326:
10805:
4941:
20:
11198:
4675:
9801:. 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.
6098:
5380:
2526:
11210:
7928:
4936:{\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}}}
32:
6283:
3758:
5878:
9029:
5171:
6103:
3528:
7721:
8813:
354:
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
6854:
8490:
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
6093:{\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}}}
9575:
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:
5375:{\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)}}}.}
209:. 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
1195:
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
9779:
7923:{\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}
8491:
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.
8809:
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
7710:
9517:
6278:{\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)}
3753:{\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} }.}
5740:
5495:
4228:
2941:
3298:
1925:
2213:
1347:
9224:
6740:
8677:
8608:
6356:
5874:
5567:
4424:
5073:
4637:
6424:
4296:
2026:
797:
9024:{\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}}}
355:
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 (
7622:
5160:
1411:
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
1763:
1647:
8818:
5883:
3520:
4555:
6497:
3879:
3820:
847:
1844:
7507:
4680:
6959:
6732:
936:
10079:
6648:
7972:
8334:
7547:
3002:
5636:
161:
2385:
426:
8217:
9582:
8054:
10814:
8122:
10010:) 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.
7053:
5805:
3366:
2830:
8276:
11075:
9077:
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:
2478:
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.)
7633:
5023:
3910:
3144:
11070:
8803:
1075:
7369:
6915:
6886:
4490:
4461:
4355:
4326:
4124:
4088:
4036:
4007:
3978:
3945:
3451:
3056:
2333:
2304:
2275:
2115:
1958:
1678:
1580:
1499:
1469:
1405:
728:
279:
246:
9039:
8485:
8418:
3400:
3206:
3175:
3087:
1193:
688:
6679:
comes naturally from the study of periodic structures. An essentially equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice
3951:
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
1439:
1135:. 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
992:
9560:
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
9348:
2617:
5095:
4996:
4966:
3422:
3109:
3027:
2476:
2454:
2407:
2355:
2060:
1791:
1705:
1158:
1037:
892:
459:
183:
9340:
8445:
7091:
8142:
6653:
This method appeals to the definition, and allows generalization to arbitrary dimensions. The cross product formula dominates introductory materials on crystallography.
7332:
7300:
7123:
5641:
5396:
4129:
2842:
10610:
1226:
1120:
960:
653:
3217:
634:
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
569:
507:
11126:
8006:
1850:
1550:
203:
7268:
7248:
7201:
7154:
6982:
6677:
4667:
4059:
2691:
2664:
2581:
2554:
2431:
2242:
2121:
1524:
1376:
1255:
303:
8759:
8536:
9557:}. 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.
2737:
2086:
9075:
8386:
8151:
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,
7448:
7421:
7401:
7221:
7174:
2757:
2711:
2637:
1246:
1095:
1013:
868:
632:
612:
590:
549:
529:
483:
10156:
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".
6849:{\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)}}}
8728:
8708:
9083:
10354:
8447:
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.
2117:
comprise a set of three primitive wavevectors or three primitive translation vectors for the reciprocal lattice, each of whose vertices takes the form
1251:
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
8613:
8544:
6292:
5810:
5503:
4360:
2693:(i.e. any reciprocal lattice vector), the resulting plane waves have the same periodicity of the lattice – that is that any translation from point
4563:
6362:
4234:
2456:
will essentially be equal for every direct lattice vertex, in conformity with the reciprocal lattice definition above. (Although any wavevector
1964:
738:
9042:
Shadow of a 118-atom faceted carbon-pentacone's intensity reciprocal-lattice lighting up red in diffraction when intersecting the Ewald sphere.
5028:
7552:
5106:
10944:
3177:
follows the periodicity of this lattice, e.g. the function describing the electronic density in an atomic crystal, it is useful to write
1711:
1587:
11131:
10856:
3459:
11022:
10314:
2759:
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.
4506:
1131:
In general, a geometric lattice is an infinite, regular array of vertices (points) in space, which can be modelled vectorially as a
6442:
3828:
3765:
804:
210:
1796:
11259:
7456:
7338:
lattice. (A lattice plane is a plane crossing lattice points.) The direction of the reciprocal lattice vector corresponds to the
10279:-based electron diffraction simulator lets you explore the intersection between reciprocal lattice and Ewald sphere during tilt.
329:
Adsorbed species on the surface with 1×2 superstructure give rise to additional spots in low-energy electron diffraction (LEED).
11121:
11113:
4557:, its reciprocal lattice can be determined by generating its two reciprocal primitive vectors, through the following formulae,
11174:
11152:
10288:
Learn easily crystallography and how the reciprocal lattice explains the diffraction phenomenon, as shown in chapters 4 and 5
6920:
6682:
5385:
Notably, in a 3D space this 2D reciprocal lattice is an infinitely extended set of Bragg rods—described by Sung et al.
4976:
uarter turn. The anti-clockwise rotation and the clockwise rotation can both be used to determine the reciprocal lattice: If
897:
11167:
11017:
10683:
10548:
10397:
10023:
6513:
11254:
11157:
11055:
10751:
10268:
7936:
2529:
Demonstration of relation between real and reciprocal lattice. A real space 2D lattice (red dots) with primitive vectors
10404:
10114:
8281:
7512:
2946:
9054:
sum of amplitudes from all points of scattering (in this case from each individual atom). This sum is denoted by the
6859:
and so on for the other primitive vectors. The crystallographer's definition has the advantage that the definition of
5572:
11269:
11179:
11037:
11007:
10936:
10233:
9046:
One path to the reciprocal lattice of an arbitrary collection of atoms comes from the idea of scattered waves in the
252:
of plane waves in the
Fourier series of a spatial function whose periodicity is the same as that of a direct lattice
133:
9774:{\displaystyle I=\sum _{j=1}^{N}\sum _{k=1}^{N}f_{j}\leftf_{k}\lefte^{2\pi i{\vec {g}}\cdot {\vec {r}}_{\!\!\;jk}}.}
2360:
374:
11214:
10889:
8154:
8340:
by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation).
6503:. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using
11162:
11085:
10959:
10558:
9802:
8011:
6984:. This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of
10997:
10919:
4357:, is itself a Bravais lattice as it is formed by integer combinations of its own primitive translation vectors
10006:. The dual lattice is then defined by all points in the linear span of the original lattice (typically all of
942:
in the three dimensional reciprocal space. (The magnitude of a wavevector is called wavenumber.) The constant
11012:
11002:
10307:
9051:
8059:
350:
arising from the
Fourier transform of a time dependent function; reciprocal space is a space over which the
325:
11136:
10784:
10409:
10387:
7705:{\displaystyle \sigma ={\begin{pmatrix}1&2&\cdots &n\\2&3&\cdots &1\end{pmatrix}},}
6993:
5745:
3306:
2770:
8222:
10688:
10442:
10337:
9953:
3884:
3114:
281:. Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of
11045:
10342:
8455:
The reciprocal lattice to an FCC lattice is the body-centered cubic (BCC) lattice, with a cube side of
1042:
9512:{\displaystyle F_{h,k,\ell }=\sum _{j=1}^{m}f_{j}\lefte^{2\pi i\left(hu_{j}+kv_{j}+\ell w_{j}\right)}}
8764:
7345:
6891:
6862:
4466:
4437:
4331:
4302:
4100:
4064:
4038:. Each plane wave in the Fourier series has the same phase (actually can be differed by a multiple of
4012:
3983:
3954:
3921:
3427:
3032:
2309:
2280:
2251:
2091:
1934:
1654:
1556:
1475:
1445:
1381:
696:
255:
222:
11244:
11060:
10989:
10447:
10437:
6988:. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed.
3374:
3180:
3149:
3061:
1167:
661:
56:
8458:
8391:
1416:
969:
11239:
11202:
10926:
10822:
10695:
10658:
10573:
10452:
10432:
10300:
9835:
9259:
7424:
5001:
2586:
366:
Reciprocal space comes into play regarding waves, both classical and quantum mechanical. Because a
5735:{\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}
5490:{\displaystyle \mathbf {R} _{n}=n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}
5078:
4979:
4949:
4223:{\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}
3405:
3092:
3010:
2936:{\displaystyle \mathbf {R} _{n}=n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}
2459:
2437:
2390:
2338:
2035:
1769:
1683:
1141:
1020:
875:
437:
166:
11249:
11050:
10894:
10839:
10588:
10553:
9995:, a lattice is a locally discrete set of points described by all integral linear combinations of
9307:
8610:(cubic, tetragonal, orthorhombic) have primitive translation vectors for the reciprocal lattice,
7058:
206:
10282:
8423:
8127:
6436:
4094:
3293:{\displaystyle \sum _{m}f_{m}e^{i\mathbf {G} _{m}\cdot \mathbf {r} }=f\left(\mathbf {r} \right)}
3209:
10804:
10746:
10563:
10020:
to have columns as the linearly independent vectors that describe the lattice, then the matrix
9047:
8349:
7451:
7305:
7273:
7096:
1920:{\displaystyle \mathbf {a} _{2}\cdot \mathbf {b} _{1}=\mathbf {a} _{3}\cdot \mathbf {b} _{1}=0}
44:
2208:{\displaystyle \mathbf {G} =m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}
1342:{\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}
1199:
1105:
945:
638:
111:
considered as a vector space, and the reciprocal lattice is the sublattice of that space that
11103:
10899:
10861:
10668:
10620:
9963:
6286:
2490:
554:
492:
367:
314:
10287:
7981:
1529:
188:
11264:
10827:
10700:
10536:
10427:
10175:
10014:
9992:
9934:
8504:
8500:
7625:
7253:
7226:
7179:
7132:
6964:
6659:
5098:
4645:
4041:
2669:
2642:
2559:
2532:
2434:(that can be possibly zero if the multiplier is zero), so the phase of the plane wave with
2413:
2220:
1506:
1354:
306:
at each direct lattice point (so essentially same phase at all the direct lattice points).
285:
52:
4426:, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the
8:
10844:
10832:
10707:
10673:
10653:
8733:
8510:
2767:
and labelling each lattice vector (a vector indicating a lattice point) by the subscript
2716:
2494:
2065:
1408:
96:
10179:
11093:
10904:
10849:
10191:
10165:
10129:
9977:
9850:
9846:
9304:
unit cells (as in the cases above) turns out to be non-zero only for integer values of
9060:
8371:
7433:
7406:
7386:
7206:
7159:
4427:
3916:
2742:
2696:
2622:
2498:
1231:
1080:
998:
853:
617:
597:
575:
534:
514:
468:
463:
217:
92:
10187:
9219:{\displaystyle F=\sum _{j=1}^{N}f_{j}\!\lefte^{2\pi i{\vec {g}}\cdot {\vec {r}}_{j}}.}
2517:
provide more abstract generalizations of reciprocal space and the reciprocal lattice.
11027:
10866:
10794:
10774:
10494:
10364:
10283:
DoITPoMS Teaching and
Learning Package on Reciprocal Space and the Reciprocal Lattice
10229:
10195:
10108:
9966:
9055:
8688:
7372:
6985:
1680:
is the unit vector perpendicular to these two adjacent wavefronts and the wavelength
486:
351:
343:
76:
60:
48:
8388:, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side
7375:
and is equal to the reciprocal of the interplanar spacing of the real space planes.
11065:
10871:
10789:
10779:
10578:
10511:
10482:
10475:
10183:
9842:
8713:
8693:
8145:
6504:
2502:
2335:
in this case. Simple algebra then shows that, for any plane wave with a wavevector
347:
10954:
10949:
10914:
10734:
10633:
10568:
10531:
10526:
10377:
10323:
10272:
10096:
9824:
8361:
6428:
4969:
3058:
is a primitive translation vector or shortly primitive vector. Taking a function
2764:
2245:
2030:
1132:
123:
88:
80:
64:
9841:
The first, which generalises directly the reciprocal lattice construction, uses
19:
10764:
10729:
10717:
10712:
10678:
10648:
10638:
10597:
10541:
10465:
10419:
10090:
9984:
consisting of all continuous characters that are equal to one at each point of
9895:
9884:
8672:{\displaystyle \left(\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right)}
8603:{\displaystyle \left(\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3}\right)}
8365:
7339:
6351:{\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}
5869:{\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}
5562:{\displaystyle \left(\mathbf {a_{1}} ,\mathbf {a} _{2},\mathbf {a} _{3}\right)}
5498:
4501:
4419:{\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}
3762:
Because equality of two
Fourier series implies equality of their coefficients,
2583:
are shown by blue and green arrows respectively. Atop, plane waves of the form
2486:
2482:
1161:
310:
127:
84:
10265:
9530:
atoms inside the unit cell whose fractional lattice indices are respectively {
9286:
For the special case of an infinite periodic crystal, the scattered amplitude
8541:
It can be proven that only the
Bravais lattices which have 90 degrees between
1931:
Cycling through the indices in turn, the same method yields three wavevectors
11233:
10909:
10722:
10521:
9960:
1039:
is a unit vector perpendicular to this wavefront. The wavefronts with phases
431:
4632:{\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}}
10615:
10605:
10499:
10382:
10123:
9930:
9922:
9814:
7126:
4431:
2514:
2248:
as it is formed by integer combinations of the primitive vectors, that are
1793:
is equal to the distance between the two wavefronts. Hence by construction
112:
16:
Fourier transform of a real-space lattice, important in solid-state physics
6656:
The above definition is called the "physics" definition, as the factor of
6419:{\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}
4291:{\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}
2021:{\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}
792:{\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{0})}
11098:
10769:
10643:
10470:
9820:
7975:
7427:
7342:
to the real space planes. The magnitude of the reciprocal lattice vector
3948:
2510:
1136:
939:
249:
7549:. The reciprocal lattice vectors are uniquely determined by the formula
10663:
10349:
10102:
9899:
6439:. This choice also satisfies the requirement of the reciprocal lattice
5807:
can be determined by generating its three reciprocal primitive vectors
5068:{\displaystyle \mathbf {Q} \,\mathbf {v} =-\mathbf {Q'} \,\mathbf {v} }
4328:
for the Fourier series of a spatial function which periodicity follows
4299:. With this form, the reciprocal lattice as the set of all wavevectors
2506:
24:
7617:{\displaystyle g(\mathbf {a} _{i},\mathbf {b} _{j})=2\pi \delta _{ij}}
5155:{\displaystyle \sigma ={\begin{pmatrix}1&2\\2&1\end{pmatrix}}}
3525:
Expressing the above instead in terms of their Fourier series we have
1101:, comprise a set of parallel planes, equally spaced by the wavelength
10372:
10135:
963:
10969:
10739:
10487:
10170:
8337:
1758:{\displaystyle \lambda _{1}=\mathbf {a} _{1}\cdot \mathbf {e} _{1}}
1642:{\displaystyle \mathbf {b} _{1}=2\pi \mathbf {e} _{1}/\lambda _{1}}
10292:
10093: – Primitive cell in the reciprocal space lattice of crystals
9283:. The Fourier phase depends on one's choice of coordinate origin.
10979:
7302:
in the reciprocal lattice corresponds to a set of lattice planes
3515:{\displaystyle f(\mathbf {r} +\mathbf {R} _{n})=f(\mathbf {r} ).}
1098:
2525:
9929:. But given an identification of the two, which is in any case
734:
In three dimensions, the corresponding plane wave term becomes
692:
hence the corresponding wavenumber in reciprocal space will be
4550:{\displaystyle \left(\mathbf {a} _{1},\mathbf {a} _{2}\right)}
2639:
is any integer combination of reciprocal lattice vector basis
2493:) of the reciprocal lattice, which plays an important role in
1552:
from the former wavefront passing the origin) passing through
10974:
7335:
6492:{\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1}
3874:{\displaystyle \mathbf {G} _{m}\cdot \mathbf {R} _{n}=2\pi N}
3815:{\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1}
2833:
842:{\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} +\varphi )}
1839:{\displaystyle \mathbf {a} _{1}\cdot \mathbf {b} _{1}=2\pi }
31:
10276:
9038:
8730:
is another simple hexagonal lattice with lattice constants
7502:{\displaystyle (\mathbf {a} _{1},\ldots ,\mathbf {a} _{n})}
8687:
The reciprocal to a simple hexagonal Bravais lattice with
8343:
6427:
as the known condition (There may be other condition.) of
370:
with unit amplitude can be written as an oscillatory term
11076:
Zeitschrift für Kristallographie – New Crystal Structures
3915:
Mathematically, the reciprocal lattice is the set of all
23:
The computer-generated reciprocal lattice of a fictional
11071:
Zeitschrift für Kristallographie – Crystalline Materials
10228:(8th ed.). John Wiley & Sons, Inc. p. 44.
4500:
For an infinite two-dimensional lattice, defined by its
1503:
and with its adjacent wavefront (whose phase differs by
894:
is the position vector of a point in real space and now
10119:
Pages displaying short descriptions of redirect targets
10111: – Energy conservation during diffraction by atoms
10081:
has columns of vectors that describe the dual lattice.
