Phonon - Knowledge

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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.

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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.

9774: 8672: 6316: 4159: 4175: 653: 6015: 2789: 3118: 488: 7241: 7459: 4439: 3403: 641:(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 4559:, 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 2355: 5741:. 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 2325: 4654:

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.

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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

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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

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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

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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

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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

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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

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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

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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

6716: 7252: 5903: 4230: 3129: 5059: 6523: 4570:, 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 3668: 3526: 6198: 4041: 4933: 5659: 3820: 4688:

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

6805: 5495: 1585: 613: 5390: 2784:{\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}}} 713:

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

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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

3946: 3113:{\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}}} 2159: 7236:{\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 }} 6382:. 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. 1056: 5952:

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

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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.

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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.

7454:{\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 }} 5684:

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

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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

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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:

6394:, 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 6358:

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.

1447: 5268: 5696:). 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 4434:{\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}}}}},} 3398:{\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}} 5761: 8552:

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).

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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

5969:. 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 4938: 1724:

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.

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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

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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.

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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.

5945:, 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. 4486: 3537: 3414: 2090:

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

6082: 3957: 4821: 8209: 6245: 5560: 1706: 3706: 1798: 1247: 6731: 6572:) 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 5395: 4755: 1462: 1296: 4657:

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

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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

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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

5525: 4782: 1878: 1752: 2320:{\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}}} 4098:

All quantum systems show wavelike and particlelike properties simultaneously. The particle-like properties of the phonon are best understood using the methods of

3849: 1851: 6402:: 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: 924: 2054:{\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}} 7656:

of phonons between the electrons. The evidence that phonons, the vibrations of the ionic lattice, are relevant for superconductivity is provided by the

4150:. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons. 6711:{\displaystyle {\mathcal {H}}={\tfrac {1}{2}}\sum _{\alpha }\left(p_{\alpha }^{2}+\omega _{\alpha }^{2}q_{\alpha }^{2}-\hbar \omega _{\alpha }\right)} 3700: + 1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is 7624:

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

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Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions gives rise to a set of vibration

8688: 7926: 6308:, and so forth. Physically, the reciprocal lattice vectors act as additional chunks of momentum which the lattice can impart to the phonon. 329: 7496: 5941:, fluctuations in displacement create an electrical polarization that couples to the electromagnetic field. Hence, they can be excited by 4146:. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like 1343: 9447: 5898:{\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).} 9685: 1337:(this requires a significant manipulation using the orthonormality and completeness relations of the discrete Fourier transform), 4142:

of the phonons. In the one-dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to

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into the wavevector space Hamiltonian, as it is defined above, and simplifying then results in the Hamiltonian taking the form:

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mode. In the optical mode two adjacent different atoms move against each other, while in the acoustic mode they move together.

6518:{\displaystyle n\left(\omega _{k,s}\right)={\frac {1}{\exp \left({\dfrac {\hbar \omega _{k,s}}{k_{\mathrm {B} }T}}\right)-1}}} 8303: 8243: 8190: 8158: 8095: 8063: 8031: 7936: 7909: 7870: 7843: 7807: 7641: 5970: 687: 451:. This is permissible as long as the atoms remain close to their equilibrium positions. Formally, this is accomplished by 8727: 8176: 6722: 5908:

This can be interpreted as the sum of 3N independent oscillator Hamiltonians, one for each wave vector and polarization.

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When measuring optical phonon energy experimentally, optical phonon frequencies are sometimes given in spectroscopic

3663:{\displaystyle {\mathcal {H}}={\frac {1}{2m}}\sum _{k}\left(\Pi _{k}\Pi _{-k}+m^{2}\omega _{k}^{2}Q_{k}Q_{-k}\right)} 8476: 7482:

annihilates one. Hence, they are respectively the creation and annihilation operators for phonons. Analogous to the

6338:. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector. 3521:{\displaystyle \omega _{k}={\sqrt {2\omega ^{2}\left(1-\cos {ka}\right)}}=2\omega \left|\sin {\frac {ka}{2}}\right|} 9664: 6193:{\displaystyle Q_{k}{\stackrel {\mathrm {def} }{=}}Q_{k+K};\quad \Pi _{k}{\stackrel {\mathrm {def} }{=}}\Pi _{k+K}} 4036:{\displaystyle {\tfrac {1}{2}}\hbar \omega ,\ {\tfrac {3}{2}}\hbar \omega ,\ {\tfrac {5}{2}}\hbar \omega \ \cdots } 7682: 7016:), the phonons are said to be non-interacting. The action of the creation and annihilation operators is given by: 6925:", is the state composed of no phonons. Hence, the energy of the ground state is 0. When a system is in the state 4928:{\displaystyle b_{k}={\sqrt {\frac {m\omega _{k}}{2\hbar }}}\left(Q_{k}+{\frac {i}{m\omega _{k}}}\Pi _{-k}\right)} 4451: 6915: 6811: 4106: 1881: 7732: 8486: 5065: 642: 7860: 5654:{\displaystyle {\mathcal {H}}=\sum _{k}\hbar \omega _{k}\left({b_{k}}^{\dagger }b_{k}+{\tfrac {1}{2}}\right)} 4705:

The above-derived Hamiltonian may look like a classical Hamiltonian function, but if it is interpreted as an

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th atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted (nn).

5734:, since any number of identical excitations can be created by repeated application of the creation operator 4087:

must be supplied to the harmonic oscillator lattice to push it to the next energy level. By analogy to the

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This is known as the second quantization technique, also known as the occupation number formulation, where

4678: 4555:(i.e. long wavelengths), the dispersion relation is almost linear, and the speed of sound is approximately 3815:{\displaystyle k=k_{n}={\frac {2\pi n}{Na}}\quad {\mbox{for }}n=0,\pm 1,\pm 2,\ldots \pm {\frac {N}{2}}.\ } 223:

of vibration. Normal modes are important because any arbitrary lattice vibration can be considered to be a

8676: 6870:. Because the difference in energy is what we measure and not the absolute value of it, the constant term 1673: 9808: 7668:

In 2019, researchers were able to isolate individual phonons without destroying them for the first time.

7483: 5742: 4070: 6800:{\displaystyle {\mathcal {H}}=\sum _{\alpha }\hbar \omega _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }} 1757: 1203: 7605: 5490:{\displaystyle \Pi _{k}=i{\sqrt {\frac {\hbar m\omega _{k}}{2}}}\left({b_{k}}^{\dagger }-b_{-k}\right)} 2125: 1580:{\displaystyle Q_{k}=A_{k}e^{i\omega _{k}t};\qquad \omega _{k}={\sqrt {{\frac {2C}{m}}(1-\cos {ka})}}.} 1065: 9756: 6867: 608:{\displaystyle \sum _{\{ij\}(\mathrm {nn} )}{\tfrac {1}{2}}m\omega ^{2}\left(R_{i}-R_{j}\right)^{2}.} 244: 168: 6022:(red) do not carry any more information than their counterparts (black) in the first Brillouin zone. 4736: 4567: 9168: 8450: 7487: 5989: 5954: 5385:{\displaystyle Q_{k}={\sqrt {\frac {\hbar }{2m\omega _{k}}}}\left({b_{k}}^{\dagger }+b_{-k}\right)} 1252: 440: 299: 260: 55: 6068:

and integer multiples thereof). For example, in the one-dimensional model, the normal coordinates

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explicitly in either classical or quantum mechanics. In order to simplify the task, two important

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This technique is readily generalized to three dimensions, where the Hamiltonian takes the form:

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forces are generally negligible. The forces between each pair of atoms may be characterized by a

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This choice retains the desired commutation relations in either real space or wavevector space

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Not every possible lattice vibration has a well-defined wavelength and frequency. However, the

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by a weak attractive force. In a conventional superconductor, this attraction is caused by an

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Reiter, D. E.; Sauer, S.; Huneke, J.; Papenkort, T.; Kuhn, T.; Vagov, A.; Axt, V. M. (2009).

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Three important properties of phonons may be deduced from this technique. First, phonons are

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between atoms. Any wavelength shorter than this can be mapped onto a wavelength longer than 2

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of the wave is given by the displacements of the atoms from their equilibrium positions. The

39: 4638: – 3. The lower figure shows the dispersion relations for several phonon modes in 3941:{\displaystyle E_{n}=\left({\tfrac {1}{2}}+n\right)\hbar \omega _{k}\qquad n=0,1,2,3\ldots } 8700: 8633: 8576: 8516: 8396: 8392: 8345: 7772: 7702: 7657: 6395: 5748: 5503: 4760: 4710: 4139: 3692:

The form of the quantization depends on the choice of boundary conditions; for simplicity,

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of the harmonic potentials, which are assumed to be the same since the lattice is regular.

105: 43: 8182: 8: 9761: 8744: 8409: 8380: 4718: 4504: 4099: 1709: 1618: 448: 291: 240: 216: 8637: 8580: 8520: 8400: 8349: 686:, due to the periodicity of the lattice. This can be thought of as a consequence of the 664:

propagating through the lattice. One such wave is shown in the figure to the right. The

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for a typical sample of a solid. Since the lattice is rigid, the atoms must be exerting

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for elastic structures of interacting particles. Phonons can be thought of as quantized

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are expelled from the material. In a superconductor, electrons are bound together into

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respectively create and destroy a single field excitation, a phonon, with an energy of

4706: 3682: 1836: 113: 79: 8554: 7660:, the dependence of the superconducting critical temperature on the mass of the ions. 4182: 9773: 9741: 9654: 9133: 8878: 8805: 8653: 8596: 8538: 8482: 8299: 8239: 8186: 8154: 8129: 8091: 8059: 8027: 8002: 7977: 7932: 7905: 7866: 7839: 7803: 7722: 7712: 7637: 6577: 6565: 6351: 4627: 4166: 4143: 4066: 1622: 1064:

Since the solutions are expected to be oscillatory, new coordinates are defined by a

1051:{\displaystyle -2Cu_{n}+C\left(u_{n+1}+u_{n-1}\right)=m{\frac {d^{2}u_{n}}{dt^{2}}}.} 729: 623: 479: 431: 200: 86: 24: 8365: 5984:

represents ordinary frequency (not angular frequency), and is expressed in units of

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In contrast to the previous section, the positions of the masses are not denoted by

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Phonon propagating through a square lattice (atom displacements greatly exaggerated)

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on one another to keep each atom near its equilibrium position. These forces may be

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There is a minimum possible wavelength, given by twice the equilibrium separation

9803: 9719: 9644: 9628: 9568: 8978: 8903: 8893: 8883: 8795: 7931:. International publishing locations: Oxford University Press. pp. 231–232. 7717: 6841: 6379: 5938: 4722: 4700: 4631: 4147: 1633: 690:, the lattice points being viewed as the "sampling points" of a continuous wave. 283: 271: 7958:. Solid State Physics. Vol. Supplement 3 (Second ed.). New York: Academic Press. 5992:. In other words, the wave-number in cm units corresponds to the inverse of the 5703:

is then given by the sum of the individual eigenvalues of the sub-Hamiltonians.

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The couplings between the position variables have been transformed away; if the

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is the occupation number of the phonons. The energy of a single phonon of type

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of quantum mechanics but instead use relations for which there exists a direct

8357: 9792: 9736: 9588: 9555: 9347: 9317: 9249: 9108: 8888: 8815: 8800: 8381:"Generation of squeezed phonon states by optical excitation of a quantum dot" 7737: 7653: 7625: 6581: 6363: 6355: 2095: 452: 435: 295: 132: 75: 67: 4733:

without directly solving the differential equations. Given the Hamiltonian,

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For a one-dimensional alternating array of two types of ion or atom of mass

9714: 9264: 9254: 9244: 9005: 8956: 8898: 8820: 8775: 8321:"Phonon–phonon interactions due to non-linear effects in a linear ion trap" 7862:

Stalin's great science : the times and adventures of Soviet physicists

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in the lattice, is given by the slope of the acoustic dispersion relation,

2099: 709: 120: 8611: 8058:. Physics and Materials Properties (4th ed.). Springer. p. 111. 1597:

represents an independent vibrational mode of the lattice with wavenumber

9729: 9497: 9400: 9395: 9312: 9307: 9237: 9191: 9148: 9113: 9072: 8964: 8928: 8790: 8340: 7649: 6371: 4730: 4563:, i.e. short wavelengths, due to the microscopic details of the lattice. 1602: 694: 267: 220: 98: 90: 8451:"Researchers suggest phonons may have mass and perhaps negative gravity" 6319:

Brillouin zones, (a) in a square lattice, and (b) in a hexagonal lattice

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The following commutators can be easily obtained by substituting in the

9471: 9365: 9355: 9337: 9227: 9128: 9063: 8780: 8705: 8555:"Strong Electronic Winds Blowing under Liquid Flows on Carbon Surfaces" 6387: 6035: 5993: 5985: 5977: 4571: 4121: 4116:

This may be generalized to a three-dimensional lattice. The wavenumber

2343: 2335: 2103: 1626: 798:···o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o··· 669: 3829:

comes from the minimum wavelength, which is twice the lattice spacing

398:{\displaystyle {\frac {1}{2}}\sum _{i\neq j}V\left(r_{i}-r_{j}\right)} 9613: 9603: 9573: 9466: 9432: 9425: 9302: 9292: 9287: 9259: 9027: 8810: 8052:"Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes" 7777: 7551:{\displaystyle N=\sum _{\alpha }{a_{\alpha }}^{\dagger }a_{\alpha }.} 6258:

is thus equivalent to an infinite family of phonons with wavenumbers

5917: 5752:, it can be shown that phonons act as waves of lattice displacement. 698: 665: 307: 212: 204: 180: 146: 124: 6892:

can be ignored without changing the equations of motion. Hence, the

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exhibit two types of phonons: acoustic phonons and optical phonons.

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can be inverted to redefine the conjugate position and momentum as:

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is quantized, the quantum of vibrational energy is called a phonon.

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is the distance between atoms when the chain is in equilibrium, and

9709: 9537: 9492: 9476: 9437: 9410: 9123: 9118: 9098: 9068: 9058: 9053: 8873: 8848: 8843: 8770: 8628: 8571: 8437: 8424: 6042: 4158: 2083: 1442:{\displaystyle 2C(\cos {ka-1})Q_{k}=m{\frac {d^{2}Q_{k}}{dt^{2}}}.} 51: 8697: 5957:. Optical phonons that interact in this way with light are called 4518:

The speed of propagation of an acoustic phonon, which is also the

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The harmonic oscillator eigenvalues or energy levels for the mode

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is a large number, say of the order of 10, or on the order of the

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Angelo Esposito, Rafael Krichevsky, and Alberto Nicolis. (2018).

7707: 7601: 6331:| in their "family". The set of all such wavevectors defines the 6315: 4174: 3685:(which they are not), the transformed Hamiltonian would describe 311: 175:

in the same way that photons represent wave-particle duality for

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It can be shown that phonons are symmetric under exchange (i.e.

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is the wavevector of the vibration related to its wavelength by

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is at its equilibrium position.) In two or more dimensions, the

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A one-dimensional quantum mechanical harmonic chain consists of

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Substitution into the equation of motion produces the following

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factor is absent in the operator formalized expression for the

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University of Cambridge Teaching and Learning Packages Library

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in vacuum that has the same frequency as the measured phonon.

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defined in the quantum treatment section above, we can define

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Interpretation of phonons using second quantization techniques

9623: 9563: 9415: 9274: 9153: 9093: 9048: 8941: 8919: 8762: 8551: 8084:"§2.1.3 Normal modes of a one-dimensional chain with a basis" 7904:(1st ed.). Oxford: Oxford University Press. p. 82. 7752: 7697: 7621: 5731: 4714: 4685: 4198:) for some waves corresponding to lattice vibrations in GaAs. 2068:

is the mass of each atom (assuming it is equal for all), and

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Maradudin, A.; Montroll, E.; Weiss, G.; Ipatova, I. (1971).

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and the total energy of a general phonon system is given by

470:. The error in ignoring higher order terms remains small if 9390: 9322: 9282: 8858: 8853: 6541:

is the frequency of the phonons (or photons) in the state,

6064:

is only determined up to addition of constant vectors (the

5706:

As with the quantum harmonic oscillator, one can show that

4639: 4170:

Optical and acoustic vibrations in a linear diatomic chain.

2091: 1167:{\displaystyle u_{n}=\sum _{Nak/2\pi =1}^{N}Q_{k}e^{ikna}.} 661: 236: 47: 8378: 8204: 8202: 6814:

in operator formalism, we have not taken into account the

5263:{\displaystyle \left=\delta _{k,k'},\quad {\Big }=\left=0} 4178:

Vibrations of the diatomic chain at different frequencies.

494:

The potential energy of the lattice may now be written as

8481:. Mineola, New York: Dover Publications, Inc. p. 9. 8318: 5988:. The value is obtained by dividing the frequency by the 914:

the mass of the atom, then the equation of motion of the

8319:

Marquet, C.; Schmidt-Kaler, F.; James, D. F. V. (2003).

7956:

Theory of lattice dynamics in the harmonic approximation

6580:

to occur and therefore cannot be explained by classical

6323:

It is usually convenient to consider phonon wavevectors

487: 8199: 458:

about its equilibrium value to quadratic order, giving

211:

of atoms or molecules uniformly oscillates at a single

6613: 6052:

is not actually a physical momentum; it is called the

5876: 5635: 4462: 4010: 3986: 3962: 3872: 3754: 3531:

The Hamiltonian may be written in wavevector space as

3329: 3218: 3134: 538: 278:

particles. These particles may be atoms or molecules.

8257: 8255: 7499: 7255: 7025: 6734: 6601: 6459: 6411: 6212: 6085: 5764: 5563: 5533: 5506: 5398: 5296: 5077: 4941: 4824: 4790: 4763: 4739: 4454: 4233: 3960: 3852: 3709: 3540: 3417: 3132: 2803: 2358: 2162: 1893: 1859: 1839: 1806: 1760: 1733: 1676: 1465: 1346: 1304: 1255: 1206: 1080: 927: 503: 332: 302:, and others, all of which are ultimately due to the 7972:

A guide to Feynman Diagrams in the many-body problem

7678: 4677:

Many phonon dispersion curves have been measured by

1800:

as measured from their equilibrium positions. (I.e.

1183:

corresponds and devolves to the continuous variable

8612:"Secret of Flow-Induced Electric Currents Revealed" 8425:

