How do we construct a Hamiltonian for a classical system of particles?
The state of a a classical system at a particular time <math> t_0 </math> is typically determined by the position and momentum variables <math>x(t_0)</math> and <math>p(t_0)</math>. All physical quantities of the system can be expressed as functions of these variables. The total energy of the system is determined by its classical Hamiltonian:
<center>
<br>
<math>H(x,p,t)=\frac
+V(x,t)</math>
<br>
</center>
How do we derive the equations of motion for a classical system?
The state of a system at any time <math>t</math> can be completely determined by the equations of motion, which are known as Hamilton's equations
.
<center>
<br>
<math>\frac
=\frac
</math>
<br>
<math>\frac
=-\frac
</math>
<br>
</center>
Information below is a paraphrase of the wikipedia article "Hamiltonian mechanics"
<p>
</p>
Hamiltonian mechanics is a re-formulation of of a re-formulation of classical mechanics. A starting point is Lagrangian mechanics. The equations of motion of Lagrangian mechanics are based on generalized coordinates and velocities, written below.
<center>
<br>
<math>L(q_j, \dot
_j, t)</math>
<br>
<math>\mbox
\left{\, \dot
_j | j=1, \ldots ,N \,\right}</math>
<br>
<math>\mbox
\left{\, \dot
_j | j=1, \ldots ,N \,\right}</math>
<br>
</center>
Hamiltonian mechanics replaces generalized velocity variables with momentum variables, which are physical momenta in Cartesian coordinates. There is a corresponding conjugate momentum for each generalized momentum, and it is defined in the equation below.
<center>
<br>
<math>p_j = {\partial L \over \partial \dot
_j}</math>
<br>
</center>
Legendre transform the Lagrangian to obtain the Hamiltonian:
<center>
<br>
<math>H\left(q_j,p_j,t\right) = \sum_i \dot
_i p_i - L(q_j,\dot
_j,t)</math>
<br>
<math>
dH = \sum_i \left[\left(
\right) \mathrm
q_i + \left(
\right) \mathrm
p_i \right] + \left(
\right) dt
</math>
<br>
<math>
dH = \sum_i \left[\dot
_i\, \mathrm
p_i + p_i\, \mathrm
\dot
_i - \left(
\right) \mathrm
q_i - \left({\partial L \over \partial \dot
_i}\right) \mathrm
\dot
_i \right] - \left(
\right) dt
</math>
<br>
</center>
Substitute defintion of conjugate momenta and match coefficients to obtain Hamilton's equations.
The Hamiltonian analysis of lattice vibrations in a monoatomic lattice
The system
<p>
</p>
Ions on a lattice are defined by a lattice vector <math>x = sa</math>. Ions deviate from lattice point by distance <math>u</math>, which is smaller than the lattice constant, <math>a</math>.
<p>
</p>
The Hamiltonian
<p>
</p>
Assume that elastic energy is quadratic in displacement. Write Hamiltonian where system parameterized by coordinate <math>s</math> and displacement vector <math>u_s</math>.
<center>
<br>
<math>H = \sum_s \frac
+ \frac
K \sum_s (u_s - u_
)^2</math>
<br>
</center>
Derive equations of motion
<p>
</p>
Derive the equations of motion through the application of Hamiltonian's equations. Look at the summation above and focus on terms that contain <math>u_s</math> and <math>p_s</math>.
<center>
<br>
<math>H_s = \frac
+ \frac
K (u_s - u_
)^2 + \frac
K (u_
- u_s)^2 +...</math>
<br>
<math>F_s = K(u_
- u_s) + K(u_
- u_s)</math>
<br>
<math>M \frac
=K(u_
+ u_
-2 u_s)</math>
<br>
</center>
Substitute a solution into the above equation and obtain a difference equation
<center>
<br>
<math>u_s \propto e^{- \imath \omega t}</math>
<br>
<math>\frac
=- \omega^2 u_s</math>
<br>
<math>M \omega^2 u_s = K(u_
+ u_
-2 u_s)</math>
</center>
The form of the solution of the difference equation is below, where the lattice constant is <math>a</math> and the wavevector is <math>k</math>.
<center>
<br>
<math>u_{s \pm 1 = ue^{\imath(s\pm1)ka</math>
<br>
</center>
The solution per atom <math>s</math> is written below.
<center>
<br>
<math>u_k(s,t) = ue^
</math>
<br>
<math>-M \omega^2 e^
= K(e^
+ e^
-2 e^
)</math>
<br>
<math>-M \omega^2 = K(e^
+ e^{- \imath k a} -2 )</math>
<br>
<math>\omega^2 (k) = \frac
(1 - \cos ka)</math>
<br>
<math>\omega(k) = \sqrt{\frac
{M}} \left | \sin \frac
\right |</math>
</center>
The graphical representation of solutions -��� dispersion relations
What is the physical significance of the shape of the dispersion relations?
Look at the physical characteristics of solutions
<p>
</p>
- Unique solutions are only for those values of <math>k</math> that are restricted to the first Brillouin zone.
<center>
<br>
<math>-\frac
\le k \le \frac
</math>
</center>
<br>
<center>
<br>
<math>c = \frac
</math>
</center>
<br>
<center>
<br>
<math>v_g = \frac
</math>
<br>
<math>v_g = \sqrt{\frac
{M}} \cos \frac
</math>
<br>
</center>
- Recognize that standing waves occur where the derivative is equal to zero; namely, at wavevector = pi/a. As the wavevector approaches zero, the group velocity becomes independent of frequency.
What is the difference between optical and acoustic modes and in what types of crystals do they emerge?
