- #1

- 49

- 0

## Homework Statement

A chain of atoms are connected by identical springs of force constant

*k*. Suppose teh atoms of mass

*m*alternate with atoms of mass

*M*. Thus the crystal consists of a sequence ...

*MkmkMkmMkmk*... which is the periodic repetition of unit cells

*Mkmk*. The size of the unit cell is

*a*(this is the equilibrium distance between two neighboring

*m*atoms).

(a) In the

*n*

^{th}unit cell let

*x*

_{n}and

*y*

_{n}be the longitudinal displacement from equilibrium of the atoms

*m*and

*M*respectively. Set up the Lagrangian and derive the equations of motion for the x and y coordinates.

(b)Determine the dispersion relation

*w(q)*. HINT: consider traveling wave solutions of the form

[itex]x_{n} = Xe^{i(qna-wt)}[/itex] and [itex] y_{n} = Ye^{i(qna-wt)}[/itex]

(c) Find the dispersion relation in the long wavelength limit (q→0) and determine the speed of sound.## Homework Equations

[itex]x^{0}_{n+1}-x^{0}_{n} = a [/itex] where x

^{0}denotes equilibrium position.

[itex]y^{0}_{n+1}-y^{0}_{n} = a [/itex]

[itex]x^{0}_{n}-y^{0}_{n} = \frac{a}{2} [/itex]

[itex]T = \frac{1}{2} m \Sigma \dot{x}^{2}_{n} + \frac{1}{2} M \Sigma \dot{y}^{2}_{n}[/itex]

## The Attempt at a Solution

Now, here is where I run into a problem; how to express the potential energy? I started with [itex]U = \frac{1}{2} k \Sigma (y_{n+1} - y_{n})^{2} + (x_{n+1} - x_{n})^{2}[/itex]. Then I realized that this makes the problem trivial and not all that interesting. My physics intuition tells me that the x's and y's must be coupled because the potential energy should depend on the neighboring particles. I think that this gives [itex]U = \frac{1}{2} k \Sigma (x_{n}-y_{n})^{2}+(y_{n+1}-x_{n})^{2}[/itex] And the equations of motion would then be

[itex]m\ddot{x}_{n} = -\frac{\partial U}{\partial y_{n+1}} - -\frac{\partial U}{\partial y_{n}}[/itex]

[itex]M\ddot{y}_{n} = -\frac{\partial U}{\partial x_{n+1}} - -\frac{\partial U}{\partial x_{n}}[/itex]

or[itex]M\ddot{y}_{n} = -\frac{\partial U}{\partial x_{n+1}} - -\frac{\partial U}{\partial x_{n}}[/itex]

[itex] m \ddot{x}_{n} = -2kx_{n} +k(y_{n+1}+y_{n}) [/itex]

[itex] m \ddot{y}_{n} = -2ky_{n} +k(x_{n+1}+x_{n}) [/itex]

[itex] m \ddot{y}_{n} = -2ky_{n} +k(x_{n+1}+x_{n}) [/itex]

I just want to know if I have over-thought things, or if I am on the right track with the second potential. Thanks for any suggestions!