What does this length represent?

[unscientific]

I was doing some calculations earlier and tried the ratio between a metal's fermi temperature $T_F$ and debye temperature $\theta_D$:
$$\frac{T_F}{\theta_D} = (6 \pi^2)^{\frac{1}{3}} \left( \frac{\lambda}{a} \right)$$

where $\lambda = \frac{\hbar}{2 m_e c}$ and lattice spacing is $a$.

I tried approximating fermi momentum $p_F \approx m_e c$ and we get $\lambda \approx \frac{1}{2k_F}$. Does this mean anything?

 

[nasu]

What is c?

 

[unscientific]

What is c?

Averaged speed of sound

 

[nasu]

Oh, then it does not make sense to associate mc with the Fermi momentum. The electrons near the Fermi sphere have much higher speeds. At least one order of magnitude if not two.

How come that electron concentration does not show up in the result? There is no free electron concentration in the Debye temperature.

 

[unscientific]

Oh, then it does not make sense to associate mc with the Fermi momentum. The electrons near the Fermi sphere have much higher speeds. At least one order of magnitude if not two.

How come that electron concentration does not show up in the result? There is no free electron concentration in the Debye temperature.

Oh I assumed it was a FCC lattice, so $n = \frac{N}{a^3} = \frac{4}{a^3}$.

 

[nasu]

N would be the number of atoms, right?
The number of free electrons is not necessarily equal to the same N. It may be, though.

 

[unscientific]

N would be the number of atoms, right?
The number of free electrons is not necessarily equal to the same N. It may be, though.

I think we can make that approximation for a mono-valent atom. I saw a similar expression in my textbook too, but they never explained what $lambda$ was which is why I'm trying to find out.

 

[unscientific]

bumpp