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A crystal structure specifies the chemical species and spatial arrangement of the chemical atoms or ions in a crystal. For an ideal periodic structure, separate two ingredients: a Bravais lattice specifies the translations that repeat the arrangement, and an atomic basis of a crystal specifies the chemical motif repeated at every lattice point. The outward shape, or crystal habit, is a different description: the internal arrangement is what this article describes.

For a concrete two-dimensional model, choose primitive lattice translation vectors $a_1=(2,0)$ and $a_2=(0,1.5)$ in arbitrary length units. At every lattice point $R=m{a}_1+n{a}_2$, with integers $m,n$, place species A at $R$ and a different species B at $R+(0.65,0.55)$. Translating by $a_1$ or $a_2$ preserves both the positions and their chemical labels. Translating by $(0.65,0.55)$ sends one A position to a B position, so it is not a translation symmetry of the labeled structure. A picture of dots with the labels erased can therefore conceal essential chemical information.

A primitive unit cell contains one lattice point, but may contain several chemical atoms. A conventional unit cell can contain several lattice points to display the symmetry more clearly. Thus the conventional cube of a face-centered cubic lattice contains four lattice points, whereas a body-centered cubic lattice has two. Neither lattice name alone specifies which chemical species occupies the repeated motif. The NaCl conventional cell and coordination example makes the distinction explicit.

This separation is useful experimentally: interplanar spacings constrain the repeating geometry, while relative wave intensities also depend on the atomic basis of a crystal through the crystal basis structure factor. The Bragg crystal reflection experiment of 1913 describes the X-ray spectrometer, angular scans and detector checks in detail. Its conversion from angle to length is developed under Bragg's law; a geometrically possible reflection can still have zero wave intensity.

William Lawrence Bragg's 1913 structure determination supplied a complementary test. He directed X-rays through sodium chloride sections cut parallel to a cube face and recorded spots on photographic plates. Comparing section thicknesses tested how transmission altered the pattern: the relative spot strengths changed. Comparing related salts led him to two interpenetrating face-centered cubic lattices, one for each kind of constituent. He explicitly distinguished evidence for that arrangement from the still-needed identification of a single chemical atom with each scattering center. The photographs were evidence for a structural model, not direct pictures of labeled ions.^[ref]

A one-atom close-packed structure with a face-centered cubic lattice and $ABCABC\dots$ close-packed stacking sequence.

An $A{B}_2$ crystal structure with A on a face-centered cubic lattice and B occupying all eight tetrahedral sites of its conventional cubic unit cell. The coordination numbers are eight for A and four for B. The cubic cell contains four $A{B}_2$ formula units.

For identical touching atoms represented by hard spheres of radius $r$, a face-centered cubic lattice has conventional side $a_F=2\sqrt{2}r$ and four atoms per conventional cell. A body-centered cubic lattice has side $a_B=4r/\sqrt{3}$ and two atoms per conventional cell. Their primitive unit cell volumes are respectivelyEach primitive cell contains one atom in the monatomic model, so its mass density is atomic mass divided by $\Omega$. These are contact-geometry estimates; a fixed hard-sphere radius does not encode thermal expansion or phase-dependent electronic bonding.

The interplanar spacing $d$ is the perpendicular distance between parallel planes in a specified family. Measuring along an oblique line gives a larger distance, so the direction of measurement matters.

There is a short reciprocal lattice derivation. With the angular convention $a_i\cdot b_j=2\pi {\delta }_{ij}$, put $G=h{b}_1+k{b}_2+l{b}_3$. Then $G\cdot r=2\pi m$ is the same plane equation as $hx+ky+lz=m$. Thus $G$ is a normal vector. Moving between consecutive sheets along this normal changes the dot product by $2\pi$, givingFor perpendicular cubic axes of length $a$, this becomesFor example, choose a primitive cubic lattice with $a=0.600\mathrm{nm}$. Its $(210)$ lattice planes have $d_{210}=0.600/\sqrt{5}\simeq 0.268328\mathrm{nm}$. This is a chosen geometry, not an experimental value. The distance from a point to a plane gives the same result: the planes $2X+Y=0$ and $2X+Y=a$ in ordinary Cartesian length coordinates are separated by $a/\sqrt{5}$.

Specify whether $d_{hkl}$ denotes spacing between successive occupied lattice planes or the formal spacing assigned to a reflection label. In a primitive cubic lattice, $(200)$ has formal spacing $a/2$ but every other sheet contains no lattice points; its reflection is the second diffraction order of $(100)$. In a face-centered cubic lattice with conventional cubic axes, the $(200)$ sheets at half-integer $X/a$ do contain lattice points. Here $a/2$ is the actual spacing of the complete family perpendicular to that cube edge. The Primitive cubic Bragg peak positions example explains how to avoid counting a reflection twice.

