337:
10816:
4952:
31:
11209:
4686:
9812:. 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.
6109:
5391:
2537:
11221:
7939:
4947:{\displaystyle {\begin{aligned}\mathbf {b} _{1}&=2\pi {\frac {-\mathbf {Q} \,\mathbf {a} _{2}}{-\mathbf {a} _{1}\cdot \mathbf {Q} \,\mathbf {a} _{2}}}=2\pi {\frac {\mathbf {Q} \,\mathbf {a} _{2}}{\mathbf {a} _{1}\cdot \mathbf {Q} \,\mathbf {a} _{2}}}\\\mathbf {b} _{2}&=2\pi {\frac {\mathbf {Q} \,\mathbf {a} _{1}}{\mathbf {a} _{2}\cdot \mathbf {Q} \,\mathbf {a} _{1}}}\end{aligned}}}
43:
6294:
3769:
5889:
9040:
5182:
6114:
3539:
7732:
8824:
365:
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 as real space. In physical applications, such as crystallography, both real and reciprocal space will often each be two or
6865:
8501:
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
6104:{\displaystyle {\begin{aligned}\mathbf {b} _{1}&={\frac {2\pi }{V}}\ \mathbf {a} _{2}\times \mathbf {a} _{3}\\\mathbf {b} _{2}&={\frac {2\pi }{V}}\ \mathbf {a} _{3}\times \mathbf {a} _{1}\\\mathbf {b} _{3}&={\frac {2\pi }{V}}\ \mathbf {a} _{1}\times \mathbf {a} _{2}\end{aligned}}}
9586:
is 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:
5386:{\displaystyle \mathbf {b} _{n}=2\pi {\frac {\mathbf {Q} \,\mathbf {a} _{\sigma (n)}}{\mathbf {a} _{n}\cdot \mathbf {Q} \,\mathbf {a} _{\sigma (n)}}}=2\pi {\frac {\mathbf {Q} '\,\mathbf {a} _{\sigma (n)}}{\mathbf {a} _{n}\cdot \mathbf {Q} '\,\mathbf {a} _{\sigma (n)}}}.}
220:. 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 represent
1206:
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
9790:
7934:{\displaystyle \mathbf {b} _{i}=2\pi {\frac {\varepsilon _{\sigma ^{1}i\ldots \sigma ^{n}i}}{\omega (\mathbf {a} _{1},\ldots ,\mathbf {a} _{n})}}g^{-1}(\mathbf {a} _{\sigma ^{n-1}i}\,\lrcorner \ldots \mathbf {a} _{\sigma ^{1}i}\,\lrcorner \,\omega )\in V}
8502:
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.
8820:
axis 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 are
7721:
9528:
6289:{\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)}
3764:{\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} }.}
5751:
5506:
4239:
2952:
3309:
1936:
2224:
1358:
9235:
6751:
8688:
8619:
6367:
5885:
5578:
4435:
5084:
4648:
6435:
4307:
2037:
808:
9035:{\displaystyle {\begin{aligned}a_{1}&={\frac {\sqrt {3}}{2}}a{\hat {x}}+{\frac {1}{2}}a{\hat {y}},\\a_{2}&=-{\frac {\sqrt {3}}{2}}a{\hat {x}}+{\frac {1}{2}}a{\hat {y}},\\a_{3}&=c{\hat {z}}.\end{aligned}}}
366:
three dimensional. Whereas the number of spatial dimensions of these two associated spaces will be the same, the spaces will differ in their quantity dimension, so that when the real space has the dimension length (
7633:
5171:
1422:
primitive 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 origin
1774:
1658:
8829:
5894:
3531:
4566:
6508:
3890:
3831:
858:
1855:
7518:
4691:
6970:
6743:
947:
10090:
6659:
7983:
8345:
7558:
3013:
5647:
172:
2396:
437:
8228:
9593:
8065:
10825:
8133:
10021:) 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.
7064:
5816:
3377:
2841:
8287:
11086:
9088:
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:
2489:
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.)
7644:
5034:
3921:
3155:
11081:
8814:
1086:
7380:
6926:
6897:
4501:
4472:
4366:
4337:
4135:
4099:
4047:
4018:
3989:
3956:
3462:
3067:
2344:
2315:
2286:
2126:
1969:
1689:
1591:
1510:
1480:
1416:
739:
290:
257:
9050:
8496:
8429:
3411:
3217:
3186:
3098:
1204:
699:
6690:
comes naturally from the study of periodic structures. An essentially equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice
3962:
of 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 vectors
1450:
1146:. 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 of
1003:
9571:
Whether the array of atoms is finite or infinite, one can also imagine an "intensity reciprocal lattice" I, which relates to the amplitude lattice F via the usual relation
9359:
2628:
5106:
5007:
4977:
3433:
3120:
3038:
2487:
2465:
2418:
2366:
2071:
1802:
1716:
1169:
1048:
903:
470:
194:
9351:
8456:
7102:
8153:
6664:
This method appeals to the definition, and allows generalization to arbitrary dimensions. The cross product formula dominates introductory materials on crystallography.
7343:
7311:
7134:
5652:
5407:
4140:
2853:
10621:
1237:
1131:
971:
664:
3228:
645:
leads to their visualization within complementary spaces (the real space and the reciprocal space). The spatial periodicity of this wave is defined by its wavelength
580:
518:
11137:
8017:
1861:
1561:
214:
7279:
7259:
7212:
7165:
6993:
6688:
4678:
4070:
2702:
2675:
2592:
2565:
2442:
2253:
2132:
1535:
1387:
1266:
314:
8770:
8547:
9568:}. 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.
2748:
2097:
9086:
8397:
8162:
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,
7459:
7432:
7412:
7232:
7185:
2768:
2722:
2648:
1257:
1106:
1024:
879:
643:
623:
601:
560:
540:
494:
10167:
Sung, S.H.; Schnitzer, N.; Brown, L.; Park, J.; Hovden, R. (2019-06-25). "Stacking, strain, and twist in 2D materials quantified by 3D electron diffraction".
6860:{\displaystyle \mathbf {b} _{1}={\frac {\mathbf {a} _{2}\times \mathbf {a} _{3}}{\mathbf {a} _{1}\cdot \left(\mathbf {a} _{2}\times \mathbf {a} _{3}\right)}}}
8739:
8719:
9094:
10365:
8458:
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.
2128:
comprise a set of three primitive wavevectors or three primitive translation vectors for the reciprocal lattice, each of whose vertices takes the form
1262:
One heuristic approach to constructing the reciprocal lattice in three dimensions is to write the position vector of a vertex of the direct lattice as
8624:
8555:
6303:
5821:
5514:
4371:
2704:(i.e. any reciprocal lattice vector), the resulting plane waves have the same periodicity of the lattice – that is that any translation from point
4574:
6373:
4245:
2467:
will essentially be equal for every direct lattice vertex, in conformity with the reciprocal lattice definition above. (Although any wavevector
1975:
749:
9053:
Shadow of a 118-atom faceted carbon-pentacone's intensity reciprocal-lattice lighting up red in diffraction when intersecting the Ewald sphere.
5039:
7563:
5117:
10955:
3188:
follows the periodicity of this lattice, e.g. the function describing the electronic density in an atomic crystal, it is useful to write
1722:
1598:
11142:
10867:
3470:
11033:
10325:
2770:
shown red), the value of the plane wave is the same. These plane waves can be added together and the above relation will still apply.
4517:
1142:
In general, a geometric lattice is an infinite, regular array of vertices (points) in space, which can be modelled vectorially as a
6453:
3839:
3776:
815:
221:
1807:
11270:
7467:
7349:
lattice. (A lattice plane is a plane crossing lattice points.) The direction of the reciprocal lattice vector corresponds to the
10290:-based electron diffraction simulator lets you explore the intersection between reciprocal lattice and Ewald sphere during tilt.
340:
Adsorbed species on the surface with 1×2 superstructure give rise to additional spots in low-energy electron diffraction (LEED).
11132:
11124:
4568:, its reciprocal lattice can be determined by generating its two reciprocal primitive vectors, through the following formulae,
11185:
11163:
10299:
Learn easily crystallography and how the reciprocal lattice explains the diffraction phenomenon, as shown in chapters 4 and 5
6931:
6693:
5396:
Notably, in a 3D space this 2D reciprocal lattice is an infinitely extended set of Bragg rods—described by Sung et al.
4987:
uarter turn. The anti-clockwise rotation and the clockwise rotation can both be used to determine the reciprocal lattice: If
908:
11178:
11028:
10694:
10559:
10408:
10034:
6524:
11265:
11168:
11066:
10762:
10279:
7947:
2540:
Demonstration of relation between real and reciprocal lattice. A real space 2D lattice (red dots) with primitive vectors
10415:
10125:
8292:
7523:
2957:
9065:
sum of amplitudes from all points of scattering (in this case from each individual atom). This sum is denoted by the
6870:
and so on for the other primitive vectors. The crystallographer's definition has the advantage that the definition of
5583:
11280:
11190:
11048:
11018:
10947:
10244:
9057:
One path to the reciprocal lattice of an arbitrary collection of atoms comes from the idea of scattered waves in the
263:
of plane waves in the
Fourier series of a spatial function whose periodicity is the same as that of a direct lattice
144:
9785:{\displaystyle I=\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}\leftf_{k}\lefte^{2\pi i{\vec {g}}\cdot {\vec {r}}_{\!\!\;jk}}.}
2371:
385:
11225:
10900:
8165:
17:
8351:
by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation).
6514:. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using
11173:
11096:
10970:
10569:
9813:
8022:
6995:. This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of
11008:
10930:
4368:, is itself a Bravais lattice as it is formed by integer combinations of its own primitive translation vectors
10017:. The dual lattice is then defined by all points in the linear span of the original lattice (typically all of
953:
in the three dimensional reciprocal space. (The magnitude of a wavevector is called wavenumber.) The constant
11023:
11013:
10318:
9062:
8070:
361:
arising from the
Fourier transform of a time dependent function; reciprocal space is a space over which the
336:
11147:
10795:
10420:
10398:
7716:{\displaystyle \sigma ={\begin{pmatrix}1&2&\cdots &n\\2&3&\cdots &1\end{pmatrix}},}
7004:
5756:
3317:
2781:
8233:
10699:
10453:
10348:
9964:
3895:
3125:
292:. Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of
11056:
10353:
8466:
The reciprocal lattice to an FCC lattice is the body-centered cubic (BCC) lattice, with a cube side of
1053:
9523:{\displaystyle F_{h,k,\ell }=\sum _{j=1}^{m}f_{j}\lefte^{2\pi i\left(hu_{j}+kv_{j}+\ell w_{j}\right)}}
8775:
7356:
6902:
6873:
4477:
4448:
4342:
4313:
4111:
4075:
4049:. Each plane wave in the Fourier series has the same phase (actually can be differed by a multiple of
4023:
3994:
3965:
3932:
3438:
3043:
2320:
2291:
2262:
2102:
1945:
1665:
1567:
1486:
1456:
1392:
707:
266:
233:
11255:
11071:
11000:
10458:
10448:
6999:. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed.
3385:
3191:
3160:
3072:
1178:
672:
67:
8469:
8402:
1427:
980:
11250:
11213:
10937:
10833:
10706:
10669:
10584:
10463:
10443:
10311:
9846:
9270:
7435:
5012:
2597:
377:
Reciprocal space comes into play regarding waves, both classical and quantum mechanical. Because a
5746:{\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}
5501:{\displaystyle \mathbf {R} _{n}=n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}
5089:
4990:
4960:
4234:{\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}
3416:
3103:
3021:
2947:{\displaystyle \mathbf {R} _{n}=n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}
2470:
2448:
2401:
2349:
2046:
1780:
1694:
1152:
1031:
886:
448:
177:
11260:
11061:
10905:
10850:
10599:
10564:
10006:, a lattice is a locally discrete set of points described by all integral linear combinations of
9318:
8621:(cubic, tetragonal, orthorhombic) have primitive translation vectors for the reciprocal lattice,
7069:
217:
10293:
8434:
8138:
6447:
4105:
3304:{\displaystyle \sum _{m}f_{m}e^{i\mathbf {G} _{m}\cdot \mathbf {r} }=f\left(\mathbf {r} \right)}
3220:
10815:
10757:
10574:
10031:
to have columns as the linearly independent vectors that describe the lattice, then the matrix
9058:
8360:
7462:
7316:
7284:
7107:
1931:{\displaystyle \mathbf {a} _{2}\cdot \mathbf {b} _{1}=\mathbf {a} _{3}\cdot \mathbf {b} _{1}=0}
55:
2219:{\displaystyle \mathbf {G} =m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}
1353:{\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}
1210:
1116:
956:
649:
122:
considered as a vector space, and the reciprocal lattice is the sublattice of that space that
11114:
10910:
10872:
10679:
10631:
9974:
6297:
2501:
565:
503:
378:
325:
10298:
7992:
1540:
199:
11275:
10838:
10711:
10547:
10438:
10186:
10025:
10003:
9945:
8515:
8511:
7636:
7264:
7237:
7190:
7143:
6975:
6670:
5109:
4656:
4052:
2680:
2653:
2570:
2543:
2445:(that can be possibly zero if the multiplier is zero), so the phase of the plane wave with
2424:
2231:
1517:
1365:
317:
at each direct lattice point (so essentially same phase at all the direct lattice points).
296:
63:
4437:, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the
8:
10855:
10843:
10718:
10684:
10664:
8744:
8521:
2778:
and labelling each lattice vector (a vector indicating a lattice point) by the subscript
2727:
2505:
2076:
1419:
107:
10190:
11104:
10915:
10860:
10202:
10176:
10140:
9988:
9861:
9857:
9315:
unit cells (as in the cases above) turns out to be non-zero only for integer values of
9071:
8382:
7444:
7417:
7397:
7217:
7170:
4438:
3927:
2753:
2707:
2633:
2509:
1242:
1091:
1009:
864:
628:
608:
586:
545:
525:
479:
474:
228:
103:
10198:
9230:{\displaystyle F=\sum _{j=1}^{N}f_{j}\!\lefte^{2\pi i{\vec {g}}\cdot {\vec {r}}_{j}}.}
2528:
provide more abstract generalizations of reciprocal space and the reciprocal lattice.
11038:
10877:
10805:
10785:
10505:
10375:
10294:
DoITPoMS Teaching and
Learning Package on Reciprocal Space and the Reciprocal Lattice
10240:
10206:
10119:
9977:
9066:
8699:
7383:
6996:
1691:
is the unit vector perpendicular to these two adjacent wavefronts and the wavelength
497:
362:
354:
87:
71:
59:
8399:, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side
7386:
and is equal to the reciprocal of the interplanar spacing of the real space planes.
11076:
10882:
10800:
10790:
10589:
10522:
10493:
10486:
10194:
9853:
8724:
8704:
8156:
6515:
2513:
2346:
in this case. Simple algebra then shows that, for any plane wave with a wavevector
358:
10965:
10960:
10925:
10745:
10644:
10579:
10542:
10537:
10388:
10334:
10283:
10107:
9835:
8372:
6439:
4980:
3069:
is a primitive translation vector or shortly primitive vector. Taking a function
2775:
2256:
2041:
1143:
134:
99:
91:
75:
9852:
The first, which generalises directly the reciprocal lattice construction, uses
30:
10775:
10740:
10728:
10723:
10689:
10659:
10649:
10608:
10552:
10476:
10430:
10101:
9995:
consisting of all continuous characters that are equal to one at each point of
9906:
9895:
8683:{\displaystyle \left(\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right)}
8614:{\displaystyle \left(\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3}\right)}
8376:
7350:
6362:{\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}
5880:{\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}
5573:{\displaystyle \left(\mathbf {a_{1}} ,\mathbf {a} _{2},\mathbf {a} _{3}\right)}
5509:
4512:
4430:{\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}
3773:
Because equality of two
Fourier series implies equality of their coefficients,
2594:
are shown by blue and green arrows respectively. Atop, plane waves of the form
2497:
2493:
1172:
321:
138:
95:
10276:
9541:
atoms inside the unit cell whose fractional lattice indices are respectively {
9297:
For the special case of an infinite periodic crystal, the scattered amplitude
8552:
It can be proven that only the
Bravais lattices which have 90 degrees between
1942:
Cycling through the indices in turn, the same method yields three wavevectors
11244:
10920:
10733:
10532:
9971:
1050:
is a unit vector perpendicular to this wavefront. The wavefronts with phases
442:
4643:{\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}}
10626:
10616:
10510:
10393:
10134:
9941:
9933:
9825:
7137:
4442:
2525:
2259:
as it is formed by integer combinations of the primitive vectors, that are
1804:
is equal to the distance between the two wavefronts. Hence by construction
123:
27:
Fourier transform of a real-space lattice, important in solid-state physics
6667:
The above definition is called the "physics" definition, as the factor of
6430:{\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}
4302:{\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}
2032:{\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}
803:{\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{0})}
11109:
10780:
10654:
10481:
9831:
7986:
7438:
7353:
to the real space planes. The magnitude of the reciprocal lattice vector
3959:
2521:
1147:
950:
260:
7560:. The reciprocal lattice vectors are uniquely determined by the formula
10674:
10360:
10113:
9910:
6450:. This choice also satisfies the requirement of the reciprocal lattice
5818:
can be determined by generating its three reciprocal primitive vectors
5079:{\displaystyle \mathbf {Q} \,\mathbf {v} =-\mathbf {Q'} \,\mathbf {v} }
4339:
for the Fourier series of a spatial function which periodicity follows
4310:. With this form, the reciprocal lattice as the set of all wavevectors
2517:
35:
7628:{\displaystyle g(\mathbf {a} _{i},\mathbf {b} _{j})=2\pi \delta _{ij}}
5166:{\displaystyle \sigma ={\begin{pmatrix}1&2\\2&1\end{pmatrix}}}
3536:
Expressing the above instead in terms of their Fourier series we have
1112:, comprise a set of parallel planes, equally spaced by the wavelength
10383:
10146:
974:
10980:
10750:
10498:
10181:
8348:
1769:{\displaystyle \lambda _{1}=\mathbf {a} _{1}\cdot \mathbf {e} _{1}}
1653:{\displaystyle \mathbf {b} _{1}=2\pi \mathbf {e} _{1}/\lambda _{1}}
10303:
10104: – Primitive cell in the reciprocal space lattice of crystals
9294:. The Fourier phase depends on one's choice of coordinate origin.
10990:
7313:
in the reciprocal lattice corresponds to a set of lattice planes
3526:{\displaystyle f(\mathbf {r} +\mathbf {R} _{n})=f(\mathbf {r} ).}
1109:
2536:
9940:. But given an identification of the two, which is in any case
745:
In three dimensions, the corresponding plane wave term becomes
703:
hence the corresponding wavenumber in reciprocal space will be
4561:{\displaystyle \left(\mathbf {a} _{1},\mathbf {a} _{2}\right)}
2650:
is any integer combination of reciprocal lattice vector basis
2504:) of the reciprocal lattice, which plays an important role in
1563:
from the former wavefront passing the origin) passing through
10985:
7346:
6503:{\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1}
3885:{\displaystyle \mathbf {G} _{m}\cdot \mathbf {R} _{n}=2\pi N}
3826:{\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1}
2844:
853:{\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} +\varphi )}
1850:{\displaystyle \mathbf {a} _{1}\cdot \mathbf {b} _{1}=2\pi }
42:
10287:
9049:
8741:
is another simple hexagonal lattice with lattice constants
7513:{\displaystyle (\mathbf {a} _{1},\ldots ,\mathbf {a} _{n})}
8698:
The reciprocal to a simple hexagonal Bravais lattice with
8354:
6438:
as the known condition (There may be other condition.) of
381:
with unit amplitude can be written as an oscillatory term
11087:
Zeitschrift für Kristallographie – New Crystal Structures
3926:
Mathematically, the reciprocal lattice is the set of all
34:
The computer-generated reciprocal lattice of a fictional
11082:
Zeitschrift für Kristallographie – Crystalline Materials
10239:(8th ed.). John Wiley & Sons, Inc. p. 44.
4511:
For an infinite two-dimensional lattice, defined by its
1514:
and with its adjacent wavefront (whose phase differs by
905:
is the position vector of a point in real space and now
10130:
Pages displaying short descriptions of redirect targets
10122: – Energy conservation during diffraction by atoms
10092:
has columns of vectors that describe the dual lattice.
9891:, in a different vector space (of the same dimension).
6965:{\displaystyle \mathbf {a} _{2}\times \mathbf {a} _{3}}
6745:. which changes the reciprocal primitive vectors to be
6738:{\displaystyle \mathbf {K} _{m}=\mathbf {G} _{m}/2\pi }
942:{\displaystyle \mathbf {k} =2\pi \mathbf {e} /\lambda }
353:-space) provides a way to visualize the results of the
10975:
10166:
9871:
is again a real vector space, and its closed subgroup
8778:
8747:
8727:
8707:
8524:
8472:
8437:
8405:
7659:
5132:
10085:{\displaystyle A=B\left(B^{\mathsf {T}}B\right)^{-1}}
10037:
9596:
9362:
9321:
9097:
9074:
8827:
8627:
8558:
8385:
8295:
8236:
8168:
8141:
8073:
8025:
7995:
7950:
7735:
7647:
7566:
7526:
7470:
7447:
7420:
7400:
7359:
7319:
7287:
7267:
7240:
7220:
7193:
7173:
7146:
7110:
7072:
7007:
6978:
6934:
6905:
6876:
6754:
6696:
6673:
6527:
6456:
6376:
6306:
6117:
5892:
5824:
5759:
5655:
5586:
5517:
5410:
5185:
5120:
5092:
5042:
5015:
4993:
4963:
4689:
4659:
4577:
4520:
4480:
4451:
4374:
4345:
4316:
4248:
4143:
4114:
4078:
4055:
4026:
3997:
3968:
3935:
3898:
3842:
3779:
3542:
3473:
3441:
3419:
3388:
3320:
3231:
3194:
3163:
3128:
3106:
3075:
3046:
3024:
2960:
2856:
2784:
2756:
2730:
2710:
2683:
2656:
2636:
2600:
2573:
2546:
2473:
2451:
2427:
2404:
2374:
2352:
2323:
2294:
2265:
2234:
2135:
2105:
2079:
2049:
1978:
1948:
1864:
1810:
1783:
1725:
1697:
1668:
1601:
1570:
1543:
1520:
1489:
1459:
1430:
1395:
1368:
1269:
1245:
1213:
1181:
1155:
1119:
1094:
1056:
1034:
1012:
983:
959:
911:
889:
867:
818:
752:
710:
675:
652:
631:
611:
589:
568:
548:
528:
506:
482:
451:
388:
370:), its reciprocal space will have inverse length, so
299:
269:
236:
202:
180:
147:
9061:(long-distance or lens back-focal-plane) limit as a
8505:
8461:
6654:{\displaystyle \left^{\mathsf {T}}=2\pi \left^{-1}.}
3413:
follows the periodicity of the lattice, translating
357:
of a spatial function. It is similar in role to the
46:
A two-dimensional crystal and its reciprocal lattice
9819:
7978:{\displaystyle \omega \colon V^{n}\to \mathbf {R} }
7726:they can be determined with the following formula:
10137: – Notation system for crystal lattice planes
10084:
9834:of the abstract dual lattice concept, for a given
9784:
9522:
9345:
9229:
9080:
9034:
8808:
8764:
8733:
8713:
8682:
8613:
8541:
8490:
8450:
8423:
8391:
8339:
8281:
8222:
8147:
8127:
8059:
8011:
7977:
7933:
7715:
7627:
7552:
7512:
7453:
7426:
7406:
7374:
7337:
7305:
7273:
7253:
7226:
7206:
7179:
7159:
7128:
7096:
7058:
6987:
6964:
6920:
6891:
6859:
6737:
6682:
6653:
6502:
6429:
6361:
6288:
6103:
5879:
5810:
5745:
5641:
5572:
5500:
5385:
5165:
5100:
5078:
5028:
5001:
4971:
4946:
4672:
4642:
4560:
4495:
4466:
4429:
4360:
4331:
4301:
4233:
4129:
4093:
4064:
4041:
4012:
3983:
3950:
3915:
3884:
3825:
3763:
3525:
3456:
3427:
3405:
3371:
3303:
3211:
3180:
3149:
3114:
3092:
3061:
3032:
3007:
2946:
2835:
2762:
2742:
2716:
2696:
2669:
2642:
2622:
2586:
2559:
2481:
2459:
2436:
2412:
2390:
2360:
2338:
2309:
2280:
2247:
2218:
2120:
2091:
2065:
2031:
1963:
1930:
1849:
1796:
1768:
1710:
1683:
1652:
1585:
1555:
1529:
1504:
1474:
1444:
1410:
1381:
1352:
1251:
1231:
1198:
1163:
1125:
1100:
1080:
1042:
1018:
997:
965:
941:
897:
873:
852:
802:
733:
693:
658:
637:
617:
595:
574:
554:
534:
512:
488:
464:
431:
308:
284:
251:
208:
188:
166:
78:associated with the arrangement of the atoms. The
9767:
9766:
9153:
8340:{\displaystyle R\in {\text{SO}}(2)\subset L(V,V)}
7553:{\displaystyle g\colon V\times V\to \mathbf {R} }
3008:{\displaystyle n_{1},n_{2},n_{3}\in \mathbb {Z} }
2368:on the reciprocal lattice, the total phase shift
977:(a plane of a constant phase) through the origin
562:(and the time-varying part as a function of both
11242:
10143: – Experimental method in X-ray diffraction
9816:) effects may be important to consider as well.
9044:
5642:{\displaystyle n=\left(n_{1},n_{2},n_{3}\right)}
10263:(Addison-Wesley, Reading MA/Dover, Mineola NY).
8019:is the inverse of the vector space isomorphism
2255:are integers. The reciprocal lattice is also a
167:{\displaystyle \mathbf {p} =\hbar \mathbf {k} }
58:, and plays a major role in many areas such as
10149: – High symmetry orientation of a crystal
10110: – Scientific study of crystal structures
9894:The other aspect is seen in the presence of a
2391:{\displaystyle \mathbf {G} \cdot \mathbf {R} }
432:{\displaystyle \cos(kx-\omega t+\varphi _{0})}
10319:
8223:{\displaystyle \omega (u,v,w)=g(u\times v,w)}
6511:
70:of electrons in a solid. It emerges from the
9963:In mathematics, the dual lattice of a given
9932:is not intrinsic; it depends on a choice of
11156:
10277:http://newton.umsl.edu/run//nano/known.html
9952:allows one to speak to the dual lattice to
8693:
8060:{\displaystyle {\hat {g}}\colon V\to V^{*}}
2531:
10326:
10312:
9768:
6442:for the reciprocal lattice derived in the
5404:For an infinite three-dimensional lattice
10180:
10160:
7918:
7914:
7885:
6413:
5355:
5306:
5259:
5215:
5070:
5048:
4924:
4889:
4836:
4801:
4768:
4730:
4503:results in the same reciprocal lattice.)
4285:
3906:
3727:
3026:
3001:
2015:
1389:are integers defining the vertex and the
522:it can be regarded as a function of both
331:
227:The reciprocal lattice is the set of all
137:, reciprocal space is closely related to
9048:
8690:, parallel to their real-space vectors.
2630:are plotted. From this we see that when
2535:
1594:. Its angular wavevector takes the form
335:
102:). The reciprocal lattice exists in the
41:
29:
8366:
8355:Reciprocal lattices of various crystals
8128:{\displaystyle {\hat {g}}(v)(w)=g(v,w)}
6443:
2420:on the direct lattice is a multiple of
14:
11243:
10429:
10234:
10059:
9856:. It may be stated simply in terms of
9804:is the vector separation between atom
7414:dimensions can be derived assuming an
6577:
1453:contains the direct lattice points at
54:is a term associated with solids with
10307:
10219:
10128: – Patterns formed by scattering
7059:{\displaystyle m=(m_{1},m_{2},m_{3})}
5811:{\displaystyle m=(m_{1},m_{2},m_{3})}
3372:{\displaystyle m=(m_{1},m_{2},m_{3})}
3122:is a position vector from the origin
2836:{\displaystyle n=(n_{1},n_{2},n_{3})}
1137:
11220:
10560:Phase transformation crystallography
8282:{\displaystyle \omega (v,w)=g(Rv,w)}
7389:
6899:is just the reciprocal magnitude of
11067:Journal of Chemical Crystallography
10333:
10237:Introduction to Solid State Physics
9909:it allows an identification of the
9830:There are actually two versions in
5399:
5009:is the anti-clockwise rotation and
344:
222:covariant and contravariant vectors
24:
10126:Kikuchi line (solid state physics)
3916:{\displaystyle N\in \mathbb {Z} .}
3150:{\displaystyle \mathbf {R} _{n}=0}
1259:) at every direct lattice vertex.
25:
11292:
10270:
10199:10.1103/PhysRevMaterials.3.064003
8809:{\textstyle 4\pi /(a{\sqrt {3}})}
8506:Body-centered cubic (BCC) lattice
8462:Face-centered cubic (FCC) lattice
4506:
2398:between the origin and any point
1081:{\displaystyle \varphi +(2\pi )n}
156:
141:according to the proportionality
11219:
11208:
11207:
10814:
10013:linearly independent vectors in
9820:Generalization of a dual lattice
8665:
8650:
8635:
8596:
8581:
8566:
7971:
7894:
7859:
7825:
7804:
7738:
7590:
7575:
7546:
7497:
7476:
7375:{\displaystyle \mathbf {K} _{m}}
7362:
6952:
6937:
6921:{\displaystyle \mathbf {a} _{1}}
6908:
6892:{\displaystyle \mathbf {b} _{1}}
6879:
6839:
6824:
6804:
6790:
6775:
6757:
6714:
6699:
6624:
6612:
6600:
6560:
6548:
6536:
6482:
6467:
6448:multi-dimensional Fourier series
6394:
6379:
6344:
6329:
6318:
6314:
6271:
6256:
6236:
6216:
6201:
6181:
6161:
6146:
6126:
6087:
6072:
6035:
6019:
6004:
5967:
5951:
5936:
5899:
5862:
5847:
5836:
5832:
5733:
5708:
5683:
5658:
5555:
5540:
5529:
5525:
5488:
5463:
5438:
5413:
5358:
5347:
5332:
5309:
5298:
5262:
5255:
5241:
5218:
5211:
5188:
5094:
5072:
5062:
5050:
5044:
5018:
4995:
4965:
4927:
4920:
4906:
4892:
4885:
4858:
4839:
4832:
4818:
4804:
4797:
4771:
4764:
4750:
4733:
4726:
4696:
4630:
4605:
4580:
4543:
4528:
4496:{\displaystyle \mathbf {G} _{m}}
4483:
4467:{\displaystyle \mathbf {G} _{m}}
4454:
4412:
4397:
4386:
4382:
4361:{\displaystyle \mathbf {R} _{n}}
4348:
4332:{\displaystyle \mathbf {G} _{m}}
4319:
4266:
4251:
4221:
4196:
4171:
4146:
4130:{\displaystyle \mathbf {G} _{m}}
4117:
4106:multi-dimensional Fourier series
4094:{\displaystyle \mathbf {R} _{n}}
4081:
4042:{\displaystyle \mathbf {R} _{n}}
4029:
4013:{\displaystyle \mathbf {G} _{m}}
4000:
3984:{\displaystyle \mathbf {R} _{n}}
3971:
3951:{\displaystyle \mathbf {G} _{m}}
3938:
3860:
3845:
3805:
3790:
3752:
3738:
3715:
3700:
3652:
3643:
3626:
3587:
3573:
3513:
3490:
3481:
3457:{\displaystyle \mathbf {R} _{n}}
3444:
3421:
3396:
3293:
3276:
3262:
3221:multi-dimensional Fourier series
3202:
3171:
3131:
3108:
3083:
3062:{\displaystyle \mathbf {a} _{i}}
3049:
2934:
2909:
2884:
2859:
2475:
2453:
2406:
2384:
2376:
2354:
2339:{\displaystyle \mathbf {b} _{3}}
2326:
2310:{\displaystyle \mathbf {b} _{2}}
2297:
2281:{\displaystyle \mathbf {b} _{1}}
2268:
2206:
2181:
2156:
2137:
2121:{\displaystyle \mathbf {b} _{j}}
2108:
1996:
1981:
1964:{\displaystyle \mathbf {b} _{j}}
1951:
1912:
1897:
1882:
1867:
1828:
1813:
1756:
1741:
1684:{\displaystyle \mathbf {e} _{1}}
1671:
1625:
1604:
1586:{\displaystyle \mathbf {a} _{1}}
1573:
1505:{\displaystyle \mathbf {a} _{3}}
1492:
1475:{\displaystyle \mathbf {a} _{2}}
1462:
1432:
1411:{\displaystyle \mathbf {a} _{i}}
1398:
1340:
1315:
1290:
1271:
1189:
1157:
1036:
985:
927:
913:
891:
837:
829:
771:
763:
734:{\displaystyle k=2\pi /\lambda }
285:{\displaystyle \mathbf {R} _{n}}
272:
252:{\displaystyle \mathbf {G} _{m}}
239:
182:
160:
149:
9887:^ is the natural candidate for
9290:is the vector position of atom
8491:{\textstyle {\frac {4\pi }{a}}}
8424:{\textstyle {\frac {2\pi }{a}}}
3406:{\displaystyle f(\mathbf {r} )}
3212:{\displaystyle f(\mathbf {r} )}
3181:{\displaystyle f(\mathbf {r} )}
3093:{\displaystyle f(\mathbf {r} )}
1199:{\displaystyle f(\mathbf {r} )}
694:{\displaystyle k\lambda =2\pi }
11271:Synchrotron-related techniques
11009:Bilbao Crystallographic Server
10253:
10228:
10213:
10116: – Linear algebra concept
9759:
9743:
9713:
9683:
9615:
9609:
9600:
9340:
9322:
9210:
9194:
9164:
9116:
9110:
9101:
9019:
8980:
8952:
8898:
8870:
8816:rotated through 90° about the
8803:
8790:
8334:
8322:
8313:
8307:
8276:
8261:
8252:
8240:
8217:
8199:
8190:
8172:
8122:
8110:
8101:
8095:
8092:
8086:
8080:
8044:
8032:
7967:
7922:
7854:
7835:
7799:
7600:
7570:
7542:
7507:
7471:
7332:
7320:
7300:
7288:
7123:
7111:
7091:
7073:
7053:
7014:
5805:
5766:
5580:and the subscript of integers
5372:
5366:
5323:
5317:
5276:
5270:
5232:
5226:
4445:. (There may be other form of
4020:satisfy this equality for all
3662:
3639:
3517:
3509:
3500:
3477:
3400:
3392:
3366:
3327:
3206:
3198:
3175:
3167:
3087:
3079:
2830:
2791:
1445:{\displaystyle \mathbf {R} =0}
1223:
1214:
1193:
1185:
1072:
1063:
998:{\displaystyle \mathbf {r} =0}
847:
825:
797:
759:
426:
395:
349:Reciprocal space (also called
13:
1:
10153:
10024:Furthermore, if we allow the
9879:turns out to be a lattice in
9045:Arbitrary collection of atoms
8518:lattice, with a cube side of
7066:is conventionally written as
6440:primitive translation vectors
5029:{\displaystyle \mathbf {Q'} }
4137:can be chosen in the form of
3464:we get the same value, hence
2774:Assuming a three-dimensional
2623:{\displaystyle e^{iG\cdot r}}
8510:The reciprocal lattice to a
8359:Reciprocal lattices for the
5101:{\displaystyle \mathbf {v} }
5002:{\displaystyle \mathbf {Q} }
4972:{\displaystyle \mathbf {Q} }
3428:{\displaystyle \mathbf {r} }
3115:{\displaystyle \mathbf {r} }
3033:{\displaystyle \mathbb {Z} }
2482:{\displaystyle \mathbf {G} }
2460:{\displaystyle \mathbf {G} }
2413:{\displaystyle \mathbf {R} }
2361:{\displaystyle \mathbf {G} }
2066:{\displaystyle \delta _{ij}}
1797:{\displaystyle \lambda _{1}}
1711:{\displaystyle \lambda _{1}}
1164:{\displaystyle \mathbf {k} }
1043:{\displaystyle \mathbf {e} }
898:{\displaystyle \mathbf {r} }
465:{\displaystyle \varphi _{0}}
374:(the reciprocal of length).
189:{\displaystyle \mathbf {p} }
7:
11057:Crystal Growth & Design
10349:Timeline of crystallography
10095:
9346:{\displaystyle (h,k,\ell )}
9256:in crystallographer units,
9252:) is the scattering vector
8451:{\textstyle {\frac {1}{a}}}
7097:{\displaystyle (h,k,\ell )}
5753:with the integer subscript
5036:is the clockwise rotation,
4072:) at all the lattice point
3379:, so this is a triple sum.
3040:is the set of integers and
2099:and is zero otherwise. The
605:This complementary role of
328:of the reciprocal lattice.
196:is the momentum vector and
66:diffraction as well as the
10:
11297:
11266:Neutron-related techniques
10868:Nuclear magnetic resonance
9823:
8148:{\displaystyle \lrcorner }
2724:(shown orange) to a point
11203:
11123:
11095:
11072:Journal of Crystal Growth
11047:
10999:
10946:
10893:
10824:
10812:
10607:
10598:
10521:
10374:
10341:
10259:B. E. Warren (1969/1990)
10169:Physical Review Materials
7338:{\displaystyle (hk\ell )}
7306:{\displaystyle (hk\ell )}
7129:{\displaystyle (hk\ell )}
6972:, dropping the factor of
5649:, its reciprocal lattice
11281:Condensed matter physics
10938:Single particle analysis
10796:Hermann–Mauguin notation
10235:Kittel, Charles (2005).
9271:atomic scattering factor
9260:is the number of atoms,
8694:Simple hexagonal lattice
6444:heuristic approach above
4104:As shown in the section
3833:, which only holds when
3314:where now the subscript
2532:Mathematical description
1232:{\displaystyle (2\pi )n}
1126:{\displaystyle \lambda }
966:{\displaystyle \varphi }
659:{\displaystyle \lambda }
11062:Crystallography Reviews
10906:Isomorphous replacement
10700:Lomer–Cottrell junction
10224:. Springer. p. 69.
10220:Audin, Michèle (2003).
9841:in a real vector space
8230:and in two dimensions,
6510:mathematically derived
4979:represents a 90 degree
575:{\displaystyle \omega }
513:{\displaystyle \omega }
218:reduced Planck constant
118:, which is the dual of
10575:Spinodal decomposition
10086:
9786:
9662:
9641:
9524:
9408:
9347:
9277:and scattering vector
9231:
9142:
9082:
9054:
9036:
8810:
8766:
8735:
8715:
8684:
8615:
8543:
8492:
8452:
8425:
8393:
8341:
8283:
8224:
8149:
8129:
8061:
8013:
8012:{\displaystyle g^{-1}}
7979:
7935:
7717:
7629:
7554:
7514:
7455:
7428:
7408:
7376:
7339:
7307:
7275:
7255:
7228:
7208:
7181:
7161:
7130:
7098:
7060:
6989:
6966:
6922:
6893:
6861:
6739:
6684:
6655:
6504:
6431:
6363:
6300:. The choice of these
6290:
6105:
5881:
5812:
5747:
5643:
5574:
5502:
5387:
5167:
5102:
5080:
5030:
5003:
4973:
4948:
4674:
4644:
4562:
4497:
4468:
4431:
4362:
4333:
4303:
4235:
4131:
4095:
4066:
4043:
4014:
3985:
3952:
3917:
3886:
3827:
3765:
3527:
3458:
3435:by any lattice vector
3429:
3407:
3373:
3305:
3213:
3182:
3151:
3116:
3094:
3063:
3034:
3009:
2948:
2837:
2771:
2764:
2744:
2718:
2698:
2671:
2644:
2624:
2588:
2561:
2483:
2461:
2438:
2414:
2392:
2362:
2340:
2311:
2282:
2249:
2220:
2122:
2093:
2067:
2033:
1965:
1932:
1851:
1798:
1770:
1712:
1685:
1654:
1587:
1557:
1556:{\displaystyle -2\pi }
1531:
1506:
1476:
1446:
1412:
1383:
1354:
1253:
1233:
1200:
1171:of plane waves in the
1165:
1127:
1102:
1082:
1044:
1020:
999:
967:
943:
899:
875:
854:
804:
735:
695:
660:
639:
619:
597:
576:
556:
536:
514:
490:
466:
433:
341:
332:Wave-based description
310:
286:
253:
210:
209:{\displaystyle \hbar }
190:
168:
56:translational symmetry
47:
39:
11115:Gregori Aminoff Prize
10911:Molecular replacement
10087:
9956:while staying within
9787:
9642:
9621:
9525:
9388:
9348:
9232:
9122:
9083:
9052:
9037:
8811:
8767:
8736:
8716:
8685:
8616:
8544:
8493:
8453:
8426:
8394:
8342:
8284:
8225:
8150:
8130:
8062:
8014:
7980:
7936:
7718:
7630:
7555:
7520:and an inner product
7515:
7456:
7429:
7409:
7377:
7340:
7308:
7281:. Each lattice point
7276:
7274:{\displaystyle \ell }
7256:
7254:{\displaystyle m_{3}}
7229:
7209:
7207:{\displaystyle m_{2}}
7182:
7162:
7160:{\displaystyle m_{1}}
7131:
7099:
7061:
6990:
6988:{\displaystyle 2\pi }
6967:
6923:
6894:
6862:
6740:
6685:
6683:{\displaystyle 2\pi }
6656:
6505:
6432:
6364:
6298:scalar triple product
6291:
6106:
5882:
5813:
5748:
5644:
5575:
5503:
5388:
5168:
5103:
5081:
5031:
5004:
4974:
4949:
4675:
4673:{\displaystyle m_{i}}
4645:
4563:
4498:
4469:
4432:
4363:
4334:
4304:
4236:
4132:
4096:
4067:
4065:{\displaystyle 2\pi }
4044:
4015:
3986:
3953:
3918:
3887:
3828:
3766:
3528:
3459:
3430:
3408:
3374:
3306:
3214:
3183:
3152:
3117:
3095:
3064:
3035:
3010:
2949:
2838:
2765:
2745:
2719:
2699:
2697:{\displaystyle b_{2}}
2672:
2670:{\displaystyle b_{1}}
2645:
2625:
2589:
2587:{\displaystyle a_{2}}
2562:
2560:{\displaystyle a_{1}}
2539:
2500:(more specifically a
2484:
2462:
2439:
2437:{\displaystyle 2\pi }
2415:
2393:
2363:
2341:
2312:
2283:
2250:
2248:{\displaystyle m_{j}}
2221:
2123:
2094:
2068:
2034:
1966:
1933:
1852:
1799:
1771:
1713:
1686:
1655:
1588:
1558:
1532:
1530:{\displaystyle 2\pi }
1507:
1477:
1447:
1413:
1384:
1382:{\displaystyle n_{i}}
1355:
1254:
1234:
1201:
1166:
1128:
1103:
1083:
1045:
1021:
1000:
968:
944:
900:
876:
855:
805:
736:
696:
661:
640:
620:
598:
577:
557:
537:
515:
491:
467:
434:
379:sinusoidal plane wave
339:
311:
309:{\displaystyle 2\pi }
287:
254:
211:
191:
169:
45:
33:
10421:Structure prediction
10035:
10004:discrete mathematics
9936:(volume element) on
9594:
9360:
9319:
9095:
9072:
8825:
8776:
8765:{\textstyle 2\pi /c}
8745:
8725:
8705:
8625:
8556:
8542:{\textstyle 4\pi /a}
8522:
8470:
8435:
8403:
8383:
8367:Simple cubic lattice
8361:cubic crystal system
8293:
8234:
8166:
8157:inner multiplication
8139:
8071:
8023:
7993:
7948:
7733:
7645:
7564:
7524:
7468:
7445:
7418:
7398:
7357:
7317:
7285:
7265:
7238:
7218:
7191:
7171:
7144:
7108:
7070:
7005:
6976:
6932:
6928:in the direction of
6903:
6874:
6752:
6694:
6671:
6525:
6454:
6374:
6304:
6115:
5890:
5822:
5757:
5653:
5584:
5515:
5408:
5183:
5118:
5090:
5040:
5013:
4991:
4961:
4687:
4657:
4575:
4518:
4478:
4474:. Any valid form of
4449:
4441:of their respective
4372:
4343:
4314:
4246:
4141:
4112:
4076:
4053:
4024:
3995:
3966:
3933:
3896:
3840:
3777:
3540:
3471:
3439:
3417:
3386:
3318:
3229:
3192:
3161:
3157:to any position, if
3126:
3104:
3073:
3044:
3022:
2958:
2854:
2782:
2754:
2728:
2708:
2681:
2654:
2634:
2598:
2571:
2544:
2471:
2449:
2425:
2402:
2372:
2350:
2321:
2292:
2263:
2232:
2133:
2103:
2077:
2047:
1976:
1946:
1862:
1808:
1781:
1723:
1695:
1666:
1599:
1568:
1541:
1518:
1487:
1457:
1428:
1420:linearly independent
1393:
1366:
1267:
1243:
1211:
1179:
1153:
1117:
1092:
1054:
1032:
1010:
981:
973:is the phase of the
957:
909:
887:
865:
816:
812:which simplifies to
750:
708:
673:
650:
629:
609:
587:
566:
546:
526:
504:
480:
449:
386:
297:
267:
234:
200:
178:
145:
10685:Cottrell atmosphere
10665:Partial dislocation
10409:Restriction theorem
10191:2019PhRvM...3f4003S
2743:{\displaystyle R+r}
2506:solid state physics
2092:{\displaystyle i=j}
108:spatial frequencies
11105:Carl Hermann Medal
10916:Molecular dynamics
10763:Defects in diamond
10758:Stone–Wales defect
10404:Reciprocal lattice
10366:Biocrystallography
10282:2020-08-31 at the
10141:Powder diffraction
10082:
9948:, the presence of
9924:. The relation of
9858:Pontryagin duality
9782:
9520:
9343:
9227:
9078:
9055:
9032:
9030:
8806:
8762:
8731:
8711:
8680:
8611:
8539:
8488:
8448:
8421:
8389:
8337:
8279:
8220:
8145:
8125:
8057:
8009:
7975:
7931:
7713:
7704:
7625:
7550:
7510:
7451:
7424:
7404:
7372:
7335:
7303:
7271:
7251:
7224:
7204:
7177:
7157:
7126:
7094:
7056:
6985:
6962:
6918:
6889:
6857:
6735:
6680:
6651:
6500:
6427:
6359:
6286:
6101:
6099:
5877:
5808:
5743:
5639:
5570:
5498:
5383:
5163:
5157:
5108:. Thus, using the
5098:
5076:
5026:
4999:
4969:
4944:
4942:
4680:is an integer and
4670:
4640:
4558:
4493:
4464:
4439:Pontryagin duality
4427:
4358:
4329:
4299:
4231:
4127:
4091:
4062:
4039:
4010:
3981:
3948:
3913:
3882:
3823:
3761:
3679:
3605:
3552:
3523:
3454:
3425:
3403:
3369:
3301:
3241:
3209:
3178:
3147:
3112:
3090:
3059:
3030:
3005:
2944:
2833:
2772:
2760:
2740:
2714:
2694:
2667:
2640:
2620:
2584:
2557:
2479:
2457:
2434:
2410:
2388:
2358:
2336:
2307:
2278:
2245:
2216:
2118:
2089:
2063:
2029:
1961:
1928:
1847:
1794:
1766:
1708:
1681:
1650:
1583:
1553:
1527:
1502:
1472:
1442:
1408:
1379:
1350:
1249:
1229:
1196:
1161:
1138:Reciprocal lattice
1123:
1098:
1078:
1040:
1016:
995:
963:
939:
895:
871:
850:
800:
731:
691:
656:
635:
615:
593:
572:
552:
532:
510:
486:
475:angular wavenumber
462:
429:
342:
306:
282:
249:
206:
186:
164:
104:mathematical space
52:reciprocal lattice
48:
40:
11235:
11234:
11199:
11198:
10806:Thermal ellipsoid
10771:
10770:
10680:Frank–Read source
10640:
10639:
10506:Aperiodic crystal
10472:
10471:
10354:Crystallographers
10261:X-ray diffraction
9978:topological group
9762:
9746:
9716:
9686:
9612:
9213:
9197:
9167:
9113:
9081:{\displaystyle F}
9067:complex amplitude
9022:
8983:
8969:
8955:
8941:
8937:
8901:
8887:
8873:
8859:
8855:
8801:
8700:lattice constants
8486:
8446:
8419:
8392:{\displaystyle a}
8371:The simple cubic
8305:
8083:
8035:
7839:
7454:{\displaystyle V}
7427:{\displaystyle n}
7407:{\displaystyle n}
7390:Higher dimensions
7384:reciprocal length
7227:{\displaystyle k}
7180:{\displaystyle h}
7167:is replaced with
6997:spatial frequency
6855:
6069:
6065:
6001:
5997:
5933:
5929:
5510:primitive vectors
5508:, defined by its
5378:
5282:
4938:
4850:
4782:
4513:primitive vectors
3670:
3596:
3543:
3232:
2763:{\displaystyle R}
2717:{\displaystyle r}
2643:{\displaystyle G}
2502:Wigner–Seitz cell
1252:{\displaystyle n}
1101:{\displaystyle n}
1019:{\displaystyle t}
874:{\displaystyle t}
638:{\displaystyle x}
618:{\displaystyle k}
596:{\displaystyle t}
555:{\displaystyle x}
535:{\displaystyle k}
498:angular frequency
489:{\displaystyle k}
363:Fourier transform
355:Fourier transform
326:Wigner–Seitz cell
88:periodic function
72:Fourier transform
16:(Redirected from
11288:
11256:Fourier analysis
11223:
11222:
11211:
11210:
11154:
11153:
11077:Kristallografija
10931:Gerchberg–Saxton
10826:Characterisation
10818:
10801:Structure factor
10605:
10604:
10590:Ostwald ripening
10427:
10426:
10372:
10371:
10328:
10321:
10314:
10305:
10304:
10264:
10257:
10251:
10250:
10232:
10226:
10225:
10217:
10211:
10210:
10184:
10164:
10131:
10091:
10089:
10088:
10083:
10081:
10080:
10072:
10068:
10064:
10063:
10062:
10012:
9983:is the subgroup
9854:Fourier analysis
9847:finite dimension
9791:
9789:
9788:
9783:
9778:
9777:
9776:
9775:
9764:
9763:
9755:
9748:
9747:
9739:
9722:
9718:
9717:
9709:
9702:
9701:
9692:
9688:
9687:
9679:
9672:
9671:
9661:
9656:
9640:
9635:
9614:
9613:
9605:
9529:
9527:
9526:
9521:
9519:
9518:
9517:
9513:
9512:
9511:
9496:
9495:
9480:
9479:
9448:
9444:
9443:
9418:
9417:
9407:
9402:
9384:
9383:
9352:
9350:
9349:
9344:
9251:
9236:
9234:
9233:
9228:
9223:
9222:
9221:
9220:
9215:
9214:
9206:
9199:
9198:
9190:
9173:
9169:
9168:
9160:
9152:
9151:
9141:
9136:
9115:
9114:
9106:
9087:
9085:
9084:
9079:
9041:
9039:
9038:
9033:
9031:
9024:
9023:
9015:
9002:
9001:
8985:
8984:
8976:
8970:
8962:
8957:
8956:
8948:
8942:
8933:
8932:
8920:
8919:
8903:
8902:
8894:
8888:
8880:
8875:
8874:
8866:
8860:
8851:
8850:
8841:
8840:
8815:
8813:
8812:
8807:
8802:
8797:
8789:
8771:
8769:
8768:
8763:
8758:
8740:
8738:
8737:
8732:
8720:
8718:
8717:
8712:
8689:
8687:
8686:
8681:
8679:
8675:
8674:
8673:
8668:
8659:
8658:
8653:
8644:
8643:
8638:
8620:
8618:
8617:
8612:
8610:
8606:
8605:
8604:
8599:
8590:
8589:
8584:
8575:
8574:
8569:
8548:
8546:
8545:
8540:
8535:
8497:
8495:
8494:
8489:
8487:
8482:
8474:
8457:
8455:
8454:
8449:
8447:
8439:
8430:
8428:
8427:
8422:
8420:
8415:
8407:
8398:
8396:
8395:
8390:
8363:are as follows.
