# What does this length represent?

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1. Apr 21, 2015

### 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?

2. Apr 21, 2015

### nasu

What is c?

3. Apr 21, 2015

### unscientific

Averaged speed of sound

4. Apr 21, 2015

### 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.

5. Apr 21, 2015

### unscientific

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

6. Apr 21, 2015

### 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.

7. Apr 21, 2015

### unscientific

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.

8. Apr 30, 2015

bumpp