# Fermi energy for two spin states equal in equilibrium?

• lampCable
In summary, the conversation discusses the concept of conduction electrons in a magnetic field and their spin states. The textbook claims that in thermal equilibrium, the Fermi energy for both spin states must be equal, which is explained by the system being in its lowest energy state. This is because if the Fermi energies are not equal, electrons would spontaneously change spin state to lower the energy and increase entropy. This leads to the conclusion that at equilibrium, the Fermi energies must be equal.
lampCable

## Homework Statement

Let's consider conduction electrons (at T=0K) that are put in a magnetic field. The electrons can have spin that is parallel or antiparallel to the magnetic field. Below is the density of occupied states for such a system (horizontally) as a function of energy (vertically), similar to a figure shown in my textbook.

My textbook claims that, in thermal equilibrium, the Fermi energy for both spin states must be equal, but I do not fully understand why this is the case. I think it has to do with the system being in its lowest energy state, and I reason as follows:

The Fermi energy is the energy required to add another electron to the given spin state. So if the Fermi energies are not equal then some electrons in one of the spin states would have energy greater than the Fermi energy of the other spin state. These electrons would spontaneously change spin state because it would lower the energy in the system, thus increasing the total entropy. As more electrons change spin state the Fermi energies are "evened out", so at equilibrium the Fermi energies must be equal.

Is this a correct interpretation as to why the Fermi energies are equal?

## The Attempt at a Solution

I think your argument is good.

lampCable

## 1. What is the Fermi energy for two spin states equal in equilibrium?

The Fermi energy for two spin states equal in equilibrium is the energy level at which there is a 50/50 probability of finding an electron with either spin state. It is also known as the chemical potential.

## 2. How is the Fermi energy determined for two spin states equal in equilibrium?

The Fermi energy for two spin states equal in equilibrium is determined by the number of electrons in the system. It is proportional to the number of electrons and inversely proportional to the volume of the system.

## 3. What is the significance of the Fermi energy for two spin states equal in equilibrium?

The Fermi energy for two spin states equal in equilibrium is an important concept in understanding the behavior of electrons in a system. It helps explain phenomena such as electrical conductivity and the properties of metals.

## 4. How does temperature affect the Fermi energy for two spin states equal in equilibrium?

As temperature increases, the Fermi energy for two spin states equal in equilibrium also increases. This is because at higher temperatures, more energy is available for electrons to occupy higher energy states.

## 5. Can the Fermi energy for two spin states equal in equilibrium change?

Yes, the Fermi energy for two spin states equal in equilibrium can change if the number of electrons or the volume of the system changes. It can also change with changes in temperature or external factors such as applied electric or magnetic fields.

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