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A balanced Dirichlet hyperbola method split takes real thresholds $A=B=\sqrt{N}$. An asymmetric split takes $1\le A\le N$ and $B=N/A$. Every lattice point with $ab\le N$ then has $a\le A$ or $b\le B$, and every point in their overlap rectangle is allowed because $AB=N$. For arithmetic functions $f,g$, define their partial sums by $F(t)={\sum }_{a\le t}f(a)$ and $G(t)={\sum }_{b\le t}g(b)$, always over positive integers. The same inclusion-exclusion principle, now with weight $f(a)g(b)$, yieldsNo multiplicativity assumption is needed: this is a finite rearrangement of Dirichlet convolution. The constant-one arithmetic function in both positions recovers the divisor summatory function. For $N=12$, $A=2$, $B=6$ givesIt uses two columns and six rows, with a twelve-point overlap.

For general positive real thresholds, $AB\ge N$ is sufficient for coverage, since $a>A$ and $b>B$ would imply $ab>AB\ge N$. If $AB>N$, however, the overlap can include points above the hyperbola. The correct subtraction is then ${\sum }_{a\le A,b\le B,ab\le N}f(a)g(b)$; the product $F(A)G(B)$ is justified only when the whole integer rectangle lies below it. For example, $N=12$, $A=B=4$ covers every allowed pair, but its rectangle contains the forbidden pair $(4,4)$, so subtracting all sixteen points would give $34$ instead of $35$.

Conversely, the balanced integer thresholds $r,r$ have $r^2\le N$ and still cover every allowed pair because $(r+1{)}^2>N$. More generally, positive integer thresholds $U,V$ give coverage exactly when $(U+1)(V+1)>N$, and their overlap is a full rectangle exactly when $UV\le N$. These discrete conditions explain why the square-root identity works even when $r^2<N$.

An atomic basis of a crystal is the motif of chemical species $s_j$ and relative positions $r_j$ attached identically to every lattice point $R$ of a Bravais lattice $\Lambda$. The ideal periodic structure is the labeled setThe primitive lattice translation vectors generate $R$ by integer combinations; the atomic basis lists what is repeated at $R$. These are different meanings of the word basis. In particular, the offsets $r_j$ need not be linearly independent vectors, and one of them may be zero. A primitive unit cell represents $N$ motif sites, even though it represents only one lattice point.

Changing the origin, choosing another primitive unit cell, or moving a motif site by a Bravais lattice translation changes the coordinates used to describe the same infinite crystal structure. The grouping into a motif does not, by itself, identify bonded molecules. In solid sodium chloride, grouping one sodium cation and one chlorine anion into an atomic basis does not create a collection of discrete NaCl molecules; the ionic crystal has an extended network of equivalent neighboring ions.

Consider $N$ noninteracting bosons in one internal quantum state, at thermal equilibrium in the three-dimensional harmonic oscillator potentialHere $m$ is the mass, the positive ${\omega }_i$ are angular frequencies, and $\bar{\omega }$ is their geometric mean. Measure the one-particle energy $ϵ$ and chemical potential $\mu$ from the ground-state energy. The quantum harmonic oscillator levels then have $ϵ=\hslash ({\omega }_x{n}_x+{\omega }_y{n}_y+{\omega }_z{n}_z)$ with nonnegative integer $n_i$ and reduced Planck constant $\hslash =h/(2\pi )$.

When $ϵ\gg \hslash {\omega }_i$, counting lattice points can be approximated by the volume of a tetrahedron in the positive octant of $(n_x,n_y,n_z)$:This is a density of states, with units of inverse energy. The factor $1/6$ is the unit tetrahedron volume; differentiating produces the factor $1/2$. Treat the single ground state separately: its macroscopic population would be lost by replacing every level with a smooth density of states.

For $k_{B}T\gg \hslash {\omega }_i$, integrate the Bose-Einstein distribution over excited energy eigenstates. As $\mu$ approaches zero from below, their maximum population isHere $k_B$ is the Boltzmann constant and $\zeta$ the Riemann zeta function. A short proof uses $1/(e^x-1)={\sum }_{\ell \ge 1}{e}^{-\ell x}$ and ${\int }_0^{\infty }{x}^2{e}^{-\ell x}dx=2/{\ell }^3$. The nonnegative terms permit exchanging summation and integration by Tonelli's theorem, giving ${\int }_0^{\infty }{x}^2/(e^x-1)dx=2\zeta (3)$. Near zero the integrand is proportional to $x$, so its integral converges. That finite excited-state capacity is the mechanism of Bose-Einstein condensation.