When a crystal has two atoms or more per primtive basis, such as in the rocksalt structure of <math>NaCl</math> (FCC with two atom basis), diamond structure of Si (FCC with two atom basis), Ge, diamond carbon or <math>\alpha-Sn</math>, each polarization mode develops two branches, the acoustic and optical branches. If there are <math>p</math> atoms in the primitive cell, there are <math>3p</math> branches to the dispersion relations; there are <math>3</math> acoustic branches and <math>3p-3</math> optical branches. Examine limiting cases of the dispersion relations.
<p>
</p>
Case I
<p>
</p>
The long wavelength limit occurs when <math>ka \ll \pi</math>
<center>
<br>
<table cellpadding=5>
<tr>
<td>
<math>\mbox
</math>
</td>
<td>
<math>\omega^2 \approx 2 \left ( \frac
\right )</math>
</td>
<td>
<math>u_1=-u_2</math>
</td>
</tr>
<tr>
<td>
<math>\mbox
</math>
</td>
<td>
<math>\omega^2 \approx \frac
k^2a^2</math>
</td>
<td>
<math>u_1=u_2</math>
</td>
</tr>
</table>
</center>
In the long wavelength limit, the motion of every primitive cell is identical. In the acoustic mode the ions within a cell move together, and they move <math>180^\circ</math> out of phase in the optical mode.
<p>
</p>
Case II
<p>
</p>
Consider when <math>ka = \pi</math>.
<br>
<center>
<table cellpadding=5>
<tr>
<td>
<math>\mbox
</math>
</td>
<td>
<math>\omega^2 \left (k= \frac
\right ) = \frac
</math>
</td>
<td>
<math>u_1=-u_2</math>
</td>
</tr>
<tr>
<td>
<math>\mbox
</math>
</td>
<td>
<math>\omega^2 \left (k= \frac
\right ) = \frac
</math>
</td>
<td>
<math>u_1=u_2</math>
</td>
</tr>
</table>
</center>
Motion changes by <math>180^\circ</math> from cell to cell. The ions within each cell move in phase in the acoustic mode and <math>180^\circ</math> out of phase in the optical mode. Note that if the springs were identical the motion would be the same in both cases. This is why the two branches become degenerate at the edges of the zone when <math>K=G</math>
<p>
</p>
Case III
<p>
</p>
Consider when <math>K \gg G</math>.
<br>
<center>
<table cellpadding=5>
<tr>
<td>
<math>\mbox
</math>
</td>
<td>
<math>\omega^2 \approx \frac
</math>
</td>
<td>
<math>u_1=-u_2</math>
</td>
</tr>
<tr>
<td>
<math>\mbox
</math>
</td>
<td>
<math>\omega^2 \approx \frac
\left | \sin \frac
\right |</math>
</td>
<td>
<math>u_1=u_2</math>
</td>
</tr>
</table>
</center>
<p>
</p>
The optical branch is <math>k</math> independent and is approximately equal to that of the frequency of vibration of a diatomic molecule. This leads to an additional insight into the physical interpretation of the optical branch and the distinction between it and the acoustic branch. This branch is a band of frequencies which results from the broadening associated with the coupling between individual oscillators. The motion originates from the diatomic motion. In the acoustic mode the ions in the cell are moving in the same direction and thus the motion is essentially a collective motion.
How do we use Mathematica to solve systems of differential equations and plot the solutions?
How is the spring constant related to the elastic modulii?
The spring constant, which is related to the bond stiffness, can be related in a simplified way to the Elastic moduli, <math>E</math>, which is a macroscopically measured quantity. In the above harmonic analysis, elastic energy was assumed to be proportional to the displacement squared. A restoring force was proportional to the displacement and opposite to it in direction.
<center>
<br>
<math>U=\frac
u^2</math>
<br>
<math>F=-Ku</math>
<br>
</center>
The constant, <math>K</math>, is related to the bond stiffness. A force applied to a cross section of a solid stretches <math>N=\frac
</math> bonds. The stress is the force applied per unit area.
<center>
<br>
<math>u=x-x_0</math>
<br>
<math>\sigma=NK(x-x_0)</math>
<br>
<math>\sigma= \frac
\frac
</math>
<br>
<math>\epsilon = \frac
</math>
<br>
<math>E= \frac
</math>
</center>
How is the elastic modulii measured in a material?
One of the best methods for measuring Young's modulus of materials is through the velocity of sound. Below is an equation of the velocity of a longitudinal wave in the [100] direction of a cubic crystal. The term <math>C_
</math> is one of the three independent elastic stiffness coefficients of a cubic crystal.
<center>
<br>
<math>v_1=\left ( \frac
\right )^
</math>
<br>
<math>v_1=\left ( \frac{C_
}
\right )^
</math>
<br>
</center>
One typically attaches a piezoelectric crystal to one end and measures the time it takes the excitation to reach the other end.
What are the Roman and Brillouin effects?
Raman scattering involves inelastic scattering of photons by a crystal. The difference between incoming and outgoing photon energy and momentum is made up by a phonon or magnon. When the phonon mode is optical the effect is referred to as Raman scattering, and it is called Brillouin scattering when the phonon is acoustic.
How are they measured?
The following information is from wikipedia article about Raman spectroscopy.
Monochromatic light is generally produced from a laser in the visible, near infrared, or near ultraviolet range. The laser photons are shifted up or down through absorption or emission of phonons or other excitations in the system. Light is collected in a lens and directed into a monochromator. Wavelengths in a spectral window away from the laser line are dispersed into a detector.