96
5 Mechanical Properties
Fig. 5.1 Schematic linear
chain models with a same
masses and spring
constants, b different
masses and same spring
constants, c same masses
and different spring
constants, d different
masses and different spring
constants. Mirror operation
X (S) exchanges the A–B
atoms in the dimer (the
spring constants)
5.2.1 Monoatomic Linear Chain
The essential physics of lattice vibrations can best be seen from a one-dimensional model that is called
the linear chain model. The mechanical vibrations will also be called phonons, although technically this
term is reserved for the quantized lattice vibrations resulting from the quantum-mechanical treatment.
In the monoatomic linear chain the atoms of mass M are positioned along a line (x-axis) with a
period (lattice constant) a at the positions x n 0 = na. This represents a one-dimensional Bravais lattice.
The Brillouin zone of this system is [−π/a, π/a].
The atoms will interact with a harmonic potential, i.e. the energy is proportional to the displacement
u n = x n − x n 0 to the second power. This can be thought of as if the atoms are coupled by massless
springs (Fig. 5.1a). The total (mechanical) energy of the system is then:
U =
1
2
C
n
(u n − u n+1 )
2
.
(5.1)
The model assumes that the mass points are connected via massless, ideal springs with a spring constant
C. If φ(x) is the interaction energy between two atoms, C is given by C = φ
(a). Again, the harmonic
approximation is only valid for small displacements, i.e. u n a. The displacement of the atoms can
be along the chain (longitudinal wave) or perpendicular to the chain (transverse wave), see Fig. 5.2.
We note that for these two types of waves the elastic constant C must not be the same.
When the sum in (5.1) has a finite number of terms (n = 0, . . . , N − 1), the boundary conditions
have to be considered. There are typically two possibilities: The boundary atoms are fixed, i.e. u 0 =
u N −1 = 0, the boundary conditions are periodic (Born–von Karman), i.e. u i = u N +i . If N 1, the
boundary conditions play no significant role anyway, thus those with the greatest ease for subsequent
math are chosen. In solid-state physics typically periodic boundary conditions are used. Boundary
phenomena, such as at surfaces, are then treated separately (see Sect. 11.6.1).
The equations of motion derived from (5.1) are
M ¨
u n = F n = −
∂U
∂u n
= −C (2u n − u n−1 − u n+1 ) .
(5.2)
We solve for solutions that are periodic in time (harmonic waves), i.e. u n (x, t) = u n exp(−i ωt). Then
the time derivative can be executed immediately as ¨
u n = −ω
2 u n and we obtain:
5 Mechanical Properties
Fig. 5.1 Schematic linear
chain models with a same
masses and spring
constants, b different
masses and same spring
constants, c same masses
and different spring
constants, d different
masses and different spring
constants. Mirror operation
X (S) exchanges the A–B
atoms in the dimer (the
spring constants)
5.2.1 Monoatomic Linear Chain
The essential physics of lattice vibrations can best be seen from a one-dimensional model that is called
the linear chain model. The mechanical vibrations will also be called phonons, although technically this
term is reserved for the quantized lattice vibrations resulting from the quantum-mechanical treatment.
In the monoatomic linear chain the atoms of mass M are positioned along a line (x-axis) with a
period (lattice constant) a at the positions x n 0 = na. This represents a one-dimensional Bravais lattice.
The Brillouin zone of this system is [−π/a, π/a].
The atoms will interact with a harmonic potential, i.e. the energy is proportional to the displacement
u n = x n − x n 0 to the second power. This can be thought of as if the atoms are coupled by massless
springs (Fig. 5.1a). The total (mechanical) energy of the system is then:
U =
1
2
C
n
(u n − u n+1 )
2
.
(5.1)
The model assumes that the mass points are connected via massless, ideal springs with a spring constant
C. If φ(x) is the interaction energy between two atoms, C is given by C = φ
(a). Again, the harmonic
approximation is only valid for small displacements, i.e. u n a. The displacement of the atoms can
be along the chain (longitudinal wave) or perpendicular to the chain (transverse wave), see Fig. 5.2.
We note that for these two types of waves the elastic constant C must not be the same.
When the sum in (5.1) has a finite number of terms (n = 0, . . . , N − 1), the boundary conditions
have to be considered. There are typically two possibilities: The boundary atoms are fixed, i.e. u 0 =
u N −1 = 0, the boundary conditions are periodic (Born–von Karman), i.e. u i = u N +i . If N 1, the
boundary conditions play no significant role anyway, thus those with the greatest ease for subsequent
math are chosen. In solid-state physics typically periodic boundary conditions are used. Boundary
phenomena, such as at surfaces, are then treated separately (see Sect. 11.6.1).
The equations of motion derived from (5.1) are
M ¨
u n = F n = −
∂U
∂u n
= −C (2u n − u n−1 − u n+1 ) .
(5.2)
We solve for solutions that are periodic in time (harmonic waves), i.e. u n (x, t) = u n exp(−i ωt). Then
the time derivative can be executed immediately as ¨
u n = −ω
2 u n and we obtain: