(a) (i) What is the origin of paramagnetism? Discuss the temperature dependence of paramagnetic susceptibility. (ii) Calculate…

(a)(i) Origin of paramagnetism and its temperature dependence

Paramagnetism arises in materials whose atoms, ions or molecules carry a permanent magnetic dipole moment, which requires an unpaired electron: the moment is the vector sum of the spin and orbital moments of the partly filled shell, μ = g μB √(J(J+1)) for a free ion with total angular momentum J. Transition-metal and rare-earth ions, free radicals, O₂ and NO are paramagnetic. In zero field the moments point at random and the net magnetisation is zero; an applied field B tends to align them, since a moment has energy −μ·B, while thermal agitation tends to randomise them. The magnetisation is therefore small, positive and field-dependent, and the susceptibility is positive but much smaller than one.

Treating the moments classically, the average moment along the field is μL(x), L the Langevin function, x = μB/kT; for x ≪ 1, L(x) ≈ x/3, so M = Nμ²B/(3kT) and χ = μ₀Nμ²/(3kT) = C/T. Quantum mechanically the 2J + 1 orientations give the Brillouin function and the same low-field result with μ² replaced by g²μB²J(J+1). This inverse dependence on temperature is Curie's law, χ = C/T: raising the temperature increases the thermal disorder and lowers the susceptibility, so a plot of 1/χ against T is a straight line through the origin. In materials whose moments interact weakly the line is shifted, χ = C/(T − θ), the Curie–Weiss law, θ positive for ferromagnetic and negative for antiferromagnetic coupling. Two cases do not obey Curie's law: the Pauli paramagnetism of conduction electrons in metals, where only the electrons within kT of the Fermi level can respond, giving a small susceptibility nearly independent of temperature, and Van Vleck paramagnetism from field-induced mixing of excited states, also temperature-independent.

(a)(ii) Diamagnetic susceptibility of atomic hydrogen at STP

The Langevin diamagnetic susceptibility of N atoms per unit volume with Z electrons each is

χ = −μ₀ N Z e² ⟨r²⟩/(6m),

where ⟨r²⟩ is the mean square distance of the electrons from the nucleus. Here Z = 1 and ⟨r²⟩ = 3a₀² = 3 × (0.529 × 10⁻¹⁰)² = 8.39 × 10⁻²¹ m². At STP an ideal gas contains N = 2.69 × 10²⁵ atoms/m³ (Loschmidt's number).

χ = −(4π × 10⁻⁷ × 2.69 × 10²⁵ × (1.6 × 10⁻¹⁹)² × 8.39 × 10⁻²¹)/(6 × 9.11 × 10⁻³¹) = −(7.3 × 10⁻³⁹)/(5.47 × 10⁻³⁰) = −1.3 × 10⁻⁹.

The volume susceptibility of atomic hydrogen at STP is about −1.3 × 10⁻⁹ (SI, dimensionless), negative and tiny, as expected for an induced moment in a dilute gas; per mole it is about −2.4 × 10⁻⁶ cm³/mol in CGS units.

(b)(i) Effective mass of an electron

An electron in a crystal moves under the periodic potential of the lattice as well as any applied force. Its response to an external force F can still be written in Newton's form, dv/dt = F/m*, provided the free-electron mass is replaced by the effective mass

1/m* = (1/ħ²) d²E/dk²,

which follows from the group velocity v = (1/ħ) dE/dk and ħ dk/dt = F. The effective mass is a property of the band structure: it is small where the band is sharply curved, large where the band is flat, and negative near the top of a band where d²E/dk² < 0, meaning the electron accelerates opposite to the force because the lattice pushes back more than the field pushes forward. In a real crystal m* is a tensor. Near the top of a nearly full band the collective behaviour of the electrons is described more simply by the missing electron, the hole, which has positive charge and positive effective mass −m*e.

(b)(ii) Effective mass of the hole

E(k) = −1 × 10⁻²⁶ k² erg (k in cm⁻¹), so d²E/dk² = −2 × 10⁻²⁶ erg cm², and for an electron in this band

m*e = ħ²/(d²E/dk²) = (1.055 × 10⁻²⁷)²/(−2 × 10⁻²⁶) = 1.113 × 10⁻⁵⁴/(−2 × 10⁻²⁶) = −5.6 × 10⁻²⁹ g.

The electron near the top of the valence band has negative effective mass, as expected for a band curving downwards. Removing it from the orbital k = 10⁷ k̂ₓ cm⁻¹ leaves a hole with wave vector −10⁷ k̂ₓ cm⁻¹, positive charge and effective mass

m*h = −m*e = +5.6 × 10⁻²⁹ g ≈ 0.061 mₑ.

The sign is positive and the magnitude 5.6 × 10⁻²⁹ g, about six per cent of the free-electron mass; because the band is parabolic the value does not depend on which k the electron was removed from.

(c) Frequencies at which the gain falls to 80

For an RC-coupled amplifier the gain at low frequency is A = Am/√(1 + (f₁/f)²) and at high frequency A = Am/√(1 + (f/f₂)²), where f₁ = 20 Hz and f₂ = 50 kHz are the lower and upper half-power (cut-off) frequencies. Gain 80 means A/Am = 0.8, so √(1 + x²) = 1.25, x² = 0.5625, x = 0.75.

Low-frequency side: f₁/f = 0.75, f = 20/0.75 = 26.7 Hz.

High-frequency side: f/f₂ = 0.75, f = 0.75 × 50 kHz = 37.5 kHz.

The gain is 80 or more (within 2 dB of the mid-band value) between 26.7 Hz and 37.5 kHz; outside this band it falls further, reaching 70.7 at the two cut-off frequencies.