8346:
8344:
8343:
8338:
8306:
8303:
8288:
8286:
8285:
8280:
8229:
8227:
8226:
8221:
8154:
8152:
8151:
8146:
8134:
8132:
8131:
8126:
8085:
8084:
8076:
8066:
8064:
8063:
8058:
8056:
8055:
8037:
8036:
8028:
8018:
8016:
8015:
8010:
8008:
8007:
7984:
7982:
7981:
7976:
7974:
7966:
7965:
7940:
7938:
7937:
7932:
7913:
7912:
7908:
7907:
7897:
7884:
7883:
7879:
7878:
7862:
7853:
7852:
7840:
7838:
7834:
7833:
7828:
7813:
7812:
7807:
7794:
7793:
7789:
7788:
7773:
7772:
7758:
7747:
7746:
7741:
7722:
7720:
7719:
7714:
7709:
7708:
7634:
7632:
7631:
7626:
7624:
7623:
7599:
7598:
7593:
7584:
7583:
7578:
7559:
7557:
7556:
7551:
7549:
7519:
7517:
7516:
7511:
7506:
7505:
7500:
7485:
7484:
7479:
7460:
7458:
7457:
7452:
7433:
7431:
7430:
7425:
7413:
7411:
7410:
7405:
7394:The formula for
7381:
7379:
7378:
7373:
7371:
7370:
7365:
7344:
7342:
7341:
7336:
7312:
7310:
7309:
7304:
7280:
7278:
7277:
7272:
7260:
7258:
7257:
7252:
7250:
7249:
7233:
7231:
7230:
7225:
7213:
7211:
7210:
7205:
7203:
7202:
7186:
7184:
7183:
7178:
7166:
7164:
7163:
7158:
7156:
7155:
7135:
7133:
7132:
7127:
7103:
7101:
7100:
7095:
7065:
7063:
7062:
7057:
7052:
7051:
7039:
7038:
7026:
7025:
6994:
6992:
6991:
6986:
6971:
6969:
6968:
6963:
6961:
6960:
6955:
6946:
6945:
6940:
6927:
6925:
6924:
6919:
6917:
6916:
6911:
6898:
6896:
6895:
6890:
6888:
6887:
6882:
6866:
6864:
6863:
6858:
6856:
6854:
6853:
6849:
6848:
6847:
6842:
6833:
6832:
6827:
6813:
6812:
6807:
6800:
6799:
6798:
6793:
6784:
6783:
6778:
6771:
6766:
6765:
6760:
6744:
6742:
6741:
6736:
6728:
6723:
6722:
6717:
6708:
6707:
6702:
6689:
6687:
6686:
6681:
6660:
6658:
6657:
6652:
6647:
6646:
6638:
6634:
6633:
6632:
6627:
6621:
6620:
6615:
6609:
6608:
6603:
6582:
6581:
6580:
6574:
6570:
6569:
6568:
6563:
6557:
6556:
6551:
6545:
6544:
6539:
6516:matrix inversion
6509:
6507:
6506:
6501:
6493:
6492:
6491:
6490:
6485:
6476:
6475:
6470:
6446:and the section
6437:
6436:
6434:
6433:
6428:
6426:
6425:
6403:
6402:
6397:
6388:
6387:
6382:
6368:
6366:
6365:
6360:
6358:
6354:
6353:
6352:
6347:
6338:
6337:
6332:
6323:
6322:
6321:
6295:
6293:
6292:
6287:
6285:
6281:
6280:
6279:
6274:
6265:
6264:
6259:
6245:
6244:
6239:
6230:
6226:
6225:
6224:
6219:
6210:
6209:
6204:
6190:
6189:
6184:
6175:
6171:
6170:
6169:
6164:
6155:
6154:
6149:
6135:
6134:
6129:
6110:
6108:
6107:
6102:
6100:
6096:
6095:
6090:
6081:
6080:
6075:
6067:
6066:
6061:
6053:
6044:
6043:
6038:
6028:
6027:
6022:
6013:
6012:
6007:
5999:
5998:
5993:
5985:
5976:
5975:
5970:
5960:
5959:
5954:
5945:
5944:
5939:
5931:
5930:
5925:
5917:
5908:
5907:
5902:
5886:
5884:
5883:
5878:
5876:
5872:
5871:
5870:
5865:
5856:
5855:
5850:
5841:
5840:
5839:
5817:
5815:
5814:
5809:
5804:
5803:
5791:
5790:
5778:
5777:
5752:
5750:
5749:
5744:
5742:
5741:
5736:
5730:
5729:
5717:
5716:
5711:
5705:
5704:
5692:
5691:
5686:
5680:
5679:
5667:
5666:
5661:
5648:
5646:
5645:
5640:
5638:
5634:
5633:
5632:
5620:
5619:
5607:
5606:
5579:
5577:
5576:
5571:
5569:
5565:
5564:
5563:
5558:
5549:
5548:
5543:
5534:
5533:
5532:
5507:
5505:
5504:
5499:
5497:
5496:
5491:
5485:
5484:
5472:
5471:
5466:
5460:
5459:
5447:
5446:
5441:
5435:
5434:
5422:
5421:
5416:
5400:Three dimensions
5392:
5390:
5389:
5384:
5379:
5377:
5376:
5375:
5361:
5354:
5350:
5341:
5340:
5335:
5328:
5327:
5326:
5312:
5305:
5301:
5294:
5283:
5281:
5280:
5279:
5265:
5258:
5250:
5249:
5244:
5237:
5236:
5235:
5221:
5214:
5208:
5197:
5196:
5191:
5172:
5170:
5169:
5164:
5162:
5161:
5107:
5105:
5104:
5099:
5097:
5086:for all vectors
5085:
5083:
5082:
5077:
5075:
5069:
5068:
5053:
5047:
5035:
5033:
5032:
5027:
5025:
5024:
5008:
5006:
5005:
5000:
4998:
4978:
4976:
4975:
4970:
4968:
4953:
4951:
4950:
4945:
4943:
4939:
4937:
4936:
4935:
4930:
4923:
4915:
4914:
4909:
4902:
4901:
4900:
4895:
4888:
4882:
4867:
4866:
4861:
4851:
4849:
4848:
4847:
4842:
4835:
4827:
4826:
4821:
4814:
4813:
4812:
4807:
4800:
4794:
4783:
4781:
4780:
4779:
4774:
4767:
4759:
4758:
4753:
4743:
4742:
4741:
4736:
4729:
4720:
4705:
4704:
4699:
4679:
4677:
4676:
4671:
4669:
4668:
4649:
4647:
4646:
4641:
4639:
4638:
4633:
4627:
4626:
4614:
4613:
4608:
4602:
4601:
4589:
4588:
4583:
4567:
4565:
4564:
4559:
4557:
4553:
4552:
4551:
4546:
4537:
4536:
4531:
4502:
4500:
4499:
4494:
4492:
4491:
4486:
4473:
4471:
4470:
4465:
4463:
4462:
4457:
4436:
4434:
4433:
4428:
4426:
4422:
4421:
4420:
4415:
4406:
4405:
4400:
4391:
4390:
4389:
4367:
4365:
4364:
4359:
4357:
4356:
4351:
4338:
4336:
4335:
4330:
4328:
4327:
4322:
4309:
4308:
4306:
4305:
4300:
4298:
4297:
4275:
4274:
4269:
4260:
4259:
4254:
4240:
4238:
4237:
4232:
4230:
4229:
4224:
4218:
4217:
4205:
4204:
4199:
4193:
4192:
4180:
4179:
4174:
4168:
4167:
4155:
4154:
4149:
4136:
4134:
4133:
4128:
4126:
4125:
4120:
4100:
4098:
4097:
4092:
4090:
4089:
4084:
4071:
4069:
4068:
4063:
4048:
4046:
4045:
4040:
4038:
4037:
4032:
4019:
4017:
4016:
4011:
4009:
4008:
4003:
3990:
3988:
3987:
3982:
3980:
3979:
3974:
3957:
3955:
3954:
3949:
3947:
3946:
3941:
3922:
3920:
3919:
3914:
3909:
3891:
3889:
3888:
3883:
3869:
3868:
3863:
3854:
3853:
3848:
3832:
3830:
3829:
3824:
3816:
3815:
3814:
3813:
3808:
3799:
3798:
3793:
3770:
3768:
3767:
3762:
3757:
3756:
3755:
3747:
3746:
3741:
3726:
3725:
3724:
3723:
3718:
3709:
3708:
3703:
3689:
3688:
3678:
3666:
3665:
3661:
3660:
3655:
3646:
3635:
3634:
3629:
3615:
3614:
3604:
3592:
3591:
3590:
3582:
3581:
3576:
3562:
3561:
3551:
3532:
3530:
3529:
3524:
3516:
3499:
3498:
3493:
3484:
3463:
3461:
3460:
3455:
3453:
3452:
3447:
3434:
3432:
3431:
3426:
3424:
3412:
3410:
3409:
3404:
3399:
3378:
3376:
3375:
3370:
3365:
3364:
3352:
3351:
3339:
3338:
3310:
3308:
3307:
3302:
3300:
3296:
3281:
3280:
3279:
3271:
3270:
3265:
3251:
3250:
3240:
3218:
3216:
3215:
3210:
3205:
3187:
3185:
3184:
3179:
3174:
3156:
3154:
3153:
3148:
3140:
3139:
3134:
3121:
3119:
3118:
3113:
3111:
3099:
3097:
3096:
3091:
3086:
3068:
3066:
3065:
3060:
3058:
3057:
3052:
3039:
3037:
3036:
3031:
3029:
3014:
3012:
3011:
3006:
3004:
2996:
2995:
2983:
2982:
2970:
2969:
2953:
2951:
2950:
2945:
2943:
2942:
2937:
2931:
2930:
2918:
2917:
2912:
2906:
2905:
2893:
2892:
2887:
2881:
2880:
2868:
2867:
2862:
2842:
2840:
2839:
2834:
2829:
2828:
2816:
2815:
2803:
2802:
2769:
2767:
2766:
2761:
2749:
2747:
2746:
2741:
2723:
2721:
2720:
2715:
2703:
2701:
2700:
2695:
2693:
2692:
2676:
2674:
2673:
2668:
2666:
2665:
2649:
2647:
2646:
2641:
2629:
2627:
2626:
2621:
2619:
2618:
2593:
2591:
2590:
2585:
2583:
2582:
2566:
2564:
2563:
2558:
2556:
2555:
2514:pure mathematics
2488:
2486:
2485:
2480:
2478:
2466:
2464:
2463:
2458:
2456:
2444:
2443:
2441:
2440:
2435:
2419:
2417:
2416:
2411:
2409:
2397:
2395:
2394:
2389:
2387:
2379:
2367:
2365:
2364:
2359:
2357:
2345:
2343:
2342:
2337:
2335:
2334:
2329:
2316:
2314:
2313:
2308:
2306:
2305:
2300:
2287:
2285:
2284:
2279:
2277:
2276:
2271:
2254:
2252:
2251:
2246:
2244:
2243:
2227:
2225:
2223:
2222:
2217:
2215:
2214:
2209:
2203:
2202:
2190:
2189:
2184:
2178:
2177:
2165:
2164:
2159:
2153:
2152:
2140:
2127:
2125:
2124:
2119:
2117:
2116:
2111:
2098:
2096:
2095:
2090:
2073:equals one when
2072:
2070:
2069:
2064:
2062:
2061:
2039:
2038:
2036:
2035:
2030:
2028:
2027:
2005:
2004:
1999:
1990:
1989:
1984:
1970:
1968:
1967:
1962:
1960:
1959:
1954:
1939:
1937:
1935:
1934:
1929:
1921:
1920:
1915:
1906:
1905:
1900:
1891:
1890:
1885:
1876:
1875:
1870:
1856:
1854:
1853:
1848:
1837:
1836:
1831:
1822:
1821:
1816:
1803:
1801:
1800:
1795:
1793:
1792:
1776:
1775:
1773:
1772:
1767:
1765:
1764:
1759:
1750:
1749:
1744:
1735:
1734:
1717:
1715:
1714:
1709:
1707:
1706:
1690:
1688:
1687:
1682:
1680:
1679:
1674:
1661:
1659:
1657:
1656:
1651:
1649:
1648:
1639:
1634:
1633:
1628:
1613:
1612:
1607:
1593:
1592:
1590:
1589:
1584:
1582:
1581:
1576:
1562:
1560:
1559:
1554:
1536:
1534:
1533:
1528:
1513:
1511:
1509:
1508:
1503:
1501:
1500:
1495:
1481:
1479:
1478:
1473:
1471:
1470:
1465:
1452:
1451:
1449:
1448:
1443:
1435:
1417:
1415:
1414:
1409:
1407:
1406:
1401:
1388:
1386:
1385:
1380:
1378:
1377:
1361:
1359:
1357:
1356:
1351:
1349:
1348:
1343:
1337:
1336:
1324:
1323:
1318:
1312:
1311:
1299:
1298:
1293:
1287:
1286:
1274:
1258:
1256:
1255:
1250:
1239:with an integer
1238:
1236:
1235:
1230:
1205:
1203:
1202:
1197:
1192:
1175:of any function
1170:
1168:
1167:
1162:
1160:
1134:
1132:
1130:
1129:
1124:
1107:
1105:
1104:
1099:
1087:
1085:
1084:
1079:
1049:
1047:
1046:
1041:
1039:
1027:
1025:
1023:
1022:
1017:
1004:
1002:
1001:
996:
988:
972:
970:
969:
964:
948:
946:
945:
940:
935:
930:
916:
904:
902:
901:
896:
894:
882:
880:
878:
877:
872:
860:at a fixed time
859:
857:
856:
851:
840:
832:
811:
809:
807:
806:
801:
796:
795:
774:
766:
742:
740:
738:
737:
732:
727:
702:
700:
698:
697:
692:
667:
665:
663:
662:
657:
644:
642:
641:
636:
624:
622:
621:
616:
604:
602:
600:
599:
594:
581:
579:
578:
573:
561:
559:
558:
553:
541:
539:
538:
533:
521:
519:
517:
516:
511:
495:
493:
492:
487:
473:
471:
469:
468:
463:
461:
460:
440:
438:
436:
435:
430:
425:
424:
359:frequency domain
352:
345:Reciprocal space
316:
315:
313:
312:
307:
291:
289:
288:
283:
281:
280:
275:
258:
256:
255:
250:
248:
247:
242:
224:, respectively.