Mutual Interactions of Phonons, Rotons, and Gravity

6374:has an energy that is not constant, but fluctuates 4661:The velocity of the wave also is given in terms of 8693:Phonons in a One Dimensional Microfluidic Crystal 8252: 8119: 8117: 8115: 8113: 8111: 8109: 8107: 7969: 7928:Statistical mechanics: algorithms and computations 7865:. London: Imperial College Press. pp. 64–69. 7550: 7453: 7235: 6799: 6710: 6517: 6370:, and contains no phonons. A lattice at a nonzero 6239: 6192: 5897: 5653: 5546: 5519: 5489: 5384: 5262: 5053: 4927: 4803: 4776: 4749: 4480: 4433: 4035: 3940: 3814: 3662: 3520: 3397: 3112: 2783: 2319: 2053: 1872: 1845: 1825: 1792: 1746: 1700: 1579: 1441: 1322: 1290: 1241: 1166: 1050: 607: 397: 179:. Solids with more than one atom in the smallest 8128:. Saunders College Publishing. pp. 780–783. 7446: 7357: 7333: 7268: 7228: 7142: 7112: 7047: 6048:; however, this is not strictly correct, because 5965:can also interact indirectly with light, through 5948:Optical phonons have a non-zero frequency at the 5190: 5155: 4566:For a crystal that has at least two atoms in its 4491:The connection between frequency and wavevector, 4131:is now associated with three normal coordinates. 188: 184: 119:The concept of phonons was introduced in 1930 by 9790: 7918: 8104: 7999:Theoretical Mechanics of Particles and Continua 7997:Fetter, Alexander; Walecka, John (2003-12-16). 5916:Solids with more than one atom in the smallest 5911: 4697:fluids, which only apply to high frequencies). 4107:Canonical quantization § Real scalar field 3696:boundary conditions are imposed, defining the ( 239:phenomena in classical mechanics, phonons have 159:, because long-wavelength phonons give rise to 8503:"Detecting the softest sounds in the Universe" 8123: 7838:. Cambridge University Press. pp. 78–96. 7003: +.... As there are no cross terms (e.g. 163:. The name emphasizes the analogy to the word 8721: 7996: 466:and the elastic force simply proportional to 8423:Alberto Nicolis and Riccardo Penco. (2017). 8238:(3 ed.). India: Elsevier. p. 201. 8233: 8124:Ashcroft, Neil W.; Mermin, N. David (1976). 7829: 7827: 7825: 7823: 7821: 7819: 5500:Directly substituting these definitions for 4481:{\displaystyle k={\tfrac {2\pi }{\lambda }}} 4111: 1992: 1983: 518: 509: 7858: 7620:Recent research has shown that phonons and 6366:temperature, a crystal lattice lies in its 474:remains close to the equilibrium position. 427:is the potential energy between two atoms. 8728: 8714: 8294:. Reading, MA: Benjamin-Cummings. p. 8045: 8043: 7833: 4220:, connected by springs of spring constant 4162:Dispersion curves in linear diatomic chain 1712:, an assembly of independent oscillators. 910:is the elastic constant of the spring and 704: 8627: 8570: 8528: 8408: 8339: 8153:. Princeton: Princeton University Press. 8049: 7816: 7640:is a state of electronic matter in which 5809: 4507:. The plus sign results in the so-called 4102:and operator techniques described later. 2146:defined as the Fourier transforms of the 478:phonons. (For other common lattices, see 255:The equations in this section do not use 8735: 8609: 8427:, Arxiv.org, Retrieved November 27, 2018 8291:Statistical Mechanics, A Set of Lectures 8077: 8075: 6314: 6013: 4181: 4173: 4165: 4157: 708: 697:do possess well-defined wavelengths and 651: 243:properties too, in a way related to the 8474: 8287: 8040: 7967: 7802:(4th ed.). Springer. p. 253. 7797: 7615: 1452:These are the equations for decoupled 903:th atom from its equilibrium position. 9791: 9199: 8495: 8173: 8050:Yu, Peter Y.; Cardona, Manuel (2010). 7924: 4622: ≥ 2 different atoms in the 4153: 722: 16:Quasiparticle of mechanical vibrations 8709: 8385:Journal of Physics: Conference Series 8148: 8142: 8081: 8072: 8021: 7899: 7834:Girvin, Steven M.; Yang, Kun (2019). 6587: 6240:{\displaystyle K={\frac {2n\pi }{a}}} 4757:, as well as the conjugate position, 8015: 7950: 7948: 7632: 7588:), so therefore they are considered 6312:obey a similar set of restrictions. 4216:repeated periodically at a distance 1715: 1701:{\displaystyle \omega (k)\propto ka} 1061:This is a set of coupled equations. 8177:Introduction to Solid State Physics 6921:The ground state, also called the " 6723:creation and annihilation operators 6592:The phonon Hamiltonian is given by 6564:Phonons have been shown to exhibit 6559: 6327:which have the smallest magnitude | 6003: 5937:because in ionic crystals, such as 4813:creation and annihilation operators 4626:exhibits three acoustic modes: one 4120:is replaced by a three-dimensional 250: 13: 6737: 6604: 6489: 6175: 6166: 6163: 6160: 6141: 6112: 6109: 6106: 5767: 5566: 5535: 5400: 5037: 4908: 4792: 4742: 4729:, is a means of extracting energy 4634:. The number of optical modes is 3 3592: 3582: 3543: 3094: 3084: 2752: 2739: 2448: 2240: 2124:may be introduced, defined as the 2002: 1999: 1896: 1793:{\displaystyle x_{1},x_{2},\dots } 1708:. This amounts to classical free 1242:{\displaystyle \phi _{k}=e^{ikna}} 633:is the position coordinate of the 528: 525: 14: 9820: 8664: 8478:Introduction to Superconductivity 7945: 7663: 6755: 6690: 6341: 6034:, phonons have been treated with 5806: 5584: 4986: 4860: 4224:, two modes of vibration result: 4021: 3997: 3973: 3951:The levels are evenly spaced at: 3894: 2660: 2595: 2406: 2086:operators, respectively, for the 462:proportional to the displacement 9772: 9665:Timeline of particle discoveries 8670: 8610:Schirber, Michael (2023-02-17). 7681: 4511:mode, and the minus sign to the 3689:uncoupled harmonic oscillators. 688:Nyquist–Shannon sampling theorem 647: 486: 8603: 8545: 8468: 8443: 8430: 8417: 8372: 8312: 8281: 8227: 8167: 7836:Modern Condensed Matter Physics 7595: 6139: 5152: 3907: 3752: 1515: 8410:10.1088/1742-6596/193/1/012121 8151:Condensed Matter in a Nutshell 8090:. Academic Press. p. 44. 