An ordered cubic $A_{3}B$ structure with B at cube corners and A at face centres. The chemically labelled Bravais lattice is primitive cubic; its superlattice reflections distinguish it from a chemically disordered face-centered cubic lattice.

Ordinary solid sodium chloride has the rock salt structure. Choose a conventional cube of side $a$ with chlorine anions at the corners and face centers, and sodium cations at the edge centers and cube center. Corner sites are shared by eight cubes, face sites by two and edge sites by four. The unit cell counts are thereforeThere are four formula units per conventional unit cell, rather than four separate NaCl molecules.^[ref]

The Bravais lattice is a face-centered cubic lattice. One choice of primitive lattice translation vectors and atomic basis isThe determinant gives a primitive unit cell volume of $a^3/4$; that cell represents one cation and one anion. Four such volumes make the conventional cube. The union of all ion positions resembles a primitive cubic lattice of step $a/2$ only if the chemical labels are discarded: translation by $a(1,0,0)/2$ exchanges unlike ions and fails to preserve the chemical structure.

Each ion has six nearest unlike neighbors at distance $a/2$, along the positive and negative directions of the three cube axes. Thus the coordination number is six. Within either same-species face-centered cubic lattice, the nearest same-ion neighbors instead number twelve: their displacements are permutations of $(\pm a/2,\pm a/2,0)$, all of length $a/\sqrt{2}$. Since $a/\sqrt{2}>a/2$, those twelve are a more distant shell in the complete sodium chloride structure. The number twelve describes the same-ion FCC lattice neighborhood; six describes the nearest unlike-ion neighborhood.

An overlap estimate must use nearest magnetic-ion separations. For the face-centered cubic lattice of Mn, $d_{\mathrm{Mn}}=a_{\mathrm{Mn}}/\sqrt{2}=3.147\mathring{\mathrm{A}}$; for the body-centered cubic lattice of Fe, $d_{\mathrm{Fe}}=\sqrt{3}{a}_{\mathrm{Fe}}/2=2.477\mathring{\mathrm{A}}$. The supplied radial functions have inverse decay lengths ${\kappa }_Z=Z/(3a_B)$. Their intersite overlap therefore contains approximately $e^{-{\kappa }_{Z}d}$, up to polynomial and angular factors. The decisive exponent difference isThe exponential orbital-tail estimate of exchange thus gives a hopping suppression approximately $e^{-8.99}\simeq 1.25\times 10^{-4}$ before polynomial corrections. Keeping the supplied $Z^{7/2}{r}^2$ radial prefactor in a midpoint overlap proxy givesThis proxy is not an evaluated hopping integral, but confirms that the polynomial factors do not undo the exponential loss. If comparable Coulomb denominators and nonexponential prefactors are assumed, a kinetic-exchange scale proportional to $t^2/U$ is suppressed by roughly $10^{-8}$–$10^{-7}$. Direct intersite Coulomb exchange likewise contains products of the small tails. Coordination-number differences cannot compensate for many orders of magnitude.

By contrast, the measured ordering-temperature ratio supplied in the question is $122/1043\simeq 0.117$, not an exponentially tiny number. Within this overlap model, direct Mn–Mn exchange cannot explain MnO's substantial magnetic ordering scale by extrapolating the Fe mechanism. Another exchange path is required.

This is an order-of-magnitude argument, not an exact relation between an itinerant Curie temperature and a localized-spin Néel temperature. The hydrogenic functions use the printed bare $Z$ values; actual solid-state orbitals are screened and hybridized. Their literal absolute exchange scales should not be treated as quantitative predictions for Fe or MnO.

In a face-centered cubic lattice, nearest neighbours touch along a face diagonal. Four radii span that diagonal, so $\sqrt{2}a=4r$ andThusThe primitive vectors enclose volume $a^3/4$, containing one atom; the conventional cubic cell of volume $a^3$ contains four. Keeping these cell conventions distinct avoids a factor-of-four error in the electron density.

Use the total scattering angle $\psi =2\theta$. For a cubic unit cell, Bragg's law and the Miller indices giveThe four measured values of ${\sin }^2\theta$, normalized to the first, are $1,1.330,2.655,3.649$. Multiplying by three gives approximately $3,4,8,11$, the initial allowed sequence for a face-centered cubic lattice:The FCC lattice factor is $1+(-1{)}^{h+k}+(-1{)}^{h+l}+(-1{)}^{k+l}$; it vanishes unless all three indices have the same parity. Thus the initial ratios differ from those of a primitive or body-centered cubic lattice. This is the conventional minimal cubic indexing; arbitrarily hiding additional weak reflections could make a short list alone ambiguous.