215:
213:
212:
207:
195:
193:
192:
187:
185:
173:
171:
170:
165:
163:
152:
112:reciprocal space
21:
18:Reciprocal space
11296:
11295:
11291:
11290:
11289:
11287:
11286:
11285:
11251:Crystallography
11241:
11240:
11238:
11236:
11231:
11195:
11152:
11119:
11091:
11043:
10995:
10966:CrystalExplorer
10942:
10926:Phase retrieval
10889:
10820:
10819:
10810:
10767:
10746:Schottky defect
10645:Perfect crystal
10636:
10632:Abnormal growth
10594:
10580:Supersaturation
10543:Miscibility gap
10524:
10517:
10468:
10425:
10389:Bravais lattice
10370:
10337:
10335:Crystallography
10332:
10284:Wayback Machine
10273:
10268:
10267:
10258:
10254:
10247:
10233:
10229:
10218:
10214:
10165:
10161:
10156:
10129:
10108:Crystallography
10098:
10073:
10058:
10057:
10053:
10052:
10048:
10047:
10036:
10033:
10032:
10007:
9975:locally compact
9828:
9822:
9803:
9765:
9754:
9753:
9752:
9738:
9737:
9727:
9723:
9708:
9707:
9703:
9697:
9693:
9678:
9677:
9673:
9667:
9663:
9657:
9646:
9636:
9625:
9604:
9603:
9595:
9592:
9591:
9567:
9558:
9549:
9537: = 1,
9533:when there are
9507:
9503:
9491:
9487:
9475:
9471:
9467:
9463:
9453:
9449:
9427:
9423:
9419:
9413:
9409:
9403:
9392:
9367:
9363:
9361:
9358:
9357:
9320:
9317:
9316:
9309:
9289:
9268:
9249:
9216:
9205:
9204:
9203:
9189:
9188:
9178:
9174:
9159:
9158:
9154:
9147:
9143:
9137:
9126:
9105:
9104:
9096:
9093:
9092:
9073:
9070:
9069:
9047:
9029:
9028:
9014:
9013:
9003:
8997:
8993:
8990:
8989:
8975:
8974:
8961:
8947:
8946:
8931:
8921:
8915:
8911:
8908:
8907:
8893:
8892:
8879:
8865:
8864:
8849:
8842:
8836:
8832:
8828:
8826:
8823:
8822:
8796:
8785:
8777:
8774:
8773:
8754:
8746:
8743:
8742:
8726:
8723:
8722:
8706:
8703:
8702:
8696:
8669:
8664:
8663:
8654:
8649:
8648:
8639:
8634:
8633:
8632:
8628:
8626:
8623:
8622:
8600:
8595:
8594:
8585:
8580:
8579:
8570:
8565:
8564:
8563:
8559:
8557:
8554:
8553:
8531:
8523:
8520:
8519:
8514:lattice is the
8508:
8475:
8473:
8471:
8468:
8467:
8464:
8438:
8436:
8433:
8432:
8408:
8406:
8404:
8401:
8400:
8384:
8381:
8380:
8373:Bravais lattice
8369:
8357:
8302:
8294:
8291:
8290:
8235:
8232:
8231:
8167:
8164:
8163:
8140:
8137:
8136:
8075:
8074:
8072:
8069:
8068:
8051:
8047:
8027:
8026:
8024:
8021:
8020:
8000:
7996:
7994:
7991:
7990:
7970:
7961:
7957:
7949:
7946:
7945:
7903:
7899:
7898:
7893:
7892:
7868:
7864:
7863:
7858:
7857:
7845:
7841:
7829:
7824:
7823:
7808:
7803:
7802:
7795:
7784:
7780:
7768:
7764:
7763:
7759:
7757:
7742:
7737:
7736:
7734:
7731:
7730:
7703:
7702:
7697:
7692:
7687:
7681:
7680:
7675:
7670:
7665:
7655:
7654:
7646:
7643:
7642:
7616:
7612:
7594:
7589:
7588:
7579:
7574:
7573:
7565:
7562:
7561:
7545:
7525:
7522:
7521:
7501:
7496:
7495:
7480:
7475:
7474:
7469:
7466:
7465:
7446:
7443:
7442:
7419:
7416:
7415:
7399:
7396:
7395:
7392:
7366:
7361:
7360:
7358:
7355:
7354:
7318:
7315:
7314:
7286:
7283:
7282:
7266:
7263:
7262:
7245:
7241:
7239:
7236:
7235:
7219:
7216:
7215:
7198:
7194:
7192:
7189:
7188:
7172:
7169:
7168:
7151:
7147:
7145:
7142:
7141:
7109:
7106:
7105:
7071:
7068:
7067:
7047:
7043:
7034:
7030:
7021:
7017:
7006:
7003:
7002:
6977:
6974:
6973:
6956:
6951:
6950:
6941:
6936:
6935:
6933:
6930:
6929:
6912:
6907:
6906:
6904:
6901:
6900:
6883:
6878:
6877:
6875:
6872:
6871:
6843:
6838:
6837:
6828:
6823:
6822:
6821:
6817:
6808:
6803:
6802:
6801:
6794:
6789:
6788:
6779:
6774:
6773:
6772:
6770:
6761:
6756:
6755:
6753:
6750:
6749:
6724:
6718:
6713:
6712:
6703:
6698:
6697:
6695:
6692:
6691:
6672:
6669:
6668:
6639:
6628:
6623:
6622:
6616:
6611:
6610:
6604:
6599:
6598:
6597:
6593:
6592:
6576:
6575:
6564:
6559:
6558:
6552:
6547:
6546:
6540:
6535:
6534:
6533:
6529:
6528:
6526:
6523:
6522:
6486:
6481:
6480:
6471:
6466:
6465:
6461:
6457:
6455:
6452:
6451:
6418:
6414:
6398:
6393:
6392:
6383:
6378:
6377:
6375:
6372:
6371:
6370:
6348:
6343:
6342:
6333:
6328:
6327:
6317:
6313:
6312:
6311:
6307:
6305:
6302:
6301:
6275:
6270:
6269:
6260:
6255:
6254:
6253:
6249:
6240:
6235:
6234:
6220:
6215:
6214:
6205:
6200:
6199:
6198:
6194:
6185:
6180:
6179:
6165:
6160:
6159:
6150:
6145:
6144:
6143:
6139:
6130:
6125:
6124:
6116:
6113:
6112:
6098:
6097:
6091:
6086:
6085:
6076:
6071:
6070:
6054:
6052:
6045:
6039:
6034:
6033:
6030:
6029:
6023:
6018:
6017:
6008:
6003:
6002:
5986:
5984:
5977:
5971:
5966:
5965:
5962:
5961:
5955:
5950:
5949:
5940:
5935:
5934:
5918:
5916:
5909:
5903:
5898:
5897:
5893:
5891:
5888:
5887:
5866:
5861:
5860:
5851:
5846:
5845:
5835:
5831:
5830:
5829:
5825:
5823:
5820:
5819:
5799:
5795:
5786:
5782:
5773:
5769:
5758:
5755:
5754:
5737:
5732:
5731:
5725:
5721:
5712:
5707:
5706:
5700:
5696:
5687:
5682:
5681:
5675:
5671:
5662:
5657:
5656:
5654:
5651:
5650:
5628:
5624:
5615:
5611:
5602:
5598:
5597:
5593:
5585:
5582:
5581:
5559:
5554:
5553:
5544:
5539:
5538:
5528:
5524:
5523:
5522:
5518:
5516:
5513:
5512:
5492:
5487:
5486:
5480:
5476:
5467:
5462:
5461:
5455:
5451:
5442:
5437:
5436:
5430:
5426:
5417:
5412:
5411:
5409:
5406:
5405:
5402:
5362:
5357:
5356:
5346:
5345:
5336:
5331:
5330:
5329:
5313:
5308:
5307:
5297:
5296:
5295:
5293:
5266:
5261:
5260:
5254:
5245:
5240:
5239:
5238:
5222:
5217:
5216:
5210:
5209:
5207:
5192:
5187:
5186:
5184:
5181:
5180:
5156:
5155:
5150:
5144:
5143:
5138:
5128:
5127:
5119:
5116:
5115:
5093:
5091:
5088:
5087:
5071:
5061:
5060:
5049:
5043:
5041:
5038:
5037:
5017:
5016:
5014:
5011:
5010:
4994:
4992:
4989:
4988:
4981:rotation matrix
4964:
4962:
4959:
4958:
4941:
4940:
4931:
4926:
4925:
4919:
4910:
4905:
4904:
4903:
4896:
4891:
4890:
4884:
4883:
4881:
4868:
4862:
4857:
4856:
4853:
4852:
4843:
4838:
4837:
4831:
4822:
4817:
4816:
4815:
4808:
4803:
4802:
4796:
4795:
4793:
4775:
4770:
4769:
4763:
4754:
4749:
4748:
4744:
4737:
4732:
4731:
4725:
4721:
4719:
4706:
4700:
4695:
4694:
4690:
4688:
4685:
4684:
4664:
4660:
4658:
4655:
4654:
4634:
4629:
4628:
4622:
4618:
4609:
4604:
4603:
4597:
4593:
4584:
4579:
4578:
4576:
4573:
4572:
4547:
4542:
4541:
4532:
4527:
4526:
4525:
4521:
4519:
4516:
4515:
4509:
4487:
4482:
4481:
4479:
4476:
4475:
4458:
4453:
4452:
4450:
4447:
4446:
4416:
4411:
4410:
4401:
4396:
4395:
4385:
4381:
4380:
4379:
4375:
4373:
4370:
4369:
4352:
4347:
4346:
4344:
4341:
4340:
4323:
4318:
4317:
4315:
4312:
4311:
4290:
4286:
4270:
4265:
4264:
4255:
4250:
4249:
4247:
4244:
4243:
4242:
4225:
4220:
4219:
4213:
4209:
4200:
4195:
4194:
4188:
4184:
4175:
4170:
4169:
4163:
4159:
4150:
4145:
4144:
4142:
4139:
4138:
4121:
4116:
4115:
4113:
4110:
4109:
4085:
4080:
4079:
4077:
4074:
4073:
4054:
4051:
4050:
4033:
4028:
4027:
4025:
4022:
4021:
4004:
3999:
3998:
3996:
3993:
3992:
3975:
3970:
3969:
3967:
3964:
3963:
3942:
3937:
3936:
3934:
3931:
3930:
3905:
3897:
3894:
3893:
3864:
3859:
3858:
3849:
3844:
3843:
3841:
3838:
3837:
3809:
3804:
3803:
3794:
3789:
3788:
3784:
3780:
3778:
3775:
3774:
3751:
3742:
3737:
3736:
3732:
3728:
3719:
3714:
3713:
3704:
3699:
3698:
3694:
3690:
3684:
3680:
3674:
3656:
3651:
3650:
3642:
3630:
3625:
3624:
3620:
3616:
3610:
3606:
3600:
3586:
3577:
3572:
3571:
3567:
3563:
3557:
3553:
3547:
3541:
3538:
3537:
3512:
3494:
3489:
3488:
3480:
3472:
3469:
3468:
3448:
3443:
3442:
3440:
3437:
3436:
3420:
3418:
3415:
3414:
3395:
3387:
3384:
3383:
3360:
3356:
3347:
3343:
3334:
3330:
3319:
3316:
3315:
3292:
3288:
3275:
3266:
3261:
3260:
3256:
3252:
3246:
3242:
3236:
3230:
3227:
3226:
3201:
3193:
3190:
3189:
3170:
3162:
3159:
3158:
3135:
3130:
3129:
3127:
3124:
3123:
3107:
3105:
3102:
3101:
3082:
3074:
3071:
3070:
3053:
3048:
3047:
3045:
3042:
3041:
3025:
3023:
3020:
3019:
3000:
2991:
2987:
2978:
2974:
2965:
2961:
2959:
2956:
2955:
2938:
2933:
2932:
2926:
2922:
2913:
2908:
2907:
2901:
2897:
2888:
2883:
2882:
2876:
2872:
2863:
2858:
2857:
2855:
2852:
2851:
2824:
2820:
2811:
2807:
2798:
2794:
2783:
2780:
2779:
2776:Bravais lattice
2755:
2752:
2751:
2729:
2726:
2725:
2709:
2706:
2705:
2688:
2684:
2682:
2679:
2678:
2661:
2657:
2655:
2652:
2651:
2635:
2632:
2631:
2605:
2601:
2599:
2596:
2595:
2578:
2574:
2572:
2569:
2568:
2551:
2547:
2545:
2542:
2541:
2534:
2510:Bloch's theorem
2474:
2472:
2469:
2468:
2452:
2450:
2447:
2446:
2426:
2423:
2422:
2421:
2405:
2403:
2400:
2399:
2383:
2375:
2373:
2370:
2369:
2353:
2351:
2348:
2347:
2330:
2325:
2324:
2322:
2319:
2318:
2301:
2296:
2295:
2293:
2290:
2289:
2272:
2267:
2266:
2264:
2261:
2260:
2257:Bravais lattice
2239:
2235:
2233:
2230:
2229:
2210:
2205:
2204:
2198:
2194:
2185:
2180:
2179:
2173:
2169:
2160:
2155:
2154:
2148:
2144:
2136:
2134:
2131:
2130:
2129:
2112:
2107:
2106:
2104:
2101:
2100:
2078:
2075:
2074:
2054:
2050:
2048:
2045:
2044:
2042:Kronecker delta
2020:
2016:
2000:
1995:
1994:
1985:
1980:
1979:
1977:
1974:
1973:
1972:
1955:
1950:
1949:
1947:
1944:
1943:
1916:
1911:
1910:
1901:
1896:
1895:
1886:
1881:
1880:
1871:
1866:
1865:
1863:
1860:
1859:
1858:
1832:
1827:
1826:
1817:
1812:
1811:
1809:
1806:
1805:
1788:
1784:
1782:
1779:
1778:
1760:
1755:
1754:
1745:
1740:
1739:
1730:
1726:
1724:
1721:
1720:
1719:
1702:
1698:
1696:
1693:
1692:
1675:
1670:
1669:
1667:
1664:
1663:
1644:
1640:
1635:
1629:
1624:
1623:
1608:
1603:
1602:
1600:
1597:
1596:
1595:
1577:
1572:
1571:
1569:
1566:
1565:
1564:
1542:
1539:
1538:
1519:
1516:
1515:
1496:
1491:
1490:
1488:
1485:
1484:
1483:
1466:
1461:
1460:
1458:
1455:
1454:
1431:
1429:
1426:
1425:
1424:
1402:
1397:
1396:
1394:
1391:
1390:
1373:
1369:
1367:
1364:
1363:
1344:
1339:
1338:
1332:
1328:
1319:
1314:
1313:
1307:
1303:
1294:
1289:
1288:
1282:
1278:
1270:
1268:
1265:
1264:
1263:
1244:
1241:
1240:
1212:
1209:
1208:
1188:
1180:
1177:
1176:
1156:
1154:
1151:
1150:
1144:Bravais lattice
1140:
1118:
1115:
1114:
1113:
1108:represents any
1093:
1090:
1089:
1055:
1052:
1051:
1035:
1033:
1030:
1029:
1011:
1008:
1007:
1006:
984:
982:
979:
978:
958:
955:
954:
931:
926:
912:
910:
907:
906:
890:
888:
885:
884:
866:
863:
862:
861:
836:
828:
817:
814:
813:
791:
787:
770:
762:
751:
748:
747:
746:
723:
709:
706:
705:
704:
674:
671:
670:
669:
651:
648:
647:
646:
630:
627:
626:
610:
607:
606:
588:
585:
584:
583:
567:
564:
563:
547:
544:
543:
527:
524:
523:
505:
502:
501:
500:
481:
478:
477:
456:
452:
450:
447:
446:
445:
420:
416:
387:
384:
383:
382:
350:
347:
334:
298:
295:
294:
293:
276:
271:
270:
268:
265:
264:
243:
238:
237:
235:
232:
231:
201:
198:
197:
181:
179:
176:
175:
159:
148:
146:
143:
142:
135:quantum physics
100:Bravais lattice
28:
23:
22:
15:
12:
11:
5:
11294:
11284:
11283:
11278:
11273:
11268:
11263:
11261:Lattice points
11258:
11253:
11233:
11232:
11230:
11229:
11217:
11204:
11201:
11200:
11197:
11196:
11194:
11193:
11188:
11183:
11182:
11181:
11176:
11171:
11160:
11158:
11151:
11150:
11145:
11140:
11135:
11129:
11127:
11121:
11120:
11118:
11117:
11112:
11107:
11101:
11099:
11093:
11092:
11090:
11089:
11084:
11079:
11074:
11069:
11064:
11059:
11053:
11051:
11045:
11044:
11042:
11041:
11036:
11031:
11026:
11021:
11016:
11011:
11005:
11003:
10997:
10996:
10994:
10993:
10988:
10983:
10978:
10973:
10968:
10963:
10958:
10952:
10950:
10944:
10943:
10941:
10940:
10935:
10934:
10933:
10923:
10918:
10913:
10908:
10903:
10901:Direct methods
10897:
10895:
10891:
10890:
10888:
10887:
10886:
10885:
10880:
10870:
10865:
10864:
10863:
10858:
10848:
10847:
10846:
10841:
10830:
10828:
10822:
10821:
10813:
10811:
10809:
10808:
10803:
10798:
10793:
10788:
10786:Ewald's sphere
10783:
10778:
10772:
10769:
10768:
10766:
10765:
10760:
10755:
10754:
10753:
10748:
10738:
10737:
10736:
10731:
10729:Frenkel defect
10726:
10724:Bjerrum defect
10716:
10715:
10714:
10704:
10703:
10702:
10697:
10692:
10690:Peierls stress
10687:
10682:
10677:
10672:
10667:
10662:
10660:Burgers vector
10652:
10650:Stacking fault
10647:
10641:
10638:
10637:
10635:
10634:
10629:
10624:
10619:
10613:
10611:
10609:Grain boundary
10602:
10596:
10595:
10593:
10592:
10587:
10582:
10577:
10572:
10567:
10562:
10557:
10556:
10555:
10553:Liquid crystal
10550:
10545:
10540:
10529:
10527:
10519:
10518:
10516:
10515:
10514:
10513:
10503:
10502:
10501:
10491:
10490:
10489:
10484:
10473:
10470:
10469:
10467:
10466:
10461:
10456:
10451:
10446:
10441:
10435:
10433:
10424:
10423:
10418:
10416:Periodic table
10413:
10412:
10411:
10406:
10401:
10396:
10391:
10380:
10378:
10369:
10368:
10363:
10358:
10357:
10356:
10345:
10343:
10339:
10338:
10331:
10330:
10323:
10316:
10308:
10302:
10301:
10296:
10291:
10272:
10271:External links
10269:
10266:
10265:
10252:
10245:
10227:
10212:
10158:
10157:
10155:
10152:
10151:
10150:
10144:
10138:
10132:
10123:
10120:Ewald's sphere
10117:
10111:
10105:
10102:Brillouin zone
10097:
10094:
10079:
10076:
10071:
10067:
10061:
10056:
10051:
10046:
10043:
10040:
9907:non-degenerate
9896:quadratic form
9883:^. Therefore,
9824:Main article:
9821:
9818:
9799:
9793:
9792:
9781:
9774:
9771:
9761:
9758:
9751:
9745:
9742:
9736:
9733:
9730:
9726:
9721:
9715:
9712:
9706:
9700:
9696:
9691:
9685:
9682:
9676:
9670:
9666:
9660:
9655:
9652:
9649:
9645:
9639:
9634:
9631:
9628:
9624:
9620:
9617:
9611:
9608:
9602:
9599:
9563:
9554:
9545:
9531:
9530:
9516:
9510:
9506:
9502:
9499:
9494:
9490:
9486:
9483:
9478:
9474:
9470:
9466:
9462:
9459:
9456:
9452:
9447:
9442:
9439:
9436:
9433:
9430:
9426:
9422:
9416:
9412:
9406:
9401:
9398:
9395:
9391:
9387:
9382:
9379:
9376:
9373:
9370:
9366:
9342:
9339:
9336:
9333:
9330:
9327:
9324:
9307:
9285:
9264:
9238:
9237:
9226:
9219:
9212:
9209:
9202:
9196:
9193:
9187:
9184:
9181:
9177:
9172:
9166:
9163:
9157:
9150:
9146:
9140:
9135:
9132:
9129:
9125:
9121:
9118:
9112:
9109:
9103:
9100:
9077:
9046:
9043:
9027:
9021:
9018:
9012:
9009:
9006:
9004:
9000:
8996:
8992:
8991:
8988:
8982:
8979:
8973:
8968:
8965:
8960:
8954:
8951:
8945:
8940:
8936:
8930:
8927:
8924:
8922:
8918:
8914:
8910:
8909:
8906:
8900:
8897:
8891:
8886:
8883:
8878:
8872:
8869:
8863:
8858:
8854:
8848:
8845:
8843:
8839:
8835:
8831:
8830:
8805:
8800:
8795:
8792:
8788:
8784:
8781:
8761:
8757:
8753:
8750:
8734:{\textstyle c}
8730:
8714:{\textstyle a}
8710:
8695:
8692:
8678:
8672:
8667:
8662:
8657:
8652:
8647:
8642:
8637:
8631:
8609:
8603:
8598:
8593:
8588:
8583:
8578:
8573:
8568:
8562:
8538:
8534:
8530:
8527:
8507:
8504:
8485:
8481:
8478:
8463:
8460:
8445:
8442:
8418:
8414:
8411:
8388:
8377:primitive cell
8368:
8365:
8356:
8353:
8336:
8333:
8330:
8327:
8324:
8321:
8318:
8315:
8312:
8309:
8301:
8298:
8278:
8275:
8272:
8269:
8266:
8263:
8260:
8257:
8254:
8251:
8248:
8245:
8242:
8239:
8219:
8216:
8213:
8210:
8207:
8204:
8201:
8198:
8195:
8192:
8189:
8186:
8183:
8180:
8177:
8174:
8171:
8144:
8124:
8121:
8118:
8115:
8112:
8109:
8106:
8103:
8100:
8097:
8094:
8091:
8088:
8082:
8079:
8054:
8050:
8046:
8043:
8040:
8034:
8031:
8006:
8003:
7999:
7973:
7969:
7964:
7960:
7956:
7953:
7942:
7941:
7930:
7927:
7924:
7921:
7917:
7911:
7906:
7902:
7896:
7891:
7888:
7882:
7877:
7874:
7871:
7867:
7861:
7856:
7851:
7848:
7844:
7837:
7832:
7827:
7822:
7819:
7816:
7811:
7806:
7801:
7798:
7792:
7787:
7783:
7779:
7776:
7771:
7767:
7762:
7756:
7753:
7750:
7745:
7740:
7724:
7723:
7712:
7707:
7701:
7698:
7696:
7693:
7691:
7688:
7686:
7683:
7682:
7679:
7676:
7674:
7671:
7669:
7666:
7664:
7661:
7660:
7658:
7653:
7650:
7622:
7619:
7615:
7611:
7608:
7605:
7602:
7597:
7592:
7587:
7582:
7577:
7572:
7569:
7548:
7544:
7541:
7538:
7535:
7532:
7529:
7509:
7504:
7499:
7494:
7491:
7488:
7483:
7478:
7473:
7450:
7423:
7403:
7391:
7388:
7369:
7364:
7334:
7331:
7328:
7325:
7322:
7302:
7299:
7296:
7293:
7290:
7270:
7261:replaced with
7248:
7244:
7223:
7214:replaced with
7201:
7197:
7176:
7154:
7150:
7138:Miller indices
7125:
7122:
7119:
7116:
7113:
7093:
7090:
7087:
7084:
7081:
7078:
7075:
7055:
7050:
7046:
7042:
7037:
7033:
7029:
7024:
7020:
7016:
7013:
7010:
6984:
6981:
6959:
6954:
6949:
6944:
6939:
6915:
6910:
6886:
6881:
6868:
6867:
6852:
6846:
6841:
6836:
6831:
6826:
6820:
6816:
6811:
6806:
6797:
6792:
6787:
6782:
6777:
6769:
6764:
6759:
6734:
6731:
6727:
6721:
6716:
6711:
6706:
6701:
6679:
6676:
6662:
6661:
6650:
6645:
6642:
6637:
6631:
6626:
6619:
6614:
6607:
6602:
6596:
6591:
6588:
6585:
6579:
6573:
6567:
6562:
6555:
6550:
6543:
6538:
6532:
6499:
6496:
6489:
6484:
6479:
6474:
6469:
6464:
6460:
6424:
6421:
6417:
6412:
6409:
6406:
6401:
6396:
6391:
6386:
6381:
6369:is to satisfy
6357:
6351:
6346:
6341:
6336:
6331:
6326:
6320:
6316:
6310:
6284:
6278:
6273:
6268:
6263:
6258:
6252:
6248:
6243:
6238:
6233:
6229:
6223:
6218:
6213:
6208:
6203:
6197:
6193:
6188:
6183:
6178:
6174:
6168:
6163:
6158:
6153:
6148:
6142:
6138:
6133:
6128:
6123:
6120:
6094:
6089:
6084:
6079:
6074:
6064:
6060:
6057:
6051:
6048:
6046:
6042:
6037:
6032:
6031:
6026:
6021:
6016:
6011:
6006:
5996:
5992:
5989:
5983:
5980:
5978:
5974:
5969:
5964:
5963:
5958:
5953:
5948:
5943:
5938:
5928:
5924:
5921:
5915:
5912:
5910:
5906:
5901:
5896:
5895:
5875:
5869:
5864:
5859:
5854:
5849:
5844:
5838:
5834:
5828:
5807:
5802:
5798:
5794:
5789:
5785:
5781:
5776:
5772:
5768:
5765:
5762:
5740:
5735:
5728:
5724:
5720:
5715:
5710:
5703:
5699:
5695:
5690:
5685:
5678:
5674:
5670:
5665:
5660:
5637:
5631:
5627:
5623:
5618:
5614:
5610:
5605:
5601:
5596:
5592:
5589:
5568:
5562:
5557:
5552:
5547:
5542:
5537:
5531:
5527:
5521:
5495:
5490:
5483:
5479:
5475:
5470:
5465:
5458:
5454:
5450:
5445:
5440:
5433:
5429:
5425:
5420:
5415:
5401:
5398:
5394:
5393:
5382:
5374:
5371:
5368:
5365:
5360:
5353:
5349:
5344:
5339:
5334:
5325:
5322:
5319:
5316:
5311:
5304:
5300:
5292:
5289:
5286:
5278:
5275:
5272:
5269:
5264:
5257:
5253:
5248:
5243:
5234:
5231:
5228:
5225:
5220:
5213:
5206:
5203:
5200:
5195:
5190:
5174:
5173:
5160:
5154:
5151:
5149:
5146:
5145:
5142:
5139:
5137:
5134:
5133:
5131:
5126:
5123:
5096:
5074:
5067:
5064:
5059:
5056:
5052:
5046:
5023:
5020:
4997:
4967:
4955:
4954:
4934:
4929:
4922:
4918:
4913:
4908:
4899:
4894:
4887:
4880:
4877:
4874:
4871:
4869:
4865:
4860:
4855:
4854:
4846:
4841:
4834:
4830:
4825:
4820:
4811:
4806:
4799:
4792:
4789:
4786:
4778:
4773:
4766:
4762:
4757:
4752:
4747:
4740:
4735:
4728:
4724:
4718:
4715:
4712:
4709:
4707:
4703:
4698:
4693:
4692:
4667:
4663:
4651:
4650:
4637:
4632:
4625:
4621:
4617:
4612:
4607:
4600:
4596:
4592:
4587:
4582:
4556:
4550:
4545:
4540:
4535:
4530:
4524:
4508:
4507:Two dimensions
4505:
4490:
4485:
4461:
4456:
4425:
4419:
4414:
4409:
4404:
4399:
4394:
4388:
4384:
4378:
4355:
4350:
4326:
4321:
4296:
4293:
4289:
4284:
4281:
4278:
4273:
4268:
4263:
4258:
4253:
4228:
4223:
4216:
4212:
4208:
4203:
4198:
4191:
4187:
4183:
4178:
4173:
4166:
4162:
4158:
4153:
4148:
4124:
4119:
4088:
4083:
4061:
4058:
4036:
4031:
4007:
4002:
3978:
3973:
3945:
3940:
3924:
3923:
3912:
3908:
3904:
3901:
3881:
3878:
3875:
3872:
3867:
3862:
3857:
3852:
3847:
3822:
3819:
3812:
3807:
3802:
3797:
3792:
3787:
3783:
3760:
3754:
3750:
3745:
3740:
3735:
3731:
3722:
3717:
3712:
3707:
3702:
3697:
3693:
3687:
3683:
3677:
3673:
3669:
3664:
3659:
3654:
3649:
3645:
3641:
3638:
3633:
3628:
3623:
3619:
3613:
3609:
3603:
3599:
3595:
3589:
3585:
3580:
3575:
3570:
3566:
3560:
3556:
3550:
3546:
3534:
3533:
3522:
3519:
3515:
3511:
3508:
3505:
3502:
3497:
3492:
3487:
3483:
3479:
3476:
3451:
3446:
3423:
3402:
3398:
3394:
3391:
3368:
3363:
3359:
3355:
3350:
3346:
3342:
3337:
3333:
3329:
3326:
3323:
3312:
3311:
3299:
3295:
3291:
3287:
3284:
3278:
3274:
3269:
3264:
3259:
3255:
3249:
3245:
3239:
3235:
3208:
3204:
3200:
3197:
3177:
3173:
3169:
3166:
3146:
3143:
3138:
3133:
3110:
3089:
3085:
3081:
3078:
3056:
3051:
3028:
3016:
3015:
3003:
2999:
2994:
2990:
2986:
2981:
2977:
2973:
2968:
2964:
2941:
2936:
2929:
2925:
2921:
2916:
2911:
2904:
2900:
2896:
2891:
2886:
2879:
2875:
2871:
2866:
2861:
2832:
2827:
2823:
2819:
2814:
2810:
2806:
2801:
2797:
2793:
2790:
2787:
2759:
2739:
2736:
2733:
2713:
2691:
2687:
2664:
2660:
2639:
2617:
2614:
2611:
2608:
2604:
2581:
2577:
2554:
2550:
2533:
2530:
2498:primitive cell
2494:Brillouin zone
2477:
2455:
2433:
2430:
2408:
2386:
2382:
2378:
2356:
2333:
2328:
2304:
2299:
2275:
2270:
2242:
2238:
2213:
2208:
2201:
2197:
2193:
2188:
2183:
2176:
2172:
2168:
2163:
2158:
2151:
2147:
2143:
2139:
2115:
2110:
2088:
2085:
2082:
2060:
2057:
2053:
2026:
2023:
2019:
2014:
2011:
2008:
2003:
1998:
1993:
1988:
1983:
1958:
1953:
1927:
1924:
1919:
1914:
1909:
1904:
1899:
1894:
1889:
1884:
1879:
1874:
1869:
1846:
1843:
1840:
1835:
1830:
1825:
1820:
1815:
1791:
1787:
1763:
1758:
1753:
1748:
1743:
1738:
1733:
1729:
1705:
1701:
1678:
1673:
1647:
1643:
1638:
1632:
1627:
1622:
1619:
1616:
1611:
1606:
1580:
1575:
1552:
1549:
1546:
1526:
1523:
1499:
1494:
1469:
1464:
1441:
1438:
1434:
1405:
1400:
1376:
1372:
1347:
1342:
1335:
1331:
1327:
1322:
1317:
1310:
1306:
1302:
1297:
1292:
1285:
1281:
1277:
1273:
1248:
1228:
1225:
1222:
1219:
1216:
1195:
1191:
1187:
1184:
1173:Fourier series
1159:
1139:
1136:
1122:
1097:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1038:
1015:
994:
991:
987:
962:
938:
934:
929:
925:
922:
919:
915:
893:
870:
849:
846:
843:
839:
835:
831:
827:
824:
821:
799:
794:
790:
786:
783:
780:
777:
773:
769:
765:
761:
758:
755:
730:
726:
722:
719:
716:
713:
690:
687:
684:
681:
678:
655:
634:
614:
592:
571:
551:
531:
509:
485:
459:
455:
428:
423:
419:
415:
412:
409:
406:
403:
400:
397:
394:
391:
346:
343:
333:
330:
322:Brillouin zone
305:
302:
279:
274:
246:
241:
205:
184:
162:
158:
155:
151:
139:momentum space
128:direct lattice
120:physical space
96:crystal system
92:physical space
80:direct lattice
26:
9:
6:
4:
3:
2:
11293:
11282:
11279:
11277:
11274:
11272:
11269:
11267:
11264:
11262:
11259:
11257:
11254:
11252:
11249:
11248:
11246:
11239:
11228:
11227:
11218:
11216:
11215:
11206:
11205:
11202:
11192:
11189:
11187:
11184:
11180:
11177:
11175:
11172:
11170:
11167:
11166:
11165:
11162:
11161:
11159:
11155:
11149:
11146:
11144:
11141:
11139:
11136:
11134:
11131:
11130:
11128:
11126:
11122:
11116:
11113:
11111:
11108:
11106:
11103:
11102:
11100:
11098:
11094:
11088:
11085:
11083:
11080:
11078:
11075:
11073:
11070:
11068:
11065:
11063:
11060:
11058:
11055:
11054:
11052:
11050:
11046:
11040:
11037:
11035:
11032:
11030:
11027:
11025:
11022:
11020:
11017:
11015:
11012:
11010:
11007:
11006:
11004:
11002:
10998:
10992:
10989:
10987:
10984:
10982:
10979:
10977:
10974:
10972:
10969:
10967:
10964:
10962:
10959:
10957:
10954:
10953:
10951:
10949:
10945:
10939:
10936:
10932:
10929:
10928:
10927:
10924:
10922:
10921:Patterson map
10919:
10917:
10914:
10912:
10909:
10907:
10904:
10902:
10899:
10898:
10896:
10892:
10884:
10881:
10879:
10876:
10875:
10874:
10871:
10869:
10866:
10862:
10859:
10857:
10854:
10853:
10852:
10849:
10845:
10842:
10840:
10837:
10836:
10835:
10832:
10831:
10829:
10827:
10823:
10817:
10807:
10804:
10802:
10799:
10797:
10794:
10792:
10791:Friedel's law
10789:
10787:
10784:
10782:
10779:
10777:
10774:
10773:
10764:
10761:
10759:
10756:
10752:
10749:
10747:
10744:
10743:
10742:
10739:
10735:
10734:Wigner effect
10732:
10730:
10727:
10725:
10722:
10721:
10720:
10719:Interstitials
10717:
10713:
10710:
10709:
10708:
10705:
10701:
10698:
10696:
10693:
10691:
10688:
10686:
10683:
10681:
10678:
10676:
10673:
10671:
10668:
10666:
10663:
10661:
10658:
10657:
10656:
10653:
10651:
10648:
10646:
10643:
10642:
10633:
10630:
10628:
10625:
10623:
10620:
10618:
10615:
10614:
10612:
10610:
10606:
10603:
10601:
10597:
10591:
10588:
10586:
10583:
10581:
10578:
10576:
10573:
10571:
10568:
10566:
10565:Precipitation
10563:
10561:
10558:
10554:
10551:
10549:
10546:
10544:
10541:
10539:
10536:
10535:
10534:
10533:Phase diagram
10531:
10530:
10528:
10526:
10520:
10512:
10509:
10508:
10507:
10504:
10500:
10497:
10496:
10495:
10492:
10488:
10485:
10483:
10480:
10479:
10478:
10475:
10474:
10465:
10462:
10460:
10457:
10455:
10452:
10450:
10447:
10445:
10442:
10440:
10437:
10436:
10434:
10432:
10428:
10422:
10419:
10417:
10414:
10410:
10407:
10405:
10402:
10400:
10397:
10395:
10392:
10390:
10387:
10386:
10385:
10382:
10381:
10379:
10377:
10373:
10367:
10364:
10362:
10359:
10355:
10352:
10351:
10350:
10347:
10346:
10344:
10340:
10336:
10329:
10324:
10322:
10317:
10315:
10310:
10309:
10306:
10300:
10297:
10295:
10292:
10289:
10285:
10281:
10278:
10275:
10274:
10262:
10256:
10248:
10246:0-471-41526-X
10242:
10238:
10231:
10223:
10216:
10208:
10204:
10200:
10196:
10192:
10188:
10183:
10178:
10175:(6): 064003.