9880:, in a different vector space (of the same dimension).
6954:{\displaystyle \mathbf {a} _{2}\times \mathbf {a} _{3}}
6734:. which changes the reciprocal primitive vectors to be
6727:{\displaystyle \mathbf {K} _{m}=\mathbf {G} _{m}/2\pi }
931:{\displaystyle \mathbf {k} =2\pi \mathbf {e} /\lambda }
342:-space) provides a way to visualize the results of the
10964:
10155:
9860:
is again a real vector space, and its closed subgroup
8767:
8736:
8716:
8696:
8513:
8461:
8426:
8394:
7648:
5121:
10074:{\displaystyle A=B\left(B^{\mathsf {T}}B\right)^{-1}}
10026:
9585:
9351:
9310:
9086:
9063:
8816:
8616:
8547:
8374:
8284:
8225:
8157:
8130:
8062:
8014:
7984:
7939:
7724:
7636:
7555:
7515:
7459:
7436:
7409:
7389:
7348:
7308:
7276:
7256:
7229:
7209:
7182:
7162:
7135:
7099:
7061:
6996:
6967:
6923:
6894:
6865:
6743:
6685:
6662:
6516:
6445:
6365:
6295:
6106:
5881:
5813:
5748:
5644:
5575:
5506:
5399:
5174:
5109:
5081:
5031:
5004:
4982:
4952:
4678:
4648:
4566:
4509:
4469:
4440:
4363:
4334:
4305:
4237:
4132:
4103:
4067:
4044:
4015:
3986:
3957:
3924:
3887:
3831:
3768:
3531:
3462:
3430:
3408:
3377:
3309:
3220:
3183:
3152:
3117:
3095:
3064:
3035:
3013:
2949:
2845:
2773:
2745:
2719:
2699:
2672:
2645:
2625:
2589:
2562:
2535:
2462:
2440:
2416:
2393:
2363:
2341:
2312:
2283:
2254:
2223:
2124:
2094:
2068:
2038:
1967:
1937:
1853:
1799:
1772:
1714:
1686:
1657:
1590:
1559:
1532:
1509:
1478:
1448:
1419:
1384:
1357:
1258:
1234:
1202:
1170:
1144:
1108:
1083:
1045:
1023:
1001:
972:
948:
900:
878:
856:
807:
741:
699:
664:
641:
620:
600:
578:
557:
537:
517:
495:
471:
440:
377:
359:), its reciprocal space will have inverse length, so
288:
258:
225:
191:
169:
136:
9050:(long-distance or lens back-focal-plane) limit as a
8494:
8450:
6643:{\displaystyle \left^{\mathsf {T}}=2\pi \left^{-1}.}
3402:
follows the periodicity of the lattice, translating
346:
of a spatial function. It is similar in role to the
35:
A two-dimensional crystal and its reciprocal lattice
9808:
7967:{\displaystyle \omega \colon V^{n}\to \mathbf {R} }
7715:they can be determined with the following formula:
10126: – Notation system for crystal lattice planes
10073:
9823:of the abstract dual lattice concept, for a given
9773:
9511:
9334:
9218:
9069:
9023:
8797:
8753:
8722:
8702:
8671:
8602:
8530:
8479:
8439:
8412:
8380:
8328:
8270:
8211:
8136:
8116:
8048:
8000:
7966:
7922:
7704:
7616:
7541:
7501:
7442:
7415:
7395:
7363:
7326:
7294:
7262:
7242:
7215:
7195:
7168:
7148:
7117:
7085:
7047:
6976:
6953:
6909:
6880:
6848:
6726:
6671:
6642:
6491:
6418:
6350:
6277:
6092:
5868:
5799:
5734:
5630:
5561:
5489:
5374:
5154:
5089:
5067:
5017:
4990:
4960:
4935:
4661:
4631:
4549:
4484:
4455:
4418:
4349:
4320:
4290:
4222:
4118:
4082:
4053:
4030:
4001:
3972:
3939:
3904:
3873:
3814:
3752:
3514:
3445:
3416:
3394:
3360:
3292:
3200:
3169:
3138:
3103:
3081:
3050:
3021:
2996:
2935:
2824:
2751:
2731:
2705:
2685:
2658:
2631:
2611:
2575:
2548:
2470:
2448:
2425:
2401:
2379:
2349:
2327:
2298:
2269:
2236:
2207:
2109:
2080:
2054:
2020:
1952:
1919:
1838:
1785:
1757:
1699:
1672:
1641:
1574:
1544:
1518:
1493:
1463:
1433:
1399:
1370:
1341:
1240:
1220:
1187:
1152:
1114:
1089:
1069:
1031:
1007:
986:
954:
930:
886:
862:
841:
791:
722:
682:
647:
626:
606:
584:
563:
543:
523:
501:
477:
453:
420:
297:
273:
240:
197:
177:
155:
67:associated with the arrangement of the atoms. The
9756:
9755:
9142:
8329:{\displaystyle R\in {\text{SO}}(2)\subset L(V,V)}
7542:{\displaystyle g\colon V\times V\to \mathbf {R} }
2997:{\displaystyle n_{1},n_{2},n_{3}\in \mathbb {Z} }
2357:on the reciprocal lattice, the total phase shift
966:(a plane of a constant phase) through the origin
551:(and the time-varying part as a function of both
11231:
10132: – Experimental method in X-ray diffraction
9805:) effects may be important to consider as well.
9033:
5631:{\displaystyle n=\left(n_{1},n_{2},n_{3}\right)}
10252:(Addison-Wesley, Reading MA/Dover, Mineola NY).
8008:is the inverse of the vector space isomorphism
2244:are integers. The reciprocal lattice is also a
156:{\displaystyle \mathbf {p} =\hbar \mathbf {k} }
47:, and plays a major role in many areas such as
10138: – High symmetry orientation of a crystal
10099: – Scientific study of crystal structures
9883:The other aspect is seen in the presence of a
2380:{\displaystyle \mathbf {G} \cdot \mathbf {R} }
421:{\displaystyle \cos(kx-\omega t+\varphi _{0})}
10308:
8212:{\displaystyle \omega (u,v,w)=g(u\times v,w)}
6500:
59:of electrons in a solid. It emerges from the
9952:In mathematics, the dual lattice of a given
9921:is not intrinsic; it depends on a choice of
11145:
10266:http://newton.umsl.edu/run//nano/known.html
9941:allows one to speak to the dual lattice to
8682:
8049:{\displaystyle {\hat {g}}\colon V\to V^{*}}
2520:
10315:
10301:
9757:
6431:for the reciprocal lattice derived in the
5393:For an infinite three-dimensional lattice
10169:
10149:
7907:
7903:
7874:
6402:
5344:
5295:
5248:
5204:
5059:
5037:
4913:
4878:
4825:
4790:
4757:
4719:
4492:results in the same reciprocal lattice.)
4274:
3895:
3716:
3015:
2990:
2004:
1378:are integers defining the vertex and the
511:it can be regarded as a function of both
320:
216:The reciprocal lattice is the set of all
126:, reciprocal space is closely related to
9037:
8679:, parallel to their real-space vectors.
2619:are plotted. From this we see that when
2524:
1583:. Its angular wavevector takes the form
324:
91:). The reciprocal lattice exists in the
30:
18:
8355:
8344:Reciprocal lattices of various crystals
8117:{\displaystyle {\hat {g}}(v)(w)=g(v,w)}
6432:
2409:on the direct lattice is a multiple of
11232:
10418:
10223:
10048:
9845:. It may be stated simply in terms of
9793:is the vector separation between atom
7403:dimensions can be derived assuming an
6566:
1442:contains the direct lattice points at
43:is a term associated with solids with
10296:
10208:
10117: – Patterns formed by scattering
7048:{\displaystyle m=(m_{1},m_{2},m_{3})}
5800:{\displaystyle m=(m_{1},m_{2},m_{3})}
3361:{\displaystyle m=(m_{1},m_{2},m_{3})}
3111:is a position vector from the origin
2825:{\displaystyle n=(n_{1},n_{2},n_{3})}
1126:
11209:
10549:Phase transformation crystallography
8271:{\displaystyle \omega (v,w)=g(Rv,w)}
7378:
6888:is just the reciprocal magnitude of
11056:Journal of Chemical Crystallography
10322:
10226:Introduction to Solid State Physics
9898:it allows an identification of the
9819:There are actually two versions in
5388:
4998:is the anti-clockwise rotation and
333:
211:covariant and contravariant vectors
13:
10115:Kikuchi line (solid state physics)
3905:{\displaystyle N\in \mathbb {Z} .}
3139:{\displaystyle \mathbf {R} _{n}=0}
1248:) at every direct lattice vertex.
14:
11281:
10259:
10188:10.1103/PhysRevMaterials.3.064003
8798:{\textstyle 4\pi /(a{\sqrt {3}})}
8495:Body-centered cubic (BCC) lattice
8451:Face-centered cubic (FCC) lattice
4495:
2387:between the origin and any point
1070:{\displaystyle \varphi +(2\pi )n}
145:
130:according to the proportionality
11208:
11197:
11196:
10803:
10002:linearly independent vectors in
9809:Generalization of a dual lattice
8654:
8639:
8624:
8585:
8570:
8555:
7960:
7883:
7848:
7814:
7793:
7727:
7579:
7564:
7535:
7486:
7465:
7364:{\displaystyle \mathbf {K} _{m}}
7351:
6941:
6926:
6910:{\displaystyle \mathbf {a} _{1}}
6897:
6881:{\displaystyle \mathbf {b} _{1}}
6868:
6828:
6813:
6793:
6779:
6764:
6746:
6703:
6688:
6613:
6601:
6589:
6549:
6537:
6525:
6471:
6456:
6437:multi-dimensional Fourier series
6383:
6368:
6333:
6318:
6307:
6303:
6260:
6245:
6225:
6205:
6190:
6170:
6150:
6135:
6115:
6076:
6061:
6024:
6008:
5993:
5956:
5940:
5925:
5888:
5851:
5836:
5825:
5821:
5722:
5697:
5672:
5647:
5544:
5529:
5518:
5514:
5477:
5452:
5427:
5402:
5347:
5336:
5321:
5298:
5287:
5251:
5244:
5230:
5207:
5200:
5177:
5083:
5061:
5051:
5039:
5033:
5007:
4984:
4954:
4916:
4909:
4895:
4881:
4874:
4847:
4828:
4821:
4807:
4793:
4786:
4760:
4753:
4739:
4722:
4715:
4685:
4619:
4594:
4569:
4532:
4517:
4485:{\displaystyle \mathbf {G} _{m}}
4472:
4456:{\displaystyle \mathbf {G} _{m}}
4443:
4401:
4386:
4375:
4371:
4350:{\displaystyle \mathbf {R} _{n}}
4337:
4321:{\displaystyle \mathbf {G} _{m}}
4308:
4255:
4240:
4210:
4185:
4160:
4135:
4119:{\displaystyle \mathbf {G} _{m}}
4106:
4095:multi-dimensional Fourier series
4083:{\displaystyle \mathbf {R} _{n}}
4070:
4031:{\displaystyle \mathbf {R} _{n}}
4018:
4002:{\displaystyle \mathbf {G} _{m}}
3989:
3973:{\displaystyle \mathbf {R} _{n}}
3960:
3940:{\displaystyle \mathbf {G} _{m}}
3927:
3849:
3834:
3794:
3779:
3741:
3727:
3704:
3689:
3641:
3632:
3615:
3576:
3562:
3502:
3479:
3470:
3446:{\displaystyle \mathbf {R} _{n}}
3433:
3410:
3385:
3282:
3265:
3251:
3210:multi-dimensional Fourier series
3191:
3160:
3120:
3097:
3072:
3051:{\displaystyle \mathbf {a} _{i}}
3038:
2923:
2898:
2873:
2848:
2464:
2442:
2395:
2373:
2365:
2343:
2328:{\displaystyle \mathbf {b} _{3}}
2315:
2299:{\displaystyle \mathbf {b} _{2}}
2286:
2270:{\displaystyle \mathbf {b} _{1}}
2257:
2195:
2170:
2145:
2126:
2110:{\displaystyle \mathbf {b} _{j}}
2097:
1985:
1970:
1953:{\displaystyle \mathbf {b} _{j}}
1940:
1901:
1886:
1871:
1856:
1817:
1802:
1745:
1730:
1673:{\displaystyle \mathbf {e} _{1}}
1660:
1614:
1593:
1575:{\displaystyle \mathbf {a} _{1}}
1562:
1494:{\displaystyle \mathbf {a} _{3}}
1481:
1464:{\displaystyle \mathbf {a} _{2}}
1451:
1421:
1400:{\displaystyle \mathbf {a} _{i}}
1387:
1329:
1304:
1279:
1260:
1178:
1146:
1025:
974:
916:
902:
880:
826:
818:
760:
752:
723:{\displaystyle k=2\pi /\lambda }
274:{\displaystyle \mathbf {R} _{n}}
261:
241:{\displaystyle \mathbf {G} _{m}}
228:
171:
149:
138:
9876:^ is the natural candidate for
9279:is the vector position of atom
8480:{\textstyle {\frac {4\pi }{a}}}
8413:{\textstyle {\frac {2\pi }{a}}}
3395:{\displaystyle f(\mathbf {r} )}
3201:{\displaystyle f(\mathbf {r} )}
3170:{\displaystyle f(\mathbf {r} )}
3082:{\displaystyle f(\mathbf {r} )}
1188:{\displaystyle f(\mathbf {r} )}
683:{\displaystyle k\lambda =2\pi }
11260:Synchrotron-related techniques
10998:Bilbao Crystallographic Server
10242:
10217:
10202:
10105: – Linear algebra concept
9748:
9732:
9702:
9672:
9604:
9598:
9589:
9329:
9311:
9199:
9183:
9153:
9105:
9099:
9090:
9008:
8969:
8941:
8887:
8859:
8805:rotated through 90° about the
8792:
8779:
8323:
8311:
8302:
8296:
8265:
8250:
8241:
8229:
8206:
8188:
8179:
8161:
8111:
8099:
8090:
8084:
8081:
8075:
8069:
8033:
8021:
7956:
7911:
7843:
7824:
7788:
7589:
7559:
7531:
7496:
7460:
7321:
7309:
7289:
7277:
7112:
7100:
7080:
7062:
7042:
7003:
5794:
5755:
5569:and the subscript of integers
5361:
5355:
5312:
5306:
5265:
5259:
5221:
5215:
4434:. (There may be other form of
4009:satisfy this equality for all
3651:
3628:
3506:
3498:
3489:
3466:
3389:
3381:
3355:
3316:
3195:
3187:
3164:
3156:
3076:
3068:
2819:
2780:
1434:{\displaystyle \mathbf {R} =0}
1212:
1203:
1182:
1174:
1061:
1052:
987:{\displaystyle \mathbf {r} =0}
836:
814:
786:
748:
415:
384:
338:Reciprocal space (also called
1:
10142:
10013:Furthermore, if we allow the
9868:turns out to be a lattice in
9034:Arbitrary collection of atoms
8507:lattice, with a cube side of
7055:is conventionally written as
6429:primitive translation vectors
5018:{\displaystyle \mathbf {Q'} }
4126:can be chosen in the form of
3453:we get the same value, hence
2763:Assuming a three-dimensional
2612:{\displaystyle e^{iG\cdot r}}
8499:The reciprocal lattice to a
8348:Reciprocal lattices for the
5090:{\displaystyle \mathbf {v} }
4991:{\displaystyle \mathbf {Q} }
4961:{\displaystyle \mathbf {Q} }
3417:{\displaystyle \mathbf {r} }
3104:{\displaystyle \mathbf {r} }
3022:{\displaystyle \mathbb {Z} }
2471:{\displaystyle \mathbf {G} }
2449:{\displaystyle \mathbf {G} }
2402:{\displaystyle \mathbf {R} }
2350:{\displaystyle \mathbf {G} }
2055:{\displaystyle \delta _{ij}}
1786:{\displaystyle \lambda _{1}}
1700:{\displaystyle \lambda _{1}}
1153:{\displaystyle \mathbf {k} }
1032:{\displaystyle \mathbf {e} }
887:{\displaystyle \mathbf {r} }
454:{\displaystyle \varphi _{0}}
363:(the reciprocal of length).
178:{\displaystyle \mathbf {p} }
7:
11046:Crystal Growth & Design
10338:Timeline of crystallography
10084:
9335:{\displaystyle (h,k,\ell )}
9245:in crystallographer units,
9241:) is the scattering vector
8440:{\textstyle {\frac {1}{a}}}
7086:{\displaystyle (h,k,\ell )}
5742:with the integer subscript
5025:is the clockwise rotation,
4061:) at all the lattice point
3368:, so this is a triple sum.
3029:is the set of integers and
2088:and is zero otherwise. The
594:This complementary role of
317:of the reciprocal lattice.
185:is the momentum vector and
55:diffraction as well as the
10:
11286:
11255:Neutron-related techniques
10857:Nuclear magnetic resonance
9812:
8137:{\displaystyle \lrcorner }
2713:(shown orange) to a point
11192:
11112:
11084:
11061:Journal of Crystal Growth
11036:
10988:
10935:
10882:
10813:
10801:
10596:
10587:
10510:
10363:
10330:
10248:B. E. Warren (1969/1990)
10158:Physical Review Materials
7327:{\displaystyle (hk\ell )}
7295:{\displaystyle (hk\ell )}
7118:{\displaystyle (hk\ell )}
6961:, dropping the factor of
5638:, its reciprocal lattice
11270:Condensed matter physics
10927:Single particle analysis
10785:Hermann–Mauguin notation
10224:Kittel, Charles (2005).
9260:atomic scattering factor
9249:is the number of atoms,
8683:Simple hexagonal lattice
6433:heuristic approach above
4093:As shown in the section
3822:, which only holds when
3303:where now the subscript
2521:Mathematical description
1221:{\displaystyle (2\pi )n}
1115:{\displaystyle \lambda }
955:{\displaystyle \varphi }
648:{\displaystyle \lambda }
11051:Crystallography Reviews
10895:Isomorphous replacement
10689:Lomer–Cottrell junction
10213:. Springer. p. 69.
10209:Audin, Michèle (2003).
9830:in a real vector space
8219:and in two dimensions,
6499:mathematically derived
4968:represents a 90 degree
564:{\displaystyle \omega }
502:{\displaystyle \omega }
207:reduced Planck constant
107:, which is the dual of
10564:Spinodal decomposition
10075:
9775:
9651:
9630:
9513:
9397:
9336:
9266:and scattering vector
9220:
9131:
9071:
9043:
9025:
8799:
8755:
8724:
8704:
8673:
8604:
8532:
8481:
8441:
8414:
8382:
8330:
8272:
8213:
8138:
8118:
8050:
8002:
8001:{\displaystyle g^{-1}}
7968:
7924:
7706:
7618:
7543:
7503:
7444:
7417:
7397:
7365:
7328:
7296:
7264:
7244:
7217:
7197:
7170:
7150:
7119:
7087:
7049:
6978:
6955:
6911:
6882:
6850:
6728:
6673:
6644:
6493:
6420:
6352:
6289:. The choice of these
6279:
6094:
5870:
5801:
5736:
5632:
5563:
5491:
5376:
5156:
5091:
5069:
5019:
4992:
4962:
4937:
4663:
4633:
4551:
4486:
4457:
4420:
4351:
4322:
4292:
4224:
4120:
4084:
4055:
4032:
4003:
3974:
3941:
3906:
3875:
3816:
3754:
3516:
3447:
3424:by any lattice vector
3418:
3396:
3362:
3294:
3202:
3171:
3140:
3105:
3083:
3052:
3023:
2998:
2937:
2826:
2760:
2753:
2733:
2707:
2687:
2660:
2633:
2613:
2577:
2550:
2472:
2450:
2427:
2403:
2381:
2351:
2329:
2300:
2271:
2238:
2209:
2111:
2082:
2056:
2022:
1954:
1921:
1840:
1787:
1759:
1701:
1674:
1643:
1576:
1546:
1545:{\displaystyle -2\pi }
1520:
1495:
1465:
1435:
1401:
1372:
1343:
1242:
1222:
1189:
1160:of plane waves in the
1154:
1116:
1091:
1071:
1033:
1009:
988:
956:
932:
888:
864:
843:
793:
724:
684:
649:
628:
608:
586:
565:
545:
525:
503:
479:
455:
422:
330:
321:Wave-based description
299:
275:
242:
199:
198:{\displaystyle \hbar }
179:
157:
45:translational symmetry
36:
28:
11104:Gregori Aminoff Prize
10900:Molecular replacement
10076:
9945:while staying within
9776:
9631:
9610:
9514:
9377:
9337:
9221:
9111:
9072:
9041:
9026:
8800:
8756:
8725:
8705:
8674:
8605:
8533:
8482:
8442:
8415:
8383:
8331:
8273:
8214:
8139:
8119:
8051:
8003:
7969:
7925:
7707:
7619:
7544:
7509:and an inner product
7504:
7445:
7418:
7398:
7366:
7329:
7297:
7270:. Each lattice point
7265:
7263:{\displaystyle \ell }
7245:
7243:{\displaystyle m_{3}}
7218:
7198:
7196:{\displaystyle m_{2}}
7171:
7151:
7149:{\displaystyle m_{1}}
7120:
7088:
7050:
6979:
6977:{\displaystyle 2\pi }
6956:
6912:
6883:
6851:
6729:
6674:
6672:{\displaystyle 2\pi }
6645:
6494:
6421:
6353:
6287:scalar triple product
6280:
6095:
5871:
5802:
5737:
5633:
5564:
5492:
5377:
5157:
5092:
5070:
5020:
4993:
4963:
4938:
4664:
4662:{\displaystyle m_{i}}
4634:
4552:
4487:
4458:
4421:
4352:
4323:
4293:
4225:
4121:
4085:
4056:
4054:{\displaystyle 2\pi }
4033:
4004:
3975:
3942:
3907:
3876:
3817:
3755:
3517:
3448:
3419:
3397:
3363:
3295:
3203:
3172:
3141:
3106:
3084:
3053:
3024:
2999:
2938:
2827:
2754:
2734:
2708:
2688:
2686:{\displaystyle b_{2}}
2661:
2659:{\displaystyle b_{1}}
2634:
2614:
2578:
2576:{\displaystyle a_{2}}
2551:
2549:{\displaystyle a_{1}}
2528:
2489:(more specifically a
2473:
2451:
2428:
2426:{\displaystyle 2\pi }
2404:
2382:
2352:
2330:
2301:
2272:
2239:
2237:{\displaystyle m_{j}}
2210:
2112:
2083:
2057:
2023:
1955:
1922:
1841:
1788:
1760:
1702:
1675:
1644:
1577:
1547:
1521:
1519:{\displaystyle 2\pi }
1496:
1466:
1436:
1402:
1373:
1371:{\displaystyle n_{i}}
1344:
1243:
1223:
1190:
1155:
1117:
1092:
1072:
1034:
1010:
989:
957:
933:
889:
865:
844:
794:
725:
685:
650:
629:
609:
587:
566:
546:
526:
504:
480:
456:
423:
368:sinusoidal plane wave
328:
300:
298:{\displaystyle 2\pi }
276:
243:
200:
180:
158:
34:
22:
10410:Structure prediction
10024:
9993:discrete mathematics
9925:(volume element) on
9583:
9349:
9308:
9084:
9061:
8814:
8765:
8754:{\textstyle 2\pi /c}
8734:
8714:
8694:
8614:
8545:
8531:{\textstyle 4\pi /a}
8511:
8459:
8424:
8392:
8372:
8356:Simple cubic lattice
8350:cubic crystal system
8282:
8223:
8155:
8146:inner multiplication
8128:
8060:
8012:
7982:
7937:
7722:
7634:
7553:
7513:
7457:
7434:
7407:
7387:
7346:
7306:
7274:
7254:
7227:
7207:
7180:
7160:
7133:
7097:
7059:
6994:
6965:
6921:
6917:in the direction of
6892:
6863:
6741:
6683:
6660:
6514:
6443:
6363:
6293:
6104:
5879:
5811:
5746:
5642:
5573:
5504:
5397:
5172:
5107:
5079:
5029:
5002:
4980:
4950:
4676:
4646:
4564:
4507:
4467:
4463:. Any valid form of
4438:
4430:of their respective
4361:
4332:
4303:
4235:
4130:
4101:
4065:
4042:
4013:
3984:
3955:
3922:
3885:
3829:
3766:
3529:
3460:
3428:
3406:
3375:
3307:
3218:
3181:
3150:
3146:to any position, if
3115:
3093:
3062:
3033:
3011:
2947:
2843:
2771:
2743:
2717:
2697:
2670:
2643:
2623:
2587:
2560:
2533:
2460:
2438:
2414:
2391:
2361:
2339:
2310:
2281:
2252:
2221:
2122:
2092:
2066:
2036:
1965:
1935:
1851:
1797:
1770:
1712:
1684:
1655:
1588:
1557:
1530:
1507:
1476:
1446:
1417:
1409:linearly independent
1382:
1355:
1256:
1232:
1200:
1168:
1142:
1106:
1081:
1043:
1021:
999:
970:
962:is the phase of the
946:
898:
876:
854:
805:
801:which simplifies to
739:
697:
662:
639:
618:
598:
576:
555:
535:
515:
493:
469:
438:
375:
286:
256:
223:
189:
167:
134:
10674:Cottrell atmosphere
10654:Partial dislocation
10398:Restriction theorem
10180:2019PhRvM...3f4003S
2732:{\displaystyle R+r}
2495:solid state physics
2081:{\displaystyle i=j}
97:spatial frequencies
11094:Carl Hermann Medal
10905:Molecular dynamics
10752:Defects in diamond
10747:Stone–Wales defect
10393:Reciprocal lattice
10355:Biocrystallography
10271:2020-08-31 at the
10130:Powder diffraction
10071:
9937:, the presence of
9913:. The relation of
9847:Pontryagin duality
9771:
9509:
9332:
9216:
9067:
9044:
9021:
9019:
8795:
8751:
8720:
8700:
8669:
8600:
8528:
8477:
8437:
8410:
8378:
8326:
8268:
8209:
8134:
8114:
8046:
7998:
7964:
7920:
7702:
7693:
7614:
7539:
7499:
7440:
7413:
7393:
7361:
7324:
7292:
7260:
7240:
7213:
7193:
7166:
7146:
7115:
7083:
7045:
6974:
6951:
6907:
6878:
6846:
6724:
6669:
6640:
6489:
6416:
6348:
6275:
6090:
6088:
5866:
5797:
5732:
5628:
5559:
5487:
5372:
5152:
5146:
5097:. Thus, using the
5087:
5065:
5015:
4988:
4958:
4933:
4931:
4669:is an integer and
4659:
4629:
4547:
4482:
4453:
4428:Pontryagin duality
4416:
4347:
4318:
4288:
4220:
4116:
4080:
4051:
4028:
3999:
3970:
3937:
3902:
3871:
3812:
3750:
3668:
3594:
3541:
3512:
3443:
3414:
3392:
3358:
3290:
3230:
3198:
3167:
3136:
3101:
3079:
3048:
3019:
2994:
2933:
2822:
2761:
2749:
2729:
2703:
2683:
2656:
2629:
2609:
2573:
2546:
2468:
2446:
2423:
2399:
2377:
2347:
2325:
2296:
2267:
2234:
2205:
2107:
2078:
2052:
2018:
1950:
1917:
1836:
1783:
1755:
1697:
1670:
1639:
1572:
1542:
1516:
1491:
1461:
1431:
1397:
1368:
1339:
1238:
1218:
1185:
1150:
1127:Reciprocal lattice
1112:
1087:
1067:
1029:
1005:
984:
952:
928:
884:
860:
839:
789:
720:
680:
645:
624:
604:
582:
561:
541:
521:
499:
475:
464:angular wavenumber
451:
418:
331:
295:
271:
238:
195:
175:
153:
93:mathematical space
41:reciprocal lattice
37:
29:
11224:
11223:
11188:
11187:
10795:Thermal ellipsoid
10760:
10759:
10669:Frank–Read source
10629:
10628:
10495:Aperiodic crystal
10461:
10460:
10343:Crystallographers
10250:X-ray diffraction
9967:topological group
9751:
9735:
9705:
9675:
9601:
9202:
9186:
9156:
9102:
9070:{\displaystyle F}
9056:complex amplitude
9011:
8972:
8958:
8944:
8930:
8926:
8890:
8876:
8862:
8848:
8844:
8790:
8689:lattice constants
8475:
8435:
8408:
8381:{\displaystyle a}
8360:The simple cubic
8294:
8072:
8024:
7828:
7443:{\displaystyle V}
7416:{\displaystyle n}
7396:{\displaystyle n}
7379:Higher dimensions
7373:reciprocal length
7216:{\displaystyle k}
7169:{\displaystyle h}
7156:is replaced with
6986:spatial frequency
6844:
6058:
6054:
5990:
5986:
5922:
5918:
5499:primitive vectors
5497:, defined by its
5367:
5271:
4927:
4839:
4771:
4502:primitive vectors
3659:
3585:
3532:
3221:
2752:{\displaystyle R}
2706:{\displaystyle r}
2632:{\displaystyle G}
2491:Wigner–Seitz cell
1241:{\displaystyle n}
1090:{\displaystyle n}
1008:{\displaystyle t}
863:{\displaystyle t}
627:{\displaystyle x}
607:{\displaystyle k}
585:{\displaystyle t}
544:{\displaystyle x}
524:{\displaystyle k}
487:angular frequency
478:{\displaystyle k}
352:Fourier transform
344:Fourier transform
315:Wigner–Seitz cell
77:periodic function
61:Fourier transform
11277:
11245:Fourier analysis
11212:
11211:
11200:
11199:
11143:
11142:
11066:Kristallografija
10920:Gerchberg–Saxton
10815:Characterisation
10807:
10790:Structure factor
10594:
10593:
10579:Ostwald ripening
10416:
10415:
10361:
10360:
10317:
10310:
10303:
10294:
10293:
10253:
10246:
10240:
10239:
10221:
10215:
10214:
10206:
10200:
10199:
10173:
10153:
10120:
10080:
10078:
10077:
10072:
10070:
10069:
10061:
10057:
10053:
10052:
10051:
10001:
9972:is the subgroup
9843:Fourier analysis
9836:finite dimension
9780:
9778:
9777:
9772:
9767:
9766:
9765:
9764:
9753:
9752:
9744:
9737:
9736:
9728:
9711:
9707:
9706:
9698:
9691:
9690:
9681:
9677:
9676:
9668:
9661:
9660:
9650:
9645:
9629:
9624:
9603:
9602:
9594:
9518:
9516:
9515:
9510:
9508:
9507:
9506:
9502:
9501:
9500:
9485:
9484:
9469:
9468:
9437:
9433:
9432:
9407:
9406:
9396:
9391:
9373:
9372:
9341:
9339:
9338:
9333:
9240:
9225:
9223:
9222:
9217:
9212:
9211:
9210:
9209:
9204:
9203:
9195:
9188:
9187:
9179:
9162:
9158:
9157:
9149:
9141:
9140:
9130:
9125:
9104:
9103:
9095:
9076:
9074:
9073:
9068:
9030:
9028:
9027:
9022:
9020:
9013:
9012:
9004:
8991:
8990:
8974:
8973:
8965:
8959:
8951:
8946:
8945:
8937:
8931:
8922:
8921:
8909:
8908:
8892:
8891:
8883:
8877:
8869:
8864:
8863:
8855:
8849:
8840:
8839:
8830:
8829:
8804:
8802:
8801:
8796:
8791:
8786:
8778:
8760:
8758:
8757:
8752:
8747:
8729:
8727:
8726:
8721:
8709:
8707:
8706:
8701:
8678:
8676:
8675:
8670:
8668:
8664:
8663:
8662:
8657:
8648:
8647:
8642:
8633:
8632:
8627:
8609:
8607:
8606:
8601:
8599:
8595:
8594:
8593:
8588:
8579:
8578:
8573:
8564:
8563:
8558:
8537:
8535:
8534:
8529:
8524:
8486:
8484:
8483:
8478:
8476:
8471:
8463:
8446:
8444:
8443:
8438:
8436:
8428:
8419:
8417:
8416:
8411:
8409:
8404:
8396:
8387:
8385:
8384:
8379:
8352:are as follows.
8335:
8333:
8332:
8327:
8295:
8292:
8277:
8275:
8274:
8269:
8218:
8216:
8215:
8210:
8143:
8141:
8140:
8135:
8123:
8121:
8120:
8115:
8074:
8073:
8065:
8055:
8053:
8052:
8047:
8045:
8044:
8026:
8025:
8017:
8007:
8005:
8004:
7999:
7997:
7996:
7973:
7971:
7970:
7965:
7963:
7955:
7954:
7929:
7927:
7926:
7921:
7902:
7901:
7897:
7896:
7886:
7873:
7872:
7868:
7867:
7851:
7842:
7841:
7829:
7827:
7823:
7822:
7817:
7802:
7801:
7796:
7783:
7782:
7778:
7777:
7762:
7761:
7747:
7736:
7735:
7730:
7711:
7709:
7708:
7703:
7698:
7697:
7623:
7621:
7620:
7615:
7613:
7612:
7588:
7587:
7582:
7573:
7572:
7567:
7548:
7546:
7545:
7540:
7538:
7508:
7506:
7505:
7500:
7495:
7494:
7489:
7474:
7473:
7468:
7449:
7447:
7446:
7441:
7422:
7420:
7419:
7414:
7402:
7400:
7399:
7394:
7383:The formula for
7370:
7368:
7367:
7362:
7360:
7359:
7354:
7333:
7331:
7330:
7325:
7301:
7299:
7298:
7293:
7269:
7267:
7266:
7261:
7249:
7247:
7246:
7241:
7239:
7238:
7222:
7220:
7219:
7214:
7202:
7200:
7199:
7194:
7192:
7191:
7175:
7173:
7172:
7167:
7155:
7153:
7152:
7147:
7145:
7144:
7124:
7122:
7121:
7116:
7092:
7090:
7089:
7084:
7054:
7052:
7051:
7046:
7041:
7040:
7028:
7027:
7015:
7014:
6983:
6981:
6980:
6975:
6960:
6958:
6957:
6952:
6950:
6949:
6944:
6935:
6934:
6929:
6916:
6914:
6913:
6908:
6906:
6905:
6900:
6887:
6885:
6884:
6879:
6877:
6876:
6871:
6855:
6853:
6852:
6847:
6845:
6843:
6842:
6838:
6837:
6836:
6831:
6822:
6821:
6816:
6802:
6801:
6796:
6789:
6788:
6787:
6782:
6773:
6772:
6767:
6760:
6755:
6754:
6749:
6733:
6731:
6730:
6725:
6717:
6712:
6711:
6706:
6697:
6696:
6691:
6678:
6676:
6675:
6670:
6649:
6647:
6646:
6641:
6636:
6635:
6627:
6623:
6622:
6621:
6616:
6610:
6609:
6604:
6598:
6597:
6592:
6571:
6570:
6569:
6563:
6559:
6558:
6557:
6552:
6546:
6545:
6540:
6534:
6533:
6528:
6505:matrix inversion
6498:
6496:
6495:
6490:
6482:
6481:
6480:
6479:
6474:
6465:
6464:
6459:
6435:and the section
6426:
6425:
6423:
6422:
6417:
6415:
6414:
6392:
6391:
6386:
6377:
6376:
6371:
6357:
6355:
6354:
6349:
6347:
6343:
6342:
6341:
6336:
6327:
6326:
6321:
6312:
6311:
6310:
6284:
6282:
6281:
6276:
6274:
6270:
6269:
6268:
6263:
6254:
6253:
6248:
6234:
6233:
6228:
6219:
6215:
6214:
6213:
6208:
6199:
6198:
6193:
6179:
6178:
6173:
6164:
6160:
6159:
6158:
6153:
6144:
6143:
6138:
6124:
6123:
6118:
6099:
6097:
6096:
6091:
6089:
6085:
6084:
6079:
6070:
6069:
6064:
6056:
6055:
6050:
6042:
6033:
6032:
6027:
6017:
6016:
6011:
6002:
6001:
5996:
5988:
5987:
5982:
5974:
5965:
5964:
5959:
5949:
5948:
5943:
5934:
5933:
5928:
5920:
5919:
5914:
5906:
5897:
5896:
5891:
5875:
5873:
5872:
5867:
5865:
5861:
5860:
5859:
5854:
5845:
5844:
5839:
5830:
5829:
5828:
5806:
5804:
5803:
5798:
5793:
5792:
5780:
5779:
5767:
5766:
5741:
5739:
5738:
5733:
5731:
5730:
5725:
5719:
5718:
5706:
5705:
5700:
5694:
5693:
5681:
5680:
5675:
5669:
5668:
5656:
5655:
5650:
5637:
5635:
5634:
5629:
5627:
5623:
5622:
5621:
5609:
5608:
5596:
5595:
5568:
5566:
5565:
5560:
5558:
5554:
5553:
5552:
5547:
5538:
5537:
5532:
5523:
5522:
5521:
5496:
5494:
5493:
5488:
5486:
5485:
5480:
5474:
5473:
5461:
5460:
5455:
5449:
5448:
5436:
5435:
5430:
5424:
5423:
5411:
5410:
5405:
5389:Three dimensions
5381:
5379:
5378:
5373:
5368:
5366:
5365:
5364:
5350:
5343:
5339:
5330:
5329:
5324:
5317:
5316:
5315:
5301:
5294:
5290:
5283:
5272:
5270:
5269:
5268:
5254:
5247:
5239:
5238:
5233:
5226:
5225:
5224:
5210:
5203:
5197:
5186:
5185:
5180:
5161:
5159:
5158:
5153:
5151:
5150:
5096:
5094:
5093:
5088:
5086:
5075:for all vectors
5074:
5072:
5071:
5066:
5064:
5058:
5057:
5042:
5036:
5024:
5022:
5021:
5016:
5014:
5013:
4997:
4995:
4994:
4989:
4987:
4967:
4965:
4964:
4959:
4957:
4942:
4940:
4939:
4934:
4932:
4928:
4926:
4925:
4924:
4919:
4912:
4904:
4903:
4898:
4891:
4890:
4889:
4884:
4877:
4871:
4856:
4855:
4850:
4840:
4838:
4837:
4836:
4831:
4824:
4816:
4815:
4810:
4803:
4802:
4801:
4796:
4789:
4783:
4772:
4770:
4769:
4768:
4763:
4756:
4748:
4747:
4742:
4732:
4731:
4730:
4725:
4718:
4709:
4694:
4693:
4688:
4668:
4666:
4665:
4660:
4658:
4657:
4638:
4636:
4635:
4630:
4628:
4627:
4622:
4616:
4615:
4603:
4602:
4597:
4591:
4590:
4578:
4577:
4572:
4556:
4554:
4553:
4548:
4546:
4542:
4541:
4540:
4535:
4526:
4525:
4520:
4491:
4489:
4488:
4483:
4481:
4480:
4475:
4462:
4460:
4459:
4454:
4452:
4451:
4446:
4425:
4423:
4422:
4417:
4415:
4411:
4410:
4409:
4404:
4395:
4394:
4389:
4380:
4379:
4378:
4356:
4354:
4353:
4348:
4346:
4345:
4340:
4327:
4325:
4324:
4319:
4317:
4316:
4311:
4298:
4297:
4295:
4294:
4289:
4287:
4286:
4264:
4263:
4258:
4249:
4248:
4243:
4229:
4227:
4226:
4221:
4219:
4218:
4213:
4207:
4206:
4194:
4193:
4188:
4182:
4181:
4169:
4168:
4163:
4157:
4156:
4144:
4143:
4138:
4125:
4123:
4122:
4117:
4115:
4114:
4109:
4089:
4087:
4086:
4081:
4079:
4078:
4073:
4060:
4058:
4057:
4052:
4037:
4035:
4034:
4029:
4027:
4026:
4021:
4008:
4006:
4005:
4000:
3998:
3997:
3992:
3979:
3977:
3976:
3971:
3969:
3968:
3963:
3946:
3944:
3943:
3938:
3936:
3935:
3930:
3911:
3909:
3908:
3903:
3898:
3880:
3878:
3877:
3872:
3858:
3857:
3852:
3843:
3842:
3837:
3821:
3819:
3818:
3813:
3805:
3804:
3803:
3802:
3797:
3788:
3787:
3782:
3759:
3757:
3756:
3751:
3746:
3745:
3744:
3736:
3735:
3730:
3715:
3714:
3713:
3712:
3707:
3698:
3697:
3692:
3678:
3677:
3667:
3655:
3654:
3650:
3649:
3644:
3635:
3624:
3623:
3618:
3604:
3603:
3593:
3581:
3580:
3579:
3571:
3570:
3565:
3551:
3550:
3540:
3521:
3519:
3518:
3513:
3505:
3488:
3487:
3482:
3473:
3452:
3450:
3449:
3444:
3442:
3441:
3436:
3423:
3421:
3420:
3415:
3413:
3401:
3399:
3398:
3393:
3388:
3367:
3365:
3364:
3359:
3354:
3353:
3341:
3340:
3328:
3327:
3299:
3297:
3296:
3291:
3289:
3285:
3270:
3269:
3268:
3260:
3259:
3254:
3240:
3239:
3229:
3207:
3205:
3204:
3199:
3194:
3176:
3174:
3173:
3168:
3163:
3145:
3143:
3142:
3137:
3129:
3128:
3123:
3110:
3108:
3107:
3102:
3100:
3088:
3086:
3085:
3080:
3075:
3057:
3055:
3054:
3049:
3047:
3046:
3041:
3028:
3026:
3025:
3020:
3018:
3003:
3001:
3000:
2995:
2993:
2985:
2984:
2972:
2971:
2959:
2958:
2942:
2940:
2939:
2934:
2932:
2931:
2926:
2920:
2919:
2907:
2906:
2901:
2895:
2894:
2882:
2881:
2876:
2870:
2869:
2857:
2856:
2851:
2831:
2829:
2828:
2823:
2818:
2817:
2805:
2804:
2792:
2791:
2758:
2756:
2755:
2750:
2738:
2736:
2735:
2730:
2712:
2710:
2709:
2704:
2692:
2690:
2689:
2684:
2682:
2681:
2665:
2663:
2662:
2657:
2655:
2654:
2638:
2636:
2635:
2630:
2618:
2616:
2615:
2610:
2608:
2607:
2582:
2580:
2579:
2574:
2572:
2571:
2555:
2553:
2552:
2547:
2545:
2544:
2503:pure mathematics
2477:
2475:
2474:
2469:
2467:
2455:
2453:
2452:
2447:
2445:
2433:
2432:
2430:
2429:
2424:
2408:
2406:
2405:
2400:
2398:
2386:
2384:
2383:
2378:
2376:
2368:
2356:
2354:
2353:
2348:
2346:
2334:
2332:
2331:
2326:
2324:
2323:
2318:
2305:
2303:
2302:
2297:
2295:
2294:
2289:
2276:
2274:
2273:
2268:
2266:
2265:
2260:
2243:
2241:
2240:
2235:
2233:
2232:
2216:
2214:
2212:
2211:
2206:
2204:
2203:
2198:
2192:
2191:
2179:
2178:
2173:
2167:
2166:
2154:
2153:
2148:
2142:
2141:
2129:
2116:
2114:
2113:
2108:
2106:
2105:
2100:
2087:
2085:
2084:
2079:
2062:equals one when
2061:
2059:
2058:
2053:
2051:
2050:
2028:
2027:
2025:
2024:
2019:
2017:
2016:
1994:
1993:
1988:
1979:
1978:
1973:
1959:
1957:
1956:
1951:
1949:
1948:
1943:
1928:
1926:
1924:
1923:
1918:
1910:
1909:
1904:
1895:
1894:
1889:
1880:
1879:
1874:
1865:
1864:
1859:
1845:
1843:
1842:
1837:
1826:
1825:
1820:
1811:
1810:
1805:
1792:
1790:
1789:
1784:
1782:
1781:
1765:
1764:
1762:
1761:
1756:
1754:
1753:
1748:
1739:
1738:
1733:
1724:
1723:
1706:
1704:
1703:
1698:
1696:
1695:
1679:
1677:
1676:
1671:
1669:
1668:
1663:
1650:
1648:
1646:
1645:
1640:
1638:
1637:
1628:
1623:
1622:
1617:
1602:
1601:
1596:
1582:
1581:
1579:
1578:
1573:
1571:
1570:
1565:
1551:
1549:
1548:
1543:
1525:
1523:
1522:
1517:
1502:
1500:
1498:
1497:
1492:
1490:
1489:
1484:
1470:
1468:
1467:
1462:
1460:
1459:
1454:
1441:
1440:
1438:
1437:
1432:
1424:
1406:
1404:
1403:
1398:
1396:
1395:
1390:
1377:
1375:
1374:
1369:
1367:
1366:
1350:
1348:
1346:
1345:
1340:
1338:
1337:
1332:
1326:
1325:
1313:
1312:
1307:
1301:
1300:
1288:
1287:
1282:
1276:
1275:
1263:
1247:
1245:
1244:
1239:
1228:with an integer
1227:
1225:
1224:
1219:
1194:
1192:
1191:
1186:
1181:
1164:of any function
1159:
1157:
1156:
1151:
1149:
1123:
1121:
1119:
1118:
1113:
1096:
1094:
1093:
1088:
1076:
1074:
1073:
1068:
1038:
1036:
1035:
1030:
1028:
1016:
1014:
1012:
1011:
1006:
993:
991:
990:
985:
977:
961:
959:
958:
953:
937:
935:
934:
929:
924:
919:
905:
893:
891:
890:
885:
883:
871:
869:
867:
866:
861:
849:at a fixed time
848:
846:
845:
840:
829:
821:
800:
798:
796:
795:
790:
785:
784:
763:
755:
731:
729:
727:
726:
721:
716:
691:
689:
687:
686:
681:
656:
654:
652:
651:
646:
633:
631:
630:
625:
613:
611:
610:
605:
593:
591:
589:
588:
583:
570:
568:
567:
562:
550:
548:
547:
542:
530:
528:
527:
522:
510:
508:
506:
505:
500:
484:
482:
481:
476:
462:
460:
458:
457:
452:
450:
449:
429:
427:
425:
424:
419:
414:
413:
348:frequency domain
341:
334:Reciprocal space
305:
304:
302:
301:
296:
280:
278:
277:
272:
270:
269:
264:
247:
245:
244:
239:
237:
236:
231:
213:, respectively.
