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Sites at 000 and one half of each cubic coordinate give $F=f_A+(-1{)}^{h+k+l}{f}_B$. Identical sites produce a body-centered cubic lattice with absent odd-parity reflections. Distinct chemical species generally preserve only the simple cubic lattice translations, so those same odd reflections need not vanish. A geometric body center is not by itself a symmetry translation of the decorated crystal.

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]

The cubic crystal system has cubic point symmetry. Its three cubic Bravais lattices are primitive, body-centered, and face-centered. The body-centered cubic lattice is one member of that classification.

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.

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.

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 two sites give phases one and $\exp [i\pi (h+k+l)]=(-1{)}^{h+k+l}$. ThusFor real nonresonant form factors, even $h+k+l$ gives the square of the sum and odd $h+k+l$ the square of the difference. The factor $K_{hkl}$ represents the remaining experimental corrections, not another basis-interference term. Since the two ions differ, a half-body-diagonal translation does not preserve the decorated crystal. Its Bravais lattice is therefore simple cubic, with a two-ion basis; CsCl is not a monatomic body-centered cubic Bravais lattice.

Replacing both sites by identical Fe atoms changes the amplitude toNow the body-centering translation does preserve the structure. It is a body-centered cubic lattice with one Fe atom per primitive cell, or two per conventional cubic cell.

For 111 the parity is odd: CsCl generally has a nonzero amplitude $f_{\mathrm{C}{\mathrm{s}}^{+}}-f_{\mathrm{C}{\mathrm{l}}^{-}}$, whereas the identical-Fe 111 reflection is systematically absent. A numerical absolute CsCl intensity requires the angle-dependent form factors and experimental normalization; no such values are supplied. Forward electron counts, 54 for Cs+ and 18 for Cl-, explain why their amplitudes are substantially unequal, but are not substituted for the actual finite-angle factors without an additional approximation.

Use the integrated constant-compressibility model from c, with $\kappa p=0.0059\times 100=0.59$:ThereforeCompression reduces the spacing and moves the same allowed reflection to larger deflection, as the positive pressure derivative predicts. The input compressibility corresponds to a constant bulk modulus $1/\kappa \simeq 169.5$ GPa in this ideal model.

This is the stipulated unchanged cubic-phase, constant-compressibility extrapolation, rather than a reliable prediction for real Fe at 100 GPa. Real Fe exhibits a pressure-induced body-centered-cubic to hexagonal-close-packed transformation, and compressibility can change with pressure.^[ref] A real high-pressure diffraction interpretation would require its actual phase and equation of state. The source gives neither, so its numerical exercise is answered within its explicitly stated idealization.

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.

In the one-band electron Hall effect model, $R_H=-1/(ne)$, with $e>0$. The measured Hall coefficient therefore givesA monovalent body-centered cubic lattice has two conduction electrons per conventional unit cell. Thus $n{a}^3=2$, andCounting two spin states within the Fermi sphere gives $k_F=(3{\pi }^{2}n{)}^{1/3}=9.0418\times 10^9{\mathrm{m}}^{-1}$. Using the specified effective mass in the dispersion,There is one mole of conduction electrons per mole of sodium. Its molar density of states at $E_F$ is $3N_A/(2E_F)$, givingUsing the volume density of states in c givesThe effective mass changes the dispersion and density of states; the electron Bohr magneton used for the spin coupling retains the bare electron mass. Replacing it by $e\hslash /(2m^{\ast })$ would be an additional, unjustified magnetic-moment assumption. A Hall factor or a multiband metal would also invalidate the simple density inference; the stated calculation uses the requested one-band approximation.

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.

A Bravais lattice is the set of all integer combinations of linearly independent primitive translation vectors. Its reciprocal lattice consists of $g$ satisfying $g\cdot l\in 2\pi \mathbb{Z}$ for every translation $l$; equivalently $e^{ig\cdot l}=1$.