Set $N_{\mathrm{ex}}^{\mathrm{sat}}(T_c)=N$. Particle conservation gives, to this continuum accuracy,Above $T_c$, the macroscopic condensate fraction is zero and the chemical potential adjusts below zero. This cubic power belongs to a three-dimensional harmonic trap; it is not the homogeneous three-dimensional ideal gas exponent. For $N=10^5$, $\bar{\omega }/(2\pi )=100\mathrm{Hz}$ and $\zeta (3)\simeq 1.202056903$, the model gives $T_c\simeq 209.5\mathrm{nK}$. At $T=0.8T_c\simeq 167.6\mathrm{nK}$, $N_0/N=0.488$, or approximately $48800$ condensed bosons. These are chosen parameters and calculated values.

Finite $N$ gives a smooth crossover rather than a sharp phase transition. The exact ideal gas sum over excited quantum harmonic oscillator levels differs from the continuum integral, especially when $k_{B}T$ is not large relative to every $\hslash {\omega }_i$. A sharp trapped-gas limit takes $N\to \infty$, $\bar{\omega }\to 0$ with $N{\bar{\omega }}^3$ fixed and fixed frequency ratios. Interactions also alter the critical temperature, condensate fraction and release profile; a dilute gas can have appreciable interaction energy compared with the trap level spacing. The cubic law is a useful baseline, not a fit guaranteed for either 1995 experiment.

A Bravais lattice with three perpendicular conventional axes and lattice points at $(0,0,0)$ and $(1/2,1/2,0)$. A translation by half a diagonal of an $ab$ face preserves the complete crystal motif; apparent centering of unlabelled molecular centers alone is insufficient.

A crystallographic unit cell chosen to display the symmetry or a standard setting of a crystal structure. It tiles the periodic arrangement but need not be primitive. A conventional FCC cube contains four lattice points and a BCC cube contains two; a primitive unit cell contains one. These are lattice-point counts, not a universal count of atoms in a chemical motif.

The arrangement of atoms or other constituents attached identically to every lattice point of a Bravais lattice. The lattice specifies translations; the motif specifies the contents repeated by those translations. A multi-atom motif does not make its atoms separate lattice points.

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 Dirichlet hyperbola method turns a long summation over a Dirichlet convolution into two short ones by covering the lattice points below a hyperbola with two overlapping strips. Its simplest application is an exact count: for a positive integer $N$, every positive integer divisor $a$ of $n\le N$ corresponds to the ordered pair $(a,n/a)$. Thus the divisor summatory function counts precisely the ordered pairs of positive integers with $ab\le N$:Here $\tau$ is the divisor function, and the floor function gives the number of allowed values of $b$ for each $a$. The axes are excluded, while points on $ab=N$ are included.

Set $r=\lfloor \sqrt{N}\rfloor$, using the floor function and the positive square root. Every counted pair has $a\le r$ or $b\le r$. Otherwise $ab\ge (r+1{)}^2>N$. The first strip contains ${\sum }_{a=1}^r\lfloor N/a\rfloor$ points; swapping the two coordinates gives the same count for the second. Their overlap contains all $r^2$ pairs with $1\le a,b\le r$, since $r^2\le N$. The inclusion-exclusion principle therefore gives the exact identityThe subtraction is for the whole overlap square, rather than only the diagonal. The same Dirichlet hyperbola method identity holds for real $x\ge 1$, with $D(x)={\sum }_{n\le x}\tau (n)$ and $r=\lfloor \sqrt{x}\rfloor$. ^[ref]

The divisor summatory function adds the divisor function up to a cutoff $x\ge 1$. It counts all ordered pairs of positive integers whose product is at most $x$. For $N=12$, the twelve individual divisor function values areCounting columns of lattice points instead gives $12+6+4+3+2+2+1+1+1+1+1+1=35$. The Dirichlet hyperbola method needs only the first three column counts:The arithmetic work of the Dirichlet hyperbola method is much smaller even though the answer is identical.

This is useful for elementary averages: individual divisor function values fluctuate, but their summation has the regular behavior given by the divisor summatory asymptotic. Dividing that estimate by a positive integer $N$ giveswhere $\log$ is the natural logarithm, $\gamma$ is the Euler--Mascheroni constant, and $O$ uses big O notation. This describes an average, rather than an estimate for every individual divisor function value. ^[ref]

A floor sum adds values of a floor function, often in the form $F(n,a,b,m)={\sum }_{i=0}^{n-1}\lfloor (ai+b)/m\rfloor$ for integers $n\ge 0$, $m>0$, $a$, and $b$. If $a,b\ge 0$, it counts lattice points $(i,j)$ with $0\le i<n$ and $1\le j\le (ai+b)/m$. Splitting $a$ and $b$ into quotients and remainders isolates an arithmetic progression and a constant contribution; swapping the remaining counting axes leads to an Euclidean algorithm reduction.

