3702:
7072:, the Korringa–Kohn–Rostoker coherent potential approximation (KKR-CPA) is obtained. The KKR-CPA is used to calculate the electronic states for substitutional solid-solution alloys, for which Bloch's theorem does not hold. The electronic states for an even wider range of condensed matter structures can be found using the locally self-consistent multiple scattering (LSMS) method, which is also based on the single-particle Green's function.
98:
3990:
1858:
2462:
3391:
5909:
5107:
3282:
3714:
5712:
4594:
5526:
2145:
1625:
2671:. A plane wave impinges on a site, and a spherical wave exits it. The outgoing wave from the crystal is determined by the constructive interference of the waves from the sites. Advances to this theory involve the inclusion of higher-order terms in the total scattering matrix
1415:
2904:
4739:
5305:
2322:
1982:
3697:{\displaystyle {G_{0+}}\left({{\bf {r}}-{\bf {r'}}}\right)=-i\alpha \sum \limits _{l',m'}{{Y_{l',m'}}\left({{\bf {r}}_{i}}\right)h_{_{l'}}^{+}\left({\alpha {r_{i}}}\right){j_{l'}}\left({\alpha {{r'}_{i}}}\right)Y_{l',m'}^{*}\left({{\bf {r'}}_{i}}\right)}}
2306:
4358:
556:
4865:
658:
2551:
738:
363:
251:
6766:
It is now known that the KKR method can be used with space-filling non-spherical potentials. It can be extended to treat crystals with any number of atoms in a unit cell. There are versions of the theory that can be used to calculate
3113:
30:
traveling through porous media, light scattering from water droplets in a cloud, or x-rays scattering from a crystal. A more recent application is to the propagation of quantum matter waves like electrons or neutrons through a solid.
3985:{\displaystyle -i\alpha {Y_{l'm'}}\left({{\bf {r}}_{j}}\right)h_{l'}^{+}\left({r_{j}}\right)=\sum \limits _{l,m}{{Y_{lm}}\left({{\bf {r}}_{i}}\right){j_{l}}\left({\alpha {r_{i}}}\right){g_{lm,l'm'}}\left({E,{{\bf {R}}_{ij}}}\right)}}
7025:
6676:
5717:
6757:
can be used in many applications. The muffin-tin approximation is adequate for most metals in a close-packed arrangement. It cannot be used for calculating forces between atoms, or for important systems like semiconductors.
2771:
4462:
1518:
1617:
1853:{\displaystyle T=\sum \limits _{i}{t_{i}}+\sum \limits _{i}{t_{i}}{G_{0+}}\sum \limits _{j\neq i}{t_{j}}+\sum \limits _{i}{t_{i}}{G_{0+}}\sum \limits _{j\neq i}{{t_{j}}{G_{0+}}\sum \limits _{k\neq j}{t_{k}}+}...}
2039:
4897:
482:
88:
The terms "multiple scattering" and "multiple scattering theory" are often used in other contexts. For example, Molière's theory of the scattering of fast charged particles in matter is described in that way.
5993:
5370:
2955:
are bounded in space and do not overlap, there is an interstitial region between them within which the potential is a constant, usually taken to be zero. In this region, the Schrödinger equation becomes
6117:
3383:
3041:
1167:
842:
2807:
6505:
In the original derivations of the KKR method, spherically symmetric muffin-tin potentials were used. Such potentials have the advantage that the inverse of the scattering matrix is diagonal in
6324:
5535:
2607:
2457:{\displaystyle \left|\psi \right\rangle =\left|\phi \right\rangle +{G_{0+}}T\left|\phi \right\rangle =\left|\phi \right\rangle +\sum \limits _{j=1}^{N}{\left|{\phi _{j}^{out}}\right\rangle }}
1099:
941:
6221:
6859:
105:
The MST equations can be derived with different wave equations, but one of the simplest and most useful ones is the Schrödinger equation for an electron moving in a solid. With the help of
6061:
3085:
2008:
that we have chosen to focus on can be any of the sites in the cluster. The incoming wave on this site is the incoming wave on the cluster and the outgoing waves from all the other sites
6720:
6171:
1894:
5165:
420:
5378:
4421:
6500:
6375:
2953:
6722:
is the scattering phase shift that appears in the partial wave analysis in scattering theory. It is also easier to visualize the waves scattering from one atom to another, and
6805:
2205:
4454:
994:
7438:
Yang Wang; G. M. Stocks; W. A. Shelton; D. M. C. Nicholson; Z. Szotek; W. M. Temmerman (1995). "Order-N Multiple
Scattering Approach to Electronic Structure Calculations".
6755:
487:
7062:
5173:
287:
6449:
6270:
4029:
3321:
6017:
4889:
5921:(KKR method) for band theory calculations. Ewald derived a mathematically sophisticated summation process that makes it possible to calculate the structure constants,
561:
6420:
2669:
2640:
1023:
392:
4364:
which is a solution in principle of the multiple scattering equations for stationary states. This theory is very important for studies in condensed matter physics.
3277:{\displaystyle \phi _{i}^{in}\left({{\bf {r}}_{i}}\right)=\sum \nolimits _{l,m}{{Y_{lm}}\left({{\bf {r}}_{i}}\right){j_{l}}\left({\alpha {r_{i}}}\right)d_{lm}^{i}}}
2467:
2201:
667:
4605:
2035:
484:. A formal solution of the inhomogeneous equation is the sum of the solution of the homogeneous equation with a particular solution of the inhomogeneous equation
295:
115:
7403:
G. M. Stocks; W. M. Temmerman; B. L. Gyorffy (1978). "Complete
Solution of the Korringa-Kohn-Rostoker Coherent-Potential-Approximation Equations: Cu-Ni Alloys".
6523:
6395:
6241:
3105:
2799:
2781:
In 1947, Korringa pointed out that the multiple scattering equations can be used to calculate stationary states in a crystal for which the number of scatterers
2689:
2169:
2006:
1886:
1159:
1139:
1119:
1043:
961:
885:
865:
5904:{\displaystyle {A_{lm,l'm'}}\left({E,{\bf {k}}}\right)=\sum \limits _{j}{e^{i{\bf {k}}\cdot {{\bf {R}}_{ij}}}}{g_{lm,l'm'}}\left({E,{{\bf {R}}_{ij}}}\right)}
6223:. The dimension of these matrix equations is technically infinite, but by ignoring all contributions that correspond to an angular momentum quantum number
4589:{\displaystyle {\psi _{\bf {k}}}\left({{\bf {r}}+{{\bf {R}}_{i}}}\right)={e^{i{\bf {k}}\cdot {{\bf {R}}_{i}}}}{\psi _{\bf {k}}}\left({\bf {r}}\right)}
2801:
goes to infinity. Setting the incoming wave on the cluster and the outgoing wave from the cluster to zero, he wrote the first multiple scattering as
2694:
6867:
39:
7069:
5918:
4456:
form a periodic array. Bloch's theorem holds for such a system, which means that the solutions of the Schrödinger equation may be written as a
1424:
70:
7240:
A. A. Bednyakov (2014). "On the Molière theory of multiple scattering of charged particles (1947–1948) and its critique in subsequent years".
7224:
2140:{\displaystyle \left|{\phi _{i}^{in}}\right\rangle =\left|\phi \right\rangle +\sum \limits _{j\neq i}{\left|{\phi _{j}^{out}}\right\rangle }}
5102:{\displaystyle M_{lm,l'm'}^{ij}=m_{lm,l'm'}^{i}{\delta _{ij}}-\left({1-{\delta _{ij}}}\right){g_{lm,l'm'}}\left({E,{{\bf {R}}_{ij}}}\right)}
4748:
1523:
6530:
425:
50:, portions of which were published in German in the Proceedings of the Royal Academy of Sciences in Amsterdam under the sponsorship of
7196:
4036:
7360:
L. Szunyogh; B. Újfalussy; P. Weinberger; J. Kollár (1994). "Self-consistent localized KKR scheme for surfaces and interfaces".
6066:
2316:
1410:{\displaystyle \sum \limits _{i}{Q_{i}}=\sum \limits _{i}{{v_{i}}(1+{G_{0+}}\sum \limits _{j}{Q_{j}}})=\sum \limits _{i}{\left}}
661:
3326:
59:
2959:
101:
Simulation of a pulse multiply scattered by a randomly arranged set of obstacles (real part and modulus squared of the field).
5924:
2899:{\displaystyle \left|{\phi _{i}^{in}}\right\rangle =\sum \limits _{j\neq i}^{\infty }{\left|{\phi _{j}^{out}}\right\rangle }}
743:
5707:{\displaystyle {M_{lm,l'm'}}\left({E,{\bf {k}}}\right)={m_{lm,l'm'}}\left(E\right)-{A_{lm,l'm'}}\left({E,{\bf {k}}}\right)}
6275:
5310:
2556:
1865:
1048:
890:
6176:
1977:{\displaystyle \left|\psi \right\rangle =\left|{\phi _{i}^{in}}\right\rangle +\left|{\phi _{i}^{out}}\right\rangle }
7492:
7487:
6022:
4372:
The calculation of stationary states is simplified considerably for periodic solids in which all of the potentials
3046:
7065:
6685:
6122:
5521:{\displaystyle \sum \limits _{l'm'}{{M_{lm,l'm'}}\left({E,{\bf {k}}}\right){c_{l'm'}}\left(E,{\bf {k}}\right)=0}}
106:
6810:
397:
82:
4375:
26:
formalism that is used to describe the propagation of a wave through a collection of scatterers. Examples are
4599:
It is more convenient to deal with a symmetric matrix for the coefficients, and this can be done by defining
2910:
A simple description of this process is that the electrons scatter from one atom to the other ad infinitum.
51:
2301:{\displaystyle \left|{\phi _{i}^{out}}\right\rangle ={G_{0+}}{t_{i}}\left|{\phi _{i}^{in}}\right\rangle }
2916:
6777:
943:. The physical meaning of this is that it describes the interaction of the electron with a cluster of
551:{\displaystyle \left|\psi \right\rangle =\left|\phi \right\rangle +{G_{0+}}V\left|\psi \right\rangle }
7447:
7412:
7369:
7334:
7295:
7249:
7137:
7098:
5300:{\displaystyle c_{l'm'}^{j}={e^{-i{\bf {k}}\cdot {{\bf {R}}_{j}}}}{c_{l'm'}}\left(E,{\bf {k}}\right)}
5116:
4426:
966:
78:
66:
55:
6807:
can also be used to formulate a multiple scattering version of the single-particle Green's function
2773:. These terms are particularly important in the scattering of charged particles treated by Molière.
