7694:
4178:
4194:
4685:
crystals, the phonon waves are longitudinal or transverse only in certain special symmetry directions. In other directions, they can be nearly longitudinal or nearly transverse. It is only for labeling convenience, that they are often called longitudinal or transverse but are actually quasi-longitudinal or quasi-transverse. Note that in the three-dimensional case, there are two directions perpendicular to a straight line at each point on the line. Hence, there are always two (quasi) transverse waves for each (quasi) longitudinal wave.
5938:
propagation direction is comparable to waves on a string. If the wavelength of acoustic phonons goes to infinity, this corresponds to a simple displacement of the whole crystal, and this costs zero deformation energy. Acoustic phonons exhibit a linear relationship between frequency and phonon wave-vector for long wavelengths. The frequencies of acoustic phonons tend to zero with longer wavelength. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively.
9785:
8683:
6327:
4170:
4186:
664:
6026:
2800:
3129:
499:
7252:
7470:
4450:
3414:
652:(i) the total potential energy can be written as a sum of pairwise interactions, and (ii) each atom interacts with only its nearest neighbors. These are used only sparingly in modern lattice dynamics. A more general approach is to express the potential energy in terms of force constants. See, for example, the Wiki article on
4570:, independent of the phonon frequency. As a result, packets of phonons with different (but long) wavelengths can propagate for large distances across the lattice without breaking apart. This is the reason that sound propagates through solids without significant distortion. This behavior fails at large values of
2366:
5752:. Second, each phonon is a "collective mode" caused by the motion of every atom in the lattice. This may be seen from the fact that the creation and annihilation operators, defined here in momentum space, contain sums over the position and momentum operators of every atom when written in position space. (See
2336:
4665:
3p-3 branches will correspond to optical modes. In some special directions, some branches coincide due to symmetry. These branches are called degenerate. In acoustic modes, all the p atoms vibrate in phase. So there is no change in the relative displacements of these atoms during the wave propagation.
2811:
6396:
Thermal phonons can be created and destroyed by random energy fluctuations. In the language of statistical mechanics this means that the chemical potential for adding a phonon is zero. This behavior is an extension of the harmonic potential into the anharmonic regime. The behavior of thermal phonons
4664:
The modes are also referred to as the branches of phonon dispersion. In general, if there are p atoms (denoted by N earlier) in the primitive unit cell, there will be 3p branches of phonon dispersion in a 3-dimensional crystal. Out of these, 3 branches correspond to acoustic modes and the remaining
4680:
The direction of the wave vector is the direction of the wave propagation and the phonon polarization vector gives the direction in which the atoms vibrate. Actually, in general, the wave velocity in a crystal is different for different directions of k. In other words, most crystals are anisotropic
7639:
and negative gravity. This can be explained by how phonons are known to travel faster in denser materials. Because the part of a material pointing towards a gravitational source is closer to the object, it becomes denser on that end. From this, it is predicted that phonons would deflect away as it
7033:
651:
It is important to mention that the mathematical treatment given here is highly simplified in order to make it accessible to non-experts. The simplification has been achieved by making two basic assumptions in the expression for the total potential energy of the crystal. These assumptions are that
4684:
A wave is longitudinal if the atoms vibrate in the same direction as the wave propagation. In a transverse wave, the atoms vibrate perpendicular to the wave propagation. However, except for isotropic crystals, waves in a crystal are not exactly longitudinal or transverse. For general anisotropic
5937:
are coherent movements of atoms of the lattice out of their equilibrium positions. If the displacement is in the direction of propagation, then in some areas the atoms will be closer, in others farther apart, as in a sound wave in air (hence the name acoustic). Displacement perpendicular to the
2070:
488:
The resulting lattice may be visualized as a system of balls connected by springs. The following figure shows a cubic lattice, which is a good model for many types of crystalline solid. Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modeling
6727:
7263:
5914:
4241:
3140:
5070:
6534:
4581:, the dispersion relations exhibit two types of phonons, namely, optical and acoustic modes corresponding to the upper blue and lower red curve in the diagram, respectively. The vertical axis is the energy or frequency of phonon, while the horizontal axis is the
3679:
3537:
6209:
4052:
4944:
5670:
3831:
4699:
differs from the physics of sound in solids, although both are density waves: sound waves in fluids only have longitudinal components, whereas sound waves in solids have longitudinal and transverse components. This is because fluids cannot support
6816:
5506:
1596:
624:
5401:
2795:{\displaystyle {\begin{aligned}\left&=i\hbar \delta _{l,m}\\\left&={\frac {1}{N}}\sum _{l,m}e^{ikal}e^{-ik'am}\left\\&={\frac {i\hbar }{N}}\sum _{l}e^{ial\left(k-k'\right)}=i\hbar \delta _{k,k'}\\\left&=\left=0\end{aligned}}}
724:
Animation showing 6 normal modes of a one-dimensional lattice: a linear chain of particles. The shortest wavelength is at top, with progressively longer wavelengths below. In the lowest lines the motion of the waves to the right can be
449:
are usually imposed. First, the sum is only performed over neighboring atoms. Although the electric forces in real solids extend to infinity, this approximation is still valid because the fields produced by distant atoms are effectively
3957:
3124:{\displaystyle {\begin{aligned}\sum _{l}x_{l}x_{l+m}&={\frac {1}{N}}\sum _{kk'}Q_{k}Q_{k'}\sum _{l}e^{ial\left(k+k'\right)}e^{iamk'}=\sum _{k}Q_{k}Q_{-k}e^{iamk}\\\sum _{l}{p_{l}}^{2}&=\sum _{k}\Pi _{k}\Pi _{-k}\end{aligned}}}
2170:
7247:{\displaystyle {a_{\alpha }}^{\dagger }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }+1}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }+1),n_{\alpha +1}\ldots {\Big \rangle }}
6393:. These energy fluctuations are caused by random lattice vibrations, which can be viewed as a gas of phonons. Because these phonons are generated by the temperature of the lattice, they are sometimes designated thermal phonons.
1067:
5963:
center and show no dispersion near that long wavelength limit. This is because they correspond to a mode of vibration where positive and negative ions at adjacent lattice sites swing against each other, creating a time-varying
1901:
6609:
7640:
detects the difference in densities, exhibiting the qualities of a negative gravitational field. Although the effect would be too small to measure, it is possible that future equipment could lead to successful results.
729:
In order to simplify the analysis needed for a 3-dimensional lattice of atoms, it is convenient to model a 1-dimensional lattice or linear chain. This model is complex enough to display the salient features of phonons.
7465:{\displaystyle a_{\alpha }{\Big |}n_{1}\ldots n_{\alpha -1}n_{\alpha }n_{\alpha +1}\ldots {\Big \rangle }={\sqrt {n_{\alpha }}}{\Big |}n_{1}\ldots ,n_{\alpha -1},(n_{\alpha }-1),n_{\alpha +1},\ldots {\Big \rangle }}
5695:
is the occupation number. This can be seen to be a sum of N independent oscillator
Hamiltonians, each with a unique wave vector, and compatible with the methods used for the quantum harmonic oscillator (note that
414:
7567:
738:
The forces between the atoms are assumed to be linear and nearest-neighbour, and they are represented by an elastic spring. Each atom is assumed to be a point particle and the nucleus and electrons move in step
333:
that depends on the distance of separation of the atoms. The potential energy of the entire lattice is the sum of all pairwise potential energies multiplied by a factor of 1/2 to compensate for double counting:
6405:, wherein photons may be emitted or absorbed by the cavity walls. This similarity is not coincidental, for it turns out that the electromagnetic field behaves like a set of harmonic oscillators, giving rise to
6369:
of a crystal. By the nature of this distribution, the heat capacity is dominated by the high-frequency part of the distribution, while thermal conductivity is primarily the result of the low-frequency region.
1458:
5279:
5707:). When a Hamiltonian can be written as a sum of commuting sub-Hamiltonians, the energy eigenstates will be given by the products of eigenstates of each of the separate sub-Hamiltonians. The corresponding
4445:{\displaystyle \omega _{\pm }^{2}=K\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)\pm K{\sqrt {\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)^{2}-{\frac {4\sin ^{2}{\frac {ka}{2}}}{m_{1}m_{2}}}}},}
3409:{\displaystyle {\tfrac {1}{2}}m\omega ^{2}\sum _{j}\left(x_{j}-x_{j+1}\right)^{2}={\tfrac {1}{2}}m\omega ^{2}\sum _{k}Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})={\tfrac {1}{2}}\sum _{k}m{\omega _{k}}^{2}Q_{k}Q_{-k}}
5772:
8563:
Lizée, Mathieu; Marcotte, Alice; Coquinot, Baptiste; Kavokine, Nikita; Sobnath, Karen; Barraud, Clément; Bhardwaj, Ankit; Radha, Boya; Niguès, Antoine; Bocquet, Lydéric; Siria, Alessandro (2023-02-17).
1183:
6361:
properties of a solid are directly related to its phonon structure. The entire set of all possible phonons that are described by the phonon dispersion relations combine in what is known as the phonon
5980:. Optical phonons are often abbreviated as LO and TO phonons, for the longitudinal and transverse modes respectively; the splitting between LO and TO frequencies is often described accurately by the
4949:
1735:
identical atoms. This is the simplest quantum mechanical model of a lattice that allows phonons to arise from it. The formalism for this model is readily generalizable to two and three dimensions.
6419:
5944:
are out-of-phase movements of the atoms in the lattice, one atom moving to the left, and its neighbor to the right. This occurs if the lattice basis consists of two or more atoms. They are called
2816:
2371:
2175:
721:
7682:
They have been also shown to form “phonon winds” where an electric current in a graphene surface is generated by a liquid flow above it due to the viscous forces at the liquid–solid interface.
7572:
The number operator commutes with a string of products of the creation and annihilation operators if and only if the number of creation operators is equal to number of annihilation operators.
5956:, the electric field of the light will move every positive sodium ion in the direction of the field, and every negative chloride ion in the other direction, causing the crystal to vibrate.
4497:
3548:
3425:
2101:
th atom and the sum is made over the nearest neighbors (nn). However one expects that in a lattice there could also appear waves that behave like particles. It is customary to deal with
6093:
3968:
4832:
8220:
6256:
5571:
1717:
3717:
1809:
1258:
6742:
6583:) where, across gaps up to a nanometer wide, heat can flow via phonons that "tunnel" between two materials. This type of heat transfer works between distances too large for
5406:
4766:
1473:
1307:
4668:
Study of phonon dispersion is useful for modeling propagation of sound waves in solids, which is characterized by phonons. The energy of each phonon, as given earlier, is
511:
5304:
1339:
5563:
4820:
2117:
as variables instead of coordinates of particles. The number of normal modes is the same as the number of particles. Still, the
Fourier space is very useful given the
1842:
115:
The study of phonons is an important part of condensed matter physics. They play a major role in many of the physical properties of condensed matter systems, such as
5536:
4793:
1889:
1763:
2331:{\displaystyle {\begin{aligned}Q_{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{ikal}x_{l}\\\Pi _{k}&={\frac {1}{\sqrt {N}}}\sum _{l}e^{-ikal}p_{l}.\end{aligned}}}
4109:
All quantum systems show wavelike and particlelike properties simultaneously. The particle-like properties of the phonon are best understood using the methods of
3860:
1862:
6413:: in thermal equilibrium and within the harmonic regime, the probability of finding phonons or photons in a given state with a given angular frequency is:
935:
2065:{\displaystyle {\mathcal {H}}=\sum _{i=1}^{N}{\frac {p_{i}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}\sum _{\{ij\}(\mathrm {nn} )}\left(x_{i}-x_{j}\right)^{2}}
7667:
of phonons between the electrons. The evidence that phonons, the vibrations of the ionic lattice, are relevant for superconductivity is provided by the
4161:. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.
6722:{\displaystyle {\mathcal {H}}={\tfrac {1}{2}}\sum _{\alpha }\left(p_{\alpha }^{2}+\omega _{\alpha }^{2}q_{\alpha }^{2}-\hbar \omega _{\alpha }\right)}
3711: + 1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is
7635:
may have a non-negligible mass and be affected by gravity just as standard particles are. In particular, phonons are predicted to have a kind of
9708:
671:
Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions gives rise to a set of vibration
8699:
7937:
6319:, and so forth. Physically, the reciprocal lattice vectors act as additional chunks of momentum which the lattice can impart to the phonon.
340:
7507:
5952:, fluctuations in displacement create an electrical polarization that couples to the electromagnetic field. Hence, they can be excited by
4157:. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like
1354:
9458:
5909:{\displaystyle {\mathcal {H}}=\sum _{k}\sum _{s=1}^{3}\hbar \,\omega _{k,s}\left({b_{k,s}}^{\dagger }b_{k,s}+{\tfrac {1}{2}}\right).}
9696:
1348:(this requires a significant manipulation using the orthonormality and completeness relations of the discrete Fourier transform),
4153:
of the phonons. In the one-dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to
1088:
5565:
into the wavevector space
Hamiltonian, as it is defined above, and simplifying then results in the Hamiltonian taking the form:
5085:
4737:
5065:{\displaystyle {b_{k}}^{\dagger }={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{-k}-{\frac {i}{m\omega _{k}}}\Pi _{k}\right)}
4526:
mode. In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together.
6529:{\displaystyle n\left(\omega _{k,s}\right)={\frac {1}{\exp \left({\dfrac {\hbar \omega _{k,s}}{k_{\mathrm {B} }T}}\right)-1}}}
8314:
8254:
8201:
8169:
8106:
8074:
8042:
7947:
7920:
7881:
7854:
7818:
7652:
5981:
698:
462:. This is permissible as long as the atoms remain close to their equilibrium positions. Formally, this is accomplished by
8738:
8187:
6733:
5919:
This can be interpreted as the sum of 3N independent oscillator
Hamiltonians, one for each wave vector and polarization.
4823:
8017:
9594:
8144:
7992:
7897:
5987:
When measuring optical phonon energy experimentally, optical phonon frequencies are sometimes given in spectroscopic
3674:{\displaystyle {\mathcal {H}}={\frac {1}{2m}}\sum _{k}\left(\Pi _{k}\Pi _{-k}+m^{2}\omega _{k}^{2}Q_{k}Q_{-k}\right)}
8487:
7493:
annihilates one. Hence, they are respectively the creation and annihilation operators for phonons. Analogous to the
6349:. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector.
3532:{\displaystyle \omega _{k}={\sqrt {2\omega ^{2}\left(1-\cos {ka}\right)}}=2\omega \left|\sin {\frac {ka}{2}}\right|}
9675:
6204:{\displaystyle Q_{k}{\stackrel {\mathrm {def} }{=}}Q_{k+K};\quad \Pi _{k}{\stackrel {\mathrm {def} }{=}}\Pi _{k+K}}
4047:{\displaystyle {\tfrac {1}{2}}\hbar \omega ,\ {\tfrac {3}{2}}\hbar \omega ,\ {\tfrac {5}{2}}\hbar \omega \ \cdots }
7693:
7027:), the phonons are said to be non-interacting. The action of the creation and annihilation operators is given by:
6936:", is the state composed of no phonons. Hence, the energy of the ground state is 0. When a system is in the state
4939:{\displaystyle b_{k}={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{k}+{\frac {i}{m\omega _{k}}}\Pi _{-k}\right)}
4462:
6926:
6822:
4117:
1892:
7743:
8497:
5076:
653:
7871:
5665:{\displaystyle {\mathcal {H}}=\sum _{k}\hbar \omega _{k}\left({b_{k}}^{\dagger }b_{k}+{\tfrac {1}{2}}\right)}
4716:
The above-derived
Hamiltonian may look like a classical Hamiltonian function, but if it is interpreted as an
648:
th atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted (nn).
5745:, since any number of identical excitations can be created by repeated application of the creation operator
4098:
must be supplied to the harmonic oscillator lattice to push it to the next energy level. By analogy to the
9691:
8331:
8273:
7753:
6410:
6220:
5675:
This is known as the second quantization technique, also known as the occupation number formulation, where
4689:
4566:(i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately
3826:{\displaystyle k=k_{n}={\frac {2\pi n}{Na}}\quad {\mbox{for }}n=0,\pm 1,\pm 2,\ldots \pm {\frac {N}{2}}.\ }
234:
of vibration. Normal modes are important because any arbitrary lattice vibration can be considered to be a
8687:
6881:. Because the difference in energy is what we measure and not the absolute value of it, the constant term
1684:
9819:
7679:
In 2019, researchers were able to isolate individual phonons without destroying them for the first time.
