 x Bragg's law Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, Bragg's law gives the angles for coherent and incoherent scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

from a crystal lattice. When X-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

s are incident on an atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

, they make the electronic cloud
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

move as does any electromagnetic wave. The movement
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

of these charges
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

WAVES
The WAVES were a World War II-era division of the U.S. Navy that consisted entirely of women. The name of this group is an acronym for "Women Accepted for Volunteer Emergency Service" ; the word "emergency" implied that the acceptance of women was due to the unusual circumstances of the war and...

with the same frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

(blurred slightly due to a variety of effects); this phenomenon is known as Rayleigh scattering
Rayleigh scattering
Rayleigh scattering, named after the British physicist Lord Rayleigh, is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. The particles may be individual atoms or molecules. It can occur when light travels through...

(or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible. A similar process
Process (science)
In science, a process is every sequence of changes of a real object/body which is observable using the scientific method. Therefore, all sciences analyze and model processes....

occurs upon scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

neutron waves from the nuclei
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

or by a coherent
Coherence (physics)
In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....

spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

interaction with an unpaired electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

. These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern
Pattern
A pattern, from the French patron, is a type of theme of recurring events or objects, sometimes referred to as elements of a set of objects.These elements repeat in a predictable manner...

is the basis of diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

analysis. This analysis is called Bragg diffraction.

Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William Lawrence Bragg
William Lawrence Bragg
Sir William Lawrence Bragg CH OBE MC FRS was an Australian-born British physicist and X-ray crystallographer, discoverer of the Bragg law of X-ray diffraction, which is basic for the determination of crystal structure. He was joint winner of the Nobel Prize for Physics in 1915. He was knighted...

and William Henry Bragg
William Henry Bragg
Sir William Henry Bragg OM, KBE, PRS was a British physicist, chemist, mathematician and active sportsman who uniquely shared a Nobel Prize with his son William Lawrence Bragg - the 1915 Nobel Prize in Physics...

in 1913 in response to their discovery that crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

line solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). They found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation (known as Bragg peaks). The concept of Bragg diffraction applies equally to neutron diffraction
Neutron diffraction
Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material: A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of...

and electron diffraction
Electron diffraction
Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern...

processes. Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale
Length scale
In physics, length scale is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple...

. W. L. Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter d. It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered constructively. The interference is constructive when the phase shift is a multiple of 2π; this condition can be expressed by Bragg's law, Where n is an integer, λ is the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

of incident wave, d is the spacing between the planes in the atomic lattice, and θ is the angle between the incident ray and the scattering planes. Note that moving particles, including electrons, protons and neutrons, have an associated De Broglie wavelength.
Bragg's Law was derived by physicist Sir William Lawrence Bragg
William Lawrence Bragg
Sir William Lawrence Bragg CH OBE MC FRS was an Australian-born British physicist and X-ray crystallographer, discoverer of the Bragg law of X-ray diffraction, which is basic for the determination of crystal structure. He was joint winner of the Nobel Prize for Physics in 1915. He was knighted...

in 1912 and first presented on 11 November 1912 to the Cambridge Philosophical Society
Cambridge Philosophical Society
The Cambridge Philosophical Society is a scientific society at University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of theology and medicine...

. Although simple, Bragg's law confirmed the existence of real particle
Subatomic particle
In physics or chemistry, subatomic particles are the smaller particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles...

s at the atomic scale, as well as providing a powerful new tool for studying crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

s in the form of X-ray and neutron diffraction. William Lawrence Bragg and his father, Sir William Henry Bragg
William Henry Bragg
Sir William Henry Bragg OM, KBE, PRS was a British physicist, chemist, mathematician and active sportsman who uniquely shared a Nobel Prize with his son William Lawrence Bragg - the 1915 Nobel Prize in Physics...

, were awarded the Nobel Prize
Nobel Prize
The Nobel Prizes are annual international awards bestowed by Scandinavian committees in recognition of cultural and scientific advances. The will of the Swedish chemist Alfred Nobel, the inventor of dynamite, established the prizes in 1895...

in physics in 1915 for their work in determining crystal structures beginning with NaCl
Sodium chloride
Sodium chloride, also known as salt, common salt, table salt or halite, is an inorganic compound with the formula NaCl. Sodium chloride is the salt most responsible for the salinity of the ocean and of the extracellular fluid of many multicellular organisms...

, ZnS
Zinc sulfide
Zinc sulfide is a inorganic compound with the formula ZnS. ZnS is the main form of zinc in nature, where it mainly occurs as the mineral sphalerite...