8082:Misra, Prasanta Kumar (2010). 8056:Fundamentals of Semiconductors 7990: 7961: 7893: 7852: 7791: 7416: 7397: 7201: 7182: 6866:terms will add up yielding an 6018:k-vectors exceeding the first 5066:canonical commutation relation 4750:{\displaystyle {\mathcal {H}}} 3322: 3275: 2006: 1995: 1686: 1680: 1569: 1546: 1376: 1353: 1285: 1276: 532: 521: 430:It is difficult to solve this 266:For example: a rigid regular, 1: 8263:"Tunneling across a tiny gap" 8210:"Non-metals: thermal phonons" 7925:Krauth, Werner (April 2006). 7902:The Oxford solid state basics 7885:: CS1 maint: date and year ( 7784: 5971:Lyddane–Sachs–Teller relation 3123:The potential energy term is 1291:{\displaystyle k=2\pi j/(Na)} 1068:, in order to decouple them. 643:multiscale Green's functions. 203:description of an elementary 194: 9681:History of subatomic physics 8288:Feynman, Richard P. (1982). 7743:Relativistic heat conduction 5912:Acoustic and optical phonons 4727:quantum harmonic oscillators 4679:inelastic neutron scattering 4642:as a function of wavevector 1187:of scalar field theory. The 167:, in that phonons represent 147: 7: 8515:(7763): 8–9. July 1, 2019. 8440:Retrieved November 11, 2018 8088:Physics of Condensed Matter 7859:Kozhevnikov, A. B. (2004). 7674: 7604:, phonons can interact via 7484:quantum harmonic oscillator 6576:to occur but too small for 6254:. A phonon with wavenumber 5980:notation, where the symbol 5961:. Optical phonons that are 5743:position and momentum space 4071:quantum harmonic oscillator 2126:discrete Fourier transforms 1880:are vector quantities. The 443:. Secondly, the potentials 10: 9825: 8689:Optical and acoustic modes 8646:10.1103/PhysRevX.13.011020 8589:10.1103/PhysRevX.13.011020 8530:10.1038/d41586-019-02009-5 8001:. Dover Books on Physics. 7800:Advanced Quantum Mechanics 7606:parametric down conversion 6066:reciprocal lattice vectors 6007: 5273:Using this, the operators 4721:technique, similar to the 4628:longitudinal acoustic mode 4104: 1323:{\displaystyle j=1\dots N} 1200:for continuum field modes 1066:discrete Fourier transform 882:th atom out of a total of 235:). While normal modes are 141: 112:, as well as in models of 18: 9770: 9673: 9637: 9554: 9515: 9485: 9459: 9455: 9446: 9378: 9346: 9273: 9208: 9190: 9086: 9041: 9013: 9004: 8995: 8977: 8955: 8927: 8918: 8834: 8761: 8752: 8743: 8475:Tinkham, Michael (1996). 8358:10.1007/s00340-003-1097-7 7900:Simon, Steven H. (2013). 7471:creates a phonon of type 4784:, and conjugate momentum 4632:transverse acoustic modes 4112:Three-dimensional lattice 1608:The second equation, for 300:electrostatic attractions 274:) lattice is composed of 9698:mathematical formulation 9293:Eta and eta prime mesons 8174:Kittel, Charles (2004). 7610:squeezed coherent states 7488:particle number operator 6810:Here, in expressing the 6400:Bose–Einstein statistics 5990:speed of light in vacuum 5955:electrical dipole moment 5547:{\displaystyle \Pi _{k}} 4804:{\displaystyle \Pi _{k}} 4684:The physics of sound in 4670:for phonon propagation. 2794:From the general result 1456:which have the solution 899:the displacement of the 263:in classical mechanics. 19:Not to be confused with 9360:Double-charm tetraquark 8234:Pathria; Beale (2011). 7798:Schwabl, Franz (2008). 7464:The creation operator, 4650:of its Brillouin zone. 4614:are those of the first 1826:{\displaystyle x_{i}=0} 1670:, a scalar field, and 1590:Each normal coordinate 705:One-dimensional lattice 151:), which translates to 110:electrical conductivity 8675:Quotations related to 8149:Mahan, Gerald (2010). 8026:. New York: Springer. 7552: 7455: 7237: 6801: 6712: 6519: 6398:. Both gases obey the 6392:electromagnetic cavity 6320: 6241: 6194: 6023: 5899: 5805: 5745:.) Finally, using the 5655: 5548: 5521: 5491: 5386: 5264: 5055: 4929: 4805: 4778: 4751: 4709:, then it describes a 4482: 4435: 4199: 4179: 4171: 4163: 4037: 3942: 3833:, as discussed above. 3816: 3664: 3522: 3399: 3114: 2785: 2321: 2055: 1924: 1874: 1847: 1827: 1794: 1748: 1702: 1601:, which is known as a 1581: 1443: 1324: 1292: 1243: 1168: 1131: 1052: 715: 657: 609: 399: 247:of quantum mechanics. 9757:Wave–particle duality 9747:Relativistic particle 8884:Electron antineutrino 8236:Statistical Mechanics 8024:Many-Particle Physics 8022:Mahan, G. D. (1981). 7763:Surface acoustic wave 7642:electrical resistance 7553: 7456: 7238: 6802: 6725:, these are given by 6713: 6520: 6354:which determines the 6318: 6242: 6195: 6017: 5900: 5785: 5656: 5549: 5522: 5520:{\displaystyle Q_{k}} 5492: 5387: 5265: 5056: 4930: 4806: 4779: 4777:{\displaystyle Q_{k}} 4752: 4574:. The boundaries at − 4483: 4436: 4185: 4177: 4169: 4161: 4093:electromagnetic field 4038: 3943: 3817: 3665: 3523: 3400: 3115: 2786: 2338:of the phonon, i.e. 2 2322: 2117:"normal coordinates" 2082:are the position and 2056: 1904: 1875: 1873:{\displaystyle x_{i}} 1848: 1828: 1795: 1749: 1747:{\displaystyle u_{i}} 1703: 1582: 1444: 1325: 1293: 1244: 1169: 1094: 1053: 712: 655: 610: 400: 245:wave–particle duality 231:vibration modes (cf. 169:wave-particle duality 116:and related effects. 40:collective excitation 8987:Faddeev–Popov ghosts 8737:Particles in physics 8393:Institute of Physics 7968:Mattuck, R. (1976). 7773:Thermal conductivity 7703:Brillouin scattering 7616:Predicted properties 7497: 7486:case, we can define 7253: 7023: 6732: 6599: 6556:is the temperature. 