Reciprocal lattice

Reciprocal latticeis a concept associated with solids withtranslational symmetrywhich plays a major role in many areas such asX-rayandelectrondiffraction as well as theenergiesof electrons in a solid. It emerges from theFourier transformof thelatticeassociated with the arrangement of the atoms. Thedirect latticeorreal latticeis aperiodic functioninphysical space, such as acrystal system(usually aBravais lattice). The reciprocal lattice exists in themathematical spaceofspatial frequenciesorwavenumbersk, known asreciprocal spaceorkspace; it is the dual of physical space considered as avector space. In other words, the reciprocal lattice is thesublatticewhichis dualto the direct lattice.
The reciprocal lattice is the set of allvectorsGm{\displaystyle \mathbf {G} _{m}}, that arewavevectorskof plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct latticeRn{\displaystyle \mathbf {R} _{n}}. Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of2π{\displaystyle 2\pi }at each direct lattice point (so essentially same phase at all the direct lattice points).
The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors representcovariant and contravariant vectors, respectively.
TheBrillouin zoneis aWigner–Seitz cellof the reciprocal lattice.
Reciprocal space (also calledk-space) provides a way to visualize the results of theFourier transformof a spatial function. It is similar in role to thefrequency domainarising from the Fourier transform of a time dependent function; reciprocal space is a space over which the Fourier transform of a spatial function is represented at spatial frequencies or wavevectors of plane waves of the Fourier transform. The domain of the spatial function itself is often referred to asspatial domainor real space. In physical applications, such as crystallography, both real and reciprocal space will often each be two or three dimensional. Whereas the number of spatial dimensions of these two associated spaces will be the same, the spaces will differ in theirquantity dimension, so that when the real space has the dimension length (L), its reciprocal space will haveinverse length, soL(the reciprocal of length).
Reciprocal space comes into play regarding waves, both classical and quantum mechanical. Because asinusoidal plane wavewith unit amplitude can be written as an oscillatory termcos⁡(kx−ωt+φ0){\displaystyle \cos(kx-\omega t+\varphi _{0})},with initialphaseφ0{\displaystyle \varphi _{0}},angular wavenumberk{\displaystyle k}andangular frequencyω{\displaystyle \omega },it can be regarded as a function of bothk{\displaystyle k}andx{\displaystyle x}(and the time-varying part as a function of bothω{\displaystyle \omega }andt{\displaystyle t}).This complementary role ofk{\displaystyle k}andx{\displaystyle x}leads to their visualization within complementary spaces (the real space and the reciprocal space). The spatial periodicity of this wave is defined by its wavelengthλ{\displaystyle \lambda },wherekλ=2π{\displaystyle k\lambda =2\pi };hence the corresponding wavenumber in reciprocal space will bek=2π/λ{\displaystyle k=2\pi /\lambda }.
In three dimensions, the corresponding plane wave term becomescos⁡(k⋅r−ωt+φ0){\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{0})},which simplifies tocos⁡(k⋅r+φ){\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} +\varphi )}at a fixed timet{\displaystyle t},wherer{\displaystyle \mathbf {r} }is the position vector of a point in real space and nowk=2πe/λ{\displaystyle \mathbf {k} =2\pi \mathbf {e} /\lambda }is thewavevectorin the three dimensional reciprocal space. (The magnitude of a wavevector is called wavenumber.) The constantφ{\displaystyle \varphi }is the phase of thewavefront(a plane of a constant phase) through the originr=0{\displaystyle \mathbf {r} =0}at timet{\displaystyle t},ande{\displaystyle \mathbf {e} }is a unitnormal vectorto this wavefront. The wavefronts with phasesφ+(2π)n{\displaystyle \varphi +(2\pi )n}, wheren{\displaystyle n}represents anyinteger, comprise a set of parallel planes, equally spaced by the wavelengthλ{\displaystyle \lambda }.