204:
202:
201:
196:
184:
182:
181:
176:
174:
162:
160:
159:
154:
152:
141:
101:reciprocal space
11285:
11284:
11280:
11279:
11278:
11276:
11275:
11274:
11240:Crystallography
11230:
11229:
11227:
11225:
11220:
11184:
11141:
11108:
11080:
11032:
10984:
10955:CrystalExplorer
10931:
10915:Phase retrieval
10878:
10809:
10808:
10799:
10756:
10735:Schottky defect
10634:Perfect crystal
10625:
10621:Abnormal growth
10583:
10569:Supersaturation
10532:Miscibility gap
10513:
10506:
10457:
10414:
10378:Bravais lattice
10359:
10326:
10324:Crystallography
10321:
10273:Wayback Machine
10262:
10257:
10256:
10247:
10243:
10236:
10222:
10218:
10207:
10203:
10154:
10150:
10145:
10118:
10097:Crystallography
10087:
10062:
10047:
10046:
10042:
10041:
10037:
10036:
10025:
10022:
10021:
9996:
9964:locally compact
9817:
9811:
9792:
9754:
9743:
9742:
9741:
9727:
9726:
9716:
9712:
9697:
9696:
9692:
9686:
9682:
9667:
9666:
9662:
9656:
9652:
9646:
9635:
9625:
9614:
9593:
9592:
9584:
9581:
9580:
9556:
9547:
9538:
9526: = 1,
9522:when there are
9496:
9492:
9480:
9476:
9464:
9460:
9456:
9452:
9442:
9438:
9416:
9412:
9408:
9402:
9398:
9392:
9381:
9356:
9352:
9350:
9347:
9346:
9309:
9306:
9305:
9298:
9278:
9257:
9238:
9205:
9194:
9193:
9192:
9178:
9177:
9167:
9163:
9148:
9147:
9143:
9136:
9132:
9126:
9115:
9094:
9093:
9085:
9082:
9081:
9062:
9059:
9058:
9036:
9018:
9017:
9003:
9002:
8992:
8986:
8982:
8979:
8978:
8964:
8963:
8950:
8936:
8935:
8920:
8910:
8904:
8900:
8897:
8896:
8882:
8881:
8868:
8854:
8853:
8838:
8831:
8825:
8821:
8817:
8815:
8812:
8811:
8785:
8774:
8766:
8763:
8762:
8743:
8735:
8732:
8731:
8715:
8712:
8711:
8695:
8692:
8691:
8685:
8658:
8653:
8652:
8643:
8638:
8637:
8628:
8623:
8622:
8621:
8617:
8615:
8612:
8611:
8589:
8584:
8583:
8574:
8569:
8568:
8559:
8554:
8553:
8552:
8548:
8546:
8543:
8542:
8520:
8512:
8509:
8508:
8503:lattice is the
8497:
8464:
8462:
8460:
8457:
8456:
8453:
8427:
8425:
8422:
8421:
8397:
8395:
8393:
8390:
8389:
8373:
8370:
8369:
8362:Bravais lattice
8358:
8346:
8291:
8283:
8280:
8279:
8224:
8221:
8220:
8156:
8153:
8152:
8129:
8126:
8125:
8064:
8063:
8061:
8058:
8057:
8040:
8036:
8016:
8015:
8013:
8010:
8009:
7989:
7985:
7983:
7980:
7979:
7959:
7950:
7946:
7938:
7935:
7934:
7892:
7888:
7887:
7882:
7881:
7857:
7853:
7852:
7847:
7846:
7834:
7830:
7818:
7813:
7812:
7797:
7792:
7791:
7784:
7773:
7769:
7757:
7753:
7752:
7748:
7746:
7731:
7726:
7725:
7723:
7720:
7719:
7692:
7691:
7686:
7681:
7676:
7670:
7669:
7664:
7659:
7654:
7644:
7643:
7635:
7632:
7631:
7605:
7601:
7583:
7578:
7577:
7568:
7563:
7562:
7554:
7551:
7550:
7534:
7514:
7511:
7510:
7490:
7485:
7484:
7469:
7464:
7463:
7458:
7455:
7454:
7435:
7432:
7431:
7408:
7405:
7404:
7388:
7385:
7384:
7381:
7355:
7350:
7349:
7347:
7344:
7343:
7307:
7304:
7303:
7275:
7272:
7271:
7255:
7252:
7251:
7234:
7230:
7228:
7225:
7224:
7208:
7205:
7204:
7187:
7183:
7181:
7178:
7177:
7161:
7158:
7157:
7140:
7136:
7134:
7131:
7130:
7098:
7095:
7094:
7060:
7057:
7056:
7036:
7032:
7023:
7019:
7010:
7006:
6995:
6992:
6991:
6966:
6963:
6962:
6945:
6940:
6939:
6930:
6925:
6924:
6922:
6919:
6918:
6901:
6896:
6895:
6893:
6890:
6889:
6872:
6867:
6866:
6864:
6861:
6860:
6832:
6827:
6826:
6817:
6812:
6811:
6810:
6806:
6797:
6792:
6791:
6790:
6783:
6778:
6777:
6768:
6763:
6762:
6761:
6759:
6750:
6745:
6744:
6742:
6739:
6738:
6713:
6707:
6702:
6701:
6692:
6687:
6686:
6684:
6681:
6680:
6661:
6658:
6657:
6628:
6617:
6612:
6611:
6605:
6600:
6599:
6593:
6588:
6587:
6586:
6582:
6581:
6565:
6564:
6553:
6548:
6547:
6541:
6536:
6535:
6529:
6524:
6523:
6522:
6518:
6517:
6515:
6512:
6511:
6475:
6470:
6469:
6460:
6455:
6454:
6450:
6446:
6444:
6441:
6440:
6407:
6403:
6387:
6382:
6381:
6372:
6367:
6366:
6364:
6361:
6360:
6359:
6337:
6332:
6331:
6322:
6317:
6316:
6306:
6302:
6301:
6300:
6296:
6294:
6291:
6290:
6264:
6259:
6258:
6249:
6244:
6243:
6242:
6238:
6229:
6224:
6223:
6209:
6204:
6203:
6194:
6189:
6188:
6187:
6183:
6174:
6169:
6168:
6154:
6149:
6148:
6139:
6134:
6133:
6132:
6128:
6119:
6114:
6113:
6105:
6102:
6101:
6087:
6086:
6080:
6075:
6074:
6065:
6060:
6059:
6043:
6041:
6034:
6028:
6023:
6022:
6019:
6018:
6012:
6007:
6006:
5997:
5992:
5991:
5975:
5973:
5966:
5960:
5955:
5954:
5951:
5950:
5944:
5939:
5938:
5929:
5924:
5923:
5907:
5905:
5898:
5892:
5887:
5886:
5882:
5880:
5877:
5876:
5855:
5850:
5849:
5840:
5835:
5834:
5824:
5820:
5819:
5818:
5814:
5812:
5809:
5808:
5788:
5784:
5775:
5771:
5762:
5758:
5747:
5744:
5743:
5726:
5721:
5720:
5714:
5710:
5701:
5696:
5695:
5689:
5685:
5676:
5671:
5670:
5664:
5660:
5651:
5646:
5645:
5643:
5640:
5639:
5617:
5613:
5604:
5600:
5591:
5587:
5586:
5582:
5574:
5571:
5570:
5548:
5543:
5542:
5533:
5528:
5527:
5517:
5513:
5512:
5511:
5507:
5505:
5502:
5501:
5481:
5476:
5475:
5469:
5465:
5456:
5451:
5450:
5444:
5440:
5431:
5426:
5425:
5419:
5415:
5406:
5401:
5400:
5398:
5395:
5394:
5391:
5351:
5346:
5345:
5335:
5334:
5325:
5320:
5319:
5318:
5302:
5297:
5296:
5286:
5285:
5284:
5282:
5255:
5250:
5249:
5243:
5234:
5229:
5228:
5227:
5211:
5206:
5205:
5199:
5198:
5196:
5181:
5176:
5175:
5173:
5170:
5169:
5145:
5144:
5139:
5133:
5132:
5127:
5117:
5116:
5108:
5105:
5104:
5082:
5080:
5077:
5076:
5060:
5050:
5049:
5038:
5032:
5030:
5027:
5026:
5006:
5005:
5003:
5000:
4999:
4983:
4981:
4978:
4977:
4970:rotation matrix
4953:
4951:
4948:
4947:
4930:
4929:
4920:
4915:
4914:
4908:
4899:
4894:
4893:
4892:
4885:
4880:
4879:
4873:
4872:
4870:
4857:
4851:
4846:
4845:
4842:
4841:
4832:
4827:
4826:
4820:
4811:
4806:
4805:
4804:
4797:
4792:
4791:
4785:
4784:
4782:
4764:
4759:
4758:
4752:
4743:
4738:
4737:
4733:
4726:
4721:
4720:
4714:
4710:
4708:
4695:
4689:
4684:
4683:
4679:
4677:
4674:
4673:
4653:
4649:
4647:
4644:
4643:
4623:
4618:
4617:
4611:
4607:
4598:
4593:
4592:
4586:
4582:
4573:
4568:
4567:
4565:
4562:
4561:
4536:
4531:
4530:
4521:
4516:
4515:
4514:
4510:
4508:
4505:
4504:
4498:
4476:
4471:
4470:
4468:
4465:
4464:
4447:
4442:
4441:
4439:
4436:
4435:
4405:
4400:
4399:
4390:
4385:
4384:
4374:
4370:
4369:
4368:
4364:
4362:
4359:
4358:
4341:
4336:
4335:
4333:
4330:
4329:
4312:
4307:
4306:
4304:
4301:
4300:
4279:
4275:
4259:
4254:
4253:
4244:
4239:
4238:
4236:
4233:
4232:
4231:
4214:
4209:
4208:
4202:
4198:
4189:
4184:
4183:
4177:
4173:
4164:
4159:
4158:
4152:
4148:
4139:
4134:
4133:
4131:
4128:
4127:
4110:
4105:
4104:
4102:
4099:
4098:
4074:
4069:
4068:
4066:
4063:
4062:
4043:
4040:
4039:
4022:
4017:
4016:
4014:
4011:
4010:
3993:
3988:
3987:
3985:
3982:
3981:
3964:
3959:
3958:
3956:
3953:
3952:
3931:
3926:
3925:
3923:
3920:
3919:
3894:
3886:
3883:
3882:
3853:
3848:
3847:
3838:
3833:
3832:
3830:
3827:
3826:
3798:
3793:
3792:
3783:
3778:
3777:
3773:
3769:
3767:
3764:
3763:
3740:
3731:
3726:
3725:
3721:
3717:
3708:
3703:
3702:
3693:
3688:
3687:
3683:
3679:
3673:
3669:
3663:
3645:
3640:
3639:
3631:
3619:
3614:
3613:
3609:
3605:
3599:
3595:
3589:
3575:
3566:
3561:
3560:
3556:
3552:
3546:
3542:
3536:
3530:
3527:
3526:
3501:
3483:
3478:
3477:
3469:
3461:
3458:
3457:
3437:
3432:
3431:
3429:
3426:
3425:
3409:
3407:
3404:
3403:
3384:
3376:
3373:
3372:
3349:
3345:
3336:
3332:
3323:
3319:
3308:
3305:
3304:
3281:
3277:
3264:
3255:
3250:
3249:
3245:
3241:
3235:
3231:
3225:
3219:
3216:
3215:
3190:
3182:
3179:
3178:
3159:
3151:
3148:
3147:
3124:
3119:
3118:
3116:
3113:
3112:
3096:
3094:
3091:
3090:
3071:
3063:
3060:
3059:
3042:
3037:
3036:
3034:
3031:
3030:
3014:
3012:
3009:
3008:
2989:
2980:
2976:
2967:
2963:
2954:
2950:
2948:
2945:
2944:
2927:
2922:
2921:
2915:
2911:
2902:
2897:
2896:
2890:
2886:
2877:
2872:
2871:
2865:
2861:
2852:
2847:
2846:
2844:
2841:
2840:
2813:
2809:
2800:
2796:
2787:
2783:
2772:
2769:
2768:
2765:Bravais lattice
2744:
2741:
2740:
2718:
2715:
2714:
2698:
2695:
2694:
2677:
2673:
2671:
2668:
2667:
2650:
2646:
2644:
2641:
2640:
2624:
2621:
2620:
2594:
2590:
2588:
2585:
2584:
2567:
2563:
2561:
2558:
2557:
2540:
2536:
2534:
2531:
2530:
2523:
2499:Bloch's theorem
2463:
2461:
2458:
2457:
2441:
2439:
2436:
2435:
2415:
2412:
2411:
2410:
2394:
2392:
2389:
2388:
2372:
2364:
2362:
2359:
2358:
2342:
2340:
2337:
2336:
2319:
2314:
2313:
2311:
2308:
2307:
2290:
2285:
2284:
2282:
2279:
2278:
2261:
2256:
2255:
2253:
2250:
2249:
2246:Bravais lattice
2228:
2224:
2222:
2219:
2218:
2199:
2194:
2193:
2187:
2183:
2174:
2169:
2168:
2162:
2158:
2149:
2144:
2143:
2137:
2133:
2125:
2123:
2120:
2119:
2118:
2101:
2096:
2095:
2093:
2090:
2089:
2067:
2064:
2063:
2043:
2039:
2037:
2034:
2033:
2031:Kronecker delta
2009:
2005:
1989:
1984:
1983:
1974:
1969:
1968:
1966:
1963:
1962:
1961:
1944:
1939:
1938:
1936:
1933:
1932:
1905:
1900:
1899:
1890:
1885:
1884:
1875:
1870:
1869:
1860:
1855:
1854:
1852:
1849:
1848:
1847:
1821:
1816:
1815:
1806:
1801:
1800:
1798:
1795:
1794:
1777:
1773:
1771:
1768:
1767:
1749:
1744:
1743:
1734:
1729:
1728:
1719:
1715:
1713:
1710:
1709:
1708:
1691:
1687:
1685:
1682:
1681:
1664:
1659:
1658:
1656:
1653:
1652:
1633:
1629:
1624:
1618:
1613:
1612:
1597:
1592:
1591:
1589:
1586:
1585:
1584:
1566:
1561:
1560:
1558:
1555:
1554:
1553:
1531:
1528:
1527:
1508:
1505:
1504:
1485:
1480:
1479:
1477:
1474:
1473:
1472:
1455:
1450:
1449:
1447:
1444:
1443:
1420:
1418:
1415:
1414:
1413:
1391:
1386:
1385:
1383:
1380:
1379:
1362:
1358:
1356:
1353:
1352:
1333:
1328:
1327:
1321:
1317:
1308:
1303:
1302:
1296:
1292:
1283:
1278:
1277:
1271:
1267:
1259:
1257:
1254:
1253:
1252:
1233:
1230:
1229:
1201:
1198:
1197:
1177:
1169:
1166:
1165:
1145:
1143:
1140:
1139:
1133:Bravais lattice
1129:
1107:
1104:
1103:
1102:
1097:represents any
1082:
1079:
1078:
1044:
1041:
1040:
1024:
1022:
1019:
1018:
1000:
997:
996:
995:
973:
971:
968:
967:
947:
944:
943:
920:
915:
901:
899:
896:
895:
879:
877:
874:
873:
855:
852:
851:
850:
825:
817:
806:
803:
802:
780:
776:
759:
751:
740:
737:
736:
735:
712:
698:
695:
694:
693:
663:
660:
659:
658:
640:
637:
636:
635:
619:
616:
615:
599:
596:
595:
577:
574:
573:
572:
556:
553:
552:
536:
533:
532:
516:
513:
512:
494:
491:
490:
489:
470:
467:
466:
445:
441:
439:
436:
435:
434:
409:
405:
376:
373:
372:
371:
339:
336:
323:
287:
284:
283:
282:
265:
260:
259:
257:
254:
253:
232:
227:
226:
224:
221:
220:
190:
187:
186:
170:
168:
165:
164:
148:
137:
135:
132:
131:
124:quantum physics
89:Bravais lattice
17:
12:
11:
5:
11283:
11273:
11272:
11267:
11262:
11257:
11252:
11250:Lattice points
11247:
11242:
11222:
11221:
11219:
11218:
11206:
11193:
11190:
11189:
11186:
11185:
11183:
11182:
11177:
11172:
11171:
11170:
11165:
11160:
11149:
11147:
11140:
11139:
11134:
11129:
11124:
11118:
11116:
11110:
11109:
11107:
11106:
11101:
11096:
11090:
11088:
11082:
11081:
11079:
11078:
11073:
11068:
11063:
11058:
11053:
11048:
11042:
11040:
11034:
11033:
11031:
11030:
11025:
11020:
11015:
11010:
11005:
11000:
10994:
10992:
10986:
10985:
10983:
10982:
10977:
10972:
10967:
10962:
10957:
10952:
10947:
10941:
10939:
10933:
10932:
10930:
10929:
10924:
10923:
10922:
10912:
10907:
10902:
10897:
10892:
10890:Direct methods
10886:
10884:
10880:
10879:
10877:
10876:
10875:
10874:
10869:
10859:
10854:
10853:
10852:
10847:
10837:
10836:
10835:
10830:
10819:
10817:
10811:
10810:
10802:
10800:
10798:
10797:
10792:
10787:
10782:
10777:
10775:Ewald's sphere
10772:
10767:
10761:
10758:
10757:
10755:
10754:
10749:
10744:
10743:
10742:
10737:
10727:
10726:
10725:
10720:
10718:Frenkel defect
10715:
10713:Bjerrum defect
10705:
10704:
10703:
10693:
10692:
10691:
10686:
10681:
10679:Peierls stress
10676:
10671:
10666:
10661:
10656:
10651:
10649:Burgers vector
10641:
10639:Stacking fault
10636:
10630:
10627:
10626:
10624:
10623:
10618:
10613:
10608:
10602:
10600:
10598:Grain boundary
10591:
10585:
10584:
10582:
10581:
10576:
10571:
10566:
10561:
10556:
10551:
10546:
10545:
10544:
10542:Liquid crystal
10539:
10534:
10529:
10518:
10516:
10508:
10507:
10505:
10504:
10503:
10502:
10492:
10491:
10490:
10480:
10479:
10478:
10473:
10462:
10459:
10458:
10456:
10455:
10450:
10445:
10440:
10435:
10430:
10424:
10422:
10413:
10412:
10407:
10405:Periodic table
10402:
10401:
10400:
10395:
10390:
10385:
10380:
10369:
10367:
10358:
10357:
10352:
10347:
10346:
10345:
10334:
10332:
10328:
10327:
10320:
10319:
10312:
10305:
10297:
10291:
10290:
10285:
10280:
10261:
10260:External links
10258:
10255:
10254:
10241:
10234:
10216:
10201:
10147:
10146:
10144:
10141:
10140:
10139:
10133:
10127:
10121:
10112:
10109:Ewald's sphere
10106:
10100:
10094:
10091:Brillouin zone
10086:
10083:
10068:
10065:
10060:
10056:
10050:
10045:
10040:
10035:
10032:
10029:
9896:non-degenerate
9885:quadratic form
9872:^. Therefore,
9813:Main article:
9810:
9807:
9788:
9782:
9781:
9770:
9763:
9760:
9750:
9747:
9740:
9734:
9731:
9725:
9722:
9719:
9715:
9710:
9704:
9701:
9695:
9689:
9685:
9680:
9674:
9671:
9665:
9659:
9655:
9649:
9644:
9641:
9638:
9634:
9628:
9623:
9620:
9617:
9613:
9609:
9606:
9600:
9597:
9591:
9588:
9552:
9543:
9534:
9520:
9519:
9505:
9499:
9495:
9491:
9488:
9483:
9479:
9475:
9472:
9467:
9463:
9459:
9455:
9451:
9448:
9445:
9441:
9436:
9431:
9428:
9425:
9422:
9419:
9415:
9411:
9405:
9401:
9395:
9390:
9387:
9384:
9380:
9376:
9371:
9368:
9365:
9362:
9359:
9355:
9331:
9328:
9325:
9322:
9319:
9316:
9313:
9296:
9274:
9253:
9227:
9226:
9215:
9208:
9201:
9198:
9191:
9185:
9182:
9176:
9173:
9170:
9166:
9161:
9155:
9152:
9146:
9139:
9135:
9129:
9124:
9121:
9118:
9114:
9110:
9107:
9101:
9098:
9092:
9089:
9066:
9035:
9032:
9016:
9010:
9007:
9001:
8998:
8995:
8993:
8989:
8985:
8981:
8980:
8977:
8971:
8968:
8962:
8957:
8954:
8949:
8943:
8940:
8934:
8929:
8925:
8919:
8916:
8913:
8911:
8907:
8903:
8899:
8898:
8895:
8889:
8886:
8880:
8875:
8872:
8867:
8861:
8858:
8852:
8847:
8843:
8837:
8834:
8832:
8828:
8824:
8820:
8819:
8794:
8789:
8784:
8781:
8777:
8773:
8770:
8750:
8746:
8742:
8739:
8723:{\textstyle c}
8719:
8703:{\textstyle a}
8699:
8684:
8681:
8667:
8661:
8656:
8651:
8646:
8641:
8636:
8631:
8626:
8620:
8598:
8592:
8587:
8582:
8577:
8572:
8567:
8562:
8557:
8551:
8527:
8523:
8519:
8516:
8496:
8493:
8474:
8470:
8467:
8452:
8449:
8434:
8431:
8407:
8403:
8400:
8377:
8366:primitive cell
8357:
8354:
8345:
8342:
8325:
8322:
8319:
8316:
8313:
8310:
8307:
8304:
8301:
8298:
8290:
8287:
8267:
8264:
8261:
8258:
8255:
8252:
8249:
8246:
8243:
8240:
8237:
8234:
8231:
8228:
8208:
8205:
8202:
8199:
8196:
8193:
8190:
8187:
8184:
8181:
8178:
8175:
8172:
8169:
8166:
8163:
8160:
8133:
8113:
8110:
8107:
8104:
8101:
8098:
8095:
8092:
8089:
8086:
8083:
8080:
8077:
8071:
8068:
8043:
8039:
8035:
8032:
8029:
8023:
8020:
7995:
7992:
7988:
7962:
7958:
7953:
7949:
7945:
7942:
7931:
7930:
7919:
7916:
7913:
7910:
7906:
7900:
7895:
7891:
7885:
7880:
7877:
7871:
7866:
7863:
7860:
7856:
7850:
7845:
7840:
7837:
7833:
7826:
7821:
7816:
7811:
7808:
7805:
7800:
7795:
7790:
7787:
7781:
7776:
7772:
7768:
7765:
7760:
7756:
7751:
7745:
7742:
7739:
7734:
7729:
7713:
7712:
7701:
7696:
7690:
7687:
7685:
7682:
7680:
7677:
7675:
7672:
7671:
7668:
7665:
7663:
7660:
7658:
7655:
7653:
7650:
7649:
7647:
7642:
7639:
7611:
7608:
7604:
7600:
7597:
7594:
7591:
7586:
7581:
7576:
7571:
7566:
7561:
7558:
7537:
7533:
7530:
7527:
7524:
7521:
7518:
7498:
7493:
7488:
7483:
7480:
7477:
7472:
7467:
7462:
7439:
7412:
7392:
7380:
7377:
7358:
7353:
7323:
7320:
7317:
7314:
7311:
7291:
7288:
7285:
7282:
7279:
7259:
7250:replaced with
7237:
7233:
7212:
7203:replaced with
7190:
7186:
7165:
7143:
7139:
7127:Miller indices
7114:
7111:
7108:
7105:
7102:
7082:
7079:
7076:
7073:
7070:
7067:
7064:
7044:
7039:
7035:
7031:
7026:
7022:
7018:
7013:
7009:
7005:
7002:
6999:
6973:
6970:
6948:
6943:
6938:
6933:
6928:
6904:
6899:
6875:
6870:
6857:
6856:
6841:
6835:
6830:
6825:
6820:
6815:
6809:
6805:
6800:
6795:
6786:
6781:
6776:
6771:
6766:
6758:
6753:
6748:
6723:
6720:
6716:
6710:
6705:
6700:
6695:
6690:
6668:
6665:
6651:
6650:
6639:
6634:
6631:
6626:
6620:
6615:
6608:
6603:
6596:
6591:
6585:
6580:
6577:
6574:
6568:
6562:
6556:
6551:
6544:
6539:
6532:
6527:
6521:
6488:
6485:
6478:
6473:
6468:
6463:
6458:
6453:
6449:
6413:
6410:
6406:
6401:
6398:
6395:
6390:
6385:
6380:
6375:
6370:
6358:is to satisfy
6346:
6340:
6335:
6330:
6325:
6320:
6315:
6309:
6305:
6299:
6273:
6267:
6262:
6257:
6252:
6247:
6241:
6237:
6232:
6227:
6222:
6218:
6212:
6207:
6202:
6197:
6192:
6186:
6182:
6177:
6172:
6167:
6163:
6157:
6152:
6147:
6142:
6137:
6131:
6127:
6122:
6117:
6112:
6109:
6083:
6078:
6073:
6068:
6063:
6053:
6049:
6046:
6040:
6037:
6035:
6031:
6026:
6021:
6020:
6015:
6010:
6005:
6000:
5995:
5985:
5981:
5978:
5972:
5969:
5967:
5963:
5958:
5953:
5952:
5947:
5942:
5937:
5932:
5927:
5917:
5913:
5910:
5904:
5901:
5899:
5895:
5890:
5885:
5884:
5864:
5858:
5853:
5848:
5843:
5838:
5833:
5827:
5823:
5817:
5796:
5791:
5787:
5783:
5778:
5774:
5770:
5765:
5761:
5757:
5754:
5751:
5729:
5724:
5717:
5713:
5709:
5704:
5699:
5692:
5688:
5684:
5679:
5674:
5667:
5663:
5659:
5654:
5649:
5626:
5620:
5616:
5612:
5607:
5603:
5599:
5594:
5590:
5585:
5581:
5578:
5557:
5551:
5546:
5541:
5536:
5531:
5526:
5520:
5516:
5510:
5484:
5479:
5472:
5468:
5464:
5459:
5454:
5447:
5443:
5439:
5434:
5429:
5422:
5418:
5414:
5409:
5404:
5390:
5387:
5383:
5382:
5371:
5363:
5360:
5357:
5354:
5349:
5342:
5338:
5333:
5328:
5323:
5314:
5311:
5308:
5305:
5300:
5293:
5289:
5281:
5278:
5275:
5267:
5264:
5261:
5258:
5253:
5246:
5242:
5237:
5232:
5223:
5220:
5217:
5214:
5209:
5202:
5195:
5192:
5189:
5184:
5179:
5163:
5162:
5149:
5143:
5140:
5138:
5135:
5134:
5131:
5128:
5126:
5123:
5122:
5120:
5115:
5112:
5085:
5063:
5056:
5053:
5048:
5045:
5041:
5035:
5012:
5009:
4986:
4956:
4944:
4943:
4923:
4918:
4911:
4907:
4902:
4897:
4888:
4883:
4876:
4869:
4866:
4863:
4860:
4858:
4854:
4849:
4844:
4843:
4835:
4830:
4823:
4819:
4814:
4809:
4800:
4795:
4788:
4781:
4778:
4775:
4767:
4762:
4755:
4751:
4746:
4741:
4736:
4729:
4724:
4717:
4713:
4707:
4704:
4701:
4698:
4696:
4692:
4687:
4682:
4681:
4656:
4652:
4640:
4639:
4626:
4621:
4614:
4610:
4606:
4601:
4596:
4589:
4585:
4581:
4576:
4571:
4545:
4539:
4534:
4529:
4524:
4519:
4513:
4497:
4496:Two dimensions
4494:
4479:
4474:
4450:
4445:
4414:
4408:
4403:
4398:
4393:
4388:
4383:
4377:
4373:
4367:
4344:
4339:
4315:
4310:
4285:
4282:
4278:
4273:
4270:
4267:
4262:
4257:
4252:
4247:
4242:
4217:
4212:
4205:
4201:
4197:
4192:
4187:
4180:
4176:
4172:
4167:
4162:
4155:
4151:
4147:
4142:
4137:
4113:
4108:
4077:
4072:
4050:
4047:
4025:
4020:
3996:
3991:
3967:
3962:
3934:
3929:
3913:
3912:
3901:
3897:
3893:
3890:
3870:
3867:
3864:
3861:
3856:
3851:
3846:
3841:
3836:
3811:
3808:
3801:
3796:
3791:
3786:
3781:
3776:
3772:
3749:
3743:
3739:
3734:
3729:
3724:
3720:
3711:
3706:
3701:
3696:
3691:
3686:
3682:
3676:
3672:
3666:
3662:
3658:
3653:
3648:
3643:
3638:
3634:
3630:
3627:
3622:
3617:
3612:
3608:
3602:
3598:
3592:
3588:
3584:
3578:
3574:
3569:
3564:
3559:
3555:
3549:
3545:
3539:
3535:
3523:
3522:
3511:
3508:
3504:
3500:
3497:
3494:
3491:
3486:
3481:
3476:
3472:
3468:
3465:
3440:
3435:
3412:
3391:
3387:
3383:
3380:
3357:
3352:
3348:
3344:
3339:
3335:
3331:
3326:
3322:
3318:
3315:
3312:
3301:
3300:
3288:
3284:
3280:
3276:
3273:
3267:
3263:
3258:
3253:
3248:
3244:
3238:
3234:
3228:
3224:
3197:
3193:
3189:
3186:
3166:
3162:
3158:
3155:
3135:
3132:
3127:
3122:
3099:
3078:
3074:
3070:
3067:
3045:
3040:
3017:
3005:
3004:
2992:
2988:
2983:
2979:
2975:
2970:
2966:
2962:
2957:
2953:
2930:
2925:
2918:
2914:
2910:
2905:
2900:
2893:
2889:
2885:
2880:
2875:
2868:
2864:
2860:
2855:
2850:
2821:
2816:
2812:
2808:
2803:
2799:
2795:
2790:
2786:
2782:
2779:
2776:
2748:
2728:
2725:
2722:
2702:
2680:
2676:
2653:
2649:
2628:
2606:
2603:
2600:
2597:
2593:
2570:
2566:
2543:
2539:
2522:
2519:
2487:primitive cell
2483:Brillouin zone
2466:
2444:
2422:
2419:
2397:
2375:
2371:
2367:
2345:
2322:
2317:
2293:
2288:
2264:
2259:
2231:
2227:
2202:
2197:
2190:
2186:
2182:
2177:
2172:
2165:
2161:
2157:
2152:
2147:
2140:
2136:
2132:
2128:
2104:
2099:
2077:
2074:
2071:
2049:
2046:
2042:
2015:
2012:
2008:
2003:
2000:
1997:
1992:
1987:
1982:
1977:
1972:
1947:
1942:
1916:
1913:
1908:
1903:
1898:
1893:
1888:
1883:
1878:
1873:
1868:
1863:
1858:
1835:
1832:
1829:
1824:
1819:
1814:
1809:
1804:
1780:
1776:
1752:
1747:
1742:
1737:
1732:
1727:
1722:
1718:
1694:
1690:
1667:
1662:
1636:
1632:
1627:
1621:
1616:
1611:
1608:
1605:
1600:
1595:
1569:
1564:
1541:
1538:
1535:
1515:
1512:
1488:
1483:
1458:
1453:
1430:
1427:
1423:
1394:
1389:
1365:
1361:
1336:
1331:
1324:
1320:
1316:
1311:
1306:
1299:
1295:
1291:
1286:
1281:
1274:
1270:
1266:
1262:
1237:
1217:
1214:
1211:
1208:
1205:
1184:
1180:
1176:
1173:
1162:Fourier series
1148:
1128:
1125:
1111:
1086:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1027:
1004:
983:
980:
976:
951:
927:
923:
918:
914:
911:
908:
904:
882:
859:
838:
835:
832:
828:
824:
820:
816:
813:
810:
788:
783:
779:
775:
772:
769:
766:
762:
758:
754:
750:
747:
744:
719:
715:
711:
708:
705:
702:
679:
676:
673:
670:
667:
644:
623:
603:
581:
560:
540:
520:
498:
474:
448:
444:
417:
412:
408:
404:
401:
398:
395:
392:
389:
386:
383:
380:
335:
332:
322:
319:
311:Brillouin zone
294:
291:
268:
263:
235:
230:
194:
173:
151:
147:
144:
140:
128:momentum space
117:direct lattice
109:physical space
85:crystal system
81:physical space
69:direct lattice
15:
9:
6:
4:
3:
2:
11282:
11271:
11268:
11266:
11263:
11261:
11258:
11256:
11253:
11251:
11248:
11246:
11243:
11241:
11238:
11237:
11235:
11228:
11217:
11216:
11207:
11205:
11204:
11195:
11194:
11191:
11181:
11178:
11176:
11173:
11169:
11166:
11164:
11161:
11159:
11156:
11155:
11154:
11151:
11150:
11148:
11144:
11138:
11135:
11133:
11130:
11128:
11125:
11123:
11120:
11119:
11117:
11115:
11111:
11105:
11102:
11100:
11097:
11095:
11092:
11091:
11089:
11087:
11083:
11077:
11074:
11072:
11069:
11067:
11064:
11062:
11059:
11057:
11054:
11052:
11049:
11047:
11044:
11043:
11041:
11039:
11035:
11029:
11026:
11024:
11021:
11019:
11016:
11014:
11011:
11009:
11006:
11004:
11001:
10999:
10996:
10995:
10993:
10991:
10987:
10981:
10978:
10976:
10973:
10971:
10968:
10966:
10963:
10961:
10958:
10956:
10953:
10951:
10948:
10946:
10943:
10942:
10940:
10938:
10934:
10928:
10925:
10921:
10918:
10917:
10916:
10913:
10911:
10910:Patterson map
10908:
10906:
10903:
10901:
10898:
10896:
10893:
10891:
10888:
10887:
10885:
10881:
10873:
10870:
10868:
10865:
10864:
10863:
10860:
10858:
10855:
10851:
10848:
10846:
10843:
10842:
10841:
10838:
10834:
10831:
10829:
10826:
10825:
10824:
10821:
10820:
10818:
10816:
10812:
10806:
10796:
10793:
10791:
10788:
10786:
10783:
10781:
10780:Friedel's law
10778:
10776:
10773:
10771:
10768:
10766:
10763:
10762:
10753:
10750:
10748:
10745:
10741:
10738:
10736:
10733:
10732:
10731:
10728:
10724:
10723:Wigner effect
10721:
10719:
10716:
10714:
10711:
10710:
10709:
10708:Interstitials
10706:
10702:
10699:
10698:
10697:
10694:
10690:
10687:
10685:
10682:
10680:
10677:
10675:
10672:
10670:
10667:
10665:
10662:
10660:
10657:
10655:
10652:
10650:
10647:
10646:
10645:
10642:
10640:
10637:
10635:
10632:
10631:
10622:
10619:
10617:
10614:
10612:
10609:
10607:
10604:
10603:
10601:
10599:
10595:
10592:
10590:
10586:
10580:
10577:
10575:
10572:
10570:
10567:
10565:
10562:
10560:
10557:
10555:
10554:Precipitation
10552:
10550:
10547:
10543:
10540:
10538:
10535:
10533:
10530:
10528:
10525:
10524:
10523:
10522:Phase diagram
10520:
10519:
10517:
10515:
10509:
10501:
10498:
10497:
10496:
10493:
10489:
10486:
10485:
10484:
10481:
10477:
10474:
10472:
10469:
10468:
10467:
10464:
10463:
10454:
10451:
10449:
10446:
10444:
10441:
10439:
10436:
10434:
10431:
10429:
10426:
10425:
10423:
10421:
10417:
10411:
10408:
10406:
10403:
10399:
10396:
10394:
10391:
10389:
10386:
10384:
10381:
10379:
10376:
10375:
10374:
10371:
10370:
10368:
10366:
10362:
10356:
10353:
10351:
10348:
10344:
10341:
10340:
10339:
10336:
10335:
10333:
10329:
10325:
10318:
10313:
10311:
10306:
10304:
10299:
10298:
10295:
10289:
10286:
10284:
10281:
10278:
10274:
10270:
10267:
10264:
10263:
10251:
10245:
10237:
10235:0-471-41526-X
10231:
10227:
10220:
10212:
10205:
10197:
10193:
10189:
10185:
10181:
10177:
10172:
10167:
10164:(6): 064003.