7188:
6725:
6454:
6329:
3996:
This leads to a set of homogeneous simultaneous equations that determines the unknown coefficients
46:. Korringa and Ewald acknowledged the influence on their work of the 1903 doctoral dissertation of
7167:. Minutes of the regular meetings of the mathematics and physics division of 26 February: 460–480.
7034:
259:
7265:
6425:
6246:
5917:
and Norman
Rostoker derived this same theory using the Kohn variational method. It is called the
2610:
653:{\displaystyle {G_{0+}}={\lim _{\varepsilon \to 0}}{\left({E-{H_{0}}+i\varepsilon }\right)^{-1}}}
3999:
3291:
7163:
N. Kasterin (1898). "Concerning the dispersion of acoustic waves in a non-homogeneous medium".
5998:
4870:
7463:
7385:
7220:
7192:
2546:{\displaystyle \left|{\phi _{i}^{out}}\right\rangle ={G_{0+}}{t_{i}}\left|\phi \right\rangle }
74:
2645:
2616:
999:
733:{\displaystyle \left|\psi \right\rangle =\left({1+{G_{0+}}T}\right)\left|\phi \right\rangle }
368:
7455:
7420:
7377:
7342:
7303:
7257:
7145:
7106:
4734:{\displaystyle c_{lm}^{i}=\sum \nolimits _{l'm'}{t_{lm,l'm'}^{i}\left(E\right)d_{l'm'}^{i}}}
358:{\displaystyle \left({E-{H_{0}}}\right)\left|\psi \right\rangle =V\left|\psi \right\rangle }
246:{\displaystyle \left\psi ({\bf {r}})=i\hbar {\frac {\partial \psi ({\bf {r}})}{\partial t}}}
47:
43:
6774:
The arguments that lead to a multiple scattering solution for the single-particle orbital
2177:
27:
2014:
7451:
7416:
7373:
7338:
7299:
7253:
7141:
7102:
6400:
7089:
J. Korringa (1994). "Early history of
Multiple Scattering Theory for ordered systems".
6768:
6508:
6380:
6326:. The justification for this approximation is that the matrix elements of the t-matrix
6226:
3090:
2784:
2674:
2154:
1991:
1871:
1144:
1124:
1104:
1028:
946:
870:
850:
292:
In the Dirac notation, the wave equation can be written as an inhomogeneous equation,
7481:
7307:
7269:
7110:
7286:
J. Korringa (1947). "On the calculation of the energy of a Bloch wave in a metal".
97:
35:
23:
5170:
For a Bloch wave the coefficients depend on the site only through a phase factor,
7459:
5914:
7424:
7346:
2312:
These last two equations are the fundamental equations of multiple scattering.
7261:
7068:
that was used in the preceding discussion. With this Green's function and the
4457:
2766:{\displaystyle \sum \limits _{i}{t_{i}}{G_{0+}}\sum \limits _{j\neq i}{t_{j}}}
7381:
7149:
7020:{\displaystyle \leftG(E,{\bf {r}},{\bf {r'}})=-\delta ({\bf {r}}-{\bf {r'}})}
394:
is the kinetic energy operator. The solution of the homogeneous equation is
7467:
7389:
1619:
is the scattering matrix for one atom. Iterating this equation leads to
1513:{\displaystyle {Q_{i}}={t_{i}}(1+{G_{0+}}\sum \limits _{j\neq i}{Q_{j}})}
109:, this problem can be reduced to the solution of a one-electron equation
65:
The multiple-scattering approach is the best way to derive one-electron
2315:
To apply this theory to x-ray or neutron diffraction we go back to the
4860:{\displaystyle \sum \limits _{j,l'm'}{M_{lm,l'm'}^{ij}c_{l'm'}^{j}=0}}
1612:{\displaystyle {t_{i}}={\left({1-{v_{i}}{G_{0+}}}\right)^{-1}}{v_{i}}}
7322:
7125:
6671:{\displaystyle {m_{lm,l'm'}}=\left{\delta _{l,l'}}{\delta _{m,m'}}}
477:{\displaystyle \left({E-{H_{0}}}\right)\left|\phi \right\rangle =0}
38:, the origin of this theory can be traced back to an 1892 paper by
16:
Mathematical theory that describes the scattering of particle waves
3385:. The Green's function may be expanded in the interstitial region
96:
42:. An important mathematical formulation of the theory was made by
289:, is a functional of the density of the electrons in the system.
7323:"Green's functions in full-potential multiple-scattering theory"
5995:. The energy eigenvalues of the periodic solid for a particular
4353:{\displaystyle \sum \limits _{j,l''m''}{\leftd_{l''m''}^{j}}=0}
1868:
can thus be written as the sum of an incoming wave on any site
77:, but they are the best starting point for calculations of the
2464:. The scattering from a site is assumed to be very small, so
2613:
is used to calculate the t-matrix, which simply means that
7217:
Multiple
Scattering Theory: Electronic Structure of Solids
6112:{\displaystyle \det {\bf {M}}\left({E,{\bf {k}}}\right)=0}
7321:
A. Rusanu; G. M. Stocks; Y. Wang; J. S. Faulkner (2011).
3378:{\displaystyle {{\bf {r}}_{i}}={\bf {r}}-{{\bf {R}}_{i}}}
2777:
Multiple scattering theory of electronic states in solids
3036:{\displaystyle \left{\psi _{i}}\left({\bf {r}}\right)=0}
5988:{\displaystyle {A_{lm,l'm'}}\left({E,{\bf {k}}}\right)}
4745:
These coefficients satisfy the set of linear equations
837:{\displaystyle T=V+V{G_{0+}}V+V{G_{0+}}V{G_{0+}}V+...}
7037:
6870:
6813:
6780:
6728:
6688:
6533:
6511:
6457:
6428:
6403:
6383:
6332:
6278:
6249:
6229:
6179:
6125:
6069:
6025:
6001:
5927:
5720:
5538:
5381:
5313:
5176:
5119:
4900:
4873:
4751:
4608:
4465:
4429:
4378:
4039:
4002:
3717:
3394:
3329:
3294:
3116:
3093:
3049:
2962:
2919:
2810:
2787:
2697:
2677:
2648:
2619:
2559:
2470:
2325:
2208:
2180:
2157:
2042:
2017:
1994:
1897:
1874:
1628:
1526:
1427:
1170:
1147:
1127:
1107:
1051:
1031:
1002:
969:
949:
893:
873:
853:
746:
670:
564:
490:
428:
400:
371:
298:
262:
118:
3107:can thus be written in the position representation
7056:
7019:
6853:
6799:
6749:
6714:
6670:
6517:
6494:
6443:
6414:
6389:
6369:
6319:{\displaystyle {\left({{l_{\max }}+1}\right)^{2}}}
6318:
6264:
6235:
6215:
6165:
6119:. The eigenfunctions are found by solving for the
6111:
6055:
6011:
5987:
5903:
5706:
5520:
5365:{\displaystyle {c_{l'm'}}\left(E,{\bf {k}}\right)}
5364:
5299:
5159:
5101:
4883:
4859:
4733:
4588:
4448:
4415:
4352:
4023:
3984:
3696:
3377:
3315:
3276:
3099:
3079:
3035:
2947:
2898:
2793:
2765:
2683:
2663:
2634:
2601:
2545:
2456:
2300:
2195:
2163:
2139:
2029:
2000:
1976:
1880:
1852:
1611:
1512:
1409:
1153:
1133:
1113:
1093:
1037:
1017:
988:
955:
935:
879:
859:
836:
732:
652:
550:
476:
414:
386:
357:
281:
245:
5167:are the elements of the inverse of the t-matrix.
85:systems that cannot be treated with band theory.
6735:
6435:
6292:
6256:
6070:
3708:and the outgoing Hankel function can be written
2602:{\displaystyle T=\sum \limits _{i=1}^{N}{t_{i}}}
1094:{\displaystyle T=\sum \limits _{i=1}^{N}{Q_{i}}}
936:{\displaystyle V=\sum \limits _{i=1}^{N}{v_{i}}}
585:
6216:{\displaystyle E={E_{b}}\left({\bf {k}}\right)}
6056:{\displaystyle {E_{b}}\left({\bf {k}}\right)}
8:
7215:J.S. Faulkner; G.M. Stocks; Y. Wang (2018).
3080:{\displaystyle \alpha ={\sqrt {2mE}}/\hbar }
256:where the effective one-electron potential,
6715:{\displaystyle {\delta _{l}}\left(E\right)}
6166:{\displaystyle {c_{l,m}}\left({E,k}\right)}
7210:
7208:
7183:Antonios Gonis; William H. Butler (2000).
