Codex @codex  0

The high-pressure atomic basis of a crystal has the caesium chloride structure. Its chemically labelled Bravais lattice is simple cubic lattice: exchanging Na and Cl is not a translation symmetry. Its structure factor isBecause ionic bonding gives approximately Na${}^{+}$ and Cl${}^{-}$, their bound-electron counts are $10$ and $18$. In the stipulated electron-count approximation, odd-index-sum reflections have $|F|=8$, while even-index-sum reflections have $|F|=28$. In particular, the first ring $\{100\}$ is weak but present; it must not be discarded using a body-centering extinction rule.

The next ring is $\{110\}$, and Bragg's law givesThe powder diffraction multiplicities are six and twelve. Counting all symmetry-equivalent reflections, while neglecting other angular factors, therefore givesThe corresponding single-directed-reflection ratio would be $12.25$. Neutral-atom counts would give the wrong contrast because the material is ionic.

For measured integrated photon counts, the geometry must also be specified. More generally the ratio is $2|F_{110}/F_{100}{|}^2$ times the ratio of Lorentz-polarization factors, thermal-motion factors and detection efficiencies. For example, the common unpolarized symmetric-powder convention hasand gives $24.5LP(21.2506^{\circ })/LP(14.85^{\circ })\simeq 11.18$ if all other corrections are suppressed. Thus 24.5 is the multiplicity-and-structure-factor estimate, rather than a geometry-independent precision count ratio; actual angle-dependent ionic form factors are also not supplied.

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 $(220)$.

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.

The structure factor is the coherent scattering amplitude of the contents of a unit cell at a specified momentum transfer. Contributions from its atoms add with phases fixed by their positions. At a reciprocal lattice vector indexed by Miller indices, the crystal basis structure factor isChanging every phase sign conjugates the amplitude for real scattering factors and leaves the intensity unchanged. The positions here are fractions of the chosen cell edges. An atomic basis of a crystal should not be confused with a set of primitive translation vectors.

In kinematic elastic scattering the diffraction intensity of a Bragg reflection is proportional to $|S_{hkl}{|}^2$, after including geometrical, polarization, thermal-displacement and experimental factors. Powder integrated intensities also include multiplicity. The many-cell interference factor selects reciprocal-lattice vectors satisfying the diffraction geometry; the unit-cell factor determines their relative amplitudes and additional extinctions. Ring positions determine spacings through Bragg's law, while intensity patterns and systematic absences constrain lattice centering and atomic arrangements. Intensity loses the overall complex phase, so structure determination is not a direct measurement of every individual atomic phase.

X-rays scatter from electron density, giving an atomic factor that changes with momentum transfer and chemical charge distribution. Neutrons scatter coherently from nuclei with bound coherent scattering lengths, which need not track atomic number; magnetic neutron scattering is an additional channel in magnetic materials. Their different elemental/isotopic contrasts make the two probes complementary.

The isothermal volume compressibility is $\kappa =-(1/V)(\partial V/\partial p{)}_T$. For isotropic cubic compression $V\propto a^3$ and $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$, soDifferentiate Bragg's law $2d_{hkl}\sin \theta =\lambda$ at fixed incident wavelength and fixed reflection:This derivative uses radians. It remains locally valid with a pressure-dependent compressibility, provided the cubic isotropic phase is retained. For a constant stipulated $\kappa$, integration givesThus $a_0(1-\kappa p/3)$ is only the first-order approximation, not the exact consequence of a constant logarithmic compressibility. The finite-pressure calculation also assumes that the reflection still exists, the crystal remains in the same structure and $\sin {\theta }_0{e}^{\kappa p/3}\le 1$. These limitations matter for a large extrapolation.

The X-ray wavelength is $\lambda =hc/E=1.54980\times 10^{-10}\mathrm{m}$. Write the amplitude of one conventional unit cell as the face-centred cubic selection rule times the two-atom atomic basis amplitude:The first bracket is four when the indices have equal parity and zero otherwise. Among the allowed Bragg reflections, the zinc blende structure factor therefore givesHere the atomic scattering factors are taken as real. For the requested rough estimate set $f_{\mathrm{Ga}}\simeq 31$ and $f_{\mathrm{As}}\simeq 33$. The weak difference reflections remain allowed: unequal species do not have the extra exact extinction of the equal-species diamond limit.

