Calculating the Magnetization in Curie and Pauli Paramagnetism

1 Types of magnetism
2 Quantum mechanical origin of magnetism in an isolated atom
3 Calculation of magnetization of a system of non-interacting atoms using partition function (Curie paramagnetism)
4 Calculation of paramagnetism of a free electron gas (Pauli paramagnetism)

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http://web.iitd.ac.in/~debanjan/Module2A-Paramagnetism.pdf

Difference between Curie and Pauli Paramagnetism


The above calculation of paramagnetism (Curie Paramagnetism) assumes electrons are localized to their atoms. This assumption works for molecules and free atoms and ions, which have a net magnetic moment, but in metals there are two major discrepancies between susceptibility calculated from equation (13), also known as Curie susceptibility (κm,Curie), and susceptibility measured experimentally:

 1. Susceptibility of metals measured experimentally is orders of magnitude lower than κm,Curie. For example, susceptibilities of sodium and aluminum metals are ≈ 10−5 (Figure 3.5, Modern Magnetic Materials by Robert O’Handley)
2. Also the plot of susceptibility of metals versus temperature, at small temperature, is flat, showing that susceptibility is independent of temperature, which contradicts Curie’s Law.

This discrepancy happens because in metals magnetism often originates from conduction electrons which are not localized in individual atoms but are delocalized in the entire solid. As a result, the metal needs to be treated as free electron gas instead of an assortment of atoms which we did above. Susceptibility, calculated using the free electron gas model of atoms, is called Pauli susceptibility (κm,P auli) and the associated paramagnetism is called Pauli paramagnetism, which we discuss in the next section.

Orders of magnitude lower susceptibility in the case of Pauli paramagnetism compared to Curie paramagnetism can be explained as follows: Unlike Curie paramagnetism, here electrons are delocalized, i.e, all the electrons belong to the entire solid. A particular value of ~k or kx, ky, kz corresponds to a specific orbital state of the electron and for each such orbital state, there can be two possible spin states of the electron: spin up(↑) and spin down (↓), only one electron in each state following Pauli exclusion principle. Due to the presence of the magnetic field say in up direction, energy of the spin up states is lower than the spin down states. So first the k-states corresponding to spin up states fill up and then the same k-states corresponding to spin down. As a result after the electrons fill up all the available states up to Fermi energy, there is an imbalance due to some higher k-states, filled up only by spin up electrons. Now if a magnetic field is applied in either direction, only the spin of electrons with energy close to the the Fermi energy can flip (high k-values) can flip because for the low energy or low k- states, already there are 2 electrons one with spin up and one with spin down, and no additional electron can be allowed since it will violate the Pauli exclusion principle. As a result, the calculated value of Pauli susceptibility is orders of magnitude lower than of Curie Susceptibility.

Another thing to be noted is that unlike Curie susceptibility, Pauli susceptibility is not dependent on temperature. This happens because as explained before, only the electrons close to the Fermi surface take part in the spin flipping in Pauli paramagnetism unlike Curie paramagnetism. So Curie susceptibility has to be multiplied by the ratio of no. of electrons close to the Fermi surface to the total no. of electrons to get the Pauli susceptibility, which is equal to kBT kBTF . TF is the Fermi temperature corresponding to the Fermi energy (a constant). Since Curie susceptibility varies as 1 T Pauli susceptibility is independent of temperature T.