In general, a geometric lattice is an infinite, regular array of vertices (points) in space, which can be modelled vectorially as aBravais lattice. Some lattices may be skew, which means that their primary lines may not necessarily be at right angles. In reciprocal space, a reciprocal lattice is defined as the set ofwavevectorsk{\displaystyle \mathbf {k} }of plane waves in theFourier seriesof any functionf(r){\displaystyle f(\mathbf {r} )}whose periodicity is compatible with that of an initial direct lattice in real space. Equivalently, a wavevector is a vertex of the reciprocal lattice if it corresponds to a plane wave in real space whose phase at any given time is the same (actually differs by(2π)n{\displaystyle (2\pi )n}with an integern{\displaystyle n}) at every direct lattice vertex.
One heuristic approach to constructing the reciprocal lattice in three dimensions is to write the position vector of a vertex of the direct lattice asR=n1a1+n2a2+n3a3{\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}},where theni{\displaystyle n_{i}}are integers defining the vertex and theai{\displaystyle \mathbf {a} _{i}}arelinearly independentprimitive translation vectors (or shortly called primitive vectors) that are characteristic of the lattice. There is then a unique plane wave (up to a factor of negative one), whose wavefront through the originR=0{\displaystyle \mathbf {R} =0}contains the direct lattice points ata2{\displaystyle \mathbf {a} _{2}}anda3{\displaystyle \mathbf {a} _{3}},and with its adjacent wavefront (whose phase differs by2π{\displaystyle 2\pi }or−2π{\displaystyle -2\pi }from the former wavefront passing the origin) passing througha1{\displaystyle \mathbf {a} _{1}}. Its angular wavevector takes the formb1=2πe1/λ1{\displaystyle \mathbf {b} _{1}=2\pi \mathbf {e} _{1}/\lambda _{1}},wheree1{\displaystyle \mathbf {e} _{1}}is the unit vector perpendicular to these two adjacent wavefronts and the wavelengthλ1{\displaystyle \lambda _{1}}must satisfyλ1=a1⋅e1{\displaystyle \lambda _{1}=\mathbf {a} _{1}\cdot \mathbf {e} _{1}}, means thatλ1{\displaystyle \lambda _{1}}is equal to the distance between the two wavefronts. Hence by constructiona1⋅b1=2π{\displaystyle \mathbf {a} _{1}\cdot \mathbf {b} _{1}=2\pi }anda2⋅b1=a3⋅b1=0{\displaystyle \mathbf {a} _{2}\cdot \mathbf {b} _{1}=\mathbf {a} _{3}\cdot \mathbf {b} _{1}=0}.
Cycling through the indices in turn, the same method yields three wavevectorsbj{\displaystyle \mathbf {b} _{j}}withai⋅bj=2πδij{\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}, where theKronecker deltaδij{\displaystyle \delta _{ij}}equals one wheni=j{\displaystyle i=j}and is zero otherwise. Thebj{\displaystyle \mathbf {b} _{j}}comprise a set of three primitive wavevectors or three primitive translation vectors for the reciprocal lattice, each of whose vertices takes the formG=m1b1+m2b2+m3b3{\displaystyle \mathbf {G} =m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}},where themj{\displaystyle m_{j}}are integers. The reciprocal lattice is also aBravais latticeas it is formed by integer combinations of the primitive vectors, that areb1{\displaystyle \mathbf {b} _{1}},b2{\displaystyle \mathbf {b} _{2}}, andb3{\displaystyle \mathbf {b} _{3}}in this case. Simple algebra then shows that, for any plane wave with a wavevectorG{\displaystyle \mathbf {G} }on the reciprocal lattice, the total phase shiftG⋅R{\displaystyle \mathbf {G} \cdot \mathbf {R} }between the origin and any pointR{\displaystyle \mathbf {R} }on the direct lattice is a multiple of2π{\displaystyle 2\pi }(that can be possibly zero if the multiplier is zero), so the phase of the plane wave withG{\displaystyle \mathbf {G} }will essentially be equal for every direct lattice vertex, in conformity with the reciprocal lattice definition above. (Although any wavevectorG{\displaystyle \mathbf {G} }on the reciprocal lattice does always take this form, this derivation is motivational, rather than rigorous, because it has omitted the proof that no other possibilities exist.)
TheBrillouin zoneis aprimitive cell(more specifically aWigner–Seitz cell) of the reciprocal lattice, which plays an important role insolid state physicsdue toBloch's theorem. Inpure mathematics, thedual spaceoflinear formsand thedual latticeprovide more abstract generalizations of reciprocal space and the reciprocal lattice.
