Modeling the Driven Damped Oscillations in a Material

[luckreez]

Homework Statement

[/B]
Let us assume that neutral atoms or molecules can be modeled as harmonic oscillators in some cases. Then, the equation of the displacement between nucleus and electron cloud can be written as
$$\mu\left(\frac{d^x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x\right)=qE.$$
where $x$ is the displacement in the direction of the external electric field $E$, $\mu$ is the effective mass, $\gamma$ is the damping constant, $\omega_0$ is the oscillation frequency of the harmonic potential, and $q$ is the effective charge of the neutral atom or molecule. Assuming that E is a single frequency electric field with angular frequency $ω$, answer the following questions:

1. Obtain the amplitude of the stationary solution for the equation of the displacement.
2. Assuming that the material contains such atoms or molecules with density $n_0$ , obtain the current density.
3. Obtain the dielectric constant $\epsilon(\omega)$, which will be in complex form.
4. Obtain the real and imaginary parts of the dielectric constant. Assume that $\gamma << \omega$ and calculate approximate solutions.
5. Introducing the following constants, $\epsilon_{st} = \epsilon(\omega=0)$ and $\epsilon_\infty=\epsilon(\omega=\infty)$, draw the curves for the real and imaginary parts of the dielectric constant as functions of the frequency $\omega$.

Homework Equations

1. Tricks of finding solution to 2nd order ODE
2. Definition of current density?
3. 4. 5. Totally no idea.

The Attempt at a Solution

1. So I assume that stationary solution is the steady-state solution, which represent the particular solution of the differential equations. As the driving function is constant (not a function of the displacement) then the I choose the particular solution to be constant,
$$ x_P=\frac{qE}{\omega_0^2\mu}.$$

2. 3. 4. 5.
So what I do not understand this question is how to relate the damped oscillation model of nucleus and electron clouds to the electromagnetic variables in matter, like current density, or the dielectric constant.

I thought I could relate the current density with the driving force through drift velocity in this formula,
$$ j=nqv$$
but I cannot figure it out.

I am sorry if it seems I still did not do much, but I have spent a whole day trying to read, start from the definition of current density, complex dielectric constant in internet or EM books, but still not get the slightest idea on how to solve it.

I would be very grateful for any hints given.
Thank you in advance.

 

[ehild]

Homework Statement

[/B]
Let us assume that neutral atoms or molecules can be modeled as harmonic oscillators in some cases. Then, the equation of the displacement between nucleus and electron cloud can be written as
$$\mu\left(\frac{d^x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x\right)=qE.$$
where $x$ is the displacement in the direction of the external electric field $E$, $\mu$ is the effective mass, $\gamma$ is the damping constant, $\omega_0$ is the oscillation frequency of the harmonic potential, and $q$ is the effective charge of the neutral atom or molecule. Assuming that E is a single frequency electric field with angular frequency $ω$, answer the following questions:

1. Obtain the amplitude of the stationary solution for the equation of the displacement.
2. Assuming that the material contains such atoms or molecules with density $n_0$ , obtain the current density.
3. Obtain the dielectric constant $\epsilon(\omega)$, which will be in complex form.
4. Obtain the real and imaginary parts of the dielectric constant. Assume that $\gamma << \omega$ and calculate approximate solutions.
5. Introducing the following constants, $\epsilon_{st} = \epsilon(\omega=0)$ and $\epsilon_\infty=\epsilon(\omega=\infty)$, draw the curves for the real and imaginary parts of the dielectric constant as functions of the frequency $\omega$.

Homework Equations

1. Tricks of finding solution to 2nd order ODE
2. Definition of current density?
3. 4. 5. Totally no idea.

The Attempt at a Solution

1. So I assume that stationary solution is the steady-state solution, which represent the particular solution of the differential equations. As the driving function is constant (not a function of the displacement) then the I choose the particular solution to be constant,
$$ x_P=\frac{qE}{\omega_0^2\mu}.$$

The driving function is not constant.

Assuming that E is a single frequency electric field with angular frequency $ω$

. The driving force is eE.

 

[luckreez]

The driving function is not constant. . The driving force is eE.

Ah, sorry, thank you for pointing out. But it doesn't change the answer to the particular solution, does it? Since the driving force is still not a function of x.

 

[ehild]

Ah, sorry, thank you for pointing out. But it doesn't change the answer to the particular solution, does it? Since the driving force is still not a function of x.

t (time) is the independent variable, and you need to find the displacement x as function of t. The driving force is a harmonic function of t.

 

[luckreez]

t (time) is the independent variable, and you need to find the displacement x as function of t. The driving force is a harmonic function of t.

Thank you, I see my mistake, so here's my new answer.
So I try to write E as, $E=E_0e^{i\omega_0t}$, and assume the particular solution have this form $x_p=Ce^{i\omega_0t}$. By putting it into the differential equation, I get
$$ C=\frac{qE_0}{i\omega_0\mu} $$
which is the amplitude of the stationary solution as the question asked, is it?

 

[ehild]

Thank you, I see my mistake, so here's my new answer.
So I try to write E as, $E=E_0e^{i\omega_0t}$,

The angular frequency of the electric field is given as ω, so the electric field is E=E₀e^iωt. ω₀ is a parameter characterizing the oscillator. The amplitude of the stationary solution will depend how ω is related to ω₀.

 

[luckreez]

The angular frequency of the electric field is given as ω, so the electric field is E=E₀e^iωt. ω₀ is a parameter characterizing the oscillator. The amplitude of the stationary solution will depend how ω is related to ω₀.

OK, so then $E=E_0e^{i\omega t}$. Let the solution $x_p=Ce^{i\omega t}$, then substitute to the diff. equation I get,
$$ C\left(\omega_0^2+i\gamma\omega-\omega^2\right)=\frac{qE_0}{\mu} $$
or
$$ C= \frac{qE_0}{\mu\left(\omega_0^2+i\gamma\omega-\omega^2\right)} $$,
which is the amplitude that depend on how $\omega$ related to $\omega_0$. Is it like that?

 

[ehild]

OK, so then $E=E_0e^{i\omega t}$. Let the solution $x_p=Ce^{i\omega t}$, then substitute to the diff. equation I get,
$$ C\left(\omega_0^2+i\gamma\omega-\omega^2\right)=\frac{qE_0}{\mu} $$
or
$$ C= \frac{qE_0}{\mu\left(\omega_0^2+i\gamma\omega-\omega^2\right)} $$,
which is the amplitude that depend on how $\omega$ related to $\omega_0$. Is it like that?

Yes, it is the complex amplitude, that can be written in the form of A e(iΦ). A is the real amplitude and Φ is the phase difference between the driving signal and the oscillation of the particle. You can determine both A and Φ. You will see that the amplitude A is maximum, when the driving frequency is near to ω₀,the "natural frequency" of the molecule.