Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 1

Away from an absorption edge, the scalar atomic scattering factor is the Fourier transform of an approximately spherical, real atomic electron density. Opposite points contribute conjugate phases, so their sum is real. In Thomson scattering the electronic response is also effectively in phase with the driving field in this nonabsorbing approximation. Near an absorption edge, resonantly driven electronic excitations introduce a phase lag and absorption. The atomic scattering factor becomes $f_0+f^{\prime }+i{f}^{\prime \prime }$, with $f^{\prime \prime }$ describing the absorptive component and $f^{\prime }$ its dispersive companion. The sign of the imaginary part depends on the time and scattering-phase conventions; here it is the source's $+i{f}^{\prime \prime }$ convention. This discussion concerns scalar charge scattering, without magnetic or tensor resonant contributions.

Use fractional coordinates and the structure factor conventionFor real atomic scattering factors, $F(-\mathbf{h})=F(\mathbf{h}{)}^{\ast }$, immediately proving Friedel's law. If an inversion centre is chosen as origin, each site at ${\mathbf{r}}_j$ has an identical partner at $-{\mathbf{r}}_j$, with the same possibly complex atomic scattering factor. Each pair contributes $2f_j\cos (2\pi \mathbf{h}\cdot {\mathbf{r}}_j)$; sites fixed by inversion modulo a lattice translation also have even phase factors at reciprocal-lattice vectors. Henceeven for scalar anomalous X-ray scattering in a centrosymmetric structure. This last equality is an equality of amplitudes, not generally complex conjugacy.

Write $F_x(\mathbf{h})=f_x{S}_x(h,k,l)$, where the four-site orbit has structure factorA cyclic permutation of $h,k,l$ fixes the first term and permutes the remaining three. Thus $S_x(k,l,h)=S_x(h,k,l)$. Substituting $-h,-k,l$ exchanges the first and second exponentials and the third and fourth, with the common coefficient $(-1{)}^{h+l}$:The two forms of this phase, $e^{+i\pi (h+l)}$ and $e^{-i\pi (h+l)}$, agree for integer indices. Multiplication by a unit-modulus phase preserves intensity, irrespective of whether $f_x$ is real or complex. Since the phase is independent of $x$ and atomic species, it also multiplies the total structure factor of the crystal.

With $A=S_{x_{\mathrm{Si}}}(h,k,l)$ and $B=S_{x_{\mathrm{Mn}}}(h,k,l)$,For $l=0$, the transformation in part c maps $(h,k,0)$ onto its Friedel pair with only a phase factor. Cyclic permutations give the same result when either other index vanishes. Friedel's law therefore survives for every such reflection, including any systematic absences.

For the $(111)$ reflection set $t=2\pi x$. ThenSince the purely positional sums obey $S_x(-\mathbf{h})=S_x(\mathbf{h}{)}^{\ast }$, the scalar resonant Friedel-pair intensity difference isup to the same positive experimental scale factor. In particular,Different positional phases can therefore produce a nonzero resonant intensity difference at $(111)$, allowing the opposite structural handednesses to be distinguished. The claimed violation is generic, not an unconditional identity: the source supplies no numerical internal coordinates. For a concrete noncentrosymmetric counterexample, take $x_{\mathrm{Si}}=1/8$ and $x_{\mathrm{Mn}}=5/8$. Their orbit sums at $(111)$ are $\sqrt{2}(1-i)$ and $-\sqrt{2}(1-i)$, so the difference vanishes even if $f^{\prime \prime }\ne 0$; each generic chemical orbit is noncentrosymmetric, so this cancellation does not require inversion symmetry. Accidental cancellation, or $f^{\prime \prime }=0$, must be excluded before asserting a violation for a particular specimen.