Magnetic susceptibility

The magnetic and electric properties of molecules and solids are analogous. For instance, some molecules possess permanent magnetic dipole moments, and an applied magnetic field can induce a magnetic moment, with the result that the entire solid sample becomes magnetized. The analogue of the electric polarization, P, is the magnetization, M, the average molecular magnetic dipole moment multiplied by the number density of molecules in the sample. The magnetization induced by a field of strength H is proportional to H, and we write where χ is the dimensionless volume magnetic susceptibility. A closely related quantity is the molar magnetic susceptibility, χm:

χm=χ Vm

where Vm is the molar volume of the substance (we shall soon see why it is sensible to introduce this quantity). The magnetic flux density, B, is related to the applied field strength and the magnetization by

B=µ₀(H+M) =µ₀(1+χ) H

where µ0 is the vacuum permeability, µ0 = 4π×10⁻⁷ J C⁻² m⁻¹ s². The magnetic flux density can be thought of as the density of magnetic lines of force permeating the medium. This density is increased if M adds to H (when χ > 0), but the density is decreased if M opposes H (when χ < 0). Materials for which χ is positive are called paramagnetic. Those for which χ is negative are called diamagnetic. Just as polar molecules in fluid phases contribute a term proportional to µ2/3kT to the electric polarization of a medium (eqn 18.15), so molecules with a permanent magnetic dipole moment of magnitude mcontribute to the magnetization an amount proportional to m2/3kT. However, unlike for polar molecules, this contribution to the magnetization is obtained even for paramagnetic species trapped in solids, because the direction of the spin of the electrons is typically not coupled to the orientation of the molecular framework and so contributes even when the nuclei are stationary. An applied field can also induce a magnetic moment by stirring up currents in the electron distribution like those responsible for the chemical shift in NMR (Section 15.5). The constant of proportionality between the induced moment and the applied field is called the magnetizability, ξ (xi), and the magnetic analogue of eqn 18.15 is

We can now see why it is convenient to introduce χm, because the product of the number density N and the molar volume is Avogadro’s constant, N_A:

N Vm=

Hence

and the density dependence of the susceptibility (which occurs in eqn 20.30 via N = N_Aρ/M) has been eliminated. The expression for χm is in agreement with the empirical Curie law:

χm=A+

with A = NAµ0ξ and C = NAµ0m²/3k. As indicated above, and in contrast to electric moments, this expression applies to solids as well as fluid phases. The magnetic susceptibility is traditionally measured with a Gouy balance. This instrument consists of a sensitive balance from which the sample hangs in the form of a narrow cylinder and lies between the poles of a magnet. If the sample is paramagnetic, it is drawn into the field, and its apparent weight is greater than when the field is off. A diamagnetic sample tends to be expelled from the field and appears to weigh less when the field is turned on. The balance is normally calibrated against a sample of

known susceptibility. The modern version of the determination makes use of a super conducting quantum interference device (SQUID, Fig. 20.65). A SQUID takes advantage of the quantization of magnetic flux and the property of current loops in superconductors that, as part of the circuit, include a weakly conducting link through which electrons must tunnel. The current that flows in the loop in a magnetic field depends on the value of the magnetic flux, and a SQUID can be exploited as a very sensitive magnetometer. Table 20.6 lists some experimental values. A typical paramagnetic volume susceptibility is about 10⁻³, and a typical diamagnetic volume susceptibility is about (−)10⁻⁵. The permanent magnetic moment can be extracted from susceptibility measurements by plotting χ against 1/T.

Fig. 20.65 The arrangement used to magnetic susceptibility with a SQUID. The sample is moved upwards in small increments and the potential difference across the SQUID is measured.

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مواضيع ذات صلة

Strong electrolytes

Conductance and conductivity

Superconductors

Induced magnetic moments

Nonlinear optical phenomena

Light emission by solid-state lasers and light-emitting diodes

Light absorption by molecular solids, metallic conductors, and semiconductors

Insulators and semiconductors

The occupation of orbitals

The formation of bands

Molecular solids and covalent networks

Energetics