The original PDF's third family has a bare $j$, not $n_{2}j$. Taken literally, the listed set is not the claimed Bravais lattice: the claimed primitive vector $a(i+k)/2$ does not occur. Its half-integer first and third coordinates force it into that third family, whose second coordinate is always $a$, not zero. The intended repair is $j\mapsto n_{2}j$ in the third family. The following conclusions apply to that corrected set.

The four corrected families are exactlyThe allowed parity patterns are all even, or exactly two odd coordinates. An integer combination of the proposed primitive vectors has coordinates $(a/2)(m_2+m_3,m_1+m_3,m_1+m_2)$, so lies in this set. Conversely, solveAll three are integers precisely when $h+k+l$ is even. This proves that the corrected set is the face-centered cubic lattice with the proposed primitive basis.

Direct dot products give $a_i\cdot b_j=2\pi {\delta }_{ij}$ forThese vectors generate the entire reciprocal lattice, since the coefficients of an arbitrary reciprocal vector in this basis are its integer pairings with the $a_i$. Its vectors are $(2\pi /a)(h,k,l)$ with all three coordinates of the same parity, the body-centered cubic lattice.

In Bragg scattering, translating a scattering atom by $l$ changes the amplitude by $e^{i(k-k^{\prime })\cdot l}$. Constructive interference from every lattice translate requires this factor to be one, so $k^{\prime }-k=g$ is a reciprocal lattice vector. For elastic scattering, $|k^{\prime }|=|k|$; the chord between these two vectors has length $2|k|\sin (\theta /2)$. ThereforeThe shortest nonzero reciprocal vectors are the $(111)$ family of length $(2\pi /a)\sqrt{3}$, followed by the $(200)$ family of length $(2\pi /a)2$. Identical atoms on this Bravais lattice introduce no additional basis extinction. If both families are accessible at the incident wavelength, their scattering angles obey $\boxed{\sin ({\theta }_1/2)/\sin ({\theta }_2/2)=\sqrt{3}/2}$.

Let the Bravais lattice have primitive vectors $a_1,a_2,a_3$, and putThe reciprocal lattice has the basisThe scalar triple product and cross product identities giveConsequently every $q={\sum }_j{n}_j{b}_j$ satisfies $q\cdot l\in 2\pi \mathbb{Z}$ for every lattice point $l={\sum }_i{m}_i{a}_i$, and conversely these three conditions force the coefficients of $q$ in the reciprocal basis to be integers. This proves that the displayed vectors generate ${\Lambda }^{\ast }$.

In the Born approximation, suppose that the crystal potential is the sum of translates of one atomic potential $V_0$ over the finite set $S$ of lattice sites. Its Fourier transform at the momentum transfer $Q=k-k^{\prime }$ factors asThus the single-atom scattering amplitude is multiplied by the crystal lattice structure factor $\Delta (Q)$. Writingseparates the sum into three finite geometric series. For $l_i=-L_i/2,\dots ,L_i/2$ this givesWhen the $L_i$ are large, a factor has a sharp maximum when ${\alpha }_i=2\pi n_i$. All three factors are therefore simultaneously large precisely when $Q\in {\Lambda }^{\ast }$. These are the reciprocal-lattice peaks of a finite crystal.

For the stated body-centered cubic lattice basis,the reciprocal-basis formula givesEquivalently,The shortest nonzero reciprocal vectors have squared Euclidean norm $2(2\pi /a{)}^2$, so

In elastic scattering, $|k|=|k^{\prime }|=k$. If $q=k-k^{\prime }\in {\Lambda }^{\ast }$ and $\theta$ is the angle between $k$ and $k^{\prime }$, Euclidean geometry givesThe first possible diffraction peak therefore occurs atas $ka\to \infty$, by the small-angle approximation. This is the elastic Bragg scattering condition for the shortest reciprocal-lattice vector.

The characteristic effect is an irradiation-induced upward shift of the ductile-to-brittle transition in ferritic steel. A component previously used above its transition can then fail by cleavage fracture at the same service temperature, with much less warning from plastic deformation.

