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Arjen K. Lenstra works in computational number theory and cryptography. With Hendrik W. Lenstra and László Lovász, he authored the 1982 paper introducing the LLL algorithm and applying lattice reduction to factoring polynomials with rational number coefficients in polynomial time.^[ref]

A rational factorization can be converted to an integer-coefficient problem, then studied through modular information and short lattice vectors. Reduction helps recover an exact factor from that indirect arithmetic data. The achievement concerns polynomial factorization; it does not provide a polynomial-time algorithm for general integer factorization. His EPFL interview describes his introduction to randomized algorithms and his approach to teaching mathematical computation.^[ref]

A crystal is a solid with long-range spatial order in its constituents. In a periodic crystal, translation by a lattice vector reproduces the structure; a unit cell and its constituent arrangement describe the repeating pattern. The atomic structure is distinct from the visible external shape: an irregularly cut specimen can remain a single ordered crystal.

Diffraction probes that order because radiation scattered from many regularly arranged sites can reinforce at selected directions. The resulting pattern reveals structural spacings and orientation, rather than merely showing the specimen's outer faces. A single crystal maintains one coherent orientation across the specimen, while a polycrystalline material contains many crystal grains. Crystal orientation can make properties anisotropic, so a test along one direction need not predict response along another. Surface shape additionally depends on equilibrium surface free energy and growth kinetics.

A crystal lattice records the translation symmetry of a periodic crystal. With a primitive basis of three linearly independent lattice vectors $a_1,a_2,a_3$, its translation vectors formThe crystal structure additionally specifies the arrangement of atoms associated with each lattice point. A lattice point is therefore a symmetry marker, not necessarily one atom.^[ref]

For example, a one-dimensional chain with alternating species $A,B$ at spacing $a$ has translation period $2a$, not $a$: shifting by $a$ exchanges distinct species. It can be described by lattice $2a\mathbb{Z}$ with a two-atom basis at offsets zero and $a$. In a diffraction measurement, the translation lattice selects possible reciprocal-space peak positions, while the atomic basis affects their intensities and systematic absences. This distinction lets diffraction reveal internal structure rather than just the visible shape of the specimen.

A lattice cube is a cube whose vertices belong to a specified lattice. Its three edge vectors are mutually perpendicular lattice vectors of equal length. In the simple cubic lattice $a{\mathbb{Z}}^3$, an axis-aligned cube of side $a$ is a primitive unit cell.

Larger cubes can have lattice vertices while enclosing many primitive cells, so “lattice cube” is a geometric condition rather than a synonym for primitive cell. For a cube of side $Na$ in that example, its volume is $N^3$ times the primitive-cell volume. Consistent shared-boundary counting assigns one lattice point to the side-$a$ cell, even though eight corner points are visible in a closed drawing.

A lattice parallelepiped is a parallelepiped spanned by three linearly independent lattice vectors. If the vectors are a primitive basis of a Bravais lattice, translates of the corresponding half-open parallelepiped tile space without overlap, with one lattice point per cell under consistent boundary counting.

Its volume is $|{\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)|$. Choosing longer lattice vectors can instead produce a nonprimitive cell containing several lattice points; integer coordinates of its vertices alone do not prove primitiveness. The primitive version is useful for counting states and relating real-space volume to the volume of a reciprocal lattice cell.

A direct lattice vector is an integer linear combination of primitive translation vectors ${\mathbf{a}}_i$ of a Bravais lattice. Translating by such a vector preserves the lattice. Its dot product with every reciprocal lattice vector is an integer multiple of $2\pi$ in the angular-wavevector convention.

A magnetic flux quantum is the flux scaleassociated with a coherent degree of freedom of electric charge $q\ne 0$. Its Aharonov-Bohm phase on a loop enclosing magnetic flux $\Phi$ is $q\Phi /\hslash$; adding ${\Phi }_0$ changes that phase by one full turn.

For an electron, the magnitude of the charge is $e$, giving $h/e$. A conventional superconducting Cooper pair has charge magnitude $2e$, giving $h/(2e)$. The relevant charge must therefore be stated: a superconducting flux scale and a single-electron interference period differ by a factor of two.

In the magnetic translation algebra, translations along two lattice vectors commute when the enclosed unit-cell magnetic flux obeysThis is a condition on the algebra for the specified cell, not a claim that every externally applied magnetic flux must be quantized. The distinct magnetic flux spectrum of a quantum ring illustrates flux-periodic energy levels.

For a lattice vector $r$,acts in the symmetric gauge asIt commutes with $(p-eA{)}^2/(2m)+V(x)$ when $V(x+r)=V(x)$.