, and diamond
Diamond
In mineralogy, diamond is an allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the conversion rate from diamond to graphite is negligible at ambient conditions...

. They are the only father-son team to jointly win. W. L. Bragg was 25 years old, making him the youngest Nobel laureate.

## Bragg condition Bragg diffraction occurs when electromagnetic radiation or subatomic particle waves with wavelength comparable to atomic spacings are incident upon a crystalline sample, are scattered in a specular fashion by the atoms in the system, and undergo constructive interference in accordance to Bragg's law. For a crystalline solid, the waves are scattered from lattice planes separated by the interplanar distance d. Where the scattered waves interfere constructively, they remain in phase since the path length of each wave is equal to an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

multiple of the wavelength. The path difference between two waves undergoing constructive interference is given by 2dsinθ, where θ is the scattering angle. This leads to Bragg's law
Bragg's law
In physics, Bragg's law gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident on an atom, they make the electronic cloud move as does any electromagnetic wave...

, which describes the condition for constructive interference from successive crystallographic planes (h, k, and l, as given in Miller Notation
Miller index
Miller indices form a notation system in crystallography for planes and directions in crystal lattices.In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written , and each index denotes a plane orthogonal to a direction in the...

) of the crystalline lattice: where n is an integer determined by the order given, and λ is the wavelength. A diffraction pattern is obtained by measuring the intensity of scattered waves as a function of scattering angle. Very strong intensities known as Bragg peaks are obtained in the diffraction pattern when scattered waves satisfy the Bragg condition.

## Reciprocal space

Although the misleading common opinion reigns that Bragg's law measures atomic distances in real space, it does not. This first statement only seems to be true if it's further elaborated that distances measured during a Bragg experiment are inversely proportional to the distance d in the lattice diagram. Furthermore, the term demonstrates that it measures the number of wavelengths fitting between two rows of atoms, thus measuring reciprocal distances. Reciprocal lattice vectors describe the set of lattice planes as a normal vector to this set with length Max von Laue
Max von Laue
Max Theodor Felix von Laue was a German physicist who won the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals...

had interpreted this correctly in a vector form, the Laue equation where is a reciprocal lattice vector and and are the wave vectors of the incident and the diffracted beams.

Together with the condition for elastic scattering and the introduction of the scattering angle this leads equivalently to Bragg's equation. This is simply explained by the conservation of momentum transfer. In this system the scanning variable can be the length or the direction of the incident or exit wave vectors relating to energy- and angle-dispersive setups. The simple relationship between diffraction angle and Q-space is then: The concept of reciprocal lattice
Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly k-space, due to the relationship between the Pontryagin duals momentum and...

is the Fourier space of a crystal lattice and necessary for a full mathematical description of wave mechanics.

## Alternate derivation

Suppose that a single monochromatic wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

(of any type) is incident on aligned planes of lattice
Square lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group is known symbolically as p4m.Two...

points, with separation , at angle , as shown below. There will be a path difference between the ray
Ray (optics)
In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very...

that gets reflected along AC' and the ray that gets transmitted, then reflected, along AB and BC respectively. This path difference is The two separate waves will arrive at a point with the same phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

, and hence undergo constructive interference, if and only if this path difference is equal to any integer value of the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

, i.e. (C' needs to be defined)
where the same definition of and apply as above.

Clearly, and from which it follows that Putting everything together, which simplifies to which is Bragg's law.

## Bragg scattering of visible light by colloids

A colloidal crystal
Colloidal crystal
A colloidal crystal is an ordered array of colloid particles, analogous to a standard crystal whose repeating subunits are atoms or molecules. A natural example of this phenomenon can be found in the gem opal, where spheres of silica assume a close-packed locally periodic structure under moderate...

is a highly ordered array of particles which can be formed over a very long range (from a few millimeters to one centimeter) in length, and which appear analogous to their atomic or molecular counterparts. The periodic arrays of spherical particles make similar arrays of interstitial
Interstitial
An interstitial space or interstice is an empty space or gap between spaces full of structure or matter.In particular, interstitial may refer to:-Physical sciences:...

void
Void
-In science and engineering:*Void , the empty spaces between galaxy filaments*Lack of matter, or vacuum*Void, in boiling heat transfer, formed where there is a departure from nucleate boiling, causing a critical heat flux...

s, which act as a natural diffraction grating
Diffraction grating
In optics, a diffraction grating is an optical component with a periodic structure, which splits and diffracts light into several beams travelling in different directions. The directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as...

for visible light waves
Visible spectrum
The visible spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. Electromagnetic radiation in this range of wavelengths is called visible light or simply light. A typical human eye will respond to wavelengths from about 390 to 750 nm. In terms of...