Ferrite has a body-centered cubic lattice. At low temperature, dislocation glide becomes difficult enough that brittle crack propagation can precede substantial plastic relaxation. Irradiation hardening raises the yield strength by introducing obstacles to dislocation glide; the competition with cleavage fracture consequently shifts toward brittle failure. Austenite, with its face-centered cubic lattice, does not normally show this same sharp cleavage transition. This does not make austenitic steel immune to loss of ductility, helium embrittlement or other radiation damage in materials.

Examine the fracture surface for cleavage fracture facets and river patterns, and compare Charpy impact test energy-versus-temperature curves from exposed material and an unirradiated reference of the same composition and heat treatment. The reference is important: a low impact energy from one failed specimen cannot establish an irradiation-induced shift. Transition curves can also use lateral expansion or the proportion of ductile fracture appearance.^[ref]

A direct fracture toughness test over the transition region complements the Charpy impact test; the two measurements are not interchangeable. Combine these observations with exposure history, local temperature, microstructure and crack location. Brittle appearance supports the failure mechanism, while a measured transition shift and suitable controls support its attribution to irradiation.

For hard spheres of radius r in a face-centered cubic lattice, contact occurs along a face diagonal: $4r=\sqrt{2}a$. Four spheres belong to a conventional cell, so the hard-sphere cubic packing fractions giveFor a body-centered cubic lattice, contact occurs along a body diagonal, $4r=\sqrt{3}a$, and there are two spheres per conventional cell:Their ratio $8/(3\sqrt{6})=1.0887>1$ proves that FCC packs more densely. Coordination alone would not establish the occupied volume fraction.

For conventional cubic cell side $a$, the face-centered cubic lattice has corner and face-centre sites, giving four lattice points per cell; the body-centered cubic lattice has corners and a body centre, giving two. Their primitive unit cells each contain one lattice point. With reciprocal vectors written $\mathbf{G}=(2\pi /a)(h,k,l)$, face-centered cubic lattice allows $h,k,l$ all even or all odd, while body-centred cubic allows $h+k+l$ even. HenceHere $a^{\ast }$ is the side of the reciprocal lattice's conventional cubic cell. The parity conditions describe the reciprocal point sets; reciprocal space contains wavevectors, not a second arrangement of physical atoms.

In a conventional face-centered cubic lattice cell of side $a_F$, touching nearest-neighbour hard spheres lie along a face diagonal. Four radii span that diagonal, so $\sqrt{2}{a}_F=4r$ and $a_F=2\sqrt{2}r$. In a conventional body-centered cubic lattice cell, the contact lies on the body diagonal: $\sqrt{3}{a}_B=4r$, giving $a_B=4r/\sqrt{3}$. ThereforeThese are the nonprimitive cubic unit cells requested. They contain four and two atoms respectively, so their volume ratio should not be confused with the volume-per-atom ratio, which is half as large, $3\sqrt{6}/8\simeq 0.9186$. The common sphere radius cancels.

For a Bravais lattice with primitive translation vectors ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$, define the reciprocal lattice by vectors $\mathbf{G}$ satisfying $e^{i\mathbf{G}\cdot \mathbf{R}}=1$ for every direct-lattice translation $\mathbf{R}$. A convenient basis iswith cyclic formulas for ${\mathbf{b}}_2,{\mathbf{b}}_3$. Then ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$ and $\mathbf{G}=h{\mathbf{b}}_1+k{\mathbf{b}}_2+l{\mathbf{b}}_3$ for integers $h,k,l$. The reciprocal primitive volume is $(2\pi {)}^3/V_{\mathrm{cell}}$.

These vectors are the spatial Fourier series frequencies of a periodic lattice potential or density. Elastic coherent crystal diffraction occurs when the scattering wavevector transfer equals a reciprocal lattice vector, with intensities additionally controlled by the basis crystal lattice structure factor. This is the reciprocal-space version of Bragg's law. A lattice of points and its basis should not be confused: a basis can make allowed reciprocal-lattice diffraction intensities vanish.

