74
2 General Analysis of Vibrations of Monatomic Lattices
2.7
The Crystal as a Violation of Space Symmetry
The motion of a crystal lattice in which each atom vibrates around its equilibrium
position can be expanded in terms of motions of independ ent oscillators, i. e., normal
vibrations. The crystal energy (or its Hamiltonian function) is separated into the terms
corresponding to individual normal modes.
Separation of independent motions that may be superpositioned to compose any
complex motion o f a system of many particles (atoms) is known as the procedure of
introducing the collective excitations and relevant collective coordinates (or variables).
For small crystal vibrations, i. e., mechanically weakly excited states of a crystal body,
the collective excitations are represented by normal modes and collective coordinates
by normal coordinates.
The dispersion law of collective vibrations of a monatomic lattice has the universal
property: the frequencies of all three branches of vibrations vanish at k → 0.The
extremely long-wave vibrations ( k = 0, λ = ∞) are equivalent to the displacement of
a lattice as a whole and this property is a d irect result of the crystal-energy invariance
with respect to its translational motion as a whole. In proving the relation (2.1.6) we
proceeded from the fact that due to space homogeneity the internal state of a body is
independent of the position of its center of masses.
However, this property (i. e., the condition ω(k) → 0 as k → 0) o f the frequency
spectrum of crystal eigenvibrations can be explained in another way. Since the space
where a crystal exists is homogeneous, the movement from one point of free space to
another by an arbitrary vector, including an infinitely small one, is equal to the trans-
formation into an equivalent state. For this reason the energy of a system of interacting
atoms does not change for arbitrary translations of the whole system. The symmetry
connected with the Lagrangian function (or Hamiltonian function) invariance relative
to transformations of a continuous group of translations is inherent to any system of
particles.
However, in the crystal ground state the atoms form a space lattice whose symmetry
is lower than the initial one: the physical characteristics of an equilibrium crystal
are invariant under a discrete group of translations, since they are described by some
periodic functions reflecting the lattice periodicity.
When the symmetry of the ground state of a system is lower than that of the corre-
sponding Lagrangian function, the initial symmetry is broken spontaneously.
If the properties of the ground state of a system with a large number of degrees of
freedom break its symmetry with respect to transformations of a certain continuous
group then the collective excitations whose frequencies tend to zero at k → 0 arise in
the system (Goldstone, 1961). These excitations seem to strive to re-establish the bro-
ken symmetry of the system. The number of branches of such Goldston e excitations
is determined by the number of broken independent elements of a continuous symme-
try group of the Lagrangian function of the system (by the number of “disappeared”
generators of the initial symmetry group).