38:
489:
794:
between the center of mass of the nucleus and the electronic cloud when an electric field is present. The amount of that displacement, called the electric dipole moment, is related linearly to the applied field for small fields, but as the magnitude of the field is increased, the field-dipole moment relationship becomes nonlinear, just as in the mechanical system.
507:
817:. Studying vibrating anharmonic systems using quantum mechanics is a computationally demanding task because anharmonicity not only makes the potential experienced by each oscillator more complicated, but also introduces coupling between the oscillators. It is possible to use first-principles methods such as
533:
703:
As a result of the nonlinearity of anharmonic oscillators, the vibration frequency can change, depending upon the system's displacement. These changes in the vibration frequency result in energy being coupled from the fundamental vibration frequency to other frequencies through a process known as
793:
There are many systems throughout the physical world that can be modeled as anharmonic oscillators in addition to the nonlinear mass-spring system. For example, an atom, which consists of a positively charged nucleus surrounded by a negatively charged electronic cloud, experiences a displacement
797:
Further examples of anharmonic oscillators include the large-angle pendulum; nonequilibrium semiconductors that possess a large hot carrier population, which exhibit nonlinear behaviors of various types related to the effective mass of the carriers; and ionospheric plasmas, which also exhibit
813:. However, when the vibrational amplitudes are large, for example at high temperatures, anharmonicity becomes important. An example of the effects of anharmonicity is the thermal expansion of solids, which is usually studied within the
699:
Anharmonic oscillators, however, are characterized by the nonlinear dependence of the restorative force on the displacement x. Consequently, the anharmonic oscillator's period of oscillation may depend on its amplitude of oscillation.
979:
821:
to map the anharmonic potential experienced by the atoms in both molecules and solids. Accurate anharmonic vibrational energies can then be obtained by solving the anharmonic vibrational equations for the atoms within a
802:. In fact, virtually all oscillators become anharmonic when their pump amplitude increases beyond some threshold, and as a result it is necessary to use nonlinear equations of motion to describe their behavior.
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Monserrat, B.; Drummond, N.D.; Needs, R.J. (2013), "Anharmonic vibrational properties in periodic systems: energy, electron-phonon coupling, and stress",
827:
809:. The atoms in a molecule or a solid vibrate about their equilibrium positions. When these vibrations have small amplitudes they can be described by
114:
is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be calculated using
67:
is a good approximation for small oscillations. The red approximation treats the molecule as a harmonic oscillator, because the restoring force,
1153:
1397:
665:
In harmonic oscillators, the restoring force is proportional in magnitude (and opposite in direction) to the displacement of
1339:
Amore, Paolo; Fernández, Francisco M. (2005). "Exact and approximate expressions for the period of anharmonic oscillators".
785:
is small. For this reason, anharmonic motion can be approximated as harmonic motion as long as the oscillations are small.
428:
59:. (Imagine a marble rolling back and forth in the depression.) The blue curve is close in shape to the molecule's actual
1431:
814:
662:
increases, so does the restoring force acting on the pendulums weight that pushes it back towards its resting position.
264:
606:
An oscillator is a physical system characterized by periodic motion, such as a pendulum, tuning fork, or vibrating
818:
332:
554:
back towards the middle. This oscillator is anharmonic because the restoring force is not proportional to
111:
95:
1030:
984:
512:
The "block-on-a-spring" is a classic example of harmonic oscillation. Depending on the block's location,
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518:, it will experience a restoring force toward the middle. The restoring force is proportional to
229:
119:
805:
Anharmonicity plays a role in lattice and molecular vibrations, in quantum oscillations, and in
1149:
1110:
823:
610:. Mathematically speaking, the essential feature of an oscillator is that for some coordinate
607:
162:
139:
1162:
Filipponi, A.; Cavicchia, D. R. (2011), "Anharmonic dynamics of a mass O-spring oscillator",
209:
185:
122:
have to be used. In reality all oscillating systems are anharmonic, but most approximate the
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1179:
258:
41:
1407:
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Jung, J. O.; Benny Gerber, R. (1996), "Vibrational wave functions and spectroscopy of (H
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37:
841:
799:
469:
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60:
45:
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nonlinear behavior based on the anharmonicity of the plasma, transversal oscillating
477:
31:
27:
532:
687:
369:
134:
20:
1318:
974:{\displaystyle T={\sqrt {2m}}\int _{x_{-}}^{x_{+}}{\frac {dx}{\sqrt {E-U(x)}}}}
1142:
1133:
763:
is linear, so it will describe simple harmonic motion. Further, this function
488:
107:
1201:"The Effect of Anharmonicity on Diatomic Vibration: A Spreadsheet Simulation"
806:
304:
127:
19:
This article is about anharmonic oscillators. For the anharmonic ratio, see
1253:=2,3,4,5: Vibrational self-consistent field with correlation corrections",
1225:
1120:
64:
1353:
696:
may oscillate with any amplitude, but will always have the same period.
