Past exam of the condensed matter physics course of the University of Oxford / 2013 (2765, B3: VI)

Source paper: www-thphys.physics.ox.ac.uk/people/SteveSimon/condmat2015/b3_VI_2013.pdf

The sites form a face-centered cubic lattice decorated by an identical two-atom atomic basis of a crystal, at fractional displacements $0$ and $\mathbf{\delta }=(1/4,1/4,1/4)$. This is the diamond cubic structure; the eight atoms in its conventional cube are not eight independent Bravais-lattice points.

For identical silicon atomic scattering factors $f$, the conventional-cell structure factor separates into the FCC factor and the two-site motif factor:The first factor vanishes unless $h,k,l$ are all even or all odd. For all-even indices, the motif also vanishes if $h+k+l=4r+2$; it survives if the sum is divisible by four. All-odd indices survive, with motif modulus $\sqrt{2}$. Thus $(100)$ and $(110)$ are absent by centering, $(200)$ is absent by the diamond basis, and $(111)$ and $(220)$ are allowed.

For a cubic cell, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. At fixed X-ray wavelength, Bragg's law makes the smallest scattering angles correspond to the smallest allowed positive values of $h^2+k^2+l^2$. These start with $3$ and $8$, soThe braces denote whole powder families, including their symmetry-equivalent plane normals. The next family is $\{311\}$, with squared-index sum $11$; using the FCC rule alone would incorrectly insert a $(200)$ line before $(220)$.

Interpret the reported angles as total scattering angles $\psi =2\theta$. The photon energy givesCombining $2d\sin \theta =\lambda$ with the two index assignments yieldsThe slight difference is consistent with the rounded angular data, givingThe intensity estimate needs a stated convention. The geometric structure factors obey $|F_{111}{|}^2=32|f_{111}{|}^2$ and $|F_{220}{|}^2=64|f_{220}{|}^2$, so equal atomic factors give a single-directed-reflection ratio $1/2$. For a powder line, the powder diffraction multiplicities are eight for $\{111\}$ and twelve for $\{220\}$. Hence a multiplicity-and-structure-factor estimate isif other angle-dependent factors are neglected.

A measured integrated powder intensity also contains the Lorentz-polarization factor, atomic form factor, thermal-motion factor and instrument/sample corrections. For example, in the common unpolarized symmetric powder geometry,Including this factor but still taking equal atomic factors gives $I_{111}/I_{220}\simeq 0.9725$, rather than $1/3$. More generally,within that geometry, before other corrections. The actual form factors and measurement geometry are not supplied, so no unique precision experimental peak-height ratio follows. The values $1/2$, $1/3$ and $0.9725$ refer to distinct explicitly stated levels of approximation; a powder family must not be silently treated as one directed reflection.

At fixed radiation wavelength, differentiating Bragg's law gives $\cot \theta d\theta =-da/a$. With constant linear thermal expansion coefficient ${\alpha }_L$,Both scattering angles decrease as the lattice expands. Substitution givesThese are shifts of $2\theta$, not of the Bragg angle alone. As a finite-step check, using $a^{\prime }=a(1+0.0018)$ gives ${\psi }^{\prime }=2\arcsin [\sin (\psi /2)/(1.0018)]$. It produces shifts $-0.04151^{\circ }$ and $-0.07008^{\circ }$, consistent with the first-order estimate. A strictly constant differential coefficient would instead give $a^{\prime }/a=e^{{\alpha }_L\Delta T}$; its difference from the linear length approximation is second order in this small expansion.

Diffraction thermal-expansion thermometry needs a stable monochromatic X-ray beam, controlled collimation, an accurately calibrated angular scale, and an X-ray-sensitive detector with enough counts to fit reproducible line centroids. Crystal monochromators or appropriate energy selection are relevant here; an apparatus for visible-light interference does not directly measure these X-ray peaks. Sample displacement and angular-zero drift should be checked with a suitable reference standard.

The first-order angular sensitivities are about $6.93\times 10^{-5}$ and $1.17\times 10^{-4}$ degrees per kelvin, respectively. A one-kelvin inference therefore requires comparable centroid stability, with high count statistics and control of systematic errors; a centroid can be fitted more precisely than the line width. Fractional wavelength drift also mimics lattice expansion, so the desired temperature precision constrains $\Delta \lambda /\lambda$ relative to ${\alpha }_L\Delta T$.

The powder must be at a known uniform temperature and allowed to equilibrate. Mechanical stress, clamping, composition changes, oxidation and phase changes can also shift or broaden peaks. Accurate work requires a calibrated $a(T)$, including the actual temperature dependence of the thermal expansion coefficient, rather than assuming the supplied constant describes every temperature exactly. Using both lines provides a consistency check on a common cubic expansion.

Put $C=\alpha q^2/(4\pi {ϵ}_0)$ with $\alpha =2\ln 2$. Relative to a chosen positive ion, successive neighbours have alternating charge. The sum of its interactions with all other ions is $-C/R$. Summing this over all $2N$ ions would count every pair twice, so the total electrostatic energy is $-NC/R$, or $-C/R$ per ion pair.

