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
as the condition of elastic wave scattering by a crystal lattice, where
,
, and
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
, three vectors.
,
, and
, 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
). 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
since
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
be
primitive translation vectors (shortly called primitive vectors) of a crystal lattice , where atoms are located at lattice points described by
with
,
, and
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
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
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
, and let
be the wavevector of an outgoing (diffracted) beam or wave from
. Then the vector
, called the scattering vector or ''transferred wavevector'', measures the difference between the incoming and outgoing wavevectors.
The three conditions that the scattering vector
must satisfy, called the Laue equations, are the following:
:
:
:
where numbers
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
, 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
. Hence there are infinitely many scattering vectors that satisfy the Laue equations as there are infinitely many choices of Miller indices
. Allowed scattering vectors
form a lattice
, 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
, as each
indicates a point of
. (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
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
(and the angular frequency
) 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
:
:
where
and
are
wave vectors for the incident and outgoing plane waves,
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
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
and
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
of the crystal, where they resonate with the oscillators, so the phases of these waves must coincide. At each point
of the lattice
, we have
:
or equivalently, we must have
:
for some integer
, that depends on the point
. Since this equation holds at
,
at some integer
. So
:
(We still use
instead of
since both the notations essentially indicate some integer.) By rearranging terms, we get
:
Now, it is enough to check that this condition is satisfied at the primitive vectors
(which is exactly what the Laue equations say), because, at any lattice point
, we have
:
where
is the integer
. The claim that each parenthesis, e.g.
, is to be a multiple of
(that is each Laue equation) is justified since otherwise
does not hold for any arbitrary integers
.
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
with
,
,
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
(defined by
) in real space, we know that
with an integer
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
.) But notice that this is nothing but the Laue equations. Hence we identify
, means that allowed scattering vectors
are those equal to reciprocal lattice vectors
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.
Applying the elastic scattering condition
(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
:
:
The second equation is obtained from the first equation by using
.
The result
(also
) 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
satisfying this equation) as its equivalent equation
is a
plane equation in geometry. Another equivalent equation, that may be more easier to understand, is
(also
). This indicates the plane that is perpendicular to the straight line between the reciprocal lattice origin
and
and located at the middle of the line. Such a plane is called Bragg plane.
This plane can be understood since
for scattering to occur (It is the Laue condition, equivalent to the Laue equations.) and the elastic scattering
has been assumed so
,
, and
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
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
, and these wavefronts are coincident with parallel crystal lattice planes. This means that X-rays are seemingly "reflected" off parallel crystal lattice planes perpendicular
at the same angle as their angle of approach to the crystal
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
for the crystal lattice play as parallel mirrors for light which, together with
, incoming (to the crystal) and outgoing (from the crystal by scattering) wavevectors forms a rhombus.''
Since the angle between
and
is
, (Due to the mirror-like scattering, the angle between
and
is also
.)
. Recall,
with
as the light (typically X-ray) wavelength, and
with
as the distance between adjacent parallel crystal lattice planes and
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):
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