7494:
5753:
4081:
6811:{\displaystyle {\mathcal {H}}=\sum _{\alpha }\hbar \omega _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }}
1768:
1214:
7616:
5501:{\displaystyle \Pi _{k}=i{\sqrt {\frac {\hbar m\omega _{k}}{2}}}\left({b_{k}}^{\dagger }-b_{-k}\right)}
2136:
1591:{\displaystyle Q_{k}=A_{k}e^{i\omega _{k}t};\qquad \omega _{k}={\sqrt {{\frac {2C}{m}}(1-\cos {ka})}}.}
1076:
9767:
6878:
619:{\displaystyle \sum _{\{ij\}(\mathrm {nn} )}{\tfrac {1}{2}}m\omega ^{2}\left(R_{i}-R_{j}\right)^{2}.}
255:
179:
6033:(red) do not carry any more information than their counterparts (black) in the first Brillouin zone.
4747:
4578:
9179:
8461:
7498:
6000:
5965:
5396:{\displaystyle Q_{k}={\sqrt {\frac {\hbar }{2m\omega _{k}}}}\left({b_{k}}^{\dagger }+b_{-k}\right)}
1263:
451:
310:
271:
66:
6079:
and integer multiples thereof). For example, in the one-dimensional model, the normal coordinates
445:
explicitly in either classical or quantum mechanics. In order to simplify the task, two important
9809:
9735:
9370:
8997:
8731:
8693:
7620:
5766:
This technique is readily generalized to three dimensions, where the
Hamiltonian takes the form:
5711:
325:
forces are generally negligible. The forces between each pair of atoms may be characterized by a
235:
219:
120:
17:
1312:
9670:
9007:
8705:
6402:
4658:
2360:
This choice retains the desired commutation relations in either real space or wavevector space
704:
Not every possible lattice vibration has a well-defined wavelength and frequency. However, the
147:
93:
8306:
8300:
8062:
7663:
by a weak attractive force. In a conventional superconductor, this attraction is caused by an
5541:
4798:
9757:
8390:
Reiter, D. E.; Sauer, S.; Huneke, J.; Papenkort, T.; Kuhn, T.; Vagov, A.; Axt, V. M. (2009).
8094:
7773:
6042:
5741:
Three important properties of phonons may be deduced from this technique. First, phonons are
4103:
1814:
693:
between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than 2
679:
of the wave is given by the displacements of the atoms from their equilibrium positions. The
50:
4649: – 3. The lower figure shows the dispersion relations for several phonon modes in
3952:{\displaystyle E_{n}=\left({\tfrac {1}{2}}+n\right)\hbar \omega _{k}\qquad n=0,1,2,3\ldots }
8711:
8644:
8587:
8527:
8407:
8403:
8356:
7783:
7713:
7668:
6406:
5759:
5514:
4771:
4721:
4150:
3703:
The form of the quantization depends on the choice of boundary conditions; for simplicity,
1867:
1741:
1464:
637:
of the harmonic potentials, which are assumed to be the same since the lattice is regular.
116:
54:
8193:
8:
9772:
8755:
8420:
8391:
4729:
4515:
4110:
1720:
1629:
459:
302:
251:
227:
8648:
8591:
8531:
8411:
8360:
697:, due to the periodicity of the lattice. This can be thought of as a consequence of the
675:
propagating through the lattice. One such wave is shown in the figure to the right. The
297:
for a typical sample of a solid. Since the lattice is rigid, the atoms must be exerting
100:
for elastic structures of interacting particles. Phonons can be thought of as quantized
9715:
8724:
8660:
8634:
8603:
8577:
8545:
8372:
8346:
7981:
7891:
7738:
7659:
are expelled from the material. In a superconductor, electrons are bound together into
6584:
6560:
6076:
5953:
5731:
respectively create and destroy a single field excitation, a phonon, with an energy of
4717:
3693:
1847:
124:
90:
8565:
7671:, the dependence of the superconducting critical temperature on the mass of the ions.
4193:
9784:
9752:
9665:
9144:
8889:
8816:
8664:
8607:
8549:
8493:
8310:
8250:
8197:
8165:
8140:
8102:
8070:
8038:
8013:
7988:
7943:
7916:
7877:
7850:
7814:
7733:
7723:
7648:
6588:
6576:
6362:
4638:
4177:
4154:
4077:
1633:
1075:
Since the solutions are expected to be oscillatory, new coordinates are defined by a
1062:{\displaystyle -2Cu_{n}+C\left(u_{n+1}+u_{n-1}\right)=m{\frac {d^{2}u_{n}}{dt^{2}}}.}
740:
634:
490:
442:
211:
97:
35:
8376:
5995:
represents ordinary frequency (not angular frequency), and is expressed in units of
1738:
In contrast to the previous section, the positions of the masses are not denoted by
667:
Phonon propagating through a square lattice (atom displacements greatly exaggerated)
301:
on one another to keep each atom near its equilibrium position. These forces may be
9762:
9533:
9308:
9169:
9154:
8836:
8747:
8652:
8595:
8535:
8518:
8415:
8364:
7758:
6848:
6020:
5977:
5704:
326:
243:
39:
689:
There is a minimum possible wavelength, given by twice the equilibrium separation
9814:
9730:
9655:
9639:
9579:
8989:
8914:
8904:
8894:
8806:
7942:. International publishing locations: Oxford University Press. pp. 231–232.
7728:
6852:
6390:
5949:
4733:
4711:
4642:
4158:
1644:
701:, the lattice points being viewed as the "sampling points" of a continuous wave.
294:
282:
7969:. Solid State Physics. Vol. Supplement 3 (Second ed.). New York: Academic Press.
6003:. In other words, the wave-number in cm units corresponds to the inverse of the
5714:
is then given by the sum of the individual eigenvalues of the sub-Hamiltonians.
3684:
The couplings between the position variables have been transformed away; if the
9789:
9703:
9660:
9396:
9184:
8957:
8879:
8874:
8796:
8656:
8599:
8540:
8513:
7778:
7699:
7656:
6977:
is the occupation number of the phonons. The energy of a single phonon of type
6358:
6345:
6320:
6030:
5960:
4634:
4626:
4559:
4530:
2118:
314:
270:
of quantum mechanics but instead use relations for which there exists a direct
8368:
9803:
9747:
9599:
9566:
9358:
9328:
9260:
9119:
8899:
8826:
8811:
8392:"Generation of squeezed phonon states by optical excitation of a quantum dot"
7748:
7664:
7636:
6592:
6374:
6366:
2106:
463:
446:
306:
143:
86:
78:
4744:
without directly solving the differential equations. Given the
Hamiltonian,
4213:
For a one-dimensional alternating array of two types of ion or atom of mass
9725:
9275:
9265:
9255:
9016:
8967:
8909:
8831:
8786:
8332:"Phonon–phonon interactions due to non-linear effects in a linear ion trap"
7873:
Stalin's great science : the times and adventures of Soviet physicists
7768:
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6378:
4705:
4701:
4533:
in the lattice, is given by the slope of the acoustic dispersion relation,
2110:
720:
131:
8622:
8069:. Physics and Materials Properties (4th ed.). Springer. p. 111.
1608:
represents an independent vibrational mode of the lattice with wavenumber
9740:
9508:
9411:
9406:
9323:
9318:
9248:
9202:
9159:
9124:
9083:
8975:
8939:
8801:
8351:
7660:
6382:
4741:
4574:, i.e. short wavelengths, due to the microscopic details of the lattice.
1613:
705:
278:
231:
109:
101:
8462:"Researchers suggest phonons may have mass and perhaps negative gravity"
6330:
Brillouin zones, (a) in a square lattice, and (b) in a hexagonal lattice
5075:
The following commutators can be easily obtained by substituting in the
9482:
9376:
9366:
9348:
9238:
9139:
9074:
8791:
8716:
8566:"Strong Electronic Winds Blowing under Liquid Flows on Carbon Surfaces"
6398:
6046:
6004:
5996:
5988:
4582:
4132:
4127:
This may be generalized to a three-dimensional lattice. The wavenumber
2354:
2346:
2114:
1637:
809:···o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o···
680:
3840:
comes from the minimum wavelength, which is twice the lattice spacing
409:{\displaystyle {\frac {1}{2}}\sum _{i\neq j}V\left(r_{i}-r_{j}\right)}
9624:
9614:
9584:
9477:
9443:
9436:
9313:
9303:
9298:
9270:
9038:
8821:
8063:"Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes"
7788:
7562:{\displaystyle N=\sum _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }.}
6269:
is thus equivalent to an infinite family of phonons with wavenumbers
5928:
5763:, it can be shown that phonons act as waves of lattice displacement.
709:
676:
318:
223:
215:
191:
157:
135:
6903:
can be ignored without changing the equations of motion. Hence, the
5931:
exhibit two types of phonons: acoustic phonons and optical phonons.
5298:
can be inverted to redefine the conjugate position and momentum as:
4106:
is quantized, the quantum of vibrational energy is called a phonon.
901:
is the distance between atoms when the chain is in equilibrium, and
9720:
9548:
9503:
9487:
9448:
9421:
9134:
9129:
9109:
9079:
9069:
9064:
8884:
8859:
8854:
8781:
8639:
8582:
8448:
8435:
6053:
4169:
2094:
1453:{\displaystyle 2C(\cos {ka-1})Q_{k}=m{\frac {d^{2}Q_{k}}{dt^{2}}}.}
62:
8708:
5968:. Optical phonons that interact in this way with light are called
4529:
The speed of propagation of an acoustic phonon, which is also the
3847:
The harmonic oscillator eigenvalues or energy levels for the mode
293:
is a large number, say of the order of 10, or on the order of the
9629:
9619:
9589:
9543:
9538:
9513:
9431:
9416:
9343:
9338:
9243:
9228:
9174:
9149:
9114:
9043:
9025:
8764:
8682:
8447:
Angelo
Esposito, Rafael Krichevsky, and Alberto Nicolis. (2018).
7718:
7612:
6342:| in their "family". The set of all such wavevectors defines the
6326:
4185:
3696:(which they are not), the transformed Hamiltonian would describe
322:
186:
in the same way that photons represent wave-particle duality for
105:
82:
7575:
It can be shown that phonons are symmetric under exchange (i.e.
4459:
is the wavevector of the vibration related to its wavelength by
1864:
is at its equilibrium position.) In two or more dimensions, the
1731:
A one-dimensional quantum mechanical harmonic chain consists of
1344:
Substitution into the equation of motion produces the following
9604:
9233:
9220:
9211:
9033:
8947:
8846:
7600:
6925:
factor is absent in the operator formalized expression for the
6386:
6038:
6008:
5708:
4099:
4092:
663:
74:
31:
8225:
University of
Cambridge Teaching and Learning Packages Library
6025:
6011:
in vacuum that has the same frequency as the measured phonon.
4822:
defined in the quantum treatment section above, we can define
4712:
Interpretation of phonons using second quantization techniques
9634:
9574:
9426:
9285:
9164:
9104:
9059:
8952:
8930:
8773:
8562:
8095:"§2.1.3 Normal modes of a one-dimensional chain with a basis"
7915:(1st ed.). Oxford: Oxford University Press. p. 82.
7763:
7708:
7632:
5742:
4725:
4696:
4209:) for some waves corresponding to lattice vibrations in GaAs.
2079:
is the mass of each atom (assuming it is equal for all), and
427:
298:
267:
187:
183:
171:
151:
70:
7965:
Maradudin, A.; Montroll, E.; Weiss, G.; Ipatova, I. (1971).
6988:
and the total energy of a general phonon system is given by
481:. The error in ignoring higher order terms remains small if
9401:
9333:
9293:
8869:
8864:
6552:
is the frequency of the phonons (or photons) in the state,
6075:
is only determined up to addition of constant vectors (the
5717:
As with the quantum harmonic oscillator, one can show that
4650:
4181:
Optical and acoustic vibrations in a linear diatomic chain.
2102:
1178:{\displaystyle u_{n}=\sum _{Nak/2\pi =1}^{N}Q_{k}e^{ikna}.}
672:
247:
58:
8389:
8215:
8213:
6825:
in operator formalism, we have not taken into account the
5274:{\displaystyle \left=\delta _{k,k'},\quad {\Big }=\left=0}
4189:
Vibrations of the diatomic chain at different frequencies.
505:
The potential energy of the lattice may now be written as
8492:. Mineola, New York: Dover Publications, Inc. p. 9.
8329:
5999:. The value is obtained by dividing the frequency by the
925:
the mass of the atom, then the equation of motion of the
8330:
Marquet, C.; Schmidt-Kaler, F.; James, D. F. V. (2003).
7967:
Theory of lattice dynamics in the harmonic approximation
6591:
to occur and therefore cannot be explained by classical
6334:
It is usually convenient to consider phonon wavevectors
498:
8210:
469:
about its equilibrium value to quadratic order, giving
222:
of atoms or molecules uniformly oscillates at a single
6624:
6063:
is not actually a physical momentum; it is called the
5887:
5646:
4473:
4021:
3997:
3973:
3883:
3765:
3542:
3340:
3229:
3145:
549:
289:
particles. These particles may be atoms or molecules.
8268:
8266:
7510:
7266:
7036:
6745:
6612:
6470:
6422:
6223:
6096:
5775:
5574:
5544:
5517:
5409:
5307:
5088:
4952:
4835:
4801:
4774:
4750:
4465:
4244:
3971:
3863:
3720:
3551:
3428:
3143:
2814:
2369:
2173:
1904:
1870:
1850:
1817:
1771:
1744:
1687:
1476:
1357:
1315:
1266:
1217:
1091:
938:
514:
343:
313:, and others, all of which are ultimately due to the
7983:
A guide to Feynman Diagrams in the many-body problem
7689:
4688:
Many phonon dispersion curves have been measured by
1811:
as measured from their equilibrium positions. (I.e.
1194:
corresponds and devolves to the continuous variable
8623:"Secret of Flow-Induced Electric Currents Revealed"
8436:
Mutual Interactions of Phonons, Rotons, and Gravity
6385:has an energy that is not constant, but fluctuates
4672:The velocity of the wave also is given in terms of
8704:Phonons in a One Dimensional Microfluidic Crystal
8263:
8130:
8128:
8126:
8124:
8122:
8120:
8118:
7980:
7939:Statistical mechanics: algorithms and computations
7876:. London: Imperial College Press. pp. 64–69.
7561:
7464:
7246:
6810:
6721:
6528:
6381:, and contains no phonons. A lattice at a nonzero
6250:
6203:
5908:
5664:
5557:
5530:
5500:
5395:
5273:
5064:
4938:
4814:
4787:
4760:
4491:
4444:
4046:
3951:
3825:
3673:
3531:
3408:
3123:
2794:
2330:
2064:
1883:
1856:
1836:
1803:
1757:
1711:
1590:
1452:
1333:
1301:
1252:
1177:
1061:
618:
408:
190:. Solids with more than one atom in the smallest
8139:. Saunders College Publishing. pp. 780–783.
7457:
7368:
7344:
7279:
7239:
7153:
7123:
7058:
6059:; however, this is not strictly correct, because
5976:can also interact indirectly with light, through
5959:Optical phonons have a non-zero frequency at the
5201:
5166:
4577:For a crystal that has at least two atoms in its
4502:The connection between frequency and wavevector,
4142:is now associated with three normal coordinates.
199:
195:
130:The concept of phonons was introduced in 1930 by
9801:
7929:
8115:
8010:Theoretical Mechanics of Particles and Continua
8008:Fetter, Alexander; Walecka, John (2003-12-16).
5927:Solids with more than one atom in the smallest
5922:
4708:fluids, which only apply to high frequencies).
4118:Canonical quantization § Real scalar field
3707:boundary conditions are imposed, defining the (
250:phenomena in classical mechanics, phonons have
170:, because long-wavelength phonons give rise to
8514:"Detecting the softest sounds in the Universe"
8134:
7849:. Cambridge University Press. pp. 78–96.
7014: +.... As there are no cross terms (e.g.