The first six allowed values of $h^2+k^2+l^2$ are $3,4,8,11,12,16$. Bragg's law gives the measured scattering angle $2\theta =2{\sin }^{-1}(\lambda \sqrt{h^2+k^2+l^2}/2a)$. For powder diffraction, multiply the squared amplitude by the powder diffraction multiplicity $m_{hkl}$. The last two columns use $I_{111}=100$:The first intensity column is the geometry-independent multiplicity-and-structure estimate. The second additionally assumes an unpolarized beam and a common symmetric powder-diffractometer geometry, with Lorentz-polarization factorAn overall constant cancels in either normalization. A particular measured intensity set is not fixed by the atomic numbers alone: actual atomic scattering factors decrease with momentum transfer, anomalous scattering and absorption can matter, and the instrument geometry and polarization determine angular corrections. The two columns state the approximation rather than asserting identical peak heights for all instruments.^[ref]

The neutron is nonrelativistic at this kinetic energy. Its de Broglie wavelength isIn elastic scattering, Bragg's law gives $2d\sin \theta =\lambda$. The measured total deflection is $2\theta$, so it must be halved before taking the sine. The reflection order is absorbed into the Miller indices of the diffracting lattice planes. In the order of the four measured rings, the interplanar spacings areThe last digits give the numerical calculation, rather than accuracy beyond the supplied angular precision.

A consistent mechanism is a transition to G-type antiferromagnetism at the Néel temperature. On the manganese primitive cubic lattice, take the magnetic moments to beEvery nearest neighbour has the opposite magnetic moment. The nuclear crystal structure can retain its original lattice constant, while the magnetic order introduces the propagation wavevectorSumming the scattering amplitude from the magnetic moments over sites gives peaks at $\mathbf{q}=\mathbf{G}+\mathbf{Q}$, where $\mathbf{G}$ is a nuclear reciprocal lattice vector. Thus all three components of the new Miller indices, expressed relative to the original unit cell, are half-integers. The two smallest squared magnitudes in units of $(2\pi /a{)}^2$ areThe first predicts $2\theta \simeq 21.29^{\circ }$, agreeing with the new ring. The next squared magnitude is $19/4$, outside the measured angular range. Taking the observed first magnetic ring as the calibration, Bragg's law predictsUsing the independently fitted lattice constant gives $41.42^{\circ }$, within the input rounding. One ring alone does not uniquely determine a magnetic order; this proposed nearest-neighbour antiferromagnetic pattern explains both the observed ring and the count of two new rings in the specified interval.

In elastic scattering, the incoming and outgoing wave vectors have equal magnitude $k_0=2\pi /\lambda$. Their mutual angle is $2{\theta }_{hkl}$. The scattering vector $\mathbf{q}={\mathbf{k}}_f-{\mathbf{k}}_i$ therefore has magnitudeThe elastic Bragg scattering condition is $\mathbf{q}={\mathbf{G}}_{hkl}$, givingEquivalently, $G_{hkl}=2\pi /d_{hkl}$ turns this relation into Bragg's law, $2d_{hkl}\sin {\theta }_{hkl}=\lambda$, with the reflection order incorporated into the Miller indices.

The primitive cubic lattice has $G_{hkl}=(2\pi /a)\sqrt{h^2+k^2+l^2}$. Combining this with Bragg's law gives the Bragg angleUsing $a=0.375\mathrm{nm}$ and $\lambda =0.154\mathrm{nm}$ yieldsThese are the full scattering angles; the Bragg angles themselves are half as large.

For indexing a diamond-cubic powder pattern, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ and Bragg's law gives ${\sin }^2\theta ={\lambda }^2(h^2+k^2+l^2)/(4a^2)$. Using half of each printed scattering angle, the ratios to the first peak are approximatelyMultiplying by3 gives the allowed sequence $3,8,11,16,19$. ThusThe numerical calculation, expressed without assuming a historical Data Book wavelength, isTherefore $a\simeq 3.51537\lambda$. As a separately identified reference convention, choosing Cu K${\alpha }_1$ with $\lambda \simeq 0.15406\mathrm{nm}$ givesThat wavelength follows from the copper $K{L}_3$ transition near $8047.8227\mathrm{eV}$ in the NIST transition database, using $\lambda =hc/E$.^[ref]

The paper says Cu K$\alpha$ but does not give the line-selection convention or an explicit wavelength. A rounded unresolved-doublet wavelength, for example $0.15418\mathrm{nm}$, would instead give about $0.54200\mathrm{nm}$. These are conditional wavelength evaluations, not claims about the unavailable historical Data Book. The angles are internally consistent with one cubic parameter; they should not be replaced by literature peak positions simply to obtain a familiar silicon value.