Put $v={\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)\ne 0$. DefineThe scalar triple product gives ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$. Consequently every integer combination $\mathbf{G}={\sum }_j{m}_j{\mathbf{b}}_j$ satisfiesThis proves that the proposed vectors generate points in the reciprocal lattice, but primitiveness also requires the converse. Let $\mathbf{G}$ satisfy the exponential condition for every lattice vector, and in particular for $\mathbf{R}={\mathbf{a}}_i$. ThenSince the ${\mathbf{b}}_i$ form a basis of three-dimensional space, the vector ${\sum }_i{m}_i{\mathbf{b}}_i$ has the same dot products with each ${\mathbf{a}}_j$ as $\mathbf{G}$. Their difference is orthogonal to three independent vectors and is zero. Henceand no additional reciprocal lattice vectors lie outside that integer span. Thus the ${\mathbf{b}}_i$ are primitive, not merely three possible reciprocal translations.

Translate the integration variable by an arbitrary lattice vector and use the periodicity of $V$:Thus $(1-e^{i\mathbf{q}\cdot \mathbf{R}})V_{\mathbf{q}}=0$ for every $\mathbf{R}$. A nonzero coefficient requires $e^{i\mathbf{q}\cdot \mathbf{R}}=1$ for every lattice translation, which is precisely the defining condition for $\mathbf{q}\in {\Lambda }^{\ast }$.

For an infinite crystal the integral is a Fourier transform of a periodic distribution, rather than an ordinary convergent integral. More explicitly, the Fourier seriesgivesThis is the precise meaning of zero away from the reciprocal lattice. A finite specimen instead has broadened reciprocal-lattice peaks of a finite crystal; exact delta-function support is an infinite-periodicity idealization.

Given independent primitive lattice translation vectors ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$, defineThen ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$. The reciprocal lattice consists ofequivalently all wavevectors for which $e^{i\mathbf{G}\cdot \mathbf{R}}=1$ for every direct lattice vector $\mathbf{R}$.

For the face-centered cubic lattice, chooseTheir scalar triple product is $a^3/4$, and direct evaluation of the cross products givesEvery integer combination has coordinates $(2\pi /a)(H,K,L)$ with $H,K,L$ all even or all odd. Conversely, for such coordinates the coefficients $(K+L)/2,(H+L)/2,(H+K)/2$ are integers, so every such point occurs. The even-coordinate points are the corners of cubes of edge $4\pi /a$, and the odd-coordinate points are their body centres. ThereforeThis is a property of the Bravais lattice; the two-atom atomic basis changes structure factors rather than these reciprocal lattice translation vectors.

For the mutually orthogonal direct primitive lattice translation vectors ${\mathbf{a}}_i$, define the reciprocal lattice basis byThus $\mathbf{G}={\sum }_i{h}_i{\mathbf{b}}_i$, with integer $h_i$, satisfies $\exp (i\mathbf{G}\cdot \mathbf{R})=1$ for every lattice vector $\mathbf{R}$.

Normalize the plane waves in a periodic crystal of area or volume $\mathcal{V}$. The Fourier coefficients of the periodic potential obeyTranslation of the integral by any lattice vector $\mathbf{R}$ leaves it invariant but multiplies it by $e^{-i(\mathbf{k}-{\mathbf{k}}^{\prime })\cdot \mathbf{R}}$. A nonzero result therefore requires $\mathbf{k}-{\mathbf{k}}^{\prime }$ to be a reciprocal lattice vector.

Near an isolated Bragg scattering degeneracy, ordinary nondegenerate perturbation theory fails because the free energies coincide. Retain the resonant plane waves $|\mathbf{k}⟩$, $|\mathbf{k}+\mathbf{G}⟩$ and apply degenerate perturbation theory. Absorb the constant $V_0$ into the energy origin. Since a real potential has $V_{-\mathbf{G}}=V_{\mathbf{G}}^{\ast }$, the projected Schrödinger equation has coefficient matrixA nonzero eigenstate requiresOff-resonant plane waves give higher-order energy shifts omitted in this leading nearly-free electron model.

A reciprocal lattice vector $\mathbf{G}$ has the displayed property for every direct lattice vector $\mathbf{R}$. Thus $e^{i\mathbf{G}\cdot \mathbf{R}}=1$. Fourier components of a lattice-periodic potential carry these wavevectors and couple plane-wave electronic states whose momenta differ by $\hslash \mathbf{G}$. This is the origin of the Bragg-plane mixing in a nearly-free electron model.