7045:
7044:
7036:
7003:
7002:
6993:
6992:
6966:
6965:
6956:
6955:
6923:
6922:
6906:
6901:
6885:
6879:
6875:
6869:
6854:{\displaystyle G(E,{\bf {r}},{\bf {r'}})}
6837:
6836:
6827:
6826:
6812:
6788:
6787:
6779:
6734:
6729:
6727:
6694:
6689:
6687:
6650:
6645:
6627:
6622:
6590:
6585:
6575:
6539:
6534:
6532:
6510:
6463:
6458:
6456:
6451:, and the elements of the inverse matrix
6434:
6429:
6427:
6402:
6382:
6338:
6333:
6331:
6309:
6291:
6286:
6285:
6279:
6277:
6255:
6250:
6248:
6228:
6203:
6202:
6191:
6186:
6178:
6148:
6131:
6126:
6124:
6092:
6091:
6084:
6074:
6073:
6068:
6043:
6042:
6031:
6026:
6024:
6003:
6002:
6000:
5974:
5973:
5966:
5933:
5928:
5926:
5886:
5880:
5879:
5877:
5870:
5837:
5832:
5819:
5813:
5812:
5810:
5801:
5800:
5796:
5791:
5785:
5767:
5766:
5759:
5726:
5721:
5719:
5693:
5692:
5685:
5652:
5647:
5604:
5599:
5585:
5584:
5577:
5544:
5539:
5537:
5500:
5499:
5468:
5463:
5452:
5451:
5444:
5411:
5406:
5405:
5386:
5380:
5351:
5350:
5319:
5314:
5312:
5286:
5285:
5254:
5249:
5239:
5233:
5232:
5230:
5221:
5220:
5213:
5208:
5199:
5181:
5175:
5151:
5124:
5118:
5084:
5078:
5077:
5075:
5068:
5035:
5030:
5015:
5010:
5003:
4986:
4981:
4975:
4948:
4932:
4905:
4899:
4875:
4874:
4872:
4844:
4826:
4813:
4786:
4781:
4756:
4750:
4724:
4706:
4685:
4658:
4653:
4634:
4621:
4613:
4607:
4576:
4575:
4563:
4562:
4557:
4547:
4541:
4540:
4538:
4529:
4528:
4524:
4519:
4504:
4498:
4497:
4495:
4486:
4485:
4484:
4472:
4471:
4466:
4464:
4439:
4433:
4432:
4430:
4428:
4403:
4397:
4396:
4394:
4384:
4379:
4377:
4337:
4319:
4292:
4255:
4236:
4230:
4229:
4227:
4220:
4187:
4182:
4181:
4162:
4143:
4138:
4131:
4095:
4090:
4080:
4075:
4074:
4069:
4044:
4038:
4015:
4007:
4001:
3966:
3960:
3959:
3957:
3950:
3917:
3912:
3900:
3895:
3891:
3880:
3875:
3864:
3858:
3857:
3855:
3841:
3836:
3835:
3823:
3805:
3800:
3790:
3780:
3765:
3759:
3758:
3756:
3732:
3727:
3716:
3682:
3671:
3670:
3668:
3658:
3637:
3621:
3611:
3609:
3605:
3589:
3584:
3572:
3567:
3563:
3553:
3541:
3537:
3522:
3516:
3515:
3513:
3486:
3481:
3480:
3458:
3426:
3425:
3416:
3415:
3414:
3400:
3395:
3393:
3368:
3362:
3361:
3359:
3350:
3349:
3339:
3333:
3332:
3330:
3328:
3307:
3299:
3293:
3267:
3259:
3243:
3238:
3234:
3223:
3218:
3207:
3201:
3200:
3198:
3184:
3179:
3178:
3166:
3148:
3142:
3141:
3139:
3126:
3121:
3115:
3092:
3069:
3056:
3048:
3017:
3016:
3005:
3000:
2988:
2983:
2973:
2968:
2967:
2961:
2936:
2935:
2925:
2920:
2918:
2878:
2873:
2868:
2863:
2857:
2846:
2825:
2820:
2815:
2809:
2786:
2756:
2751:
2739:
2725:
2720:
2713:
2708:
2702:
2696:
2676:
2654:
2649:
2647:
2625:
2620:
2618:
2592:
2587:
2581:
2570:
2558:
2525:
2520:
2510:
2505:
2485:
2480:
2475:
2469:
2436:
2431:
2426:
2421:
2415:
2404:
2359:
2354:
2324:
2284:
2279:
2274:
2263:
2258:
2248:
2243:
2223:
2218:
2213:
2207:
2179:
2156:
2119:
2114:
2109:
2104:
2092:
2057:
2052:
2047:
2041:
2016:
1993:
1957:
1952:
1947:
1926:
1921:
1916:
1896:
1873:
1830:
1825:
1813:
1799:
1794:
1787:
1782:
1781:
1769:
1755:
1750:
1743:
1738:
1732:
1718:
1713:
1701:
1687:
1682:
1675:
1670:
1664:
1650:
1645:
1639:
1627:
1602:
1597:
1587:
1572:
1567:
1560:
1555:
1548:
1542:
1532:
1527:
1525:
1500:
1495:
1483:
1469:
1464:
1448:
1443:
1433:
1428:
1426:
1389:
1384:
1372:
1358:
1353:
1346:
1335:
1330:
1320:
1315:
1305:
1300:
1293:
1288:
1287:
1282:
1276:
1258:
1253:
1247:
1233:
1228:
1212:
1207:
1206:
1200:
1186:
1181:
1175:
1169:
1146:
1126:
1106:
1084:
1079:
1073:
1062:
1050:
1030:
1008:
1003:
1001:
979:
973:
972:
970:
968:
963:atoms having nuclei located at positions
948:
926:
921:
915:
904:
892:
872:
852:
809:
804:
791:
786:
767:
762:
745:
701:
696:
689:
669:
640:
619:
614:
607:
601:
588:
583:
570:
565:
563:
524:
519:
489:
445:
440:
433:
427:
415:{\displaystyle \left|\phi \right\rangle }
399:
377:
372:
370:
315:
310:
303:
297:
270:
269:
261:
223:
222:
210:
192:
191:
171:
170:
154:
149:
133:
127:
123:
117:
7178:
7176:
7174:
4423:are the same, and the nuclear positions
4416:{\displaystyle {v_{i}}({{\bf {r}}_{i}})}
7081:
6882:
4368:Periodic solids, one atom per unit cell
3074:
207:
130:
54:. The MST formalism is widely used for
7165:Royal Academy of Sciences in Amsterdam
7281:
7279:
7126:"On the foundation of crystal optics"
1888:and the outgoing wave from that site
7:
6861:which is a solution of the equation
62:, and is the subject of many books.
5782:
5383:
5372:satisfy the homogeneous equations
4753:
4631:
4159:
4041:
3820:
3455:
3163:
2843:
2736:
2699:
2567:
2401:
2089:
1810:
1766:
1729:
1698:
1661:
1636:
1480:
1369:
1273:
1244:
1197:
1172:
1059:
901:
6903:
4867:, with the elements of the matrix
3323:are undetermined coefficients and
2970:
2948:{\displaystyle {v_{i}}({\bf {r}})}
2858:
234:
213:
151:
69:. These functions differ from the
14:
7046:
7005:
6994:
6968:
6957:
6924:
6839:
6828:
6800:{\displaystyle \psi ({\bf {r}})}
6789:
6204:
6093:
6075:
6063:, are the roots of the equation
6044:
6004:
5975:
5881:
5814:
5802:
5768:
5694:
5586:
5501:
5453:
5352:
5287:
5234:
5222:
5079:
4876:
4577:
4564:
4542:
4530:
4499:
4487:
4473:
4434:
4398:
4231:
3961:
3859:
3760:
3673:
3517:
3428:
3417:
3363:
3351:
3334:
3202:
3143:
3018:
2937:
2151:The outgoing wave from the site
1101:. Inserting the expressions for
974:
271:
224:
193:
172:
7242:Physics of Particles and Nuclei
5160:{\displaystyle m_{lm,l'm'}^{i}}
4449:{\displaystyle {{\bf {R}}_{i}}}
989:{\displaystyle {{\bf {R}}_{i}}}
7051:
7041:
7014:
6989:
6977:
6946:
6929:
6919:
6848:
6817:
6794:
6784:
4410:
4391:
2942:
2932:
2190:
2181:
2024:
2018:
1507:
1455:
1266:
1219:
592:
276:
266:
229:
219:
198:
188:
177:
167:
1:
7185:Multiple Scattering in Solids
7070:Korringa–Kohn–Rostoker method
6750:{\displaystyle {l_{\max }}=3}
6495:{\displaystyle {m_{lm,l'm'}}}
6370:{\displaystyle {t_{lm,l'm'}}}
5919:Korringa–Kohn–Rostoker method
740:. The t-matrix is defined by
7308:10.1016/0031-8914(47)90013-X
7111:10.1016/0370-1573(94)90122-8
7057:{\displaystyle V({\bf {r}})}
3087:. The incoming wave on site
887:non-overlapping potentials,
664:, which can also be written
282:{\displaystyle V({\bf {r}})}
7460:10.1103/PhysRevLett.75.2867
6444:{\displaystyle {l_{\max }}}
6265:{\displaystyle {l_{\max }}}
2317:Lippmann–Schwinger equation
1866:Lippmann-Schwinger equation
847:Suppose that the potential
662:Lippmann–Schwinger equation
7509:
7425:10.1103/PhysRevLett.41.339
7347:10.1103/PhysRevB.84.035102
4024:{\displaystyle d_{lm}^{i}}
3316:{\displaystyle d_{lm}^{i}}
20:Multiple scattering theory
7262:10.1134/s1063779614050037
7066:density functional theory
6012:{\displaystyle {\bf {k}}}
4884:{\displaystyle {\bf {M}}}
107:density functional theory
7382:10.1103/PhysRevB.49.2721
7150:10.1002/andp.19163540102
6762:Extensions of the theory
1045:can be written as a sum
93:Mathematical formulation
58:calculations as well as
7440:Physical Review Letters
7405:Physical Review Letters
2664:{\displaystyle {v_{i}}}
2635:{\displaystyle {t_{i}}}
1141:into the definition of
1018:{\displaystyle {Q_{i}}}
387:{\displaystyle {H_{0}}}
7219:. IOP Publishing Ltd.