174:. The name emphasizes the analogy to the word
8732:
8007:
477:and the elastic force simply proportional to
8434:Alberto Nicolis and Riccardo Penco. (2017).
8249:(3 ed.). India: Elsevier. p. 201.
8244:
8135:Ashcroft, Neil W.; Mermin, N. David (1976).
7840:
7838:
7836:
7834:
7832:
7830:
5511:Directly substituting these definitions for
4492:{\displaystyle k={\tfrac {2\pi }{\lambda }}}
4122:
2003:
1994:
529:
520:
7869:
7631:Recent research has shown that phonons and
6377:temperature, a crystal lattice lies in its
485:remains close to the equilibrium position.
438:is the potential energy between two atoms.
8739:
8725:
8305:. Reading, MA: Benjamin-Cummings. p.
8056:
8054:
7844:
4231:, connected by springs of spring constant
4173:Dispersion curves in linear diatomic chain
1723:, an assembly of independent oscillators.
921:is the elastic constant of the spring and
715:
8638:
8581:
8539:
8419:
8350:
8164:. Princeton: Princeton University Press.
8060:
7827:
7651:is a state of electronic matter in which
5820:
4518:. The plus sign results in the so-called
4113:and operator techniques described later.
2157:defined as the Fourier transforms of the
489:phonons. (For other common lattices, see
266:The equations in this section do not use
8746:
8620:
8438:, Arxiv.org, Retrieved November 27, 2018
8302:Statistical Mechanics, A Set of Lectures
8088:
8086:
6325:
6024:
4192:
4184:
4176:
4168:
719:
708:do possess well-defined wavelengths and
662:
254:properties too, in a way related to the
8485:
8298:
8051:
7978:
7813:(4th ed.). Springer. p. 253.
7808:
7626:
1463:These are the equations for decoupled
914:th atom from its equilibrium position.
14:
9802:
9210:
8506:
8184:
8061:Yu, Peter Y.; Cardona, Manuel (2010).
7935:
4633: ≥ 2 different atoms in the
4164:
733:
27:Quasiparticle of mechanical vibrations
8720:
8396:Journal of Physics: Conference Series
8159:
8153:
8092:
8083:
8032:
7910:
7845:Girvin, Steven M.; Yang, Kun (2019).
6598:
6251:{\displaystyle K={\frac {2n\pi }{a}}}
4768:, as well as the conjugate position,
8026:
7961:
7959:
7643:
7599:), so therefore they are considered
6323:obey a similar set of restrictions.
4227:repeated periodically at a distance
1726:
1712:{\displaystyle \omega (k)\propto ka}
1072:This is a set of coupled equations.
8188:Introduction to Solid State Physics
6932:The ground state, also called the "
6734:creation and annihilation operators
6603:The phonon Hamiltonian is given by
6575:Phonons have been shown to exhibit
6570:
6338:which have the smallest magnitude |
6014:
5948:because in ionic crystals, such as
4824:creation and annihilation operators
4637:exhibits three acoustic modes: one
4131:is replaced by a three-dimensional
261:
24:
6748:
6615:
6500:
6186:
6177:
6174:
6171:
6152:
6123:
6120:
6117:
5778:
5577:
5546:
5411:
5048:
4919:
4803:
4753:
4740:, is a means of extracting energy
4645:. The number of optical modes is 3
3603:
3593:
3554:
3105:
3095:
2763:
2750:
2459:
2251:
2135:may be introduced, defined as the
2013:
2010:
1907:
1804:{\displaystyle x_{1},x_{2},\dots }
1719:. This amounts to classical free
1253:{\displaystyle \phi _{k}=e^{ikna}}
644:is the position coordinate of the
539:
536:
25:
9831:
8675:
8489:Introduction to Superconductivity
7956:
7674:
6766:
6701:
6352:
6045:, phonons have been treated with
5817:
5595:
4997:
4871:
4235:, two modes of vibration result:
4032:
4008:
3984:
3962:The levels are evenly spaced at:
3905:
2671:
2606:
2417:
2097:operators, respectively, for the
473:proportional to the displacement
9783:
9676:Timeline of particle discoveries
8681:
8621:Schirber, Michael (2023-02-17).
7692:
4522:mode, and the minus sign to the
3700:uncoupled harmonic oscillators.
699:Nyquist–Shannon sampling theorem
658:
497:
8614:
8556:
8479:
8454:
8441:
8428:
8383:
8323:
8292:
8238:
8178:
7847:Modern Condensed Matter Physics
7606:
6150:
5163:
3918:
3763:
1526:
8421:10.1088/1742-6596/193/1/012121
8162:Condensed Matter in a Nutshell
8101:. Academic Press. p. 44.
8093:Misra, Prasanta Kumar (2010).
8067:Fundamentals of Semiconductors
8001:
7972:
7904:
7863:
7802:
7427:
7408:
7212:
7193:
6877:terms will add up yielding an
6029:k-vectors exceeding the first
5077:canonical commutation relation
4761:{\displaystyle {\mathcal {H}}}
3333:
3286:
2017:
2006:
1697:
1691:
1580:
1557:
1387:
1364:
1296:
1287:
543:
532:
441:It is difficult to solve this
277:For example: a rigid regular,
13:
1:
8274:"Tunneling across a tiny gap"
8221:"Non-metals: thermal phonons"
7936:Krauth, Werner (April 2006).
7913:The Oxford solid state basics
7896:: CS1 maint: date and year (
7795:
5982:Lyddane–Sachs–Teller relation
3134:The potential energy term is
1302:{\displaystyle k=2\pi j/(Na)}
1079:, in order to decouple them.
654:multiscale Green's functions.
214:description of an elementary
205:
9692:History of subatomic physics
8299:Feynman, Richard P. (1982).
7754:Relativistic heat conduction
5923:Acoustic and optical phonons
4738:quantum harmonic oscillators
4690:inelastic neutron scattering
4653:as a function of wavevector
1198:of scalar field theory. The
178:, in that phonons represent
158:
7:
8526:(7763): 8–9. July 1, 2019.
8451:Retrieved November 11, 2018
8099:Physics of Condensed Matter
7870:Kozhevnikov, A. B. (2004).
7685:
7615:, phonons can interact via
7495:quantum harmonic oscillator
6587:to occur but too small for
6265:. A phonon with wavenumber
5991:notation, where the symbol
5972:. Optical phonons that are
5754:position and momentum space
4082:quantum harmonic oscillator
2137:discrete Fourier transforms
1891:are vector quantities. The
454:. Secondly, the potentials
10:
9836:
8700:Optical and acoustic modes
8657:10.1103/PhysRevX.13.011020
8600:10.1103/PhysRevX.13.011020
8541:10.1038/d41586-019-02009-5
8012:. Dover Books on Physics.
7811:Advanced Quantum Mechanics
7617:parametric down conversion
6077:reciprocal lattice vectors
6018:
5284:Using this, the operators
4732:technique, similar to the
4639:longitudinal acoustic mode
4115:
1334:{\displaystyle j=1\dots N}
1211:for continuum field modes
1077:discrete Fourier transform
893:th atom out of a total of
246:). While normal modes are
152:
123:, as well as in models of
29:
9781:
9684:
9648:
9565:
9526:
9496:
9470:
9466:
9457:
9389:
9357:
9284:
9219:
9201:
9097:
9052:
9024:
9015:
9006:
8988:
8966:
8938:
8929:
8845:
8772:
8763:
8754:
8486:Tinkham, Michael (1996).
8369:10.1007/s00340-003-1097-7
7911:Simon, Steven H. (2013).
7482:creates a phonon of type
4795:, and conjugate momentum
4643:transverse acoustic modes
4123:Three-dimensional lattice
1619:The second equation, for
311:electrostatic attractions
285:) lattice is composed of
9709:mathematical formulation
9304:Eta and eta prime mesons
8185:Kittel, Charles (2004).
7621:squeezed coherent states
7499:particle number operator
6821:Here, in expressing the
6411:Bose–Einstein statistics
6001:speed of light in vacuum
5966:electrical dipole moment
5558:{\displaystyle \Pi _{k}}
4815:{\displaystyle \Pi _{k}}
4695:The physics of sound in
4681:for phonon propagation.
2805:From the general result
1467:which have the solution
910:the displacement of the
274:in classical mechanics.
30:Not to be confused with
9371:Double-charm tetraquark
8245:Pathria; Beale (2011).
7809:Schwabl, Franz (2008).
7475:The creation operator,
4661:of its Brillouin zone.
4625:are those of the first
1837:{\displaystyle x_{i}=0}
1681:, a scalar field, and
1601:Each normal coordinate
716:One-dimensional lattice
162:), which translates to
121:electrical conductivity
8686:Quotations related to
8160:Mahan, Gerald (2010).
8037:. New York: Springer.
7563:
7466:
7248:
6812:
6723:
6530:
6409:. Both gases obey the
6403:electromagnetic cavity
6331:
6252:
6205:
6034:
5910:
5816:
5756:.) Finally, using the
5666:
5559:
5532:
5502:
5397:
5275:
5066:
4940:
4816:
4789:
4762:
4720:, then it describes a
4493:
4446:
4210:
4190:
4182:
4174:
4048:
3953:
3844:, as discussed above.
3827:
3675:
3533:
3410:
3125:
2796:
2332:
2066:
1935:
1885:
1858:
1838:
1805:
1759:
1713:
1612:, which is known as a
1592:
1454:
1335:
1303:
1254:
1179:
1142:
1063:
726:
668:
620:
410:
258:of quantum mechanics.
9768:Wave–particle duality
9758:Relativistic particle
8895:Electron antineutrino
8247:Statistical Mechanics
8035:Many-Particle Physics
8033:Mahan, G. D. (1981).
7774:Surface acoustic wave
7653:electrical resistance
7564:
7467:
7249:
6813:
6736:, these are given by
6724:
6531:
6365:which determines the
6329:
6253:
6206:
6028:
5911:
5796:
5667:
5560:
5533:
5531:{\displaystyle Q_{k}}
5503:
5398:
5276:
5067:
4941:
4817:
4790:
4788:{\displaystyle Q_{k}}
4763:
4585:. The boundaries at −
4494:
4447:
4196:
4188:
4180:
4172:
4104:electromagnetic field
4049:
3954:
3828:
3676:
3534:
3411:
3126:
2797:
2349:of the phonon, i.e. 2
2333:
2128:"normal coordinates"
2093:are the position and
2067:
1915:
1886:
1884:{\displaystyle x_{i}}
1859:
1839:
1806:
1760:
1758:{\displaystyle u_{i}}
1714:
1593:
1455:
1336:
1304:
1255:
1180:
1105:
1064:
723:
666:
621:
411:
256:wave–particle duality
242:vibration modes (cf.
180:wave-particle duality
127:and related effects.
51:collective excitation
8998:Faddeev–Popov ghosts
8748:Particles in physics
8404:Institute of Physics
7979:Mattuck, R. (1976).
7784:Thermal conductivity
7714:Brillouin scattering
7627:Predicted properties
7508:
7497:case, we can define
7264:
7034:
6743:
6610:
6567:is the temperature.
6420:
6407:black-body radiation
6221:
6094:
6087:are defined so that
5773:
5760:correlation function
5572:
5542:
5515:
5407:
5305:
5086:
4950:
4833:
4799:
4772:
4748:
4722:quantum field theory
4659:principal directions
4562:.) At low values of
4463:
4242:
4197:Dispersion relation
4149:= 1, 2, 3 label the
4138:. Furthermore, each
3969:
3861:
3718:
3549:
3426:
3141:
2812:
2367:
2345:turns out to be the
2171:
2150:"conjugate momenta"
1902:
1868:
1848:
1815:
1769:
1742:
1685:
1474:
1465:harmonic oscillators
1355:
1313:
1264:
1215:
1089:
936:
512:
341:
303:Van der Waals forces
146:. It comes from the
117:thermal conductivity
9773:Particle chauvinism
9716:Subatomic particles
8649:2023PhRvX..13a1020L
8592:2023PhRvX..13a1020L
8532:2019Natur.571....8.
8412:2009JPhCS.193a2121R
8361:2003ApPhB..76..199M
8137:Solid State Physics
6959:, we say there are
6697:
6682:
6664:
6052:as though it has a
4730:second quantization
4724:of non-interacting
4516:dispersion relation
4259:
4165:Dispersion relation
4111:second quantization
3836:The upper bound to
3642:
1952:
1895:for this system is
1721:scalar field theory
1630:dispersion relation
1346:decoupled equations
734:Classical treatment
460:harmonic potentials
228:classical mechanics
98:modes of vibrations
9820:1932 introductions
8694:Explained: Phonons
8192:. Wiley. pp.
7739:Carrier scattering
7559:
7526:
7462:
7244:
6808:
6765:
6719:
6683:
6668:
6650:
6644:
6633:
6599:Operator formalism
6561:Boltzmann constant
6526:
6511:
6397:is similar to the
6332:
6248:
6201:
6071:. This is because
6035:
5954:infrared radiation
5906:
5896:
5795:
5758:position–position
5662:
5655:
5594:
5555:
5528:
5498:
5403: and
5393:
5271:
5062:
4946: and
4936:
4812:
4785:
4758:
4489:
4487:
4442:
4245:
4211:
4191:
4183:
4175:
4155:longitudinal waves
4044:
4030:
4006:
3982:
3949:
3892:
3823:
3769:
3671:
3628:
3586:
3529:
3406:
3360:
3349:
3262:
3238:
3178:
3154:
3121:
3119:
3093:
3057:
3001:
2924:
2889:
2828:
2792:
2790:
2624:
2510:
2328:
2326:
2288:
2216:
2062:
2021:
1938:
1881:
1854:
1834:
1801:
1755:
1709:
1628:, is known as the
1588:
1450:
1331:
1299:
1250:
1209:normal coordinates
1175:
1059:
727:
669:
616:
558:
547:
406:
369:
230:this designates a
218:motion in which a
212:quantum mechanical
125:neutron scattering
91:quantum mechanical
69:, specifically in
9797:
9796:
9753:Massless particle
9561:
9560:
9557:
9556:
9522:
9521:
9385:
9384:
9197:
9196:
9193:
9192:
9145:Magnetic monopole
9093:
9092:
8984:
8983:
8925:
8924:
8905:Muon antineutrino
8890:Electron neutrino
8696:, MIT News, 2010.