6409: 6396:black-body radiation 6210: 6083: 6076:are defined so that 5762: 5749:correlation function 5561: 5531: 5504: 5396: 5294: 5075: 4939: 4822: 4788: 4761: 4737: 4711:quantum field theory 4648:principal directions 4551:.) At low values of 4452: 4231: 4186:Dispersion relation 4138:= 1, 2, 3 label the 4127:. Furthermore, each 3958: 3850: 3707: 3538: 3415: 3130: 2801: 2356: 2334:turns out to be the 2160: 2139:"conjugate momenta" 1891: 1857: 1837: 1804: 1758: 1731: 1674: 1463: 1454:harmonic oscillators 1344: 1302: 1253: 1204: 1078: 925: 501: 330: 292:Van der Waals forces 135:. It comes from the 106:thermal conductivity 9762:Particle chauvinism 9705:Subatomic particles 8638:2023PhRvX..13a1020L 8581:2023PhRvX..13a1020L 8521:2019Natur.571....8. 8401:2009JPhCS.193a2121R 8350:2003ApPhB..76..199M 8126:Solid State Physics 6948:, we say there are 6686: 6671: 6653: 6041:as though it has a 4719:second quantization 4713:of non-interacting 4505:dispersion relation 4248: 4154:Dispersion relation 4100:second quantization 3825:The upper bound to 3631: 1941: 1884:for this system is 1710:scalar field theory 1619:dispersion relation 1335:decoupled equations 723:Classical treatment 449:harmonic potentials 217:classical mechanics 87:modes of vibrations 9809:1932 introductions 8683:Explained: Phonons 8181:. Wiley. pp. 7728:Carrier scattering 7548: 7515: 7451: 7233: 6797: 6754: 6708: 6672: 6657: 6639: 6633: 6622: 6588:Operator formalism 6550:Boltzmann constant 6515: 6500: 6386:is similar to the 6321: 6237: 6190: 6060:. This is because 6024: 5943:infrared radiation 5895: 5885: 5784: 5747:position–position 5651: 5644: 5583: 5544: 5517: 5487: 5392: and 5382: 5260: 5051: 4935: and 4925: 4801: 4774: 4747: 4478: 4476: 4431: 4234: 4200: 4180: 4172: 4164: 4144:longitudinal waves 4033: 4019: 3995: 3971: 3938: 3881: 3812: 3758: 3660: 3617: 3575: 3518: 3395: 3349: 3338: 3251: 3227: 3167: 3143: 3110: 3108: 3082: 3046: 2990: 2913: 2878: 2817: 2781: 2779: 2613: 2499: 2317: 2315: 2277: 2205: 2051: 2010: 1927: 1870: 1843: 1823: 1790: 1744: 1698: 1617:, is known as the 1577: 1439: 1320: 1288: 1239: 1198:normal coordinates 1164: 1048: 716: 658: 605: 547: 536: 395: 358: 219:this designates a 207:motion in which a 201:quantum mechanical 114:neutron scattering 80:quantum mechanical 58:, specifically in 9786: 9785: 9742:Massless particle 9550: 9549: 9546: 9545: 9511: 9510: 9374: 9373: 9186: 9185: 9182: 9181: 9134:Magnetic monopole 9082: 9081: 8973: 8972: 8914: 8913: 8894:Muon antineutrino 8879:Electron neutrino 8685:, MIT News, 2010. 8559:Physical Review X 8438:The mass of sound 8328:Applied Physics B 8305:978-0-8053-2508-9 8245:978-93-80931-89-0 8192:978-0-471-41526-8 8160:978-0-691-14016-2 8097:978-0-12-384954-0 8065:978-3-642-00709-5 8033:978-0-306-46338-9 7938:978-0-19-851536-4 7911:978-0-19-968077-1 7872:978-1-86094-419-2 7845:978-1-107-13739-4 7809:978-3-540-85062-5 7723:Phonon scattering 7713:Linear elasticity 7638:Superconductivity 7633:Superconductivity 7506: 7353: 7138: 6836:term as, given a 6745: 6624: 6621: 6566:quantum tunneling 6513: 6499: 6352:density of states 6235: 6171: 6117: 5884: 5775: 5643: 5574: 5440: 5439: 5335: 5334: 5034: 4991: 4990: 4905: 4865: 4864: 4618:. A crystal with 4503:), is known as a 4475: 4426: 4424: 4399: 4351: 4331: 4295: 4275: 4067:zero-point energy 4029: 4018: 4008: 3994: 3984: 3970: 3880: 3811: 3804: 3757: 3750: 3566: 3564: 3511: 3476: 3340: 3337: 3242: 3226: 3158: 3142: 3073: 3037: 2981: 2904: 2861: 2859: 2808: 2604: 2602: 2484: 2482: 2268: 2266: 2265: 2196: 2194: 2193: 1978: 1963: 1950: 1846:{\displaystyle i} 1754:, but instead by 1716:Quantum treatment 1623:angular frequency 1572: 1544: 1434: 1196:are known as the 1043: 730:adiabatic theorem 624:natural frequency 546: 504: 480:crystal structure 432:many-body problem 343: 341: 131:was suggested by 74:, a phonon is an 25:Phonon (software) 9816: 9776: 9752:Virtual particle 9523:Mesonic molecule 9457: 9456: 9453: 9452: 9298:Bottom eta meson 9206: 9205: 9197: 9196: 9169:W′ and Z′ bosons 9159:Sterile neutrino 9144:Majorana fermion 9011: 9010: 9002: 9001: 8925: 8924: 8904:Tau antineutrino 8759: 8758: 8750: 8749: 8730: 8723: 8716: 8707: 8706: 8674: 8658: 8657: 8631: 8607: 8601: 8600: 8574: 8549: 8543: 8542: 8532: 8499: 8493: 8492: 8472: 8466: 8465: 8463: 8461: 8447: 8441: 8434: 8428: 8421: 8415: 8414: 8412: 8376: 8370: 8369: 8343: 8341:quant-ph/0211079 8325: 8316: 8310: 8309: 8285: 8279: 8278: 8276: 8274: 8259: 8250: 8249: 8231: 8225: 8224: 8222: 8220: 8206: 8197: 8196: 8171: 8165: 8164: 8146: 8140: 8139: 8121: 8102: 8101: 8079: 8070: 8069: 8047: 8038: 8037: 8019: 8013: 8012: 7994: 7988: 7987: 7975: 7965: 7959: 7952: 7943: 7942: 7922: 7916: 7915: 7897: 7891: 7890: 7884: 7876: 7856: 7850: 7849: 7831: 7814: 7813: 7795: 7748:Rigid unit modes 7733:Phononic crystal 7691: 7686: 7685: 7587: 7575: 7557: 7555: 7554: 7549: 7544: 7543: 7534: 7533: 7528: 7527: 7526: 7514: 7460: 7458: 7457: 7452: 7450: 7449: 7437: 7436: 7409: 7408: 7393: 7392: 7371: 7370: 7361: 7360: 7354: 7352: 7351: 7342: 7337: 7336: 7327: 7326: 7311: 7310: 7301: 7300: 7282: 7281: 7272: 7271: 7265: 7264: 7242: 7240: 7239: 7234: 7232: 7231: 7222: 7221: 7194: 7193: 7178: 7177: 7156: 7155: 7146: 7145: 7139: 7131: 7130: 7121: 7116: 7115: 7106: 7105: 7090: 7089: 7080: 7079: 7061: 7060: 7051: 7050: 7044: 7043: 7038: 7037: 7036: 6955:phonons of type 6947: 6907: 6905: 6904: 6901: 6898: 6885: 6883: 6882: 6879: 6876: 6859: 6857: 6856: 6853: 6850: 6842:infinite lattice 6829: 6827: 6826: 6823: 6820: 6806: 6804: 6803: 6798: 6796: 6795: 6786: 6785: 6780: 6779: 6778: 6767: 6766: 6753: 6741: 6740: 6721:In terms of the 6717: 6715: 6714: 6709: 6707: 6703: 6702: 6701: 6685: 6680: 6670: 6665: 6652: 6647: 6632: 6623: 6614: 6608: 6607: 6570:phonon tunneling 6560:Phonon tunneling 6524: 6522: 6521: 6516: 6514: 6512: 6505: 6501: 6498: 6494: 6493: 6492: 6481: 6480: 6479: 6460: 6443: 6438: 6434: 6433: 6307: 6305: 6304: 6299: 6296: 6295: 6282: 6280: 6279: 