Assuming a three-dimensionalBravais latticeand labelling each lattice vector (a vector indicating a lattice point) by the subscriptn=(n1,n2,n3){\displaystyle n=(n_{1},n_{2},n_{3})}as3-tupleof integers,
whereZ{\displaystyle \mathbb {Z} }is the set of integers andai{\displaystyle \mathbf {a} _{i}}is a primitive translation vector or shortly primitive vector. Taking a functionf(r){\displaystyle f(\mathbf {r} )}wherer{\displaystyle \mathbf {r} }is a position vector from the originRn=0{\displaystyle \mathbf {R} _{n}=0}to any position, iff(r){\displaystyle f(\mathbf {r} )}follows the periodicity of this lattice, e.g. the function describing the electronic density in an atomic crystal, it is useful to writef(r){\displaystyle f(\mathbf {r} )}as amulti-dimensional Fourier series
where now the subscriptm=(m1,m2,m3){\displaystyle m=(m_{1},m_{2},m_{3})}, so this is a triple sum.
Asf(r){\displaystyle f(\mathbf {r} )}follows the periodicity of the lattice, translatingr{\displaystyle \mathbf {r} }by any lattice vectorRn{\displaystyle \mathbf {R} _{n}}we get the same value, hence
Expressing the above instead in terms of their Fourier series we have∑mfmeiGm⋅r=∑mfmeiGm⋅(r+Rn)=∑mfmeiGm⋅RneiGm⋅r.{\displaystyle \sum _{m}f_{m}e^{i\mathbf {G} _{m}\cdot \mathbf {r} }=\sum _{m}f_{m}e^{i\mathbf {G} _{m}\cdot (\mathbf {r} +\mathbf {R} _{n})}=\sum _{m}f_{m}e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}\,e^{i\mathbf {G} _{m}\cdot \mathbf {r} }.}
Because equality of two Fourier series implies equality of their coefficients,eiGm⋅Rn=1{\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1}, which only holds when
Mathematically, the reciprocal lattice is the set of allvectorsGm{\displaystyle \mathbf {G} _{m}}, that arewavevectorsof plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice as the set of all direct lattice point position vectorsRn{\displaystyle \mathbf {R} _{n}}, andGm{\displaystyle \mathbf {G} _{m}}satisfy this equality for allRn{\displaystyle \mathbf {R} _{n}}. Each plane wave in the Fourier series has the same phase (actually can be differed by a multiple of2π{\displaystyle 2\pi }) at all the lattice pointRn{\displaystyle \mathbf {R} _{n}}.
As shown in the sectionmulti-dimensional Fourier series,Gm{\displaystyle \mathbf {G} _{m}}can be chosen in the form ofGm=m1b1+m2b2+m3b3{\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}whereai⋅bj=2πδij{\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}. With this form, the reciprocal lattice as the set of all wavevectorsGm{\displaystyle \mathbf {G} _{m}}for the Fourier series of a spatial function which periodicity followsRn{\displaystyle \mathbf {R} _{n}}, is itself a Bravais lattice as it is formed by integer combinations of its own primitive translation vectors(b1,b2,b3){\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}, and the reciprocal of the reciprocal lattice is the original lattice, which reveals thePontryagin dualityof their respectivevector spaces. (There may be other form ofGm{\displaystyle \mathbf {G} _{m}}. Any valid form ofGm{\displaystyle \mathbf {G} _{m}}results in the same reciprocal lattice.)
For an infinite two-dimensional lattice, defined by itsprimitive vectors(a1,a2){\displaystyle \left(\mathbf {a} _{1},\mathbf {a} _{2}\right)}, its reciprocal lattice can be determined by generating its two reciprocal primitive vectors, through the following formulae,
wheremi{\displaystyle m_{i}}is an integer and
HereQ{\displaystyle \mathbf {Q} }represents a 90 degreerotation matrix, i.e. aquarter turn. The anti-clockwise rotation and the clockwise rotation can both be used to determine the reciprocal lattice: IfQ{\displaystyle \mathbf {Q} }is the anti-clockwise rotation andQ′{\displaystyle \mathbf {Q'} }is the clockwise rotation,Qv=−Q′v{\displaystyle \mathbf {Q} \,\mathbf {v} =-\mathbf {Q'} \,\mathbf {v} }for all vectorsv{\displaystyle \mathbf {v} }. Thus, using thepermutation
we obtain
Notably, in a 3D space this 2D reciprocal lattice is an infinitely extended set of Bragg rods—described by Sung et al.