There are $2N$ nearest-neighbour bonds in a periodic chain of $N$ pairs; boundary corrections are negligible for large $N$. If $A/R^n$ is the energy of each such bond, the repulsive contribution per ion pair is $2A/R^n$. Therefore the alternating-charge chain cohesive energy isThe factor two in the short-range term counts two bonds per ion pair. If a repulsion parameter were instead defined as the whole pair's repulsive coefficient, it would equal $2A$ in this convention; changing that definition must not silently change the equilibrium formula. The absolute zero of energy is separated ions for a genuinely decaying repulsion.

For a repulsive short-range potential and a stable finite minimum, assume $A>0$ and integer $n>1$. Setting $u^{\prime }(R)=C/R^2-2nA/R^{n+1}=0$ givesAt this point $2A/R_0^n=C/(n{R}_0)$, soHere $u^{\prime \prime }$ denotes the second derivative of the energy $u$, not a displacement variable. The negative equilibrium energy is the cohesive energy per pair relative to infinite separation.

The qualifiers on $A,n$ matter: an arbitrary real coefficient and integer exponent do not automatically describe stabilizing repulsion. For instance $A<0,n>1$ gives collapse as $R\to 0$, whereas $n=1$ gives $u=(2A-C)/R$ and has no isolated finite equilibrium. The displayed minimum therefore applies to the physically repulsive, faster-than-Coulomb branch of the stated model.

For a one-dimensional free-electron gas of length $L$, allowed wave vectors have spacing $2\pi /L$. Including two spin states and occupied wave vectors from $-k_F$ to $+k_F$,The ground-state kinetic energy is found by integrating rather than assigning every electron the Fermi energy:Charge neutrality gives $N_e=ZN$ because each positive ion has charge $+Ze$. Thus $K_e=ZN{E}_F/3$. Since the ion separation is $2R$, $L\simeq 2RN$ and $n_e=Z/(2R)$. These are one-dimensional number densities and state counts; using a three-dimensional $k_F\propto n^{1/3}$ relation here would change the model. The results concern the ground-state Fermi-gas contribution, neglecting electron interactions beyond the supplied electrostatic energy.

Use the supplied effective electrostatic term and enforce charge neutrality while varying the chain length: $R=Z/(2n_e)$, with $N$ and $Z$ fixed. The total energy per positive ion becomesThe positive quadratic term supplies degeneracy pressure against compression. Differentiating gives the neutral one-dimensional metal equilibrium density,where $a_B=4\pi {ϵ}_0{\hslash }^2/(m_e{e}^2)$ is the Bohr radius. The second derivative $Z{\pi }^2{\hslash }^2/(12m_e)>0$ verifies a minimum. Within this specified toy energy, the minimizing electron density is independent of $Z$; instead the equilibrium ion spacing is $2R^{\ast }=Z/n_e^{\ast }$.

At the minimum the electrostatic contribution is minus twice the kinetic contribution, so ${\epsilon }_{\ast }=-Z{E}_F^{\ast }/3$. These follow from the given electrostatic approximation, not from a full self-consistent calculation of a real one-dimensional metal. In particular, additional Coulomb terms cannot be appended without the corresponding neutral-background subtraction that defines the supplied finite energy.

Let $a$ be the equilibrium spacing of this monatomic chain and $u_j$ the displacement from $ja$. The right nearest-neighbour bond exerts $K(u_{j+1}-u_j)$ and the left bond $K(u_{j-1}-u_j)$. The two next-nearest-neighbour bonds similarly connect $j$ to $j\pm 2$. ThusEvery uniform displacement gives zero force, as translation symmetry requires. There is one displacement coordinate per primitive cell. Adding further-neighbour springs does not introduce a second atom or turn this into a diatomic-chain problem.

Substitute the traveling normal mode $u_j=u\exp [i(kja-\omega t)]$. Combining the opposite-neighbour phases givesso the positive-frequency branch isFor positive spring constants this is a stable acoustic branch. At long wavelengths,The next-nearest-neighbour bond spans twice the distance, hence contributes the factor four. Relative to $G=0$, positive $G$ increases the sound speed by $\sqrt{1+4G/K}$. The chain still has one acoustic branch and no optical branch.

The group speed near $k=0$ and the maximum frequency need not change in the same way. The zone-edge value is always ${\omega }_0(\pi /a)=2\sqrt{K/m}$, because the next-nearest-neighbour sites then move together. For $G>K/4$ the frequency maximum moves inside the zone, with $\cos (ka)=-K/(4G)$; the small-$k$ sound speed nevertheless keeps increasing.

For a viscously damped monatomic-chain phonon, the extra force gives $m{\ddot{u}}_j+\Gamma {\dot{u}}_j$ on the left of the preceding equation. Keep a real wave number $k$ and permit complex frequency in the convention $u_j\propto e^{i(kja-\omega t)}$. The dispersion equation isWriting $\gamma =\Gamma /(2m)$, its two roots areThe negative imaginary part corresponds to decay with the chosen time convention. Here $\Gamma$ has units of mass per time, while $\gamma$ is a decay rate. A driven experiment at specified real frequency would instead solve for a generally complex wave number; it is a different way of specifying the response, not a contradictory sign choice.