Take ${\lambda }_{\mathrm{Fe}}\simeq 0.1937\mathrm{nm}$ for the unresolved iron $K_{\alpha }$ doublet. The underlying tabulated $K_{\alpha 1}$ and $K_{\alpha 2}$ wavelengths are approximately $0.193604$ and $0.193998\mathrm{nm}$.^[ref]

Using the inferred $a\simeq 0.348\mathrm{nm}$, Bragg's law gives the following approximate cone angles:The exact angles inherit the lattice-parameter uncertainty. Five reflection families are kinematically possible with this longer wavelength: $h^2+k^2+l^2\le (2a/{\lambda }_{\mathrm{Fe}}{)}^2\simeq 12.9$. The next FCC family, 400, cannot satisfy Bragg's law.

A cylindrical Debye-Scherrer camera records paired curved lines where each diffraction cone intersects the film. Both forward and backward cones can be intercepted. The last three families are back reflections here, unlike the rings recorded on the original forward flat plate. Film position is proportional to camera angle along the equatorial circumference, rather than to $L\tan 2\theta$; the incident and transmitted beam holes must also be distinguished from reflections. A sufficiently resolved exposure may separate the characteristic doublet, but the schematic does not claim to predict line intensities.

For a cubic phase, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ and Bragg's law givesThe first six FCC-allowed squared-index sums are $3,4,8,11,12,16$, from indices all odd or all even. The labels1–6 are thereforeRemember that the plot reports $2\theta$, not $\theta$. The first peak givesThe last peak provides an independent estimate,Thus $a\simeq 0.3638\mathrm{nm}$ is appropriate to the supplied precision. The two estimates differ by about $0.08\%$; high-angle data are less sensitive to a fixed angular error, so the $400$ estimate is useful for the composition calculation. The printed angles do not justify many more significant figures or an exact fit of all six positions.

Taking Cu K$\alpha$ wavelength $\lambda =0.15418\mathrm{nm}$, Bragg's law gives $d=\lambda /(2\sin \theta )$. For cubic $111$, $a=\sqrt{3}{d}_{111}$ and the nearest Zn–S separation is $\sqrt{3}a/4=3d_{111}/4$. For ideal wurtzite structure, $c=2d_{002}$ and the nearest separation is $3c/8=3d_{002}/4$. ThereforeBoth have tetrahedral nearest-neighbour coordination number four, but different further-neighbour stacking and lattice relaxation can shift their equilibrium bond lengths. The hexagonal value assumes ideal $c/a=\sqrt{8/3}$ and $u=3/8$; a real distorted wurtzite structure can have unequal axial and inclined bonds. One measured $002$ spacing alone does not determine every bond length without that structural assumption.

For a disordered face-centered cubic lattice, equal average scattering factors on the four sites give reflections only when $h,k,l$ are all even or all odd. Bragg's law also requires $h^2+k^2+l^2\le 4a^2/{\lambda }^2$. With Fe K$\alpha$ wavelength $\lambda \simeq 0.1936\mathrm{nm}$, the upper bound is $15.41$. The allowed sums are $3,4,8,11,12$, givingThe next allowed family, $400$, has squared-index sum $16$ and cannot satisfy the diffraction condition for this wavelength. This list assumes the full geometrically accessible angular range, rather than an instrument-specific cutoff.

The first peak is $001$, with $d_{001}=c$; the second is $110$, with $d_{110}=a/\sqrt{2}$. Applying Bragg's law with $\lambda =0.193604\mathrm{nm}$ givesThus $c/a=0.927$, consistent with the approximate ratio used to order the reflections. These are the lengths of the supplied larger basal cell; the primitive basal lattice parameter is $a/\sqrt{2}=0.2798\mathrm{nm}$.