For an infinite three-dimensional latticeRn=n1a1+n2a2+n3a3{\displaystyle \mathbf {R} _{n}=n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}, defined by itsprimitive vectors(a1,a2,a3){\displaystyle \left(\mathbf {a_{1}} ,\mathbf {a} _{2},\mathbf {a} _{3}\right)}and the subscript of integersn=(n1,n2,n3){\displaystyle n=\left(n_{1},n_{2},n_{3}\right)}, its reciprocal latticeGm=m1b1+m2b2+m3b3{\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}with the integer subscriptm=(m1,m2,m3){\displaystyle m=(m_{1},m_{2},m_{3})}can be determined by generating its three reciprocal primitive vectors(b1,b2,b3){\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}b1=2πVa2×a3b2=2πVa3×a1b3=2πVa1×a2{\displaystyle {\begin{aligned}\mathbf {b} _{1}&={\frac {2\pi }{V}}\ \mathbf {a} _{2}\times \mathbf {a} _{3}\\[8pt]\mathbf {b} _{2}&={\frac {2\pi }{V}}\ \mathbf {a} _{3}\times \mathbf {a} _{1}\\[8pt]\mathbf {b} _{3}&={\frac {2\pi }{V}}\ \mathbf {a} _{1}\times \mathbf {a} _{2}\end{aligned}}}whereV=a1⋅(a2×a3)=a2⋅(a3×a1)=a3⋅(a1×a2){\displaystyle V=\mathbf {a} _{1}\cdot \left(\mathbf {a} _{2}\times \mathbf {a} _{3}\right)=\mathbf {a} _{2}\cdot \left(\mathbf {a} _{3}\times \mathbf {a} _{1}\right)=\mathbf {a} _{3}\cdot \left(\mathbf {a} _{1}\times \mathbf {a} _{2}\right)}is thescalar triple product. The choice of these(b1,b2,b3){\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}is to satisfyai⋅bj=2πδij{\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}as the known condition (There may be other condition.) ofprimitive translation vectorsfor the reciprocal lattice derived in theheuristic approach aboveand the sectionmulti-dimensional Fourier series. This choice also satisfies the requirement of the reciprocal latticeeiGm⋅Rn=1{\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1}mathematically derivedabove. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten usingmatrix inversion:
This method appeals to the definition, and allows generalization to arbitrary dimensions. The cross product formula dominates introductory materials on crystallography.
The above definition is called the "physics" definition, as the factor of2π{\displaystyle 2\pi }comes naturally from the study of periodic structures. An essentially equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal latticeKm=Gm/2π{\displaystyle \mathbf {K} _{m}=\mathbf {G} _{m}/2\pi }. which changes the reciprocal primitive vectors to be
and so on for the other primitive vectors. The crystallographer's definition has the advantage that the definition ofb1{\displaystyle \mathbf {b} _{1}}is just the reciprocal magnitude ofa1{\displaystyle \mathbf {a} _{1}}in the direction ofa2×a3{\displaystyle \mathbf {a} _{2}\times \mathbf {a} _{3}}, dropping the factor of2π{\displaystyle 2\pi }. This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units ofspatial frequency. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed.
m=(m1,m2,m3){\displaystyle m=(m_{1},m_{2},m_{3})}is conventionally written as(h,k,ℓ){\displaystyle (h,k,\ell )}or(hkℓ){\displaystyle (hk\ell )}, calledMiller indices;m1{\displaystyle m_{1}}is replaced withh{\displaystyle h},m2{\displaystyle m_{2}}replaced withk{\displaystyle k}, andm3{\displaystyle m_{3}}replaced withℓ{\displaystyle \ell }. Each lattice point(hkℓ){\displaystyle (hk\ell )}in the reciprocal lattice corresponds to a set of lattice planes(hkℓ){\displaystyle (hk\ell )}in thereal spacelattice. (A lattice plane is a plane crossing lattice points.) The direction of the reciprocal lattice vector corresponds to thenormalto the real space planes. The magnitude of the reciprocal lattice vectorKm{\displaystyle \mathbf {K} _{m}}is given inreciprocal lengthand is equal to the reciprocal of the interplanar spacing of the real space planes.
