Bloch's theorem - Knowledge

8509: 7282: 8504:{\displaystyle {\begin{aligned}E_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )&={\frac {-\hbar ^{2}}{2m}}\nabla \cdot \left(i\mathbf {k} e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )+e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )\right)+U(\mathbf {x} )e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\\E_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )&={\frac {-\hbar ^{2}}{2m}}\left(i\mathbf {k} \cdot \left(i\mathbf {k} e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )+e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )\right)+i\mathbf {k} \cdot e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )+e^{i\mathbf {k} \cdot \mathbf {x} }\nabla ^{2}u_{\mathbf {k} }(\mathbf {x} )\right)+U(\mathbf {x} )e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\\E_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )&={\frac {\hbar ^{2}}{2m}}\left(\mathbf {k} ^{2}e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )-2i\mathbf {k} \cdot e^{i\mathbf {k} \cdot \mathbf {x} }\nabla u_{\mathbf {k} }(\mathbf {x} )-e^{i\mathbf {k} \cdot \mathbf {x} }\nabla ^{2}u_{\mathbf {k} }(\mathbf {x} )\right)+U(\mathbf {x} )e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\\E_{\mathbf {k} }u_{\mathbf {k} }(\mathbf {x} )&={\frac {\hbar ^{2}}{2m}}\left(-i\nabla +\mathbf {k} \right)^{2}u_{\mathbf {k} }(\mathbf {x} )+U(\mathbf {x} )u_{\mathbf {k} }(\mathbf {x} )\end{aligned}}} 11959: 10717: 2601: 11520: 10278: 2200: 11060: 11954:{\displaystyle {\frac {\partial ^{2}\varepsilon _{n}(\mathbf {k} )}{\partial k_{i}\partial k_{j}}}={\frac {\hbar ^{2}}{m}}\delta _{ij}+\left({\frac {\hbar ^{2}}{m}}\right)^{2}\sum _{n'\neq n}{\frac {\langle n\mathbf {k} |-i\nabla _{i}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{j}|n\mathbf {k} \rangle +\langle n\mathbf {k} |-i\nabla _{j}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{i}|n\mathbf {k} \rangle }{\varepsilon _{n}(\mathbf {k} )-\varepsilon _{n'}(\mathbf {k} )}}} 10712:{\displaystyle {\frac {\partial ^{2}\varepsilon _{n}(\mathbf {k} )}{\partial k_{i}\partial k_{j}}}={\frac {\hbar ^{2}}{m}}\delta _{ij}+\left({\frac {\hbar ^{2}}{m}}\right)^{2}\sum _{n'\neq n}{\frac {\langle n\mathbf {k} |-i\nabla _{i}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{j}|n\mathbf {k} \rangle +\langle n\mathbf {k} |-i\nabla _{j}|n'\mathbf {k} \rangle \langle n'\mathbf {k} |-i\nabla _{i}|n\mathbf {k} \rangle }{\varepsilon _{n}(\mathbf {k} )-\varepsilon _{n'}(\mathbf {k} )}}} 11461: 13173: 13601: 2596:{\displaystyle {\begin{aligned}u(\mathbf {r} +\mathbf {a} _{j})&=e^{-i\mathbf {k} \cdot (\mathbf {r} +\mathbf {a} _{j})}\psi (\mathbf {r} +\mathbf {a} _{j})\\&={\big (}e^{-i\mathbf {k} \cdot \mathbf {r} }e^{-i\mathbf {k} \cdot \mathbf {a} _{j}}{\big )}{\big (}e^{2\pi i\theta _{j}}\psi (\mathbf {r} ){\big )}\\&=e^{-i\mathbf {k} \cdot \mathbf {r} }e^{-2\pi i\theta _{j}}e^{2\pi i\theta _{j}}\psi (\mathbf {r} )\\&=u(\mathbf {r} ).\end{aligned}}} 38: 10729: 13625: 3141: 53: 11153: 727: 8837: 13637: 3839: 9238: 13613: 2910: 4275: 11055:{\displaystyle {\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}\varepsilon _{n}}{\partial k_{i}\partial k_{j}}}q_{i}q_{j}={\frac {\hbar ^{2}}{2m}}q^{2}+\sum _{n'\neq n}{\frac {|\int d\mathbf {r} \,u_{n\mathbf {k} }^{*}{\frac {\hbar ^{2}}{m}}\mathbf {q} \cdot (-i\nabla +\mathbf {k} )u_{n'\mathbf {k} }|^{2}}{\varepsilon _{n\mathbf {k} }-\varepsilon _{n'\mathbf {k} }}}} 3634: 8994: 9924: 9731: 8655: 11456:{\displaystyle {\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}\varepsilon _{n}}{\partial k_{i}\partial k_{j}}}q_{i}q_{j}={\frac {\hbar ^{2}}{2m}}q^{2}+\sum _{n'\neq n}{\frac {|\langle n\mathbf {k} |{\frac {\hbar ^{2}}{m}}\mathbf {q} \cdot (-i\nabla )|n'\mathbf {k} \rangle |^{2}}{\varepsilon _{n\mathbf {k} }-\varepsilon _{n'\mathbf {k} }}}} 7275: 6654: 3288: 10133: 4052: 9486: 10261: 4515: 6901: 697:

The most common example of Bloch's theorem is describing electrons in a crystal, especially in characterizing the crystal's electronic properties, such as electronic band structure. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium. For example,

3136:{\displaystyle {\begin{aligned}{\hat {\mathbf {T} }}_{\mathbf {n} }\psi (\mathbf {r} )&=\psi (\mathbf {r} +\mathbf {T} _{\mathbf {n} })\\&=\psi (\mathbf {r} +n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3})\\&=\psi (\mathbf {r} +\mathbf {A} \mathbf {n} )\end{aligned}}} 9741: 4883: 6438: 9491: 1425:

The defining property of a crystal is translational symmetry, which means that if the crystal is shifted an appropriate amount, it winds up with all its atoms in the same places. (A finite-size crystal cannot have perfect translational symmetry, but it is a useful approximation.)

7072: 1395: 8832:{\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}={\frac {\hbar ^{2}}{m}}\int d\mathbf {r} \,\psi _{n\mathbf {k} }^{*}(-i\nabla )\psi _{n\mathbf {k} }={\frac {\hbar }{m}}\langle {\hat {\mathbf {p} }}\rangle =\hbar \langle {\hat {\mathbf {v} }}\rangle } 12132: 6534: 3146: 3834:{\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n} _{1}}{\hat {\mathbf {T} }}_{\mathbf {n} _{2}}\psi (\mathbf {x} )=\psi (\mathbf {x} +\mathbf {A} \mathbf {n} _{1}+\mathbf {A} \mathbf {n} _{2})={\hat {\mathbf {T} }}_{\mathbf {n} _{1}+\mathbf {n} _{2}}\psi (\mathbf {x} )} 9233:{\displaystyle \varepsilon _{n}(\mathbf {k} +\mathbf {q} )=\varepsilon _{n}(\mathbf {k} )+\sum _{i}{\frac {\partial \varepsilon _{n}}{\partial k_{i}}}q_{i}+{\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}\varepsilon _{n}}{\partial k_{i}\partial k_{j}}}q_{i}q_{j}+O(q^{3})} 10002: 2887:

operator. This basis is what we are looking for. The wave functions in this basis are energy eigenstates (because they are eigenstates of the Hamiltonian), and they are also Bloch states (because they are eigenstates of the translation operators; see Lemma above).

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A Bloch wave function (bottom) can be broken up into the product of a periodic function (top) and a plane-wave (center). The left side and right side represent the same Bloch state broken up in two different ways, involving the wave vector

4270:{\displaystyle 1=\int _{V}|\psi (\mathbf {x} )|^{2}d\mathbf {x} =\int _{V}\left|{\hat {\mathbf {T} }}_{\mathbf {n} }\psi (\mathbf {x} )\right|^{2}d\mathbf {x} =|\lambda _{\mathbf {n} }|^{2}\int _{V}|\psi (\mathbf {x} )|^{2}d\mathbf {x} } 5781: 2048: 12243: 6975: 2195: 6529: 2679: 1141: 845: 8579: 165: 4799: 12264:

The concept of the Bloch state was developed by Felix Bloch in 1928 to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by

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operators gives three commuting cyclic subgroups (given they can be generated by only one element) which are infinite, 1-dimensional and abelian. All irreducible representations of abelian groups are one dimensional.

4794: 6262: 6047: 5929: 652: 9919:{\displaystyle \sum _{i}{\frac {\partial \varepsilon _{n}}{\partial k_{i}}}q_{i}=\sum _{i}\int d\mathbf {r} \,u_{n\mathbf {k} }^{*}{\frac {\hbar ^{2}}{m}}(-i\nabla +\mathbf {k} )_{i}q_{i}u_{n\mathbf {k} }} 5281: 9284: 12048: 3875: 4693:

Apart from the group theory technicalities this proof is interesting because it becomes clear how to generalize the Bloch theorem for groups that are not only translations. This is typically done for

3511: 5498: 9330: 984:. Therefore, the first Brillouin zone is often used to depict all of the Bloch states without redundancy, for example in a band structure, and it is used for the same reason in many calculations. 9726:{\displaystyle E_{n}=E_{n}^{0}+\int d\mathbf {r} \,\psi _{n}^{*}{\hat {V}}\psi _{n}+\sum _{n'\neq n}{\frac {|\int d\mathbf {r} \,\psi _{n}^{*}{\hat {V}}\psi _{n}|^{2}}{E_{n}^{0}-E_{n'}^{0}}}+...} 3629: 1157: 7270:{\displaystyle E_{\mathbf {k} }\left(e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\right)=\left\left(e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )\right)} 2915: 2205: 12369: 6144: 4991: 2885: 4327: 7037: 2755: 1731: 6649:{\displaystyle {\hat {P}}_{\varepsilon |{\boldsymbol {\tau }}}\psi (\mathbf {r} )=\psi (\mathbf {r} )e^{i\mathbf {k} \cdot {\boldsymbol {\tau }}}=\psi (\mathbf {r} +{\boldsymbol {\tau }})} 4520: 3283:{\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {a} _{1}&\mathbf {a} _{2}&\mathbf {a} _{3}\end{bmatrix}},\quad \mathbf {n} ={\begin{pmatrix}n_{1}\\n_{2}\\n_{3}\end{pmatrix}}} 12029: 5593: 5144: 11996: 4709:. In this proof it is also possible to notice how it is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian. 10128:{\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}={\frac {\hbar ^{2}}{m}}\int d\mathbf {r} \,u_{n\mathbf {k} }^{*}(-i\nabla +\mathbf {k} )u_{n\mathbf {k} }} 6082: 564: 510: 392: 5685: 5106: 1964: 3870: 11065: 8633: 9481:{\displaystyle {\hat {H}}_{\mathbf {k} +\mathbf {q} }={\hat {H}}_{\mathbf {k} }+{\frac {\hbar ^{2}}{m}}\mathbf {q} \cdot (-i\nabla +\mathbf {k} )+{\frac {\hbar ^{2}}{2m}}q^{2}} 6906: 4384: 4047: 2122: 10256:{\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}={\frac {\hbar ^{2}}{m}}\int d\mathbf {r} \,\psi _{n\mathbf {k} }^{*}(-i\nabla )\psi _{n\mathbf {k} }} 6443: 8608: 7061: 6999: 1072: 776: 8517: 5064: 4609: 953:(see figure at right). Therefore, wave vectors that differ by a reciprocal lattice vector are equivalent, in the sense that they characterize the same set of Bloch states. 12157: 5414: 5136: 1295: 1273: 674: 590: 532: 438: 263: 190: 8895: 4705:

and it is used for computing the band structure, spectrum and specific heats of crystals given a specific crystal group symmetry like FCC or BCC and eventually an extra

13042:"A tutorial survey on waves propagating in periodic media: Electronic, photonic and phononic crystals. Perception of the Bloch theorem in both real and Fourier domains" 4510:{\displaystyle \mathbf {{\hat {T}}_{n}} \psi (\mathbf {x} )=\psi (\mathbf {x} +\mathbf {n} \cdot \mathbf {a} )=e^{ik\mathbf {n} \cdot \mathbf {a} }\psi (\mathbf {x} ),} 3362: 1458: 1020: 5031: 13087: 6896:{\displaystyle {\hat {H}}_{\mathbf {k} }u_{\mathbf {k} }(\mathbf {r} )=\leftu_{\mathbf {k} }(\mathbf {r} )=\varepsilon _{\mathbf {k} }u_{\mathbf {k} }(\mathbf {r} )} 5509: 4888: 11515: 11488: 6254: 6227: 6200: 6173: 5810: 5441: 1067: 919: 871: 210: 4356: 1822: 12176: 3969: 3506: 3293: 1410:

Being Bloch's theorem a statement about lattice periodicity, in this proof all the symmetries are encoded as translation symmetries of the wave function itself.

9929: 6683:(as the invariants of the irreducible representations) can be treated as the fundamental building blocks instead of the irreducible representations themselves. 4731: 478: 458: 412: 311: 287: 234: 2610: 99: 12623: 5934: 5815: 3508:

and the two operators shall have a common set of eigenfunctions. Therefore, we start to look at the eigen-functions of the translation operator:

8849: 8635:. More precisely the crystal momentum is not a momentum but it stands for the momentum in the same way as the electromagnetic momentum in the 13080: 4878:{\displaystyle {\hat {\boldsymbol {\tau }}}={\hat {\boldsymbol {\tau }}}_{1}{\hat {\boldsymbol {\tau }}}_{2}{\hat {\boldsymbol {\tau }}}_{3}} 6433:{\displaystyle \chi _{k_{1}}(n_{1},a_{1})\chi _{k_{2}}(n_{2},a_{2})\chi _{k_{3}}(n_{3},a_{3})=e^{i\mathbf {k} \cdot {\boldsymbol {\tau }}}} 5009:

which in this case is a one dimensional matrix. All these subgroups, given they are cyclic, they have characters which are appropriate

12606: 12296: 1390:{\displaystyle \psi _{\mathbf {k} }(\mathbf {x} +\mathbf {a} )=e^{i\mathbf {k} \cdot \mathbf {a} }\psi _{\mathbf {k} }(\mathbf {x} ).} 13617: 12569: 4280: 595: 13073: 12127:{\displaystyle \mathbf {M} (\mathbf {k} )\mathbf {a} =\mp e\left(\mathbf {E} +\mathbf {v} (\mathbf {k} )\times \mathbf {B} \right)} 4717: 1734: 6660:

In the generalized version of the Bloch theorem, the Fourier transform, i.e. the wave function expansion, gets generalized from a

9243: 853:

is periodic with the same periodicity as the crystal lattice. The actual quantum state of the electron is entirely determined by

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As an intuitive interpretation, both of the previous two equations resemble formally and are in a semi-classical analogy with

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For a detailed example in which the consequences of Bloch's theorem are worked out in a specific situation, see the article

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Translated by A. T. Fuller from Edouard Davaux's French translation (1907) of the original Russian dissertation (1892).

6665: 6087: 5108:. Note that this is straightforward in the finite cyclic group case but in the countable infinite case of the infinite 3356: 56:

Solid line: A schematic of the real part of a typical Bloch state in one dimension. The dotted line is from the factor

5383:{\displaystyle V\left(\mathbf {r} +\sum _{i}N_{i}\mathbf {a} _{i}\right)=V(\mathbf {r} +\mathbf {L} )=V(\mathbf {r} )} 4960: 2820: 13673: 13605: 12849: 12486: 12462:

Bloch, F. (1929). Über die quantenmechanik der elektronen in kristallgittern. Zeitschrift für physik, 52(7), 555-600.

7004: 4599:{\displaystyle \psi _{\mathbf {k} }(\mathbf {x} )=e^{i\mathbf {k} \cdot \mathbf {x} }u_{\mathbf {k} }(\mathbf {x} )} 2691: 1667: 12393:

Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is applied to

3957:{\displaystyle \lambda _{\mathbf {n} _{1}}\lambda _{\mathbf {n} _{2}}=\lambda _{\mathbf {n} _{1}+\mathbf {n} _{2}}} 3584:{\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n} }\psi (\mathbf {x} )=\lambda _{\mathbf {n} }\psi (\mathbf {x} )} 12839: 12432: 12278: 715: 12001: 12511: 13641: 13586: 13197: 11967: 1232:{\displaystyle u_{\mathbf {k} }(\mathbf {x} )=u_{\mathbf {k} }(\mathbf {x} +\mathbf {n} \cdot \mathbf {a} ).} 12748:"On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon" 13132: 13023: 12427: 11144:{\displaystyle \psi _{n\mathbf {k} }=|n\mathbf {k} \rangle =e^{i\mathbf {k} \mathbf {x} }u_{n\mathbf {k} }} 6052: 537: 483: 365: 12917:

Kotani M; Sunada T. (2000). "Albanese maps and an off diagonal long time asymptotic for the heat kernel".

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For any wave function that satisfies the Schrödinger equation and for a translation of a lattice vector

13668: 13377: 13286: 12422: 12290: 6661: 4361: 4024: 681: 4678:{\displaystyle u_{\mathbf {k} }(\mathbf {x} )=u_{\mathbf {k} }(\mathbf {x} +\mathbf {A} \mathbf {n} )} 13326: 13301: 12417: 8584: 1653: 1559: 356: 290: 7042: 6980: 5001:

are the same. The character is the representation over the complex numbers of the group or also the

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In this proof all the symmetries are encoded as commutation properties of the translation operators

30:

This article is about a theorem in quantum mechanics. For the theorem used in complex analysis, see

13496: 13250: 13240: 13096: 8967:{\displaystyle {\frac {\partial ^{2}\varepsilon _{n}(\mathbf {k} )}{\partial k_{i}\partial k_{j}}}} 8640: 5036: 3441:

Given the Hamiltonian is invariant for translations it shall commute with the translation operator

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is a parameter of the Hamiltonian and therefore we arrive at a "continuous family" of eigenvalues

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is equivalent to one (and only one) vector in the first Brillouin zone. Therefore, if we restrict

657: 573: 515: 421: 246: 173: 13342: 12383: 5670:{\displaystyle \psi \left(\mathbf {r} +\sum _{i}N_{i}\mathbf {a} _{i}\right)=\psi (\mathbf {r} )} 5265:{\displaystyle \chi _{k_{1}}({\hat {\boldsymbol {\tau }}}_{1}(n_{1},a_{1}))=e^{ik_{1}n_{1}a_{1}}} 994: 12593:

The vibrational spectrum and specific heat of a face centered cubic crystal, Robert B. Leighton

6692: 5503: 1455:. If the crystal is shifted by any of these three vectors, or a combination of them of the form 9926:

where the integrations are over a primitive cell or the entire crystal, given if the integral

5141:

Given the character is a root of unity, for each subgroup the character can be then written as

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are integers. Because the crystal has translational symmetry, this operator commutes with the

13393: 13372: 13306: 5776:{\displaystyle {\hat {P}}_{\varepsilon |\tau _{i}+L_{i}}={\hat {P}}_{\varepsilon |\tau _{i}}} 5016: 2813:. Moreover, every such translation operator commutes with every other. Therefore, there is a 2043:{\displaystyle \psi (\mathbf {r} +\mathbf {a} _{j})=e^{2\pi i\theta _{j}}\psi (\mathbf {r} )} 12238:{\displaystyle \hbar {\dot {k}}=-e\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)} 6256:. In this context, the wave vector serves as a quantum number for the translation operator. 13362: 13137: 12926: 12709: 12594: 11493: 11466: 6977:

Given this is defined in a finite volume we expect an infinite family of eigenvalues; here

6232: 6205: 6178: 6151: 5788: 5419: 5006: 2810: 1052: 957: 904: 856: 195: 12558: 6970:{\displaystyle u_{\mathbf {k} }(\mathbf {r} )=u_{\mathbf {k} }(\mathbf {r} +\mathbf {R} )} 2190:{\displaystyle u(\mathbf {r} )=e^{-i\mathbf {k} \cdot \mathbf {r} }\psi (\mathbf {r} )\,.} 8: 13316: 13296: 13281: 13230: 12743: 12554: 12437: 12266: 6524:{\displaystyle {\hat {P}}_{R}\psi _{j}=\sum _{\alpha }\psi _{\alpha }\chi _{\alpha j}(R)} 12930: 12713: 1554:

are three integers, then the atoms end up in the same set of locations as they started.

13463: 13453: 13202: 12942: 12899: 12816: 12725: 12407: 12274: 2814: 2674:{\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} ),} 1136:{\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} ),} 950: 840:{\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} ),} 766: 567: 463: 443: 397: 296: 272: 219: 8574:{\displaystyle {\hat {\mathbf {p} }}_{\text{eff}}=-i\hbar \nabla +\hbar \mathbf {k} ,} 1005:

of an electron can be calculated based on how the energy of a Bloch state varies with

160:{\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r} )} 13624: 13571: 13536: 13443: 13367: 13010: 13000: 12979: 12973: 12946: 12845: 12729: 12507: 12482: 12394: 1819:

which is an eigenstate of all the translation operators. As a special case of this,

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From here we can see that also the character shall be invariant by a translation of

13541: 13433: 13413: 13408: 13403: 13398: 13255: 13235: 13192: 13157: 13127: 13057: 13053: 12934: 12891: 12879: 12796: 12759: 12717: 12442: 12387: 8636: 6680: 6669: 4721: 998: 711: 707: 703: 266: 13040:

J. Gazalet; S. Dupont; J.C. Kastelik; Q. Rolland & B. Djafari-Rouhani (2013).

3434:{\displaystyle {\hat {H}}={\frac {{\hat {\mathbf {p} }}^{2}}{2m}}+U(\mathbf {x} )} 1540:{\displaystyle n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3},} 13566: 13521: 13187: 13104: 12969: 12382:

is a periodic potential. Specific periodic one-dimensional equations include the

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Finally, we are ready for the main proof of Bloch's theorem which is as follows.