The low-pressure rock salt structure has an FCC Bravais lattice with one NaCl formula unit per primitive unit cell, or four per conventional cube. A possible atomic basis of a crystal is Na at zero and Cl at $(1/2,0,0)$ in conventional fractional coordinates. The two species have unequal atomic scattering factors, so both the odd and even FCC families can occur.

The four independent lattice constant estimates areTheir small scatter reflects the rounded angular data. The appropriate reported result is

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 $(220)$.

Use fractional coordinates relative to conventional cubic vectors $(a\hat{x},a\hat{y},a\hat{z})$. A body-centered cubic lattice has representative pointsso its conventional cell contains two lattice points: eight shared corners contribute one and the interior point contributes one. A face-centered cubic lattice hasso its conventional cell contains four: one from the shared corners and three from six half-shared faces.

For $N$ atoms at fractional positions $(x_j,y_j,z_j)$, the geometric structure factor and the atomic-factor-weighted amplitude are respectivelyFor identical atoms $F=fG$. Applying the conventional-cell lattice factors givesandThese are lattice selection rules; an additional nontrivial crystal basis structure factor can extinguish further allowed reflections.

Of the eight equally represented parity triples, four satisfy the BCC condition and two the FCC condition. Thus the fractions among integer index triples are $1/2$ and $1/4$, respectively. This reflects the two and four equivalent lattice points in the conventional cells: their reciprocal lattices occupy corresponding subsets of the conventional reciprocal grids. In primitive reciprocal coordinates every integer triple denotes a reciprocal-lattice vector; lattice-centering absences are then built into the grid rather than an extra rule. A multi-atom chemical basis can still cause genuine additional absences. If permutations means merely reordering one fixed triple, its allowedness never changes: the quoted fractions concern all index triples, not reordering a chosen $(h,k,l)$.

The conventional face-centered cubic lattice cell contains four atoms, soThe preceding free-electron and Debye expressions giveAt low temperature the electronic and lattice heat-capacity crossover in a monovalent metal is found fromThus the electronic term dominates for $0<T\lesssim 3.4$ K within the normal-metal model. Both low-temperature approximations are well controlled at that crossover. At exactly zero temperature both heat capacities vanish; below sufficiently small finite-size energy scales a continuum model also needs revision.

Room temperature is not $T\ll {\theta }_D$: using the cubic formula at 300 K would incorrectly predict $C_{\mathrm{ph}}\simeq 142n{k}_B$, far above the harmonic high-temperature limit $3n{k}_B$. Instead use the Dulong-Petit law as an estimate:Keeping the finite cutoff in the Debye heat capacity givesAt 300 K this is $2.8006n{k}_B$, whereas $C_e=0.017994n{k}_B$, giving $\boxed{C_e/C_{\mathrm{ph}}=6.42\times 10^{-3}}$. Either appropriate room-temperature estimate shows that the lattice contribution dominates. The electronic approximation remains excellent because $300\ll T_F$.

For the four conventional face-centered cubic lattice sites $000$, $0\frac{1}{2}\frac{1}{2}$, $\frac{1}{2}0\frac{1}{2}$ and $\frac{1}{2}\frac{1}{2}0$, their centering factor isThis is the face-centred cubic selection rule. Enumerating the nonzero representatives with $h^2+k^2+l^2\le 16$ givesSymmetry-related permutations and signs belong to the same powder ring, not additional rings. For Al, the basis contains one atom, $S=F_{\mathrm{fcc}}{f}_{\mathrm{Al}}$. ThusFor the stated KCl basis,The all-even reflections add the two atomic factors; all-odd reflections subtract them. Ionic $K^{+}$ and $C{l}^{-}$ each have 18 electrons, giving similar X-ray amplitudes. Their difference is much smaller than their sum, so the 111 and 311 rings are weak. Consequently the four strong rings areThis is an accidental diffraction extinction from equal atomic scattering factors in the equal-factor limit, not an extra exact fcc selection rule. Isoelectronic ions can have different radial electron distributions, so real odd reflections need not vanish exactly. Even the cruder neutral-atom estimate $f_K\simeq 19$, $f_{\mathrm{Cl}}\simeq 17$ makes the odd/even basis-intensity ratio only $(2/36{)}^2\simeq 0.0031$, before other ring-intensity factors.

For Si's displaced identical basis atoms,For all-odd indices, the bracket is $1\pm i$, with squared magnitude 2. For all-even indices it vanishes when $h+k+l=2(\mathrm{mod}4)$, and has squared magnitude 4 when the sum is a multiple of four. Thus 200 and 222 vanish, whereas 111, 220, 311 and 400 survive:These are the basis extinctions of the ideal diamond cubic structure. Intensity strengths within each surviving set can still differ because of atomic factors, multiplicity and geometry.