, especially when the interstitial spacing is of the same order of magnitude
Order of magnitude
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

as the incident
Angle of incidence
Angle of incidence is a measure of deviation of something from "straight on", for example:* in the approach of a ray to a surface, or* the angle at which the wing or horizontal tail of an airplane is installed on the fuselage, measured relative to the axis of the fuselage.-Optics:In geometric...

lightwave.

Thus, it has been known for many years that, due to repulsive
Repulsive force
Repulsive force may refer to:* A repulsive force of an Accelerating universe, which according to certain theories causes planets and matter to get farther and farther apart* Like charges repelling according to Coulomb's law...

Coulombic interactions, electrically charged macromolecules in an aqueous environment can exhibit long-range crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

-like correlations with interparticle separation distances often being considerably greater than the individual particle diameter. In all of these cases in nature, the same brilliant iridescence
Iridescence
Iridescence is generally known as the property of certain surfaces which appear to change color as the angle of view or the angle of illumination changes...

(or play of colours) can be attributed to the diffraction and constructive interference of visible lightwaves which satisfy Bragg’s law, in a matter analogous to the scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

of X-rays in crystalline solid.

## Selection rules and practical crystallography

Bragg's law, as stated above, can be used to obtain the lattice spacing of a particular cubic system through the following relation: where is the lattice spacing of the cubic crystal, and , , and are the Miller indices of the Bragg plane. Combining this relation with Bragg's law: One can derive selection rules for the Miller indices for different cubic Bravais lattices; here, selection rules for several will be given as is.
Selection rules for the Miller indices
Bravais lattice Example compounds Allowed reflections Forbidden reflections
Simple cubic Po, KCl Any h, k, l None
Body-centered cubic Fe, W, Ta, Cr h + k + l even h + k + l odd
Face-centered cubic Cu, Al, Ni, NaCl, LiH, PbS h, k, l all odd or all even h, k, l mixed odd and even
Diamond F.C.C. ZnSe, CuCl, AgI, CuF, Si, Ge all: odd, or even & h+k+l = 4n above, or even & h+k+l ≠ 4n
Triangular lattice Ti, Zr, Cd, Be l even, h + 2k ≠ 3n h + 2k = 3n for odd l

These selection rules can be used for any crystal with the given crystal structure. KCl exhibits a fcc cubic structure. However, the K+ and the Cl+ ion have the same number of electrons and are quite close in size, so that the diffraction pattern becomes essentially the same as for a simple cubic structure with half the lattice parameter. Selection rules for other structures can be referenced elsewhere, or derived
Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly k-space, due to the relationship between the Pontryagin duals momentum and...

.

• Crystal lattice
• Diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

• Distributed Bragg reflector
Distributed Bragg reflector
A distributed Bragg reflector is a reflector used in waveguides, such as optical fibers. It is a structure formed from multiple layers of alternating materials with varying refractive index, or by periodic variation of some characteristic of a dielectric waveguide, resulting in periodic variation...

• Fiber Bragg grating
Fiber Bragg grating
A fiber Bragg grating is a type of distributed Bragg reflector constructed in a short segment of optical fiber that reflects particular wavelengths of light and transmits all others. This is achieved by adding a periodic variation to the refractive index of the fiber core, which generates a...

• Dynamical theory of diffraction
Dynamical theory of diffraction
The dynamical theory of diffraction describes the interaction of waves with a regular lattice. The wave fields traditionally described are X-rays, neutrons or electrons and the regular lattice, atomic crystal structures or nanometer scaled multi-layers or self arranged systems...

• Henderson limit
Henderson limit
The value of the Henderson limit is defined as 2 × 107 Gy .Although generalizable, the limit is defined in the context of biomolecular X-ray crystallography, where a typical experiment consists of exposing a single frozen crystal of a macromolecule to an intense X-ray beam...

• Laue conditions
• Powder diffraction
Powder diffraction
Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials.-Explanation:...

• Structure factor
Structure factor
In condensed matter physics and crystallography, the static structure factor is a mathematical description of how a material scatters incident radiation...

• William Lawrence Bragg
William Lawrence Bragg
Sir William Lawrence Bragg CH OBE MC FRS was an Australian-born British physicist and X-ray crystallographer, discoverer of the Bragg law of X-ray diffraction, which is basic for the determination of crystal structure. He was joint winner of the Nobel Prize for Physics in 1915. He was knighted...

• X-ray crystallography
X-ray crystallography
X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a... 