418:, which is present because there are many different wave functions with the same 13041: 13629: 13576: 13458: 13347: 12835: 12780: 12472: 12282: 12270: 12040: 12032: 6664:

which is applicable only for cyclic groups, and therefore translations, into a

5583:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )} 5010: 5002: 4948:{\displaystyle {\hat {\boldsymbol {\tau }}}_{i}=n_{i}{\hat {\mathbf {a} }}_{i}} 4049:

if we use the normalization condition over a single primitive cell of volume V

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are integers). The following fact is helpful for the proof of Bloch's theorem:

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each of these wave functions is a Bloch state, meaning that this wave function

1002: 677: 337: 333: 329: 314: 45: 13657: 13501: 13483: 13468: 13448: 13352: 13321: 13152: 13014: 12965: 12622:

Group Representations and Harmonic Analysis from Euler to Langlands, Part II

12253: 4725: 1885:{\displaystyle \psi (\mathbf {r} +\mathbf {a} _{j})=C_{j}\psi (\mathbf {r} )} 1800: 213: 8514:

This shows how the effective momentum can be seen as composed of two parts,

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and from the last equation we get for each dimension a periodic condition:

5109: 4014:{\displaystyle \lambda _{\mathbf {n} }=e^{s\mathbf {n} \cdot \mathbf {a} }} 3348:{\displaystyle U(\mathbf {x} +\mathbf {T} _{\mathbf {n} })=U(\mathbf {x} )} 1154:

has the same periodicity as the atomic structure of the crystal, such that

13039: 13024:"Periodic Potentials and Bloch's Theorem – lectures in "Semiconductors I"" 12938: 9983:{\displaystyle \int d\mathbf {r} \,u_{n\mathbf {k} }^{*}u_{n\mathbf {k} }} 8885:{\displaystyle {\frac {\partial \varepsilon _{n}}{\partial \mathbf {k} }}} 4789:{\displaystyle {\boldsymbol {\tau }}=\sum _{i=1}^{3}n_{i}\mathbf {a} _{i}} 3841:

If we substitute here the eigenvalue equation and dividing both sides for

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with the property that no two of them are equivalent, yet every possible

415: 241: 87: 12785:"Sur les équations différentielles linéaires à coefficients périodiques" 12289:). The general form of a one-dimensional periodic potential equation is 13473: 13311: 13147: 12903: 12801: 12784: 12764: 12721: 12277:(1892). As a result, a variety of nomenclatures are common: applied to 1907: 699: 90:, who discovered the theorem in 1929. Mathematically, they are written 79: 41: 37: 12882:(1987). "Homology and closed geodesics in a compact Riemann surface". 12700:(1928). "Über die Quantenmechanik der Elektronen in Kristallgittern". 13531: 13357: 13142: 12994: 12895: 12770:

This work was initially published and distributed privately in 1877.

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First order semi-classical equation of motion for electron in a band

6042:{\displaystyle k_{1}n_{1}a_{1}=k_{1}(n_{1}a_{1}+L_{1})-2\pi m_{1}} 1421:

Preliminaries: Crystal symmetries, lattice, and reciprocal lattice

1243:

A second and equivalent way to state the theorem is the following

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vector. In all plots, blue is real part and red is imaginary part.

13581: 13548: 13526: 13506: 5924:{\displaystyle e^{ik_{1}n_{1}a_{1}}=e^{ik_{1}(n_{1}a_{1}+L_{1})}} 4997:

Given they are one dimensional the matrix representation and the

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are the reciprocal lattice vectors (see above). Finally, define

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to the first Brillouin zone, then every Bloch state has a unique

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The description of electrons in terms of Bloch functions, termed

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and thus at the basic concept of an electronic band structure.

726: 647:{\displaystyle \psi _{n\mathbf {k} }=\psi _{n(\mathbf {k+K} )}} 12815: 8974:

given they are the coefficients of the following expansion in

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identify the irreducible representation in the same manner as

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is a macroscopic periodic length of the crystal in direction

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Group theory: application to the physics of condensed matter

12506:(2nd ed.). Cambridge University Press. pp. 17–20. 5679:

And for each dimension a translation operator with a period

13035:. Texts in Mathematics. Edinburgh: Scottish Academic Press. 12031:

and we can use it to write a semi-classical equation for a

9279:{\displaystyle \varepsilon _{n}(\mathbf {k} +\mathbf {q} )} 1803:

of all of the translation operators (simultaneously), then

1021:

Particle in a one-dimensional lattice (periodic potential)

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We can consider the following perturbation problem in q:

5493:{\textstyle \mathbf {L} =\sum _{i}N_{i}\mathbf {a} _{i}} 12559:"Applications of Group Theory to the Physics of Solids" 6440:

and the generic formula for the wave function becomes:

5112:(i.e. the translation group here) there is a limit for 4728:. Translations can be written in terms of unit vectors 1586:(with units of inverse length), with the property that 13033:

The Spectral Theory of Periodic Differential Equations

9325:{\displaystyle {\hat {H}}_{\mathbf {k} +\mathbf {q} }} 5449: 3231: 3163: 714:. It is generally treated in the various forms of the 12299: 12179: 12143: 12051: 12039:

Second order semi-classical equation of motion for a

12004: 11970: 11523: 11496: 11469: 11156: 11068: 10732: 10281: 10141: 10005: 9932: 9744: 9494: 9338: 9292: 9246: 8997: 8898: 8852: 8658: 8616: 8587: 8520: 7285: 7075: 7045: 7007: 6983: 6909: 6701: 6537: 6446: 6265: 6235: 6208: 6181: 6154: 6090: 6055: 5937: 5818: 5791: 5688: 5596: 5512: 5422: 5400: 5284: 5147: 5118: 5072: 5039: 5019: 4963: 4891: 4802: 4734: 4612: 4523: 4392: 4364: 4335: 4283: 4055: 4027: 3972: 3878: 3847: 3637: 3597: 3514: 3447: 3365: 3296: 3149: 2913: 2823: 2694: 2613: 2203: 2125: 1967: 1825: 1670: 1461: 1303: 1281: 1259: 1160: 1075: 1055: 1046:

each of these wave functions is an energy eigenstate,

1042:

of wave functions with the following two properties:

907: 859: 779: 660: 598: 576: 540: 518: 486: 466: 446: 424: 400: 368: 299: 275: 249: 222: 198: 176: 102: 12877: 9488:

Perturbation theory of the second order states that

3624:{\displaystyle {\hat {\mathbf {T} }}_{\mathbf {n} }} 1659: 12789:

Annales Scientifiques de l'École Normale Supérieure

12742: 3290:We use the hypothesis of a mean periodic potential 12992: 12363: 12237: 12151: 12126: 12023: 11990: 11953: 11509: 11482: 11455: 11143: 11054: 10711: 10255: 10127: 9982: 9918: 9725: 9480: 9324: 9278: 9232: 8966: 8884: 8831: 8627: 8602: 8573: 8503: 7269: 7055: 7031: 6993: 6969: 6895: 6648: 6523: 6432: 6248: 6221: 6194: 6167: 6138: 6076: 6041: 5923: 5804: 5775: 5669: 5582: 5492: 5435: 5408: 5382: 5264: 5130: 5100: 5058: 5025: 4985: 4947: 4877: 4788: 4677: 4598: 4509: 4378: 4350: 4321: 4269: 4041: 4013: 3956: 3864: 3833: 3623: 3583: 3500: 3433: 3347: 3282: 3135: 2879: 2749: 2673: 2595: 2189: 2042: 1884: 1725: 1539: 1389: 1289: 1267: 1231: 1135: 1061: 913: 865: 839: 668: 646: 584: 558: 526: 504: 472: 452: 432: 406: 386: 305: 281: 257: 228: 204: 184: 159: 12916: 12866:Floquet theory for partial differential equations 12549: 12547: 12364:{\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0,} 6139:{\displaystyle k_{1}={\frac {2\pi m_{1}}{L_{1}}}} 4401: 2876: 362:These eigenstates are written with subscripts as 86:. The theorem is named after the Swiss physicist 13655: 12259: 11964:The quantity on the right multiplied by a factor 10135:and we can reinsert the complete wave functions 4986:{\displaystyle {\hat {\boldsymbol {\tau }}}_{i}} 2880:{\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}\!} 687: 4322:{\displaystyle 1=|\lambda _{\mathbf {n} }|^{2}} 2817:of the Hamiltonian operator and every possible 1923:in a different form, by choosing three numbers 1557:Another helpful ingredient in the proof is the 1038:For electrons in a perfect crystal, there is a 27:Fundamental theorem in condensed matter physics 13030: 12964: 12828: 12821:The General Problem of the Stability of Motion 12779: 12684: 12672: 12660: 12648: 12633: 12544: 12538: 12526: 9990:is normalized across the cell or the crystal. 7032:{\displaystyle \varepsilon _{n}(\mathbf {k} )} 6686: 5590:induces a periodicity with the wave function: 5394:is a macroscopic periodicity in the direction 2761:that shifts every wave function by the amount 2750:{\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}} 1737:that shifts every wave function by the amount 1726:{\displaystyle {\hat {T}}_{n_{1},n_{2},n_{3}}} 240:with the same periodicity as the crystal, the 13081: 12834: 4796:We can think of these as commuting operators 2681:that proves that the state is a Bloch state. 2452: 2405: 2398: 2332: 13095: 11893: 11843: 11840: 11790: 11784: 11734: 11731: 11681: 11395: 11321: 11100: 10651: 10601: 10598: 10548: 10542: 10492: 10489: 10439: 8826: 8809: 8800: 8783: 1405: 78:in a periodic potential can be expressed as 12696: 12553: 13171: 13088: 13074: 12024:{\displaystyle \mathbf {M} (\mathbf {k} )} 5502:This substituting in the time independent 2607:has the periodicity of the lattice. Since 710:, and a periodic acoustic medium leads to 12800: 12763: 10908: 10202: 10066: 9944: 9823: 9618: 9537: 8719: 8646:For the effective velocity we can derive 6070: 4372: 4035: 2183: 2057:are three numbers which do not depend on 1011:; for more details see crystal momentum. 440:(each has a different periodic component 12978:. New York: Holt, Rinehart and Winston. 6259:We can generalize this for 3 dimensions 4517:which is true for a Bloch wave i.e. for 773:Suppose an electron is in a Bloch state 725: 51: 36: 11991:{\displaystyle {\frac {1}{\hbar ^{2}}}} 6639: 6615: 6561: 6531:i.e. specializing it for a translation 6424: 5416:that can also be seen as a multiple of 5172: 4968: 4896: 4860: 4841: 4822: 4806: 4736: 14: 13656: 12495: 12471: 7039:dependent on the continuous parameter 1429:A three-dimensional crystal has three 13069: 12644: 12642: 12501: 6695:to the Bloch wave function we obtain 6679:Also here is possible to see how the 6077:{\displaystyle m_{1}\in \mathbb {Z} } 4688: 1916:. It is helpful to write the numbers 559:{\displaystyle \psi _{n\mathbf {k} }} 505:{\displaystyle \psi _{n\mathbf {k} }} 387:{\displaystyle \psi _{n\mathbf {k} }} 48:of a Bloch state in a silicon lattice 13612: 12575:from the original on 1 November 2019 12163:. This equation is analogous to the 8986:is considered small with respect to 6656:and we have proven Bloch’s theorem. 5506:with a simple effective Hamiltonian 5138:where the character remains finite. 5101:{\displaystyle \chi (\gamma )^{n}=1} 1815:Assume that we have a wave function 885:directly. This is important because 320:Functions of this form are known as 62:. The light circles represent atoms. 13636: 13021: 12604: 12478:Introduction to Solid State Physics 6672:are given from the specific finite 3865:{\displaystyle \psi (\mathbf {x} )} 2907:We define the translation operator 1275:, there exists at least one vector 1014: 32:Bloch's theorem (complex variables) 24: 12957: 12678: 12666: 12654: 12639: 12627: 12504:Principles of the theory of solids 11871: 11813: 11762: 11704: 11573: 11560: 11528: 11371: 11218: 11205: 11184: 10963: 10794: 10781: 10760: 10629: 10571: 10520: 10462: 10331: 10318: 10286: 10232: 10160: 10145: 10096: 10024: 10009: 9870: 9773: 9758: 9429: 9172: 9159: 9138: 9082: 9067: 8948: 8935: 8903: 8871: 8856: 8749: 8677: 8662: 8628:{\displaystyle \hbar \mathbf {k} } 8597: 8554: 8412: 8232: 8179: 7885: 7832: 7764: 7473: 7382: 7180: 6788: 5554: 5276:Born–von Karman boundary condition 5125: 5013:. In fact they have one generator 3357:independent electron approximation 2892: 460:). Within a band (i.e., for fixed 25: 13685: 12532: 12520: 8806: 8650:mean velocity of a Bloch electron 8594: 8560: 8551: 6691:If we apply the time-independent 4379:{\displaystyle k\in \mathbb {R} } 4042:{\displaystyle s\in \mathbb {C} } 1660:Lemma about translation operators 960:is a restricted set of values of 13635: 13623: 13611: 13600: 13599: 12226: 12218: 12210: 12145: 12115: 12104: 12096: 12088: 12069: 12061: 12053: 12014: 12006: 11998:is called effective mass tensor 11941: 11912: 11889: 11855: 11836: 11797: 11780: 11746: 11727: 11688: 11551: 11444: 11421: 11391: 11355: 11328: 11135: 11120: 11115: 11096: 11078: 11043: 11020: 10991: 10970: 10947: 10918: 10904: 10699: 10670: 10647: 10613: 10594: 10555: 10538: 10504: 10485: 10446: 10309: 10247: 10212: 10198: 10164: 10119: 10103: 10076: 10062: 10028: 9974: 9954: 9940: 9910: 9877: 9833: 9819: 9614: 9533: 9436: 9413: 9386: 9362: 9354: 9316: 9308: 9269: 9261: 9044: 9020: 9012: 8926: 8875: 8816: 8790: 8764: 8729: 8715: 8681: 8621: 8564: 8526: 8490: 8480: 8467: 8450: 8440: 8419: 8364: 8354: 8342: 8325: 8315: 8303: 8295: 8279: 8257: 8247: 8225: 8217: 8198: 8188: 8173: 8165: 8149: 8132: 8122: 8110: 8102: 8083: 8040: 8030: 8018: 8010: 7995: 7978: 7968: 7956: 7948: 7932: 7910: 7900: 7878: 7870: 7851: 7841: 7826: 7818: 7802: 7783: 7773: 7758: 7750: 7731: 7721: 7709: 7701: 7688: 7672: 7622: 7612: 7600: 7592: 7577: 7560: 7550: 7538: 7530: 7514: 7492: 7482: 7467: 7459: 7440: 7430: 7418: 7410: 7397: 7341: 7331: 7319: 7311: 7296: 7255: 7245: 7233: 7225: 7199: 7132: 7122: 7110: 7102: 7082: 7048: 7022: 6986: 6960: 6952: 6942: 6926: 6916: 6886: 6876: 6864: 6848: 6838: 6820: 6795: 6739: 6729: 6717: 6631: 6607: 6591: 6574: 6416: 5660: 5635: 5606: 5573: 5480: 5451: 5402: 5373: 5356: 5348: 5323: 5294: 4929: 4776: 4668: 4663: 4655: 4645: 4629: 4619: 4589: 4579: 4567: 4559: 4540: 4530: 4497: 4484: 4476: 4454: 4446: 4438: 4421: 4408: 4398: 4301: 4263: 4240: 4200: 4182: 4160: 4147: 4135: 4107: 4084: 4005: 3997: 3979: 3942: 3927: 3905: 3886: 3855: 3824: 3805: 3790: 3777: 3756: 3750: 3736: 3730: 3722: 3705: 3686: 3673: 3656: 3643: 3615: 3603: 3574: 3561: 3545: 3532: 3520: 3483: 3471: 3424: 3388: 3338: 3319: 3313: 3304: 3219: 3196: 3182: 3168: 3151: 3122: 3117: 3109: 3079: 3054: 3029: 3010: 2984: 2978: 2969: 2948: 2935: 2923: 2661: 2648: 2640: 2621: 2579: 2555: 2487: 2479: 2443: 2384: 2375: 2357: 2349: 2307: 2298: 2276: 2267: 2256: 2224: 2215: 2176: 2163: 2155: 2133: 2033: 1984: 1975: 1875: 1842: 1833: 1524: 1499: 1474: 1377: 1367: 1355: 1347: 1328: 1320: 1310: 1283: 1261: 1219: 1211: 1203: 1193: 1177: 1167: 1123: 1110: 1102: 1083: 827: 814: 806: 787: 692: 662: 635: 632: 629: 608: 578: 566:is unique only up to a constant 550: 520: 496: 426: 414:is a discrete index, called the 378: 251: 178: 150: 137: 129: 110: 12910: 12871: 12858: 12809: 12773: 12736: 12690: 12433:Symmetries in quantum mechanics 12279:ordinary differential equations 12252:for an electron in an external 8603:{\displaystyle -i\hbar \nabla } 6668:of the wave function where the 3217: 1414:Proof Using lattice periodicity 716:dynamical theory of diffraction 676:can be restricted to the first 13058:10.1016/j.wavemoti.2012.12.010 12817:Alexander Mihailovich Lyapunov 12616: 12598: 12587: 12465: 12456: 12346: 12340: 12108: 12100: 12065: 12057: 12018: 12010: 11945: 11937: 11916: 11908: 11881: 11860: 11823: 11802: 11772: 11751: 11714: 11693: 11555: 11547: 11400: 11378: 11374: 11362: 11333: 11317: 11088: 10999: 10974: 10954: 10893: 10703: 10695: 10674: 10666: 10639: 10618: 10581: 10560: 10530: 10509: 10472: 10451: 10313: 10305: 10235: 10223: 10107: 10087: 9882: 9861: 9733:To compute to linear order in 9658: 9640: 9603: 9559: 9440: 9420: 9378: 9346: 9300: 9273: 9257: 9227: 9214: 9048: 9040: 9024: 9008: 8930: 8922: 8820: 8794: 8752: 8740: 8530: 8494: 8486: 8471: 8463: 8454: 8446: 8368: 8360: 8329: 8321: 8283: 8275: 8261: 8253: 8202: 8194: 8136: 8128: 8044: 8036: 7982: 7974: 7936: 7928: 7914: 7906: 7855: 7847: 7787: 7779: 7735: 7727: 7626: 7618: 7564: 7556: 7518: 7510: 7496: 7488: 7444: 7436: 7345: 7337: 7259: 7251: 7203: 7195: 7136: 7128: 7056:{\displaystyle {\mathbf {k} }} 7026: 7018: 6994:{\displaystyle {\mathbf {k} }} 6964: 6948: 6930: 6922: 6890: 6882: 6852: 6844: 6824: 6816: 6743: 6735: 6709: 6643: 6627: 6595: 6587: 6578: 6570: 6556: 6545: 6518: 6512: 6454: 6401: 6375: 6355: 6329: 6309: 6283: 6017: 5981: 5916: 5880: 5757: 5746: 5707: 5696: 5664: 5656: 5577: 5569: 5519: 5377: 5369: 5360: 5344: 5216: 5213: 5187: 5175: 5165: 5122: 5083: 5076: 5066:, and therefore the character 4971: 4933: 4899: 4863: 4844: 4825: 4809: 4672: 4651: 4633: 4625: 4593: 4585: 4544: 4536: 4501: 4493: 4458: 4434: 4425: 4417: 4309: 4291: 4249: 4244: 4236: 4229: 4208: 4190: 4164: 4156: 4139: 4093: 4088: 4080: 4073: 3859: 3851: 3828: 3820: 3781: 3766: 3718: 3709: 3701: 3677: 3647: 3607: 3578: 3570: 3549: 3541: 3524: 3489: 3475: 3457: 3448: 3428: 3420: 3392: 3372: 3342: 3334: 3325: 3300: 3126: 3105: 3089: 3006: 2990: 2965: 2952: 2944: 2927: 2831: 2702: 2665: 2657: 2625: 2617: 2583: 2575: 2559: 2551: 2447: 2439: 2317: 2294: 2286: 2263: 2234: 2211: 2180: 2172: 2137: 2129: 2037: 2029: 1994: 1971: 1879: 1871: 1852: 1829: 1678: 1381: 1373: 1332: 1316: 1223: 1199: 1181: 1173: 1127: 1119: 1087: 1079: 921:can be written as above using 831: 823: 791: 783: 721: 639: 625: 154: 146: 114: 106: 13: 1: 13664:Eponymous theorems of physics 13198:Spontaneous symmetry breaking 12844:. Courier Dover. p. 11. 12823:. London: Taylor and Francis. 12449: 12260:History and related equations 5059:{\displaystyle \gamma ^{n}=1} 4697:which are a combination of a 688:Applications and consequences 654:. Therefore, the wave vector 355:), underlies the concept of 74:states that solutions to the 12612:. University of Puget Sound. 12428:Periodic boundary conditions 12152:{\displaystyle \mathbf {a} } 8846:We evaluate the derivatives 5409:{\displaystyle \mathbf {a} } 5131:{\displaystyle n\to \infty } 1290:{\displaystyle \mathbf {k} } 1268:{\displaystyle \mathbf {a} } 1026: 669:{\displaystyle \mathbf {k} } 585:{\displaystyle \mathbf {K} } 534:, as does its energy. Also, 527:{\displaystyle \mathbf {k} } 433:{\displaystyle \mathbf {k} } 258:{\displaystyle \mathbf {k} } 185:{\displaystyle \mathbf {r} } 7: 12993:Dresselhaus, M. S. (2010). 12868:, RUSS MATH SURV., 37, 1–60 12400: 6687:Velocity and effective mass 4713:Proof with character theory 1069:can be written in the form 997:, it equals the electron's 10: 13690: 13378:Spin gapless semiconductor 13287:Nearly free electron model 12685:Ashcroft & Mermin 1976 12673:Ashcroft & Mermin 1976 12661:Ashcroft & Mermin 1976 12649:Ashcroft & Mermin 1976 12634:Ashcroft & Mermin 1976 12605:Roy, Ricky (May 2, 2010). 12539:Ashcroft & Mermin 1976 12527:Ashcroft & Mermin 1976 12423:Nearly free electron model 6662:discrete Fourier transform 1563:. These are three vectors 1560:reciprocal lattice vectors 682:without loss of generality 680:of the reciprocal lattice 357:electronic band structures 328:, and serve as a suitable 29: 13595: 13557: 13482: 13426: 13386: 13335: 13327:Density functional theory 13302:electronic band structure 13269: 13218: 13211: 13180: 13169: 13103: 13026:. The University of Kiel. 12418:Electronic band structure 6903:with boundary conditions 4957:The commutativity of the 3631:is an additive operator 1910:) which do not depend on 1654:reciprocal lattice vector 1431:primitive lattice vectors 1406:Using lattice periodicity 951:reciprocal lattice vector 901:unique. Specifically, if 749:(right). The difference ( 512:varies continuously with 13674:Condensed matter physics 13497:Bogoliubov quasiparticle 13241:Quantum spin Hall effect 13133:Bose–Einstein condensate 13097:Condensed matter physics 12651:, p. 765 Appendix E 12287:Lyapunov–Floquet theorem 8641:canonical transformation 1400: 68:condensed matter physics 13031:M.S.P. Eastham (1973). 12607:"Representation Theory" 12413:Bloch wave – MoM method 8610:and a crystal momentum 5026:{\displaystyle \gamma } 2815:simultaneous eigenbasis 1906:are three numbers (the 1641:. (For the formula for 1001:. Related to this, the 995:reduced Planck constant 267:crystal momentum vector 12702:Zeitschrift für Physik 12365: 12239: 12167:type of approximation 12153: 12128: 12025: 11992: 11955: 11511: 11484: 11457: 11145: 11056: 10726:The second order term 10713: 10273:effective mass theorem 10257: 10129: 9984: 9920: 9727: 9482: 9326: 9280: 9234: 8968: 8886: 8833: 8629: 8604: 8575: 8505: 7271: 7057: 7033: 6995: 6971: 6897: 6650: 6525: 6434: 6250: 6223: 6196: 6169: 6140: 6078: 6043: 5925: 5806: 5777: 5671: 5584: 5494: 5437: 5410: 5384: 5266: 5132: 5102: 5060: 5027: 4987: 4949: 4879: 4790: 4763: 4679: 4600: 4511: 4380: 4352: 4323: 4271: 4043: 4015: 3958: 3866: 3835: 3625: 3585: 3502: 3435: 3349: 3284: 3137: 2881: 2751: 2675: 2597: 2191: 2044: 1886: 1727: 1541: 1391: 1291: 1269: 1233: 1137: 1063: 915: 867: 841: 770: 670: 648: 586: 560: 528: 506: 474: 454: 434: 408: 388: 307: 283: 259: 230: 206: 186: 161: 63: 49: 13373:Topological insulator 13307:Anderson localization 12939:10.1007/s002200050033 12838:; Winkler, S (2004). 12502:Ziman, J. M. (1972). 12366: 12285:(or occasionally the 12240: 12154: 12129: 12026: 11993: 11956: 11512: 11510:{\displaystyle q_{j}} 11485: 11483:{\displaystyle q_{i}} 11458: 11146: 11057: 10714: 10258: 10130: 9993:We can simplify over 9985: 9921: 9728: 9483: 9327: 9281: 9235: 8969: 8887: 8834: 8630: 8605: 8576: 8506: 7272: 7058: 7034: 6996: 6972: 6898: 6651: 6526: 6435: 6251: 6249:{\displaystyle a_{1}} 6224: 6222:{\displaystyle L_{1}} 6197: 6195:{\displaystyle m_{1}} 6170: 6168:{\displaystyle k_{1}} 6141: 6079: 6044: 5926: 5807: 5805:{\displaystyle L_{i}} 5778: 5672: 5585: 5495: 5438: 5436:{\displaystyle a_{i}} 5411: 5385: 5267: 5133: 5103: 5061: 5028: 4988: 4950: 4880: 4791: 4743: 4680: 4601: 4512: 4381: 4353: 4324: 4272: 4044: 4016: 3959: 3867: 3836: 3626: 3586: 3503: 3436: 3350: 3285: 3138: 2901:Proof using operators 2882: 2752: 2676: 2598: 2192: 2045: 1887: 1728: 1542: 1392: 1292: 1270: 1234: 1138: 1064: 1062:{\displaystyle \psi } 993:is multiplied by the 916: 914:{\displaystyle \psi } 868: 866:{\displaystyle \psi } 842: 729: 671: 649: 587: 561: 529: 507: 475: 455: 435: 409: 389: 308: 284: 260: 231: 207: 205:{\displaystyle \psi } 187: 162: 55: 40: 13251:Aharonov–Bohm effect 13138:Fermionic condensate 12864:Kuchment, P.