In many areas of science,Bragg's law,Wulff–Bragg's condition, orLaue–Bragg interference are a special case ofLaue diffraction, giving the angles for coherentscattering of waves from a large crystal lattice. It describes how the superposition of wave fronts scattered by lattice planes leads to a strict relation between the wavelength and scattering angle. This law was initially formulated for X-rays, but it also applies to all types ofmatter waves including neutron and electron waves if there are a large number of atoms, as well as visible light with artificial periodic microscale lattices.
Bragg diffraction (also referred to as theBragg formulation of X-ray diffraction) was first proposed byLawrence Bragg and his father,William Henry Bragg, in 1913[1] after their discovery thatcrystalline solids produced surprising patterns of reflectedX-rays (in contrast to those produced with, for instance, a liquid). They found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation.
Lawrence Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameterd. He proposed that the incident X-ray radiation would produce a Bragg peak if reflections off the various planes interfered constructively. The interference is constructive when the phase difference between the wave reflected off different atomic planes is a multiple of2π; this condition (seeBragg condition section below) was first presented by Lawrence Bragg on 11 November 1912 to theCambridge Philosophical Society.[2] Although simple, Bragg's law confirmed the existence of realparticles at the atomic scale, as well as providing a powerful new tool for studyingcrystals. Lawrence Bragg and his father, William Henry Bragg, were awarded theNobel Prize in physics in 1915 for their work in determining crystal structures beginning withNaCl,ZnS, anddiamond.[3] They are the only father-son team to jointly win.
The concept of Bragg diffraction applies equally toneutron diffraction[4] and approximately toelectron diffraction.[5] In both cases the wavelengths are comparable with inter-atomic distances (~ 150 pm). Many other types ofmatter waves have also been shown to diffract,[6][7] and also light from objects with a larger ordered structure such asopals.[8]
Bragg diffraction occurs when radiation of awavelengthλ comparable to atomic spacings is scattered in aspecular fashion (mirror-like reflection) by planes of atoms in a crystalline material, and undergoes constructive interference.[10] When the scattered waves are incident at a specific angle, they remain in phase and constructivelyinterfere. Theglancing angleθ (see figure on the right, and note that this differs from the convention inSnell's law whereθ is measured from the surface normal), the wavelengthλ, and the "grating constant"d of the crystal are connected by the relation:[11]: 1026 where is thediffraction order ( is first order, is second order,[10]: 221 is third order[11]: 1028 ). This equation, Bragg's law, describes the condition onθ for constructive interference.[12]
A map of the intensities of the scattered waves as a function of their angle is called a diffraction pattern. Strong intensities known as Bragg peaks are obtained in the diffraction pattern when the scattering angles satisfy Bragg condition. This is a special case of the more generalLaue equations, and the Laue equations can be shown to reduce to the Bragg condition with additional assumptions.[13]
In Bragg's original paper he describes his approach as aHuygens' construction for a reflected wave.[14]: 46 Suppose that aplane wave (of any type) is incident on planes oflattice points, with separation, at an angle as shown in the Figure. PointsA andC are on one plane, andB is on the plane below. PointsABCC' form aquadrilateral.[15]: 69
There will be a path difference between theray that gets reflected alongAC' and the ray that gets transmitted alongAB, then reflected alongBC. This path difference is
The two separate waves will arrive at a point (infinitely far from these lattice planes) with the samephase, and hence undergoconstructive interference, if and only if this path difference is equal to any integer value of thewavelength, i.e.
where and are an integer and the wavelength of the incident wave respectively.
Therefore, from the geometry
from which it follows that
Putting everything together,
which simplifies to which is Bragg's law shown above.
If only two planes of atoms were diffracting, as shown in the Figure then the transition from constructive to destructive interference would be gradual as a function of angle, with gentlemaxima at the Bragg angles. However, since many atomic planes are participating in most real materials, sharp peaks are typical.[5][13]
A rigorous derivation from the more general Laue equations is available (see page:Laue equations).