10163:
10159:
10152:
10148:
10137:
10134:
10131:
10128:
10125:
10122:
10116:
10113:
10110:
10107:
10104:
10101:
10098:
10095:
10092:
10089:
10088:
10082:
10066:
10063:
10058:
10054:
10043:
10038:
10033:
10030:
10027:
10019:
10016:
10011:
10009:
10005:
10000:
9994:
9989:
9987:
9983:
9979:
9975:
9971:
9968:
9965:
9962:
9958:
9955:
9950:
9948:
9944:
9940:
9936:
9932:
9928:
9924:
9920:
9916:
9912:
9908:
9904:
9901:
9897:
9893:
9889:
9886:
9881:
9879:
9875:
9871:
9867:
9863:
9859:
9855:
9852:
9848:
9844:
9839:
9837:
9833:
9829:
9826:
9822:
9816:
9806:
9804:
9800:
9796:
9791:
9787:
9768:
9761:
9758:
9745:
9738:
9729:
9723:
9720:
9717:
9713:
9708:
9699:
9693:
9687:
9683:
9678:
9669:
9663:
9657:
9653:
9647:
9642:
9639:
9636:
9632:
9626:
9621:
9618:
9615:
9611:
9607:
9595:
9586:
9579:
9578:
9577:
9574:
9570:
9567:
9563:
9558:
9555:
9551:
9546:
9542:
9537:
9533:
9529:
9525:
9503:
9497:
9493:
9489:
9486:
9481:
9477:
9473:
9470:
9465:
9461:
9457:
9453:
9449:
9446:
9443:
9439:
9434:
9429:
9426:
9423:
9420:
9417:
9413:
9409:
9403:
9399:
9393:
9388:
9385:
9382:
9378:
9374:
9369:
9366:
9363:
9360:
9357:
9353:
9345:
9344:
9343:
9326:
9323:
9320:
9317:
9314:
9303:
9299:
9293:
9289:
9284:
9282:
9277:
9273:
9269:
9265:
9261:
9256:
9252:
9248:
9244:
9236:
9232:
9213:
9206:
9196:
9189:
9180:
9174:
9171:
9168:
9164:
9159:
9150:
9144:
9137:
9133:
9127:
9122:
9119:
9116:
9112:
9108:
9096:
9087:
9080:
9079:
9078:
9064:
9057:
9053:
9052:Huygens-style
9049:
9040:
9031:
9014:
9005:
8999:
8996:
8994:
8987:
8983:
8975:
8966:
8960:
8955:
8952:
8947:
8938:
8932:
8927:
8923:
8917:
8914:
8912:
8905:
8901:
8893:
8884:
8878:
8873:
8870:
8865:
8856:
8850:
8845:
8841:
8835:
8833:
8826:
8822:
8808:
8787:
8782:
8775:
8771:
8768:
8748:
8744:
8740:
8737:
8717:
8697:
8690:
8680:
8665:
8659:
8649:
8644:
8634:
8629:
8618:
8596:
8590:
8580:
8575:
8565:
8560:
8549:
8539:
8525:
8521:
8517:
8514:
8506:
8502:
8492:
8488:
8472:
8468:
8465:
8448:
8432:
8429:
8405:
8401:
8398:
8375:
8367:
8364:, with cubic
8363:
8353:
8351:
8341:
8339:
8320:
8317:
8314:
8308:
8305:
8299:
8288:
8285:
8262:
8259:
8256:
8253:
8247:
8244:
8238:
8235:
8232:
8226:
8203:
8200:
8197:
8194:
8191:
8185:
8182:
8176:
8173:
8170:
8167:
8164:
8158:
8149:
8147:
8131:
8108:
8105:
8102:
8096:
8093:
8087:
8078:
8066:
8041:
8037:
8030:
8027:
8018:
7993:
7990:
7986:
7977:
7951:
7947:
7943:
7940:
7917:
7914:
7908:
7904:
7898:
7893:
7889:
7878:
7875:
7869:
7864:
7861:
7858:
7854:
7838:
7835:
7831:
7819:
7809:
7806:
7803:
7798:
7785:
7779:
7774:
7770:
7766:
7763:
7758:
7754:
7749:
7743:
7740:
7737:
7732:
7718:
7717:
7716:
7699:
7694:
7688:
7683:
7678:
7673:
7666:
7661:
7656:
7651:
7645:
7640:
7637:
7630:
7629:
7628:
7627:
7609:
7606:
7602:
7598:
7595:
7592:
7584:
7574:
7569:
7556:
7528:
7525:
7522:
7519:
7516:
7491:
7481:
7478:
7475:
7470:
7453:
7437:
7430:vector space
7429:
7426:
7410:
7390:
7376:
7374:
7356:
7341:
7337:
7318:
7315:
7312:
7286:
7283:
7280:
7257:
7235:
7231:
7210:
7188:
7184:
7163:
7141:
7137:
7128:
7109:
7106:
7103:
7077:
7074:
7071:
7068:
7065:
7037:
7033:
7029:
7024:
7020:
7016:
7011:
7007:
7000:
6997:
6989:
6987:
6971:
6968:
6946:
6936:
6931:
6902:
6873:
6839:
6833:
6823:
6818:
6807:
6803:
6798:
6784:
6774:
6769:
6756:
6751:
6737:
6736:
6735:
6721:
6718:
6714:
6708:
6698:
6693:
6666:
6663:
6654:
6637:
6632:
6629:
6624:
6618:
6606:
6594:
6583:
6578:
6575:
6572:
6560:
6554:
6542:
6530:
6519:
6510:
6509:
6508:
6506:
6502:
6486:
6483:
6476:
6466:
6461:
6451:
6447:
6438:
6434:
6430:
6411:
6408:
6404:
6399:
6396:
6393:
6388:
6378:
6373:
6344:
6338:
6328:
6323:
6313:
6297:
6288:
6271:
6265:
6255:
6250:
6239:
6235:
6230:
6220:
6216:
6210:
6200:
6195:
6184:
6180:
6175:
6165:
6161:
6155:
6145:
6140:
6129:
6125:
6120:
6110:
6107:
6081:
6071:
6066:
6051:
6047:
6044:
6038:
6036:
6029:
6013:
6003:
5998:
5983:
5979:
5976:
5970:
5968:
5961:
5945:
5935:
5930:
5915:
5911:
5908:
5902:
5900:
5893:
5862:
5856:
5846:
5841:
5831:
5815:
5789:
5785:
5781:
5776:
5772:
5768:
5763:
5759:
5752:
5749:
5727:
5715:
5711:
5707:
5702:
5690:
5686:
5682:
5677:
5665:
5661:
5657:
5652:
5624:
5618:
5614:
5610:
5605:
5601:
5597:
5592:
5588:
5583:
5579:
5576:
5555:
5549:
5539:
5534:
5524:
5508:
5500:
5482:
5470:
5466:
5462:
5457:
5445:
5441:
5437:
5432:
5420:
5416:
5412:
5407:
5386:
5369:
5358:
5352:
5340:
5331:
5326:
5309:
5303:
5291:
5279:
5276:
5273:
5262:
5256:
5240:
5235:
5218:
5212:
5193:
5190:
5187:
5182:
5168:
5167:
5166:
5147:
5141:
5136:
5129:
5124:
5118:
5113:
5110:
5103:
5102:
5101:
5100:
5054:
5046:
5043:
5010:
4975:
4971:
4921:
4905:
4900:
4886:
4867:
4864:
4861:
4859:
4852:
4833:
4817:
4812:
4798:
4779:
4776:
4773:
4765:
4749:
4744:
4734:
4727:
4711:
4705:
4702:
4699:
4697:
4690:
4672:
4671:
4670:
4654:
4650:
4624:
4612:
4608:
4604:
4599:
4587:
4583:
4579:
4574:
4560:
4559:
4558:
4543:
4537:
4527:
4522:
4511:
4503:
4493:
4477:
4448:
4433:
4432:vector spaces
4429:
4412:
4406:
4396:
4391:
4381:
4365:
4342:
4313:
4283:
4280:
4276:
4271:
4268:
4265:
4260:
4250:
4245:
4215:
4203:
4199:
4195:
4190:
4178:
4174:
4170:
4165:
4153:
4149:
4145:
4140:
4111:
4096:
4091:
4075:
4048:
4045:
4023:
3994:
3965:
3950:
3932:
3918:
3899:
3891:
3888:
3868:
3865:
3862:
3859:
3854:
3844:
3839:
3825:
3824:
3823:
3809:
3806:
3799:
3789:
3784:
3774:
3770:
3760:
3747:
3737:
3732:
3722:
3718:
3709:
3699:
3694:
3684:
3680:
3674:
3670:
3664:
3660:
3656:
3646:
3636:
3625:
3620:
3610:
3606:
3600:
3596:
3590:
3586:
3582:
3572:
3567:
3557:
3553:
3547:
3543:
3537:
3533:
3509:
3495:
3492:
3484:
3474:
3463:
3456:
3455:
3454:
3438:
3378:
3369:
3350:
3346:
3342:
3337:
3333:
3329:
3324:
3320:
3313:
3310:
3286:
3278:
3274:
3271:
3261:
3256:
3246:
3242:
3236:
3232:
3226:
3222:
3214:
3213:
3212:
3211:
3184:
3153:
3133:
3130:
3125:
3065:
3043:
2986:
2981:
2977:
2973:
2968:
2964:
2960:
2955:
2951:
2928:
2916:
2912:
2908:
2903:
2891:
2887:
2883:
2878:
2866:
2862:
2858:
2853:
2839:
2838:
2837:
2836:of integers,
2835:
2814:
2810:
2806:
2801:
2797:
2793:
2788:
2784:
2777:
2774:
2766:
2746:
2726:
2723:
2720:
2700:
2678:
2674:
2651:
2647:
2626:
2604:
2601:
2598:
2595:
2591:
2568:
2564:
2541:
2537:
2527:
2518:
2516:
2512:
2508:
2504:
2500:
2496:
2492:
2488:
2484:
2479:
2420:
2417:
2369:
2320:
2291:
2262:
2247:
2229:
2225:
2200:
2188:
2184:
2180:
2175:
2163:
2159:
2155:
2150:
2138:
2134:
2130:
2102:
2075:
2072:
2069:
2047:
2044:
2040:
2032:
2013:
2010:
2006:
2001:
1998:
1995:
1990:
1980:
1975:
1945:
1929:
1914:
1911:
1906:
1896:
1891:
1881:
1876:
1866:
1861:
1833:
1830:
1827:
1822:
1812:
1807:
1778:
1774:
1766:, means that
1750:
1740:
1735:
1725:
1720:
1716:
1707:must satisfy
1692:
1688:
1665:
1634:
1630:
1625:
1619:
1609:
1606:
1603:
1598:
1567:
1539:
1536:
1533:
1513:
1510:
1486:
1456:
1428:
1425:
1410:
1392:
1363:
1359:
1334:
1322:
1318:
1314:
1309:
1297:
1293:
1289:
1284:
1272:
1268:
1264:
1249:
1235:
1215:
1209:
1206:
1171:
1163:
1138:
1134:
1124:
1109:
1100:
1084:
1064:
1058:
1055:
1049:
1046:
1002:
981:
978:
965:
949:
941:
925:
921:
912:
909:
906:
857:
833:
830:
822:
811:
808:
781:
777:
773:
770:
767:
764:
756:
745:
742:
732:
717:
713:
709:
706:
703:
700:
677:
674:
671:
668:
665:
642:
621:
601:
579:
558:
538:
518:
496:
488:
472:
465:
446:
442:
433:
430:with initial
410:
406:
402:
399:
396:
393:
390:
387:
381:
378:
369:
364:
362:
358:
353:
349:
345:
327:
318:
316:
312:
307:
292:
289:
266:
251:
233:
219:
214:
212:
208:
192:
142:
129:
125:
120:
118:
114:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
33:
26:
21:
11226:
11213:
11201:
11146:Associations
11114:Organisation
10606:Disclination
10537:Polymorphism
10500:Quasicrystal
10443:Orthorhombic
10392:
10383:Miller index
10331:Key concepts
10249:
10244:
10225:
10219:
10210:
10204:
10161:
10157:
10151:
10124:Miller index
10017:
10012:
10007:
10003:
9998:
9990:
9985:
9981:
9973:
9969:
9956:
9951:
9946:
9942:
9938:
9931:well-defined
9926:
9923:Haar measure
9918:
9914:
9910:
9906:
9902:
9891:
9887:
9882:
9878:dual lattice
9877:
9873:
9869:
9865:
9861:
9857:
9853:
9840:
9831:
9827:
9818:
9815:Dual lattice
9798:
9794:
9789:
9785:
9783:
9572:
9568:
9565:
9561:
9559:
9553:
9549:
9544:
9540:
9535:
9531:
9527:
9523:
9521:
9301:
9294:
9291:
9287:
9285:
9280:
9275:
9271:
9267:
9263:
9254:
9250:
9246:
9242:
9234:
9230:
9228:
9045:
8806:
8686:
8540:
8498:
8489:
8454:
8359:
8347:
8150:
8144:denotes the
7932:
7714:
7624:. Using the
7382:
7371:is given in
6990:
6858:
6655:
6652:
5392:
5384:
5164:
4973:
4945:
4641:
4499:
4092:
3914:
3761:
3524:
3370:
3302:
3006:
2762:
2515:dual lattice
2511:linear forms
2480:
2029:, where the
1930:
1250:
1130:
733:
365:
360:
356:
337:
308:
215:
121:
116:
108:
104:
100:
83:, such as a
73:real lattice
72:
68:
40:
38:
11265:Diffraction
11099:Ewald Prize
10867:Diffraction
10845:Diffraction
10828:Diffraction
10770:Bragg plane
10765:Bragg's law
10644:Dislocation
10559:Segregation
10471:Crystallite
10388:Point group
9894:; if it is
9821:mathematics
8056:defined by
7976:volume form
7626:permutation
7425:dimensional
5099:permutation
3949:wavevectors
3947:, that are
1137:wavevectors
250:wavevectors
248:, that are
99:, known as
87:(usually a
27:3D crystal.
11234:Categories
10883:Algorithms
10872:Scattering
10850:Scattering
10833:Scattering
10701:Slip bands
10664:Cross slip
10514:transition
10448:Tetragonal
10438:Monoclinic
10350:Metallurgy
10171:1905.11354
10143:References
10103:Dual basis
9978:dual group
9900:dual space
9864:^ dual to
9851:dual group
9048:Fraunhofer
7336:real space
5165:we obtain
2507:dual space
2217:where the
1351:where the
940:wavevector
25:monoclinic
10990:Databases
10453:Triclinic
10433:Hexagonal
10373:Unit cell
10365:Structure
10196:166228311
10136:Zone axis
10064:−
9803:dynamical
9797:and atom
9749:→
9739:⋅
9733:→
9721:π
9703:→
9673:→
9633:∑
9612:∑
9599:→
9490:ℓ
9447:π
9430:ℓ
9379:∑
9370:ℓ
9327:ℓ
9262:for atom
9200:→
9190:⋅
9184:→
9172:π
9154:→
9113:∑
9100:→
9009:^
8970:^
8942:^
8918:−
8888:^
8860:^
8772:π
8741:π
8518:π
8469:π
8402:π
8306:⊂
8289:∈
8227:ω
8195:×
8159:ω
8132:⌟
8070:^
8042:∗
8034:→
8028::
8022:^
7991:−
7957:→
7944::
7941:ω
7915:∈
7909:ω
7905:⌟
7890:σ
7879:…
7876:⌟
7862:−
7855:σ
7836:−
7807:…
7786:ω
7771:σ
7767:…
7755:σ
7750:ε
7744:π
7684:⋯
7662:⋯
7638:σ
7603:δ
7599:π
7532:→
7526:×
7520::
7479:…
7319:ℓ
7287:ℓ
7258:ℓ
7125:, called
7110:ℓ
7078:ℓ
6972:π
6937:×
6824:×
6804:⋅
6775:×
6722:π
6667:π
6630:−
6579:π
6467:⋅
6405:δ
6400:π
6379:⋅
6256:×
6236:⋅
6201:×
6181:⋅
6146:×
6126:⋅
6072:×
6048:π
6004:×
5980:π
5936:×
5912:π
5353:σ
5332:⋅
5304:σ
5280:π
5257:σ
5241:⋅
5213:σ
5194:π
5111:σ
5047:−
4972:, i.e. a
4906:⋅
4868:π
4818:⋅
4780:π
4750:⋅
4735:−
4712:−
4706:π
4277:δ
4272:π
4251:⋅
4049:π
3892:∈
3866:π
3845:⋅
3790:⋅
3738:⋅
3700:⋅
3661:∑
3626:⋅
3587:∑
3573:⋅
3534:∑
3262:⋅
3223:∑
2987:∈
2602:⋅
2421:π
2370:⋅
2041:δ
2007:δ
2002:π
1981:⋅
1897:⋅
1867:⋅
1834:π
1813:⋅
1775:λ
1741:⋅
1717:λ
1689:λ
1631:λ
1610:π
1540:π
1534:−
1514:π
1210:π
1110:λ
1059:π
1047:φ
964:wavefront
950:φ
926:λ
913:π
834:φ
823:⋅
812:
778:φ
768:ω
765:−
757:⋅
746:
718:λ
710:π
678:π
669:λ
643:λ
559:ω
497:ω
443:φ
407:φ
397:ω
394:−
382:
293:π
193:ℏ
146:ℏ
11203:Category
11038:Journals
10970:OctaDist
10965:JANA2020
10937:Software
10823:Electron
10740:F-center
10527:Eutectic
10488:Fiveling
10483:Twinning
10476:Equiaxed
10269:Archived
10211:Geometry
10085:See also
9933:up to a
9342:, where
9270:, while
8368:of side
8338:rotation
8278:, where
5341:′
5292:′
5055:′
5011:′
2513:and the
1077:, where
994:at time
163:, where
63:of the
57:energies
53:electron
11215:Commons
11163:Germany
10840:Neutron
10730:Vacancy
10589:Defects
10574:GP-zone
10420:Systems
10176:Bibcode
9976:of the
9961:abelian
9954:lattice
9825:lattice
9258:is the
8336:is the
7974:is the
7450:with a
7334:in the
6285:is the
3917:vectors
2834:3-tuple
2497:due to
1099:integer
938:is the
218:vectors
205:is the
115:to the
113:is dual
105:k space
65:lattice
11158:France
11153:Europe
11086:Awards
10616:Growth
10466:Growth
10232:
10194:
10015:matrix
9997:dim =
9959:in an
9935:scalar
9849:. The
9571:where
7933:Here,
7340:normal
7223:, and
6100:where
6057:
5989:
5921:
4642:where
4230:where
3980:, and
3881:where
3089:where
3007:where
2943:where
2505:, the
2306:, and
1651:where
872:where
657:where
11180:Japan
11127:IOBCr
10980:SHELX
10975:Olex2
10862:X-ray
10512:Phase
10428:Cubic
10192:S2CID
10166:arXiv
9909:with
9856:^ to
9834:, of
9784:Here
9300:from
9297:h,k,ℓ
9229:Here
7452:basis
6501:above
4946:Here
3208:as a
2501:. In
2485:is a
1960:with
432:phase
313:is a
75:is a
49:X-ray
11122:IUCr
11023:ICDD
11018:ICSD
11003:CCDC
10950:Coot
10945:CCP4
10696:Slip
10659:Kink
10277:Jmol
10230:ISBN
8761:and
8710:and
8420:(or
8124:and
7428:real
2666:and
2556:and
2481:The
1846:and
1471:and
1407:are
1017:and
614:and
571:and
531:and
485:and
309:The
51:and
39:The
11137:DMG
11132:RAS
11028:PDB
11013:COD
11008:CIF
10960:DSR
10684:GND
10611:CSL
10184:doi
9991:In
9980:of
9917:to
9905:of
9890:on
9237:/(2
8505:FCC
8501:BCC
7093:or
3371:As
2832:as
2509:of
1526:or
809:cos
743:cos
379:cos
122:In
103:or
95:of
79:in
71:or
11236::
11175:US
11168:UK
10275:–
10190:.