6274: 6271: 6270: 6250:for any integer 6246: 6244: 6243: 6238: 6236: 6231: 6220: 6199: 6197: 6196: 6191: 6189: 6188: 6173: 6172: 6170: 6169: 6157: 6152: 6149: 6148: 6135: 6134: 6119: 6118: 6116: 6115: 6103: 6098: 6095: 6094: 6054:crystal momentum 6010:Crystal momentum 6004:Crystal momentum 5967:Raman scattering 5924:Acoustic phonons 5904: 5902: 5901: 5896: 5891: 5887: 5886: 5877: 5871: 5870: 5855: 5854: 5849: 5848: 5847: 5825: 5824: 5804: 5799: 5783: 5771: 5770: 5660: 5658: 5657: 5652: 5650: 5646: 5645: 5636: 5630: 5629: 5620: 5619: 5614: 5613: 5612: 5596: 5595: 5582: 5570: 5569: 5553: 5551: 5550: 5545: 5543: 5542: 5526: 5524: 5523: 5518: 5516: 5515: 5496: 5494: 5493: 5488: 5486: 5482: 5481: 5480: 5465: 5464: 5459: 5458: 5457: 5441: 5435: 5434: 5433: 5417: 5416: 5408: 5407: 5391: 5389: 5388: 5383: 5381: 5377: 5376: 5375: 5360: 5359: 5354: 5353: 5352: 5336: 5333: 5332: 5331: 5312: 5311: 5306: 5305: 5269: 5267: 5266: 5261: 5253: 5249: 5248: 5247: 5242: 5241: 5240: 5239: 5221: 5220: 5215: 5214: 5213: 5194: 5193: 5187: 5186: 5185: 5169: 5168: 5159: 5158: 5148: 5147: 5146: 5124: 5120: 5119: 5118: 5113: 5112: 5111: 5110: 5092: 5091: 5060: 5058: 5057: 5052: 5050: 5046: 5045: 5044: 5035: 5033: 5032: 5031: 5015: 5010: 5009: 4992: 4989: 4981: 4980: 4979: 4966: 4965: 4960: 4959: 4954: 4953: 4952: 4934: 4932: 4931: 4926: 4924: 4920: 4919: 4918: 4906: 4904: 4903: 4902: 4886: 4881: 4880: 4866: 4863: 4855: 4854: 4853: 4840: 4839: 4834: 4833: 4810: 4808: 4807: 4802: 4800: 4799: 4783: 4781: 4780: 4775: 4773: 4772: 4756: 4754: 4753: 4748: 4746: 4745: 4725:method used for 4613: 4611: 4610: 4605: 4602: 4601: 4593: 4591: 4590: 4585: 4582: 4581: 4546: 4544: 4543: 4537: 4534: 4487: 4485: 4484: 4479: 4477: 4471: 4463: 4440: 4438: 4437: 4432: 4427: 4425: 4423: 4422: 4421: 4412: 4411: 4401: 4400: 4395: 4387: 4382: 4381: 4368: 4363: 4362: 4357: 4353: 4352: 4350: 4349: 4337: 4332: 4330: 4329: 4317: 4309: 4301: 4297: 4296: 4294: 4293: 4281: 4276: 4274: 4273: 4261: 4247: 4242: 4148:transverse waves 4134:The new indices 4061: 4059: 4058: 4055: 4052: 4042: 4040: 4039: 4034: 4027: 4020: 4011: 4006: 3996: 3987: 3982: 3972: 3963: 3947: 3945: 3944: 3939: 3906: 3905: 3893: 3889: 3882: 3873: 3862: 3861: 3821: 3819: 3818: 3813: 3809: 3805: 3797: 3759: 3755: 3751: 3749: 3741: 3730: 3725: 3724: 3669: 3667: 3666: 3661: 3659: 3655: 3654: 3653: 3641: 3640: 3630: 3625: 3616: 3615: 3603: 3602: 3590: 3589: 3574: 3565: 3563: 3552: 3547: 3546: 3527: 3525: 3524: 3519: 3517: 3513: 3512: 3507: 3499: 3477: 3475: 3471: 3470: 3445: 3444: 3432: 3427: 3426: 3404: 3402: 3401: 3396: 3394: 3393: 3381: 3380: 3371: 3370: 3365: 3364: 3363: 3348: 3339: 3330: 3321: 3320: 3299: 3298: 3274: 3273: 3261: 3260: 3250: 3241: 3240: 3228: 3219: 3213: 3212: 3207: 3203: 3202: 3201: 3183: 3182: 3166: 3157: 3156: 3144: 3135: 3119: 3117: 3116: 3111: 3109: 3105: 3104: 3092: 3091: 3081: 3065: 3064: 3059: 3058: 3057: 3045: 3032: 3031: 3013: 3012: 3000: 2999: 2989: 2977: 2976: 2975: 2953: 2952: 2951: 2947: 2946: 2912: 2903: 2902: 2901: 2888: 2887: 2877: 2876: 2860: 2852: 2843: 2842: 2827: 2826: 2816: 2790: 2788: 2787: 2782: 2780: 2770: 2766: 2765: 2764: 2763: 2747: 2746: 2725: 2721: 2720: 2719: 2718: 2702: 2701: 2683: 2682: 2681: 2653: 2652: 2651: 2647: 2646: 2612: 2603: 2598: 2590: 2582: 2578: 2574: 2573: 2572: 2560: 2559: 2545: 2544: 2537: 2518: 2517: 2498: 2483: 2475: 2466: 2462: 2461: 2460: 2459: 2443: 2442: 2424: 2423: 2395: 2391: 2390: 2389: 2377: 2376: 2341: 2326: 2324: 2323: 2318: 2316: 2309: 2308: 2299: 2298: 2276: 2267: 2261: 2257: 2248: 2247: 2234: 2233: 2224: 2223: 2204: 2195: 2189: 2185: 2176: 2175: 2060: 2058: 2057: 2052: 2050: 2049: 2044: 2040: 2039: 2038: 2026: 2025: 2009: 2005: 1977: 1976: 1964: 1956: 1951: 1949: 1940: 1935: 1926: 1923: 1918: 1900: 1899: 1879: 1877: 1876: 1871: 1869: 1868: 1852: 1850: 1849: 1844: 1832: 1830: 1829: 1824: 1816: 1815: 1799: 1797: 1796: 1791: 1783: 1782: 1770: 1769: 1753: 1751: 1750: 1745: 1743: 1742: 1707: 1705: 1704: 1699: 1669: 1658: 1649: 1643: 1639: 1616: 1600: 1586: 1584: 1583: 1578: 1573: 1568: 1545: 1540: 1532: 1530: 1525: 1524: 1511: 1510: 1506: 1505: 1488: 1487: 1475: 1474: 1448: 1446: 1445: 1440: 1435: 1433: 1432: 1431: 1418: 1417: 1416: 1407: 1406: 1396: 1388: 1387: 1375: 1329: 1327: 1326: 1321: 1297: 1295: 1294: 1289: 1275: 1248: 1246: 1245: 1240: 1238: 1237: 1216: 1215: 1195: 1186: 1182: 1173: 1171: 1170: 1165: 1160: 1159: 1141: 1140: 1130: 1125: 1112: 1090: 1089: 1057: 1055: 1054: 1049: 1044: 1042: 1041: 1040: 1027: 1026: 1025: 1016: 1015: 1005: 997: 993: 992: 991: 973: 972: 946: 945: 917: 913: 902: 898: 889: 885: 881: 877: 847: 838: 823: 819: 779: 773: 769: 762: 756: 614: 612: 611: 606: 601: 600: 595: 591: 590: 589: 577: 576: 561: 560: 548: 539: 535: 531: 490: 453:Taylor expanding 404: 402: 401: 396: 394: 390: 389: 388: 376: 375: 357: 342: 334: 316:potential energy 251:Lattice dynamics 233:Fourier analysis 199:A phonon is the 150: 144: 143: 56:condensed matter 29:Phonon (company) 9824: 9823: 9819: 9818: 9817: 9815: 9814: 9813: 9789: 9788: 9787: 9782: 9766: 9720:Nuclear physics 9669: 9633: 9569:Davydov soliton 9542: 9507: 9481: 9442: 9370: 9342: 9269: 9178: 9078: 9037: 8991: 8969: 8951: 8910: 8830: 8739: 8734: 8699:with movies in 8667: 8662: 8661: 8608: 8604: 8550: 8546: 8501: 8500: 8496: 8489: 8473: 8469: 8459: 8457: 8449: 8448: 8444: 8435: 8431: 8422: 8418: 8377: 8373: 8323: 8317: 8313: 8306: 8286: 8282: 8272: 8270: 8261: 8260: 8253: 8246: 8232: 8228: 8218: 8216: 8208: 8207: 8200: 8193: 8172: 8168: 8161: 8147: 8143: 8136: 8122: 8105: 8098: 8080: 8073: 8066: 8048: 8041: 8034: 8020: 8016: 8009: 7995: 7991: 7984: 7976:. McGraw-Hill. 7966: 7962: 7953: 7946: 7939: 7923: 7919: 7912: 7898: 7894: 7878: 7877: 7873: 7857: 