For passive friction $\Gamma >0$, the time dependence of each spatial normal coordinate follows its damping regime:

For any fixed positive damping, sufficiently small nonzero $k$ is overdamped because ${\omega }_0\simeq c_s|k|$. The slow branch then has $\omega \simeq -i(m{c}_s^2/\Gamma )k^2$, a diffusive long-wavelength relaxation. At exactly $k=0$, uniform translation has no restoring force, so $u_0(t)=A+B{e}^{-\Gamma t/m}$ and the constant displacement need not decay. Thus saying every amplitude simply decays as $e^{-\Gamma t/(2m)}$ is valid only for the underdamped envelope, not for all wave numbers.

In the SI linear-response convention $\mathbf{M}={\chi }_H\mathbf{H}$, a diamagnetic contribution has negative susceptibility: the induced moment opposes the applied field. Orbital motion supplies a familiar microscopic mechanism, and closed-shell atoms can be diamagnetic even without permanent moments. Paramagnetism has positive susceptibility, with induced magnetization along the field; thermal alignment of permanent localized moments is one mechanism, while conduction-electron spin response is another.

Neither sign alone is a definition of ferromagnetic ordering. An ordinary paramagnetic phase has no spontaneous zero-field magnetization. A diamagnetic background can coexist with a larger paramagnetic contribution in the same specimen. Curie's inverse-temperature law describes an appropriate localized-moment model, not every possible paramagnet.

Interpret the source's $\mathbf{J}$ as the dimensionless angular-momentum operator, with $J_z|J,m_J⟩=m_J|J,m_J⟩$. If angular momentum is written with physical units, the magnetic-moment formula instead contains ${\mathbf{J}}_{\mathrm{phys}}/\hslash$; this is a convention, not an extra factor in the requested energy.

The field fixes the quantization axis, soThere are $2J+1$ Zeeman levels equally spaced in energy by $g{\mu }_{B}B$. For positive $gB$, the state with largest positive moment has lowest energy. The given sign defines $\mathbf{J}$ through the specified moment relation; inserting an additional electron-charge minus sign into it would change the source convention.

Let $x=g{\mu }_{B}B/(k_{B}T)$. The single-ion partition function and mean projected magnetic moment areFor $J>0$, the geometric sum also givesIn the high-temperature, weak-polarization regime $|x|J\ll 1$, expand the Boltzmann factors. The numerator's zero-order term vanishes because $\sum m_J=0$, and the first-order term uses $\sum m_J^2=J(J+1)(2J+1)/3$. The denominator is $2J+1+O(x^2)$. ThereforeThis fixed-field per-ion Curie coefficient describes mean moment per ion at fixed field, so it includes $B$. It is not the field-independent volume-susceptibility Curie constant. For number density $n$ and $B\simeq {\mu }_{0}H$, the latter is $C_{\chi }=n{\mu }_0(g{\mu }_B{)}^{2}J(J+1)/(3k_B)$, giving ${\chi }_H=C_{\chi }/T$. At $J=0$, the specified moment is zero and both expressions vanish.

Linear Curie behavior fails gradually when the Zeeman energy of a substantial moment is comparable to thermal energy. Its appropriate scale isAt $T=300\mathrm{K}$, $k_{B}T/{\mu }_B=446.6\mathrm{T}$. Thus $B_{\mathrm{nl}}\sim 446.6/(gJ)\mathrm{T}$: about $223\mathrm{T}$ for $g=2,J=1$, or $447\mathrm{T}$ for a spin-one-half moment with $g=2$. Larger moments saturate at smaller fields. The source does not specify numerical $g,J$, so it determines this parametric scale rather than one universal room-temperature threshold. At sufficiently large field the moment tends toward $g{\mu }_{B}J$ and cannot keep increasing linearly with $B$.

Room-temperature iron is ferromagnetic: interactions among its electronic moments produce spontaneous local order. The independent-ion Curie law therefore needs an interaction and an ordered-state description. In a Weiss mean-field approximation, replace the applied field by $H_{\mathrm{eff}}=H+\Lambda M$. Above the ordering temperature, linearizing the single-ion response yieldsThis is the Curie–Weiss approximation on the paramagnetic side. Merely substituting room temperature below $T_C$ into that high-temperature formula would produce a negative response and is not a valid model of stable iron.

Below $T_C$, use the full nonlinear moment response to the effective field; it has nonzero zero-field solutions. On a chosen ordered branch the local small-field response has the form $M\simeq M_s(T)+{\chi }_{\mathrm{diff}}(T)H$, rather than $M\propto H/T$ through the origin. Macroscopic magnetic domains and hysteresis further determine the measured net moment. Exchange interactions, arising from electronic quantum statistics and Coulomb correlations, supply the alignment energy; bare dipolar fields alone do not explain ordinary iron's strong ferromagnetism. A localized-ion Weiss model describes the qualitative ordering, while iron's itinerant electronic structure limits a literal independent-fixed-ion interpretation.