(1982), 12297: 12177: 12141: 12049: 12002: 11968: 11521: 11517:we have the theorem 11494: 11467: 11154: 11066: 10730: 10279: 10139: 10003: 9930: 9742: 9492: 9336: 9290: 9244: 8995: 8896: 8850: 8656: 8614: 8585: 8581:a standard momentum 8518: 7283: 7073: 7043: 7005: 6981: 6907: 6699: 6693:Schrödinger equation 6535: 6444: 6263: 6233: 6206: 6179: 6152: 6088: 6053: 5935: 5816: 5789: 5686: 5594: 5510: 5504:Schrödinger equation 5447: 5420: 5398: 5282: 5274:If we introduce the 5145: 5116: 5070: 5037: 5033:which shall obey to 5017: 4961: 4889: 4800: 4732: 4610: 4521: 4390: 4362: 4351:{\displaystyle s=ik} 4333: 4281: 4053: 4025: 3970: 3876: 3845: 3635: 3595: 3512: 3445: 3363: 3359:with an Hamiltonian 3294: 3147: 2911: 2821: 2811:Hamiltonian operator 2759:translation operator 2692: 2611: 2201: 2123: 1965: 1823: 1735:translation operator 1668: 1459: 1301: 1279: 1257: 1158: 1073: 1053: 958:first Brillouin zone 905: 857: 777: 658: 596: 574: 538: 516: 484: 464: 444: 422: 398: 366: 297: 273: 247: 220: 196: 174: 100: 76:Schrödinger equation 13642:Physics WikiProject 13317:tight binding model 13297:Fermi liquid theory 13282:Free electron model 13231:Quantum Hall effect 13212:Electrons in solids 12999:. Springer-Verlag. 12975:Solid State Physics 12931:2000CMaPh.209..633K 12744:George William Hill 12714:1929ZPhy...52..555B 12481:. New York: Wiley. 12438:Tight-binding model 12384:Kronig–Penney model 12267:George William Hill 12250:Newton's second law 10928: 10222: 10086: 9964: 9843: 9707: 9684: 9633: 9552: 9522: 9286:are eigenvalues of 8739: 8639:, and as part of a 6666:character expansion 1795:If a wave function 1793: — 1251: — 1036: — 18:Bloch surface waves 13203:Critical phenomena 12802:10.24033/asens.220 12765:10.1007/BF02417081 12722:10.1007/BF01339455 12555:Dresselhaus, M. S. 12408:Bloch oscillations 12388:Mathieu's equation 12361: 12275:Alexander Lyapunov 12235: 12149: 12124: 12021: 11988: 11951: 11677: 11507: 11480: 11453: 11312: 11179: 11141: 11052: 10909: 10888: 10755: 10724: 10709: 10435: 10253: 10203: 10125: 10067: 9980: 9945: 9916: 9824: 9811: 9754: 9723: 9688: 9670: 9619: 9598: 9538: 9508: 9478: 9322: 9276: 9230: 9133: 9063: 8964: 8882: 8844: 8829: 8720: 8625: 8600: 8571: 8501: 8499: 7267: 7068: 7053: 7029: 6991: 6967: 6893: 6646: 6521: 6488: 6430: 6246: 6219: 6192: 6165: 6136: 6084:is an integer and 6074: 6039: 5921: 5802: 5773: 5667: 5622: 5580: 5490: 5467: 5433: 5406: 5380: 5310: 5278:on the potential: 5262: 5128: 5098: 5056: 5023: 4983: 4945: 4875: 4786: 4714: 4689:Using group theory 4675: 4596: 4507: 4376: 4348: 4319: 4267: 4039: 4011: 3966:This is true for 3954: 3862: 3831: 3621: 3581: 3501:{\displaystyle =0} 3498: 3431: 3345: 3280: 3274: 3208: 3133: 3131: 2902: 2877: 2747: 2671: 2593: 2591: 2187: 2040: 1882: 1813: 1807:is a Bloch state. 1791: 1723: 1537: 1415: 1387: 1287: 1265: 1249: 1229: 1133: 1059: 1034: 911: 863: 837: 771: 767:reciprocal lattice 666: 644: 582: 568:reciprocal lattice 556: 524: 502: 470: 450: 430: 404: 384: 342:crystalline solids 303: 279: 255: 226: 202: 182: 157: 84:periodic functions 64: 50: 13669:Quantum mechanics 13651: 13650: 13537:Exciton-polariton 13422: 13421: 13394:Thermoelectricity 13006:978-3-642-06945-1 12985:978-0-03-083993-1 12395:spectral geometry 12332: 12192: 11986: 11949: 11657: 11645: 11607: 11587: 11451: 11352: 11292: 11277: 11232: 11167: 11165: 11050: 10944: 10868: 10853: 10808: 10743: 10741: 10722: 10707: 10415: 10403: 10365: 10345: 10189: 10169: 10053: 10033: 9859: 9802: 9787: 9745: 9709: 9643: 9578: 9562: 9466: 9410: 9381: 9349: 9303: 9186: 9121: 9119: 9096: 9054: 8962: 8880: 8842: 8823: 8797: 8781: 8706: 8686: 8643:of the momentum. 8539: 8533: 8398: 8074: 7661: 7380: 7177: 7066: 6774: 6712: 6548: 6479: 6457: 6134: 5749: 5699: 5613: 5551: 5522: 5458: 5301: 5178: 4974: 4936: 4902: 4866: 4847: 4828: 4812: 4712: 4404: 4142: 3784: 3680: 3650: 3610: 3527: 3478: 3460: 3412: 3395: 3375: 2930: 2900: 2834: 2705: 2603:This proves that 1811: 1789: 1681: 1413: 1247: 1032: 931:be written using 712:phononic crystals 708:photonic crystals 473:{\displaystyle n} 453:{\displaystyle u} 407:{\displaystyle n} 306:{\displaystyle i} 282:{\displaystyle e} 238:periodic function 229:{\displaystyle u} 16:(Redirected from 13681: 13639: 13638: 13627: 13615: 13614: 13603: 13602: 13542:Phonon polariton 13434:Amorphous magnet 13414:Electrostriction 13409:Flexoelectricity 13404:Ferroelectricity 13399:Piezoelectricity 13256:Josephson effect 13236:Spin Hall effect 13216: 13215: 13193:Phase transition 13175: 13158:Luttinger liquid 13105:States of matter 13090: 13083: 13076: 13067: 13066: 13061: 13036: 13027: 13018: 12989: 12970:Mermin, N. David 12951: 12950: 12919:Comm. Math. Phys 12914: 12908: 12907: 12875: 12869: 12862: 12856: 12855: 12832: 12826: 12824: 12813: 12807: 12806: 12804: 12777: 12771: 12769: 12767: 12740: 12734: 12733: 12708:(7–8): 555–600. 12694: 12688: 12682: 12676: 12670: 12664: 12658: 12652: 12646: 12637: 12631: 12625: 12620: 12614: 12613: 12611: 12602: 12596: 12591: 12585: 12584: 12582: 12580: 12574: 12563: 12551: 12542: 12536: 12530: 12524: 12518: 12517: 12499: 12493: 12492: 12469: 12463: 12460: 12443:Wannier function 12381: 12370: 12368: 12367: 12362: 12333: 12331: 12330: 12329: 12316: 12312: 12311: 12301: 12244: 12242: 12241: 12236: 12234: 12230: 12229: 12221: 12213: 12194: 12193: 12185: 12158: 12156: 12155: 12150: 12148: 12133: 12131: 12130: 12125: 12123: 12119: 12118: 12107: 12099: 12091: 12072: 12064: 12056: 12030: 12028: 12027: 12022: 12017: 12009: 11997: 11995: 11994: 11989: 11987: 11985: 11984: 11972: 11960: 11958: 11957: 11952: 11950: 11948: 11944: 11936: 11935: 11934: 11915: 11907: 11906: 11896: 11892: 11884: 11879: 11878: 11863: 11858: 11853: 11839: 11834: 11826: 11821: 11820: 11805: 11800: 11783: 11775: 11770: 11769: 11754: 11749: 11744: 11730: 11725: 11717: 11712: 11711: 11696: 11691: 11679: 11676: 11669: 11656: 11655: 11650: 11646: 11641: 11640: 11631: 11621: 11620: 11608: 11603: 11602: 11593: 11588: 11586: 11585: 11584: 11572: 11571: 11558: 11554: 11546: 11545: 11536: 11535: 11525: 11516: 11514: 11513: 11508: 11506: 11505: 11489: 11487: 11486: 11481: 11479: 11478: 11462: 11460: 11459: 11454: 11452: 11450: 11449: 11448: 11447: 11442: 11426: 11425: 11424: 11410: 11409: 11408: 11403: 11394: 11389: 11381: 11358: 11353: 11348: 11347: 11338: 11336: 11331: 11320: 11314: 11311: 11304: 11288: 11287: 11278: 11276: 11268: 11267: 11258: 11253: 11252: 11243: 11242: 11233: 11231: 11230: 11229: 11217: 11216: 11203: 11202: 11201: 11192: 11191: 11181: 11178: 11166: 11158: 11150: 11148: 11147: 11142: 11140: 11139: 11138: 11125: 11124: 11123: 11118: 11099: 11091: 11083: 11082: 11081: 11061: 11059: 11058: 11053: 11051: 11049: 11048: 11047: 11046: 11041: 11025: 11024: 11023: 11009: 11008: 11007: 11002: 10996: 10995: 10994: 10989: 10973: 10950: 10945: 10940: 10939: 10930: 10927: 10922: 10921: 10907: 10896: 10890: 10887: 10880: 10864: 10863: 10854: 10852: 10844: 10843: 10834: 10829: 10828: 10819: 10818: 10809: 10807: 10806: 10805: 10793: 10792: 10779: 10778: 10777: 10768: 10767: 10757: 10754: 10742: 10734: 10718: 10716: 10715: 10710: 10708: 10706: 10702: 10694: 10693: 10692: 10673: 10665: 10664: 10654: 10650: 10642: 10637: 10636: 10621: 10616: 10611: 10597: 10592: 10584: 10579: 10578: 10563: 10558: 10541: 10533: 10528: 10527: 10512: 10507: 10502: 10488: 10483: 10475: 10470: 10469: 10454: 10449: 10437: 10434: 10427: 10414: 10413: 10408: 10404: 10399: 10398: 10389: 10379: 10378: 10366: 10361: 10360: 10351: 10346: 10344: 10343: 10342: 10330: 10329: 10316: 10312: 10304: 10303: 10294: 10293: 10283: 10262: 10260: 10259: 10254: 10252: 10251: 10250: 10221: 10216: 10215: 10201: 10190: 10185: 10184: 10175: 10170: 10168: 10167: 10158: 10157: 10156: 10143: 10134: 10132: 10131: 10126: 10124: 10123: 10122: 10106: 10085: 10080: 10079: 10065: 10054: 10049: 10048: 10039: 10034: 10032: 10031: 10022: 10021: 10020: 10007: 9998: 9989: 9987: 9986: 9981: 9979: 9978: 9977: 9963: 9958: 9957: 9943: 9925: 9923: 9922: 9917: 9915: 9914: 9913: 9900: 9899: 9890: 9889: 9880: 9860: 9855: 9854: 9845: 9842: 9837: 9836: 9822: 9810: 9798: 9797: 9788: 9786: 9785: 9784: 9771: 9770: 9769: 9756: 9753: 9738: 9732: 9730: 9729: 9724: 9710: 9708: 9706: 9701: 9700: 9683: 9678: 9668: 9667: 9666: 9661: 9655: 9654: 9645: 9644: 9636: 9632: 9627: 9617: 9606: 9600: 9597: 9590: 9574: 9573: 9564: 9563: 9555: 9551: 9546: 9536: 9521: 9516: 9504: 9503: 9487: 9485: 9484: 9479: 9477: 9476: 9467: 9465: 9457: 9456: 9447: 9439: 9416: 9411: 9406: 9405: 9396: 9391: 9390: 9389: 9383: 9382: 9374: 9367: 9366: 9365: 9357: 9351: 9350: 9342: 9331: 9329: 9328: 9323: 9321: 9320: 9319: 9311: 9305: 9304: 9296: 9285: 9283: 9282: 9277: 9272: 9264: 9256: 9255: 9239: 9237: 9236: 9231: 9226: 9225: 9207: 9206: 9197: 9196: 9187: 9185: 9184: 9183: 9171: 9170: 9157: 9156: 9155: 9146: 9145: 9135: 9132: 9120: 9112: 9107: 9106: 9097: 9095: 9094: 9093: 9080: 9079: 9078: 9065: 9062: 9047: 9039: 9038: 9023: 9015: 9007: 9006: 8991: 8985: 8979: 8973: 8971: 8970: 8965: 8963: 8961: 8960: 8959: 8947: 8946: 8933: 8929: 8921: 8920: 8911: 8910: 8900: 8891: 8889: 8888: 8883: 8881: 8879: 8878: 8869: 8868: 8867: 8854: 8838: 8836: 8835: 8830: 8825: 8824: 8819: 8814: 8799: 8798: 8793: 8788: 8782: 8774: 8769: 8768: 8767: 8738: 8733: 8732: 8718: 8707: 8702: 8701: 8692: 8687: 8685: 8684: 8675: 8674: 8673: 8660: 8637:minimal coupling 8634: 8632: 8631: 8626: 8624: 8609: 8607: 8606: 8601: 8580: 8578: 8577: 8572: 8567: 8541: 8540: 8537: 8535: 8534: 8529: 8524: 8510: 8508: 8507: 8502: 8500: 8493: 8485: 8484: 8483: 8470: 8453: 8445: 8444: 8443: 8433: 8432: 8427: 8423: 8422: 8399: 8397: 8389: 8388: 8379: 8367: 8359: 8358: 8357: 8347: 8346: 8345: 8328: 8320: 8319: 8318: 8308: 8307: 8306: 8298: 8282: 8268: 8264: 8260: 8252: 8251: 8250: 8240: 8239: 8230: 8229: 8228: 8220: 8201: 8193: 8192: 8191: 8178: 8177: 8176: 8168: 8152: 8135: 8127: 8126: 8125: 8115: 8114: 8113: 8105: 8092: 8091: 8086: 8075: 8073: 8065: 8064: 8055: 8043: 8035: 8034: 8033: 8023: 8022: 8021: 8013: 8000: 7999: 7998: 7981: 7973: 7972: 7971: 7961: 7960: 7959: 7951: 7935: 7921: 7917: 7913: 7905: 7904: 7903: 7893: 7892: 7883: 7882: 7881: 7873: 7854: 7846: 7845: 7844: 7831: 7830: 7829: 7821: 7805: 7794: 7790: 7786: 7778: 7777: 7776: 7763: 7762: 7761: 7753: 7734: 7726: 7725: 7724: 7714: 7713: 7712: 7704: 7691: 7675: 7662: 7660: 7652: 7651: 7650: 7637: 7625: 7617: 7616: 7615: 7605: 7604: 7603: 7595: 7582: 7581: 7580: 7563: 7555: 7554: 7553: 7543: 7542: 7541: 7533: 7517: 7503: 7499: 7495: 7487: 7486: 7485: 7472: 7471: 7470: 7462: 7443: 7435: 7434: 7433: 7423: 7422: 7421: 7413: 7400: 7381: 7379: 7371: 7370: 7369: 7356: 7344: 7336: 7335: 7334: 7324: 7323: 7322: 7314: 7301: 7300: 7299: 7276: 7274: 7273: 7268: 7266: 7262: 7258: 7250: 7249: 7248: 7238: 7237: 7236: 7228: 7210: 7206: 7202: 7188: 7187: 7178: 7176: 7168: 7167: 7166: 7153: 7143: 7139: 7135: 7127: 7126: 7125: 7115: 7114: 7113: 7105: 7087: 7086: 7085: 7062: 7060: 7059: 7054: 7052: 7051: 7038: 7036: 7035: 7030: 7025: 7017: 7016: 7000: 6998: 6997: 6992: 6990: 6989: 6976: 6974: 6973: 6968: 6963: 6955: 6947: 6946: 6945: 6929: 6921: 6920: 6919: 6902: 6900: 6899: 6894: 6889: 6881: 6880: 6879: 6869: 6868: 6867: 6851: 6843: 6842: 6841: 6831: 6827: 6823: 6809: 6808: 6803: 6799: 6798: 6775: 6773: 6765: 6764: 6755: 6742: 6734: 6733: 6732: 6722: 6721: 6720: 6714: 6713: 6705: 6655: 6653: 6652: 6647: 6642: 6634: 6620: 6619: 6618: 6610: 6594: 6577: 6566: 6565: 6564: 6559: 6550: 6549: 6541: 6530: 6528: 6527: 6522: 6511: 6510: 6498: 6497: 6487: 6475: 6474: 6465: 6464: 6459: 6458: 6450: 6439: 6437: 6436: 6431: 6429: 6428: 6427: 6419: 6400: 6399: 6387: 6386: 6374: 6373: 6372: 6371: 6354: 6353: 6341: 6340: 6328: 6327: 6326: 6325: 6308: 6307: 6295: 6294: 6282: 6281: 6280: 6279: 6255: 6253: 6252: 6247: 6245: 6244: 6228: 6226: 6225: 6220: 6218: 6217: 6201: 6199: 6198: 6193: 6191: 6190: 6174: 6172: 6171: 6166: 6164: 6163: 6148:The wave vector 6145: 6143: 6142: 6137: 6135: 6133: 6132: 6123: 6122: 6121: 6105: 6100: 6099: 6083: 6081: 6080: 6075: 6073: 6065: 6064: 6048: 6046: 6045: 6040: 6038: 6037: 6016: 6015: 6003: 6002: 5993: 5992: 5980: 5979: 5967: 5966: 5957: 5956: 5947: 5946: 5930: 5928: 5927: 5922: 5920: 5919: 5915: 5914: 5902: 5901: 5892: 5891: 5879: 5878: 5858: 5857: 5856: 5855: 5846: 5845: 5836: 5835: 5811: 5809: 5808: 5803: 5801: 5800: 5782: 5780: 5779: 5774: 5772: 5771: 5770: 5769: 5760: 5751: 5750: 5742: 5735: 5734: 5733: 5732: 5720: 5719: 5710: 5701: 5700: 5692: 5676: 5674: 5673: 5668: 5663: 5649: 5645: 5644: 5643: 5638: 5632: 5631: 5621: 5609: 5589: 5587: 5586: 5581: 5576: 5562: 5561: 5552: 5550: 5542: 5541: 5532: 5524: 5523: 5515: 5499: 5497: 5496: 5491: 5489: 5488: 5483: 5477: 5476: 5466: 5454: 5442: 5440: 5439: 5434: 5432: 5431: 5415: 5413: 5412: 5407: 5405: 5389: 5387: 5386: 5381: 5376: 5359: 5351: 5337: 5333: 5332: 5331: 5326: 5320: 5319: 5309: 5297: 5271: 5269: 5268: 5263: 5261: 5260: 5259: 5258: 5249: 5248: 5239: 5238: 5212: 5211: 5199: 5198: 5186: 5185: 5180: 5179: 5171: 5164: 5163: 5162: 5161: 5137: 5135: 5134: 5129: 5107: 5105: 5104: 5099: 5091: 5090: 5065: 5063: 5062: 5057: 5049: 5048: 5032: 5030: 5029: 5024: 4992: 4990: 4989: 4984: 4982: 4981: 4976: 4975: 4967: 4954: 4952: 4951: 4946: 4944: 4943: 4938: 4937: 4932: 4927: 4923: 4922: 4910: 4909: 4904: 4903: 4895: 4884: 4882: 4881: 4876: 4874: 4873: 4868: 4867: 4859: 4855: 4854: 4849: 4848: 4840: 4836: 4835: 4830: 4829: 4821: 4814: 4813: 4805: 4795: 4793: 4792: 4787: 4785: 4784: 4779: 4773: 4772: 4762: 4757: 4739: 4684: 4682: 4681: 4676: 4671: 4666: 4658: 4650: 4649: 4648: 4632: 4624: 4623: 4622: 4605: 4603: 4602: 4597: 4592: 4584: 4583: 4582: 4572: 4571: 4570: 4562: 4543: 4535: 4534: 4533: 4516: 4514: 4513: 4508: 4500: 4489: 4488: 4487: 4479: 4457: 4449: 4441: 4424: 4413: 4412: 4411: 4406: 4405: 4397: 4385: 4383: 4382: 4377: 4375: 4357: 4355: 4354: 4349: 4328: 4326: 4325: 4320: 4318: 4317: 4312: 4306: 4305: 4304: 4294: 4276: 4274: 4273: 4268: 4266: 4258: 4257: 4252: 4243: 4232: 4227: 4226: 4217: 4216: 4211: 4205: 4204: 4203: 4193: 4185: 4177: 4176: 4171: 4167: 4163: 4152: 4151: 4150: 4144: 4143: 4138: 4133: 4123: 4122: 4110: 4102: 4101: 4096: 4087: 4076: 4071: 4070: 4048: 4046: 4045: 4040: 4038: 4020: 4018: 4017: 4012: 4010: 4009: 4008: 4000: 3984: 3983: 3982: 3963: 3961: 3960: 3955: 3953: 3952: 3951: 3950: 3945: 3936: 3935: 3930: 3916: 3915: 3914: 3913: 3908: 3897: 3896: 3895: 3894: 3889: 3871: 3869: 3868: 3863: 3858: 3840: 3838: 3837: 3832: 3827: 3816: 3815: 3814: 3813: 3808: 3799: 3798: 3793: 3786: 3785: 3780: 3775: 3765: 3764: 3759: 3753: 3745: 3744: 3739: 3733: 3725: 3708: 3697: 3696: 3695: 3694: 3689: 3682: 3681: 3676: 3671: 3667: 3666: 3665: 3664: 3659: 3652: 3651: 3646: 3641: 3630: 3628: 3627: 3622: 3620: 3619: 3618: 3612: 3611: 3606: 3601: 3590: 3588: 3587: 3582: 3577: 3566: 3565: 3564: 3548: 3537: 3536: 3535: 3529: 3528: 3523: 3518: 3507: 3505: 3504: 3499: 3488: 3487: 3486: 3480: 3479: 3474: 3469: 3462: 3461: 3453: 3440: 3438: 3437: 3432: 3427: 3413: 3411: 3403: 3402: 3397: 3396: 3391: 3386: 3382: 3377: 3376: 3368: 3354: 3352: 3351: 3346: 3341: 3324: 3323: 3322: 3316: 3307: 3289: 3287: 3286: 3281: 3279: 3278: 3271: 3270: 3257: 3256: 3243: 3242: 3222: 3213: 3212: 3205: 3204: 3199: 3191: 3190: 3185: 3177: 3176: 3171: 3154: 3142: 3140: 3139: 3134: 