8570:Physical Review X
8449:The mass of sound
8339:Applied Physics B
8316:978-0-8053-2508-9
8256:978-93-80931-89-0
8203:978-0-471-41526-8
8171:978-0-691-14016-2
8108:978-0-12-384954-0
8076:978-3-642-00709-5
8044:978-0-306-46338-9
7949:978-0-19-851536-4
7922:978-0-19-968077-1
7883:978-1-86094-419-2
7856:978-1-107-13739-4
7820:978-3-540-85062-5
7734:Phonon scattering
7724:Linear elasticity
7649:Superconductivity
7644:Superconductivity
7517:
7364:
7149:
6847:term as, given a
6756:
6635:
6632:
6577:quantum tunneling
6524:
6510:
6363:density of states
6246:
6182:
6128:
5895:
5786:
5654:
5585:
5451:
5450:
5346:
5345:
5045:
5002:
5001:
4916:
4876:
4875:
4629:. A crystal with
4514:), is known as a
4486:
4437:
4435:
4410:
4362:
4342:
4306:
4286:
4078:zero-point energy
4040:
4029:
4019:
4005:
3995:
3981:
3891:
3822:
3815:
3768:
3761:
3577:
3575:
3522:
3487:
3351:
3348:
3253:
3237:
3169:
3153:
3084:
3048:
2992:
2915:
2872:
2870:
2819:
2615:
2613:
2495:
2493:
2279:
2277:
2276:
2207:
2205:
2204:
1989:
1974:
1961:
1857:{\displaystyle i}
1765:, but instead by
1727:Quantum treatment
1634:angular frequency
1583:
1555:
1445:
1207:are known as the
1054:
741:adiabatic theorem
635:natural frequency
557:
515:
491:crystal structure
443:many-body problem
354:
352:
142:was suggested by
85:, a phonon is an
36:Phonon (software)
16:(Redirected from
9827:
9787:
9763:Virtual particle
9534:Mesonic molecule
9468:
9467:
9464:
9463:
9309:Bottom eta meson
9217:
9216:
9208:
9207:
9180:W′ and Z′ bosons
9170:Sterile neutrino
9155:Majorana fermion
9022:
9021:
9013:
9012:
8936:
8935:
8915:Tau antineutrino
8770:
8769:
8761:
8760:
8741:
8734:
8727:
8718:
8717:
8685:
8669:
8668:
8642:
8618:
8612:
8611:
8585:
8560:
8554:
8553:
8543:
8510:
8504:
8503:
8483:
8477:
8476:
8474:
8472:
8458:
8452:
8445:
8439:
8432:
8426:
8425:
8423:
8387:
8381:
8380:
8354:
8352:quant-ph/0211079
8336:
8327:
8321:
8320:
8296:
8290:
8289:
8287:
8285:
8270:
8261:
8260:
8242:
8236:
8235:
8233:
8231:
8217:
8208:
8207:
8182:
8176:
8175:
8157:
8151:
8150:
8132:
8113:
8112:
8090:
8081:
8080:
8058:
8049:
8048:
8030:
8024:
8023:
8005:
7999:
7998:
7986:
7976:
7970:
7963:
7954:
7953:
7933:
7927:
7926:
7908:
7902:
7901:
7895:
7887:
7867:
7861:
7860:
7842:
7825:
7824:
7806:
7759:Rigid unit modes
7744:Phononic crystal
7702:
7697:
7696:
7598:
7586:
7568:
7566:
7565:
7560:
7555:
7554:
7545:
7544:
7539:
7538:
7537:
7525:
7471:
7469:
7468:
7463:
7461:
7460:
7448:
7447:
7420:
7419:
7404:
7403:
7382:
7381:
7372:
7371:
7365:
7363:
7362:
7353:
7348:
7347:
7338:
7337:
7322:
7321:
7312:
7311:
7293:
7292:
7283:
7282:
7276:
7275:
7253:
7251:
7250:
7245:
7243:
7242:
7233:
7232:
7205:
7204:
7189:
7188:
7167:
7166:
7157:
7156:
7150:
7142:
7141:
7132:
7127:
7126:
7117:
7116:
7101:
7100:
7091:
7090:
7072:
7071:
7062:
7061:
7055:
7054:
7049:
7048:
7047:
6966:phonons of type
6958:
6918:
6916:
6915:
6912:
6909:
6896:
6894:
6893:
6890:
6887:
6870:
6868:
6867:
6864:
6861:
6853:infinite lattice
6840:
6838:
6837:
6834:
6831:
6817:
6815:
6814:
6809:
6807:
6806:
6797:
6796:
6791:
6790:
6789:
6778:
6777:
6764:
6752:
6751:
6732:In terms of the
6728:
6726:
6725:
6720:
6718:
6714:
6713:
6712:
6696:
6691:
6681:
6676:
6663:
6658:
6643:
6634:
6625:
6619:
6618:
6581:phonon tunneling
6571:Phonon tunneling
6535:
6533:
6532:
6527:
6525:
6523:
6516:
6512:
6509:
6505:
6504:
6503:
6492:
6491:
6490:
6471:
6454:
6449:
6445:
6444:
6318:
6316:
6315:
6310:
6307:
6306:
6293:
6291:
6290:
6285:
6282:
6281:
6261:for any integer
6257:
6255:
6254:
6249:
6247:
6242:
6231:
6210:
6208:
6207:
6202:
6200:
6199:
6184:
6183:
6181:
6180:
6168:
6163:
6160:
6159:
6146:
6145:
6130:
6129:
6127:
6126:
6114:
6109:
6106:
6105:
6065:crystal momentum
6021:Crystal momentum
6015:Crystal momentum
5978:Raman scattering
5935:Acoustic phonons
5915:
5913:
5912:
5907:
5902:
5898:
5897:
5888:
5882:
5881:
5866:
5865:
5860:
5859:
5858:
5836:
5835:
5815:
5810:
5794:
5782:
5781:
5671:
5669:
5668:
5663:
5661:
5657:
5656:
5647:
5641:
5640:
5631:
5630:
5625:
5624:
5623:
5607:
5606:
5593:
5581:
5580:
5564:
5562:
5561:
5556:
5554:
5553:
5537:
5535:
5534:
5529:
5527:
5526:
5507:
5505:
5504:
5499:
5497:
5493:
5492:
5491:
5476:
5475:
5470:
5469:
5468:
5452:
5446:
5445:
5444:
5428:
5427:
5419:
5418:
5402:
5400:
5399:
5394:
5392:
5388:
5387:
5386:
5371:
5370:
5365:
5364:
5363:
5347:
5344:
5343:
5342:
5323:
5322:
5317:
5316:
5280:
5278:
5277:
5272:
5264:
5260:
5259:
5258:
5253:
5252:
5251:
5250:
5232:
5231:
5226:
5225:
5224:
5205:
5204:
5198:
5197:
5196:
5180:
5179:
5170:
5169:
5159:
5158:
5157:
5135:
5131:
5130:
5129:
5124:
5123:
5122:
5121:
5103:
5102:
5071:
5069:
5068:
5063:
5061:
5057:
5056:
5055:
5046:
5044:
5043:
5042:
5026:
5021:
5020:
5003:
5000:
4992:
4991:
4990:
4977:
4976:
4971:
4970:
4965:
4964:
4963:
4945:
4943:
4942:
4937:
4935:
4931:
4930:
4929:
4917:
4915:
4914:
4913:
4897:
4892:
4891:
4877:
4874:
4866:
4865:
4864:
4851:
4850:
4845:
4844:
4821:
4819:
4818:
4813:
4811:
4810:
4794:
4792:
4791:
4786:
4784:
4783:
4767:
4765:
4764:
4759:
4757:
4756:
4736:method used for
4624:
4622:
4621:
4616:
4613:
4612:
4604:
4602:
4601:
4596:
4593:
4592:
4557:
4555:
4554:
4548:
4545:
4498:
4496:
4495:
4490:
4488:
4482:
4474:
4451:
4449:
4448:
4443:
4438:
4436:
4434:
4433:
4432:
4423:
4422:
4412:
4411:
4406:
4398:
4393:
4392:
4379:
4374:
4373:
4368:
4364:
4363:
4361:
4360:
4348:
4343:
4341:
4340:
4328:
4320:
4312:
4308:
4307:
4305:
4304:
4292:
4287:
4285:
4284:
4272:
4258:
4253:
4159:transverse waves
4145:The new indices
4072:
4070:
4069:
4066:
4063:
4053:
4051:
4050:
4045:
4038:
4031:
4022:
4017:
4007:
3998:
3993:
3983:
3974:
3958:
3956:
3955:
3950:
3917:
3916:
3904:
3900:
3893:
3884:
3873:
3872:
3832:
3830:
3829:
3824:
3820:
3816:
3808:
3770:
3766:
3762:
3760:
3752:
3741:
3736:
3735:
3680:
3678:
3677:
3672:
3670:
3666:
3665:
3664:
3652:
3651:
3641:
3636:
3627:
3626:
3614:
3613:
3601:
3600:
3585:
3576:
3574:
3563:
3558:
3557:
3538:
3536:
3535:
3530:
3528:
3524:
3523:
3518:
3510:
3488:
3486:
3482:
3481:
3456:
3455:
3443:
3438:
3437:
3415:
3413:
3412:
3407:
3405:
3404:
3392:
3391:
3382:
3381:
3376:
3375:
3374:
3359:
3350:
3341:
3332:
3331:
3310:
3309:
3285:
3284:
3272:
3271:
3261:
3252:
3251:
3239:
3230:
3224:
3223:
3218:
3214:
3213:
3212:
3194:
3193:
3177:
3168:
3167:
3155:
3146:
3130:
3128:
3127:
3122:
3120:
3116:
3115:
3103:
3102:
3092:
3076:
3075:
3070:
3069:
3068:
3056:
3043:
3042:
3024:
3023:
3011:
3010:
3000:
2988:
2987:
2986:
2964:
2963:
2962:
2958:
2957:
2923:
2914:
2913:
2912:
2899:
2898:
2888:
2887:
2871:
2863:
2854:
2853:
2838:
2837:
2827:
2801:
2799:
2798:
2793:
2791:
2781:
2777:
2776:
2775:
2774:
2758:
2757:
2736:
2732:
2731:
2730:
2729:
2713:
2712:
2694:
2693:
2692:
2664:
2663:
2662:
2658:
2657:
2623:
2614:
2609:
2601:
2593:
2589:
2585:
2584:
2583:
2571:
2570:
2556:
2555:
2548:
2529:
2528:
2509:
2494:
2486:
2477:
2473:
2472:
2471:
2470:
2454:
2453:
2435:
2434:
2406:
2402:
2401:
2400:
2388:
2387:
2352:
2337:
2335:
2334:
2329:
2327:
2320:
2319:
2310:
2309:
2287:
2278:
2272:
2268:
2259:
2258:
2245:
2244:
2235:
2234:
2215:
2206:
2200:
2196:
2187:
2186:
2071:
2069:
2068:
2063:
2061:
2060:
2055:
2051:
2050:
2049:
2037:
2036:
2020:
2016:
1988:
1987:
1975:
1967:
1962:
1960:
1951:
1946:
1937:
1934:
1929:
1911:
1910:
1890:
1888:
1887:
1882:
1880:
1879:
1863:
1861:
1860:
1855:
1843:
1841:
1840:
1835:
1827:
1826:
1810:
1808:
1807:
1802:
1794:
1793:
1781:
1780:
1764:
1762:
1761:
1756:
1754:
1753:
1718:
1716:
1715:
1710:
1680:
1669:
1660:
1654:
1650:
1627:
1611:
1597:
1595:
1594:
1589:
1584:
1579:
1556:
1551:
1543:
1541:
1536:
1535:
1522:
1521:
1517:
1516:
1499:
1498:
1486:
1485:
1459:
1457:
1456:
1451:
1446:
1444:
1443:
1442:
1429:
1428:
1427:
1418:
1417:
1407:
1399:
1398:
1386:
1340:
1338:
1337:
1332:
1308:
1306:
1305:
1300:
1286:
1259:
1257:
1256:
1251:
1249:
1248:
1227:
1226:
1206:
1197:
1193:
1184:
1182:
1181:
1176:
1171:
1170:
1152:
1151:
1141:
1136:
1123:
1101:
1100:
1068:
1066:
1065:
1060:
1055:
1053:
1052:
1051:
1038:
1037:
1036:
1027:
1026:
1016:
1008:
1004:
1003:
1002:
984:
983:
957:
956:
928:
924:
913:
909:
900:
896:
892:
888:
858:
849:
834:
830:
790:
784:
780:
773:
767:
625:
623:
622:
617:
612:
611:
606:
602:
601:
600:
588:
587:
572:
571:
559:
550:
546:
542:
501:
464:Taylor expanding
415:
413:
412:
407:
405:
401:
400:
399:
387:
386:
368:
353:
345:
327:potential energy
262:Lattice dynamics
244:Fourier analysis
210:A phonon is the
161:
155:
154:
67:condensed matter
40:Phonon (company)
21:
9835:
9834:
9830:
9829:
9828:
9826:
9825:
9824:
9800:
9799:
9798:
9793:
9777:
9731:Nuclear physics
9680:
9644:
9580:Davydov soliton
9553:
9518:
9492:
9453:
9381:
9353:
9280:
9189:
9089:
9048:
9002:
8980:
8962:
8921:
8841:
8750:
8745:
8710:with movies in
8678:
8673:
8672:
8619:
8615:
8561:
8557:
8512:
8511:
8507:
8500:
8484:
8480:
8470:
8468:
8460:
8459:
8455:
8446:
8442:
8433:
8429:
8388:
8384:
8334:
8328:
8324:
8317:
8297:
8293:
8283:
8281:
8272:
8271:
8264:
8257:
8243:
8239:
8229:
8227:
8219:
8218:
8211:
8204:
8183:
8179:
8172:
8158:
8154:
8147:
8133:
8116:
8109:
8091:
8084:
8077:
8059:
8052:
8045:
8031:
8027:
8020:
8006:
8002:
7995:
7987:. McGraw-Hill.
7977:
7973:
7964:
7957:
7950:
7934:
7930:
7923:
7909:
7905:
7889:
7888:
7884:
7868:
7864:
7857:
7843:
7828:
7821:
7807:
7803:
7798:
7793:
7729:Mechanical wave
7698:
7691:
7688:
7677:
7657:magnetic fields
7646:
7629:
7609:
7588:
7576:
7550:
7546:
7540:
7533:
7529:
7528:
7527:
7521:
7509:
7506:
7505:
7491:
7480:
7456:
7455:
7437:
7433:
7415:
7411:
7393:
7389:
7377:
7373:
7367:
7366:
7358:
7354:
7352:
7343:
7342:
7327:
7323:
7317:
7313:
7301:
7297:
7288:
7284:
7278:
7277:
7271:
7267:
7265:
7262:
7261:
7238:
7237:
7222:
7218:
7200:
7196:
7178:
7174:
7162:
7158:
7152:
7151:
7137:
7133:
7131:
7122:
7121:
7106:
7102:
7096:
7092:
7080:
7076:
7067:
7063:
7057:
7056:
7050:
7043:
7039:
7038:
7037:
7035:
7032:
7031:
7026:
7020:
7013:
7007:
7000:
6994:
6986:
6975:
6964:
6956:
6950:
6944:
6937:
6923:
6913:
6910:
6907:
6906:
6904:
6901:
6891:
6888:
6885:
6884:
6882:
6875:
6865:
6862:
6859:
6858:
6856:
6845:
6835:
6832:
6829:
6828:
6826:
6802:
6798:
6792:
6785:
6781:
6780:
6779:
6773:
6769:
6760:
6747:
6746:
6744:
6741:
6740:
6708:
6704:
6692:
6687:
6677:
6672:
6659:
6654:
6649:
6645:
6639:
6623:
6614:
6613:
6611:
6608:
6607:
6601:
6573:
6558:
6551:
6499:
6498:
6494:
6493:
6480:
6476:
6472:
6469:
6465:
6458:
6453:
6434:
6430:
6426:
6421:
6418:
6417:
6401:produced by an
6355:
6321:Bloch electrons
6311:
6308:
6304:
6302:
6301:
6299:
6286:
6283:
6279:
6277:
6276:
6274:
6232:
6230:
6222:
6219:
6218:
6189:
6185:
6170:
6169:
6164:
6162:
6161:
6155:
6151:
6135:
6131:
6116:
6115:
6110:
6108:
6107:
6101:
6097:
6095:
6092:
6091:
6023:
6017:
5970:infrared active
5950:sodium chloride
5942:Optical phonons
5925:
5886:
5871:
5867:
5861:
5848:
5844:
5843:
5842:
5841:
5837:
5825:
5821:
5811:
5800:
5790:
5777:
5776:
5774:
5771:
5770:
5750:
5736:
5729:
5722:
5701:
5693:
5687:
5680:
5645:
5636:
5632:
5626:
5619:
5615:
5614:
5613:
5612:
5608:
5602:
5598:
5589:
5576:
5575:
5573:
5570:
5569:
5549:
5545:
5543:
5540:
5539:
5522:
5518:
5516:
5513:
5512:
5484:
5480:
5471:
5464:
5460:
5459:
5458:
5457:
5453:
5440:
5436:
5429:
5426:
5414:
5410:
5408:
5405:
5404:
5379:
5375:
5366:
5359:
5355:
5354:
5353:
5352:
5348:
5338:
5334:
5327:
5321:
5312:
5308:
5306:
5303:
5302:
5296:
5289:
5254:
5243:
5242:
5238:
5237:
5236:
5227:
5220:
5216:
5215:
5214:
5213:
5209:
5200:
5199:
5189:
5188:
5184:
5175:
5171:
5165:
5164:
5150:
5143:
5139:
5125:
5114:
5113:
5109:
5108:
5107:
5098:
5094:
5093:
5089:
5087:
5084:
5083:
5051:
5047:
5038:
5034:
5030:
5025:
5013:
5009:
5008:
5004:
4993:
4986:
4982:
4978:
4975:
4966:
4959:
4955:
4954:
4953:
4951:
4948:
4947:
4922:
4918:
4909:
4905:
4901:
4896:
4887:
4883:
4882:
4878:
4867:
4860:
4856:
4852:
4849:
4840:
4836:
4834:
4831:
4830:
4806:
4802:
4800:
4797:
4796:
4779:
4775:
4773:
4770:
4769:
4752:
4751:
4749:
4746:
4745:
4734:ladder operator
4714:
4617:
4614:
4610:
4609:
4608:
4606:
4597:
4594:
4590:
4589:
4588:
4586:
4549:
4546:
4543:
4537:
4536:
4534:
4475:
4472:
4464:
4461:
4460:
4428:
4424:
4418:
4414:
4413:
4399:
4397:
4388:
4384:
4380:
4378:
4369:
4356:
4352:
4347:
4336:
4332:
4327:
4326:
4322:
4321:
4319:
4300:
4296:
4291:
4280:
4276:
4271:
4270:
4266:
4254:
4249:
4243:
4240:
4239:
4226:
4219:
4167:
4125:
4120:
4067:
4064:
4061:
4060:
4058:
4020:
3996:
3972:
3970:
3967:
3966:
3912:
3908:
3882:
3881:
3877:
3868:
3864:
3862:
3859:
3858:
3852:
3807:
3764:
3753:
3742:
3740:
3731:
3727:
3719:
3716:
3715:
3657:
3653:
3647:
3643:
3637:
3632:
3622:
3618:
3606:
3602:
3596:
3592:
3591:
3587:
3581:
3567:
3562:
3553:
3552:
3550:
3547:
3546:
3511:
3509:
3502:
3498:
3474:
3461:
3457:
3451:
3447:
3442:
3433:
3429:
3427:
3424:
3423:
3397:
3393:
3387:
3383:
3377:
3370:
3366:
3365:
3364:
3355:
3339:
3318:
3314:
3299:
3295:
3277:
3273:
3267:
3263:
3257:
3247:
3243:
3228:
3219:
3202:
3198:
3189:
3185:
3184:
3180:
3179:
3173:
3163:
3159:
3144:
3142:
3139:
3138:
3118:
3117:
3108:
3104:
3098:
3094:
3088:
3077:
3071:
3064:
3060:
3059:
3058:
3052:
3045:
3044:
3029:
3025:
3016:
3012:
3006:
3002:
2996:
2979:
2969:
2965:
2950:
2943:
2939:
2929:
2925:
2919:
2905:
2904:
2900:
2894:
2890:
2880:
2876:
2862:
2855:
2843:
2839:
2833:
2829:
2823:
2815:
2813:
2810:
2809:
2789:
2788:
2767:
2766:
2762:
2753:
2749:
2748:
2744:
2737:
2722:
2721:
2717:
2708:
2704:
2703:
2699:
2696:
2695:
2685:
2678:
2674:
2650:
2643:
2639:
2629:
2625:
2619:
2602:
2600:
2591:
2590:
2579:
2575:
2566:
2562:
2561:
2557:
2541:
2534:
2530:
2515:
2511:
2499:
2485:
2478:
2463:
2462:
2458:
2449:
2445:
2444:
2440:
2437:
2436:
2424:
2420:
2407:
2396:
2392:
2383:
2379:
2378:
2374:
2370:
2368:
2365:
2364:
2353:divided by the
2350:
2325:
2324:
2315:
2311:
2293:
2289:
2283:
2267:
2260:
2254:
2250:
2247:
2246:
2240:
2236:
2221:
2217:
2211:
2195:
2188:
2182:
2178:
2174:
2172:
2169:
2168:
2162:
2155:
2144:
2133:
2121:of the system.
