Low frequency dielectric constant

Starting from Lorentz model:

md²x/dt²+mΓdx/dt+mω₀²x=qE

which is the equation of motion for a bond electric charge q with mass m in an oscillating field E, where ω₀ is the frequency of the Harmonic potential of the charge and Γ is the dampling parameter.

Take the oscillating form of x and E:

x=x₀e^-iωt and
E=E₀e^-iωt

one get the solution:

x=q/m/(ω₀²-ω²-iΓω).

The the dielectric constant will be (notice that P=xq):

ε=1+P/(Eε₀)=1+q²/mε₀/(ω₀²-ω²-iΓω)

For a low frequency measurement ω₀>>ω,which is valid for normal dielectric measurement because the lowest ω₀ is normally at the 10¹² Hz range.

In this case, we can ignore the high order term ω²,

ε=1+q²/mε₀/(ω₀²-iΓω).

Imaginary part:

The imaginary part is

ε₂=q²Γω/mε₀/(ω₀⁴+Γ²ω²).

Experimentally, a maximum is always observed for the temperature dependence of ε₂, here we can calculate the maximum by solving

dε₂/dT=0.

Note that the only temperature dependent parameter is Γ.

Then one gets:

Γ=ω₀²/ω at dε₂/dT=0.

Now let's look at the parameter Γ. It is a dampling parameter, which should be proportional to relaxation time τ, which satisfies the Arhenious relation:

τ=τ₀e^E_a/(k_BT)

where Ea is activation energy (or energy barrier to overcome the).

We can even define:

τ=Γ/ω₀².

Then

τ₀e^E_a/(k_BT)=1/ω should give us the temprature where the maximum ε₂ is, which is

E_a/(k_BT_max)=-log(τ₀ω).

Experimentally, fitting the the relation between T_max and ω, one gets the activation energy E_a, which is what most paper shows.

Real part:

The real part dielectric constant is

ε₁=q²ω₀²/mε₀/(ω₀⁴+Γ²ω²).

One can see that it increases monotonic with temperature simply because of the temperature dependence of Γ.