Many solids that have small but positive magnetic susceptibilities are called paramagnetic. A magnetic dipole can minimize its potential energy by lining up with the magnetic field. Such alignment makes a positive contribution to the magnetization. The atoms in a solid may have permanent magnetic moments. Since paramagnetism depends upon magnetic moments of atoms or ions, the origin of these magnetic moments is important.
There are two chief origins for the moments, one from the orbital motion of the electrons, the other from the electron spin:
(i) The orbital moment arises in principle because an electron in an orbit about an atom can be considered to be a small circulating current about the nucleus. The magnetic moment is related to the angular momentum by an important relation derived as- The primitive magnetic moment pm of a circulating loop of charge is given by pm = iA, where A is area enclosed by current i carrying loop.
Imagine, for simplicity, that the current is caused by a single electron in a circular orbit of radius r about the central nucleus. Then the current around the orbit is the amount of charge which passes any point in the orbit per unit time, i.e.,
where, v is the velocity of the electron in the orbit. The magnetic moment then has magnitude
The angular momentum of the electron in the circular orbit of radius r is equal to-
L = mv × r … (2)
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(Recall that m is the mass of the electron). Hence L = mvr, so that-
Notice that pm and L is oppositely directed vectors, since the charge on the electron is negative. Thus, the relation between the magnetic moment and the orbital moment is
For this simple charge distribution, the magnetic moment and the angular momentum are simply related through a constant │e│/2m termed the gyromagnetic ratio.
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The angular momentum of an electron in its orbit was found to exist in multiples of the unit h/2. The magnetic moment of an electron with angular momentum h/2 is called the Bohr magneton β, defined as-
The orbital moment in Eq. 3 can be related, of course, to the quantum numbers. Recall that the quantum numbers I and ml refer to the angular momentum of the electron in its orbit. The number l refers to the total angular momentum by means of the equation-
The number ml refers to the component of angular momentum in the z direction; it satisfies the following relation-
Combining Eqs. (3), (4b) and (4c), we can write orbital moment as-
(ii) A second source of permanent moments is the electron spin. We should not envision the electron spin in the same way as we do the orbital motion because the electron spin is a relativistic effect and is not analogous to the motion of a spinning body. The term spin is used to refer to the fact that electrons possess intrinsic angular moments and magnetic moments like those expected of charged body.
The spin angular momentum of the electron is ±1/2 (h/2), and the spin magnetic moment is ±µB. The spin angular momentum and magnetic moment can be aligned up or down. Since any quantum state can be filled by two electrons of opposite spin, the net spin magnetic moment of a completely filled atomic shell is also zero.
Open shells may fill in a complicated manner, but if we know which states are occupied and how, we may be able to combine the spin and orbital moments and obtain the net atomic magnetic moment. A third very small contribution, of the order of 10-3 Bohr magnetons per atom (can be neglected in comparison to electronic contribution) arises from the spin of the atomic nucleus. It can only be observed by sensitive measurement.
The magnetic moment of a multi-electron atom is simply the sum of the magnetic moments of all the electrons, including both orbital and spin moments. Each electron contributes an independent vector quantity to the total magnetic moment of the atom. Since all filled shells have zero total angular momentum, they also have zero total magnetic moment. In particular, atoms or ions which possess only filled shells have no permanent moments, and hence they cannot be paramagnetic.
No exceptions to this result have been observed. The inert gases He, Ar, Kr, etc., and ions such as Na+ and Cl– are all diamagnetic. Also many gases such as H2, etc. are diamagnetic because the electrons are all completely paired. Other free atoms show paramagnetism if there are unpaired spins or a net orbital angular momentum.
The net spin angular momentum and spin magnetic moment to be expected of a partly filled shell are not obvious. However, Hund’s rule states that spins of the electrons of a shell always add together in such a way as to contribute the maximum angular momentum and magnetic moment. Consider, for example, the electrons of a d shell. There are 10 states, 5 with spin up and 5 with spin down.
If a state contains 2 electrons, they have spins in the same direction, say, spin up. This gives the maximum spin for the 2 electrons, since if their spins were opposite, the total spin would be zero. If the state contains 5 electrons, they all have spin up (or down), which is the maximum spin angular momentum a d shell can have, since a sixth electron must have spin down (or up), cancelling one of the first 5. Again, when 10 electrons are present, the total spin is zero, with 5 up and 5 down.
Since the atoms in a solid may have permanent magnetic moments, we can consider the behaviour of these moments by assuming that they are free to rotate. The potential energy of a single dipole in a magnetic field is given by U = -pm H cos θ. It is function of the angle between the dipole and the field. The dipole tends to lower its energy by lining up with the applied field and therefore gives rise to a positive susceptibility, xm. However, it is necessary to reckon with the effect of temperature.
According to the Maxwell-Boltzmann distribution, the number of dipoles having energy, E, is proportional to e-E/kT. Employing potential energy of the dipole, this expression becomes e-Pm.H/kT. At temperature T, this assumes that the dipoles make different angles with the field, with the probable number at any angle proportional to the Boltzmann factor. More dipoles are aligned with, rather than against, the field. Hence a small positive susceptibility results. The paramagnetic susceptibility of N dipoles of moment pm is-
Where, µ0 is the permeability of free space. Equation 5, called the Langevin equation, is correct for pm of about a Bohr magneton, H less than 106 amp/meter, and T at or above room temperature. The linear dependence of the susceptibility xm on the reciprocal of the temperature is called the Curie law and the constant C in Equation 5, the curie constant.
The Langevin equation does not apply to conduction electrons for they obey the Fermi-Dirac distribution. The Fermi-Dirac distribution gives a paramagnetic susceptibility for the conduction electrons which is 50 percent larger than the observed susceptibility. The reason is that the applied field also changes the states of the conduction electrons in a manner similar to the diamagnetic changes in the atomic states.
A separate diamagnetic contribution equal to 1/3 the paramagnetic contribution arises which cancels the excess predicted. The resulting net susceptibility for N conduction electrons is-
Where, EF is the Fermi level. This type of paramagnetism is independent of temperature. Generally, the susceptibility of paramagnetic solids is between 10-3 and 10-5. Paramagnetism usually masks the atomic diamagnetism present in solids.