10174:
10170:
10163:
10159:
10148:
10145:
10142:
10139:
10136:
10133:
10127:
10124:
10121:
10118:
10115:
10112:
10109:
10106:
10103:
10100:
10099:
10093:
10077:
10074:
10069:
10065:
10054:
10049:
10044:
10041:
10038:
10030:
10027:
10022:
10020:
10016:
10011:
10005:
10000:
9998:
9994:
9990:
9986:
9982:
9979:
9976:
9973:
9969:
9966:
9961:
9959:
9955:
9951:
9947:
9943:
9939:
9935:
9931:
9927:
9923:
9919:
9915:
9912:
9908:
9904:
9900:
9897:
9892:
9890:
9886:
9882:
9878:
9874:
9870:
9866:
9863:
9859:
9855:
9850:
9848:
9844:
9840:
9837:
9833:
9827:
9817:
9815:
9811:
9807:
9802:
9798:
9779:
9772:
9769:
9756:
9749:
9740:
9734:
9731:
9728:
9724:
9719:
9710:
9704:
9698:
9694:
9689:
9680:
9674:
9668:
9664:
9658:
9653:
9650:
9647:
9643:
9637:
9632:
9629:
9626:
9622:
9618:
9606:
9597:
9590:
9589:
9588:
9585:
9581:
9578:
9574:
9569:
9566:
9562:
9557:
9553:
9548:
9544:
9540:
9536:
9514:
9508:
9504:
9500:
9497:
9492:
9488:
9484:
9481:
9476:
9472:
9468:
9464:
9460:
9457:
9454:
9450:
9445:
9440:
9437:
9434:
9431:
9428:
9424:
9420:
9414:
9410:
9404:
9399:
9396:
9393:
9389:
9385:
9380:
9377:
9374:
9371:
9368:
9364:
9356:
9355:
9354:
9337:
9334:
9331:
9328:
9325:
9314:
9310:
9304:
9300:
9295:
9293:
9288:
9284:
9280:
9276:
9272:
9267:
9263:
9259:
9255:
9247:
9243:
9224:
9217:
9207:
9200:
9191:
9185:
9182:
9179:
9175:
9170:
9161:
9155:
9148:
9144:
9138:
9133:
9130:
9127:
9123:
9119:
9107:
9098:
9091:
9090:
9089:
9075:
9068:
9064:
9063:Huygens-style
9060:
9051:
9042:
9025:
9016:
9010:
9007:
9005:
8998:
8994:
8986:
8977:
8971:
8966:
8963:
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8949:
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8928:
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8798:
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8779:
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8748:
8728:
8708:
8701:
8691:
8676:
8670:
8660:
8655:
8645:
8640:
8629:
8607:
8601:
8591:
8586:
8576:
8571:
8560:
8550:
8536:
8532:
8528:
8525:
8517:
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8503:
8499:
8483:
8479:
8476:
8459:
8443:
8440:
8416:
8412:
8409:
8386:
8378:
8375:, with cubic
8374:
8364:
8362:
8352:
8350:
8331:
8328:
8325:
8319:
8316:
8310:
8299:
8296:
8273:
8270:
8267:
8264:
8258:
8255:
8249:
8246:
8243:
8237:
8214:
8211:
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8205:
8202:
8196:
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8169:
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8158:
8142:
8119:
8116:
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8107:
8104:
8098:
8089:
8077:
8052:
8048:
8041:
8038:
8029:
8004:
8001:
7997:
7988:
7962:
7958:
7954:
7951:
7928:
7925:
7919:
7915:
7909:
7904:
7900:
7889:
7886:
7880:
7875:
7872:
7869:
7865:
7849:
7846:
7842:
7830:
7820:
7817:
7814:
7809:
7796:
7790:
7785:
7781:
7777:
7774:
7769:
7765:
7760:
7754:
7751:
7748:
7743:
7729:
7728:
7727:
7710:
7705:
7699:
7694:
7689:
7684:
7677:
7672:
7667:
7662:
7656:
7651:
7648:
7641:
7640:
7639:
7638:
7620:
7617:
7613:
7609:
7606:
7603:
7595:
7585:
7580:
7567:
7539:
7536:
7533:
7530:
7527:
7502:
7492:
7489:
7486:
7481:
7464:
7448:
7441:vector space
7440:
7437:
7421:
7401:
7387:
7385:
7367:
7352:
7348:
7329:
7326:
7323:
7297:
7294:
7291:
7268:
7246:
7242:
7221:
7199:
7195:
7174:
7152:
7148:
7139:
7120:
7117:
7114:
7088:
7085:
7082:
7079:
7076:
7048:
7044:
7040:
7035:
7031:
7027:
7022:
7018:
7011:
7008:
7000:
6998:
6982:
6979:
6957:
6947:
6942:
6913:
6884:
6850:
6844:
6834:
6829:
6818:
6814:
6809:
6795:
6785:
6780:
6767:
6762:
6748:
6747:
6746:
6732:
6729:
6725:
6719:
6709:
6704:
6677:
6674:
6665:
6648:
6643:
6640:
6635:
6629:
6617:
6605:
6594:
6589:
6586:
6583:
6571:
6565:
6553:
6541:
6530:
6521:
6520:
6519:
6517:
6513:
6497:
6494:
6487:
6477:
6472:
6462:
6458:
6449:
6445:
6441:
6422:
6419:
6415:
6410:
6407:
6404:
6399:
6389:
6384:
6355:
6349:
6339:
6334:
6324:
6308:
6299:
6282:
6276:
6266:
6261:
6250:
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6241:
6231:
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6221:
6211:
6206:
6195:
6191:
6186:
6176:
6172:
6166:
6156:
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6140:
6136:
6131:
6121:
6118:
6092:
6082:
6077:
6062:
6058:
6055:
6049:
6047:
6040:
6024:
6014:
6009:
5994:
5990:
5987:
5981:
5979:
5972:
5956:
5946:
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5922:
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5913:
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5873:
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5826:
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5787:
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5770:
5763:
5760:
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5718:
5713:
5701:
5697:
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5688:
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5668:
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5629:
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5621:
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5608:
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5594:
5590:
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5566:
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5550:
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5535:
5519:
5511:
5493:
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5456:
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5448:
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5431:
5427:
5423:
5418:
5397:
5380:
5369:
5363:
5351:
5342:
5337:
5320:
5314:
5302:
5290:
5287:
5284:
5273:
5267:
5251:
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5229:
5223:
5204:
5201:
5198:
5193:
5179:
5178:
5177:
5158:
5152:
5147:
5140:
5135:
5129:
5124:
5121:
5114:
5113:
5112:
5111:
5065:
5057:
5054:
5021:
4986:
4982:
4932:
4916:
4911:
4897:
4878:
4875:
4872:
4870:
4863:
4844:
4828:
4823:
4809:
4790:
4787:
4784:
4776:
4760:
4755:
4745:
4738:
4722:
4716:
4713:
4710:
4708:
4701:
4683:
4682:
4681:
4665:
4661:
4635:
4623:
4619:
4615:
4610:
4598:
4594:
4590:
4585:
4571:
4570:
4569:
4554:
4548:
4538:
4533:
4522:
4514:
4504:
4488:
4459:
4444:
4443:vector spaces
4440:
4423:
4417:
4407:
4402:
4392:
4376:
4353:
4324:
4294:
4291:
4287:
4282:
4279:
4276:
4271:
4261:
4256:
4226:
4214:
4210:
4206:
4201:
4189:
4185:
4181:
4176:
4164:
4160:
4156:
4151:
4122:
4107:
4102:
4086:
4059:
4056:
4034:
4005:
3976:
3961:
3943:
3929:
3910:
3902:
3899:
3879:
3876:
3873:
3870:
3865:
3855:
3850:
3836:
3835:
3834:
3820:
3817:
3810:
3800:
3795:
3785:
3781:
3771:
3758:
3748:
3743:
3733:
3729:
3720:
3710:
3705:
3695:
3691:
3685:
3681:
3675:
3671:
3667:
3657:
3647:
3636:
3631:
3621:
3617:
3611:
3607:
3601:
3597:
3593:
3583:
3578:
3568:
3564:
3558:
3554:
3548:
3544:
3520:
3506:
3503:
3495:
3485:
3474:
3467:
3466:
3465:
3449:
3389:
3380:
3361:
3357:
3353:
3348:
3344:
3340:
3335:
3331:
3324:
3321:
3297:
3289:
3285:
3282:
3272:
3267:
3257:
3253:
3247:
3243:
3237:
3233:
3225:
3224:
3223:
3222:
3195:
3164:
3144:
3141:
3136:
3076:
3054:
2997:
2992:
2988:
2984:
2979:
2975:
2971:
2966:
2962:
2939:
2927:
2923:
2919:
2914:
2902:
2898:
2894:
2889:
2877:
2873:
2869:
2864:
2850:
2849:
2848:
2847:of integers,
2846:
2825:
2821:
2817:
2812:
2808:
2804:
2799:
2795:
2788:
2785:
2777:
2757:
2737:
2734:
2731:
2711:
2689:
2685:
2662:
2658:
2637:
2615:
2612:
2609:
2606:
2602:
2579:
2575:
2552:
2548:
2538:
2529:
2527:
2523:
2519:
2515:
2511:
2507:
2503:
2499:
2495:
2490:
2431:
2428:
2380:
2331:
2302:
2273:
2258:
2240:
2236:
2211:
2199:
2195:
2191:
2186:
2174:
2170:
2166:
2161:
2149:
2145:
2141:
2113:
2086:
2083:
2080:
2058:
2055:
2051:
2043:
2024:
2021:
2017:
2012:
2009:
2006:
2001:
1991:
1986:
1956:
1940:
1925:
1922:
1917:
1907:
1902:
1892:
1887:
1877:
1872:
1844:
1841:
1838:
1833:
1823:
1818:
1789:
1785:
1777:, means that
1761:
1751:
1746:
1736:
1731:
1727:
1718:must satisfy
1703:
1699:
1676:
1645:
1641:
1636:
1630:
1620:
1617:
1614:
1609:
1578:
1550:
1547:
1544:
1524:
1521:
1497:
1467:
1439:
1436:
1421:
1403:
1374:
1370:
1345:
1333:
1329:
1325:
1320:
1308:
1304:
1300:
1295:
1283:
1279:
1275:
1260:
1246:
1226:
1220:
1217:
1182:
1174:
1149:
1145:
1135:
1120:
1111:
1095:
1075:
1069:
1066:
1060:
1057:
1013:
992:
989:
976:
960:
952:
936:
932:
923:
920:
917:
868:
844:
841:
833:
822:
819:
792:
788:
784:
781:
778:
775:
767:
756:
753:
743:
728:
724:
720:
717:
714:
711:
688:
685:
682:
679:
676:
653:
632:
612:
590:
569:
549:
529:
507:
499:
483:
476:
457:
453:
444:
441:with initial
421:
417:
413:
410:
407:
404:
401:
398:
392:
389:
380:
375:
373:
369:
364:
360:
356:
338:
329:
327:
323:
318:
303:
300:
277:
262:
244:
230:
225:
223:
219:
203:
153:
140:
136:
131:
129:
125:
121:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
44:
37:
32:
19:
11237:
11224:
11212:
11157:Associations
11125:Organisation
10617:Disclination
10548:Polymorphism
10511:Quasicrystal
10454:Orthorhombic
10403:
10394:Miller index
10342:Key concepts
10260:
10255:
10236:
10230:
10221:
10215:
10172:
10168:
10162:
10135:Miller index
10028:
10023:
10018:
10014:
10009:
10001:
9996:
9992:
9984:
9980:
9967:
9962:
9957:
9953:
9949:
9942:well-defined
9937:
9934:Haar measure
9929:
9925:
9921:
9917:
9913:
9902:
9898:
9893:
9889:dual lattice
9888:
9884:
9880:
9876:
9872:
9868:
9864:
9851:
9842:
9838:
9829:
9826:Dual lattice
9809:
9805:
9800:
9796:
9794:
9583:
9579:
9576:
9572:
9570:
9564:
9560:
9555:
9551:
9546:
9542:
9538:
9534:
9532:
9312:
9305:
9302:
9298:
9296:
9291:
9286:
9282:
9278:
9274:
9265:
9261:
9257:
9253:
9245:
9241:
9239:
9056:
8817:
8697:
8551:
8509:
8500:
8465:
8370:
8358:
8161:
8155:denotes the
7943:
7725:
7635:. Using the
7393:
7382:is given in
7001:
6869:
6666:
6663:
5403:
5395:
5175:
4984:
4956:
4652:
4510:
4103:
3925:
3772:
3535:
3381:
3313:
3017:
2773:
2526:dual lattice
2522:linear forms
2491:
2040:, where the
1941:
1261:
1141:
744:
376:
371:
367:
348:
319:
226:
132:
127:
119:
115:
111:
94:, such as a
84:real lattice
83:
79:
51:
49:
11276:Diffraction
11110:Ewald Prize
10878:Diffraction
10856:Diffraction
10839:Diffraction
10781:Bragg plane
10776:Bragg's law
10655:Dislocation
10570:Segregation
10482:Crystallite
10399:Point group
9905:; if it is
9832:mathematics
8067:defined by
7987:volume form
7637:permutation
7436:dimensional
5110:permutation
3960:wavevectors
3958:, that are
1148:wavevectors
261:wavevectors
259:, that are
110:, known as
98:(usually a
38:3D crystal.