690:
over time, with a period of oscillation that is inherent to the system.
425:, ... anharmonicity results in additional oscillations with frequencies
649:
may represent the displacement of a pendulum from its resting position
1183:
48:. When the molecules are too close or too far away, they experience a
1274:
472:
of the resonance curve, leading to interesting phenomena such as the
99:
72:
725:
of the displacement of x from its natural position, we may replace
1301:
36:
16:
Deviation of a physical system from being a harmonic oscillator
542:
harmonic oscillator. Depending on the mass's angular position
628:
away from extreme values and back toward some central value
707:
Treating the nonlinear restorative force as a function
492:
2 DOF elastic pendulum exhibiting anharmonic behavior.
206:
of the oscillator, appear. Furthermore, the frequency
1033:
987:
893:
864:
844:
458:{\displaystyle \omega _{\alpha }\pm \omega _{\beta }}
431:
404:
377:
335:
312:
267:
232:
212:
188:
165:
142:
1199:Lim, Kieran F.; Coleman, William F. (August 2005),
680:. The resulting differential equation implies that
1141:
1083:
1019:
973:
879:
850:
616:of the system, a force whose magnitude depends on
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417:
390:
357:
318:
296:{\displaystyle \Delta \omega =\omega -\omega _{0}}
295:
245:
218:
194:
174:
151:
756:at zero displacement. The approximating function
303:is proportional to the square of the oscillation
599:as a harmonic oscillator for small oscillations.
524:, so the system exhibits simple harmonic motion.
261:. As a first approximation, the frequency shift
981:where the extremes of the motion are given by
118:. If the anharmonicity is large, then other
8:
643:to oscillate between extremes. For example,
358:{\displaystyle \Delta \omega \propto A^{2}}
887:. The oscillation period may be derived
1352:
1300:
1224:
1066:
1044:
1032:
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992:
986:
939:
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44:of a diatomic molecule as a function of
1191:
830:to go beyond the mean-field formalism.
253:of the harmonic oscillations. See also
548:, a restoring force pushes coordinate
7:
1396:Elmer, Franz-Josef (July 20, 1998),
578:approximates the nonlinear function
1084:{\displaystyle U(x_{-})=U(x_{+})=E}
1020:{\displaystyle x_{-}<x<x_{+}}
499:Harmonic vs. Anharmonic Oscillators
828:Møller–Plesset perturbation theory
336:
268:
14:
826:. Finally, it is possible to use
391:{\displaystyle \omega _{\alpha }}
1148:(3rd ed.), Pergamon Press,
531:
505:
468:Anharmonicity also modifies the
418:{\displaystyle \omega _{\beta }}
368:In a system of oscillators with
133:As a result, oscillations with
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1059:
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1037:
965:
959:
874:
868:
568:. Because the linear function
1:
815:quasi-harmonic approximation
731:by its linear approximation
595:is small, the system can be
226:deviates from the frequency
1341:European Journal of Physics
1164:American Journal of Physics
858:moving in a potential well
656:. As the absolute value of
246:{\displaystyle \omega _{0}}
110:that is not oscillating in
1448:
1371:10.1088/0143-0807/26/4/004
1319:10.1103/PhysRevB.87.144302
671:from its natural position
25:
18:
819:density-functional theory
175:{\displaystyle 3\omega }
152:{\displaystyle 2\omega }
26:Not to be confused with
538:A pendulum is a simple
219:{\displaystyle \omega }
195:{\displaystyle \omega }
130:of the oscillation is.
1085:
1021:
975:
881:
852:
834:Period of oscillations
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704:parametric coupling.