7058:
7021:
6855:
6801:
6751:
6716:
6672:
6519:
6496:
6445:
6416:
6391:
6371:
6320:
6272:, they have dimension
6266:
6237:
6217:
6167:
6113:
6057:
6013:
5989:
5905:
5708:
5522:
5366:
5301:
5161:
5103:
4885:
4861:
4735:
4590:
4450:
4417:
4354:
4025:
3986:
3698:
3379:
3317:
3278:
3101:
3081:
3037:
2949:
2900:
2862:
2795:
2767:
2685:
2665:
2636:
2603:
2586:
2547:
2458:
2420:
2302:
2197:
2165:
2141:
2031:
2002:
1978:
1882:
1854:
1613:
1514:
1411:
1155:
1135:
1115:
1095:
1078:
1039:
1019:
990:
957:
937:
920:
881:
861:
838:
734:
654:
552:
478:
416:
388:
359:
283:
247:
102:
52:Heike Kamerlingh Onnes
7064:is the same one from
7059:
7022:
6856:
6802:
6752:
6717:
6673:
6520:
6497:
6446:
6417:
6392:
6372:
6321:
6267:
6238:
6218:
6168:
6114:
6058:
6014:
5990:
5906:
5709:
5523:
5367:
5302:
5162:
5104:
4886:
4862:
4736:
4591:
4451:
4418:
4355:
4026:
3987:
3699:
3380:
3318:
3279:
3102:
3082:
3038:
2950:
2901:
2842:
2796:
2768:
2686:
2666:
2637:
2604:
2566:
2548:
2459:
2400:
2303:
2198:
2166:
2142:
2032:
2003:
1979:
1883:
1855:
1614:
1515:
1412:
1156:
1136:
1116:
1096:
1058:
1040:
1020:
996:. Define an operator
991:
958:
938:
900:
882:
862:
839:
735:
655:
553:
479:
417:
389:
360:
284:
248:
100:
7124:P. P. Ewald (1916).
7035:
6868:
6811:
6778:
6726:
6686:
6531:
6509:
6455:
6426:
6401:
6381:
6377:are very small when
6330:
6276:
6247:
6227:
6177:
6123:
6067:
6023:
5999:
5925:
5718:
5536:
5379:
5311:
5174:
5117:
4898:
4871:
4749:
4606:
4463:
4427:
4376:
4037:
4000:
3715:
3392:
3327:
3292:
3114:
3091:
3047:
2960:
2917:
2808:
2785:
2695:
2675:
2646:
2617:
2557:
2468:
2323:
2206:
2196:{\displaystyle (II)}
2178:
2155:
2040:
2015:
1992:
1895:
1872:
1864:The solution of the
1626:
1524:
1425:
1168:
1145:
1125:
1105:
1049:
1029:
1000:
967:
947:
891:
871:
851:
744:
668:
562:
488:
426:
398:
369:
296:
260:
116:
79:electronic structure
56:electronic structure
7452:1995PhRvL..75.2867W
7417:1978PhRvL..41..339S
7374:1994PhRvB..49.2721S
7339:2011PhRvB..84c5102R
7300:1947Phy....13..392K
7254:2014PPN....45..991B
7142:1916AnP...354....1E
7103:1994PhR...238..341K
5204:
5156:
4980:
4940:
4849:
4821:
4729:
4690:
4626:
4342:
4297:
4020:
3795:
3663:
3558:
3312:
3272:
3134:
2889:
2833:
2496:
2447:
2292:
2234:
2130:
2065:
2030:{\displaystyle (I)}
1968:
1934:
7130:Annalen der Physik
7054:
7017:
6851:
6797:
6747:
6712:
6668:
6515:
6492:
6441:
6415:{\displaystyle l'}
6412:
6387:
6367:
6316:
6262:
6233:
6213:
6163:
6109:
6053:
6009:
5985:
5901:
5790:
5704:
5518:
5404:
5362:
5297:
5177:
5157:
5120:
5099:
4944:
4901:
4881:
4857:
4822:
4782:
4780:
4731:
4702:
4654:
4609:
4586:
4446:
4413:
4350:
4315:
4251:
4180:
4068:
4021:
4003:
3982:
3834:
3776:
3694:
3633:
3533:
3479:
3375:
3313:
3295:
3274:
3255:
3117:
3097:
3077:
3033:
2945:
2896:
2869:
2816:
2791:
2763:
2750:
2707:
2681:
2661:
2632:
2611:Born approximation
2599:
2543:
2476:
2454:
2427:
2298:
2275:
2214:
2193:
2161:
2137:
2110:
2103:
2048:
2027:
1998:
1974:
1948:
1917:
1878:
1850:
1824:
1780:
1737:
1712:
1669:
1644:
1609:
1510:
1494:
1407:
1383:
1281:
1252:
1205:
1180:
1151:
1131:
1111:
1091:
1035:
1015:
986:
953:
933:
877:
857:
834:
730:
650:
599:
548:
474:
412:
384:
355:
279:
243:
103:
73:used to treat the
60:diffraction theory
34:As pointed out by
7493:Quantum mechanics
7488:Scattering theory
7446:(15): 2867–2870.
7362:Physical Review B
7327:Physical Review B
7226:978-0-7503-1490-9
6899:
6518:{\displaystyle l}
6422:are greater than
6390:{\displaystyle l}
6236:{\displaystyle l}
5781:
5382:
4752:
4158:
4040:
3819:
3454:
3100:{\displaystyle i}
3067:
2794:{\displaystyle N}
2735:
2698:
2684:{\displaystyle T}
2642:is replaced with
2164:{\displaystyle i}
2088:
2001:{\displaystyle i}
1881:{\displaystyle i}
1809:
1765:
1728:
1697:
1660:
1635:
1479:
1368:
1272:
1243:
1196:
1171:
1154:{\displaystyle T}
1134:{\displaystyle T}
1114:{\displaystyle V}
1038:{\displaystyle T}
956:{\displaystyle N}
880:{\displaystyle N}
860:{\displaystyle V}
584:
241:
147:
75:many-body problem
71:Green's functions
67:Green's functions
7500:
7472:
7471:
7435:
7429:
7428:
7400:
7394:
7393:
7368:(4): 2721–2729.
7357:
7351:
7350:
7318:
7312:
7311:
7283:
7274:
7273:
7237:
7231:
7230:
7212:
7203:
7202:
7180:
7169:
7168:
7160:
7154:
7153:
7121:
7115:
7114:
7086:
7063:
7061:
7060:
7055:
7050:
7049:
7026:
7024:
7023:
7018:
7013:
7012:
7011:
6998:
6997:
6976:
6975:
6974:
6961:
6960:
6942:
6938:
6928:
6927:
6912:
6911:
6910:
6900:
6898:
6890:
6889:
6880:
6860:
6858:
6857:
6852:
6847:
6846:
6845:
6832:
6831:
6806:
6804:
6803:
6798:
6793:
6792:
6756:
6754:
6753:
6748:
6740:
6739:
6738:
6721:
6719:
6718:
6713:
6711:
6700:
6699:
6698:
6677:
6675:
6674:
6669:
6667:
6666:
6665:
6664:
6644:
6643:
6642:
6641:
6621:
6617:
6607:
6596:
6595:
6594:
6567:
6566:
6565:
6564:
6556:
6524:
6522:
6521:
6516:
6502:are very large.
6501:
6499:
6498:
6493:
6491:
6490:
6489:
6488:
6480:
6450:
6448:
6447:
6442:
6440:
6439:
6438:
6421:
6419:
6418:
6413:
6411:
6396:
6394:
6393:
6388:
6376:
6374:
6373:
6368:
6366:
6365:
6364:
6363:
6355:
6325:
6323:
6322:
6317:
6315:
6314:
6313:
6308:
6304:
6297:
6296:
6295:
6271:
6269:
6268:
6263:
6261:
6260:
6259:
6242:
6240:
6239:
6234:
6222:
6220:
6219:
6214:
6212:
6208:
6207:
6197:
6196:
6195:
6172:
6170:
6169:
6164:
6162:
6158:
6143:
6142:
6141:
6118:
6116:
6115:
6110:
6102:
6098:
6097:
6096:
6079:
6078:
6062:
6060:
6059:
6054:
6052:
6048:
6047:
6037:
6036:
6035:
6018:
6016:
6015:
6010:
6008:
6007:
5994:
5992:
5991:
5986:
5984:
5980:
5979:
5978:
5961:
5960:
5959:
5958:
5950:
5910:
5908:
5907:
5902:
5900:
5896:
5895:
5894:
5893:
5885:
5884:
5865:
5864:
5863:
5862:
5854:
5831:
5830:
5829:
5828:
5827:
5826:
5818:
5817:
5806:
5805:
5789:
5777:
5773:
5772:
5771:
5754:
5753:
5752:
5751:
5743:
5713:
5711:
5710:
5705:
5703:
5699:
5698:
5697:
5680:
5679:
5678:
5677:
5669:
5643:
5632:
5631:
5630:
5629:
5621:
5595:
5591:
5590:
5589:
5572:
5571:
5570:
5569:
5561:
5527:
5525:
5524:
5519:
5517:
5510:
5506:
5505:
5504:
5487:
5486:
5485:
5484:
5476:
5462:
5458:
5457:
5456:
5439:
5438:
5437:
5436:
5428:
5403:
5402:
5394:
5371:
5369:
5368:
5363:
5361:
5357:
5356:
5355:
5338:
5337:
5336:
5335:
5327:
5306:
5304:
5303:
5298:
5296:
5292:
5291:
5290:
5273:
5272:
5271:
5270:
5262:
5248:
5247:
5246:
5245:
5244:
5243:
5238:
5237:
5226:
5225:
5203:
5198:
5197:
5189:
5166:
5164:
5163:
5158:
5155:
5150:
5149:
5141:
5108:
5106:
5105:
5100:
5098:
5094:
5093:
5092:
5091:
5083:
5082:
5063:
5062:
5061:
5060:
5052:
5029:
5025:
5024:
5023:
5022:
4995:
4994:
4993:
4979:
4974:
4973:
4965:
4939:
4931:
4930:
4922:
4890:
4888:
4887:
4882:
4880:
4879:
4866:
4864:
4863:
4858:
4856:
4848:
4843:
4842:
4834:
4820:
4812:
4811:
4803:
4779:
4778:
4770:
4740:
4738:
4737:
4732:
4730:
4728:
4723:
4722:
4714:
4701:
4689:
4684:
4683:
4675:
4652:
4651:
4650:
4642:
4625:
4620:
4595:
4593:
4592:
4587:
4585:
4581:
4580:
4570:
4569:
4568:
4567:
4556:
4555:
4554:
4553:
4552:
4551:
4546:
4545:
4534:
4533:
4515:
4511:
4510:
4509:
4508:
4503:
4502:
4491:
4490:
4479:
4478:
4477:
4476:
4455:
4453:
4452:
4447:
4445:
4444:
4443:
4438:
4437:
4422:
4420:
4419:
4414:
4409:
4408:
4407:
4402:
4401:
4390:
4389:
4388:
4359:
4357:
4356:
4351:
4343:
4341:
4336:
4335:
4327:
4314:
4310:
4309:
4308:
4296:
4291:
4290:
4282:
4271:
4263:
4250:
4246:
4245:
4244:
4243:
4235:
4234:
4215:
4214:
4213:
4212:
4204:
4179:
4178:
4170:
4157:
4153:
4152:
4151:
4150:
4123:
4122:
4121:
4120:
4112:
4089:
4088:
4087:
4067:
4066:
4058:
4030:
4028:
4027:
4022:
4019:
4014:
3991:
3989:
3988:
3983:
3981:
3980:
3976:
3975:
3974:
3973:
3965:
3964:
3945:
3944:
3943:
3942:
3934:
3911:
3907:
3906:
3905:
3904:
3886:
3885:
3884:
3874:
3870:
3869:
3868:
3863:
3862:
3850:
3849:
3848:
3833:
3815:
3811:
3810:
3809:
3794:
3789:
3788:
3775:
3771:
3770:
3769:
3764:
3763:
3751:
3750:
3749:
3748:
3740:
3703:
3701:
3700:
3695:
3693:
3692:
3688:
3687:
3686:
3681:
3680:
3679:
3662:
3657:
3656:
3645:
3632:
3628:
3627:
3626:
3625:
3620:
3619:
3600:
3599:
3598:
3597:
3583:
3579:
3578:
3577:
3576:
3557:
3552:
3551:
3550:
3549:
3532:
3528:
3527:
3526:
3521:
3520:
3508:
3507:
3506:
3505:
3494:
3478:
3477:
3466:
3441:
3437:
3436:
3435:
3434:
3421:
3420:
3409:
3408:
3407:
3384:
3382:
3381:
3376:
3374:
3373:
3372:
3367:
3366:
3355:
3354:
3345:
3344:
3343:
3338:
3337:
3322:
3320:
3319:
3314:
3311:
3306:
3283:
3281:
3280:
3275:
3273:
3271:
3266:
3254:
3250:
3249:
3248:
3247:
3229:
3228:
3227:
3217:
3213:
3212:
3211:
3206:
3205:
3193:
3192:
3191:
3177:
3176:
3158:
3154:
3153:
3152:
3147:
3146:
3133:
3125:
3106:
3104:
3103:
3098:
3086:
3084:
3083:
3078:
3073:
3068:
3057:
3042:
3040:
3039:
3034:
3026:
3022:
3021:
3011:
3010:
3009:
2999:
2995:
2994:
2993:
2992:
2979:
2978:
2977:
2954:
2952:
2951:
2946:
2941:
2940:
2931:
2930:
2929:
2905:
2903:
2902:
2897:
2895:
2894:
2890:
2888:
2877:
2861:
2856:
2838:
2834:
2832:
2824:
2800:
2798:
2797:
2792:
2772:
2770:
2769:
2764:
2762:
2761:
2760:
2749:
2734:
2733:
2732:
2719:
2718:
2717:
2706:
2690:
2688:
2687:
2682:
2670:
2668:
2667:
2662:
2660:
2659:
2658:
2641:
2639:
2638:
2633:
2631:
2630:
2629:
2608:
2606:
2605:
2600:
2598:
2597:
2596:
2585:
2580:
2552:
2550:
2549:
2544:
2542:
2531:
2530:
2529:
2519:
2518:
2517:
2501:
2497:
2495:
2484:
2463:
2461:
2460:
2455:
2453:
2452:
2448:
2446:
2435:
2419:
2414:
2396:
2382:
2368:
2367:
2366:
2350:
2336:
2307:
2305:
2304:
2299:
2297:
2293:
2291:
2283:
2269:
2268:
2267:
2257:
2256:
2255:
2239:
2235:
2233:
2222:
2202:
2200:
2199:
2194:
2170:
2168:
2167:
2162:
2146:
2144:
2143:
2138:
2136:
2135:
2131:
2129:
2118:
2102:
2084:
2070:
2066:
2064:
2056:
2036:
2034:
2033:
2028:
2007:
2005:
2004:
1999:
1983:
1981:
1980:
1975:
1973:
1969:
1967:
1956:
1939:
1935:
1933:
1925:
1908:
1887:
1885:
1884:
1879:
1859:
1857:
1856:
1851:
1840:
1836:
1835:
1834:
1823:
1808:
1807:
1806:
1793:
1792:
1791:
1779:
1764:
1763:
1762:
1749:
1748:
1747:
1736:
1724:
1723:
1722:
1711:
1696:
1695:
1694:
1681:
1680:
1679:
1668:
1656:
1655:
1654:
1643:
1618:
1616:
1615:
1610:
1608:
1607:
1606:
1596:
1595:
1594:
1586:
1582:
1581:
1580:
1579:
1566:
1565:
1564:
1538:
1537:
1536:
1519:
1517:
1516:
1511:
1506:
1505:
1504:
1493:
1478:
1477:
1476:
1454:
1453:
1452:
1439:
1438:
1437:
1416:
1414:
1413:
1408:
1406:
1405:
1401:
1400:
1396:
1395:
1394:
1393:
1382:
1367:
1366:
1365:
1341:
1340:
1339:
1326:
1325:
1324:
1314:
1313:
1312:
1299:
1298:
1297:
1280:
1265:
1264:
1263:
1262:
1251:
1242:
1241:
1240:
1218:
1217:
1216:
1204:
1192:
1191:
1190:
1179:
1160:
1158:
1157:
1152:
1140:
1138:
1137:
1132:
1120:
1118:
1117:
1112:
1100:
1098:
1097:
1092:
1090:
1089:
1088:
1077:
1072:
1044:
1042:
1041:
1036:
1024:
1022:
1021:
1016:
1014:
1013:
1012:
995:
993:
992:
987:
985:
984:
983:
978:
977:
962:
960:
959:
954:
942:
940:
939:
934:
932:
931:
930:
919:
914:
886:
884:
883:
878:
866:
864:
863:
858:
843:
841:
840:
835:
818:
817:
816:
800:
799:
798:
776:
775:
774:
739:
737:
736:
731:
729:
718:
714:
710:
709:
708:
681:
659:
657:
656:
651:
649:
648:
647:
639:
635:
625:
624:
623:
600:
598:
579:
578:
577:
557:
555:
554:
549:
547:
533:
532:
531:
515:
501:
483:
481:
480:
475:
467:
456:
452:
451:
450:
449:
421:
419:
418:
413:
411:
393:
391:
390:
385:
383:
382:
381:
364:
362:
361:
356:
354:
337:
326:
322:
321:
320:
319:
288:
286:
285:
280:
275:
274:
252:
250:
249:
244:
242:
240:
232:
228:
227:
211:
197:
196:
184:
180:
176:
175:
160:
159:
158:
148:
146:
138:
137:
128:
83:condensed matter
48:Nikolai Kasterin
44:Paul Peter Ewald
28:acoustical waves
7508:
7507:
7503:
7502:
7501:
7499:
7498:
7497:
7478:
7477:
7476:
7475:
7437:
7436:
7432:
7402:
7401:
7397:
7359:
7358:
7354:
7320:
7319:
7315:
7285:
7284:
7277:
7239:
7238:
7234:
7227:
7214:
7213:
7206:
7199:
7182:
7181:
7172:
7162:
7161:
7157:
7123:
7122:
7118:
7091:Physics Reports
7088:
7087:
7083:
7078:
7033:
7032:
7004:
6967:
6902:
6891:
6881:
6871:
6866:
6865:
6838:
6809:
6808:
6776:
6775:
6764:
6730:
6724:
6723:
6701:
6690:
6684:
6683:
6657:
6646:
6634:
6623:
6597:
6586:
6571:
6557:
6549:
6535:
6529:
6528:
6507:
6506:
6481:
6473:
6459:
6453:
6452:
6430:
6424:
6423:
6404:
6399:
6398:
6379:
6378:
6356:
6348:
6334:
6328:
6327:
6287:
6281:
6280:
6274:
6273:
6251:
6245:
6244:
6225:
6224:
6198:
6187:
6175:
6174:
6144:
6127:
6121:
6120:
6080:
6065:
6064:
6038:
6027:
6021:
6020:
5997:
5996:
5962:
5951:
5943:
5929:
5923:
5922:
5878:
5866:
5855:
5847:
5833:
5811:
5792:
5755:
5744:
5736:
5722:
5716:
5715:
5681:
5670:
5662:
5648:
5633:
5622:
5614:
5600:
5573:
5562:
5554:
5540:
5534:
5533:
5492:
5488:
5477:
5469:
5464:
5440:
5429:
5421:
5407:
5395:
5387:
5377:
5376:
5343:
5339:
5328:
5320:
5315:
5309:
5308:
5278:
5274:
5263:
5255:
5250:
5231:
5209:
5190:
5182:
5172:
5171:
5142:
5134:
5115:
5114:
5076:
5064:
5053:
5045:
5031:
5011:
4999:
4982:
4966:
4958:
4923:
4915:
4896:
4895:
4869:
4868:
4835:
4827:
4804:
4796:
4771:
4763:
4747:
4746:
4715:
4707:
4691:
4676:
4668:
4643:
4635:
4630:
4604:
4603:
4571:
4558:
4539:
4520:
4496:
4480:
4467:
4461:
4460:
4431:
4425:
4424:
4395:
4380:
4374:
4373:
4370:
4328:
4320:
4298:
4283:
4275:
4264:
4256:
4228:
4216:
4205:
4197:
4183:
4171:
4163:
4139:
4127:
4113:
4105:
4091:
4076:
4070:
4059:
4051:
4035:
4034:
3998:
3997:
3958:
3946:
3935:
3927:
3913:
3896:
3887:
3876:
3856:
3851:
3837:
3801:
3796:
3781:
3757:
3752:
3741:
3733:
3728:
3713:
3712:
3672:
3669:
3664:
3649:
3638:
3612:
3610:
3601:
3590:
3585:
3568:
3559:
3542:
3538:
3514:
3509:
3498:
3487:
3482:
3470:
3459:
3427:
3410:
3396:
3390:
3389:
3360:
3331:
3325:
3324:
3290:
3289:
3239:
3230:
3219:
3199:
3194:
3180:
3162:
3140:
3135:
3112:
3111:
3089:
3088:
3045:
3044:
3012:
3001:
2984:
2969:
2963:
2958:
2957:
2921:
2915:
2914:
2864:
2811:
2806:
2805:
2783:
2782:
2779:
2752:
2721:
2709:
2693:
2692:
2673:
2672:
2650:
2644:
2643:
2621:
2615:
2614:
2588:
2555:
2554:
2532:
2521:
2506:
2471:
2466:
2465:
2422:
2386:
2372:
2355:
2340:
2326:
2321:
2320:
2270:
2259:
2244:
2209:
2204:
2203:
2176:
2175:
2153:
2152:
2105:
2074:
2043:
2038:
2037:
2013:
2012:
1990:
1989:
1943:
1912:
1898:
1893:
1892:
1870:
1869:
1826:
1795:
1783:
1751:
1739:
1714:
1683:
1671:
1646:
1624:
1623:
1598:
1568:
1556:
1544:
1543:
1528:
1522:
1521:
1496:
1465:
1444:
1429:
1423:
1422:
1385:
1354:
1342:
1331:
1316:
1301:
1289:
1283:
1254:
1229:
1208:
1182:
1166:
1165:
1143:
1142:
1123:
1122:
1103:
1102:
1080:
1047:
1046:
1027:
1026:
1004:
998:
997:
971:
965:
964:
945:
944:
922:
889:
888:
869:
868:
849:
848:
805:
787:
763:
742:
741:
719:
697:
685:
671:
666:
665:
615:
603:
602:
566:
560:
559:
537:
520:
505:
491:
486:
485:
457:
441:
429:
424:
423:
401:
396:
395:
373:
367:
366:
344:
327:
311:
299:
294:
293:
258:
257:
233:
212:
150:
139:
129:
119:
114:
113:
95:
17:
12:
11:
5:
7506:
7504:
7496:
7495:
7490:
7480:
7479:
7474:
7473:
7430:
7411:(5): 339–343.
