Page 68 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Vibrating Uniform Lattice
Box 2.3. A Brief Note on Hamiltonian and Lagrangian Mechanics.
Hamilton’s Principle. Hamilton’s principle states (sinks) of energy in terms of generalized forces
that the variation of the action A, also called the Ξ j and the generalized displacements ξ j .
variational indicator, is a minimum over the path Generalized Displacements and Velocities. For
chosen by a mechanical system when proceeding a system, we establish the m independent scalar
from one known configuration to another degrees of freedom d ∈ ℜ m required to describe
its motion in general. Then we impose the p con-
t 2
δA = 0 (B 2.3.1)
t 1 straint equations Bd• = 0 that specify the
The Action. The action is defined in terms of the required kinematics (the admissible path in ℜ m ).
lagrangian and the generalized energy sources This reduces the number of degrees of freedom by
L
Ξ ξ of the system p, and we obtain the n = m – p generalized sca-
j j
n
n lar displacements ξ ∈ ℜ ⊃ ℜ m of the system.
t 2
A = ∫ L + ∑ Ξ ξ d t (B 2.3.2) The generalized velocities are simply the time rate
j j
t 1
j = 1 of change of the generalized displacements,
where Ξ is the generalized force and ξ the gen- ˙
⁄
j j ξ = dξ dt .
eralized displacement
Lagrange’s Equations. An immediate conse-
The Lagrangian. The mechanical Lagrangian of
quence of (B 2.3.2) is that the following equations
a system is the difference between its kinetic co- hold for the motion
energy T ∗ ξ ξ t,( ˙ , ) (the kinetic energy expressed
j j

in terms of the system’s velocities) and its poten- d ∂L   – ∂L = Ξ (B 2.3.6)

tial energy U ξ t,( ) t d ∂  ˙ ξ j  ∂ ξ j j
j
˙
L ( ξ ξ t) = T ∗ ξ ξ t,( ˙ , ) U ξ t) (B 2.3.3) These are known as the Lagrange equations of
,
(
,
,
–
M j j j j j
motion. For a continuum, we can rewrite (B 2.3.6)
In general, the lagrangian for a crystal is written in
for the Lagrange density as
terms of three contributions: the mechanical (or
d
∂L
d ∂L
∂L
matter), the electromagnetic field and the field- ------------- = -------- – ---------------------------------------- (B 2.3.7)
matter interaction dt∂x˙ j ∂x j dX ∂∂x ⁄( j ∂X )
k
k
In the continuum crystal lagrangian the variables
d
L = ∫ ( L + L + L ) V (B 2.3.4) X represent the material coordinates of the unde-
M
F
I
V
x
formed crystal; the spatial coordinates of the
1
•
•
L M = --- ρu˙ +( ε C ε) deformed crystal. The Lagrange equations are
2
(B 2.3.5) most convenient, because they allow us to add
1  A ˙ 
L = --- E +( B) L = ---- – qψ ˙ detail to the energy expressions, so as to derive the
F 2 I  c 
equations of motion thereafter in a standard way.
Generalized Energy Sources. This term groups
all external or non-conservative internal sources
• The valence electrons form a uniform cloud of negative space charge
that interacts with the atoms.
Semiconductors for Micro and Nanosystem Technology 65