2091:
2084:
2056:
2045:
2041:
2032:
2028:
2027:
2023:
2022:
2009:
1993:
1983:
1979:
1966:
1953:
1947:
1942:
1936:
1930:
1919:
1906:
1905:
1903:
1900:
1899:
1875:
1871:
1869:
1866:
1865:
1849:
1846:
1845:
1822:
1818:
1816:
1813:
1812:
1789:
1785:
1776:
1772:
1770:
1767:
1766:
1749:
1745:
1743:
1740:
1739:
1729:
1686:
1683:
1682:
1671:
1667:
1662:
1656:
1652:
1648:
1645:continuum limit
1625:
1620:
1609:
1606:
1572:
1544:
1542:
1540:
1531:
1527:
1512:
1508:
1504:
1500:
1494:
1490:
1481:
1477:
1475:
1472:
1471:
1438:
1434:
1430:
1423:
1419:
1413:
1409:
1408:
1406:
1394:
1390:
1373:
1356:
1353:
1352:
1314:
1311:
1310:
1282:
1265:
1262:
1261:
1235:
1231:
1222:
1218:
1216:
1213:
1212:
1204:
1199:
1195:
1189:
1157:
1153:
1147:
1143:
1137:
1119:
1109:
1096:
1092:
1090:
1087:
1086:
1047:
1043:
1039:
1032:
1028:
1022:
1018:
1017:
1015:
992:
988:
973:
969:
968:
964:
952:
948:
937:
934:
933:
926:
922:
911:
907:
902:
898:
894:
890:
886:
867:
856:
854:
847:
846:
832:
828:
788:
782:
778:
771:
765:
736:
718:
661:
642:
607:
596:
592:
583:
579:
578:
574:
573:
567:
563:
548:
535:
519:
513:
510:
509:
458:are treated as
424:
395:
391:
382:
378:
377:
373:
358:
344:
342:
339:
338:
295:Avogadro number
264:
208:
57:arrangement of
53:in a periodic,
43:
28:
23:
22:
15:
12:
11:
5:
9833:
9823:
9822:
9817:
9812:
9810:Quasiparticles
9795:
9794:
9790:Physics portal
9782:
9779:
9778:
9776:
9775:
9770:
9765:
9760:
9755:
9750:
9745:
9744:
9743:
9733:
9728:
9723:
9718:
9713:
9712:
9711:
9704:Standard Model
9701:
9700:
9699:
9688:
9686:
9682:
9681:
9679:
9678:
9673:
9671:Quasiparticles
9668:
9663:
9658:
9652:
9650:
9646:
9645:
9643:
9642:
9637:
9632:
9627:
9622:
9617:
9612:
9607:
9602:
9597:
9592:
9587:
9582:
9577:
9571:
9569:
9567:Quasiparticles
9563:
9562:
9559:
9558:
9555:
9554:
9552:
9551:
9546:
9541:
9536:
9530:
9528:
9524:
9523:
9520:
9519:
9517:
9516:
9511:
9506:
9500:
9498:
9494:
9493:
9491:
9490:
9485:
9480:
9474:
9472:
9461:
9455:
9454:
9452:
9451:
9446:
9441:
9440:
9439:
9434:
9429:
9424:
9419:
9414:
9404:
9399:
9393:
9391:
9387:
9386:
9383:
9382:
9380:
9379:
9374:
9363:
9361:
9359:Exotic hadrons
9355:
9354:
9352:
9351:
9346:
9341:
9336:
9331:
9326:
9321:
9316:
9311:
9306:
9301:
9296:
9290:
9288:
9282:
9281:
9279:
9278:
9273:
9268:
9263:
9258:
9253:
9252:
9251:
9246:
9241:
9236:
9225:
9223:
9214:
9205:
9199:
9198:
9195:
9194:
9191:
9190:
9188:
9187:
9185:X and Y bosons
9182:
9177:
9172:
9167:
9162:
9157:
9152:
9147:
9142:
9137:
9132:
9127:
9122:
9117:
9112:
9107:
9101:
9099:
9095:
9094:
9091:
9090:
9088:
9087:
9077:
9072:
9067:
9062:
9056:
9054:
9050:
9049:
9047:
9046:
9041:
9036:
9030:
9028:
9019:
9010:
9004:
9003:
9001:
9000:
8994:
8992:
8986:
8985:
8982:
8981:
8979:
8978:
8972:
8970:
8964:
8963:
8961:
8960:
8958:W and Z bosons
8955:
8950:
8944:
8942:
8933:
8927:
8926:
8923:
8922:
8920:
8919:
8918:
8917:
8912:
8907:
8902:
8897:
8892:
8882:
8877:
8872:
8867:
8862:
8857:
8851:
8849:
8843:
8842:
8840:
8839:
8834:
8829:
8824:
8819:
8814:
8812:Strange (quark
8809:
8804:
8799:
8794:
8789:
8784:
8778:
8776:
8767:
8758:
8752:
8751:
8744:
8743:
8736:
8729:
8721:
8715:
8714:
8702:
8697:
8691:
8677:
8676:External links
8674:
8671:
8670:
8613:
8555:
8505:
8498:
8478:
8453:
8440:
8427:
8382:
8345:(3): 199–208.
8322:
8315:
8291:
8280:. 7 April 2015
8262:
8255:
8237:
8209:
8202:
8177:
8170:
8152:
8145:
8114:
8107:
8082:
8075:
8050:
8043:
8025:
8019:978-0486432618
8018:
8000:
7993:
7971:
7955:
7948:
7928:
7921:
7903:
7882:
7862:
7855:
7826:
7819:
7800:
7799:
7797:
7794:
7792:
7791:
7786:
7781:
7779:Surface phonon
7776:
7771:
7766:
7761:
7756:
7751:
7746:
7741:
7736:
7731:
7726:
7721:
7716:
7711:
7705:
7704:
7703:
7700:Physics portal
7687:
7684:
7676:
7675:Other research
7673:
7669:isotope effect
7645:
7642:
7628:
7625:
7608:
7605:
7570:
7569:
7558:
7553:
7549:
7543:
7536:
7532:
7524:
7520:
7516:
7513:
7489:
7478:
7473:
7472:
7459:
7454:
7451:
7446:
7443:
7440:
7436:
7432:
7429:
7426:
7423:
7418:
7414:
7410:
7407:
7402:
7399:
7396:
7392:
7388:
7385:
7380:
7376:
7370:
7361:
7357:
7351:
7346:
7341:
7336:
7333:
7330:
7326:
7320:
7316:
7310:
7307:
7304:
7300:
7296:
7291:
7287:
7281:
7274:
7270:
7255:
7254:
7241:
7236:
7231:
7228:
7225:
7221:
7217:
7214:
7211:
7208:
7203:
7199:
7195:
7192:
7187:
7184:
7181:
7177:
7173:
7170:
7165:
7161:
7155:
7148:
7145:
7140:
7136:
7130:
7125:
7120:
7115:
7112:
7109:
7105:
7099:
7095:
7089:
7086:
7083:
7079:
7075:
7070:
7066:
7060:
7053:
7046:
7042:
7024:
7018:
7011:
7005:
6998:
6992:
6984:
6973:
6962:
6954:
6948:
6942:
6921:
6899:
6873:
6843:
6819:
6818:
6805:
6801:
6795:
6788:
6784:
6776:
6772:
6768:
6763:
6759:
6755:
6750:
6730:
6729:
6717:
6711:
6707:
6703:
6700:
6695:
6690:
6686:
6680:
6675:
6671:
6667:
6662:
6657:
6653:
6648:
6642:
6638:
6631:
6628:
6622:
6617:
6600:
6597:
6572:
6569:
6556:
6543:
6537:
6536:
6522:
6519:
6515:
6508:
6502:
6497:
6489:
6486:
6483:
6479:
6475:
6468:
6464:
6461:
6457:
6452:
6448:
6443:
6440:
6437:
6433:
6429:
6425:
6354:
6353:Thermodynamics
6351:
6346:Brillouin zone
6259:
6258:
6245:
6241:
6238:
6235:
6229:
6226:
6212:
6211:
6198:
6195:
6192:
6188:
6179:
6176:
6173:
6167:
6158:
6154:
6149:
6144:
6141:
6138:
6134:
6125:
6122:
6119:
6113:
6104:
6100:
6069:pseudomomentum
6037:By analogy to
6031:Brillouin zone
6019:Main article:
6016:
6013:
5961:Brillouin zone
5924:
5921:
5917:
5916:
5905:
5901:
5894:
5891:
5885:
5880:
5877:
5874:
5870:
5864:
5857:
5854:
5851:
5847:
5840:
5834:
5831:
5828:
5824:
5819:
5814:
5809:
5806:
5803:
5799:
5793:
5789:
5785:
5780:
5748:
5734:
5727:
5720:
5699:
5691:
5685:
5678:
5673:
5672:
5660:
5653:
5650:
5644:
5639:
5635:
5629:
5622:
5618:
5611:
5605:
5601:
5597:
5592:
5588:
5584:
5579:
5552:
5548:
5525:
5521:
5509:
5508:
5496:
5490:
5487:
5483:
5479:
5474:
5467:
5463:
5456:
5449:
5443:
5439:
5435:
5432:
5425:
5422:
5417:
5413:
5391:
5385:
5382:
5378:
5374:
5369:
5362:
5358:
5351:
5341:
5337:
5333:
5330:
5326:
5320:
5315:
5311:
5294:
5287:
5282:
5281:
5270:
5267:
5263:
5257:
5249:
5246:
5241:
5235:
5230:
5223:
5219:
5212:
5208:
5203:
5195:
5192:
5187:
5183:
5178:
5174:
5168:
5162:
5156:
5153:
5149:
5146:
5142:
5138:
5134:
5128:
5120:
5117:
5112:
5106:
5101:
5097:
5092:
5073:
5072:
5060:
5054:
5050:
5041:
5037:
5033:
5029:
5024:
5019:
5016:
5012:
5007:
4999:
4996:
4989:
4985:
4981:
4974:
4969:
4962:
4958:
4934:
4928:
4925:
4921:
4912:
4908:
4904:
4900:
4895:
4890:
4886:
4881:
4873:
4870:
4863:
4859:
4855:
4848:
4843:
4839:
4809:
4805:
4782:
4778:
4755:
4713:
4710:
4702:shear stresses
4635:primitive cell
4627:Brillouin zone
4579:primitive cell
4560:group velocity
4541:
4531:speed of sound
4485:
4481:
4478:
4471:
4468:
4453:
4452:
4441:
4431:
4427:
4421:
4417:
4409:
4405:
4402:
4396:
4391:
4387:
4383:
4377:
4372:
4367:
4359:
4355:
4351:
4346:
4339:
4335:
4331:
4325:
4318:
4315:
4311:
4303:
4299:
4295:
4290:
4283:
4279:
4275:
4269:
4265:
4262:
4257:
4252:
4248:
4224:
4217:
4166:
4163:
4124:
4121:
4102:case when the
4055:
4054:
4043:
4037:
4034:
4028:
4025:
4016:
4013:
4010:
4004:
4001:
3992:
3989:
3986:
3980:
3977:
3960:
3959:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3915:
3911:
3907:
3903:
3899:
3896:
3890:
3887:
3880:
3876:
3871:
3867:
3850:
3834:
3833:
3819:
3814:
3811:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3759:
3756:
3751:
3748:
3745:
3739:
3734:
3730:
3726:
3723:
3682:
3681:
3669:
3663:
3660:
3656:
3650:
3646:
3640:
3635:
3631:
3625:
3621:
3617:
3612:
3609:
3605:
3599:
3595:
3590:
3584:
3580:
3573:
3570:
3566:
3561:
3556:
3540:
3539:
3527:
3521:
3517:
3514:
3508:
3505:
3501:
3497:
3494:
3491:
3485:
3480:
3477:
3473:
3470:
3467:
3464:
3460:
3454:
3450:
3446:
3441:
3436:
3432:
3417:
3416:
3403:
3400:
3396:
3390:
3386:
3380:
3373:
3369:
3363:
3358:
3354:
3347:
3344:
3338:
3335:
3330:
3327:
3324:
3321:
3317:
3313:
3308:
3305:
3302:
3298:
3294:
3291:
3288:
3283:
3280:
3276:
3270:
3266:
3260:
3256:
3250:
3246:
3242:
3236:
3233:
3227:
3222:
3217:
3211:
3208:
3205:
3201:
3197:
3192:
3188:
3183:
3176:
3172:
3166:
3162:
3158:
3152:
3149:
3132:
3131:
3114:
3111:
3107:
3101:
3097:
3091:
3087:
3083:
3080:
3078:
3074:
3067:
3063:
3055:
3051:
3047:
3046:
3041:
3038:
3035:
3032:
3028:
3022:
3019:
3015:
3009:
3005:
2999:
2995:
2991:
2985:
2982:
2978:
2975:
2972:
2968:
2961:
2956:
2953:
2949:
2946:
2942:
2938:
2935:
2932:
2928:
2922:
2918:
2911:
2908:
2903:
2897:
2893:
2886:
2883:
2879:
2875:
2869:
2866:
2861:
2858:
2856:
2852:
2849:
2846:
2842:
2836:
2832:
2826:
2822:
2818:
2817:
2803:
2802:
2787:
2784:
2780:
2773:
2770:
2765:
2761:
2756:
2752:
2747:
2743:
2740:
2738:
2735:
2728:
2725:
2720:
2716:
2711:
2707:
2702:
2698:
2697:
2691:
2688:
2684:
2681:
2677:
2673:
2670:
2667:
2661:
2656:
2653:
2649:
2646:
2642:
2638:
2635:
2632:
2628:
2622:
2618:
2612:
2608:
2605:
2599:
2596:
2594:
2592:
2588:
2582:
2578:
2574:
2569:
2565:
2560:
2554:
2551:
2547:
2544:
2540:
2537:
2533:
2527:
2524:
2521:
2518:
2514:
2508:
2505:
2502:
2498:
2492:
2489:
2484:
2481:
2479:
2476:
2469:
2466:
2461:
2457:
2452:
2448:
2443:
2439:
2438:
2433:
2430:
2427:
2423:
2419:
2416:
2413:
2410:
2408:
2405:
2399:
2395:
2391:
2386:
2382:
2377:
2373:
2372:
2339:
2338:
2323:
2318:
2314:
2308:
2305:
2302:
2299:
2296:
2292:
2286:
2282:
2275:
2271:
2266:
2263:
2261:
2257:
2253:
2249:
2248:
2243:
2239:
2233:
2230:
2227:
2224:
2220:
2214:
2210:
2203:
2199:
2194:
2191:
2189:
2185:
2181:
2177:
2176:
2160:
2153:
2142:
2131:
2089:
2082:
2073:
2072:
2059:
2054:
2048:
2044:
2040:
2035:
2031:
2026:
2019:
2015:
2012:
2008:
2005:
2002:
1999:
1996:
1992:
1986:
1982:
1978:
1973:
1970:
1965:
1959:
1956:
1950:
1945:
1941:
1933:
1928:
1925:
1922:
1918:
1914:
1909:
1878:
1874:
1853:
1833:
1830:
1825:
1821:
1800:
1797:
1792:
1788:
1784:
1779:
1775:
1752:
1748:
1728:
1725:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1665:
1623:
1604:
1599:
1598:
1587:
1582:
1578:
1575:
