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In
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
and
solid state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
, the Laue equations relate incoming waves to outgoing waves in the process of
elastic scattering Elastic scattering is a form of particle scattering in scattering theory, nuclear physics and particle physics. In this process, the kinetic energy of a particle is conserved in the center-of-mass frame, but its direction of propagation is modi ...
, where the photon energy or light temporal frequency does not change by scattering, by a
crystal lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
. They are named after physicist
Max von Laue Max Theodor Felix von Laue (; 9 October 1879 – 24 April 1960) was a German physicist who received the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals. In addition to his scientific endeavors with cont ...
(1879–1960). The Laue equations can be written as \mathbf= \mathbf_ - \mathbf_ = \mathbf as the condition of elastic wave scattering by a crystal lattice, where \mathbf_, \mathbf k_, and \mathbf are an incoming (to the crystal)
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
, an outgoing (from the crystal by scattering) wavevector, and a
reciprocal lattice vector In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial f ...
for the crystal respectively. Due to elastic scattering , \mathbf_, ^2=, \mathbf_, ^2, three vectors. \mathbf, \mathbf_, and -\mathbf_ , form a
rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
if the equation is satisfied. If the scattering satisfies this equation, all the crystal lattice points scatter the incoming wave toward the scattering direction (the direction along \mathbf k_). If the equation is not satisfied, then for any scattering direction, only some lattice points scatter the incoming wave. (This physical interpretation of the equation is based on the assumption that scattering at a lattice point is made in a way that the scattering wave and the incoming wave have the same phase at the point.) It also can be seen as the conservation of momentums as \hbar \mathbf_ = \hbar \mathbf_ + \hbar \mathbf since \mathbf is the wavevector for a plane wave associated with parallel crystal lattice planes. (Wavefronts of the plane wave are coincident with these lattice planes.) The equations are equivalent to
Bragg's law In physics and chemistry , Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposition of wave ...
; the Laue equations are vector equations while Bragg's law is in a form that is easier to solve, but these tell the same content.


The Laue equations

Let \mathbf\, ,\mathbf\, ,\mathbf be primitive translation vectors (shortly called primitive vectors) of a crystal lattice L , where atoms are located at lattice points described by \mathbf x = p\,\mathbf a+q\,\mathbf b+r\,\mathbf c with p, q, and r as any
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
. (So \mathbf x indicating each lattice point is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
linear combination of the primitive vectors.) Let \mathbf_ be the
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
of an incoming (incident) beam or wave toward the crystal lattice L , and let \mathbf k_ be the wavevector of an outgoing (diffracted) beam or wave from L . Then the vector \mathbf k_ - \mathbf k_ = \mathbf, called the scattering vector or ''transferred wavevector'', measures the difference between the incoming and outgoing wavevectors. The three conditions that the scattering vector \mathbf must satisfy, called the Laue equations, are the following: : \mathbf\cdot \mathbf =2\pi h :\mathbf\cdot \mathbf =2\pi k :\mathbf\cdot \mathbf =2\pi l where numbers h, k, l are
integer numbers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
. Each choice of integers (h,k,l), called
Miller indices Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''â ...
, determines a scattering vector \mathbf. Hence there are infinitely many scattering vectors that satisfy the Laue equations as there are infinitely many choices of Miller indices (h,k,l). Allowed scattering vectors \mathbf form a lattice L^*, called the
reciprocal lattice In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
of the crystal lattice L , as each \mathbf indicates a point of L^*. (This is the meaning of the Laue equations as shown below.) This condition allows a single incident beam to be diffracted in infinitely many directions. However, the beams corresponding to high Miller indices are very weak and can't be observed. These equations are enough to find a basis of the reciprocal lattice (since each observed \mathbf indicates a point of the reciprocal lattice of the crystal under the measurement), from which the crystal lattice can be determined. This is the principle of
x-ray crystallography X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
.