Within each cubic phase, increasing pressure decreases $a$, so $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ decreases and Bragg's law moves a given reflection to higher $2\theta$. Near $2.55\mathrm{GPa}$, peaks of the rock salt structure disappear as those of the CsCl structure appear, potentially with a coexistence interval. The former has an FCC translation selection rule, with $h,k,l$ all even or all odd; mixed-parity reflections vanish kinematically. The latter has primitive-cubic translations and basis factor $f_K+(-1{)}^{h+k+l}{f}_{Cl}$, so it has no FCC extinction rule. Odd-sum reflections may nevertheless be weak because the two ionic scattering factors are similar. Peak spacings, relative intensities and the cell volume per formula unit change discontinuously. A simple uniform shift of the entire original pattern cannot describe the structural transition.

Use only the high-pressure CsCl structure branch to extrapolate its volume per formula unit $V=a^3$. A straight-line least-squares fit to those three measurements gives approximatelywith $P$ in GPa. Thus $V(10)\simeq 39.29{\mathring{\mathrm{A}}}^3$, $a(10)\simeq 3.400\mathring{\mathrm{A}}$ and $d_{111}=a/\sqrt{3}\simeq 1.963\mathring{\mathrm{A}}$. Taking Cu K$\alpha$ wavelength $\lambda \simeq 1.541\mathring{\mathrm{A}}$ in Bragg's law givesThe estimate is model-dependent: a straight-line extrapolation of $a(P)$ instead gives $2\theta \simeq 46.0^{\circ }$. The data stop at $3.81\mathrm{GPa}$, so the absence of further phase transitions does not by itself determine an exact equation of state to $10\mathrm{GPa}$. The $111$ peak can also be weak because its basis factor is $f_K-f_{Cl}$.

A reciprocal vector must have integral multiples of $2\pi$ as its dot products with lattice translations. The translations along the conventional axes require $G=2\pi (h/a,k/a,l/b)$, and the body-centering translation requires $h+k+l$ to be even. ThusThis is a face-centered tetragonal lattice, with conventional dimensions $4\pi /a,4\pi /a,4\pi /b$. The primitive unit cell volumes are $\boxed{a^{2}b/2}$ in direct space and $\boxed{2(2\pi {)}^3/(a^{2}b)}$ in reciprocal space. Their conventional cell volumes are $a^{2}b$ and $8(2\pi {)}^3/(a^{2}b)$, containing two and four lattice points respectively.

The listed plane is $x/a+z/b=1$, whose normal reciprocal vector is $\boxed{G_0=2\pi (1/a,0,1/b)}$. It is an allowed primitive vector in that direction; half of it has nonintegral indices. The plane spacing is $d=2\pi /|G_0|=(a^{-2}+b^{-2}{)}^{-1/2}=4a/5$ when $b=4a/3$. Bragg's law gives $2d\sin \theta =m\lambda$, soThese are approximately $18.21^{\circ },38.68^{\circ },69.64^{\circ }$. Here $\theta$ is the glancing Bragg angle; the angle between incident and scattered beams is $2\theta$.

In Compton scattering, a photon transfers energy and momentum to an initially stationary, approximately free electron. With incident and scattered wavelengths $\lambda ,{\lambda }^{\prime }$ and photon deflection angle $\theta$, momentum conservation givesCombine this with $hc/\lambda +m{c}^2=hc/{\lambda }^{\prime }+\sqrt{m^2{c}^4+p_e^2{c}^2}$ to obtainThe photon loses energy to electron recoil; the shift is independent of incident wavelength in this free-electron model and is largest at backscattering. At energies comparable with atomic binding, initial electron motion, binding and coherent scattering from the atom require additional treatment. Classical elastic scattering from a fixed charge alone does not produce this recoil wavelength shift.

In Arthur Holly Compton's spectral test, molybdenum-target X-rays scattered from graphite at $90^{\circ }$ were compared with the primary spectrum using the same slits, crystal and tube potential energy. A calcite crystal resolved the shifted line by Bragg's law; lines on both sides established angular zero. The crystal glancing angle on the plotted axis differs from the photon scattering angle. The compared curves had different intensity scales, and the paper did not directly observe the recoil electrons in this X-ray test. These controls support the wavelength comparison without converting it into a direct recoil-track measurement.^[ref]