The formula forn{\displaystyle n}dimensions can be derived assuming ann{\displaystyle n}-dimensionalrealvector spaceV{\displaystyle V}with abasis(a1,…,an){\displaystyle (\mathbf {a} _{1},\ldots ,\mathbf {a} _{n})}and an inner productg:V×V→R{\displaystyle g\colon V\times V\to \mathbf {R} }. The reciprocal lattice vectors are uniquely determined by the formulag(ai,bj)=2πδij{\displaystyle g(\mathbf {a} _{i},\mathbf {b} _{j})=2\pi \delta _{ij}}. Using thepermutation
they can be determined with the following formula:
Here,ω:Vn→R{\displaystyle \omega \colon V^{n}\to \mathbf {R} }is thevolume form,g−1{\displaystyle g^{-1}}is the inverse of the vector space isomorphismg^:V→V∗{\displaystyle {\hat {g}}\colon V\to V^{*}}defined byg^(v)(w)=g(v,w){\displaystyle {\hat {g}}(v)(w)=g(v,w)}and⌟{\displaystyle \lrcorner }denotes theinner multiplication.
One can verify that this formula is equivalent to the known formulas for the two- and three-dimensional case by using the following facts: In three dimensions,ω(u,v,w)=g(u×v,w){\displaystyle \omega (u,v,w)=g(u\times v,w)}and in two dimensions,ω(v,w)=g(Rv,w){\displaystyle \omega (v,w)=g(Rv,w)}, whereR∈SO(2)⊂L(V,V){\displaystyle R\in {\text{SO}}(2)\subset L(V,V)}is therotationby 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation).
Reciprocal lattices for thecubic crystal systemare as follows.
The simple cubicBravais lattice, with cubicprimitive cellof sidea{\displaystyle a}, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side2πa{\textstyle {\frac {2\pi }{a}}}(or1a{\textstyle {\frac {1}{a}}}in the crystallographer's definition). The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space.
The reciprocal lattice to an FCC lattice is the body-centered cubic (BCC) lattice, with a cube side of4πa{\textstyle {\frac {4\pi }{a}}}.
Consider an FCC compound unit cell. Locate a primitive unit cell of the FCC; i.e., a unit cell with one lattice point. Now take one of the vertices of the primitive unit cell as the origin. Give the basis vectors of the real lattice. Then from the known formulae, you can calculate the basis vectors of the reciprocal lattice. These reciprocal lattice vectors of the FCC represent the basis vectors of a BCC real lattice. The basis vectors of a real BCC lattice and the reciprocal lattice of an FCC resemble each other in direction but not in magnitude.
The reciprocal lattice to aBCClattice is theFCClattice, with a cube side of4π/a{\textstyle 4\pi /a}.
It can be proven that only the Bravais lattices which have 90 degrees between(a1,a2,a3){\displaystyle \left(\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3}\right)}(cubic, tetragonal, orthorhombic) have primitive translation vectors for the reciprocal lattice,(b1,b2,b3){\displaystyle \left(\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right)}, parallel to their real-space vectors.