Mathematical derivation

For an incident
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
at a single frequency \displaystyle f (and the angular frequency \displaystyle \omega =2\pi f) on a crystal, the diffracted waves from the crystal can be thought as the sum of outgoing plane waves from the crystal. (In fact, any wave can be represented as the sum of plane waves, see
Fourier Optics Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or '' superposition'', of plane waves. It has some parallels to the Huygens–Fresnel pr ...
.) The incident wave and one of plane waves of the diffracted wave are represented as : \displaystyle f_(t,\mathbf )=A_\cos(\omega \,t-\mathbf _\cdot \mathbf +\varphi _), :\displaystyle f_(t,\mathbf )=A_\cos(\omega \,t-\mathbf _\cdot \mathbf +\varphi _), where \displaystyle \mathbf _ and \displaystyle \mathbf _ are wave vectors for the incident and outgoing plane waves, \displaystyle \mathbf is the
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s ...
, and \displaystyle t is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
representing time, and \varphi _ and \varphi _ are initial phases for the waves. For simplicity we take waves as
scalars Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
here, even though the main case of interest is an electromagnetic field, which is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
. We can think these scalar waves as components of vector waves along a certain axis (''x'', ''y'', or ''z'' axis) of the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
. The incident and diffracted waves propagate through space independently, except at points of the lattice L of the crystal, where they resonate with the oscillators, so the phases of these waves must coincide. At each point \mathbf x = p\,\mathbf a+q\,\mathbf b+r\,\mathbf c of the lattice L , we have : \cos (\omega\,t-\mathbf k_\cdot\mathbf x+\varphi _) = \cos (\omega\,t-\mathbf k_\cdot\mathbf x+\varphi _), or equivalently, we must have : \omega\,t-\mathbf k_\cdot\mathbf x + \varphi _= \omega\,t-\mathbf k_\cdot\mathbf x + \varphi _ + 2\pi n, for some integer n , that depends on the point \mathbf x . Since this equation holds at \mathbf x=0 , \varphi _ = \varphi _+2\pi n' at some integer n'. So : \omega\,t-\mathbf k_\cdot\mathbf x = \omega\,t-\mathbf k_\cdot\mathbf x + 2\pi n. (We still use n instead of (n-n') since both the notations essentially indicate some integer.) By rearranging terms, we get : \mathbf\cdot \mathbf x = (\mathbf k_-\mathbf k_)\cdot \mathbf x = 2\pi n. Now, it is enough to check that this condition is satisfied at the primitive vectors \mathbf a,\mathbf b,\mathbf c (which is exactly what the Laue equations say), because, at any lattice point \mathbf x = p\,\mathbf a+q\,\mathbf b+r\,\mathbf c , we have : \mathbf \cdot \mathbf x = \mathbf\cdot (p\,\mathbf a+q\,\mathbf b+r\,\mathbf c) = p(\mathbf\cdot \mathbf a) + q(\mathbf\cdot \mathbf b) + r(\mathbf\cdot \mathbf c) = p\,(2\pi h) + q\,(2\pi k) + r\,(2\pi l) = 2\pi(ph+qk+rl)=2\pi n, where n is the integer ph+qk+rl . The claim that each parenthesis, e.g. (\mathbf \cdot \mathbf), is to be a multiple of 2\pi (that is each Laue equation) is justified since otherwise p(\mathbf\cdot \mathbf a) + q(\mathbf\cdot \mathbf b) + r(\mathbf\cdot \mathbf c) = 2\pi n does not hold for any arbitrary integers p, q, r. This ensures that if the Laue equations are satisfied, then the incoming and outgoing (diffracted) wave have the same phase at each point of the crystal lattice, so the oscillations of atoms of the crystal, that follows the incoming wave, can at the same time generate the outgoing wave at the same phase of the incoming wave.


