Angular momentum magnetism

Quantum numbers The four quantum numbers—principal, angular momentum, magnetic, and spin—arise from solutions to the wave equation and govern the electron configuration of atoms. 

The atom comprises a nucleus surrounded by electrons. Each element has a specific number of electrons that are bound to the atomic nucleus in an orbital structure which is unique for each element. In the structure of the atom, each electron is characterised by the principal, angular momentum, magnetic and spin quantum numbers that define its energy level at no time do they simultaneously have the same value for two electrons of the same atom (Pauli s exclusion principle). 

Principal Angular momentum Magnetic Spin n f m s Any positive integer 0,. n — 1 in integer steps 0 -I-Zmintcgcr steps i 4.1 2- +2... 

K L ko) Atomic absorption edge Atomic absorption edge Fourier transform of total atomic orbital angular momentum Magnetic modulation wave vector... 

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

If the angular momentum of a free electron is represented by a spin vector S=(S, S, S ), the magnetic moment... 

An effect of space quantization of orbital angular momentum may be observed if a magnetic field is introduced along what we now identify as the z axis. The orbital angular momentum vector P, of magnitude Pi, may take up only certain orientations such that the component (Pi) along the z axis is given by... 

For 5 states there is no orbital angular momentum and therefore no resulting magnetic field to couple S to the intemuclear axis. The result is that a 5 state has only one component, whatever the multiplicity. 

The nuclei of many isotopes possess an angular momentum, called spin, whose magnitude is described by the spin quantum number / (also called the nuclear spin). This quantity, which is characteristic of the nucleus, may have integral or halfvalues thus / = 0, 5, 1, f,. . . The isotopes C and 0 both have / = 0 hence, they have no magnetic properties. H, C, F, and P are important nuclei having / = 5, whereas and N have / = 1. 

According to quantum mechanics, the maximum observable component of the angular momentum is Ih/lir, where h is Planck s constant. A nucleus can assume only 21+1 energy states. Associated with each of these states is a magnetic quantum number m. where m has the values I, I — I, I —2,, —1+ 1, —I. 

A nucleus having spin generates a magnetic moment pi. which is proportional to the angular momentum. Theory is not capable of calculating pi, so it is commonly expressed as Eq. (4-42), where 7 is called the magnetogyric ratio. 

Consider a nucleus with magnetic moment pi in a magnetic field Ho- According to classical mechanics the rate of change of the angular momentum G is the torque T. 

The third quantum number m is called the magnetic quantum number for it is only in an applied magnetic field that it is possible to define a direction within the atom with respect to which the orbital can be directed. In general, the magnetic quantum number can take up 2/ + 1 values (i.e. 0, 1,. .., /) thus an s electron (which is spherically symmetrical and has zero orbital angular momentum) can have only one orientation, but a p electron can have three (frequently chosen to be the jc, y, and z directions in Cartesian coordinates). Likewise there are five possibilities for d orbitals and seven for f orbitals. 

Any particle with charge Q, mass m and non-zero angular momentum 1 is a magnetic dipole. 

The Stern-Gerlach experiment demonstrated that electrons have an intrinsic angular momentum in addition to their orbital angular momentum, and the unfortunate term electron spin was coined to describe this pure quantum-mechanical phenomenon. Many nuclei also possess an internal angular momentum, referred to as nuclear spin. As in classical mechanics, there is a relationship between the angular momentum and the magnetic moment. For electrons, we write... 

Atomic and molecular magnetic dipoles have to obey the angular momentum laws of quantum mechanics, since they are proportional to angular momenta. Each dipole can therefore make just a number of orientations with an applied magnetic induction B. Each allowed orientation corresponds to a different potential energy, and absorption of a photon with suitable energy may cause a change in orientation. 

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