7395:
7352:
7313:
7294:(6): 392–400.
7275:
7248:(5): 991–999.
7232:
7225:
7204:
7198:978-0387988535
7197:
7170:
7155:
7116:
7097:(6): 341–360.
7080:
7079:
7077:
7074:
7053:
7048:
7043:
7040:
7031:The potential
7029:
7028:
7016:
7010:
7007:
7001:
6996:
6991:
6988:
6985:
6982:
6979:
6973:
6970:
6964:
6959:
6954:
6951:
6948:
6945:
6941:
6937:
6934:
6931:
6926:
6921:
6918:
6915:
6909:
6905:
6897:
6894:
6888:
6884:
6878:
6874:
6850:
6844:
6841:
6835:
6830:
6825:
6822:
6819:
6816:
6796:
6791:
6786:
6783:
6769:surface states
6763:
6760:
6746:
6743:
6737:
6733:
6710:
6707:
6704:
6697:
6693:
6680:
6679:
6663:
6660:
6656:
6653:
6649:
6640:
6637:
6633:
6630:
6626:
6620:
6616:
6613:
6610:
6606:
6603:
6600:
6593:
6589:
6584:
6581:
6578:
6574:
6570:
6563:
6560:
6555:
6552:
6548:
6545:
6542:
6538:
6514:
6487:
6484:
6479:
6476:
6472:
6469:
6466:
6462:
6437:
6433:
6410:
6407:
6386:
6362:
6359:
6354:
6351:
6347:
6344:
6341:
6337:
6312:
6307:
6303:
6300:
6294:
6290:
6284:
6258:
6254:
6232:
6211:
6206:
6201:
6194:
6190:
6185:
6182:
6161:
6157:
6154:
6151:
6147:
6140:
6137:
6134:
6130:
6108:
6105:
6101:
6095:
6090:
6087:
6083:
6077:
6072:
6051:
6046:
6041:
6034:
6030:
6006:
5983:
5977:
5972:
5969:
5965:
5957:
5954:
5949:
5946:
5942:
5939:
5936:
5932:
5899:
5892:
5889:
5883:
5876:
5873:
5869:
5861:
5858:
5853:
5850:
5846:
5843:
5840:
5836:
5825:
5822:
5816:
5809:
5804:
5799:
5795:
5788:
5784:
5780:
5776:
5770:
5765:
5762:
5758:
5750:
5747:
5742:
5739:
5735:
5732:
5729:
5725:
5702:
5696:
5691:
5688:
5684:
5676:
5673:
5668:
5665:
5661:
5658:
5655:
5651:
5646:
5642:
5639:
5636:
5628:
5625:
5620:
5617:
5613:
5610:
5607:
5603:
5598:
5594:
5588:
5583:
5580:
5576:
5568:
5565:
5560:
5557:
5553:
5550:
5547:
5543:
5530:
5529:
5516:
5513:
5509:
5503:
5498:
5495:
5491:
5483:
5480:
5475:
5472:
5467:
5461:
5455:
5450:
5447:
5443:
5435:
5432:
5427:
5424:
5420:
5417:
5414:
5410:
5401:
5398:
5393:
5390:
5385:
5360:
5354:
5349:
5346:
5342:
5334:
5331:
5326:
5323:
5318:
5295:
5289:
5284:
5281:
5277:
5269:
5266:
5261:
5258:
5253:
5242:
5236:
5229:
5224:
5219:
5216:
5212:
5207:
5202:
5196:
5193:
5188:
5185:
5180:
5154:
5148:
5145:
5140:
5137:
5133:
5130:
5127:
5123:
5111:
5110:
5097:
5090:
5087:
5081:
5074:
5071:
5067:
5059:
5056:
5051:
5048:
5044:
5041:
5038:
5034:
5028:
5021:
5018:
5014:
5009:
5006:
5002:
4998:
4992:
4989:
4985:
4978:
4972:
4969:
4964:
4961:
4957:
4954:
4951:
4947:
4943:
4938:
4935:
4929:
4926:
4921:
4918:
4914:
4911:
4908:
4904:
4878:
4855:
4852:
4847:
4841:
4838:
4833:
4830:
4825:
4819:
4816:
4810:
4807:
4802:
4799:
4795:
4792:
4789:
4785:
4777:
4774:
4769:
4766:
4762:
4759:
4755:
4743:
4742:
4727:
4721:
4718:
4713:
4710:
4705:
4700:
4697:
4694:
4688:
4682:
4679:
4674:
4671:
4667:
4664:
4661:
4657:
4649:
4646:
4641:
4638:
4633:
4629:
4624:
4619:
4616:
4612:
4584:
4579:
4574:
4566:
4561:
4550:
4544:
4537:
4532:
4527:
4523:
4518:
4514:
4507:
4501:
4494:
4489:
4483:
4475:
4470:
4442:
4436:
4412:
4406:
4400:
4393:
4387:
4383:
4369:
4366:
4362:
4361:
4349:
4346:
4340:
4334:
4331:
4326:
4323:
4318:
4313:
4307:
4304:
4301:
4295:
4289:
4286:
4281:
4278:
4274:
4270:
4267:
4262:
4259:
4254:
4249:
4242:
4239:
4233:
4226:
4223:
4219:
4211:
4208:
4203:
4200:
4196:
4193:
4190:
4186:
4177:
4174:
4169:
4166:
4161:
4156:
4149:
4146:
4142:
4137:
4134:
4130:
4126:
4119:
4116:
4111:
4108:
4104:
4101:
4098:
4094:
4086:
4083:
4079:
4073:
4065:
4062:
4057:
4054:
4050:
4047:
4043:
4018:
4013:
4010:
4006:
3994:
3993:
3979:
3972:
3969:
3963:
3956:
3953:
3949:
3941:
3938:
3933:
3930:
3926:
3923:
3920:
3916:
3910:
3903:
3899:
3894:
3890:
3883:
3879:
3873:
3867:
3861:
3854:
3847:
3844:
3840:
3832:
3829:
3826:
3822:
3818:
3814:
3808:
3804:
3799:
3793:
3787:
3784:
3779:
3774:
3768:
3762:
3755:
3747:
3744:
3739:
3736:
3731:
3726:
3723:
3720:
3706:
3705:
3691:
3685:
3678:
3675:
3667:
3661:
3655:
3652:
3648:
3644:
3641:
3636:
3631:
3624:
3618:
3615:
3608:
3604:
3596:
3593:
3588:
3582:
3575:
3571:
3566:
3562:
3556:
3548:
3545:
3540:
3536:
3531:
3525:
3519:
3512:
3504:
3501:
3497:
3493:
3490:
3485:
3476:
3473:
3469:
3465:
3462:
3457:
3453:
3450:
3447:
3444:
3440:
3433:
3430:
3424:
3419:
3413:
3406:
3403:
3399:
3371:
3365:
3358:
3353:
3348:
3342:
3336:
3310:
3305:
3302:
3298:
3286:
3285:
3270:
3265:
3262:
3258:
3253:
3246:
3242:
3237:
3233:
3226:
3222:
3216:
3210:
3204:
3197:
3190:
3187:
3183:
3175:
3172:
3169:
3165:
3161:
3157:
3151:
3145:
3138:
3132:
3129:
3124:
3120:
3096:
3076:
3072:
3066:
3063:
3060:
3055:
3052:
3032:
3029:
3025:
3020:
3015:
3008:
3004:
2998:
2991:
2987:
2982:
2976:
2972:
2966:
2944:
2939:
2934:
2928:
2924:
2908:
2907:
2893:
2887:
2884:
2881:
2876:
2872:
2867:
2860:
2855:
2852:
2849:
2845:
2841:
2837:
2831:
2828:
2823:
2819:
2814:
2790:
2778:
2775:
2759:
2755:
2748:
2745:
2742:
2738:
2731:
2728:
2724:
2716:
2712:
2705:
2701:
2680:
2657:
2653:
2628:
2624:
2595:
2591:
2584:
2579:
2576:
2573:
2569:
2565:
2562:
2541:
2538:
2535:
2528:
2524:
2516:
2513:
2509:
2504:
2500:
2494:
2491:
2488:
2483:
2479:
2474:
2451:
2445:
2442:
2439:
2434:
2430:
2425:
2418:
2413:
2410:
2407:
2403:
2399:
2395:
2392:
2389:
2385:
2381:
2378:
2375:
2371:
2365:
2362:
2358:
2353:
2349:
2346:
2343:
2339:
2335:
2332:
2329:
2310:
2309:
2296:
2290:
2287:
2282:
2278:
2273:
2266:
2262:
2254:
2251:
2247:
2242:
2238:
2232:
2229:
2226:
2221:
2217:
2212:
2192:
2189:
2186:
2183:
2171:is defined as
2160:
2149:
2148:
2134:
2128:
2125:
2122:
2117:
2113:
2108:
2101:
2098:
2095:
2091:
2087:
2083:
2080:
2077:
2073:
2069:
2063:
2060:
2055:
2051:
2046:
2026:
2023:
2020:
1997:
1986:
1985:
1972:
1966:
1963:
1960:
1955:
1951:
1946:
1942:
1938:
1932:
1929:
1924:
1920:
1915:
1911:
1907:
1904:
1901:
1877:
1862:
1861:
1849:
1846:
1843:
1839:
1833:
1829:
1822:
1819:
1816:
1812:
1805:
1802:
1798:
1790:
1786:
1778:
1775:
1772:
1768:
1761:
1758:
1754:
1746:
1742:
1735:
1731:
1727:
1721:
1717:
1710:
1707:
1704:
1700:
1693:
1690:
1686:
1678:
1674:
1667:
1663:
1659:
1653:
1649:
1642:
1638:
1634:
1631:
1605:
1601:
1593:
1590:
1585:
1578:
1575:
1571:
1563:
1559:
1554:
1551:
1547:
1541:
1535:
1531:
1509:
1503:
1499:
1492:
1489:
1486:
1482:
1475:
1472:
1468:
1463:
1460:
1457:
1451:
1447:
1442:
1436:
1432:
1419:
1418:
1404:
1399:
1392:
1388:
1381:
1378:
1375:
1371:
1364:
1361:
1357:
1352:
1349:
1345:
1338:
1334:
1329:
1323:
1319:
1311:
1308:
1304:
1296:
1292:
1286:
1279:
1275:
1271:
1268:
1261:
1257:
1250:
1246:
1239:
1236:
1232:
1227:
1224:
1221:
1215:
1211:
1203:
1199:
1195:
1189:
1185:
1178:
1174:
1150:
1130:
1110:
1087:
1083:
1076:
1071:
1068:
1065:
1061:
1057:
1054:
1034:
1011:
1007:
982:
976:
952:
929:
925:
918:
913:
910:
907:
903:
899:
896:
876:
867:is the sum of
856:
833:
830:
827:
824:
821:
815:
812:
808:
803:
797:
794:
790:
785:
782:
779:
773:
770:
766:
761:
758:
755:
752:
749:
728:
725:
722:
717:
713:
707:
704:
700:
695:
692:
688:
684:
680:
677:
674:
660:. This is the
646:
643:
638:
634:
631:
628:
622:
618:
613:
610:
606:
597:
594:
591:
587:
582:
576:
573:
569:
546:
543:
540:
536:
530:
527:
523:
518:
514:
511:
508:
504:
500:
497:
494:
473:
470:
466:
463:
460:
455:
448:
444:
439:
436:
432:
410:
407:
404:
380:
376:
353:
350:
347:
343:
340:
336:
333:
330:
325:
318:
314:
309:
306:
302:
278:
273:
268:
265:
254:
253:
239:
236:
231:
226:
221:
218:
215:
209:
206:
203:
200:
195:
190:
187:
183:
179:
174:
169:
166:
163:
157:
153:
145:
142:
136:
132:
126:
122:
94:
91:
15:
13:
10:
9:
6:
4:
3:
2:
7505:
7494:
7491:
7489:
7486:
7485:
7483:
7469:
7465:
7461:
7457:
7453:
7449:
7445:
7441:
7434:
7431:
7426:
7422:
7418:
7414:
7410:
7406:
7399:
7396:
7391:
7387:
7383:
7379:
7375:
7371:
7367:
7363:
7356:
7353:
7348:
7344:
7340:
7336:
7333:(3): 035102.
7332:
7328:
7324:
7317:
7314:
7309:
7305:
7301:
7297:
7293:
7289:
7282:
7280:
7276:
7271:
7267:
7263:
7259:
7255:
7251:
7247:
7243:
7236:
7233:
7228:
7222:
7218:
7211:
7209:
7205:
7200:
7194:
7190:
7186:
7179:
7177:
7175:
7171:
7166:
7159:
7156:
7151:
7147:
7143:
7139:
7135:
7131:
7127:
7120:
7117:
7112:
7108:
7104:
7100:
7096:
7092:
7085:
7082:
7075:
7073:
7071:
7067:
7038:
7008:
6999:
6986:
6983:
6980:
6971:
6962:
6952:
6949:
6943:
6939:
6935:
6932:
6916:
6913:
6907:
6895:
6892:
6886:
6876:
6872:
6864:
6863:
6862:
6842:
6833:
6823:
6820:
6814:
6781:
6772:
6770:
6761:
6759:
6744:
6741:
6731:
6708:
6705:
6702:
6695:
6691:
6661:
6658:
6654:
6651:
6647:
6638:
6635:
6631:
6628:
6624:
6618:
6614:
6611:
6608:
6604:
6601:
6598:
6591:
6587:
6582:
6579:
6576:
6572:
6568:
6561:
6558:
6553:
6550:
6546:
6543:
6540:
6536:
6527:
6526:
6525:
6512:
6503:
6485:
6482:
6477:
6474:
6470:
6467:
6464:
6460:
6431:
6408:
6405:
6384:
6360:
6357:
6352:
6349:
6345:
6342:
6339:
6335:
6310:
6305:
6301:
6298:
6288:
6282:
6252:
6243:greater than
6230:
6209:
6199:
6192:
6188:
6183:
6180:
6159:
6155:
6152:
6149:
6145:
6138:
6135:
6132:
6128:
6106:
6103:
6099:
6088:
6085:
6081:
6049:
6039:
6032:
6028:
5981:
5970:
5967:
5963:
5955:
5952:
5947:
5944:
5940:
5937:
5934:
5930:
5920:
5916:
5912:
5897:
5890:
5887:
5874:
5871:
5867:
5859:
5856:
5851:
5848:
5844:
5841:
5838:
5834:
5823:
5820:
5807:
5797:
5793:
5786:
5778:
5774:
5763:
5760:
5756:
5748:
5745:
5740:
5737:
5733:
5730:
5727:
5723:
5700:
5689:
5686:
5682:
5674:
5671:
5666:
5663:
5659:
5656:
5653:
5649:
5644:
5640:
5637:
5634:
5626:
5623:
5618:
5615:
5611:
5608:
5605:
5601:
5596:
5592:
5581:
5578:
5574:
5566:
5563:
5558:
5555:
5551:
5548:
5545:
5541:
5514:
5511:
5507:
5496:
5493:
5489:
5481:
5478:
5473:
5470:
5465:
5459:
5448:
5445:
5441:
5433:
5430:
5425:
5422:
5418:
5415:
5412:
5408:
5399:
5396:
5391:
5388:
5375:
5374:
5373:
5358:
5347:
5344:
5340:
5332:
5329:
5324:
5321:
5316:
5293:
5282:
5279:
5275:
5267:
5264:
5259:
5256:
5251:
5240:
5227:
5217:
5214:
5210:
5205:
5200:
5194:
5191:
5186:
5183:
5178:
5168:
5152:
5146:
5143:
5138:
5135:
5131:
5128:
5125:
5121:
5095:
5088:
5085:
5072:
5069:
5065:
5057:
5054:
5049:
5046:
5042:
5039:
5036:
5032:
5026:
5019:
5016:
5012:
5007:
5004:
5000:
4996:
4990:
4987:
4983:
4976:
4970:
4967:
4962:
4959:
4955:
4952:
4949:
4945:
4941:
4936:
4933:
4927:
4924:
4919:
4916:
4912:
4909:
4906:
4902:
4894:
4893:
4892:
4853:
4850:
4845:
4839:
4836:
4831:
4828:
4823:
4817:
4814:
4808:
4805:
4800:
4797:
4793:
4790:
4787:
4783:
4775:
4772:
4767:
4764:
4760:
4757:
4725:
4719:
4716:
4711:
4708:
4703:
4698:
4695:
4692:
4686:
4680:
4677:
4672:
4669:
4665:
4662:
4659:
4655:
4647:
4644:
4639:
4636:
4627:
4622:
4617:
4614:
4610:
4602:
4601:
4600:
4597:
4582:
4572:
4559:
4548:
4535:
4525:
4521:
4516:
4512:
4505:
4492:
4481:
4468:
4459:
4440:
4404:
4385:
4381:
4367:
4365:
4347:
4344:
4338:
4332:
4329:
4324:
4321:
4316:
4311:
4305:
4302:
4299:
4293:
4287:
4284:
4279:
4276:
4272:
4268:
4265:
4260:
4257:
4252:
4247:
4240:
4237:
4224:
4221:
4217:
4209:
4206:
4201:
4198:
4194:
4191:
4188:
4184:
4175:
4172:
4167:
4164:
4154:
4147:
4144:
4140:
4135:
4132:
4128:
4124:
4117:
4114:
4109:
4106:
4102:
4099:
4096:
4092:
4084:
4081:
4077:
4071:
4063:
4060:
4055:
4052:
4048:
4045:
4033:
4032:
4031:
4016:
4011:
4008:
4004:
3977:
3970:
3967:
3954:
3951:
3947:
3939:
3936:
3931:
3928:
3924:
3921:
3918:
3914:
3908:
3901:
3897:
3892:
3888:
3881:
3877:
3871:
3865:
3852:
3845:
3842:
3838:
3830:
3827:
3824:
3816:
3812:
3806:
3802:
3797:
3791:
3785:
3782:
3777:
3772:
3766:
3753:
3745:
3742:
3737:
3734:
3729:
3724:
3721:
3718:
3711:
3710:
3709:
3689:
3683:
3676:
3665:
3659:
3653:
3650:
3646:
3642:
3639:
3634:
3629:
3622:
3616:
3613:
3606:
3602:
3594:
3591:
3586:
3580:
3573:
3569:
3564:
3560:
3554:
3546:
3543:
3539:
3534:
3529:
3523:
3510:
3502:
3499:
3495:
3491:
3488:
3483:
3474:
3471:
3467:
3463:
3460:
3451:
3448:
3445:
3442:
3438:
3431:
3422:
3411:
3404:
3401:
3397:
3388:
3387:
3386:
3369:
3356:
3346:
3340:
3308:
3303:
3300:
3296:
3268:
3263:
3260:
3256:
3251:
3244:
3240:
3235:
3231:
3224:
3220:
3214:
3208:
3195:
3188:
3185:
3181:
3173:
3170:
3167:
3159:
3155:
3149:
3136:
3130:
3127:
3122:
3118:
3110:
3109:
3108:
3094:
3070:
3064:
3061:
3058:
3053:
3050:
3030:
3027:
3023:
3013:
3006:
3002:
2996:
2989:
2985:
2980:
2974:
2964:
2926:
2922:
2911:
2891:
2885:
2882:
2879:
2874:
2870:
2865:
2853:
2850:
2847:
2839:
2835:
2829:
2826:
2821:
2817:
2812:
2804:
2803:
2802:
2788:
2776:
2774:
2757:
2753:
2746:
2743:
2740:
2729:
2726:
2722:
2714:
2710:
2703:
2678:
2655:
2651:
2626:
2622:
2612:
2593:
2589:
2582:
2577:
2574:
2571:
2563:
2560:
2539:
2536:
2533:
2526:
2522:
2514:
2511:
2507:
2502:
2498:
2492:
2489:
2486:
2481:
2477:
2472:
2449:
2443:
2440:
2437:
2432:
2428:
2423:
2416:
2411:
2408:
2405:
2397:
2393:
2390:
2387:
2383:
2379:
2376:
2373:
2369:
2363:
2360:
2356:
2351:
2347:
2344:
2341:
2337:
2333:
2330:
2327:
2318:
2313:
2294:
2288:
2285:
2280:
2276:
2271:
2264:
2260:
2252:
2249:
2245:
2240:
2236:
2230:
2227:
2224:
2219:
2215:
2210:
2187:
2184:
2174:
2173:
2172:
2158:
2132:
2126:
2123:
2120:
2115:
2111:
2106:
2099:
2096:
2093:
2085:
2081:
2078:
2075:
2071:
2067:
2061:
2058:
2053:
2049:
2044:
2021:
2011:
2010:
2009:
1995:
1970:
1964:
1961:
1958:
1953:
1949:
1944:
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1930:
1927:
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1913:
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1891:
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1397:
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927:
923:
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908:
905:
897:
894:
874:
854:
845:
831:
828:
825:
822:
819:
813:
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801:
795:
792:
788:
783:
780:
777:
771:
768:
764:
759:
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747:
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698:
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541:
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471:
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408:
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378:
374:
351:
348:
345:
341:
338:
334:
331:
328:
323:
316:
312:
307:
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300:
290:
263:
237:
216:
204:
201:
185:
181:
164:
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155:
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112:
111:
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108:
99:
92:
90:
86:
84:
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76:
72:
68:
63:
61:
57:
53:
49:
45:
41:
40:Lord Rayleigh
37:
32:
29:
25:
22:(MST) is the
21:
7443:
7439:
7433:
7408:
7404:
7398:
7365:
7361:
7355:
7330:
7326:
7316:
7291:
7287:
7245:
7241:
7235:
7216:
7184:
7164:
7158:
7133:
7129:
7119:
7094:
7090:
7084:
7030:
6773:
6765:
6681:
6504:
5913:
5531:
5169:
5112:
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4598:
4371:
4363:
3995:
3707:
3287:
2912:
2909:
2780:
2314:
2311:
2150:
1987:
1863:
1420:
846:
291:
255:
104:
87:
64:
36:Jan Korringa
33:
24:mathematical
19:
18:
7136:(1): 1–38.
5915:Walter Kohn
7482:Categories
7076:References
5307:, and the
4458:Bloch wave
3288:where the
2913:Since the
7270:122407525
7000:−
6987:δ
6984:−
6933:−
6904:∇
6883:ℏ
6877:−
6782:ψ
6692:δ
6648:δ
6625:δ
6615:α
6609:−
6588:δ
6583:
6577:α
5808:⋅
5783:∑
5645:−
5384:∑
5228:⋅
5215:−
5013:δ
5008:−
4997:−
4984:δ
4754:∑
4632:∑
4560:ψ
4536:⋅
4469:ψ
4160:∑
4141:δ
4136:−
4125:−
4093:δ
4078:δ
4042:∑
3893:α
3821:∑
3725:α
3719:−
3660:∗
3607:α
3565:α
3456:∑
3452:α
3446:−
3423:−
3357:−
3236:α
3164:∑
3119:ϕ
3075:ℏ
3051:α
3003:ψ
2986:α
2971:∇
2871:ϕ
2859:∞
2851:≠
2844:∑
2818:ϕ
2744:≠
2737:∑
2700:∑
2691:, such as
2568:∑
2537:ϕ
2478:ϕ
2429:ϕ
2402:∑
2391:ϕ
2377:ϕ
2345:ϕ
2331:ψ
2277:ϕ
2216:ϕ
2112:ϕ
2097:≠
2090:∑
2079:ϕ
2050:ϕ
1988:The site
1950:ϕ
1919:ϕ
1903:ψ
1818:≠
1811:∑
1774:≠
1767:∑
1730:∑
1706:≠
1699:∑
1662:∑
1637:∑
1589:−
1553:−
1488:≠
1481:∑
1377:≠
1370:∑
1274:∑
1245:∑
1198:∑
1173:∑
1161:leads to
1060:∑
902:∑
724:ϕ
676:ψ
642:−
633:ε
612:−
593:→
590:ε
542:ψ
510:ϕ
496:ψ
462:ϕ
438:−
406:ϕ
349:ψ
332:ψ
308:−
235:∂
217:ψ
214:∂
208:ℏ
186:ψ
152:∇
131:ℏ
125:−
7468:10059425
7390:10011105
7189:Springer
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6972:′
6843:′
6662:′
6639:′
6562:′
6554:′
6486:′
6478:′
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5434:′
5426:′
5400:′
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5113:and the
5058:′
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4928:′
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4832:′
4809:′
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3654:′
3643:′
3617:′
3595:′
3547:′
3503:′
3492:′
3475:′
3464:′
3432:′
3043:, where
2892:⟩
2836:⟩
2540:⟩
2499:⟩
2450:⟩
2394:⟩
2380:⟩
2348:⟩
2334:⟩
2295:⟩
2237:⟩
2133:⟩
2082:⟩
2068:⟩
1971:⟩
1937:⟩
1906:⟩
1520:, where
1025:so that
727:⟩
679:⟩
558:, where
545:⟩
513:⟩
499:⟩
465:⟩
422:, where
409:⟩
365:, where
352:⟩
335:⟩
7448:Bibcode
7413:Bibcode
7370:Bibcode
7335:Bibcode
7296:Bibcode
7288:Physica
7250:Bibcode
7138:Bibcode
7099:Bibcode
7466:
7388:
7268:
7223:
7195:
6682:where
5532:where
4891:being
2609:. The
7266:S2CID
6173:with
7464:PMID
7386:PMID
7221:ISBN
7193:ISBN
6397:and
5714:and
1121:and
7456:doi
7421:doi
7378:doi
7343:doi
7304:doi
7258:doi
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7134:354
7107:doi
7095:238
6736:max
6580:cot
6436:max
6293:max
6257:max
6071:det
2553:or
1421:so
586:lim
81:of
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7047:r
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4851:=
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1685:G
1677:i
1673:t
1666:i
1658:+
1652:i
1648:t
1641:i
1633:=
1630:T
1604:i
1600:v
1592:1
1584:)
1577:+
1574:0
1570:G
1562:i
1558:v
1550:1
1546:(
1540:=
1534:i
1530:t
1508:)
1502:j
1498:Q
1491:i
1485:j
1474:+
1471:0
1467:G
1462:+
1459:1
1456:(
1450:i
1446:t
1441:=
1435:i
1431:Q
1417:,
1403:]
1398:)
1391:j
1387:Q
1380:i
1374:j
1363:+
1360:0
1356:G
1351:+
1348:1
1344:(
1337:i
1333:v
1328:+
1322:i
1318:Q
1310:+
1307:0
1303:G
1295:i
1291:v
1285:[
1278:i
1270:=
1267:)
1260:j
1256:Q
1249:j
1238:+
1235:0
1231:G
1226:+
1223:1
1220:(
1214:i
1210:v
1202:i
1194:=
1188:i
1184:Q
1177:i
1149:T
1129:T
1109:V
1086:i
1082:Q
1075:N
1070:1
1067:=
1064:i
1056:=
1053:T
1033:T
1010:i
1006:Q
981:i
975:R
951:N
928:i
924:v
917:N
912:1
909:=
906:i
898:=
895:V
875:N
855:V
832:.
829:.
826:.
823:+
820:V
814:+
811:0
807:G
802:V
796:+
793:0
789:G
784:V
781:+
778:V
772:+
769:0
765:G
760:V
757:+
754:V
751:=
748:T
721:|
716:)
712:T
706:+
703:0
699:G
694:+
691:1
687:(
683:=
673:|
645:1
637:)
630:i
627:+
621:0
617:H
609:E
605:(
596:0
581:=
575:+
572:0
568:G
539:|
535:V
529:+
526:0
522:G
517:+
507:|
503:=
493:|
472:0
469:=
459:|
454:)
447:0
443:H
435:E
431:(
403:|
379:0
375:H
346:|
342:V
339:=
329:|
324:)
317:0
313:H
305:E
301:(
277:)
272:r
267:(
264:V
238:t
230:)
225:r
220:(
205:i
202:=
199:)
194:r
189:(
182:]
178:)
173:r
168:(
165:V
162:+
156:2
144:m
141:2
135:2
121:[
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