3132: 3125: 3120: 3112: 3095: 3088: 3087: 3082: 3076: 3075: 3063: 3062: 3057: 3051: 3050: 3038: 3037: 3032: 3026: 3025: 3013: 2996: 2989: 2988: 2987: 2981: 2972: 2951: 2940: 2939: 2938: 2932: 2931: 2926: 2921: 2886: 2884: 2883: 2878: 2875: 2874: 2873: 2872: 2860: 2859: 2847: 2846: 2836: 2835: 2827: 2808: 2801: 2756: 2754: 2753: 2748: 2746: 2745: 2744: 2743: 2731: 2730: 2718: 2717: 2707: 2706: 2698: 2680: 2678: 2677: 2672: 2664: 2653: 2652: 2651: 2643: 2624: 2606: 2602: 2600: 2599: 2594: 2592: 2582: 2565: 2558: 2547: 2546: 2545: 2544: 2521: 2520: 2519: 2518: 2492: 2491: 2490: 2482: 2460: 2456: 2455: 2446: 2435: 2434: 2433: 2432: 2409: 2408: 2402: 2401: 2395: 2394: 2393: 2392: 2387: 2378: 2362: 2361: 2360: 2352: 2336: 2335: 2323: 2316: 2315: 2310: 2301: 2290: 2289: 2285: 2284: 2279: 2270: 2259: 2233: 2232: 2227: 2218: 2196: 2194: 2193: 2188: 2179: 2168: 2167: 2166: 2158: 2136: 2118: 2107: 2062: 2056: 2049: 2047: 2046: 2041: 2036: 2025: 2024: 2023: 2022: 1993: 1992: 1987: 1978: 1960: 1945: 1922: 1915: 1905: 1898: 1891: 1889: 1888: 1883: 1878: 1867: 1866: 1851: 1850: 1845: 1836: 1818: 1806: 1798: 1794: 1784: 1777: 1732: 1730: 1729: 1724: 1722: 1721: 1720: 1719: 1707: 1706: 1694: 1693: 1683: 1682: 1674: 1651: 1640: 1630: 1609: 1585: 1553: 1546: 1544: 1543: 1538: 1533: 1532: 1527: 1521: 1520: 1508: 1507: 1502: 1496: 1495: 1483: 1482: 1477: 1471: 1470: 1454: 1396: 1394: 1393: 1388: 1380: 1372: 1371: 1370: 1360: 1359: 1358: 1350: 1331: 1323: 1315: 1314: 1313: 1296: 1294: 1293: 1288: 1286: 1274: 1272: 1271: 1266: 1264: 1252: 1238: 1236: 1235: 1230: 1222: 1214: 1206: 1198: 1197: 1196: 1180: 1172: 1171: 1170: 1153: 1142: 1140: 1139: 1134: 1126: 1115: 1114: 1113: 1105: 1086: 1068: 1066: 1065: 1060: 1037: 1015:Detailed example 1010: 999:crystal momentum 992: 983: 977: 971: 965: 948: 942: 926: 920: 918: 917: 912: 896: 890: 884: 878: 872: 870: 869: 864: 852: 846: 844: 843: 838: 830: 819: 818: 817: 809: 790: 764: 748: 739: 704:electromagnetism 675: 673: 672: 667: 665: 653: 651: 650: 645: 643: 642: 638: 613: 612: 611: 591: 589: 588: 583: 581: 565: 563: 562: 557: 555: 554: 553: 533: 531: 530: 525: 523: 511: 509: 508: 503: 501: 500: 499: 479: 477: 476: 471: 459: 457: 456: 451: 439: 437: 436: 431: 429: 413: 411: 410: 405: 393: 391: 390: 385: 383: 382: 381: 340:of electrons in 312: 310: 309: 304: 288: 286: 285: 280: 264: 262: 261: 256: 254: 235: 233: 232: 227: 211: 209: 208: 203: 191: 189: 188: 183: 181: 166: 164: 163: 158: 153: 142: 141: 140: 132: 113: 61: 21: 13689: 13688: 13684: 13683: 13682: 13680: 13679: 13678: 13654: 13653: 13652: 13647: 13591: 13572:Granular matter 13567:Amorphous solid 13553: 13478: 13464:Antiferromagnet 13454:Superparamagnet 13427:Magnetic phases 13418: 13382: 13331: 13292:Bloch's theorem 13265: 13207: 13188:Order parameter 13181:Phase phenomena 13176: 13167: 13099: 13094: 13064: 13007: 12986: 12960: 12958:Further reading 12955: 12954: 12915: 12911: 12896:10.2307/2374542 12876: 12872: 12863: 12859: 12852: 12841:Hill's Equation 12833: 12829: 12814: 12810: 12778: 12774: 12741: 12737: 12695: 12691: 12683: 12679: 12671: 12667: 12659: 12655: 12647: 12640: 12632: 12628: 12621: 12617: 12609: 12603: 12599: 12592: 12588: 12578: 12576: 12572: 12561: 12552: 12545: 12537: 12533: 12525: 12521: 12514: 12500: 12496: 12489: 12473:Kittel, Charles 12470: 12466: 12461: 12457: 12452: 12447: 12403: 12372: 12325: 12321: 12317: 12307: 12303: 12302: 12300: 12298: 12295: 12294: 12291:Hill's equation 12281:, it is called 12262: 12246: 12225: 12217: 12209: 12208: 12204: 12184: 12183: 12178: 12175: 12174: 12165:de Broglie wave 12144: 12142: 12139: 12138: 12135: 12114: 12103: 12095: 12087: 12086: 12082: 12068: 12060: 12052: 12050: 12047: 12046: 12013: 12005: 12003: 12000: 11999: 11980: 11976: 11971: 11969: 11966: 11965: 11962: 11940: 11927: 11926: 11922: 11911: 11902: 11898: 11897: 11888: 11880: 11874: 11870: 11859: 11854: 11846: 11835: 11827: 11822: 11816: 11812: 11801: 11796: 11779: 11771: 11765: 11761: 11750: 11745: 11737: 11726: 11718: 11713: 11707: 11703: 11692: 11687: 11680: 11678: 11662: 11661: 11651: 11636: 11632: 11630: 11626: 11625: 11613: 11609: 11598: 11594: 11592: 11580: 11576: 11567: 11563: 11559: 11550: 11541: 11537: 11531: 11527: 11526: 11524: 11522: 11519: 11518: 11501: 11497: 11495: 11492: 11491: 11474: 11470: 11468: 11465: 11464: 11443: 11435: 11434: 11430: 11420: 11416: 11412: 11411: 11404: 11399: 11398: 11390: 11382: 11377: 11354: 11343: 11339: 11337: 11332: 11327: 11316: 11315: 11313: 11297: 11296: 11283: 11279: 11269: 11263: 11259: 11257: 11248: 11244: 11238: 11234: 11225: 11221: 11212: 11208: 11204: 11197: 11193: 11187: 11183: 11182: 11180: 11171: 11157: 11155: 11152: 11151: 11134: 11130: 11126: 11119: 11114: 11110: 11106: 11095: 11087: 11077: 11073: 11069: 11067: 11064: 11063: 11042: 11034: 11033: 11029: 11019: 11015: 11011: 11010: 11003: 10998: 10997: 10990: 10982: 10981: 10977: 10969: 10946: 10935: 10931: 10929: 10923: 10917: 10913: 10903: 10892: 10891: 10889: 10873: 10872: 10859: 10855: 10845: 10839: 10835: 10833: 10824: 10820: 10814: 10810: 10801: 10797: 10788: 10784: 10780: 10773: 10769: 10763: 10759: 10758: 10756: 10747: 10733: 10731: 10728: 10727: 10720: 10698: 10685: 10684: 10680: 10669: 10660: 10656: 10655: 10646: 10638: 10632: 10628: 10617: 10612: 10604: 10593: 10585: 10580: 10574: 10570: 10559: 10554: 10537: 10529: 10523: 10519: 10508: 10503: 10495: 10484: 10476: 10471: 10465: 10461: 10450: 10445: 10438: 10436: 10420: 10419: 10409: 10394: 10390: 10388: 10384: 10383: 10371: 10367: 10356: 10352: 10350: 10338: 10334: 10325: 10321: 10317: 10308: 10299: 10295: 10289: 10285: 10284: 10282: 10280: 10277: 10276: 10264: 10246: 10242: 10238: 10217: 10211: 10207: 10197: 10180: 10176: 10174: 10163: 10159: 10152: 10148: 10144: 10142: 10140: 10137: 10136: 10118: 10114: 10110: 10102: 10081: 10075: 10071: 10061: 10044: 10040: 10038: 10027: 10023: 10016: 10012: 10008: 10006: 10004: 10001: 10000: 9994: 9973: 9969: 9965: 9959: 9953: 9949: 9939: 9931: 9928: 9927: 9909: 9905: 9901: 9895: 9891: 9885: 9881: 9876: 9850: 9846: 9844: 9838: 9832: 9828: 9818: 9806: 9793: 9789: 9780: 9776: 9772: 9765: 9761: 9757: 9755: 9749: 9743: 9740: 9739: 9734: 9702: 9693: 9692: 9679: 9674: 9669: 9662: 9657: 9656: 9650: 9646: 9635: 9634: 9628: 9623: 9613: 9602: 9601: 9599: 9583: 9582: 9569: 9565: 9554: 9553: 9547: 9542: 9532: 9517: 9512: 9499: 9495: 9493: 9490: 9489: 9472: 9468: 9458: 9452: 9448: 9446: 9435: 9412: 9401: 9397: 9395: 9385: 9384: 9373: 9372: 9371: 9361: 9353: 9352: 9341: 9340: 9339: 9337: 9334: 9333: 9315: 9307: 9306: 9295: 9294: 9293: 9291: 9288: 9287: 9268: 9260: 9251: 9247: 9245: 9242: 9241: 9221: 9217: 9202: 9198: 9192: 9188: 9179: 9175: 9166: 9162: 9158: 9151: 9147: 9141: 9137: 9136: 9134: 9125: 9111: 9102: 9098: 9089: 9085: 9081: 9074: 9070: 9066: 9064: 9058: 9043: 9034: 9030: 9019: 9011: 9002: 8998: 8996: 8993: 8992: 8987: 8981: 8975: 8955: 8951: 8942: 8938: 8934: 8925: 8916: 8912: 8906: 8902: 8901: 8899: 8897: 8894: 8893: 8874: 8870: 8863: 8859: 8855: 8853: 8851: 8848: 8847: 8840: 8815: 8813: 8812: 8789: 8787: 8786: 8773: 8763: 8759: 8755: 8734: 8728: 8724: 8714: 8697: 8693: 8691: 8680: 8676: 8669: 8665: 8661: 8659: 8657: 8654: 8653: 8620: 8615: 8612: 8611: 8586: 8583: 8582: 8563: 8536: 8525: 8523: 8522: 8521: 8519: 8516: 8515: 8512: 8498: 8497: 8489: 8479: 8478: 8474: 8466: 8449: 8439: 8438: 8434: 8428: 8418: 8405: 8401: 8400: 8390: 8384: 8380: 8378: 8371: 8363: 8353: 8352: 8348: 8341: 8340: 8336: 8333: 8332: 8324: 8314: 8313: 8309: 8302: 8294: 8290: 8286: 8278: 8256: 8246: 8245: 8241: 8235: 8231: 8224: 8216: 8212: 8208: 8197: 8187: 8186: 8182: 8172: 8164: 8160: 8156: 8148: 8131: 8121: 8120: 8116: 8109: 8101: 8097: 8093: 8087: 8082: 8081: 8080: 8076: 8066: 8060: 8056: 8054: 8047: 8039: 8029: 8028: 8024: 8017: 8009: 8005: 8001: 7994: 7993: 7989: 7986: 7985: 7977: 7967: 7966: 7962: 7955: 7947: 7943: 7939: 7931: 7909: 7899: 7898: 7894: 7888: 7884: 7877: 7869: 7865: 7861: 7850: 7840: 7839: 7835: 7825: 7817: 7813: 7809: 7801: 7782: 7772: 7771: 7767: 7757: 7749: 7745: 7741: 7730: 7720: 7719: 7715: 7708: 7700: 7696: 7692: 7687: 7683: 7679: 7671: 7667: 7663: 7653: 7646: 7642: 7638: 7636: 7629: 7621: 7611: 7610: 7606: 7599: 7591: 7587: 7583: 7576: 7575: 7571: 7568: 7567: 7559: 7549: 7548: 7544: 7537: 7529: 7525: 7521: 7513: 7491: 7481: 7480: 7476: 7466: 7458: 7454: 7450: 7439: 7429: 7428: 7424: 7417: 7409: 7405: 7401: 7396: 7392: 7388: 7372: 7365: 7361: 7357: 7355: 7348: 7340: 7330: 7329: 7325: 7318: 7310: 7306: 7302: 7295: 7294: 7290: 7286: 7284: 7281: 7280: 7279:We remain with 7254: 7244: 7243: 7239: 7232: 7224: 7220: 7216: 7215: 7211: 7198: 7183: 7179: 7169: 7162: 7158: 7154: 7152: 7151: 7147: 7131: 7121: 7120: 7116: 7109: 7101: 7097: 7093: 7092: 7088: 7081: 7080: 7076: 7074: 7071: 7070: 7047: 7046: 7044: 7041: 7040: 7021: 7012: 7008: 7006: 7003: 7002: 6985: 6984: 6982: 6979: 6978: 6959: 6951: 6941: 6940: 6936: 6925: 6915: 6914: 6910: 6908: 6905: 6904: 6885: 6875: 6874: 6870: 6863: 6862: 6858: 6847: 6837: 6836: 6832: 6819: 6804: 6794: 6781: 6777: 6776: 6766: 6760: 6756: 6754: 6753: 6749: 6738: 6728: 6727: 6723: 6716: 6715: 6704: 6703: 6702: 6700: 6697: 6696: 6689: 6658: 6638: 6630: 6614: 6606: 6602: 6598: 6590: 6573: 6560: 6555: 6551: 6540: 6539: 6538: 6536: 6533: 6532: 6503: 6499: 6493: 6489: 6483: 6470: 6466: 6460: 6449: 6448: 6447: 6445: 6442: 6441: 6423: 6415: 6411: 6407: 6395: 6391: 6382: 6378: 6367: 6363: 6362: 6358: 6349: 6345: 6336: 6332: 6321: 6317: 6316: 6312: 6303: 6299: 6290: 6286: 6275: 6271: 6270: 6266: 6264: 6261: 6260: 6240: 6236: 6234: 6231: 6230: 6213: 6209: 6207: 6204: 6203: 6186: 6182: 6180: 6177: 6176: 6159: 6155: 6153: 6150: 6149: 6128: 6124: 6117: 6113: 6106: 6104: 6095: 6091: 6089: 6086: 6085: 6069: 6060: 6056: 6054: 6051: 6050: 6033: 6029: 6011: 6007: 5998: 5994: 5988: 5984: 5975: 5971: 5962: 5958: 5952: 5948: 5942: 5938: 5936: 5933: 5932: 5910: 5906: 5897: 5893: 5887: 5883: 5874: 5870: 5866: 5862: 5851: 5847: 5841: 5837: 5831: 5827: 5823: 5819: 5817: 5814: 5813: 5796: 5792: 5790: 5787: 5786: 5765: 5761: 5756: 5752: 5741: 5740: 5739: 5728: 5724: 5715: 5711: 5706: 5702: 5691: 5690: 5689: 5687: 5684: 5683: 5659: 5639: 5634: 5633: 5627: 5623: 5617: 5605: 5604: 5600: 5595: 5592: 5591: 5572: 5557: 5553: 5543: 5537: 5533: 5531: 5514: 5513: 5511: 5508: 5507: 5484: 5479: 5478: 5472: 5468: 5462: 5450: 5448: 5445: 5444: 5427: 5423: 5421: 5418: 5417: 5401: 5399: 5396: 5395: 5372: 5355: 5347: 5327: 5322: 5321: 5315: 5311: 5305: 5293: 5292: 5288: 5283: 5280: 5279: 5254: 5250: 5244: 5240: 5234: 5230: 5226: 5222: 5207: 5203: 5194: 5190: 5181: 5170: 5169: 5168: 5157: 5153: 5152: 5148: 5146: 5143: 5142: 5117: 5114: 5113: 5086: 5082: 5071: 5068: 5067: 5044: 5040: 5038: 5035: 5034: 5018: 5015: 5014: 4977: 4966: 4965: 4964: 4962: 4959: 4958: 4939: 4928: 4926: 4925: 4924: 4918: 4914: 4905: 4894: 4893: 4892: 4890: 4887: 4886: 4869: 4858: 4857: 4856: 4850: 4839: 4838: 4837: 4831: 4820: 4819: 4818: 4804: 4803: 4801: 4798: 4797: 4780: 4775: 4774: 4768: 4764: 4758: 4747: 4735: 4733: 4730: 4729: 4691: 4686: 4667: 4662: 4654: 4644: 4643: 4639: 4628: 4618: 4617: 4613: 4611: 4608: 4607: 4588: 4578: 4577: 4573: 4566: 4558: 4554: 4550: 4539: 4529: 4528: 4524: 4522: 4519: 4518: 4496: 4483: 4475: 4468: 4464: 4453: 4445: 4437: 4420: 4407: 4396: 4395: 4394: 4393: 4391: 4388: 4387: 4371: 4363: 4360: 4359: 4334: 4331: 4330: 4313: 4308: 4307: 4300: 4299: 4295: 4290: 4282: 4279: 4278: 4262: 4253: 4248: 4247: 4239: 4228: 4222: 4218: 4212: 4207: 4206: 4199: 4198: 4194: 4189: 4181: 4172: 4159: 4146: 4145: 4134: 4132: 4131: 4130: 4129: 4125: 4124: 4118: 4114: 4106: 4097: 4092: 4091: 4083: 4072: 4066: 4062: 4054: 4051: 4050: 4034: 4026: 4023: 4022: 4004: 3996: 3992: 3988: 3978: 3977: 3973: 3971: 3968: 3967: 3946: 3941: 3940: 3931: 3926: 3925: 3924: 3920: 3909: 3904: 3903: 3902: 3898: 3890: 3885: 3884: 3883: 3879: 3877: 3874: 3873: 3854: 3846: 3843: 3842: 3823: 3809: 3804: 3803: 3794: 3789: 3788: 3787: 3776: 3774: 3773: 3772: 3760: 3755: 3754: 3749: 3740: 3735: 3734: 3729: 3721: 3704: 3690: 3685: 3684: 3683: 3672: 3670: 3669: 3668: 3660: 3655: 3654: 3653: 3642: 3640: 3639: 3638: 3636: 3633: 3632: 3614: 3613: 3602: 3600: 3599: 3598: 3596: 3593: 3592: 3573: 3560: 3559: 3555: 3544: 3531: 3530: 3519: 3517: 3516: 3515: 3513: 3510: 3509: 3482: 3481: 3470: 3468: 3467: 3466: 3452: 3451: 3446: 3443: 3442: 3423: 3404: 3398: 3387: 3385: 3384: 3383: 3381: 3367: 3366: 3364: 3361: 3360: 3337: 3318: 3317: 3312: 3311: 3303: 3295: 3292: 3291: 3273: 3272: 3266: 3262: 3259: 3258: 3252: 3248: 3245: 3244: 3238: 3234: 3227: 3226: 3218: 3207: 3206: 3200: 3195: 3194: 3192: 3186: 3181: 3180: 3178: 3172: 3167: 3166: 3159: 3158: 3150: 3148: 3145: 3144: 3130: 3129: 3121: 3116: 3108: 3093: 3092: 3083: 3078: 3077: 3071: 3067: 3058: 3053: 3052: 3046: 3042: 3033: 3028: 3027: 3021: 3017: 3009: 2994: 2993: 2983: 2982: 2977: 2976: 2968: 2955: 2947: 2934: 2933: 2922: 2920: 2919: 2918: 2914: 2912: 2909: 2908: 2895: 2893:Using operators 2890: 2868: 2864: 2855: 2851: 2842: 2838: 2837: 2826: 2825: 2824: 2822: 2819: 2818: 2807: 2803: 2800: 2794: 2787: 2781: 2774: 2768: 2762: 2739: 2735: 2726: 2722: 2713: 2709: 2708: 2697: 2696: 2695: 2693: 2690: 2689: 2683: 2660: 2647: 2639: 2635: 2631: 2620: 2612: 2609: 2608: 2604: 2590: 2589: 2578: 2563: 2562: 2554: 2540: 2536: 2526: 2522: 2514: 2510: 2497: 2493: 2486: 2478: 2471: 2467: 2458: 2457: 2451: 2450: 2442: 2428: 2424: 2414: 2410: 2404: 2403: 2397: 2396: 2388: 2383: 2382: 2374: 2367: 2363: 2356: 2348: 2341: 2337: 2331: 2330: 2321: 2320: 2311: 2306: 2305: 2297: 2280: 2275: 2274: 2266: 2255: 2248: 2244: 2237: 2228: 2223: 2222: 2214: 2204: 2202: 2199: 2198: 2175: 2162: 2154: 2147: 2143: 2132: 2124: 2121: 2120: 2117: 2109: 2106: 2100: 2093: 2087: 2080: 2074: 2064: 2058: 2055: 2051: 2032: 2018: 2014: 2004: 2000: 1988: 1983: 1982: 1974: 1966: 1963: 1962: 1959: 1947: 1944: 1937: 1930: 1924: 1921: 1917: 1911: 1904: 1900: 1893: 1874: 1862: 1858: 1846: 1841: 1840: 1832: 1824: 1821: 1820: 1816: 1809: 1804: 1796: 1792: 1783: 1779: 1776: 1770: 1763: 1757: 1750: 1744: 1738: 1715: 1711: 1702: 1698: 1689: 1685: 1684: 1673: 1672: 1671: 1669: 1666: 1665: 1662: 1650: 1642: 1632: 1628: 1619: 1611: 1604: 1595: 1587: 1584: 1577: 1570: 1564: 1552: 1548: 1528: 1523: 1522: 1516: 1512: 1503: 1498: 1497: 1491: 1487: 1478: 1473: 1472: 1466: 1462: 1460: 1457: 1456: 1453: 1446: 1439: 1433: 1423: 1408: 1403: 1398: 1376: 1366: 1365: 1361: 1354: 1346: 1342: 1338: 1327: 1319: 1309: 1308: 1304: 1302: 1299: 1298: 1282: 1280: 1277: 1276: 1260: 1258: 1255: 1254: 1250: 1248:Bloch's theorem 1241: 1218: 1210: 1202: 1192: 1191: 1187: 1176: 1166: 1165: 1161: 1159: 1156: 1155: 1144: 1122: 1109: 1101: 1097: 1093: 1082: 1074: 1071: 1070: 1054: 1051: 1050: 1035: 1033:Bloch's theorem 1029: 1017: 1006: 988: 979: 973: 967: 961: 944: 932: 922: 906: 903: 902: 892: 886: 880: 874: 858: 855: 854: 848: 826: 813: 805: 801: 797: 786: 778: 775: 774: 763: 756: 750: 747: 741: 738: 732: 724: 695: 690: 661: 659: 656: 655: 628: 621: 617: 607: 603: 599: 597: 594: 593: 577: 575: 572: 571: 549: 545: 541: 539: 536: 535: 519: 517: 514: 513: 495: 491: 487: 485: 482: 481: 465: 462: 461: 445: 442: 441: 425: 423: 420: 419: 399: 396: 395: 377: 373: 369: 367: 364: 363: 351:(or less often 349:Bloch electrons 322:Bloch functions 298: 295: 294: 274: 271: 270: 250: 248: 245: 244: 221: 218: 217: 197: 194: 193: 177: 175: 172: 171: 168: 149: 136: 128: 124: 120: 109: 101: 98: 97: 72:Bloch's theorem 57: 35: 28: 23: 22: 15: 12: 11: 5: 13687: 13677: 13676: 13671: 13666: 13649: 13648: 13646: 13645: 13633: 13630:Physics Portal 13621: 13609: 13596: 13593: 13592: 13590: 13589: 13584: 13579: 13577:Liquid crystal 13574: 13569: 13563: 13561: 13555: 13554: 13552: 13551: 13546: 13545: 13544: 13539: 13529: 13524: 13519: 13514: 13509: 13504: 13499: 13494: 13488: 13486: 13484:Quasiparticles 13480: 13479: 13477: 13476: 13471: 13466: 13461: 13456: 13451: 13446: 13444:Superdiamagnet 13441: 13436: 13430: 13428: 13424: 13423: 13420: 13419: 13417: 13416: 13411: 13406: 13401: 13396: 13390: 13388: 13384: 13383: 13381: 13380: 13375: 13370: 13368:Superconductor 13365: 13360: 13355: 13350: 13348:Mott insulator 13345: 13339: 13337: 13333: 13332: 13330: 13329: 13324: 13319: 13314: 13309: 13304: 13299: 13294: 13289: 13284: 13279: 13273: 13271: 13267: 13266: 13264: 13263: 13258: 13253: 13248: 13243: 13238: 13233: 13228: 13222: 13220: 13213: 13209: 13208: 13206: 13205: 13200: 13195: 13190: 13184: 13182: 13178: 13177: 13170: 13168: 13166: 13165: 13160: 13155: 13150: 13145: 13140: 13135: 13130: 13125: 13120: 13115: 13109: 13107: 13101: 13100: 13093: 13092: 13085: 13078: 13070: 13063: 13062: 13052:(3): 619–654. 