The first Brillouin zone is the Wigner-Seitz cell around the reciprocal origin: it consists of points closer to $\mathbf{G}=0$ than to any other reciprocal point. Its faces lie on the perpendicular bisectorsTo see the condition, equate $|\mathbf{k}{|}^2$ and $|\mathbf{k}-\mathbf{G}{|}^2$. These are also free-electron Bragg-degeneracy planes, where a periodic potential can open a band gap. Higher zones classify regions beyond successive Bragg planes; the reduced-zone description folds them into the first zone while retaining a band label.

In one dimension with spacing $a$, reciprocal points occur at $2\pi j/a$, and the first zone is $[-\pi /a,\pi /a]$. A square lattice of spacing $a$ has reciprocal basis $2\pi \hat{\mathbf{x}}/a,2\pi \hat{\mathbf{y}}/a$, and a square first zone with edges at $k_x,k_y=\pm \pi /a$.

The reciprocal of a face-centered cubic lattice is a body-centered cubic lattice and vice versa, with the scale fixed by the primitive-vector definition. In a Bloch state, wavevectors differing by $\mathbf{G}$ describe equivalent crystal momenta modulo the lattice periodicity. A crystal with $N$ primitive cells has $N$ allowed wavevectors in one Brillouin zone, hence $N$ states in each band per spin. This state count links reciprocal geometry to band filling, metallic conduction and band insulating behaviour. Reciprocal space also labels phonons and makes conservation of crystal momentum modulo $\hslash \mathbf{G}$ transparent.

Interpret the sphere as an atom of iron, and assume neighbouring atoms touch without overlap. Its radius is $r=0.124\mathrm{nm}=1.24\times 10^{-10}\mathrm{m}$, and its mass is $m_{\mathrm{Fe}}=56u\simeq 9.30\times 10^{-26}\mathrm{kg}$, using atomic mass unit $u=1.66054\times 10^{-27}\mathrm{kg}$.

In the face-centered cubic lattice, atoms touch along a face diagonal: $\sqrt{2}{a}_F=4r$, so $a_F=2\sqrt{2}r=0.3507\mathrm{nm}$. Four atoms occupy a conventional cube. The hard-sphere atomic volume in cubic lattices therefore givesIn the body-centered cubic lattice, atoms touch along the body diagonal: $\sqrt{3}{a}_B=4r$, so $a_B=4r/\sqrt{3}=0.2864\mathrm{nm}$. There are two atoms per conventional cube, givingThese are primitive unit cell volumes, not the volumes of the conventional cubes or of the atomic spheres. The denser FCC contact packing gives the higher mass density in this fixed-radius model. Actual hot phases have temperature-dependent dimensions and bonding, so these numbers should not be treated as exact experimental phase densities.

The direct body-centered cubic lattice has a face-centred cubic reciprocal lattice, whereas the direct face-centered cubic lattice has a body-centred cubic reciprocal lattice. Thus the reciprocal lattice change in a structural phase transition reverses the two cubic types: the reciprocal lattice changes from FCC to BCC when the real material changes from BCC to FCC. If the direct conventional side is $a$, the reciprocal conventional side is $4\pi /a$ in either case, with the appropriate phase-specific $a$.

Within one phase, ordinary thermal expansion increases direct spacings and decreases reciprocal spacings. At the structural phase transition, reciprocal points acquire the new symmetry and allowed positions. A first-order transformation may show both phases simultaneously and may create differently oriented domains; it need not continuously preserve one perfect reciprocal array throughout.

Using the fixed-radius estimates from part (b), ${\Omega }_F/{\Omega }_B\simeq 0.919$, so the reciprocal primitive cell volume would increase by about $8.9\%$. That comparison belongs to the ideal contact model. The question supplies neither the actual thermal expansion nor a crystallographic orientation relationship, so it does not determine a unique trajectory of individual reciprocal points during the transition.