velocities of the particles. The eigenvectors not
only give information about the polarization of the mode, but also allow one
to calculate mechanical properties of the system under study. They can also
be used to calculate infrared (IR) or Raman active spectra of molecules if
information on the induced charge distribution (dipole moment in the case
of IR, and polarizability in case of Raman) under that mode are available.
From the frequencies, one can deduce a criterion for mechanical stability: if
the lowest eigenvalue is small, this means the period of the oscillations for
that mode is large, and the corresponding increase of the energy is small
(since it depends on m!2e2/2), or, in other words, the mode is soft. The
softening of a mode (! ! 0) is a signature of its becoming mechanically
unstable, and thus leading to a phase transition.
Quantum theory of vibrations of a cluster
It is also possible to obtain the vibrational frequencies of a cluster by using the
quantum formalism and start from the Hamiltonian of the system. For the
sake of completeness, we will derive phonon1 frequencies from both methods.
Again, using the Harmonic approximation, we keep up to second order terms
in the potential energy of the particles, and write the Hamiltonian as follows:
The index , as before, labels the atom and its cartesian component and
goes from 1 to 3N. This is a system of 3N coupled harmonic oscillators.
To solve it, we can first make the change of variable The
corresponding conjugate momenta become , and the Hamiltonian, in terms of becomes:
1
The word phonon is actually mostly used for quanta of vibrations in crystals.
As before, one can diagonalize the matrix and write
the Hamiltonian in the basis of its eigenvectors, which are real since D is
symmetric:
Vibration modes of a crystal
The treatment of a crystal is very similar to that of a cluster except that the
number of atoms, or degrees of freedom becomes infinite. In this case, use
has to be made of symmetry properties of the crystal, namely translational
invariance, to simplify the decoupling problem. As mentioned in the case
2To consider purely vibrational states for a cluster, we need to exclude the 6 rotational
and translational modes with frequency zero, and strictly sum _ from 7 to 3N
of a cluster, in the harmonic approximation, the Hamiltonian is quadratic,
and therefore exactly solvable. This is called the non-interacting problem,
because in principle, one can reduce it to a set of uncoupled harmonic oscillators.
The interacting problem of lattice dynamics will consist in including
higher anharmonic terms in the potential energy of the lattice. Perturbation
theory is the most widely used method to treat this problem, but we will not
dicuss these methods here.
Classical theory of lattice vibrations
As in the case of a cluster, one can get rid of unequal masses by making a
firstchange of variables from the displacements u to , at the cost of
modifying the force constant matrix from Formally one
can write this change of variable as:
where pM is the matrix with atomic masses on its diagonal. Next, using
invariance of the crystal under discrete translation by vectors R, we have
, where each atom is identified by the unit cell it is
in (denoted by the translation vector R), and its label within the unit cell
Now we can use Bloch's theorem and go to new variables. The displacements
u about the equilibrium position for an atom in the cell defined by
the translation vector R can be written as
the sum over k being restricted to the first Brillouin zone (FBZ). Note that
the above definition of u_k has the periodicity of the reciprocal space: u_k =
u_k+G. Substituting this into the expansion of the potential energy in powers
of the displacements about the equilibrium positions, and truncating the sum
at the second powers of u (harmonic approximation) , we find
=
The second line was obtained using the translational invariance of the forceconstants matrix implying that it depends only on the distance between
the two cells, and the last line maybe taken as the definition of the dynamical
matrix D. Nothe that this definition of D is very similar to the definition of
the Hamiltonian matrix in the tight-binding formalism??, which contained
a sum of the neighboring cells of a short-ranged matrix. In the case of TB,
the short-ranged matrix was that of the Hamiltonian, and in the case of
lattice dynamics, it is that of the force constants which also extends to the
neighboring atoms only. Note that similar to the TB case, implying, since both have the same set of eigenvalues, that the eigenvalues
!_k are even functions of k. Note that here again, using the "Bloch" transformation,
one was able to do the uncoupling in the k space and change the
force constants matrix to the dynamical matrix which is block-diagonal in
the k space, meaning that it does not couple k to any other vector k0 within
the first Brillouin zone. A last remark concerns the correspondance between
the eigenvalues: where both have a quadratic dispersion near
k = 0, implying that the phonon dispersion at the point is linear in k for
acoustic modes. This will be discussed in more detail in the next section
where an example will illustrate better this theory.
Now that modes of different k are separated, and one can diagonalize the
hermitian matrix for each k in the FBZ:
where, similar to the case of a crystal, we have a completeness and orthonormality
constraint on the eigenvectors of D, which are now complex. They
form therefore a unitary matrix, as opposed to a orthogonal one in the case of a cluster. The two constraints can be written as:
Recall that the vector is the transpose and conjugate of , and
that changing k to −k transforms e to One can finally write the potential
energy in the new basis as:
where u_ The potential energy is now diagonal in and can be combined with the transformed kinetic energy to yield a system of 3N uncoupled one-dimentional harmonic oscillators. Alternatively, one may
write down the equation of motion for u_k(t) from the Newtonian equations
for It is easy to verify that this equation is:
After multiplying both sides by and summing over , we recover the
uncoupled set of 1D harmonic oscillator equations for each u_k with frequency
The dynamical matrix D for a crystal is the analog of the force-constants
matrix for a cluster (within a mass factor in the denominator). Their
eigenvalues are the squares of the vibrational frequencies of the system. As
we mentioned this whole treatment is, very analogous to the tight-binding
formalism. The hopping matrix is replaced by the force constants matrix and
the electronic energy levels by the square of the phonon frequencies. They
would follow the same symmetry rules if the orbitals are of p (L=1) symmetry,
otherwise the TB Hamiltonian can have different symmetry transformation
properties if (L6= 1).
The general displacement of an atom can now be written in terms of the
normal modes:
where the 6N integration constants can be found from the initial conditions.
Quantum theory of phonons in a crystal
Now we can also quantize the theory, and consider the displacement u as an
operator. In the uncoupled case, we have :
This expression can be used when computing thermal or ground state averages
of quantities such as which appear in the calculation
of structure factors etc...
Einstein model
The simplest possible model for atoms vibrating about their equilibrium position
in a cluster or solid is to assume that each atom is oscillating independent
of all the other ones with a characteristic (constant) frequency !o. This
approximation is Einstein's model, and assumes that while the atom id oscillating,
all others are effectively fixed. It ignores correlations between motion
of neighboring atoms, and is a reasonably good model for optical phonons.
In this case, the sums are reduced to a multiplicative factor, which is the
number of vibrational degrees of freedom4 (3(N − N0) in three dimensions,
where N0 is the number of unit cells in the crystal, and therefore 3N0 is the
number of acoustic branches for which the Debye model could be used):
CESAR A. HERNANDEZ E.
19.502.806
ELECTRONICA DE ESTADOS SOLIDOS
REFERENCIA BIBLIOGRAFICA:
http://physics.ucsc.edu/~keivan/CM231/phonons.pdf
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