13037: 13028: 13019: 13005: 12990: 12984: 12966:Ashcroft, Neil 12961: 12959: 12956: 12953: 12952: 12925:(3): 633–670. 12909: 12890:(1): 145–156. 12870: 12857: 12850: 12827: 12808: 12781:Gaston Floquet 12772: 12735: 12689: 12677: 12665: 12653: 12638: 12626: 12615: 12597: 12586: 12543: 12531: 12519: 12512: 12494: 12487: 12464: 12454: 12453: 12451: 12448: 12446: 12445: 12440: 12435: 12430: 12425: 12420: 12415: 12410: 12404: 12402: 12399: 12360: 12357: 12354: 12351: 12348: 12345: 12342: 12339: 12336: 12328: 12324: 12320: 12315: 12310: 12306: 12283:Floquet theory 12271:Gaston Floquet 12261: 12258: 12233: 12228: 12224: 12220: 12216: 12212: 12207: 12203: 12200: 12197: 12191: 12188: 12182: 12169: 12147: 12122: 12117: 12113: 12110: 12106: 12102: 12098: 12094: 12090: 12085: 12081: 12078: 12075: 12071: 12067: 12063: 12059: 12055: 12041:charge carrier 12037: 12033:charge carrier 12020: 12016: 12012: 12008: 11983: 11979: 11975: 11947: 11943: 11939: 11933: 11930: 11925: 11921: 11918: 11914: 11910: 11905: 11901: 11895: 11891: 11887: 11883: 11877: 11873: 11869: 11866: 11862: 11857: 11852: 11849: 11845: 11842: 11838: 11833: 11830: 11825: 11819: 11815: 11811: 11808: 11804: 11799: 11795: 11792: 11789: 11786: 11782: 11778: 11774: 11768: 11764: 11760: 11757: 11753: 11748: 11743: 11740: 11736: 11733: 11729: 11724: 11721: 11716: 11710: 11706: 11702: 11699: 11695: 11690: 11686: 11683: 11675: 11672: 11668: 11665: 11660: 11654: 11649: 11644: 11639: 11635: 11629: 11624: 11619: 11616: 11612: 11606: 11601: 11597: 11591: 11583: 11579: 11575: 11570: 11566: 11562: 11557: 11553: 11549: 11544: 11540: 11534: 11530: 11504: 11500: 11477: 11473: 11446: 11441: 11438: 11433: 11429: 11423: 11419: 11415: 11407: 11402: 11397: 11393: 11388: 11385: 11380: 11376: 11373: 11370: 11367: 11364: 11361: 11357: 11351: 11346: 11342: 11335: 11330: 11326: 11323: 11319: 11310: 11307: 11303: 11300: 11295: 11291: 11286: 11282: 11275: 11272: 11266: 11262: 11256: 11251: 11247: 11241: 11237: 11228: 11224: 11220: 11215: 11211: 11207: 11200: 11196: 11190: 11186: 11177: 11174: 11170: 11164: 11161: 11137: 11133: 11129: 11122: 11117: 11113: 11109: 11105: 11102: 11098: 11094: 11090: 11086: 11080: 11076: 11072: 11045: 11040: 11037: 11032: 11028: 11022: 11018: 11014: 11006: 11001: 10993: 10988: 10985: 10980: 10976: 10972: 10968: 10965: 10962: 10959: 10956: 10953: 10949: 10943: 10938: 10934: 10926: 10920: 10916: 10912: 10906: 10902: 10899: 10895: 10886: 10883: 10879: 10876: 10871: 10867: 10862: 10858: 10851: 10848: 10842: 10838: 10832: 10827: 10823: 10817: 10813: 10804: 10800: 10796: 10791: 10787: 10783: 10776: 10772: 10766: 10762: 10753: 10750: 10746: 10740: 10737: 10721: 10705: 10701: 10697: 10691: 10688: 10683: 10679: 10676: 10672: 10668: 10663: 10659: 10653: 10649: 10645: 10641: 10635: 10631: 10627: 10624: 10620: 10615: 10610: 10607: 10603: 10600: 10596: 10591: 10588: 10583: 10577: 10573: 10569: 10566: 10562: 10557: 10553: 10550: 10547: 10544: 10540: 10536: 10532: 10526: 10522: 10518: 10515: 10511: 10506: 10501: 10498: 10494: 10491: 10487: 10482: 10479: 10474: 10468: 10464: 10460: 10457: 10453: 10448: 10444: 10441: 10433: 10430: 10426: 10423: 10418: 10412: 10407: 10402: 10397: 10393: 10387: 10382: 10377: 10374: 10370: 10364: 10359: 10355: 10349: 10341: 10337: 10333: 10328: 10324: 10320: 10315: 10311: 10307: 10302: 10298: 10292: 10288: 10271: 10268:effective mass 10249: 10245: 10241: 10237: 10234: 10231: 10228: 10225: 10220: 10214: 10210: 10206: 10200: 10196: 10193: 10188: 10183: 10179: 10173: 10166: 10162: 10155: 10151: 10147: 10121: 10117: 10113: 10109: 10105: 10101: 10098: 10095: 10092: 10089: 10084: 10078: 10074: 10070: 10064: 10060: 10057: 10052: 10047: 10043: 10037: 10030: 10026: 10019: 10015: 10011: 9976: 9972: 9968: 9962: 9956: 9952: 9948: 9942: 9938: 9935: 9912: 9908: 9904: 9898: 9894: 9888: 9884: 9879: 9875: 9872: 9869: 9866: 9863: 9858: 9853: 9849: 9841: 9835: 9831: 9827: 9821: 9817: 9814: 9809: 9805: 9801: 9796: 9792: 9783: 9779: 9775: 9768: 9764: 9760: 9752: 9748: 9722: 9719: 9716: 9713: 9705: 9699: 9696: 9691: 9687: 9682: 9677: 9673: 9665: 9660: 9653: 9649: 9642: 9639: 9631: 9626: 9622: 9616: 9612: 9609: 9605: 9596: 9593: 9589: 9586: 9581: 9577: 9572: 9568: 9561: 9558: 9550: 9545: 9541: 9535: 9531: 9528: 9525: 9520: 9515: 9511: 9507: 9502: 9498: 9475: 9471: 9464: 9461: 9455: 9451: 9445: 9442: 9438: 9434: 9431: 9428: 9425: 9422: 9419: 9415: 9409: 9404: 9400: 9394: 9388: 9380: 9377: 9370: 9364: 9360: 9356: 9348: 9345: 9318: 9314: 9310: 9302: 9299: 9275: 9271: 9267: 9263: 9259: 9254: 9250: 9229: 9224: 9220: 9216: 9213: 9210: 9205: 9201: 9195: 9191: 9182: 9178: 9174: 9169: 9165: 9161: 9154: 9150: 9144: 9140: 9131: 9128: 9124: 9118: 9115: 9110: 9105: 9101: 9092: 9088: 9084: 9077: 9073: 9069: 9061: 9057: 9053: 9050: 9046: 9042: 9037: 9033: 9029: 9026: 9022: 9018: 9014: 9010: 9005: 9001: 8958: 8954: 8950: 8945: 8941: 8937: 8932: 8928: 8924: 8919: 8915: 8909: 8905: 8877: 8873: 8866: 8862: 8858: 8841: 8828: 8822: 8818: 8811: 8808: 8805: 8802: 8796: 8792: 8785: 8780: 8777: 8772: 8766: 8762: 8758: 8754: 8751: 8748: 8745: 8742: 8737: 8731: 8727: 8723: 8717: 8713: 8710: 8705: 8700: 8696: 8690: 8683: 8679: 8672: 8668: 8664: 8648: 8623: 8619: 8599: 8596: 8593: 8590: 8570: 8566: 8562: 8559: 8556: 8553: 8550: 8547: 8544: 8532: 8528: 8496: 8492: 8488: 8482: 8477: 8473: 8469: 8465: 8462: 8459: 8456: 8452: 8448: 8442: 8437: 8431: 8426: 8421: 8417: 8414: 8411: 8408: 8404: 8396: 8393: 8387: 8383: 8377: 8374: 8372: 8370: 8366: 8362: 8356: 8351: 8344: 8339: 8335: 8334: 8331: 8327: 8323: 8317: 8312: 8305: 8301: 8297: 8293: 8289: 8285: 8281: 8277: 8274: 8271: 8267: 8263: 8259: 8255: 8249: 8244: 8238: 8234: 8227: 8223: 8219: 8215: 8211: 8207: 8204: 8200: 8196: 8190: 8185: 8181: 8175: 8171: 8167: 8163: 8159: 8155: 8151: 8147: 8144: 8141: 8138: 8134: 8130: 8124: 8119: 8112: 8108: 8104: 8100: 8096: 8090: 8085: 8079: 8072: 8069: 8063: 8059: 8053: 8050: 8048: 8046: 8042: 8038: 8032: 8027: 8020: 8016: 8012: 8008: 8004: 7997: 7992: 7988: 7987: 7984: 7980: 7976: 7970: 7965: 7958: 7954: 7950: 7946: 7942: 7938: 7934: 7930: 7927: 7924: 7920: 7916: 7912: 7908: 7902: 7897: 7891: 7887: 7880: 7876: 7872: 7868: 7864: 7860: 7857: 7853: 7849: 7843: 7838: 7834: 7828: 7824: 7820: 7816: 7812: 7808: 7804: 7800: 7797: 7793: 7789: 7785: 7781: 7775: 7770: 7766: 7760: 7756: 7752: 7748: 7744: 7740: 7737: 7733: 7729: 7723: 7718: 7711: 7707: 7703: 7699: 7695: 7690: 7686: 7682: 7678: 7674: 7670: 7666: 7659: 7656: 7649: 7645: 7641: 7635: 7632: 7630: 7628: 7624: 7620: 7614: 7609: 7602: 7598: 7594: 7590: 7586: 7579: 7574: 7570: 7569: 7566: 7562: 7558: 7552: 7547: 7540: 7536: 7532: 7528: 7524: 7520: 7516: 7512: 7509: 7506: 7502: 7498: 7494: 7490: 7484: 7479: 7475: 7469: 7465: 7461: 7457: 7453: 7449: 7446: 7442: 7438: 7432: 7427: 7420: 7416: 7412: 7408: 7404: 7399: 7395: 7391: 7387: 7384: 7378: 7375: 7368: 7364: 7360: 7354: 7351: 7349: 7347: 7343: 7339: 7333: 7328: 7321: 7317: 7313: 7309: 7305: 7298: 7293: 7289: 7288: 7265: 7261: 7257: 7253: 7247: 7242: 7235: 7231: 7227: 7223: 7219: 7214: 7209: 7205: 7201: 7197: 7194: 7191: 7186: 7182: 7175: 7172: 7165: 7161: 7157: 7150: 7146: 7142: 7138: 7134: 7130: 7124: 7119: 7112: 7108: 7104: 7100: 7096: 7091: 7084: 7079: 7065: 7050: 7028: 7024: 7020: 7015: 7011: 6988: 6966: 6962: 6958: 6954: 6950: 6944: 6939: 6935: 6932: 6928: 6924: 6918: 6913: 6892: 6888: 6884: 6878: 6873: 6866: 6861: 6857: 6854: 6850: 6846: 6840: 6835: 6830: 6826: 6822: 6818: 6815: 6812: 6807: 6802: 6797: 6793: 6790: 6787: 6784: 6780: 6772: 6769: 6763: 6759: 6752: 6748: 6745: 6741: 6737: 6731: 6726: 6719: 6711: 6708: 6688: 6685: 6645: 6641: 6637: 6633: 6629: 6626: 6623: 6617: 6613: 6609: 6605: 6601: 6597: 6593: 6589: 6586: 6583: 6580: 6576: 6572: 6569: 6563: 6558: 6554: 6547: 6544: 6520: 6517: 6514: 6509: 6506: 6502: 6496: 6492: 6486: 6482: 6478: 6473: 6469: 6463: 6456: 6453: 6426: 6422: 6418: 6414: 6410: 6406: 6403: 6398: 6394: 6390: 6385: 6381: 6377: 6370: 6366: 6361: 6357: 6352: 6348: 6344: 6339: 6335: 6331: 6324: 6320: 6315: 6311: 6306: 6302: 6298: 6293: 6289: 6285: 6278: 6274: 6269: 6243: 6239: 6216: 6212: 6189: 6185: 6162: 6158: 6131: 6127: 6120: 6116: 6112: 6109: 6103: 6098: 6094: 6072: 6068: 6063: 6059: 6036: 6032: 6028: 6025: 6022: 6019: 6014: 6010: 6006: 6001: 5997: 5991: 5987: 5983: 5978: 5974: 5970: 5965: 5961: 5955: 5951: 5945: 5941: 5918: 5913: 5909: 5905: 5900: 5896: 5890: 5886: 5882: 5877: 5873: 5869: 5865: 5861: 5854: 5850: 5844: 5840: 5834: 5830: 5826: 5822: 5799: 5795: 5768: 5764: 5759: 5755: 5748: 5745: 5738: 5731: 5727: 5723: 5718: 5714: 5709: 5705: 5698: 5695: 5666: 5662: 5658: 5655: 5652: 5648: 5642: 5637: 5630: 5626: 5620: 5616: 5612: 5608: 5603: 5599: 5579: 5575: 5571: 5568: 5565: 5560: 5556: 5549: 5546: 5540: 5536: 5530: 5527: 5521: 5518: 5487: 5482: 5475: 5471: 5465: 5461: 5457: 5453: 5430: 5426: 5404: 5379: 5375: 5371: 5368: 5365: 5362: 5358: 5354: 5350: 5346: 5343: 5340: 5336: 5330: 5325: 5318: 5314: 5308: 5304: 5300: 5296: 5291: 5287: 5257: 5253: 5247: 5243: 5237: 5233: 5229: 5225: 5221: 5218: 5215: 5210: 5206: 5202: 5197: 5193: 5189: 5184: 5177: 5174: 5167: 5160: 5156: 5151: 5127: 5124: 5121: 5097: 5094: 5089: 5085: 5081: 5078: 5075: 5055: 5052: 5047: 5043: 5022: 5011:roots of unity 5007:representation 4980: 4973: 4970: 4942: 4935: 4931: 4921: 4917: 4913: 4908: 4901: 4898: 4872: 4865: 4862: 4853: 4846: 4843: 4834: 4827: 4824: 4817: 4811: 4808: 4783: 4778: 4771: 4767: 4761: 4756: 4753: 4750: 4746: 4742: 4738: 4711: 4690: 4687: 4674: 4670: 4665: 4661: 4657: 4653: 4647: 4642: 4638: 4635: 4631: 4627: 4621: 4616: 4595: 4591: 4587: 4581: 4576: 4569: 4565: 4561: 4557: 4553: 4549: 4546: 4542: 4538: 4532: 4527: 4506: 4503: 4499: 4495: 4492: 4486: 4482: 4478: 4474: 4471: 4467: 4463: 4460: 4456: 4452: 4448: 4444: 4440: 4436: 4433: 4430: 4427: 4423: 4419: 4416: 4410: 4403: 4400: 4374: 4370: 4367: 4347: 4344: 4341: 4338: 4316: 4311: 4303: 4298: 4293: 4289: 4286: 4277:and therefore 4265: 4261: 4256: 4251: 4246: 4242: 4238: 4235: 4231: 4225: 4221: 4215: 4210: 4202: 4197: 4192: 4188: 4184: 4180: 4175: 4170: 4166: 4162: 4158: 4155: 4149: 4141: 4137: 4128: 4121: 4117: 4113: 4109: 4105: 4100: 4095: 4090: 4086: 4082: 4079: 4075: 4069: 4065: 4061: 4058: 4037: 4033: 4030: 4007: 4003: 3999: 3995: 3991: 3987: 3981: 3976: 3949: 3944: 3939: 3934: 3929: 3923: 3919: 3912: 3907: 3901: 3893: 3888: 3882: 3861: 3857: 3853: 3850: 3830: 3826: 3822: 3819: 3812: 3807: 3802: 3797: 3792: 3783: 3779: 3771: 3768: 3763: 3758: 3752: 3748: 3743: 3738: 3732: 3728: 3724: 3720: 3717: 3714: 3711: 3707: 3703: 3700: 3693: 3688: 3679: 3675: 3663: 3658: 3649: 3645: 3617: 3609: 3605: 3580: 3576: 3572: 3569: 3563: 3558: 3554: 3551: 3547: 3543: 3540: 3534: 3526: 3522: 3497: 3494: 3491: 3485: 3477: 3473: 3465: 3459: 3456: 3450: 3430: 3426: 3422: 3419: 3416: 3410: 3407: 3401: 3394: 3390: 3380: 3374: 3371: 3344: 3340: 3336: 3333: 3330: 3327: 3321: 3315: 3310: 3306: 3302: 3299: 3277: 3269: 3265: 3261: 3260: 3255: 3251: 3247: 3246: 3241: 3237: 3233: 3232: 3230: 3225: 3221: 3216: 3211: 3203: 3198: 3193: 3189: 3184: 3179: 3175: 3170: 3165: 3164: 3162: 3157: 3153: 3128: 3124: 3119: 3115: 3111: 3107: 3104: 3101: 3098: 3096: 3094: 3091: 3086: 3081: 3074: 3070: 3066: 3061: 3056: 3049: 3045: 3041: 3036: 3031: 3024: 3020: 3016: 3012: 3008: 3005: 3002: 2999: 2997: 2995: 2992: 2986: 2980: 2975: 2971: 2967: 2964: 2961: 2958: 2956: 2954: 2950: 2946: 2943: 2937: 2929: 2925: 2917: 2916: 2899: 2894: 2891: 2871: 2867: 2863: 2858: 2854: 2850: 2845: 2841: 2833: 2830: 2805: 2798: 2792: 2785: 2779: 2772: 2766: 2742: 2738: 2734: 2729: 2725: 2721: 2716: 2712: 2704: 2701: 2688:As above, let 2670: 2667: 2663: 2659: 2656: 2650: 2646: 2642: 2638: 2634: 2630: 2627: 2623: 2619: 2616: 2588: 2585: 2581: 2577: 2574: 2571: 2568: 2566: 2564: 2561: 2557: 2553: 2550: 2543: 2539: 2535: 2532: 2529: 2525: 2517: 2513: 2509: 2506: 2503: 2500: 2496: 2489: 2485: 2481: 2477: 2474: 2470: 2466: 2463: 2461: 2459: 2454: 2449: 2445: 2441: 2438: 2431: 2427: 2423: 2420: 2417: 2413: 2407: 2400: 2391: 2386: 2381: 2377: 2373: 2370: 2366: 2359: 2355: 2351: 2347: 2344: 2340: 2334: 2329: 2326: 2324: 2322: 2319: 2314: 2309: 2304: 2300: 2296: 2293: 2288: 2283: 2278: 2273: 2269: 2265: 2262: 2258: 2254: 2251: 2247: 2243: 2240: 2238: 2236: 2231: 2226: 2221: 2217: 2213: 2210: 2207: 2206: 2186: 2182: 2178: 2174: 2171: 2165: 2161: 2157: 2153: 2150: 2146: 2142: 2139: 2135: 2131: 2128: 2113: 2104: 2098: 2091: 2085: 2078: 2072: 2053: 2039: 2035: 2031: 2028: 2021: 2017: 2013: 2010: 2007: 2003: 1999: 1996: 1991: 1986: 1981: 1977: 1973: 1970: 1955: 1942: 1935: 1928: 1919: 1902: 1881: 1877: 1873: 1870: 1865: 1861: 1857: 1854: 1849: 1844: 1839: 1835: 1831: 1828: 1812:Proof of Lemma 1810: 1787: 1781: 1774: 1768: 1761: 1755: 1748: 1742: 1718: 1714: 1710: 1705: 1701: 1697: 1692: 1688: 1680: 1677: 1661: 1658: 1646: 1624: 1615: 1600: 1591: 1582: 1575: 1568: 1550: 1536: 1531: 1526: 1519: 1515: 1511: 1506: 1501: 1494: 1490: 1486: 1481: 1476: 1469: 1465: 1451: 1444: 1437: 1422: 1419: 1412: 1407: 1404: 1402: 1399: 1386: 1383: 1379: 1375: 1369: 1364: 1357: 1353: 1349: 1345: 1341: 1337: 1334: 1330: 1326: 1322: 1318: 1312: 1307: 1285: 1263: 1245: 1240: 1239: 1228: 1225: 1221: 1217: 1213: 1209: 1205: 1201: 1195: 1190: 1186: 1183: 1179: 1175: 1169: 1164: 1132: 1129: 1125: 1121: 1118: 1112: 1108: 1104: 1100: 1096: 1092: 1089: 1085: 1081: 1078: 1058: 1047: 1030: 1028: 1025: 1016: 1013: 1003:group velocity 910: 862: 836: 833: 829: 825: 822: 816: 812: 808: 804: 800: 796: 793: 789: 785: 782: 761: 754: 745: 736: 723: 720: 694: 691: 689: 686: 678:Brillouin zone 664: 641: 637: 634: 631: 627: 624: 620: 616: 610: 606: 602: 580: 552: 548: 544: 522: 498: 494: 490: 469: 449: 428: 403: 380: 376: 372: 334:wave functions 315:imaginary unit 302: 291:Euler's number 278: 253: 225: 201: 180: 156: 152: 148: 145: 139: 135: 131: 127: 123: 119: 116: 112: 108: 105: 94:Bloch function 92: 46:square modulus 26: 9: 6: 4: 3: 2: 13686: 13675: 13672: 13670: 13667: 13665: 13662: 13661: 13659: 13644: 13643: 13634: 13632: 13631: 13626: 13622: 13620: 13619: 13610: 13608: 13607: 13598: 13597: 13594: 13588: 13585: 13583: 13580: 13578: 13575: 13573: 13570: 13568: 13565: 13564: 13562: 13560: 13556: 13550: 13547: 13543: 13540: 13538: 13535: 13534: 13533: 13530: 13528: 13525: 13523: 13520: 13518: 13515: 13513: 13510: 13508: 13505: 13503: 13500: 13498: 13495: 13493: 13490: 13489: 13487: 13485: 13481: 13475: 13472: 13470: 13467: 13465: 13462: 13460: 13457: 13455: 13452: 13450: 13447: 13445: 13442: 13440: 13437: 13435: 13432: 13431: 13429: 13425: 13415: 13412: 13410: 13407: 13405: 13402: 13400: 13397: 13395: 13392: 13391: 13389: 13385: 13379: 13376: 13374: 13371: 13369: 13366: 13364: 13361: 13359: 13356: 13354: 13353:Semiconductor 13351: 13349: 13346: 13344: 13341: 13340: 13338: 13334: 13328: 13325: 13323: 13322:Hubbard model 13320: 13318: 13315: 13313: 13310: 13308: 13305: 13303: 13300: 13298: 13295: 13293: 13290: 13288: 13285: 13283: 13280: 13278: 13275: 13274: 13272: 13268: 13262: 13259: 13257: 13254: 13252: 13249: 13247: 13244: 13242: 13239: 13237: 13234: 13232: 13229: 13227: 13224: 13223: 13221: 13217: 13214: 13210: 13204: 13201: 13199: 13196: 13194: 13191: 13189: 13186: 13185: 13183: 13179: 13174: 13164: 13161: 13159: 13156: 13154: 13151: 13149: 13146: 13144: 13141: 13139: 13136: 13134: 13131: 13129: 13126: 13124: 13121: 13119: 13116: 13114: 13111: 13110: 13108: 13106: 13102: 13098: 13091: 13086: 13084: 13079: 13077: 13072: 13071: 13068: 13059: 13055: 13051: 13047: 13043: 13038: 13034: 13029: 13025: 13020: 13016: 13012: 13008: 13002: 12998: 12997: 12991: 12987: 12981: 12977: 12976: 12971: 12967: 12963: 12962: 12948: 12944: 12940: 12936: 12932: 12928: 12924: 12920: 12913: 12905: 12901: 12897: 12893: 12889: 12885: 12884:Amer. J. Math 12881: 12878:Katsuda, A.; 12874: 12867: 12861: 12853: 12851:0-486-49565-5 12847: 12843: 12842: 12837: 12831: 12822: 12818: 12812: 12803: 12798: 12794: 12790: 12786: 12782: 12776: 12766: 12761: 12757: 12753: 12749: 12745: 12739: 12731: 12727: 12723: 12719: 12715: 12711: 12707: 12704:(in German). 