# Bloch Theory of electrical resistivity - Derivation

• kanonear
In summary, the conversation is about a problem with equations (9.5.14) and (9.5.17) in "Electrons and Phonons" by Ziman. The person does not understand how the first equation with a double surface integral is transformed into the second equation with only one integral over q. They believe it is not a mathematical transformation but something related to physical assumptions. Eventually, they are able to solve the problem and provide the solution, which involves using spherical coordinates and substituting values to simplify the equation.
kanonear
In "Electrons and Phonons" by Ziman which you can find here, I have a problem with equation (9.5.14) on page 360 and (9.5.17) on page 361. I don't understand how from the first equation where you have double surface integral you obtain the second equation with only one integral over q.

I don't think it's a mathematical transformation, but something with physical assumptions.

Can anybody help me with that?

Ok. I figured it out. I'll write my solution here so that it will be easy to find :).

$ρ_{L}$= $\frac{9 π \hbar}{2e^{2}mNk_{b}T}\cdot \frac{1}{R^{2}σ^{2}}\int\int \frac{(K \cdot u)^{2}(K \cdot e)^{2}ζ(K)}{(1-e^{-h\nu /k_{b}T})(e^{h\nu /k_{b}T}-1)}\frac{dσ}{\upsilon}\frac{dσ\;'}{\upsilon\;'} =$

As it is stated in the book:

$(K \cdot u)^{2}=\frac{1}{3} K^{2}$
$(K \cdot e)^{2}= K^{2}$

Of course K = k' - k , and for Normal Process we can write that K=q.

$K^{2}= 2R^{2}(1-\cos ( \theta))$
$K\;dK= R^{2}\sin ( \theta) d\theta$

In the spherical coordinates (in k space) the surface element can be written as:

$d\sigma = R^{2} \sin (\theta)d\theta d\varphi$

where R correspond to the Fermi Sphere radius.
Now we rewrite the equation for resistance using equations above:

$ρ_{L}$= $\frac{9 π \hbar}{2e^{2}mNk_{b}T}\cdot \frac{1}{R^{2}σ^{2}}\int\int\int \frac{\frac{1}{3}K^{2}K^{2}ζ(K)}{(1-e^{-h\nu /k_{b}T})(e^{h\nu /k_{b}T}-1)}\frac{R^{2} \sin (\theta)d\theta d\varphi}{\upsilon}\frac{dσ\;'}{\upsilon\;'} = \frac{9 π \hbar}{2e^{2}mNk_{b}T}\cdot \frac{1}{R^{2}σ^{2}}\int \int \int \frac{\frac{1}{3}K^{4}ζ(K)}{(1-e^{-h\nu /k_{b}T})(e^{h\nu /k_{b}T}-1)}\frac{K\;dK d\varphi}{\upsilon}\frac{dσ\;'}{\upsilon\;'}$

Now let us take $\upsilon = \upsilon \; ' = \upsilon _{F}$ and K=q :

$ρ_{L} = \frac{9 π \hbar}{2e^{2}mNk_{b}T}\cdot \frac{1}{R^{2}σ^{2}}\int \frac{\frac{1}{3}q^{5}ζ(K)\;dq }{(1-e^{-h\nu /k_{b}T})(e^{h\nu /k_{b}T}-1)}\frac{1}{\upsilon_{F}^{2}}\int d\varphi \int dσ\;'$

Integral over $d\varphi$ give us $2\pi$, and the last integral is just $\sigma$.

$ρ_{L} = \frac{9 π \hbar}{2e^{2}mNk_{b}T}\cdot \frac{1}{R^{2}σ^{2}}\cdot \frac{1}{3} \frac{1}{\upsilon_{F}^{2}}2\pi \sigma \int \frac{q^{5}ζ(q)\;dq }{(1-e^{-h\nu /k_{b}T})(e^{h\nu /k_{b}T}-1)}$

Now we can substitute $\sigma = 4\pi R^{2}$:

$ρ_{L} = \frac{3 π \hbar}{4e^{2}mNk_{b}T R^{4}\upsilon_{F}^{2}} \int \frac{q^{5}ζ(q)\;dq }{(1-e^{-h\nu /k_{b}T})(e^{h\nu /k_{b}T}-1)}$

and that is what we were looking for :).

## What is the Bloch Theory of electrical resistivity?

The Bloch Theory of electrical resistivity is a theoretical model that explains the behavior of electrical resistivity in crystalline solids. It was first proposed by Swiss physicist Felix Bloch in 1928 and has since been used to understand the electrical properties of various materials, including metals, semiconductors, and insulators.

## How is the Bloch Theory derived?

The Bloch Theory is derived from the principles of quantum mechanics and solid-state physics. It takes into account the periodicity of the crystal lattice and the wave-like behavior of electrons in a solid. The derivation involves solving the Schrödinger equation for an electron in a periodic potential, resulting in the formation of energy bands and the concept of effective mass.

## What is the role of the Fermi energy in the Bloch Theory?

The Fermi energy, also known as the Fermi level, plays a crucial role in the Bloch Theory. It is the highest energy level occupied by electrons at absolute zero temperature and determines the conductivity of a material. In the Bloch Theory, the Fermi energy is used to calculate the electrical conductivity and resistivity of a material.

## What are the assumptions made in the Bloch Theory?

The Bloch Theory makes several assumptions in order to simplify the mathematical calculations and provide a basic understanding of the behavior of electrons in a crystal. These assumptions include the neglect of electron-electron interactions, the use of a one-electron approximation, and the assumption of a perfect crystal lattice with no defects.

## How does the Bloch Theory explain the temperature dependence of electrical resistivity?

The Bloch Theory can explain the temperature dependence of electrical resistivity by considering the effects of thermal energy on the movement of electrons. As temperature increases, the thermal energy causes the electrons to scatter more frequently, leading to an increase in resistivity. This is known as the Matthiessen's rule, which states that the total resistivity is the sum of contributions from different scattering mechanisms.

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