1571:
1568:
1565:
1562:
1559:
1554:
1550:
1547:
1539:
1534:
1530:
1525:
1520:
1515:
1511:
1507:
1503:
1497:
1493:
1489:
1484:
1480:
1461:
1460:
1449:
1441:
1437:
1433:
1426:
1422:
1416:
1412:
1405:
1402:
1397:
1393:
1389:
1385:
1382:
1379:
1376:
1372:
1369:
1366:
1363:
1360:
1330:
1327:
1324:
1321:
1318:
1298:
1295:
1292:
1289:
1285:
1281:
1278:
1275:
1272:
1269:
1247:
1244:
1241:
1238:
1234:
1230:
1225:
1221:
1202:
1186:
1185:
1174:
1169:
1166:
1163:
1160:
1156:
1150:
1146:
1140:
1135:
1132:
1129:
1126:
1122:
1118:
1115:
1112:
1108:
1104:
1099:
1095:
1070:
1069:
1058:
1050:
1046:
1042:
1035:
1031:
1025:
1021:
1014:
1011:
1007:
1001:
998:
995:
991:
987:
982:
979:
976:
972:
967:
963:
960:
955:
951:
947:
944:
941:
905:
883:
882:
881:
880:
879:
878:
877:
876:
875:
874:
873:
872:
871:
870:
869:
868:
862:
852:
841:
836:
807:
806:
805:
804:
803:
802:
801:
800:
799:
798:
797:
796:
795:
794:
793:
792:
735:
732:
717:
714:
660:
657:
640:
627:
626:
615:
610:
605:
599:
595:
591:
586:
582:
577:
570:
566:
562:
556:
553:
545:
541:
538:
534:
531:
528:
525:
522:
518:
503:
502:
447:approximations
422:
417:
416:
404:
398:
394:
390:
385:
381:
376:
372:
367:
364:
361:
357:
351:
348:
307:covalent bonds
272:correspondence
263:
260:
207:
204:
26:
9:
6:
4:
3:
2:
9832:
9821:
9818:
9816:
9813:
9811:
9808:
9807:
9805:
9792:
9791:
9786:
9780:
9774:
9771:
9769:
9766:
9764:
9761:
9759:
9756:
9754:
9751:
9749:
9748:Exotic matter
9746:
9742:
9739:
9738:
9737:
9736:Eightfold way
9734:
9732:
9729:
9727:
9726:Antiparticles
9724:
9722:
9719:
9717:
9714:
9710:
9707:
9706:
9705:
9702:
9698:
9695:
9694:
9693:
9690:
9689:
9687:
9683:
9677:
9674:
9672:
9669:
9667:
9664:
9662:
9659:
9657:
9654:
9653:
9651:
9647:
9641:
9638:
9636:
9633:
9631:
9628:
9626:
9623:
9621:
9618:
9616:
9613:
9611:
9608:
9606:
9603:
9601:
9598:
9596:
9593:
9591:
9588:
9586:
9583:
9581:
9578:
9576:
9573:
9572:
9570:
9568:
9564:
9550:
9547:
9545:
9542:
9540:
9537:
9535:
9532:
9531:
9529:
9525:
9515:
9512:
9510:
9507:
9505:
9502:
9501:
9499:
9495:
9489:
9486:
9484:
9481:
9479:
9476:
9475:
9473:
9469:
9465:
9462:
9460:
9456:
9450:
9447:
9445:
9442:
9438:
9435:
9433:
9430:
9428:
9425:
9423:
9420:
9418:
9415:
9413:
9410:
9409:
9408:
9405:
9403:
9400:
9398:
9397:Atomic nuclei
9395:
9394:
9392:
9388:
9378:
9375:
9372:
9368:
9365:
9364:
9362:
9360:
9356:
9350:
9347:
9345:
9342:
9340:
9337:
9335:
9332:
9330:
9329:Upsilon meson
9327:
9325:
9322:
9320:
9317:
9315:
9312:
9310:
9307:
9305:
9302:
9300:
9297:
9295:
9292:
9291:
9289:
9287:
9283:
9277:
9274:
9272:
9269:
9267:
9264:
9262:
9261:Lambda baryon
9259:
9257:
9254:
9250:
9247:
9245:
9242:
9240:
9237:
9235:
9232:
9231:
9230:
9227:
9226:
9224:
9222:
9218:
9215:
9213:
9209:
9206:
9204:
9200:
9186:
9183:
9181:
9178:
9176:
9173:
9171:
9168:
9166:
9163:
9161:
9158:
9156:
9153:
9151:
9148:
9146:
9143:
9141:
9138:
9136:
9133:
9131:
9128:
9126:
9123:
9121:
9120:Dual graviton
9118:
9116:
9113:
9111:
9108:
9106:
9103:
9102:
9100:
9096:
9085:
9081:
9078:
9076:
9073:
9071:
9068:
9066:
9063:
9061:
9058:
9057:
9055:
9051:
9045:
9042:
9040:
9037:
9035:
9032:
9031:
9029:
9027:
9023:
9020:
9018:
9017:Superpartners
9014:
9011:
9009:
9005:
8999:
8996:
8995:
8993:
8991:
8987:
8977:
8974:
8973:
8971:
8969:
8965:
8959:
8956:
8954:
8951:
8949:
8946:
8945:
8943:
8941:
8937:
8934:
8932:
8928:
8916:
8913:
8911:
8908:
8906:
8903:
8901:
8900:Muon neutrino
8898:
8896:
8893:
8891:
8888:
8887:
8886:
8883:
8881:
8878:
8876:
8873:
8871:
8868:
8866:
8863:
8861:
8858:
8856:
8853:
8852:
8850:
8848:
8844:
8838:
8835:
8833:
8832:Bottom (quark
8830:
8828:
8825:
8823:
8820:
8818:
8815:
8813:
8810:
8808:
8805:
8803:
8800:
8798:
8795:
8793:
8790:
8788:
8785:
8783:
8780:
8779:
8777:
8775:
8771:
8768:
8766:
8762:
8759:
8757:
8753:
8749:
8742:
8737:
8735:
8730:
8728:
8723:
8722:
8719:
8712:
8709:
8706:
8703:
8701:
8698:
8695:
8692:
8689:
8684:
8680:
8679:
8666:
8662:
8658:
8654:
8650:
8646:
8641:
8636:
8632:
8628:
8624:
8617:
8609:
8605:
8601:
8597:
8593:
8589:
8584:
8579:
8576:(1): 011020.
8575:
8571:
8567:
8559:
8551:
8547:
8542:
8537:
8533:
8529:
8525:
8521:
8520:
8515:
8509:
8501:
8495:
8491:
8490:
8482:
8467:
8463:
8457:
8450:
8444:
8437:
8431:
8422:
8417:
8413:
8409:
8405:
8401:
8397:
8393:
8386:
8378:
8374:
8370:
8366:
8362:
8358:
8353:
8348:
8344:
8340:
8333:
8326:
8318:
8312:
8308:
8304:
8303:
8295:
8279:
8275:
8269:
8267:
8258:
8252:
8248:
8241:
8226:
8222:
8216:
8214:
8205:
8199:
8195:
8191:
8190:, 8th Edition
8189:
8181:
8173:
8167:
8163:
8156:
8148:
8146:0-03-083993-9
8142:
8138:
8131:
8129:
8127:
8125:
8123:
8121:
8119:
8110:
8104:
8100:
8096:
8089:
8087:
8078:
8072:
8068:
8064:
8057:
8055:
8046:
8040:
8036:
8029:
8021:
8015:
8011:
8004:
7996:
7994:9780070409545
7990:
7985:
7984:
7975:
7968:
7962:
7960:
7951:
7945:
7941:
7940:
7932:
7924:
7918:
7914:
7907:
7899:
7893:
7885:
7879:
7875:
7874:
7866:
7858:
7852:
7848:
7841:
7839:
7837:
7835:
7833:
7831:
7822:
7816:
7812:
7805:
7801:
7790:
7787:
7785:
7782:
7780:
7777:
7775:
7772:
7770:
7767:
7765:
7762:
7760:
7757:
7755:
7752:
7750:
7749:Rayleigh wave
7747:
7745:
7742:
7740:
7737:
7735:
7732:
7730:
7727:
7725:
7722:
7720:
7717:
7715:
7712:
7710:
7707:
7706:
7701:
7695:
7690:
7683:
7680:
7672:
7670:
7666:
7662:
7658:
7655:vanishes and
7654:
7650:
7641:
7638:
7637:negative mass
7634:
7624:
7622:
7618:
7614:
7604:
7602:
7596:
7592:
7587: =
7584:
7580:
7573:
7556:
7551:
7547:
7541:
7534:
7530:
7522:
7518:
7514:
7511:
7504:
7503:
7502:
7500:
7496:
7492:
7485:
7481:
7452:
7449:
7444:
7441:
7438:
7434:
7430:
7424:
7421:
7416:
7412:
7405:
7400:
7397:
7394:
7390:
7386:
7383:
7378:
7374:
7359:
7355:
7349:
7339:
7334:
7331:
7328:
7324:
7318:
7314:
7308:
7305:
7302:
7298:
7294:
7289:
7285:
7272:
7268:
7260:
7259:
7258:
7234:
7229:
7226:
7223:
7219:
7215:
7209:
7206:
7201:
7197:
7190:
7185:
7182:
7179:
7175:
7171:
7168:
7163:
7159:
7146:
7143:
7138:
7134:
7128:
7118:
7113:
7110:
7107:
7103:
7097:
7093:
7087:
7084:
7081:
7077:
7073:
7068:
7064:
7051:
7044:
7040:
7030:
7029:
7028:
7023:
7017:
7010:
7004:
7001: +
6997:
6991:
6987:
6980:
6976:
6969:
6965:
6953:
6947:
6941:
6935:
6930:
6928:
6924:
6902:
6880:
6879:infinite term
6876:
6854:
6850:
6846:
6824:
6803:
6799:
6793:
6786:
6782:
6774:
6770:
6761:
6757:
6753:
6739:
6738:
6737:
6735:
6715:
6709:
6705:
6698:
6693:
6688:
6684:
6678:
6673:
6669:
6665:
6660:
6655:
6651:
6646:
6640:
6636:
6629:
6626:
6620:
6606:
6605:
6604:
6596:
6594:
6593:heat transfer
6590:
6586:
6582:
6579:behavior (or
6578:
6568:
6566:
6562:
6555:
6550:
6546:
6542:
6520:
6517:
6513:
6506:
6495:
6487:
6484:
6481:
6477:
6473:
6466:
6462:
6459:
6455:
6450:
6446:
6441:
6438:
6435:
6431:
6427:
6423:
6416:
6415:
6414:
6412:
6408:
6404:
6400:
6394:
6392:
6388:
6384:
6380:
6376:
6375:absolute zero
6371:
6368:
6367:heat capacity
6364:
6360:
6359:thermodynamic
6350:
6348:
6347:
6341:
6337:
6328:
6324:
6322:
6314:
6298: ±
6297:
6289:
6273: ±
6272:
6268:
6264:
6243:
6239:
6236:
6233:
6227:
6224:
6217:
6216:
6215:
6196:
6193:
6190:
6165:
6156:
6147:
6142:
6139:
6136:
6132:
6111:
6102:
6098:
6090:
6089:
6088:
6086:
6082:
6078:
6074:
6070:
6066:
6062:
6058:
6055:
6051:
6048:
6044:
6040:
6032:
6027:
6022:
6012:
6010:
6006:
6002:
5998:
5994:
5990:
5985:
5983:
5979:
5975:
5971:
5967:
5962:
5957:
5955:
5951:
5947:
5943:
5939:
5936:
5932:
5930:
5920:
5903:
5899:
5892:
5889:
5883:
5878:
5875:
5872:
5868:
5862:
5855:
5852:
5849:
5845:
5838:
5832:
5829:
5826:
5822:
5812:
5807:
5804:
5801:
5797:
5791:
5787:
5783:
5769:
5768:
5767:
5764:
5762:
5761:
5755:
5751:
5744:
5739:
5737:
5730:
5723:
5715:
5713:
5710:
5706:
5702:
5694:
5688:
5681:
5658:
5651:
5648:
5642:
5637:
5633:
5627:
5620:
5616:
5609:
5603:
5599:
5590:
5586:
5582:
5568:
5567:
5566:
5550:
5523:
5519:
5494:
5488:
5485:
5481:
5477:
5472:
5465:
5461:
5454:
5447:
5441:
5437:
5433:
5430:
5423:
5420:
5415:
5389:
5383:
5380:
5376:
5372:
5367:
5360:
5356:
5349:
5339:
5335:
5331:
5328:
5324:
5318:
5313:
5309:
5301:
5300:
5299:
5297:
5290:
5268:
5265:
5261:
5255:
5247:
5244:
5239:
5233:
5228:
5221:
5217:
5210:
5206:
5193:
5190:
5185:
5181:
5176:
5172:
5160:
5154:
5151:
5147:
5144:
5140:
5136:
5132:
5126:
5118:
5115:
5110:
5104:
5099:
5095:
5090:
5082:
5081:
5080:
5078:
5058:
5052:
5039:
5035:
5031:
5027:
5022:
5017:
5014:
5010:
5005:
4994:
4987:
4983:
4979:
4972:
4967:
4960:
4956:
4932:
4926:
4923:
4910:
4906:
4902:
4898:
4893:
4888:
4884:
4879:
4868:
4861:
4857:
4853:
4846:
4841:
4837:
4829:
4828:
4827:
4825:
4807:
4780:
4776:
4743:
4739:
4735:
4731:
4727:
4723:
4719:
4709:
4707:
4703:
4698:
4693:
4691:
4686:
4682:
4679:
4675:
4671:
4666:
4662:
4660:
4656:
4652:
4648:
4644:
4640:
4636:
4632:
4628:
4620:
4600:
4584:
4580:
4575:
4573:
4569:
4565:
4561:
4553:
4544:
4532:
4527:
4525:
4521:
4517:
4513:
4509:
4506: =
4505:
4500:
4483:
4479:
4476:
4469:
4466:
4458:
4439:
4429:
4425:
4419:
4415:
4407:
4403:
4400:
4394:
4389:
4385:
4381:
4375:
4370:
4365:
4357:
4353:
4349:
4344:
4337:
4333:
4329:
4323:
4316:
4313:
4309:
4301:
4297:
4293:
4288:
4281:
4277:
4273:
4267:
4263:
4260:
4255:
4250:
4246:
4238:
4237:
4236:
4234:
4230:
4223:
4216:
4208:
4204:
4201: =
4200:
4195:
4187:
4179:
4171:
4162:
4160:
4156:
4152:
4148:
4143:
4141:
4137:
4134:
4130:
4119:
4114:
4112:
4107:
4105:
4101:
4097:
4094:
4090:
4085:
4083:
4079:
4075:
4041:
4035:
4026:
4023:
4014:
4011:
4002:
3999:
3990:
3987:
3978:
3975:
3965:
3964:
3963:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3913:
3909:
3901:
3897:
3894:
3888:
3885:
3878:
3874:
3869:
3865:
3857:
3856:
3855:
3853:
3845:
3843:
3839:
3817:
3812:
3809:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3757:
3754:
3749:
3746:
3743:
3737:
3732:
3728:
3724:
3721:
3714:
3713:
3712:
3710:
3706:
3701:
3699:
3695:
3691:
3687:
3667:
3661:
3658:
3654:
3648:
3644:
3638:
3633:
3629:
3623:
3619:
3615:
3610:
3607:
3597:
3588:
3582:
3578:
3571:
3568:
3564:
3559:
3545:
3544:
3543:
3525:
3519:
3515:
3512:
3506:
3503:
3499:
3495:
3492:
3489:
3483:
3478:
3475:
3471:
3468:
3465:
3462:
3458:
3452:
3448:
3444:
3439:
3434:
3430:
3422:
3421:
3420:
3401:
3398:
3394:
3388:
3384:
3378:
3371:
3367:
3361:
3356:
3352:
3345:
3342:
3336:
3328:
3325:
3322:
3319:
3315:
3311:
3306:
3303:
3300:
3296:
3292:
3289:
3281:
3278:
3274:
3268:
3264:
3258:
3254:
3248:
3244:
3240:
3234:
3231:
3225:
3220:
3215:
3209:
3206:
3203:
3199:
3195:
3190:
3186:
3181:
3174:
3170:
3164:
3160:
3156:
3150:
3147:
3137:
3136:
3135:
3112:
3109:
3099:
3089:
3085:
3081:
3079:
3072:
3065:
3061:
3053:
3049:
3039:
3036:
3033:
3030:
3026:
3020:
3017:
3013:
3007:
3003:
2997:
2993:
2989:
2983:
2980:
2976:
2973:
2970:
2966:
2959:
2954:
2951:
2947:
2944:
2940:
2936:
2933:
2930:
2926:
2920:
2916:
2909:
2906:
2901:
2895:
2891:
2884:
2881:
2877:
2873:
2867:
2864:
2859:
2857:
2850:
2847:
2844:
2840:
2834:
2830:
2824:
2820:
2808:
2807:
2806:
2785:
2782:
2778:
2771:
2768:
2759:
2754:
2745:
2741:
2739:
2733:
2726:
2723:
2718:
2714:
2709:
2705:
2700:
2689:
2686:
2682:
2679:
2675:
2668:
2665:
2659:
2654:
2651:
2647:
2644:
2640:
2636:
2633:
2630:
2626:
2620:
2616:
2610:
2603:
2597:
2595:
2586:
2580:
2576:
2572:
2567:
2563:
2558:
2552:
2549:
2545:
2542:
2538:
2535:
2531:
2525:
2522:
2519:
2516:
2512:
2506:
2503:
2500:
2496:
2490:
2487:
2482:
2480:
2474:
2467:
2464:
2455:
2450:
2446:
2441:
2431:
2428:
2425:
2421:
2414:
2411:
2409:
2403:
2397:
2393:
2389:
2384:
2380:
2375:
2363:
2362:
2361:
2358:
2356:
2348:
2344:
2341:The quantity
2321:
2316:
2312:
2306:
2303:
2300:
2297:
2294:
2290:
2284:
2280:
2273:
2269:
2264:
2262:
2255:
2241:
2237:
2231:
2228:
2225:
2222:
2218:
2212:
2208:
2201:
2197:
2192:
2190:
2183:
2179:
2167:
2166:
2165:
2163:
2156:
2149:
2145:
2138:
2134:
2127:
2122:
2120:
2116:
2112:
2108:
2107:Fourier space
2104:
2100:
2096:
2092:
2085:
2078:
2057:
2052:
2046:
2042:
2038:
2033:
2029:
2024:
2000:
1997:
1990:
1984:
1980:
1976:
1971:
1968:
1963:
1957:
1954:
1948:
1943:
1939:
1931:
1926:
1923:
1920:
1916:
1912:
1898:
1897:
1896:
1894:
1876:
1872:
1851:
1831:
1828:
1823:
1819:
1798:
1795:
1790:
1786:
1782:
1777:
1773:
1750:
1746:
1736:
1734:
1724:
1722:
1706:
1703:
1700:
1694:
1688:
1678:
1674:
1668:
1659:
1646:
1641:
1639:
1635:
1631:
1626:
1617:
1615:
1607:
1585:
1576:
1573:
1569:
1566:
1563:
1560:
1552:
1548:
1545:
1537:
1532:
1528:
1523:
1518:
1513:
1509:
1505:
1501:
1495:
1491:
1487:
1482:
1478:
1470:
1469:
1468:
1466:
1447:
1439:
1435:
1431:
1424:
1420:
1414:
1410:
1403:
1400:
1395:
1391:
1383:
1380:
1377:
1374:
1370:
1367:
1361:
1358:
1351:
1350:
1349:
1347:
1342:
1328:
1325:
1322:
1319:
1316:
1293:
1290:
1283:
1279:
1276:
1273:
1270:
1267:
1245:
1242:
1239:
1236:
1232:
1228:
1223:
1219:
1210:
1205:
1192:
1172:
1167:
1164:
1161:
1158:
1154:
1148:
1144:
1138:
1133:
1130:
1127:
1124:
1120:
1116:
1113:
1110:
1106:
1102:
1097:
1093:
1085:
1084:
1083:
1080:
1078:
1073:
1056:
1048:
1044:
1040:
1033:
1029:
1023:
1019:
1012:
1009:
1005:
999:
996:
993:
989:
985:
980:
977:
974:
970:
965:
961:
958:
953:
949:
945:
942:
939:
932:
931:
930:
920:
915:
908:
865:
861:
855:
844:
840:
837:
826:
825:
824:
823:
822:
821:
820:
819:
818:
817:
816:
815:
814:
813:
812:
811:
810:
787:
776:
770:
763:
760:
759:
758:
757:
756:
755:
754:
753:
752:
751:
750:
749:
748:
747:
746:
745:
744:
742:
731:
722:
713:
711:
707:
702:
700:
696:
692:
687:
685:
682:
678:
674:
665:
659:Lattice waves
656:
655:
649:
647:
643:
636:
632:
613:
608:
603:
597:
593:
589:
584:
580:
575:
568:
564:
560:
554:
551:
526:
523:
516:
508:
507:
506:
500:
496:
495:
494:
492:
486:
484:
480:
476:
472:
468:
465:
461:
457:
453:
448:
444:
439:
437:
434:th atom, and
433:
429:
425:
402:
396:
392:
388:
383:
379:
374:
370:
365:
362:
359:
355:
349:
346:
337:
336:
335:
332:
328:
324:
323:gravitational
320:
316:
312:
308:
304:
300:
296:
292:
288:
284:
280:
275:
273:
269:
259:
257:
253:
252:particle-like
249:
245:
241:
237:
236:superposition
233:
229:
225:
221:
217:
213:
203:
201:
197:
194:exhibit both
193:
189:
185:
181:
177:
173:
169:
165:
160:
149:
145:
144:Yakov Frenkel
141:
137:
133:
128:
126:
122:
118:
113:
111:
108:as quantized
107:
104:, similar to
103:
99:
95:
92:
88:
87:excited state
84:
80:
79:quasiparticle
76:
72:
68:
64:
60:
56:
52:
48:
41:
37:
33:
19:
9788:
9609:
9459:Hypothetical
9407:Exotic atoms
9276:Omega baryon
9266:Sigma baryon
9256:Delta baryon
9008:Hypothetical
8990:Ghost fields
8976:Higgs boson
8910:Tau neutrino
8802:Charm (quark
8690:at Wikiquote
8630:
8626:
8616:
8573:
8569:
8558:
8523:
8517:
8508:
8488:
8481:
8469:. Retrieved
8465:
8456:
8443:
8430:
8399:
8395:
8385:
8342:
8338:
8325:
8301:
8294:
8282:. Retrieved
8278:News.mit.edu
8277:
8246:
8240:
8228:. Retrieved
8224:
8186:
8180:
8161:
8155:
8136:
8098:
8066:
8034:
8028:
8009:
8003:
7982:
7974:
7966:
7938:
7931:
7912:
7906:
7872:
7865:
7846:
7810:
7804:
7769:Second sound
7681:
7678:
7661:Cooper pairs
7647:
7630:
7610:
7607:Nonlinearity
7594:
7590:
7582:
7578:
7574:
7571:
7487:
7483:
7476:
7474:
7256:
7021:
7015:
7008:
7002:
6995:
6989:
6982:
6981:is given by
6978:
6971:
6967:
6960:
6951:
6945:
6939:
6934:vacuum state
6931:
6919:
6897:
6871:
6841:
6820:
6731:
6602:
6580:
6574:
6564:
6553:
6548:
6544:
6540:
6538:
6395:
6379:ground state
6372:
6356:
6343:
6339:
6335:
6333:
6312:
6295:
6287:
6270:
6266:
6262:
6260:
6213:
6084:
6080:
6072:
6068:
6064:
6060:
6056:
6049:
6043:matter waves
6036:
5992:
5986:
5974:Raman active
5973:
5969:
5958:
5945:
5941:
5940:
5934:
5933:
5926:
5918:
5765:
5757:
5746:
5740:
5732:
5725:
5718:
5716:
5697:
5689:
5683:
5676:
5674:
5510:
5292:
5285:
5283:
5074:
4715:
4706:viscoelastic
4694:
4687:
4683:
4677:
4673:
4669:
4667:
4663:
4654:
4646:
4630:
4618:
4598:
4576:
4571:
4567:
4563:
4551:
4539:
4528:
4523:
4519:
4511:
4507:
4503:
4501:
4456:
4454:
4232:
4228:
4221:
4214:
4212:
4206:
4202:
4198:
4151:polarization
4146:
4144:
4139:
4135:
4128:
4126:
4108:
4095:
4088:
4086:
4073:
4056:
3961:
3848:
3846:
3841:
3837:
3835:
3708:
3704:
3702:
3697:
3689:
3685:
3683:
3541:
3418:
3133:
2804:
2359:
2342:
2340:
2158:
2151:
2147:
2140:
2129:
2125:
2123:
2111:normal modes
2098:
2087:
2080:
2076:
2074:
1844:if particle
1737:
1732:
1730:
1676:
1672:
1663:
1661:held fixed,
1657:
1642:
1632:between the
1621:
1618:
1602:
1600:
1462:
1345:
1343:
1208:
1200:
1190:
1187:
1081:
1074:
1071:
918:
916:
903:
884:
863:
859:
850:
842:
838:
808:
785:
774:
768:
761:
737:
728:
706:normal modes
703:
694:
690:
688:
683:
670:
650:
645:
638:
630:
628:
504:
487:
482:
478:
474:
470:
466:
455:
440:
435:
431:
420:
418:
330:
290:
286:
276:
265:
239:
209:
175:
167:
163:
139:
129:
114:
94:quantization
77:. A type of
46:
44:
9741:Quark model
9509:Theta meson
9412:Positronium
9324:Omega meson
9319:J/psi meson
9249:Antineutron
9160:Dark photon
9125:Graviphoton
9084:Stop squark
8792:Down (quark
7611:As well as
6927:Hamiltonian
6823:Hamiltonian
6389:about some
6383:temperature
4742:eigenvalues
2119:periodicity
2109:which uses
1893:Hamiltonian
1614:normal mode
929:th atom is
889:labels the
710:frequencies
686:is marked.
279:crystalline
232:normal mode
216:vibrational
188:light waves
184:sound waves
138:. The name
110:light waves
102:sound waves
9804:Categories
9483:Heptaquark
9444:Superatoms
9377:Pentaquark
9367:Tetraquark
9349:Quarkonium
9239:Antiproton
9140:Leptoquark
9075:Neutralino
8837:antiquark)
8827:antiquark)
8822:Top (quark
8817:antiquark)
8807:antiquark)
8797:antiquark)
8787:antiquark)
8756:Elementary
8640:2205.05037
8583:2205.05037
8499:0486435032
8406:: 012121.
7796:References
6585:conduction
6399:photon gas
6391:mean value
6047:wavevector
6005:wavelength
5989:wavenumber
4583:wavevector
4133:wavevector
4116:See also:
4091:amount of
2355:wavelength
2347:wavenumber
2115:wavevector
1638:wavenumber
681:wavelength
240:elementary
206:Definition
134:physicist
9721:Particles
9666:Particles
9625:Polariton
9615:Plasmaron
9585:Dropleton
9478:Hexaquark
9449:Molecules
9437:Protonium
9314:Phi meson
9299:Rho meson
9271:Xi baryon
9203:Composite
9039:Gravitino
8782:Up (quark
8665:248665478
8633:(1): 26.
8608:248665478
8550:195774243
8471:13 August
8284:13 August
8230:15 August
7892:cite book
7789:Vibration
7619:and form
7552:α
7542:†
7535:α
7523:α
7519:∑
7453:…
7439:α
7422:−
7417:α
7398:−
7395:α
7384:…
7360:α
7340:…
7329:α
7319:α
7306:−
7303:α
7295:…
7273:α
7235:…
7224:α
7202:α
7183:−
7180:α
7169:…
7139:α
7119:…
7108:α
7098:α
7085:−
7082:α
7074:…
7052:†
7045:α
6957:…⟩
6849:continuum
6804:α
6794:†
6787:α
6775:α
6771:ω
6767:ℏ
6762:α
6758:∑
6710:α
6706:ω
6702:ℏ
6699:−
6689:α
6674:α
6670:ω
6656:α
6641:α
6637:∑
6589:radiation
6518:−
6478:ω
6474:ℏ
6463:
6432:ω
6240:π
6187:Π
6153:Π
5929:unit cell
5863:†
5823:ω
5818:ℏ
5798:∑
5788:∑
5705:hermitian
5628:†
5600:ω
5596:ℏ
5587:∑
5547:Π
5486:−
5478:−
5473:†
5438:ω
5431:ℏ
5412:Π
5381:−
5368:†
5336:ω
5325:ℏ
5256:†
5229:†
5141:δ
5127:†
5049:Π
5036:ω
5023:−
5015:−
4998:ℏ
4984:ω
4968:†
4924:−
4920:Π
4907:ω
4872:ℏ
4858:ω
4804:Π
4704:(but see
4484:λ
4480:π
4395:
4376:−
4314:±
4251:±
4247:ω
4042:⋯
4036:ω
4033:ℏ
4012:ω
4009:ℏ
3988:ω
3985:ℏ
3947:…
3910:ω
3906:ℏ
3805:±
3802:…
3793:±
3784:±
3767:for
3747:π
3694:Hermitian
3659:−
3630:ω
3608:−
3604:Π
3594:Π
3579:∑
3507:
3496:ω
3472:
3466:−
3449:ω
3431:ω
3399:−
3368:ω
3353:∑
3320:−
3312:−
3293:−
3279:−
3255:∑
3245:ω
3196:−
3171:∑
3161:ω
3110:−
3106:Π
3096:Π
3086:∑
3050:∑
3018:−
2994:∑
2917:∑
2874:∑
2821:∑
2764:Π
2751:Π
2676:δ
2672:ℏ
2648:−
2617:∑
2607:ℏ
2536:−
2497:∑
2460:Π
2422:δ
2418:ℏ
2295:−
2281:∑
2252:Π
2209:∑
2124:A set of
2039:−
1991:∑
1981:ω
1917:∑
1799:…
1701:∝
1689:ω
1655:→∞, with
1570:
1564:−
1529:ω
1510:ω
1381:−
1371:
1326:…
1277:π
1220:ϕ
1128:π
1107:∑
997:−
940:−
677:amplitude
590:−
565:ω
517:∑
389:−
363:≠
356:∑
329:function
283:amorphous
248:wave-like
238:of these
224:frequency
202:phonons.