The reciprocal to a simple hexagonal Bravais lattice withlattice constantsa{\textstyle a}andc{\textstyle c}is another simple hexagonal lattice with lattice constants2π/c{\textstyle 2\pi /c}and4π/(a3){\textstyle 4\pi /(a{\sqrt {3}})}rotated through 90° about thecaxis with respect to the direct lattice. The simple hexagonal lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space. Primitive translation vectors for this simple hexagonal Bravais lattice vectors area1=32ax^+12ay^,a2=−32ax^+12ay^,a3=cz^.{\displaystyle {\begin{aligned}a_{1}&={\frac {\sqrt {3}}{2}}a{\hat {x}}+{\frac {1}{2}}a{\hat {y}},\\[8pt]a_{2}&=-{\frac {\sqrt {3}}{2}}a{\hat {x}}+{\frac {1}{2}}a{\hat {y}},\\[8pt]a_{3}&=c{\hat {z}}.\end{aligned}}}
One path to the reciprocal lattice of an arbitrary collection of atoms comes from the idea of scattered waves in theFraunhofer(long-distance or lens back-focal-plane) limit as aHuygens-stylesum of amplitudes from all points of scattering (in this case from each individual atom).This sum is denoted by thecomplex amplitudeF{\displaystyle F}in the equation below, because it is also the Fourier transform (as a function of spatial frequency or reciprocal distance) of an effective scattering potential in direct space:
Hereg=q/(2π) is the scattering vectorqin crystallographer units,Nis the number of atoms,fj[g] is theatomic scattering factorfor atomjand scattering vectorg, whilerjis the vector position of atomj. The Fourier phase depends on one's choice of coordinate origin.
For the special case of an infinite periodic crystal, the scattered amplitudeF=MFh,k,ℓfromMunit cells (as in the cases above) turns out to be non-zero only for integer values of(h,k,ℓ){\displaystyle (h,k,\ell )}, where
when there arej= 1,matoms inside the unit cell whose fractional lattice indices are respectively {uj,vj,wj}. To consider effects due to finite crystal size, of course, a shape convolution for each point or the equation above for a finite lattice must be used instead.
Whether the array of atoms is finite or infinite, one can also imagine an "intensity reciprocal lattice" I[g], which relates to the amplitude lattice F via the usual relationI=FFwhereFis the complex conjugate of F. Since Fourier transformation is reversible, of course, this act of conversion to intensity tosses out "all except 2nd moment" (i.e. the phase) information.  For the case of an arbitrary collection of atoms, the intensity reciprocal lattice is therefore:
Hererjkis the vector separation between atomjand atomk. One can also use this to predict the effect of nano-crystallite shape, and subtle changes in beam orientation, on detected diffraction peaks even if in some directions the cluster is only one atom thick. On the down side, scattering calculations using the reciprocal lattice basically consider an incident plane wave. Thus after a first look at reciprocal lattice (kinematic scattering) effects, beam broadening and multiple scattering (i.e.dynamical) effects may be important to consider as well.
There are actually two versions inmathematicsof the abstract dual lattice concept, for a givenlatticeLin a real vector spaceV, offinite dimension.
The first, which generalises directly the reciprocal lattice construction, usesFourier analysis. It may be stated simply in terms ofPontryagin duality. Thedual groupV^ toVis again a real vector space, and its closed subgroupL^ dual toLturns out to be a lattice inV^. Therefore,L^ is the natural candidate fordual lattice, in a different vector space (of the same dimension).
The other aspect is seen in the presence of aquadratic formQonV; if it isnon-degenerateit allows an identification of thedual spaceVofVwithV. The relation ofVtoVis not intrinsic; it depends on a choice ofHaar measure(volume element) onV. But given an identification of the two, which is in any casewell-definedup to ascalar, the presence ofQallows one to speak to the dual lattice toLwhile staying withinV.
In mathematics, the dual lattice of a givenlatticeLin anabelianlocally compacttopological groupGis the subgroupLof thedual groupofGconsisting of all continuous characters that are equal to one at each point ofL.
Indiscrete mathematics, a lattice is a locally discrete set of points described by all integral linear combinations ofdim =nlinearly independent vectors inR. The dual lattice is then defined by all points in the linear span of the original lattice (typically all ofR) with the property that an integer results from the inner product with all elements of the original lattice. It follows that the dual of the dual lattice is the original lattice.
Furthermore, if we allow thematrixBto have columns as the linearly independent vectors that describe the lattice, then the matrixA=B(BTB)−1{\displaystyle A=B\left(B^{\mathsf {T}}B\right)^{-1}}has columns of vectors that describe the dual lattice.
Inquantum physics, reciprocal space is closely related tomomentum spaceaccording to the proportionalityp=ℏk{\displaystyle \mathbf {p} =\hbar \mathbf {k} }, wherep{\displaystyle \mathbf {p} }is themomentum vectorandℏ{\displaystyle \hbar }is thereduced Planck constant.