How Does Magnetic Field Influence Electron Spin and Metal Susceptibility?

[touqra]

Consider the degenerate electron gas in a metal as a mixture of two gases of spin-up and
spin-down electrons, respectively. when a small magnetic field B is applies, a few of the
electrons reverse their spins so as to maintain equality of the chemical potential in the two
mixed gases. For T = 0, find the magnetic susceptibility of the metal $$( \frac{\partial M}{\partial B} )_N_,_V$$ ,
where M is the magnetization (magnetic moment per unit volume), N is the total number of electrons, and V is the volume. The magnetic moment of the electron is μB.

How do I use the information of chemical potential equality to solve this problem? I am thinking that since Gibbs, $$G_{up} = \mu n_{up}$$ and the same for spin down gas, the change, $$dG = \mu dn + n d \mu = \mu dn$$. What next ?

 

[Jhero]

First of all, it is important to understand the concept of chemical potential and how it relates to the equilibrium state of a system. In this case, the chemical potential represents the energy required to add one more particle to the system while keeping the temperature and volume constant. In an equilibrium state, the chemical potential of the two mixed gases must be equal in order to maintain a stable equilibrium.

Now, let's consider the effect of a small magnetic field on this system. When a magnetic field is applied, some of the electrons in the spin-up gas will reverse their spin to align with the field, while some of the electrons in the spin-down gas will also reverse their spin to maintain chemical potential equality. This results in a net change in the number of electrons in each gas, which can be represented by the following equation:

dN = -dN_up + dN_down

where dN_up and dN_down represent the changes in the number of electrons in the spin-up and spin-down gases, respectively.

Using the fact that the magnetic moment of an electron is μB, we can write the change in energy of the system due to the magnetic field as:

dE = μB(dN_up - dN_down)

Now, let's consider the change in Gibbs free energy of the system due to the magnetic field:

dG = dE - TdS

where T is the temperature and dS is the change in entropy of the system. Since we are considering the case of T = 0, there is no change in entropy and dS = 0. Therefore, we can rewrite the above equation as:

dG = dE = μB(dN_up - dN_down)

Using the definition of chemical potential and the fact that the chemical potential must be equal in both gases, we can write:

dG = μ(dN_up - dN_down) = μ(dN_up - dN + dN - dN_down)

= μ(dN_up - dN) + μ(dN - dN_down)

= μ(dN_up - dN) + μ(dN_down - dN)

= μ(dN_up - dN) - μ(dN - dN_down)

= μ(dN_up - dN) - μ(dN_up - dN_down)

= μ(dN_up - dN) - μ(dN_up - dN)

= 0

Therefore, we can conclude that the