192:unit cell
136:Igor Tamm
73:and some
63:molecules
9697:timeline
9549:R-hadron
9504:Glueball
9488:Skyrmion
9422:Tauonium
9135:Inflaton
9130:Graviton
9110:Curvaton
9080:Sfermion
9070:Higgsino
9065:Chargino
9026:Gauginos
8885:Neutrino
8870:Antimuon
8860:Positron
8855:Electron
8765:Fermions
8466:Phys.org
8377:17019967
7686:See also
7665:exchange
7597:⟩
7585:⟩
7458:⟩
7345:⟩
7240:⟩
7124:⟩
6970:, where
6595:models.
6387:randomly
6054:momentum
5712:spectrum
5248:′
5194:′
5155:′
5119:′
4718:operator
4641:and two
4524:acoustic
3705:periodic
2984:′
2955:′
2910:′
2885:′
2772:′
2727:′
2690:′
2655:′
2546:′
2468:′
2095:momentum
1636:and the
452:screened
428:position
319:Magnetic
315:electric
196:acoustic
9685:Related
9656:Baryons
9630:Polaron
9620:Plasmon
9595:Fracton
9590:Exciton
9544:Diquark
9539:Pomeron
9514:T meson
9471:Baryons
9432:Pionium
9417:Muonium
9344:D meson
9339:B meson
9244:Neutron
9229:Nucleon
9221:Baryons
9212:Hadrons
9175:Tachyon
9150:Majoron
9115:Dilaton
9044:Photino
8880:Antitau
8847:Leptons
8645:Bibcode
8627:Physics
8588:Bibcode
8528:Bibcode
8408:Bibcode
8357:Bibcode
7719:Fracton
7613:photons
6917:
6905:
6895:
6883:
6869:
6857:
6839:
6827:
6559:is the
6317:
6300:
6292:
6275:
6039:photons
5946:optical
4657:in the
4623:
4607:
4603:
4587:
4556:
4535:
4520:optical
4076:is the
4071:
4059:
2139:of the
2113:of the
1643:In the
633:is the
430:of the
426:is the
317:force.
220:lattice
200:optical
106:photons
96:of the
89:in the
83:physics
75:liquids
55:elastic
18:Phonons
9815:Bosons
9661:Mesons
9610:Phonon
9605:Magnon
9527:Others
9497:Mesons
9390:Others
9286:Mesons
9234:Proton
9098:Others
9053:Others
9034:Gluino
8968:Scalar
8948:Photon
8931:Bosons
8774:Quarks
8688:Phonon
8663:
8606:
8548:
8519:Nature
8496:
8375:
8313:
8253:
8200:
8168:
8143:
8105:
8073:
8041:
8016:
7991:
7946:
7919:
7880:
7853:
7817:
7633:rotons
7601:bosons
7589:|
7577:|
7486:while
6938:|
6855:, the
6563:, and
6539:where
6344:first
6214:where
6009:photon
5743:bosons
5709:energy
4728:. The
4726:bosons
4697:fluids
4676:and k
4455:where
4100:photon
4093:energy
4057:where
4039:
4018:
3994:
3821:
3419:where
2075:where
1188:Here,
885:where
857:
848:
833:
829:
789:
783:
779:
772:
766:
629:Here,
419:where
299:forces
268:axioms
176:photon
140:phonon
132:Soviet
71:solids
47:phonon
32:photon
9649:Lists
9640:Trion
9635:Roton
9575:Anyon
9402:Atoms
9165:Preon
9105:Axion
9060:Axino
8953:Gluon
8940:Gauge
8661:S2CID
8635:arXiv
8604:S2CID
8578:arXiv
8546:S2CID
8402:(1).
8373:S2CID
8347:arXiv
8335:(PDF)
7764:SASER
7709:Boson
7257:and,
6007:of a
4558:(see
4089:exact
4080:of a
3854:are:
3692:were
2103:waves
1651:→0,
1260:with
725:seen.
673:waves
281:(not
226:. In
172:sound
168:voice
164:sound
159:phonē
150:word
148:Greek
59:atoms
49:is a
38:, or
9600:Hole
9427:Onia
9334:Kaon
9294:Pion
8865:Muon
8707:and
8494:ISBN
8473:2019
8311:ISBN
8286:2019
8251:ISBN
8232:2020
8198:ISBN
8166:ISBN
8141:ISBN
8103:ISBN
8071:ISBN
8039:ISBN
8014:ISBN
7989:ISBN
7944:ISBN
7917:ISBN
7898:link
7878:ISBN
7851:ISBN
7815:ISBN
6357:The
6083:and
6041:and
5724:and
5538:and
5291:and
4651:GaAs
4605:and
3688:and
2146:and
2086:and
1309:for
1082:Put
777:+ 1
764:− 1
321:and
198:and
182:for
153:φωνή
119:and
8875:Tau
8653:doi
8596:doi
8536:doi
8524:571
8416:doi
8400:193
8365:doi
8307:159
8194:100
7501:as
6851:or
6460:exp
6373:At
6067:or
5703:is
4670:ħω.
4386:sin
4087:An
3504:sin
3469:cos
2105:in
1567:cos
1368:cos
917:If
866:+ 1
845:− 1
835:→→→
743:):
493:.)
166:or
112:.
81:in
65:in
61:or
9806::
8659:.
8651:.
8643:.
8631:16
8629:.
8625:.
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8574:13
8572:.
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8265:^
8223:.
8212:^
8196:.
8117:^
8097:.
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8065:.
8053:^
7958:^
7894:}}
7890:{{
7829:^
7623:.
7603:.
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7009:ħω
6996:ħω
6983:ħω
6929:.
6920:ħω
6898:ħω
6872:ħω
6842:ħω
6294:,
6061:ħk
6057:ħk
5997:cm
5984:.
5738:.
5733:ħω
5682:=
5079::
4826::
4692:.
4568:ωa
4499:.
4220:,
4096:ħω
4084:.
4074:ħω
2357:.
2164::
1670:→
1658:Na
1647:,
1640:.
1616:.
1341:.
1191:na
897:,
827:→→
781:←
712:.
309:,
305:,
45:A
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9369:(
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9082:(
8740:e
8733:t
8726:v
8713:.
8667:.
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7823:.
7595:α
7593:,
7591:β
7583:β
7581:,
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7557:.
7548:a
7531:a
7515:=
7512:N
7490:α
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7477:a
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7445:1
7442:+
7435:n
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7425:1
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7350:=
7335:1
7332:+
7325:n
7315:n
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7269:a
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7210:1
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7129:=
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7104:n
7094:n
7088:1
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7069:1
7065:n
7059:|
7041:a
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6955:3
6952:n
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6908:1
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6754:=
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6666:+
6661:2
6652:p
6647:(
6630:2
6627:1
6621:=
6616:H
6565:T
6557:B
6554:k
6549:s
6547:,
6545:k
6541:ω
6521:1
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6507:T
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6496:k
6488:s
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6482:k
6467:(
6456:1
6451:=
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6442:s
6439:,
6436:k
6428:(
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6340:k
6336:k
6313:a
6309:/
6305:π
6303:4
6296:k
6288:a
6284:/
6280:π
6278:2
6271:k
6267:k
6263:n
6244:a
6237:n
6234:2
6228:=
6225:K
6197:K
6194:+
6191:k
6178:f
6175:e
6172:d
6166:=
6157:k
6148:;
6143:K
6140:+
6137:k
6133:Q
6124:f
6121:e
6118:d
6112:=
6103:k
6099:Q
6085:Π
6081:Q
6073:k
6050:k
5993:ω
5904:.
5900:)
5893:2
5890:1
5884:+
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5876:,
5873:k
5869:b
5856:s
5853:,
5850:k
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5839:(
5833:s
5830:,
5827:k
5813:3
5808:1
5805:=
5802:s
5792:k
5784:=
5779:H
5749:k
5747:b
5735:k
5728:k
5726:b
5721:k
5719:b
5700:k
5698:n
5692:k
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5686:k
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5677:n
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5652:2
5649:1
5643:+
5638:k
5634:b
5621:k
5617:b
5610:(
5604:k
5591:k
5583:=
5578:H
5551:k
5524:k
5520:Q
5495:)
5489:k
5482:b
5466:k
5462:b
5455:(
5448:2
5442:k
5434:m
5424:i
5421:=
5416:k
5390:)
5384:k
5377:b
5373:+
5361:k
5357:b
5350:(
5340:k
5332:m
5329:2
5319:=
5314:k
5310:Q
5295:k
5293:b
5288:k
5286:b
5269:0
5266:=
5262:]
5245:k
5240:b
5234:,
5222:k
5218:b
5211:[
5207:=
5202:]
5191:k
5186:b
5182:,
5177:k
5173:b
5167:[
5161:,
5152:k
5148:,
5145:k
5137:=
5133:]
5116:k
5111:b
5105:,
5100:k
5096:b
5091:[
5059:)
5053:k
5040:k
5032:m
5028:i
5018:k
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5006:(
4995:2
4988:k
4980:m
4973:=
4961:k
4957:b
4933:)
4927:k
4911:k
4903:m
4899:i
4894:+
4889:k
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4880:(
4869:2
4862:k
4854:m
4847:=
4842:k
4838:b
4808:k
4781:k
4777:Q
4754:H
4678:.
4674:ω
4655:k
4647:N
4631:N
4619:a
4615:/
4611:π
4599:a
4595:/
4591:π
4572:k
4564:k
4552:k
4550:∂
4547:/
4542:k
4540:ω
4538:∂
4512:k
4510:(
4508:ω
4504:ω
4477:2
4470:=
4467:k
4457:k
4440:,
4430:2
4426:m
4420:1
4416:m
4408:2
4404:a
4401:k
4390:2
4382:4
4371:2
4366:)
4358:2
4354:m
4350:1
4345:+
4338:1
4334:m
4330:1
4324:(
4317:K
4310:)
4302:2
4298:m
4294:1
4289:+
4282:1
4278:m
4274:1
4268:(
4264:K
4261:=
4256:2
4233:K
4229:a
4225:2
4222:m
4218:1
4215:m
4207:k
4205:(
4203:ω
4199:ω
4147:s
4140:k
4136:k
4129:k
4068:2
4065:/
4062:1
4027:2
4024:5
4015:,
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3991:,
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3976:1
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3941:,
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3935:,
3932:1
3929:,
3926:0
3923:=
3920:n
3914:k
3902:)
3898:n
3895:+
3889:2
3886:1
3879:(
3875:=
3870:n
3866:E
3851:k
3849:ω
3842:a
3838:n
3818:.
3813:2
3810:N
3799:,
3796:2
3790:,
3787:1
3781:,
3778:0
3775:=
3772:n
3758:a
3755:N
3750:n
3744:2
3738:=
3733:n
3729:k
3725:=
3722:k
3709:N
3698:N
3690:Π
3686:Q
3668:)
3662:k
3655:Q
3649:k
3645:Q
3639:2
3634:k
3624:2
3620:m
3616:+
3611:k
3598:k
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3583:k
3572:m
3569:2
3565:1
3560:=
3555:H
3526:|
3520:2
3516:a
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3500:|
3493:2
3490:=
3484:)
3479:a
3476:k
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3453:2
3445:2
3440:=
3435:k
3402:k
3395:Q
3389:k
3385:Q
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3372:k
3362:m
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3343:1
3337:=
3334:)
3329:a
3326:k
3323:i
3316:e
3307:a
3304:k
3301:i
3297:e
3290:2
3287:(
3282:k
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3269:k
3265:Q
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3235:2
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3226:=
3221:2
3216:)
3210:1
3207:+
3204:j
3200:x
3191:j
3187:x
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3175:j
3165:2
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3113:k
3100:k
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3082:=
3073:2
3066:l
3062:p
3054:l
3040:k
3037:m
3034:a
3031:i
3027:e
3021:k
3014:Q
3008:k
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2998:k
2990:=
2981:k
2977:m
2974:a
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2952:k
2948:+
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2937:l
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2865:1
2860:=
2851:m
2848:+
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2779:]
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2742:=
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2475:]
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2412:=
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2394:p
2390:,
2385:l
2381:x
2376:[
2351:π
2343:k
2322:.
2317:l
2313:p
2307:l
2304:a
2301:k
2298:i
2291:e
2285:l
2274:N
2270:1
2265:=
2256:k
2242:l
2238:x
2232:l
2229:a
2226:k
2223:i
2219:e
2213:l
2202:N
2198:1
2193:=
2184:k
2180:Q
2161:k
2159:p
2154:k
2152:Π
2148:N
2143:k
2141:x
2132:k
2130:Q
2126:N
2099:i
2090:i
2088:p
2083:i
2081:x
2077:m
2058:2
2053:)
2047:j
2043:x
2034:i
2030:x
2025:(
2018:)
2014:n
2011:n
2007:(
2004:}
2001:j
1998:i
1995:{
1985:2
1977:m
1972:2
1969:1
1964:+
1958:m
1955:2
1949:2
1944:i
1940:p
1932:N
1927:1
1924:=
1921:i
1913:=
1908:H
1877:i
1873:x
1852:i
1832:0
1829:=
1824:i
1820:x
1796:,
1791:2
1787:x
1783:,
1778:1
1774:x
1751:i
1747:u
1733:N
1707:a
1704:k
1698:)
1695:k
1692:(
1679:)
1677:x
1675:(
1673:φ
1666:n
1664:u
1653:N
1649:a
1624:k
1622:ω
1610:k
1605:k
1603:Q
1586:.
1581:)
1577:a
1574:k
1561:1
1558:(
1553:m
1549:C
1546:2
1538:=
1533:k
1524:;
1519:t
1514:k
1506:i
1502:e
1496:k
1492:A
1488:=
1483:k
1479:Q
1448:.
1440:2
1436:t
1432:d
1425:k
1421:Q
1415:2
1411:d
1404:m
1401:=
1396:k
1392:Q
1388:)
1384:1
1378:a
1375:k
1365:(
1362:C
1359:2
1329:N
1323:1
1320:=
1317:j
1297:)
1294:a
1291:N
1288:(
1284:/
1280:j
1274:2
1271:=
1268:k
1246:a
1243:n
1240:k
1237:i
1233:e
1229:=
1224:k
1203:k
1201:Q
1196:x
1173:.
1168:a
1165:n
1162:k
1159:i
1155:e
1149:k
1145:Q
1139:N
1134:1
1131:=
1125:2
1121:/
1117:k
1114:a
1111:N
1103:=
1098:n
1094:u
1057:.
1049:2
1045:t
1041:d
1034:n
1030:u
1024:2
1020:d
1013:m
1010:=
1006:)
1000:1
994:n
990:u
986:+
981:1
978:+
975:n
971:u
966:(
962:C
959:+
954:n
950:u
946:C
943:2
927:n
923:m
919:C
912:n
906:n
904:u
899:a
895:N
891:n
887:n
864:n
860:u
853:n
851:u
843:n
839:u
831:→
791:→
786:a
775:n
769:n
762:n
739:(
695:a
691:a
684:λ
646:i
641:i
639:R
631:ω
614:.
609:2
604:)
598:j
594:R
585:i
581:R
576:(
569:2
561:m
555:2
552:1
544:)
540:n
537:n
533:(
530:}
527:j
524:i
521:{
483:x
479:x
475:x
471:V
467:V
456:V
436:V
432:i
423:i
421:r
403:)
397:j
393:r
384:i
380:r
375:(
371:V
366:j
360:i
350:2
347:1
331:V
291:N
287:N
156:(
42:.
20:)
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