Relation to reciprocal lattices and Bragg's Law

If \mathbf=h \mathbf+k \mathbf+l \mathbf with h, k, l as integers represents the
reciprocal lattice In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
for a crystal lattice L (defined by \mathbf x = p\,\mathbf a+q\,\mathbf b+r\,\mathbf c) in real space, we know that \mathbf\cdot \mathbf = \mathbf \cdot (p \mathbf+q \mathbf+r \mathbf)= 2\pi(hp+kq+lr) = 2\pi n with an integer n due to the known orthogonality between primitive vectors for the reciprocal lattice and those for the crystal lattice. (We use the physical, not crystallographer's, definition for reciprocal lattice vectors which gives the factor of 2\pi.) But notice that this is nothing but the Laue equations. Hence we identify \mathbf= \mathbf_ - \mathbf_ = \mathbf, means that allowed scattering vectors \mathbf= \mathbf_ - \mathbf_ are those equal to reciprocal lattice vectors \mathbf for a crystal in diffraction, and this is the meaning of the Laue equations. This fact is sometimes called the ''Laue condition''. In this sense, ''diffraction patterns are a way to experimentally measure the reciprocal lattice for a crystal lattice.'' The Laue condition can be rewritten as the following. \begin \mathbf &= \mathbf_ - \mathbf_\\ \rightarrow , \mathbf_, ^2 &= , \mathbf_ - \mathbf, ^2\\ \rightarrow , \mathbf_, ^2 &= , \mathbf_, ^2 - 2\mathbf_\cdot\mathbf + , \mathbf, ^2. \end Applying the elastic scattering condition , \mathbf_, ^2=, \mathbf_, ^2 (In other words, the incoming and diffracted waves are at the same (temporal) frequency. We can also say that the energy per
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
does not change.) to the above equation, we obtain : 2\mathbf_\cdot\mathbf=, \mathbf, ^2, :2\cdot (-\mathbf)=, \mathbf. The second equation is obtained from the first equation by using \mathbf_ - \mathbf_ = \mathbf. The result 2\mathbf_\cdot\mathbf=, \mathbf, ^2 (also 2\cdot (-\mathbf)=, \mathbf) is an equation for a
plane (geometry) In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
(as the set of all points indicated by \mathbf_ satisfying this equation) as its equivalent equation \mathbf\cdot (2-\mathbf)=0 is a plane equation in geometry. Another equivalent equation, that may be more easier to understand, is \cdot \widehat=\frac\left, \mathbf \ (also (-)\cdot \widehat=\frac\left, \mathbf \). This indicates the plane that is perpendicular to the straight line between the reciprocal lattice origin \mathbf=0 and \mathbf and located at the middle of the line. Such a plane is called Bragg plane. This plane can be understood since \mathbf = \mathbf_ - \mathbf_ for scattering to occur (It is the Laue condition, equivalent to the Laue equations.) and the elastic scattering , \mathbf_, ^2=, \mathbf_, ^2 has been assumed so \mathbf, \mathbf_, and -\mathbf_ form a
rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
. Each \mathbf is by definition the
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
of a plane wave in the Fourier series of a spatial function which periodicity follows the crystal lattice (e.g., the function representing the electronic density of the crystal),
wavefronts In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
of each plane wave in the Fourier series is perpendicular to the plane wave's wavevector \mathbf, and these wavefronts are coincident with parallel crystal lattice planes. This means that X-rays are seemingly "reflected" off parallel crystal lattice planes perpendicular \mathbf at the same angle as their angle of approach to the crystal \theta with respect to the lattice planes; ''in the elastic light'' (''typically X-ray'')''-crystal scattering, parallel crystal lattice planes perpendicular to a reciprocal lattice vector \mathbf for the crystal lattice play as parallel mirrors for light which, together with \mathbf, incoming (to the crystal) and outgoing (from the crystal by scattering) wavevectors forms a rhombus.'' Since the angle between \mathbf_ and \mathbf is \pi/2 - \theta, (Due to the mirror-like scattering, the angle between \mathbf_ and \mathbf is also \pi/2 - \theta.) \mathbf_\cdot\mathbf = , \mathbf_, , \mathbf, \sin\theta. Recall, , \mathbf_, = 2\pi/\lambda with \lambda as the light (typically X-ray) wavelength, and , \mathbf, = \fracn with d as the distance between adjacent parallel crystal lattice planes and n as an integer. With these, we now derive
Bragg's law In physics and chemistry , Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposition of wave ...
that is equivalent to the Laue equations (also called the Laue condition): \begin 2\mathbf_\cdot\mathbf=, \mathbf, ^2 \\ 2, \mathbf_, , \mathbf, \sin\theta =, \mathbf, ^2 \\ 2 (2\pi/\lambda) (2\pi n/d) \sin\theta =(2\pi n/d )^2 \\ 2d\sin\theta=n\lambda. \end


References

*Kittel, C. (1976). ''
Introduction to Solid State Physics ''Introduction to Solid State Physics'', known colloquially as ''Kittel'', is a classic condensed matter physics textbook written by American physicist Charles Kittel in 1953. The book has been highly influential and has seen widespread adoption ...
'', New York: John Wiley & Sons. {{ISBN, 0-471-49024-5 ;Notes Crystallography