11245:Categories
10894:Algorithms
10883:Scattering
10861:Scattering
10844:Scattering
10712:Slip bands
10675:Cross slip
10525:transition
10459:Tetragonal
10449:Monoclinic
10361:Metallurgy
10182:1905.11354
10154:References
10114:Dual basis
9989:dual group
9911:dual space
9875:^ dual to
9862:dual group
9059:Fraunhofer
7347:real space
5176:we obtain
2518:dual space
2228:where the
1362:where the
951:wavevector
36:monoclinic
11001:Databases
10464:Triclinic
10444:Hexagonal
10384:Unit cell
10376:Structure
10207:166228311
10147:Zone axis
10075:−
9814:dynamical
9808:and atom
9760:→
9750:⋅
9744:→
9732:π
9714:→
9684:→
9644:∑
9623:∑
9610:→
9501:ℓ
9458:π
9441:ℓ
9390:∑
9381:ℓ
9338:ℓ
9273:for atom
9211:→
9201:⋅
9195:→
9183:π
9165:→
9124:∑
9111:→
9020:^
8981:^
8953:^
8929:−
8899:^
8871:^
8783:π
8752:π
8529:π
8480:π
8413:π
8317:⊂
8300:∈
8238:ω
8206:×
8170:ω
8143:⌟
8081:^
8053:∗
8045:→
8039::
8033:^
8002:−
7968:→
7955::
7952:ω
7926:∈
7920:ω
7916:⌟
7901:σ
7890:…
7887:⌟
7873:−
7866:σ
7847:−
7818:…
7797:ω
7782:σ
7778:…
7766:σ
7761:ε
7755:π
7695:⋯
7673:⋯
7649:σ
7614:δ
7610:π
7543:→
7537:×
7531::
7490:…
7330:ℓ
7298:ℓ
7269:ℓ
7136:, called
7121:ℓ
7089:ℓ
6983:π
6948:×
6835:×
6815:⋅
6786:×
6733:π
6678:π
6641:−
6590:π
6478:⋅
6416:δ
6411:π
6390:⋅
6267:×
6247:⋅
6212:×
6192:⋅
6157:×
6137:⋅
6083:×
6059:π
6015:×
5991:π
5947:×
5923:π
5364:σ
5343:⋅
5315:σ
5291:π
5268:σ
5252:⋅
5224:σ
5205:π
5122:σ
5058:−
4983:, i.e. a
4917:⋅
4879:π
4829:⋅
4791:π
4761:⋅
4746:−
4723:−
4717:π
4288:δ
4283:π
4262:⋅
4060:π
3903:∈
3877:π
3856:⋅
3801:⋅
3749:⋅
3711:⋅
3672:∑
3637:⋅
3598:∑
3584:⋅
3545:∑
3273:⋅
3234:∑
2998:∈
2613:⋅
2432:π
2381:⋅
2052:δ
2018:δ
2013:π
1992:⋅
1908:⋅
1878:⋅
1845:π
1824:⋅
1786:λ
1752:⋅
1728:λ
1700:λ
1642:λ
1621:π
1551:π
1545:−
1525:π
1221:π
1121:λ
1070:π
1058:φ
975:wavefront
961:φ
937:λ
924:π
845:φ
834:⋅
823:
789:φ
779:ω
776:−
768:⋅
757:
729:λ
721:π
689:π
680:λ
654:λ
570:ω
508:ω
454:φ
418:φ
408:ω
405:−
393:
304:π
204:ℏ
157:ℏ
11214:Category
11049:Journals
10981:OctaDist
10976:JANA2020
10948:Software
10834:Electron
10751:F-center
10538:Eutectic
10499:Fiveling
10494:Twinning
10487:Equiaxed
10280:Archived
10222:Geometry
10096:See also
9944:up to a
9353:, where
9281:, while
8379:of side
8349:rotation
8289:, where
5352:′
5303:′
5066:′
5022:′
2524:and the
1088:, where
1005:at time
174:, where
74:of the
68:energies
64:electron
11226:Commons
11174:Germany
10851:Neutron
10741:Vacancy
10600:Defects
10585:GP-zone
10431:Systems
10187:Bibcode
9987:of the
9972:abelian
9965:lattice
9836:lattice
9269:is the
8347:is the
7985:is the
7461:with a
7345:in the
6296:is the
3928:vectors
2845:3-tuple
2508:due to
1110:integer
949:is the
229:vectors
216:is the
126:to the
124:is dual
116:k space
76:lattice
11169:France
11164:Europe
11097:Awards
10627:Growth
10477:Growth
10243:
10205:
10026:matrix
10008:dim =
9970:in an
9946:scalar
9860:. The
9582:where
7944:Here,
7351:normal
7234:, and
6111:where
6068:
6000:
5932:
4653:where
4241:where
3991:, and
3892:where
3100:where
3018:where
2954:where
2516:, the
2317:, and
1662:where
883:where
668:where
11191:Japan
11138:IOBCr
10991:SHELX
10986:Olex2
10873:X-ray
10523:Phase
10439:Cubic
10203:S2CID
10177:arXiv
9920:with
9867:^ to
9845:, of
9795:Here
9311:from
9308:h,k,ℓ
9240:Here
7463:basis
6512:above
4957:Here
3219:as a
2512:. In
2496:is a
1971:with
443:phase
324:is a
86:is a
60:X-ray
11133:IUCr
11034:ICDD
11029:ICSD
11014:CCDC
10961:Coot
10956:CCP4
10707:Slip
10670:Kink
10288:Jmol
10241:ISBN
8772:and
8721:and
8431:(or
8135:and
7439:real
2677:and
2567:and
2492:The
1857:and
1482:and
1418:are
1028:and
625:and
582:and
542:and
496:and
320:The
62:and
50:The
11148:DMG
11143:RAS
11039:PDB
11024:COD
11019:CIF
10971:DSR
10695:GND
10622:CSL
10195:doi
10002:In
9991:of
9928:to
9916:of
9901:on
9248:/(2
8516:FCC
8512:BCC
7104:or
3382:As
2843:as
2520:of
1537:or
820:cos
754:cos
390:cos
133:In
114:or
106:of
90:in
82:or
11247::
11186:US
11179:UK
10286:–
10201:.
10193:.
10185:.
10171:.
9999:.
9960:.
9849:.
9801:jk
9575:=
9559:,
9550:,
9301:=
9244:=
8549:.
8498:.
8304:SO
8159:.
7989:,
7187:,
7140:;
6518::
4108:,
4101:.
2288:,
603:).
130:.
10327:e
10320:t
10313:v
10249:.
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10197::
10189::
10179::
10173:3
10078:1
10070:)
10066:B
10060:T
10055:B
10050:(
10045:B
10042:=
10039:A
10029:B
10019:R
10015:R
10010:n
9997:L
9993:G
9985:L
9981:G
9968:L
9958:V
9954:L
9950:Q
9938:V
9930:V
9926:V
9922:V
9918:V
9914:V
9903:V
9899:Q
9885:L
9881:V
9877:L
9873:L
9869:V
9865:V
9843:V
9839:L
9810:k
9806:j
9797:r
9780:.
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9711:g
9705:[
9699:k
9695:f
9690:]
9681:g
9675:[
9669:j
9665:f
9659:N
9654:1
9651:=
9648:k
9638:N
9633:1
9630:=
9627:j
9619:=
9616:]
9607:g
9601:[
9598:I
9584:F
9580:F
9577:F
9573:I
9565:j
9561:w
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9552:v
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9543:u
9539:m
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9515:)
9509:j
9505:w
9498:+
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9485:k
9482:+
9477:j
9473:u
9469:h
9465:(
9461:i
9455:2
9451:e
9446:]
9438:,
9435:k
9432:,
9429:h
9425:g
9421:[
9415:j
9411:f
9405:m
9400:1
9397:=
9394:j
9386:=
9378:,
9375:k
9372:,
9369:h
9365:F
9341:)
9335:,
9332:k
9329:,
9326:h
9323:(
9313:M
9306:F
9303:M
9299:F
9292:j
9287:j
9283:r
9279:g
9275:j
9266:j
9262:f
9258:N
9254:q
9250:π
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9242:g
9225:.
9218:j
9208:r
9192:g
9186:i
9180:2
9176:e
9171:]
9162:g
9156:[
9149:j
9145:f
9139:N
9134:1
9131:=
9128:j
9120:=
9117:]
9108:g
9102:[
9099:F
9076:F
9026:.
9017:z
9011:c
9008:=
8999:3
8995:a
8987:,
8978:y
8972:a
8967:2
8964:1
8959:+
8950:x
8944:a
8939:2
8935:3
8926:=
8917:2
8913:a
8905:,
8896:y
8890:a
8885:2
8882:1
8877:+
8868:x
8862:a
8857:2
8853:3
8847:=
8838:1
8834:a
8818:c
8804:)
8799:3
8794:a
8791:(
8787:/
8780:4
8760:c
8756:/
8749:2
8729:c
8709:a
8677:)
8671:3
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8646:,
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8636:b
8630:(
8608:)
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8572:1
8567:a
8561:(
8537:a
8533:/
8526:4
8484:a
8477:4
8444:a
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8332:V
8329:,
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8323:(
8320:L
8314:)
8311:2
8308:(
8297:R
8277:)
8274:w
8271:,
8268:v
8265:R
8262:(
8259:g
8256:=
8253:)
8250:w
8247:,
8244:v
8241:(
8218:)
8215:w
8212:,
8209:v
8203:u
8200:(
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8194:=
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8188:w
8185:,
8182:v
8179:,
8176:u
8173:(
8123:)
8120:w
8117:,
8114:v
8111:(
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8105:=
8102:)
8099:w
8096:(
8093:)
8090:v
8087:(
8078:g
8049:V
8042:V
8030:g
8005:1
7998:g
7972:R
7963:n
7959:V
7929:V
7923:)
7910:i
7905:1
7895:a
7881:i
7876:1
7870:n
7860:a
7855:(
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7843:g
7836:)
7831:n
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7821:,
7815:,
7810:1
7805:a
7800:(
7791:i
7786:n
7775:i
7770:1
7752:2
7749:=
7744:i
7739:b
7711:,
7706:)
7700:1
7690:3
7685:2
7678:n
7668:2
7663:1
7657:(
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7621:j
7618:i
7607:2
7604:=
7601:)
7596:j
7591:b
7586:,
7581:i
7576:a
7571:(
7568:g
7547:R
7540:V
7534:V
7528:g
7508:)
7503:n
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7487:,
7482:1
7477:a
7472:(
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7434:-
7422:n
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7333:)
7327:k
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7321:(
7301:)
7295:k
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7149:m
7124:)
7118:k
7115:h
7112:(
7092:)
7086:,
7083:k
7080:,
7077:h
7074:(
7054:)
7049:3
7045:m
7041:,
7036:2
7032:m
7028:,
7023:1
7019:m
7015:(
7012:=
7009:m
6980:2
6958:3
6953:a
6943:2
6938:a
6914:1
6909:a
6885:1
6880:b
6851:)
6845:3
6840:a
6830:2
6825:a
6819:(
6810:1
6805:a
6796:3
6791:a
6781:2
6776:a
6768:=
6763:1
6758:b
6730:2
6726:/
6720:m
6715:G
6710:=
6705:m
6700:K
6675:2
6649:.
6644:1
6636:]
6630:3
6625:a
6618:2
6613:a
6606:1
6601:a
6595:[
6587:2
6584:=
6578:T
6572:]
6566:3
6561:b
6554:2
6549:b
6542:1
6537:b
6531:[
6498:1
6495:=
6488:n
6483:R
6473:m
6468:G
6463:i
6459:e
6423:j
6420:i
6408:2
6405:=
6400:j
6395:b
6385:i
6380:a
6356:)
6350:3
6345:b
6340:,
6335:2
6330:b
6325:,
6319:1
6315:b
6309:(
6283:)
6277:2
6272:a
6262:1
6257:a
6251:(
6242:3
6237:a
6232:=
6228:)
6222:1
6217:a
6207:3
6202:a
6196:(
6187:2
6182:a
6177:=
6173:)
6167:3
6162:a
6152:2
6147:a
6141:(
6132:1
6127:a
6122:=
6119:V
6093:2
6088:a
6078:1
6073:a
6063:V
6056:2
6050:=
6041:3
6036:b
6025:1
6020:a
6010:3
6005:a
5995:V
5988:2
5982:=
5973:2
5968:b
5957:3
5952:a
5942:2
5937:a
5927:V
5920:2
5914:=
5905:1
5900:b
5874:)
5868:3
5863:b
5858:,
5853:2
5848:b
5843:,
5837:1
5833:b
5827:(
5806:)
5801:3
5797:m
5793:,
5788:2
5784:m
5780:,
5775:1
5771:m
5767:(
5764:=
5761:m
5739:3
5734:b
5727:3
5723:m
5719:+
5714:2
5709:b
5702:2
5698:m
5694:+
5689:1
5684:b
5677:1
5673:m
5669:=
5664:m
5659:G
5636:)
5630:3
5626:n
5622:,
5617:2
5613:n
5609:,
5604:1
5600:n
5595:(
5591:=
5588:n
5567:)
5561:3
5556:a
5551:,
5546:2
5541:a
5536:,
5530:1
5526:a
5520:(
5494:3
5489:a
5482:3
5478:n
5474:+
5469:2
5464:a
5457:2
5453:n
5449:+
5444:1
5439:a
5432:1
5428:n
5424:=
5419:n
5414:R
5381:.
5373:)
5370:n
5367:(
5359:a
5348:Q
5338:n
5333:a
5324:)
5321:n
5318:(
5310:a
5299:Q
5288:2
5285:=
5277:)
5274:n
5271:(
5263:a
5256:Q
5247:n
5242:a
5233:)
5230:n
5227:(
5219:a
5212:Q
5202:2
5199:=
5194:n
5189:b
5159:)
5153:1
5148:2
5141:2
5136:1
5130:(
5125:=
5095:v
5073:v
5063:Q
5055:=
5051:v
5045:Q
5019:Q
4996:Q
4985:q
4966:Q
4933:1
4928:a
4921:Q
4912:2
4907:a
4898:1
4893:a
4886:Q
4876:2
4873:=
4864:2
4859:b
4845:2
4840:a
4833:Q
4824:1
4819:a
4810:2
4805:a
4798:Q
4788:2
4785:=
4777:2
4772:a
4765:Q
4756:1
4751:a
4739:2
4734:a
4727:Q
4714:2
4711:=
4702:1
4697:b
4666:i
4662:m
4636:2
4631:b
4624:2
4620:m
4616:+
4611:1
4606:b
4599:1
4595:m
4591:=
4586:m
4581:G
4555:)
4549:2
4544:a
4539:,
4534:1
4529:a
4523:(
4489:m
4484:G
4460:m
4455:G
4424:)
4418:3
4413:b
4408:,
4403:2
4398:b
4393:,
4387:1
4383:b
4377:(
4354:n
4349:R
4325:m
4320:G
4295:j
4292:i
4280:2
4277:=
4272:j
4267:b
4257:i
4252:a
4227:3
4222:b
4215:3
4211:m
4207:+
4202:2
4197:b
4190:2
4186:m
4182:+
4177:1
4172:b
4165:1
4161:m
4157:=
4152:m
4147:G
4123:m
4118:G
4087:n
4082:R
4057:2
4035:n
4030:R
4006:m
4001:G
3977:n
3972:R
3944:m
3939:G
3911:.
3907:Z
3900:N
3880:N
3874:2
3871:=
3866:n
3861:R
3851:m
3846:G
3821:1
3818:=
3811:n
3806:R
3796:m
3791:G
3786:i
3782:e
3759:.
3753:r
3744:m
3739:G
3734:i
3730:e
3721:n
3716:R
3706:m
3701:G
3696:i
3692:e
3686:m
3682:f
3676:m
3668:=
3663:)
3658:n
3653:R
3648:+
3644:r
3640:(
3632:m
3627:G
3622:i
3618:e
3612:m
3608:f
3602:m
3594:=
3588:r
3579:m
3574:G
3569:i
3565:e
3559:m
3555:f
3549:m
3521:.
3518:)
3514:r
3510:(
3507:f
3504:=
3501:)
3496:n
3491:R
3486:+
3482:r
3478:(
3475:f
3450:n
3445:R
3422:r
3401:)
3397:r
3393:(
3390:f
3367:)
3362:3
3358:m
3354:,
3349:2
3345:m
3341:,
3336:1
3332:m
3328:(
3325:=
3322:m
3298:)
3294:r
3290:(
3286:f
3283:=
3277:r
3268:m
3263:G
3258:i
3254:e
3248:m
3244:f
3238:m
3207:)
3203:r
3199:(
3196:f
3176:)
3172:r
3168:(
3165:f
3145:0
3142:=
3137:n
3132:R
3109:r
3088:)
3084:r
3080:(
3077:f
3055:i
3050:a
3027:Z
3002:Z
2993:3
2989:n
2985:,
2980:2
2976:n
2972:,
2967:1
2963:n
2940:3
2935:a
2928:3
2924:n
2920:+
2915:2
2910:a
2903:2
2899:n
2895:+
2890:1
2885:a
2878:1
2874:n
2870:=
2865:n
2860:R
2831:)
2826:3
2822:n
2818:,
2813:2
2809:n
2805:,
2800:1
2796:n
2792:(
2789:=
2786:n
2758:R
2750:(
2738:r
2735:+
2732:R
2712:r
2690:2
2686:b
2663:1
2659:b
2638:G
2616:r
2610:G
2607:i
2603:e
2580:2
2576:a
2553:1
2549:a
2476:G
2454:G
2429:2
2407:R
2385:R
2377:G
2355:G
2332:3
2327:b
2303:2
2298:b
2274:1
2269:b
2241:j
2237:m
2226:,
2212:3
2207:b
2200:3
2196:m
2192:+
2187:2
2182:b
2175:2
2171:m
2167:+
2162:1
2157:b
2150:1
2146:m
2142:=
2138:G
2114:j
2109:b
2087:j
2084:=
2081:i
2059:j
2056:i
2025:j
2022:i
2010:2
2007:=
2002:j
1997:b
1987:i
1982:a
1957:j
1952:b
1938:.
1926:0
1923:=
1918:1
1913:b
1903:3
1898:a
1893:=
1888:1
1883:b
1873:2
1868:a
1842:2
1839:=
1834:1
1829:b
1819:1
1814:a
1790:1
1762:1
1757:e
1747:1
1742:a
1737:=
1732:1
1704:1
1677:1
1672:e
1660:,
1646:1
1637:/
1631:1
1626:e
1618:2
1615:=
1610:1
1605:b
1579:1
1574:a
1548:2
1522:2
1512:,
1498:3
1493:a
1468:2
1463:a
1440:0
1437:=
1433:R
1404:i
1399:a
1375:i
1371:n
1360:,
1346:3
1341:a
1334:3
1330:n
1326:+
1321:2
1316:a
1309:2
1305:n
1301:+
1296:1
1291:a
1284:1
1280:n
1276:=
1272:R
1247:n
1227:n
1224:)
1218:2
1215:(
1194:)
1190:r
1186:(
1183:f
1158:k
1133:.
1096:n
1076:n
1073:)
1067:2
1064:(
1061:+
1037:e
1026:,
1014:t
993:0
990:=
986:r
933:/
928:e
921:2
918:=
914:k
892:r
881:,
869:t
848:)
842:+
838:r
830:k
826:(
810:,
798:)
793:0
785:+
782:t
772:r
764:k
760:(
741:.
725:/
718:2
715:=
712:k
701:;
686:2
683:=
677:k
666:,
633:x
613:k
591:t
550:x
530:k
520:,
484:k
472:,
458:0
439:,
427:)
422:0
414:+
411:t
402:x
399:k
396:(
372:L
368:L
351:k
301:2
278:n
273:R
245:m
240:G
183:p
161:k
154:=
150:p
20:)
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