Quantum Mechanical Considerations 13 We will see in this chapter with quantum mechanics, Why certain metals that we expect to be paramagnetic are in reality diamagnetic? Why the paramagnetic susceptibility is relatively small for most metals? Why most metals do not obey Curie-Weiss law? Better understand ferromagnetism by applying elements of quantum mechanics.

13.1 Paramagnetism and Diamagnetism

Now, we consider a parabolic distribution of energy states, H = 0 H > 0 H Potential energy: DE (Magnetization is the magnetic moment per unit volume) Only the electrons close to the Fermi energy are capable of realigning in the magnetic field direction

* Diamagnetism It is important to realize that ever-present diamagnetism makes a sizeable contribution to the overall susceptibility, so that c for metals might be positive or negative depending on which of the components predominates. For Be, the density of states at the Fermi level, is very small. Showing diamagnetic property For Cu, the Fermi energy is close to the band edge – density of states near EF is relatively small The same is true for Ag, Au, Zn, and Ga. * Intrinsic semiconductor: filled band states and small density of states at the top of valence band – showing diamagnetic Be where, Z is total number of electrons

Bohr Magneton (derive the numerical value of Bohr magnetron) Starting from a magnetic moment of an orbiting electron (angular momentum) Namely, quantum theory postulates that the angular momentum, mvr, of an electron changes in discrete amount of integer multiples of Magnetic moment of an electron due to orbital motion and spinning electron are identical – smallest magnetic moment is called as Bohr magnetron

13.2 Ferromagnetism and Antiferromagnetism The ferromagnetic metals (Fe, Co, and Ni) are characterized by unfilled d-bands which overlap the next higher s-band. More importantly, the parallel alignment of spins occurs spontaneously in small domains of 1-100 mm. – exchange energy

Compare two ferromagnetic atoms with two identical pendula that are interconnected by a spring For H2 molecule:↑↓ ↑↑ versus ↑↓ As the differences between the resulting frequencies is larger, the stronger the coupling If the two pendula vibrate in a parallel fashion – the restoring force kx is small and smaller than independent vibrations Namely two coupled and symmetrically vibrating systems may have a lower energy than two individually vibrating system

Quantum mechanics treats ferromagnetism in a similar way Quantum mechanics treats ferromagnetism in a similar way. The potential energy in the Schrodinger equation then contains the exchange forces between the nuclei a and b, the forces between two electrons 1 and 2, and the interactions between the nuclei and their neighboring electrons. Bethe-Slater curve