12703: 12699: 12693: 12687:, p. 227 12686: 12681: 12675:, p. 229 12674: 12669: 12663:, p. 228 12662: 12657: 12650: 12645: 12643: 12636:, p. 140 12635: 12630: 12624: 12619: 12608: 12601: 12595: 12590: 12571: 12567: 12560: 12556: 12550: 12548: 12541:, p. 137 12540: 12535: 12529:, p. 134 12528: 12523: 12515: 12509: 12505: 12498: 12490: 12488:0-471-14286-7 12484: 12480: 12479: 12474: 12468: 12459: 12455: 12444: 12441: 12439: 12436: 12434: 12431: 12429: 12426: 12424: 12421: 12419: 12416: 12414: 12411: 12409: 12406: 12405: 12398: 12396: 12391: 12389: 12385: 12379: 12375: 12358: 12355: 12352: 12349: 12343: 12337: 12334: 12326: 12322: 12318: 12313: 12308: 12304: 12292: 12288: 12284: 12280: 12276: 12272: 12268: 12257: 12255: 12254:Lorentz force 12251: 12245: 12231: 12222: 12214: 12205: 12201: 12198: 12195: 12189: 12186: 12180: 12172: 12168: 12166: 12162: 12134: 12120: 12111: 12092: 12083: 12079: 12076: 12073: 12044: 12042: 12036: 12034: 11981: 11977: 11973: 11961: 11931: 11928: 11923: 11919: 11903: 11899: 11885: 11875: 11867: 11864: 11850: 11847: 11831: 11828: 11817: 11809: 11806: 11793: 11787: 11776: 11766: 11758: 11755: 11741: 11738: 11722: 11719: 11708: 11700: 11697: 11684: 11673: 11670: 11666: 11663: 11658: 11652: 11647: 11642: 11637: 11633: 11627: 11622: 11617: 11614: 11610: 11604: 11599: 11595: 11589: 11581: 11577: 11568: 11564: 11542: 11538: 11532: 11502: 11498: 11475: 11471: 11439: 11436: 11431: 11427: 11417: 11413: 11405: 11386: 11383: 11368: 11365: 11359: 11349: 11344: 11340: 11324: 11308: 11305: 11301: 11298: 11293: 11289: 11284: 11280: 11273: 11270: 11264: 11260: 11254: 11249: 11245: 11239: 11235: 11226: 11222: 11213: 11209: 11198: 11194: 11188: 11175: 11172: 11168: 11162: 11159: 11131: 11127: 11111: 11107: 11103: 11092: 11084: 11074: 11070: 11038: 11035: 11030: 11026: 11016: 11012: 11004: 10986: 10983: 10978: 10966: 10960: 10957: 10951: 10941: 10936: 10932: 10924: 10914: 10910: 10900: 10897: 10884: 10881: 10877: 10874: 10869: 10865: 10860: 10856: 10849: 10846: 10840: 10836: 10830: 10825: 10821: 10815: 10811: 10802: 10798: 10789: 10785: 10774: 10770: 10764: 10751: 10748: 10744: 10738: 10735: 10719: 10689: 10686: 10681: 10677: 10661: 10657: 10643: 10633: 10625: 10622: 10608: 10605: 10589: 10586: 10575: 10567: 10564: 10551: 10545: 10534: 10524: 10516: 10513: 10499: 10496: 10480: 10477: 10466: 10458: 10455: 10442: 10431: 10428: 10424: 10421: 10416: 10410: 10405: 10400: 10395: 10391: 10385: 10380: 10375: 10372: 10368: 10362: 10357: 10353: 10347: 10339: 10335: 10326: 10322: 10300: 10296: 10290: 10274: 10270: 10269: 10263: 10243: 10239: 10229: 10226: 10218: 10208: 10204: 10194: 10191: 10186: 10181: 10177: 10171: 10153: 10149: 10115: 10111: 10099: 10093: 10090: 10082: 10072: 10068: 10058: 10055: 10050: 10045: 10041: 10035: 10017: 10013: 9997: 9991: 9970: 9966: 9960: 9950: 9946: 9936: 9933: 9906: 9902: 9896: 9892: 9886: 9873: 9867: 9864: 9856: 9851: 9847: 9839: 9829: 9825: 9815: 9812: 9807: 9803: 9799: 9794: 9790: 9781: 9777: 9766: 9762: 9750: 9746: 9737: 9720: 9717: 9714: 9711: 9703: 9697: 9694: 9689: 9685: 9680: 9675: 9671: 9663: 9651: 9647: 9637: 9629: 9624: 9620: 9610: 9607: 9594: 9591: 9587: 9584: 9579: 9575: 9570: 9566: 9556: 9548: 9543: 9539: 9529: 9526: 9523: 9518: 9513: 9509: 9505: 9500: 9496: 9473: 9469: 9462: 9459: 9453: 9449: 9443: 9432: 9426: 9423: 9417: 9407: 9402: 9398: 9392: 9375: 9368: 9358: 9343: 9312: 9297: 9265: 9252: 9248: 9222: 9218: 9211: 9208: 9203: 9199: 9193: 9189: 9180: 9176: 9167: 9163: 9152: 9148: 9142: 9129: 9126: 9122: 9116: 9113: 9108: 9103: 9099: 9090: 9086: 9075: 9071: 9059: 9055: 9051: 9035: 9031: 9027: 9016: 9003: 8999: 8990: 8984: 8978: 8956: 8952: 8943: 8939: 8917: 8913: 8907: 8864: 8860: 8839: 8803: 8778: 8775: 8770: 8760: 8756: 8746: 8743: 8735: 8725: 8721: 8711: 8708: 8703: 8698: 8694: 8688: 8670: 8666: 8651: 8647: 8644: 8642: 8638: 8617: 8591: 8588: 8568: 8557: 8548: 8545: 8542: 8511: 8475: 8460: 8457: 8435: 8429: 8424: 8415: 8409: 8406: 8402: 8394: 8391: 8385: 8381: 8375: 8373: 8349: 8337: 8310: 8299: 8291: 8287: 8272: 8269: 8265: 8242: 8236: 8221: 8213: 8209: 8205: 8183: 8169: 8161: 8157: 8153: 8145: 8142: 8139: 8117: 8106: 8098: 8094: 8088: 8077: 8070: 8067: 8061: 8057: 8051: 8049: 8025: 8014: 8006: 8002: 7990: 7963: 7952: 7944: 7940: 7925: 7922: 7918: 7895: 7889: 7874: 7866: 7862: 7858: 7836: 7822: 7814: 7810: 7806: 7798: 7795: 7791: 7768: 7754: 7746: 7742: 7738: 7716: 7705: 7697: 7693: 7684: 7680: 7676: 7668: 7664: 7657: 7654: 7647: 7643: 7639: 7633: 7631: 7607: 7596: 7588: 7584: 7572: 7545: 7534: 7526: 7522: 7507: 7504: 7500: 7477: 7463: 7455: 7451: 7447: 7425: 7414: 7406: 7402: 7393: 7389: 7385: 7376: 7373: 7366: 7362: 7358: 7352: 7350: 7326: 7315: 7307: 7303: 7291: 7277: 7263: 7240: 7229: 7221: 7217: 7212: 7207: 7192: 7189: 7184: 7173: 7170: 7163: 7159: 7155: 7148: 7144: 7140: 7117: 7106: 7098: 7094: 7089: 7077: 7064: 7013: 7009: 6956: 6937: 6933: 6911: 6871: 6859: 6855: 6833: 6828: 6813: 6810: 6805: 6800: 6791: 6785: 6782: 6778: 6770: 6767: 6761: 6757: 6750: 6746: 6724: 6706: 6694: 6684: 6682: 6677: 6675: 6671: 6667: 6663: 6657: 6635: 6624: 6621: 6611: 6603: 6599: 6584: 6581: 6567: 6552: 6542: 6515: 6507: 6504: 6500: 6494: 6490: 6484: 6480: 6476: 6471: 6467: 6461: 6451: 6420: 6412: 6408: 6404: 6396: 6392: 6388: 6383: 6379: 6368: 6364: 6359: 6350: 6346: 6342: 6337: 6333: 6322: 6318: 6313: 6304: 6300: 6296: 6291: 6287: 6276: 6272: 6267: 6257: 6241: 6237: 6214: 6210: 6187: 6183: 6160: 6156: 6146: 6129: 6125: 6118: 6114: 6110: 6107: 6101: 6096: 6092: 6066: 6061: 6057: 6034: 6030: 6026: 6023: 6020: 6012: 6008: 6004: 5999: 5995: 5989: 5985: 5976: 5972: 5968: 5963: 5959: 5953: 5949: 5943: 5939: 5911: 5907: 5903: 5898: 5894: 5888: 5884: 5875: 5871: 5867: 5863: 5859: 5852: 5848: 5842: 5838: 5832: 5828: 5824: 5820: 5797: 5793: 5783: 5766: 5762: 5753: 5743: 5736: 5729: 5725: 5721: 5716: 5712: 5703: 5693: 5682: 5677: 5653: 5650: 5646: 5640: 5628: 5624: 5618: 5614: 5610: 5601: 5597: 5566: 5563: 5558: 5547: 5544: 5538: 5534: 5528: 5525: 5516: 5505: 5500: 5485: 5473: 5469: 5463: 5459: 5455: 5428: 5424: 5393: 5366: 5363: 5352: 5341: 5338: 5334: 5328: 5316: 5312: 5306: 5302: 5298: 5289: 5285: 5277: 5272: 5255: 5251: 5245: 5241: 5235: 5231: 5227: 5223: 5219: 5208: 5204: 5200: 5195: 5191: 5182: 5158: 5154: 5149: 5139: 5119: 5111: 5095: 5092: 5087: 5079: 5073: 5053: 5050: 5045: 5041: 5020: 5012: 5008: 5004: 5000: 4995: 4978: 4955: 4940: 4919: 4915: 4911: 4906: 4870: 4851: 4832: 4815: 4781: 4769: 4765: 4759: 4754: 4751: 4748: 4744: 4740: 4727: 4723: 4719: 4710: 4708: 4704: 4700: 4696: 4685: 4659: 4640: 4636: 4614: 4574: 4563: 4555: 4551: 4547: 4525: 4504: 4490: 4480: 4472: 4469: 4465: 4461: 4450: 4442: 4431: 4428: 4414: 4368: 4365: 4345: 4342: 4339: 4336: 4314: 4296: 4287: 4284: 4259: 4254: 4233: 4223: 4219: 4213: 4195: 4186: 4178: 4173: 4168: 4153: 4126: 4119: 4115: 4111: 4103: 4098: 4077: 4067: 4063: 4059: 4056: 4031: 4028: 4001: 3993: 3989: 3985: 3974: 3964: 3947: 3937: 3932: 3921: 3917: 3910: 3899: 3891: 3880: 3848: 3817: 3810: 3800: 3795: 3769: 3761: 3746: 3741: 3726: 3715: 3712: 3698: 3691: 3661: 3567: 3556: 3552: 3538: 3495: 3492: 3463: 3454: 3417: 3414: 3408: 3405: 3399: 3378: 3369: 3358: 3331: 3328: 3308: 3297: 3275: 3267: 3263: 3253: 3249: 3239: 3235: 3228: 3223: 3214: 3209: 3201: 3187: 3173: 3160: 3155: 3113: 3102: 3099: 3097: 3084: 3072: 3068: 3064: 3059: 3047: 3043: 3039: 3034: 3022: 3018: 3014: 3003: 3000: 2998: 2973: 2962: 2959: 2957: 2941: 2905: 2898: 2889: 2869: 2865: 2861: 2856: 2852: 2848: 2843: 2839: 2828: 2816: 2812: 2797: 2791: 2784: 2778: 2771: 2765: 2760: 2740: 2736: 2732: 2727: 2723: 2719: 2714: 2710: 2699: 2686: 2682: 2668: 2654: 2644: 2636: 2632: 2628: 2614: 2586: 2572: 2569: 2567: 2548: 2541: 2537: 2533: 2530: 2527: 2523: 2515: 2511: 2507: 2504: 2501: 2498: 2494: 2483: 2475: 2472: 2468: 2464: 2462: 2436: 2429: 2425: 2421: 2418: 2415: 2411: 2389: 2379: 2371: 2368: 2364: 2353: 2345: 2342: 2338: 2327: 2325: 2312: 2302: 2291: 2281: 2271: 2260: 2252: 2249: 2245: 2241: 2239: 2229: 2219: 2208: 2184: 2169: 2159: 2151: 2148: 2144: 2140: 2126: 2116: 2112: 2103: 2097: 2090: 2084: 2077: 2071: 2067: 2061: 2026: 2019: 2015: 2011: 2008: 2005: 2001: 1997: 1989: 1979: 1968: 1958: 1954: 1950: 1941: 1934: 1927: 1914: 1909: 1896: 1868: 1863: 1859: 1855: 1847: 1837: 1826: 1808: 1802: 1786: 1773: 1767: 1760: 1754: 1747: 1741: 1736: 1716: 1712: 1708: 1703: 1699: 1695: 1690: 1686: 1675: 1657: 1655: 1649: 1645: 1639: 1635: 1627: 1623: 1618: 1614: 1608: 1603: 1599: 1594: 1590: 1581: 1574: 1567: 1562: 1561: 1555: 1534: 1529: 1517: 1513: 1509: 1504: 1492: 1488: 1484: 1479: 1467: 1463: 1450: 1443: 1436: 1432: 1427: 1418: 1411: 1397: 1384: 1362: 1351: 1343: 1339: 1335: 1324: 1305: 1244: 1226: 1215: 1207: 1188: 1184: 1162: 1151: 1147: 1130: 1116: 1106: 1098: 1094: 1090: 1076: 1056: 1048: 1045: 1044: 1043: 1041: 1024: 1022: 1012: 1009: 1004: 1000: 996: 991: 985: 982: 976: 970: 964: 959: 954: 952: 947: 940: 936: 930: 925: 908: 900: 895: 889: 883: 877: 860: 851: 834: 820: 810: 802: 798: 794: 780: 768: 760: 753: 744: 735: 728: 719: 717: 713: 709: 705: 702:structure in 701: 693:Applicability 685: 683: 679: 622: 618: 614: 604: 600: 569: 546: 542: 492: 488: 467: 447: 417: 401: 374: 370: 360: 358: 354: 350: 345: 343: 339: 335: 331: 327: 323: 318: 316: 300: 292: 276: 268: 243: 239: 223: 215: 214:wave function 199: 192:is position, 167: 143: 133: 125: 121: 117: 103: 95: 91: 89: 85: 82:modulated by 81: 77: 73: 69: 60: 54: 47: 43: 39: 33: 19: 13640: 13628: 13616: 13604: 13522:Pines' demon 13291: 13261:Kondo effect 13163:Time crystal 13049: 13045: 13032: 12995: 12974: 12922: 12918: 12912: 12887: 12883: 12873: 12865: 12860: 12840: 12830: 12820: 12811: 12792: 12788: 12775: 12755: 12751: 12738: 12705: 12701: 12692: 12680: 12668: 12656: 12629: 12618: 12600: 12589: 12579:12 September 12577:. Retrieved 12565: 12534: 12522: 12503: 12497: 12476: 12467: 12458: 12392: 12377: 12373: 12286: 12273:(1883), and 12263: 12247: 12173: 12170: 12161:acceleration 12136: 12045: 12038: 11963: 11463:Eliminating 10725: 10275: 10272: 10265: 9995: 9992: 9735: 8988: 8982: 8976: 8845: 8652: 8649: 8645: 8513: 7278: 7069: 6690: 6678: 6659: 6258: 6147: 5784: 5680: 5678: 5501: 5391: 5273: 5140: 5110:cyclic group 4996: 4956: 4718:translations 4715: 4695:space groups 4692: 3965: 2906: 2903: 2896: 2795: 2789: 2782: 2776: 2769: 2763: 2758: 2687: 2684: 2114: 2110: 2101: 2095: 2088: 2082: 2075: 2069: 2065: 2059: 1956: 1952: 1948: 1939: 1932: 1925: 1912: 1894: 1814: 1788: 1771: 1765: 1758: 1752: 1745: 1739: 1663: 1647: 1643: 1637: 1633: 1625: 1621: 1616: 1612: 1606: 1601: 1597: 1592: 1588: 1579: 1572: 1565: 1558: 1556: 1448: 1441: 1434: 1430: 1428: 1424: 1416: 1409: 1297:such that: 1246: 1242: 1149: 1145: 1031: 1018: 1007: 989: 986: 980: 974: 968: 962: 955: 945: 938: 934: 928: 923: 898: 893: 887: 881: 875: 849: 772: 758: 751: 742: 733: 696: 361: 352: 348: 346: 326:Bloch states 325: 321: 319: 169: 96: 93: 71: 65: 58: 13559:Soft matter 13459:Ferromagnet 13277:Drude model 13246:Berry phase 13226:Hall effect 13046:Wave Motion 12698:Felix Bloch 11062:Again with 6674:point group 4703:point group 4699:translation 4386:. Finally, 2050:Again, the 1908:eigenvalues 1778:(as above, 722:Wave vector 698:a periodic 353:Bloch Waves 242:wave vector 88:Felix Bloch 80:plane waves 13658:Categories 13474:Spin glass 13469:Metamagnet 13449:Paramagnet 13336:Conduction 13312:BCS theory 13153:Superfluid 13148:Supersolid 12513:0521297338 12450:References 12035:in a band 9999:to obtain 6681:characters 6670:characters 1801:eigenstate 740:(left) or 700:dielectric 416:band index 42:Isosurface 13532:Polariton 13439:Diamagnet 13387:Couplings 13363:Conductor 13358:Semimetal 13343:Insulator 13219:Phenomena 13143:Fermi gas 13022:H. Föll. 13015:692760083 12947:121065949 12880:Sunada, T 12836:Magnus, W 12795:: 47–88. 12752:Acta Math 12730:120668259 12223:× 12199:− 12190:˙ 12181:ℏ 12112:× 12077:∓ 12043:in a band 11978:ℏ 11924:ε 11920:− 11900:ε 11894:⟩ 11872:∇ 11865:− 11844:⟨ 11841:⟩ 11814:∇ 11807:− 11791:⟨ 11785:⟩ 11763:∇ 11756:− 11735:⟨ 11732:⟩ 11705:∇ 11698:− 11682:⟨ 11671:≠ 11659:∑ 11634:ℏ 11611:δ 11596:ℏ 11574:∂ 11561:∂ 11539:ε 11529:∂ 11432:ε 11428:− 11414:ε 11396:⟩ 11372:∇ 11366:− 11360:⋅ 11341:ℏ 11322:⟨ 11306:≠ 11294:∑ 11261:ℏ 11219:∂ 11206:∂ 11195:ε 11185:∂ 11169:∑ 11101:⟩ 11071:ψ 11031:ε 11027:− 11013:ε 10964:∇ 10958:− 10952:⋅ 10933:ℏ 10925:∗ 10898:∫ 10882:≠ 10870:∑ 10837:ℏ 10795:∂ 10782:∂ 10771:ε 10761:∂ 10745:∑ 10682:ε 10678:− 10658:ε 10652:⟩ 10630:∇ 10623:− 10602:⟨ 10599:⟩ 10572:∇ 10565:− 10549:⟨ 10543:⟩ 10521:∇ 10514:− 10493:⟨ 10490:⟩ 10463:∇ 10456:− 10440:⟨ 10429:≠ 10417:∑ 10392:ℏ 10369:δ 10354:ℏ 10332:∂ 10319:∂ 10297:ε 10287:∂ 10240:ψ 10233:∇ 10227:− 10219:∗ 10205:ψ 10192:∫ 10178:ℏ 10161:∂ 10150:ε 10146:∂ 10097:∇ 10091:− 10083:∗ 10056:∫ 10042:ℏ 10025:∂ 10014:ε 10010:∂ 9961:∗ 9934:∫ 9871:∇ 9865:− 9848:ℏ 9840:∗ 9813:∫ 9804:∑ 9774:∂ 9763:ε 9759:∂ 9747:∑ 9686:− 9648:ψ 9641:^ 9630:∗ 9621:ψ 9608:∫ 9592:≠ 9580:∑ 9567:ψ 9560:^ 9549:∗ 9540:ψ 9527:∫ 9450:ℏ 9430:∇ 9424:− 9418:⋅ 9399:ℏ 9379:^ 9347:^ 9301:^ 9249:ε 9173:∂ 9160:∂ 9149:ε 9139:∂ 9123:∑ 9083:∂ 9072:ε 9068:∂ 9056:∑ 9032:ε 9000:ε 8949:∂ 8936:∂ 8914:ε 8904:∂ 8872:∂ 8861:ε 8857:∂ 8827:⟩ 8821:^ 8810:⟨ 8807:ℏ 8801:⟩ 8795:^ 8784:⟨ 8776:ℏ 8757:ψ 8750:∇ 8744:− 8736:∗ 8722:ψ 8709:∫ 8695:ℏ 8678:∂ 8667:ε 8663:∂ 8618:ℏ 8598:∇ 8595:ℏ 8589:− 8561:ℏ 8555:∇ 8552:ℏ 8546:− 8531:^ 8413:∇ 8407:− 8382:ℏ 8300:⋅ 8233:∇ 8222:⋅ 8206:− 8180:∇ 8170:⋅ 8154:⋅ 8140:− 8107:⋅ 8058:ℏ 8015:⋅ 7953:⋅ 7886:∇ 7875:⋅ 7833:∇ 7823:⋅ 7807:⋅ 7765:∇ 7755:⋅ 7706:⋅ 7677:⋅ 7644:ℏ 7640:− 7597:⋅ 7535:⋅ 7474:∇ 7464:⋅ 7415:⋅ 7386:⋅ 7383:∇ 7363:ℏ 7359:− 7316:⋅ 7230:⋅ 7181:∇ 7160:ℏ 7156:− 7107:⋅ 7010:ε 6860:ε 6789:∇ 6783:− 6758:ℏ 6710:^ 6640:τ 6625:ψ 6616:τ 6612:⋅ 6585:ψ 6568:ψ 6562:τ 6553:ε 6546:^ 6505:α 6501:χ 6495:α 6491:ψ 6485:α 6481:∑ 6468:ψ 6455:^ 6425:τ 6421:⋅ 6360:χ 6314:χ 6268:χ 6111:π 6067:∈ 6027:π 6021:− 5763:τ 5754:ε 5747:^ 5713:τ 5704:ε 5697:^ 5654:ψ 5615:∑ 5598:ψ 5555:∇ 5535:ℏ 5529:− 5520:^ 5460:∑ 5303:∑ 5176:^ 5173:τ 5150:χ 5126:∞ 5123:→ 5080:γ 5074:χ 5042:γ 5021:γ 4999:character 4972:^ 4969:τ 4934:^ 4900:^ 4897:τ 4864:^ 4861:τ 4845:^ 4842:τ 4826:^ 4823:τ 4810:^ 4807:τ 4745:∑ 4737:τ 4564:⋅ 4526:ψ 4491:ψ 4481:⋅ 4451:⋅ 4432:ψ 4415:ψ 4402:^ 4369:∈ 4297:λ 4234:ψ 4220:∫ 4196:λ 4154:ψ 4140:^ 4116:∫ 4078:ψ 4064:∫ 4032:∈ 4002:⋅ 3975:λ 3922:λ 3900:λ 3881:λ 3849:ψ 3818:ψ 3782:^ 3716:ψ 3699:ψ 3678:^ 3648:^ 3608:^ 3568:ψ 3557:λ 3539:ψ 3525:^ 3476:^ 3458:^ 3393:^ 3373:^ 3103:ψ 3004:ψ 2963:ψ 2942:ψ 2928:^ 2832:^ 2757:denote a 2703:^ 2645:⋅ 2615:ψ 2549:ψ 2538:θ 2531:π 2512:θ 2505:π 2499:− 2484:⋅ 2473:− 2437:ψ 2426:θ 2419:π 2380:⋅ 2369:− 2354:⋅ 2343:− 2292:ψ 2261:⋅ 2250:− 2170:ψ 2160:⋅ 2149:− 2063:. Define 2027:ψ 2016:θ 2009:π 1969:ψ 1897:= 1, 2, 3 1869:ψ 1827:ψ 1733:denote a 1679:^ 1363:ψ 1352:⋅ 1306:ψ 1216:⋅ 1107:⋅ 1077:ψ 1057:ψ 1027:Statement 927:, it can 909:ψ 861:ψ 811:⋅ 781:ψ 706:leads to 619:ψ 601:ψ 543:ψ 489:ψ 371:ψ 200:ψ 134:⋅ 104:ψ 13606:Category 13587:Colloids 12972:(1976). 