10182:.
10174:.
10160:.
9988:.
9949:.
9838:.
9790:jk
9564:=
9548:,
9539:,
9290:=
9233:=
8538:.
8487:.
8293:SO
8148:.
7978:,
7176:,
7129:;
6507::
4097:,
4090:.
2277:,
592:).
119:.
10316:e
10309:t
10302:v
10238:.
10198:.
10186::
10178::
10168::
10162:3
10067:1
10059:)
10055:B
10049:T
10044:B
10039:(
10034:B
10031:=
10028:A
10018:B
10008:R
10004:R
9999:n
9986:L
9982:G
9974:L
9970:G
9957:L
9947:V
9943:L
9939:Q
9927:V
9919:V
9915:V
9911:V
9907:V
9903:V
9892:V
9888:Q
9874:L
9870:V
9866:L
9862:L
9858:V
9854:V
9832:V
9828:L
9799:k
9795:j
9786:r
9769:.
9762:k
9759:j
9746:r
9730:g
9724:i
9718:2
9714:e
9709:]
9700:g
9694:[
9688:k
9684:f
9679:]
9670:g
9664:[
9658:j
9654:f
9648:N
9643:1
9640:=
9637:k
9627:N
9622:1
9619:=
9616:j
9608:=
9605:]
9596:g
9590:[
9587:I
9573:F
9569:F
9566:F
9562:I
9554:j
9550:w
9545:j
9541:v
9536:j
9532:u
9528:m
9524:j
9504:)
9498:j
9494:w
9487:+
9482:j
9478:v
9474:k
9471:+
9466:j
9462:u
9458:h
9454:(
9450:i
9444:2
9440:e
9435:]
9427:,
9424:k
9421:,
9418:h
9414:g
9410:[
9404:j
9400:f
9394:m
9389:1
9386:=
9383:j
9375:=
9367:,
9364:k
9361:,
9358:h
9354:F
9330:)
9324:,
9321:k
9318:,
9315:h
9312:(
9302:M
9295:F
9292:M
9288:F
9281:j
9276:j
9272:r
9268:g
9264:j
9255:j
9251:f
9247:N
9243:q
9239:π
9235:q
9231:g
9214:.
9207:j
9197:r
9181:g
9175:i
9169:2
9165:e
9160:]
9151:g
9145:[
9138:j
9134:f
9128:N
9123:1
9120:=
9117:j
9109:=
9106:]
9097:g
9091:[
9088:F
9065:F
9015:.
9006:z
9000:c
8997:=
8988:3
8984:a
8976:,
8967:y
8961:a
8956:2
8953:1
8948:+
8939:x
8933:a
8928:2
8924:3
8915:=
8906:2
8902:a
8894:,
8885:y
8879:a
8874:2
8871:1
8866:+
8857:x
8851:a
8846:2
8842:3
8836:=
8827:1
8823:a
8807:c
8793:)
8788:3
8783:a
8780:(
8776:/
8769:4
8749:c
8745:/
8738:2
8718:c
8698:a
8666:)
8660:3
8655:b
8650:,
8645:2
8640:b
8635:,
8630:1
8625:b
8619:(
8597:)
8591:3
8586:a
8581:,
8576:2
8571:a
8566:,
8561:1
8556:a
8550:(
8526:a
8522:/
8515:4
8473:a
8466:4
8433:a
8430:1
8406:a
8399:2
8376:a
8324:)
8321:V
8318:,
8315:V
8312:(
8309:L
8303:)
8300:2
8297:(
8286:R
8266:)
8263:w
8260:,
8257:v
8254:R
8251:(
8248:g
8245:=
8242:)
8239:w
8236:,
8233:v
8230:(
8207:)
8204:w
8201:,
8198:v
8192:u
8189:(
8186:g
8183:=
8180:)
8177:w
8174:,
8171:v
8168:,
8165:u
8162:(
8112:)
8109:w
8106:,
8103:v
8100:(
8097:g
8094:=
8091:)
8088:w
8085:(
8082:)
8079:v
8076:(
8067:g
8038:V
8031:V
8019:g
7994:1
7987:g
7961:R
7952:n
7948:V
7918:V
7912:)
7899:i
7894:1
7884:a
7870:i
7865:1
7859:n
7849:a
7844:(
7839:1
7832:g
7825:)
7820:n
7815:a
7810:,
7804:,
7799:1
7794:a
7789:(
7780:i
7775:n
7764:i
7759:1
7741:2
7738:=
7733:i
7728:b
7700:,
7695:)
7689:1
7679:3
7674:2
7667:n
7657:2
7652:1
7646:(
7641:=
7610:j
7607:i
7596:2
7593:=
7590:)
7585:j
7580:b
7575:,
7570:i
7565:a
7560:(
7557:g
7536:R
7529:V
7523:V
7517:g
7497:)
7492:n
7487:a
7482:,
7476:,
7471:1
7466:a
7461:(
7438:V
7423:-
7411:n
7391:n
7357:m
7352:K
7322:)
7316:k
7313:h
7310:(
7290:)
7284:k
7281:h
7278:(
7236:3
7232:m
7211:k
7189:2
7185:m
7164:h
7142:1
7138:m
7113:)
7107:k
7104:h
7101:(
7081:)
7075:,
7072:k
7069:,
7066:h
7063:(
7043:)
7038:3
7034:m
7030:,
7025:2
7021:m
7017:,
7012:1
7008:m
7004:(
7001:=
6998:m
6969:2
6947:3
6942:a
6932:2
6927:a
6903:1
6898:a
6874:1
6869:b
6840:)
6834:3
6829:a
6819:2
6814:a
6808:(
6799:1
6794:a
6785:3
6780:a
6770:2
6765:a
6757:=
6752:1
6747:b
6719:2
6715:/
6709:m
6704:G
6699:=
6694:m
6689:K
6664:2
6638:.
6633:1
6625:]
6619:3
6614:a
6607:2
6602:a
6595:1
6590:a
6584:[
6576:2
6573:=
6567:T
6561:]
6555:3
6550:b
6543:2
6538:b
6531:1
6526:b
6520:[
6487:1
6484:=
6477:n
6472:R
6462:m
6457:G
6452:i
6448:e
6412:j
6409:i
6397:2
6394:=
6389:j
6384:b
6374:i
6369:a
6345:)
6339:3
6334:b
6329:,
6324:2
6319:b
6314:,
6308:1
6304:b
6298:(
6272:)
6266:2
6261:a
6251:1
6246:a
6240:(
6231:3
6226:a
6221:=
6217:)
6211:1
6206:a
6196:3
6191:a
6185:(
6176:2
6171:a
6166:=
6162:)
6156:3
6151:a
6141:2
6136:a
6130:(
6121:1
6116:a
6111:=
6108:V
6082:2
6077:a
6067:1
6062:a
6052:V
6045:2
6039:=
6030:3
6025:b
6014:1
6009:a
5999:3
5994:a
5984:V
5977:2
5971:=
5962:2
5957:b
5946:3
5941:a
5931:2
5926:a
5916:V
5909:2
5903:=
5894:1
5889:b
5863:)
5857:3
5852:b
5847:,
5842:2
5837:b
5832:,
5826:1
5822:b
5816:(
5795:)
5790:3
5786:m
5782:,
5777:2
5773:m
5769:,
5764:1
5760:m
5756:(
5753:=
5750:m
5728:3
5723:b
5716:3
5712:m
5708:+
5703:2
5698:b
5691:2
5687:m
5683:+
5678:1
5673:b
5666:1
5662:m
5658:=
5653:m
5648:G
5625:)
5619:3
5615:n
5611:,
5606:2
5602:n
5598:,
5593:1
5589:n
5584:(
5580:=
5577:n
5556:)
5550:3
5545:a
5540:,
5535:2
5530:a
5525:,
5519:1
5515:a
5509:(
5483:3
5478:a
5471:3
5467:n
5463:+
5458:2
5453:a
5446:2
5442:n
5438:+
5433:1
5428:a
5421:1
5417:n
5413:=
5408:n
5403:R
5370:.
5362:)
5359:n
5356:(
5348:a
5337:Q
5327:n
5322:a
5313:)
5310:n
5307:(
5299:a
5288:Q
5277:2
5274:=
5266:)
5263:n
5260:(
5252:a
5245:Q
5236:n
5231:a
5222:)
5219:n
5216:(
5208:a
5201:Q
5191:2
5188:=
5183:n
5178:b
5148:)
5142:1
5137:2
5130:2
5125:1
5119:(
5114:=
5084:v
5062:v
5052:Q
5044:=
5040:v
5034:Q
5008:Q
4985:Q
4974:q
4955:Q
4922:1
4917:a
4910:Q
4901:2
4896:a
4887:1
4882:a
4875:Q
4865:2
4862:=
4853:2
4848:b
4834:2
4829:a
4822:Q
4813:1
4808:a
4799:2
4794:a
4787:Q
4777:2
4774:=
4766:2
4761:a
4754:Q
4745:1
4740:a
4728:2
4723:a
4716:Q
4703:2
4700:=
4691:1
4686:b
4655:i
4651:m
4625:2
4620:b
4613:2
4609:m
4605:+
4600:1
4595:b
4588:1
4584:m
4580:=
4575:m
4570:G
4544:)
4538:2
4533:a
4528:,
4523:1
4518:a
4512:(
4478:m
4473:G
4449:m
4444:G
4413:)
4407:3
4402:b
4397:,
4392:2
4387:b
4382:,
4376:1
4372:b
4366:(
4343:n
4338:R
4314:m
4309:G
4284:j
4281:i
4269:2
4266:=
4261:j
4256:b
4246:i
4241:a
4216:3
4211:b
4204:3
4200:m
4196:+
4191:2
4186:b
4179:2
4175:m
4171:+
4166:1
4161:b
4154:1
4150:m
4146:=
4141:m
4136:G
4112:m
4107:G
4076:n
4071:R
4046:2
4024:n
4019:R
3995:m
3990:G
3966:n
3961:R
3933:m
3928:G
3900:.
3896:Z
3889:N
3869:N
3863:2
3860:=
3855:n
3850:R
3840:m
3835:G
3810:1
3807:=
3800:n
3795:R
3785:m
3780:G
3775:i
3771:e
3748:.
3742:r
3733:m
3728:G
3723:i
3719:e
3710:n
3705:R
3695:m
3690:G
3685:i
3681:e
3675:m
3671:f
3665:m
3657:=
3652:)
3647:n
3642:R
3637:+
3633:r
3629:(
3621:m
3616:G
3611:i
3607:e
3601:m
3597:f
3591:m
3583:=
3577:r
3568:m
3563:G
3558:i
3554:e
3548:m
3544:f
3538:m
3510:.
3507:)
3503:r
3499:(
3496:f
3493:=
3490:)
3485:n
3480:R
3475:+
3471:r
3467:(
3464:f
3439:n
3434:R
3411:r
3390:)
3386:r
3382:(
3379:f
3356:)
3351:3
3347:m
3343:,
3338:2
3334:m
3330:,
3325:1
3321:m
3317:(
3314:=
3311:m
3287:)
3283:r
3279:(
3275:f
3272:=
3266:r
3257:m
3252:G
3247:i
3243:e
3237:m
3233:f
3227:m
3196:)
3192:r
3188:(
3185:f
3165:)
3161:r
3157:(
3154:f
3134:0
3131:=
3126:n
3121:R
3098:r
3077:)
3073:r
3069:(
3066:f
3044:i
3039:a
3016:Z
2991:Z
2982:3
2978:n
2974:,
2969:2
2965:n
2961:,
2956:1
2952:n
2929:3
2924:a
2917:3
2913:n
2909:+
2904:2
2899:a
2892:2
2888:n
2884:+
2879:1
2874:a
2867:1
2863:n
2859:=
2854:n
2849:R
2820:)
2815:3
2811:n
2807:,
2802:2
2798:n
2794:,
2789:1
2785:n
2781:(
2778:=
2775:n
2747:R
2739:(
2727:r
2724:+
2721:R
2701:r
2679:2
2675:b
2652:1
2648:b
2627:G
2605:r
2599:G
2596:i
2592:e
2569:2
2565:a
2542:1
2538:a
2465:G
2443:G
2418:2
2396:R
2374:R
2366:G
2344:G
2321:3
2316:b
2292:2
2287:b
2263:1
2258:b
2230:j
2226:m
2215:,
2201:3
2196:b
2189:3
2185:m
2181:+
2176:2
2171:b
2164:2
2160:m
2156:+
2151:1
2146:b
2139:1
2135:m
2131:=
2127:G
2103:j
2098:b
2076:j
2073:=
2070:i
2048:j
2045:i
2014:j
2011:i
1999:2
1996:=
1991:j
1986:b
1976:i
1971:a
1946:j
1941:b
1927:.
1915:0
1912:=
1907:1
1902:b
1892:3
1887:a
1882:=
1877:1
1872:b
1862:2
1857:a
1831:2
1828:=
1823:1
1818:b
1808:1
1803:a
1779:1
1751:1
1746:e
1736:1
1731:a
1726:=
1721:1
1693:1
1666:1
1661:e
1649:,
1635:1
1626:/
1620:1
1615:e
1607:2
1604:=
1599:1
1594:b
1568:1
1563:a
1537:2
1511:2
1501:,
1487:3
1482:a
1457:2
1452:a
1429:0
1426:=
1422:R
1393:i
1388:a
1364:i
1360:n
1349:,
1335:3
1330:a
1323:3
1319:n
1315:+
1310:2
1305:a
1298:2
1294:n
1290:+
1285:1
1280:a
1273:1
1269:n
1265:=
1261:R
1236:n
1216:n
1213:)
1207:2
1204:(
1183:)
1179:r
1175:(
1172:f
1147:k
1122:.
1085:n
1065:n
1062:)
1056:2
1053:(
1050:+
1026:e
1015:,
1003:t
982:0
979:=
975:r
922:/
917:e
910:2
907:=
903:k
881:r
870:,
858:t
837:)
831:+
827:r
819:k
815:(
799:,
787:)
782:0
774:+
771:t
761:r
753:k
749:(
730:.
714:/
707:2
704:=
701:k
690:;
675:2
672:=
666:k
655:,
622:x
602:k
580:t
539:x
519:k
509:,
473:k
461:,
447:0
428:,
416:)
411:0
403:+
400:t
391:x
388:k
385:(
361:L
357:L
340:k
290:2
267:n
262:R
234:m
229:G
172:p
150:k
143:=
139:p
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
.svg)