The Bragg condition is correct for very large crystals. Because the scattering of X-rays and neutrons is relatively weak, in many cases quite large crystals with sizes of 100 nm or more are used. While there can be additional effects due tocrystal defects, these are often quite small. In contrast, electrons interact thousands of times more strongly with solids than X-rays,[5] and also lose energy (inelastic scattering).[16] Therefore samples used intransmission electron diffraction are much thinner. Typical diffraction patterns, for instance the Figure, show spots for different directions (plane waves) of the electrons leaving a crystal. The angles that Bragg's law predicts are still approximately right, but in general there is a lattice of spots which are close to projections of thereciprocal lattice that is at right angles to the direction of the electron beam. (In contrast, Bragg's law predicts that only one or perhaps two would be present, not simultaneously tens to hundreds.) Withlow-energy electron diffraction where the electron energies are typically 30-1000electron volts, the result is similar with the electrons reflected back from a surface.[17] Also similar isreflection high-energy electron diffraction which typically leads to rings of diffraction spots.[18]
With X-rays the effect of having small crystals is described by theScherrer equation.[13][19][20] This leads to broadening of the Bragg peaks which can be used to estimate the size of the crystals.
Acolloidal crystal is a highlyordered array of particles that forms over a long range (from a fewmillimeters to onecentimeter in length); colloidal crystals have appearance and properties roughlyanalogous to their atomic or molecular counterparts.[8] It has been known for many years that, due torepulsiveCoulombic interactions,electrically chargedmacromolecules in anaqueous environment can exhibit long-rangecrystal-like correlations, with interparticle separation distances often being considerably greater than the individual particle diameter. Periodic arrays of spherical particles give rise tointerstitial voids (the spaces between the particles), which act as a naturaldiffraction grating forvisible light waves, when the interstitial spacing is of the sameorder of magnitude as theincident lightwave.[21][22][23] In these cases brilliantiridescence (or play of colours) is attributed to the diffraction andconstructive interference of visible lightwaves according to Bragg's law, in a matter analogous to thescattering ofX-rays in crystalline solid. The effects occur at visible wavelengths because the interplanar spacingd is much larger than for true crystals. Preciousopal is one example of a colloidal crystal with optical effects.
Volume Bragg gratings (VBG) orvolume holographic gratings (VHG) consist of a volume where there is a periodic change in therefractive index. Depending on the orientation of the refractive index modulation, VBG can be used either totransmit orreflect a small bandwidth ofwavelengths.[24] Bragg's law (adapted for volume hologram) dictates which wavelength will be diffracted:[25]
wherem is the Bragg order (a positive integer),λB the diffractedwavelength, Λ the fringe spacing of the grating,θ the angle between the incident beam and the normal (N) of the entrance surface andφ the angle between the normal and the grating vector (KG). Radiation that does not match Bragg's law will pass through the VBG undiffracted. The output wavelength can be tuned over a few hundred nanometers by changing the incident angle (θ). VBG are being used to producewidely tunable laser source or perform globalhyperspectral imagery (seePhoton etc.).[25]
The measurement of the angles can be used to determine crystal structure, seex-ray crystallography for more details.[5][13] As a simple example, Bragg's law, as stated above, can be used to obtain the lattice spacing of a particularcubic system through the following relation:
where is the lattice spacing of thecubic crystal, andh,k, andℓ are theMiller indices of the Bragg plane. Combining this relation with Bragg's law gives:
One can derive selection rules for theMiller indices for different cubicBravais lattices as well as many others, a few of the selection rules are given in the table below.
| Bravais lattices | Example compounds | Allowed reflections | Forbidden reflections |
|---|---|---|---|
| Simple cubic | Po | Anyh,k,ℓ | None |
| Body-centered cubic | Fe, W, Ta, Cr | h +k +ℓ = even | h +k +ℓ = odd |
| Face-centered cubic (FCC) | Cu, Al, Ni, NaCl, LiH, PbS | h,k,ℓ all odd or all even | h,k,ℓ mixed odd and even |
| Diamond FCC | Si, Ge | All odd, or all even withh +k +ℓ = 4n | h,k,ℓ mixed odd and even, or all even withh +k +ℓ ≠ 4n |
| Triangular lattice | Ti, Zr, Cd, Be | ℓ even,h + 2k ≠ 3n | h + 2k = 3n for oddℓ |
These selection rules can be used for any crystal with the given crystal structure. KCl has a face-centered cubicBravais lattice. 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, orderived. Lattice spacing for the othercrystal systems can be foundhere.
| Key concepts |
|  | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Characterisation | ||||||||||||||
| Algorithms | ||||||||||||||
| Software | ||||||||||||||
| Databases | ||||||||||||||
| Journals | ||||||||||||||
| Awards | ||||||||||||||
| Organisation |
| |||||||||||||