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.

A lattice plane is a plane containing a two-dimensional periodic array of lattice points. Parallel lattice planes form equally spaced geometrical sheets; an atomic basis can place chemical atoms between them.

With primitive lattice translation vectors $a_i$, write $r=x{a}_1+y{a}_2+z{a}_3$. For integers $h,k,l$ with greatest common divisor one, the plane equationsgive a complete family labeled by Miller indices $(hkl)$. In a primitive cubic lattice, $(110)$ means $x+y=m$, parallel to $a_3$.^[ref]

A lattice translation moves every lattice point of a Bravais lattice onto another lattice point. Its vector is an integer combination of primitive lattice translation vectors. The Laue equations require the wave phase factor of every such translation to be one.

Using the printed $x,y$ axes, take a lower vertex of cell B as origin. One allowed pair of primitive lattice translation vectors isBoth vectors connect nearest lattice points, have length $\sqrt{2}a$, and are perpendicular. Their determinant has magnitude $2a^2$, consistent with a primitive area. Exchanging them or reversing their directions gives equivalent basis conventions; the corresponding reciprocal lattice vectors must then use the same convention.

A unit cell is a repeat region whose lattice translations tile the plane. A primitive unit cell represents exactly one lattice point and has the minimum repeat area for a lattice basis.

Cell A is nonprimitive; cell B is primitive. A represents two points and has area $4a^2$. B has four shared corner points and no additional interior point, represents one point and has area $2a^2$. Fractional corner counting refers to the tiling, not to fractional physical atoms.

The stipulated conventional cell is a square of side $2a$. Although introduced as a centered rectangular lattice, this equal-side case is a square Bravais lattice viewed with a rotated primitive basis. A corner-to-center separation iswhich is shorter than the $2a$ separation between adjacent conventional corners. Consequently the first contact of equal circles occurs atThe conventional square represents two lattice points: one from its four shared corners and one from its center. Its packing fraction is thereforeEquivalently, one circle occupies each primitive area $2a^2$. This is square circle packing; the drawing does not define a triangular close-packed lattice.

There are $L$ primitive cells and therefore $L$ lattice points. The specified reciprocal-space density $L{A}_c/(2\pi {)}^2$ counts distinct allowed wavevectors; each supports two electron spin states. Filling the disk at zero temperature givesUsing $A_c=2a^2$ and $k_F^2={\pi }^2/(2a^2)$,The circle fills a fraction $\pi /4$ of the first-zone area, while one spin-degenerate band has two states per primitive cell; these give the same result. A single-spin count would instead be $\pi /4$, omitting the physical twofold spin degeneracy of this model. The circle has only just reached the sides, so the zone corners remain unfilled: first contact does not mean filling the entire first band.

A Bravais lattice is an infinite periodic set of equivalent points $\mathbf{R}=n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+n_3{\mathbf{a}}_3$, with integer $n_i$ and independent primitive translations ${\mathbf{a}}_i$. It describes translation symmetry, without assigning a chemical species to a point. An atomic basis of a crystal is the motif of atoms, with their species and relative positions, repeated identically at every lattice point. The resulting crystal structure consists of the lattice decorated by that basis; the basis can contain several atoms even when the lattice is primitive.

A primitive unit cell tiles space under lattice translations and contains exactly one lattice point, counting shared boundary points fractionally. Its volume is $|{\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)|$. A conventional unit cell is chosen to display useful crystal symmetry and may contain several lattice points. Thus a centered conventional cell is not primitive, but the crystal still has a primitive cell. Neither cell size nor the number of dots drawn on its boundary should be confused with the number of atoms in the crystal basis.

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 $(h,k,l)$.

A Bravais lattice is a set of points generated by integer translations along three independent primitive lattice translation vectors:Every lattice point has the same environment under lattice translations. A crystal structure consists of this Bravais lattice with an atomic basis of a crystal repeated at every lattice point; the set of atomic positions need not itself be a Bravais lattice.

The two positions in the conventional cubic cell of alpha-tungsten give a body-centred cubic lattice, with one tungsten atom per primitive cell and two per conventional cell. One possible set of primitive lattice translation vectors isTheir primitive-cell volume is $a^3/2$.