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460:
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204:fundamental frequency
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154:
40:
1226:10.1021/ed082p1263.1
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985:
891:
880:{\displaystyle U(x)}
862:
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811:harmonic oscillators
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265:
230:
210:
186:
163:
140:
120:numerical techniques
75:with respect to the
1432:Classical mechanics
1404:University of Basel
1399:Nonlinear Resonance
1363:2005EJPh...26..589A
1311:2013PhRvB..87n4302M
1267:1996JChPh.10510332J
1217:2005JChEd..82.1263F
1176:2011AmJPh..79..730F
1116:Nonlinear resonance
1106:Harmonic oscillator
938:
789:Examples in physics
370:natural frequencies
124:harmonic oscillator
116:perturbation theory
104:harmonic oscillator
88:classical mechanics
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1184:10.1119/1.3579129
1155:978-0-08-021022-3
1111:Musical acoustics
969:
968:
908:
851:{\displaystyle m}
824:mean-field theory
772:is accurate when
608:diatomic molecule
484:General principle
319:{\displaystyle A}
259:combination tones
1439:
1418:
1417:
1415:
1410:on June 13, 2011
1406:, archived from
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838:Consider a mass
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126:the smaller the
63:, while the red
42:Potential energy
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1354:math-ph/0409034
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1138:Lifshitz, E. M.
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983:
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941:
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836:
791:
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764:
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738:
732:
726:
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708:
691:
686:must oscillate
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486:
474:foldover effect
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255:intermodulation
233:
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112:harmonic motion
57:
50:restoring force
35:
24:
17:
12:
11:
5:
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1390:External links
1388:
1385:
1384:
1347:(4): 589–601.
1331:
1295:(14): 144302,
1279:
1255:J. Chem. Phys.
1244:
1240:
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1205:J. Chem. Educ.
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470:energy profile
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61:potential well
55:
46:atomic spacing
15:
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1261:(23): 10332,
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1134:Landau, L. D.
1131:
1130:
1126:
1122:
1119:
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1114:
1112:
1109:
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1104:
1102:
1101:Inharmonicity
1099:
1098:
1094:
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1078:
1075:
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1056:
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541:
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490:
483:
481:
479:
478:superharmonic
475:
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383:
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346:
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143:
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131:
129:
125:
121:
117:
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109:
105:
102:from being a
101:
97:
93:
92:anharmonicity
89:
81:
78:
74:
70:
66:
62:
58:
52:back towards
51:
47:
43:
39:
33:
32:Inharmonicity
29:
28:Enharmonicity
22:
1412:, retrieved
1408:the original
1398:
1344:
1340:
1334:
1292:
1289:Phys. Rev. B
1288:
1282:
1258:
1254:
1250:
1245:
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1167:
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688:sinusoidally
682:
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182:etc., where
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77:displacement
68:
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1414:October 28,
1211:(8): 1263,
480:resonance.
135:frequencies
21:Cross-ratio
1127:References
637:, causing
622:will push
108:oscillator
1379:119615357
1327:118687212
1302:1303.0745
1144:Mechanics
1046:−
994:−
954:−
921:−
912:∫
807:acoustics
560:, but to
451:β
447:ω
443:±
438:α
434:ω
411:β
407:ω
384:α
380:ω
343:∝
340:ω
337:Δ
305:amplitude
285:ω
281:−
278:ω
272:ω
269:Δ
235:ω
214:ω
190:ω
170:ω
147:ω
128:amplitude
96:deviation
1426:Category
1140:(1976),
1121:Transmon
1095:See also
741:F′
65:parabola
1359:Bibcode
1307:Bibcode
1263:Bibcode
1213:Bibcode
1172:Bibcode
800:strings
597:modeled
202:is the
94:is the
1377:
1325:
1152:
583:= sin(
100:system
73:linear
69:-V'(u)
1375:S2CID
1349:arXiv
1323:S2CID
1297:arXiv
589:when
106:. An
98:of a
71:, is
1416:2010
1150:ISBN
1027:and
1005:<
999:<
562:sin(
476:and
257:and
159:and
1367:doi
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540:an
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898:=
895:T
875:)
872:x
869:(
866:U
846:m
782:0
779:x
775:x
769:1
766:F
760:1
758:F
754:)
752:0
749:x
747:−
745:x
737:1
734:F
728:F
723:)
721:0
718:x
714:x
712:(
710:F
693:x
683:x
677:0
674:x
668:x
659:x
652:x
646:x
640:x
634:0
631:x
625:x
619:x
613:x
592:θ
587:)
585:θ
581:y
575:θ
571:y
566:)
564:θ
557:θ
551:θ
545:θ
521:x
515:x
351:2
347:A
314:A
289:0
275:=
239:0
167:3
144:2
80:u
56:0
54:u
34:.
23:.
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