12819:(1992). 12783:(1883). 12758:: 1–36. 12746:(1886). 12570:Archived 12557:(2002). 12475:(1996). 12401:See also 12269:(1877), 11932:′ 11851:′ 11832:′ 11742:′ 11723:′ 11667:′ 11440:′ 11387:′ 11302:′ 11039:′ 10987:′ 10878:′ 10690:′ 10609:′ 10590:′ 10500:′ 10481:′ 10425:′ 10266:For the 9698:′ 9588:′ 3872:we have 3355:and the 2904:Source: 2802:, where 2108:, where 1899:, where 1417:Source: 943:, where 394:, where 332:for the 13618:Commons 13582:Polymer 13549:Polaron 13527:Plasmon 13507:Exciton 12927:Bibcode 12904:2374542 12710:Bibcode 5005:of the 4726:abelian 4722:unitary 949:is any 765:) is a 570:vector 313:is the 265:is the 212:is the 44:of the 13517:Phonon 13512:Magnon 13270:Theory 13128:Plasma 13118:Liquid 13013: 13003: 12982: 12945: 12902: 12848: 12728: 12510: 12485: 12371:where 12159:is an 12137:where 9240:Given 8980:where 6202:, and 6049:where 5443:where 5390:where 4885:where 4701:and a 4358:where 4021:where 3591:Given 1817:ψ 1805:ψ 1799:is an 1797:ψ 1652:, see 1610:, but 1547:where 1143:where 873:, not 847:where 592:, or, 338:states 293:, and 170:where 13492:Anyon 13113:Solid 12943:S2CID 12900:JSTOR 12726:S2CID 12610:(PDF) 12573:(PDF) 12562:(PDF) 10723:Proof 8843:Proof 7067:Proof 5003:trace 4707:basis 4606:with 3143:with 2197:Then 1946:with 1790:Lemma 1631:when 1401:Proof 1040:basis 987:When 330:basis 236:is a 13502:Hole 13011:OCLC 13001:ISBN 12980:ISBN 12846:ISBN 12581:2020 12508:ISBN 12483:ISBN 12386:and 11490:and 8892:and 4724:and 4720:are 4716:All 4329:and 1892:for 1664:Let 956:The 929:also 897:are 891:and 13123:Gas 13054:doi 12935:doi 12923:209 12892:doi 12888:110 12797:doi 12760:doi 12718:doi 12566:MIT 8538:eff 1656:.) 1629:= 0 1605:= 2 899:not 879:or 480:), 336:or 324:or 289:is 66:In 13660:: 13050:50 13048:. 13044:. 13009:. 12968:; 12941:. 12933:. 12921:. 12898:. 12886:. 12793:12 12791:. 12787:. 12754:. 12750:. 12724:. 12716:. 12706:52 12641:^ 12568:. 12564:. 12546:^ 12397:. 12390:. 12293:: 12256:. 6676:. 5812:: 2788:+ 2775:+ 2094:+ 2081:+ 2068:= 1961:: 1951:= 1938:, 1931:, 1764:+ 1751:+ 1636:≠ 1620:· 1596:· 1578:, 1571:, 1447:, 1440:, 1023:. 937:+ 757:− 718:. 684:. 359:. 344:. 317:. 269:, 216:, 70:, 13089:e 13082:t 13075:v 13060:. 13056:: 13017:. 12988:. 12949:. 12937:: 12929:: 12906:. 12894:: 12854:. 12805:. 12799:: 12768:. 12762:: 12756:8 12732:. 12720:: 12712:: 12583:. 12516:. 12491:. 12380:) 12378:t 12376:( 12374:f 12359:, 12356:0 12353:= 12350:y 12347:) 12344:t 12341:( 12338:f 12335:+ 12327:2 12323:t 12319:d 12314:y 12309:2 12305:d 12232:) 12227:B 12219:v 12215:+ 12211:E 12206:( 12202:e 12196:= 12187:k 12146:a 12121:) 12116:B 12109:) 12105:k 12101:( 12097:v 12093:+ 12089:E 12084:( 12080:e 12074:= 12070:a 12066:) 12062:k 12058:( 12054:M 12019:) 12015:k 12011:( 12007:M 11982:2 11974:1 11946:) 11942:k 11938:( 11929:n 11917:) 11913:k 11909:( 11904:n 11890:k 11886:n 11882:| 11876:i 11868:i 11861:| 11856:k 11848:n 11837:k 11829:n 11824:| 11818:j 11810:i 11803:| 11798:k 11794:n 11788:+ 11781:k 11777:n 11773:| 11767:j 11759:i 11752:| 11747:k 11739:n 11728:k 11720:n 11715:| 11709:i 11701:i 11694:| 11689:k 11685:n 11674:n 11664:n 11653:2 11648:) 11643:m 11638:2 11628:( 11623:+ 11618:j 11615:i 11605:m 11600:2 11590:= 11582:j 11578:k 11569:i 11565:k 11556:) 11552:k 11548:( 11543:n 11533:2 11503:j 11499:q 11476:i 11472:q 11445:k 11437:n 11422:k 11418:n 11406:2 11401:| 11392:k 11384:n 11379:| 11375:) 11369:i 11363:( 11356:q 11350:m 11345:2 11334:| 11329:k 11325:n 11318:| 11309:n 11299:n 11290:+ 11285:2 11281:q 11274:m 11271:2 11265:2 11255:= 11250:j 11246:q 11240:i 11236:q 11227:j 11223:k 11214:i 11210:k 11199:n 11189:2 11176:j 11173:i 11163:2 11160:1 11136:k 11132:n 11128:u 11121:x 11116:k 11112:i 11108:e 11104:= 11097:k 11093:n 11089:| 11085:= 11079:k 11075:n 11044:k 11036:n 11021:k 11017:n 11005:2 11000:| 10992:k 10984:n 10979:u 10975:) 10971:k 10967:+ 10961:i 10955:( 10948:q 10942:m 10937:2 10919:k 10915:n 10911:u 10905:r 10901:d 10894:| 10885:n 10875:n 10866:+ 10861:2 10857:q 10850:m 10847:2 10841:2 10831:= 10826:j 10822:q 10816:i 10812:q 10803:j 10799:k 10790:i 10786:k 10775:n 10765:2 10752:j 10749:i 10739:2 10736:1 10704:) 10700:k 10696:( 10687:n 10675:) 10671:k 10667:( 10662:n 10648:k 10644:n 10640:| 10634:i 10626:i 10619:| 10614:k 10606:n 10595:k 10587:n 10582:| 10576:j 10568:i 10561:| 10556:k 10552:n 10546:+ 10539:k 10535:n 10531:| 10525:j 10517:i 10510:| 10505:k 10497:n 10486:k 10478:n 10473:| 10467:i 10459:i 10452:| 10447:k 10443:n 10432:n 10422:n 10411:2 10406:) 10401:m 10396:2 10386:( 10381:+ 10376:j 10373:i 10363:m 10358:2 10348:= 10340:j 10336:k 10327:i 10323:k 10314:) 10310:k 10306:( 10301:n 10291:2 10248:k 10244:n 10236:) 10230:i 10224:( 10213:k 10209:n 10199:r 10195:d 10187:m 10182:2 10172:= 10165:k 10154:n 10120:k 10116:n 10112:u 10108:) 10104:k 10100:+ 10094:i 10088:( 10077:k 10073:n 10069:u 10063:r 10059:d 10051:m 10046:2 10036:= 10029:k 10018:n 9996:q 9975:k 9971:n 9967:u 9955:k 9951:n 9947:u 9941:r 9937:d 9911:k 9907:n 9903:u 9897:i 9893:q 9887:i 9883:) 9878:k 9874:+ 9868:i 9862:( 9857:m 9852:2 9834:k 9830:n 9826:u 9820:r 9816:d 9808:i 9800:= 9795:i 9791:q 9782:i 9778:k 9767:n 9751:i 9736:q 9721:. 9718:. 9715:. 9712:+ 9704:0 9695:n 9690:E 9681:0 9676:n 9672:E 9664:2 9659:| 9652:n 9638:V 9625:n 9615:r 9611:d 9604:| 9595:n 9585:n 9576:+ 9571:n 9557:V 9544:n 9534:r 9530:d 9524:+ 9519:0 9514:n 9510:E 9506:= 9501:n 9497:E 9474:2 9470:q 9463:m 9460:2 9454:2 9444:+ 9441:) 9437:k 9433:+ 9427:i 9421:( 9414:q 9408:m 9403:2 9393:+ 9387:k 9376:H 9369:= 9363:q 9359:+ 9355:k 9344:H 9317:q 9313:+ 9309:k 9298:H 9274:) 9270:q 9266:+ 9262:k 9258:( 9253:n 9228:) 9223:3 9219:q 9215:( 9212:O 9209:+ 9204:j 9200:q 9194:i 9190:q 9181:j 9177:k 9168:i 9164:k 9153:n 9143:2 9130:j 9127:i 9117:2 9114:1 9109:+ 9104:i 9100:q 9091:i 9087:k 9076:n 9060:i 9052:+ 9049:) 9045:k 9041:( 9036:n 9028:= 9025:) 9021:q 9017:+ 9013:k 9009:( 9004:n 8989:k 8983:q 8977:q 8957:j 8953:k 8944:i 8940:k 8931:) 8927:k 8923:( 8918:n 8908:2 8876:k 8865:n 8817:v 8804:= 8791:p 8779:m 8771:= 8765:k 8761:n 8753:) 8747:i 8741:( 8730:k 8726:n 8716:r 8712:d 8704:m 8699:2 8689:= 8682:k 8671:n 8622:k 8592:i 8569:, 8565:k 8558:+ 8549:i 8543:= 8527:p 8495:) 8491:x 8487:( 8481:k 8476:u 8472:) 8468:x 8464:( 8461:U 8458:+ 8455:) 8451:x 8447:( 8441:k 8436:u 8430:2 8425:) 8420:k 8416:+ 8410:i 8403:( 8395:m 8392:2 8386:2 8376:= 8369:) 8365:x 8361:( 8355:k 8350:u 8343:k 8338:E 8330:) 8326:x 8322:( 8316:k 8311:u 8304:x 8296:k 8292:i 8288:e 8284:) 8280:x 8276:( 8273:U 8270:+ 8266:) 8262:) 8258:x 8254:( 8248:k 8243:u 8237:2 8226:x 8218:k 8214:i 8210:e 8203:) 8199:x 8195:( 8189:k 8184:u 8174:x 8166:k 8162:i 8158:e 8150:k 8146:i 8143:2 8137:) 8133:x 8129:( 8123:k 8118:u 8111:x 8103:k 8099:i 8095:e 8089:2 8084:k 8078:( 8071:m 8068:2 8062:2 8052:= 8045:) 8041:x 8037:( 8031:k 8026:u 8019:x 8011:k 8007:i 8003:e 7996:k 7991:E 7983:) 7979:x 7975:( 7969:k 7964:u 7957:x 7949:k 7945:i 7941:e 7937:) 7933:x 7929:( 7926:U 7923:+ 7919:) 7915:) 7911:x 7907:( 7901:k 7896:u 7890:2 7879:x 7871:k 7867:i 7863:e 7859:+ 7856:) 7852:x 7848:( 7842:k 7837:u 7827:x 7819:k 7815:i 7811:e 7803:k 7799:i 7796:+ 7792:) 7788:) 7784:x 7780:( 7774:k 7769:u 7759:x 7751:k 7747:i 7743:e 7739:+ 7736:) 7732:x 7728:( 7722:k 7717:u 7710:x 7702:k 7698:i 7694:e 7689:k 7685:i 7681:( 7673:k 7669:i 7665:( 7658:m 7655:2 7648:2 7634:= 7627:) 7623:x 7619:( 7613:k 7608:u 7601:x 7593:k 7589:i 7585:e 7578:k 7573:E 7565:) 7561:x 7557:( 7551:k 7546:u 7539:x 7531:k 7527:i 7523:e 7519:) 7515:x 7511:( 7508:U 7505:+ 7501:) 7497:) 7493:x 7489:( 7483:k 7478:u 7468:x 7460:k 7456:i 7452:e 7448:+ 7445:) 7441:x 7437:( 7431:k 7426:u 7419:x 7411:k 7407:i 7403:e 7398:k 7394:i 7390:( 7377:m 7374:2 7367:2 7353:= 7346:) 7342:x 7338:( 7332:k 7327:u 7320:x 7312:k 7308:i 7304:e 7297:k 7292:E 7264:) 7260:) 7256:x 7252:( 7246:k 7241:u 7234:x 7226:k 7222:i 7218:e 7213:( 7208:] 7204:) 7200:x 7196:( 7193:U 7190:+ 7185:2 7174:m 7171:2 7164:2 7149:[ 7145:= 7141:) 7137:) 7133:x 7129:( 7123:k 7118:u 7111:x 7103:k 7099:i 7095:e 7090:( 7083:k 7078:E 7049:k 7027:) 7023:k 7019:( 7014:n 6987:k 6965:) 6961:R 6957:+ 6953:r 6949:( 6943:k 6938:u 6934:= 6931:) 6927:r 6923:( 6917:k 6912:u 6891:) 6887:r 6883:( 6877:k 6872:u 6865:k 6856:= 6853:) 6849:r 6845:( 6839:k 6834:u 6829:] 6825:) 6821:r 6817:( 6814:U 6811:+ 6806:2 6801:) 6796:k 6792:+ 6786:i 6779:( 6771:m 6768:2 6762:2 6751:[ 6747:= 6744:) 6740:r 6736:( 6730:k 6725:u 6718:k 6707:H 6644:) 6636:+ 6632:r 6628:( 6622:= 6608:k 6604:i 6600:e 6596:) 6592:r 6588:( 6582:= 6579:) 6575:r 6571:( 6557:| 6543:P 6519:) 6516:R 6513:( 6508:j 6477:= 6472:j 6462:R 6452:P 6417:k 6413:i 6409:e 6405:= 6402:) 6397:3 6393:a 6389:, 6384:3 6380:n 6376:( 6369:3 6365:k 6356:) 6351:2 6347:a 6343:, 6338:2 6334:n 6330:( 6323:2 6319:k 6310:) 6305:1 6301:a 6297:, 6292:1 6288:n 6284:( 6277:1 6273:k 6242:1 6238:a 6215:1 6211:L 6188:1 6184:m 6161:1 6157:k 6130:1 6126:L 6119:1 6115:m 6108:2 6102:= 6097:1 6093:k 6071:Z 6062:1 6058:m 6035:1 6031:m 6024:2 6018:) 6013:1 6009:L 6005:+ 6000:1 5996:a 5990:1 5986:n 5982:( 5977:1 5973:k 5969:= 5964:1 5960:a 5954:1 5950:n 5944:1 5940:k 5917:) 5912:1 5908:L 5904:+ 5899:1 5895:a 5889:1 5885:n 5881:( 5876:1 5872:k 5868:i 5864:e 5860:= 5853:1 5849:a 5843:1 5839:n 5833:1 5829:k 5825:i 5821:e 5798:i 5794:L 5767:i 5758:| 5744:P 5737:= 5730:i 5726:L 5722:+ 5717:i 5708:| 5694:P 5681:L 5665:) 5661:r 5657:( 5651:= 5647:) 5641:i 5636:a 5629:i 5625:N 5619:i 5611:+ 5607:r 5602:( 5578:) 5574:r 5570:( 5567:V 5564:+ 5559:2 5548:m 5545:2 5539:2 5526:= 5517:H 5486:i 5481:a 5474:i 5470:N 5464:i 5456:= 5452:L 5429:i 5425:a 5403:a 5392:L 5378:) 5374:r 5370:( 5367:V 5364:= 5361:) 5357:L 5353:+ 5349:r 5345:( 5342:V 5339:= 5335:) 5329:i 5324:a 5317:i 5313:N 5307:i 5299:+ 5295:r 5290:( 5286:V 5256:1 5252:a 5246:1 5242:n 5236:1 5232:k 5228:i 5224:e 5220:= 5217:) 5214:) 5209:1 5205:a 5201:, 5196:1 5192:n 5188:( 5183:1 5166:( 5159:1 5155:k 5120:n 5096:1 5093:= 5088:n 5084:) 5077:( 5054:1 5051:= 5046:n 4979:i 4941:i 4930:a 4920:i 4916:n 4912:= 4907:i 4871:3 4852:2 4833:1 4816:= 4782:i 4777:a 4770:i 4766:n 4760:3 4755:1 4752:= 4749:i 4741:= 4673:) 4669:n 4664:A 4660:+ 4656:x 4652:( 4646:k 4641:u 4637:= 4634:) 4630:x 4626:( 4620:k 4615:u 4594:) 4590:x 4586:( 4580:k 4575:u 4568:x 4560:k 4556:i 4552:e 4548:= 4545:) 4541:x 4537:( 4531:k 4505:, 4502:) 4498:x 4494:( 4485:a 4477:n 4473:k 4470:i 4466:e 4462:= 4459:) 4455:a 4447:n 4443:+ 4439:x 4435:( 4429:= 4426:) 4422:x 4418:( 4409:n 4399:T 4373:R 4366:k 4346:k 4343:i 4340:= 4337:s 4315:2 4310:| 4302:n 4292:| 4288:= 4285:1 4264:x 4260:d 4255:2 4250:| 4245:) 4241:x 4237:( 4230:| 4224:V 4214:2 4209:| 4201:n 4191:| 4187:= 4183:x 4179:d 4174:2 4169:| 4165:) 4161:x 4157:( 4148:n 4136:T 4127:| 4120:V 4112:= 4108:x 4104:d 4099:2 4094:| 4089:) 4085:x 4081:( 4074:| 4068:V 4060:= 4057:1 4036:C 4029:s 4006:a 3998:n 3994:s 3990:e 3986:= 3980:n 3948:2 3943:n 3938:+ 3933:1 3928:n 3918:= 3911:2 3906:n 3892:1 3887:n 3860:) 3856:x 3852:( 3829:) 3825:x 3821:( 3811:2 3806:n 3801:+ 3796:1 3791:n 3778:T 3770:= 3767:) 3762:2 3757:n 3751:A 3747:+ 3742:1 3737:n 3731:A 3727:+ 3723:x 3719:( 3713:= 3710:) 3706:x 3702:( 3692:2 3687:n 3674:T 3662:1 3657:n 3644:T 3616:n 3604:T 3579:) 3575:x 3571:( 3562:n 3553:= 3550:) 3546:x 3542:( 3533:n 3521:T 3496:0 3493:= 3490:] 3484:n 3472:T 3464:, 3455:H 3449:[ 3429:) 3425:x 3421:( 3418:U 3415:+ 3409:m 3406:2 3400:2 3389:p 3379:= 3370:H 3343:) 3339:x 3335:( 3332:U 3329:= 3326:) 3320:n 3314:T 3309:+ 3305:x 3301:( 3298:U 3276:) 3268:3 3264:n 3254:2 3250:n 3240:1 3236:n 3229:( 3224:= 3220:n 3215:, 3210:] 3202:3 3197:a 3188:2 3183:a 3174:1 3169:a 3161:[ 3156:= 3152:A 3127:) 3123:n 3118:A 3114:+ 3110:r 3106:( 3100:= 3090:) 3085:3 3080:a 3073:3 3069:n 3065:+ 3060:2 3055:a 3048:2 3044:n 3040:+ 3035:1 3030:a 3023:1 3019:n 3015:+ 3011:r 3007:( 3001:= 2991:) 2985:n 2979:T 2974:+ 2970:r 2966:( 2960:= 2953:) 2949:r 2945:( 2936:n 2924:T 2870:3 2866:n 2862:, 2857:2 2853:n 2849:, 2844:1 2840:n 2829:T 2806:i 2804:n 2799:3 2796:a 2793:3 2790:n 2786:2 2783:a 2780:2 2777:n 2773:1 2770:a 2767:1 2764:n 2741:3 2737:n 2733:, 2728:2 2724:n 2720:, 2715:1 2711:n 2700:T 2669:, 2666:) 2662:r 2658:( 2655:u 2649:r 2641:k 2637:i 2633:e 2629:= 2626:) 2622:r 2618:( 2605:u 2587:. 2584:) 2580:r 2576:( 2573:u 2570:= 2560:) 2556:r 2552:( 2542:j 2534:i 2528:2 2524:e 2516:j 2508:i 2502:2 2495:e 2488:r 2480:k 2476:i 2469:e 2465:= 2453:) 2448:) 2444:r 2440:( 2430:j 2422:i 2416:2 2412:e 2406:( 2399:) 2390:j 2385:a 2376:k 2372:i 2365:e 2358:r 2350:k 2346:i 2339:e 2333:( 2328:= 2318:) 2313:j 2308:a 2303:+ 2299:r 2295:( 2287:) 2282:j 2277:a 2272:+ 2268:r 2264:( 2257:k 2253:i 2246:e 2242:= 2235:) 2230:j 2225:a 2220:+ 2216:r 2212:( 2209:u 2185:. 2181:) 2177:r 2173:( 2164:r 2156:k 2152:i 2145:e 2141:= 2138:) 2134:r 2130:( 2127:u 2115:j 2111:b 2105:3 2102:b 2099:3 2096:θ 2092:2 2089:b 2086:2 2083:θ 2079:1 2076:b 2073:1 2070:θ 2066:k 2060:r 2054:j 2052:θ 2038:) 2034:r 2030:( 2020:j 2012:i 2006:2 2002:e 1998:= 1995:) 1990:j 1985:a 1980:+ 1976:r 1972:( 1957:j 1953:C 1949:e 1943:3 1940:θ 1936:2 1933:θ 1929:1 1926:θ 1920:j 1918:C 1913:r 1903:j 1901:C 1895:j 1880:) 1876:r 1872:( 1864:j 1860:C 1856:= 1853:) 1848:j 1843:a 1838:+ 1834:r 1830:( 1782:j 1780:n 1775:3 1772:a 1769:3 1766:n 1762:2 1759:a 1756:2 1753:n 1749:1 1746:a 1743:1 1740:n 1717:3 1713:n 1709:, 1704:2 1700:n 1696:, 1691:1 1687:n 1676:T 1648:i 1644:b 1638:j 1634:i 1626:j 1622:b 1617:i 1613:a 1607:π 1602:i 1598:b 1593:i 1589:a 1583:3 1580:b 1576:2 1573:b 1569:1 1566:b 1551:i 1549:n 1535:, 1530:3 1525:a 1518:3 1514:n 1510:+ 1505:2 1500:a 1493:2 1489:n 1485:+ 1480:1 1475:a 1468:1 1464:n 1452:3 1449:a 1445:2 1442:a 1438:1 1435:a 1385:. 1382:) 1378:x 1374:( 1368:k 1356:a 1348:k 1344:i 1340:e 1336:= 1333:) 1329:a 1325:+ 1321:x 1317:( 1311:k 1284:k 1262:a 1227:. 1224:) 1220:a 1212:n 1208:+ 1204:x 1200:( 1194:k 1189:u 1185:= 1182:) 1178:x 1174:( 1168:k 1163:u 1152:) 1150:r 1148:( 1146:u 1131:, 1128:) 1124:r 1120:( 1117:u 1111:r 1103:k 1099:i 1095:e 1091:= 1088:) 1084:r 1080:( 1008:k 990:k 981:k 975:k 969:k 963:k 946:K 941:) 939:K 935:k 933:( 924:k 894:u 888:k 882:u 876:k 850:u 835:, 832:) 828:r 824:( 821:u 815:r 807:k 803:i 799:e 795:= 792:) 788:r 784:( 762:2 759:k 755:1 752:k 746:2 743:k 737:1 734:k 663:k 640:) 636:K 633:+ 630:k 626:( 623:n 615:= 609:k 605:n 579:K 551:k 547:n 521:k 497:k 493:n 468:n 448:u 427:k 402:n 379:k 375:n 301:i 277:e 252:k 224:u 179:r 155:) 151:r 147:( 144:u 138:r 130:k 126:i 122:e 118:= 115:) 111:r 107:( 59:e 34:. 20:)

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Index