A conventional face-centered cubic cell contains four atoms. With three conduction electrons per atom,Using the bare electron mass,The Drude transport relation is $\rho =m_e/(n{e}^2\tau )$, givingFor a degenerate electron gas, transport samples electrons near the Fermi surface, so use $v_F=\hslash k_F/m_e=2.025\times 10^6$ m/s:This is a transport relaxation time and associated transport mean free path inferred from the specified resistivity. It need not equal a single-particle quantum lifetime when scattering is strongly angle dependent. The free-electron valence count also neglects the detailed band structure of aluminium, as the stipulated model requires. A classical thermal speed is not the appropriate velocity scale for this strongly degenerate metal.

With the convention ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$, the reciprocal lattice of a face-centered cubic lattice is a body-centered cubic lattice. The reciprocal conventional cube is parallel to the direct conventional cube and has side $4\pi /a$. In direct-cubic coordinates its points areThe even triples supply the corners of that reciprocal cube and the odd triples its body centres. The atomic basis can change diffraction intensities through its structure factor; it does not change the reciprocal lattice determined by the translation lattice.

Within the stipulated localized, one-moment-per-atom spin model, the number density follows directly from mass density and molar mass:The face-centered cubic lattice is consistent with four atoms per conventional unit cell; no additional factor four is needed once $n$ has been obtained from the density. The Curie constant is $C=0.71549\mathrm{K}$. Identifying the model's $\theta =\lambda C$ with the specified Curie temperature givesAt very low temperature, a fully ordered single domain has $M_s=n{\mu }_B=8.4764\times 10^5\mathrm{A}{\mathrm{m}}^{-1}$. Its Weiss molecular field isThe cancellation of $n$ in the middle expression provides a useful check. The ordinary magnetostatic scale is only ${\mu }_0{M}_s\simeq 1.065\mathrm{T}$, smaller by the factor $\lambda$. Thus $B_{\mathrm{eff}}$ is not a plausible ordinary self-generated dipolar magnetic field of the nickel moments. It represents the exchange interaction energy in field units, not an additional Maxwell field of that magnitude. This does not assert that such large externally generated fields are forbidden. Nickel is an itinerant ferromagnet; the imposed one-${\mu }_B$-per-atom model is an estimate rather than its measured microscopic moment.

The specified two-direction constraint permits magnetic moments ${\mu }_z=\pm \mu$, withwhere ${\mu }_B$ is the Bohr magneton. There are four magnetic gadolinium ions per conventional face-centered cubic lattice cell; the closed-shell nitrogen ions add no local spin moment. Hence the magnetic-ion number density isAt high temperature, neglect interactions and use $B_z\simeq {\mu }_0{H}_z$. The one-ion partition function and mean magnetic dipole moment areIn the weak-magnetic field limit, $\tanh x\simeq x$, so $M_z=n⟨{\mu }_z⟩={\mu }_{0}n{\mu }^2{H}_z/(k_{B}T)$. Thus the dimensionless SI magnetic susceptibility is $\chi =C/T$, withThe coefficient has units of kelvin. The unrestricted-spin expression proportional to $S(S+1)/3$ is inappropriate here: the source explicitly retains only the two orientations with $S_z=\pm S$. In this two-state Curie law, the squared axial moment is $(g_{J}S{\mu }_B{)}^2$.

The leading coefficient fixes the atom count:With molar mass $40\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$, the sample mass isOne atom belongs to each primitive unit cell of monatomic argon on a face-centered cubic lattice, but the conventional cube contains four atoms. If its side is $a$, $N=4V/a^3$, and nearest neighbors lie across half a face diagonal. Thereforeusing $V=21\times 10^{-6}{\mathrm{m}}^3$. The corresponding conventional lattice constant is $a\simeq 5.18\text{A˚}$. Treating the source's one-atom cell as the conventional cube would omit the factor four and give a wrong nearest-neighbor distance.

For the cubic crystal system, the Miller indices satisfyThe inverse squared interplanar spacings have ratios approximately $1:2.004:3.012:4.010$. With the lattice constant close to $0.4\mathrm{nm}$, the appropriate squared-index sums are $1,2,3,4$. Thus the four powder diffraction rings areThe braces collect all symmetry-related lattice planes with the same interplanar spacing. In particular, $(100)$ is forbidden for both a body-centered cubic lattice and a face-centered cubic lattice, but occurs for a primitive cubic lattice. These observations identify the Bravais lattice as primitive cubic lattice, with a multi-atom atomic basis, consistent with a perovskite structure.

Each ring independently gives $a_j=d_j\sqrt{j}$:Their average is $0.42413\mathrm{nm}$. Allowing for the rounded measured angles, a suitable final value isA larger centred unit cell could relabel reflections, but would conflict with the stated approximate lattice constant.