7853: 7846: 7832: 7817: 7810: 7796: 7792: 7787: 7782: 7718:Mechanical wave 7687: 7680: 7677: 7666: 7646:magnetic fields 7635: 7618: 7598: 7577: 7565: 7539: 7535: 7529: 7522: 7518: 7517: 7516: 7510: 7498: 7495: 7494: 7480: 7469: 7445: 7444: 7426: 7422: 7404: 7400: 7382: 7378: 7366: 7362: 7356: 7355: 7347: 7343: 7341: 7332: 7331: 7316: 7312: 7306: 7302: 7290: 7286: 7277: 7273: 7267: 7266: 7260: 7256: 7254: 7251: 7250: 7227: 7226: 7211: 7207: 7189: 7185: 7167: 7163: 7151: 7147: 7141: 7140: 7126: 7122: 7120: 7111: 7110: 7095: 7091: 7085: 7081: 7069: 7065: 7056: 7052: 7046: 7045: 7039: 7032: 7028: 7027: 7026: 7024: 7021: 7020: 7015: 7009: 7002: 6996: 6989: 6983: 6975: 6964: 6953: 6945: 6939: 6933: 6926: 6912: 6902: 6899: 6896: 6895: 6893: 6890: 6880: 6877: 6874: 6873: 6871: 6864: 6854: 6851: 6848: 6847: 6845: 6834: 6824: 6821: 6818: 6817: 6815: 6791: 6787: 6781: 6774: 6770: 6769: 6768: 6762: 6758: 6749: 6736: 6735: 6733: 6730: 6729: 6697: 6693: 6681: 6676: 6666: 6661: 6648: 6643: 6638: 6634: 6628: 6612: 6603: 6602: 6600: 6597: 6596: 6590: 6562: 6547: 6540: 6488: 6487: 6483: 6482: 6469: 6465: 6461: 6458: 6454: 6447: 6442: 6423: 6419: 6415: 6410: 6407: 6406: 6390:produced by an 6344: 6310:Bloch electrons 6300: 6297: 6293: 6291: 6290: 6288: 6275: 6272: 6268: 6266: 6265: 6263: 6221: 6219: 6211: 6208: 6207: 6178: 6174: 6159: 6158: 6153: 6151: 6150: 6144: 6140: 6124: 6120: 6105: 6104: 6099: 6097: 6096: 6090: 6086: 6084: 6081: 6080: 6012: 6006: 5959:infrared active 5939:sodium chloride 5931:Optical phonons 5914: 5875: 5860: 5856: 5850: 5837: 5833: 5832: 5831: 5830: 5826: 5814: 5810: 5800: 5789: 5779: 5766: 5765: 5763: 5760: 5759: 5739: 5725: 5718: 5711: 5690: 5682: 5676: 5669: 5634: 5625: 5621: 5615: 5608: 5604: 5603: 5602: 5601: 5597: 5591: 5587: 5578: 5565: 5564: 5562: 5559: 5558: 5538: 5534: 5532: 5529: 5528: 5511: 5507: 5505: 5502: 5501: 5473: 5469: 5460: 5453: 5449: 5448: 5447: 5446: 5442: 5429: 5425: 5418: 5415: 5403: 5399: 5397: 5394: 5393: 5368: 5364: 5355: 5348: 5344: 5343: 5342: 5341: 5337: 5327: 5323: 5316: 5310: 5301: 5297: 5295: 5292: 5291: 5285: 5278: 5243: 5232: 5231: 5227: 5226: 5225: 5216: 5209: 5205: 5204: 5203: 5202: 5198: 5189: 5188: 5178: 5177: 5173: 5164: 5160: 5154: 5153: 5139: 5132: 5128: 5114: 5103: 5102: 5098: 5097: 5096: 5087: 5083: 5082: 5078: 5076: 5073: 5072: 5040: 5036: 5027: 5023: 5019: 5014: 5002: 4998: 4997: 4993: 4982: 4975: 4971: 4967: 4964: 4955: 4948: 4944: 4943: 4942: 4940: 4937: 4936: 4911: 4907: 4898: 4894: 4890: 4885: 4876: 4872: 4871: 4867: 4856: 4849: 4845: 4841: 4838: 4829: 4825: 4823: 4820: 4819: 4795: 4791: 4789: 4786: 4785: 4768: 4764: 4762: 4759: 4758: 4741: 4740: 4738: 4735: 4734: 4723:ladder operator 4703: 4606: 4603: 4599: 4598: 4597: 4595: 4586: 4583: 4579: 4578: 4577: 4575: 4538: 4535: 4532: 4526: 4525: 4523: 4464: 4461: 4453: 4450: 4449: 4417: 4413: 4407: 4403: 4402: 4388: 4386: 4377: 4373: 4369: 4367: 4358: 4345: 4341: 4336: 4325: 4321: 4316: 4315: 4311: 4310: 4308: 4289: 4285: 4280: 4269: 4265: 4260: 4259: 4255: 4243: 4238: 4232: 4229: 4228: 4215: 4208: 4156: 4114: 4109: 4056: 4053: 4050: 4049: 4047: 4009: 3985: 3961: 3959: 3956: 3955: 3901: 3897: 3871: 3870: 3866: 3857: 3853: 3851: 3848: 3847: 3841: 3796: 3753: 3742: 3731: 3729: 3720: 3716: 3708: 3705: 3704: 3646: 3642: 3636: 3632: 3626: 3621: 3611: 3607: 3595: 3591: 3585: 3581: 3580: 3576: 3570: 3556: 3551: 3542: 3541: 3539: 3536: 3535: 3500: 3498: 3491: 3487: 3463: 3450: 3446: 3440: 3436: 3431: 3422: 3418: 3416: 3413: 3412: 3386: 3382: 3376: 3372: 3366: 3359: 3355: 3354: 3353: 3344: 3328: 3307: 3303: 3288: 3284: 3266: 3262: 3256: 3252: 3246: 3236: 3232: 3217: 3208: 3191: 3187: 3178: 3174: 3173: 3169: 3168: 3162: 3152: 3148: 3133: 3131: 3128: 3127: 3107: 3106: 3097: 3093: 3087: 3083: 3077: 3066: 3060: 3053: 3049: 3048: 3047: 3041: 3034: 3033: 3018: 3014: 3005: 3001: 2995: 2991: 2985: 2968: 2958: 2954: 2939: 2932: 2928: 2918: 2914: 2908: 2894: 2893: 2889: 2883: 2879: 2869: 2865: 2851: 2844: 2832: 2828: 2822: 2818: 2812: 2804: 2802: 2799: 2798: 2778: 2777: 2756: 2755: 2751: 2742: 2738: 2737: 2733: 2726: 2711: 2710: 2706: 2697: 2693: 2692: 2688: 2685: 2684: 2674: 2667: 2663: 2639: 2632: 2628: 2618: 2614: 2608: 2591: 2589: 2580: 2579: 2568: 2564: 2555: 2551: 2550: 2546: 2530: 2523: 2519: 2504: 2500: 2488: 2474: 2467: 2452: 2451: 2447: 2438: 2434: 2433: 2429: 2426: 2425: 2413: 2409: 2396: 2385: 2381: 2372: 2368: 2367: 2363: 2359: 2357: 2354: 2353: 2342:divided by the 2339: 2314: 2313: 2304: 2300: 2282: 2278: 2272: 2256: 2249: 2243: 2239: 2236: 2235: 2229: 2225: 2210: 2206: 2200: 2184: 2177: 2171: 2167: 2163: 2161: 2158: 2157: 2151: 2144: 2133: 2122: 2110:of the system. 2080: 2073: 2045: 2034: 2030: 2021: 2017: 2016: 2012: 2011: 1998: 1982: 1972: 1968: 1955: 1942: 1936: 1931: 1925: 1919: 1908: 1895: 1894: 1892: 1889: 1888: 1864: 1860: 1858: 1855: 1854: 1838: 1835: 1834: 1811: 1807: 1805: 1802: 1801: 1778: 1774: 1765: 1761: 1759: 1756: 1755: 1738: 1734: 1732: 1729: 1728: 1718: 1675: 1672: 1671: 1660: 1656: 1651: 1645: 1641: 1637: 1634:continuum limit 1614: 1609: 1598: 1595: 1561: 1533: 1531: 1529: 1520: 1516: 1501: 1497: 1493: 1489: 1483: 1479: 1470: 1466: 1464: 1461: 1460: 1427: 1423: 1419: 1412: 1408: 1402: 1398: 1397: 1395: 1383: 1379: 1362: 1345: 1342: 1341: 1303: 1300: 1299: 1271: 1254: 1251: 1250: 1224: 1220: 1211: 1207: 1205: 1202: 1201: 1193: 1188: 1184: 1178: 1146: 1142: 1136: 1132: 1126: 1108: 1098: 1085: 1081: 1079: 1076: 1075: 1036: 1032: 1028: 1021: 1017: 1011: 1007: 1006: 1004: 981: 977: 962: 958: 957: 953: 941: 937: 926: 923: 922: 915: 911: 900: 896: 891: 887: 883: 879: 875: 856: 845: 843: 836: 835: 821: 817: 777: 771: 767: 760: 754: 725: 707: 650: 631: 596: 585: 581: 572: 568: 567: 563: 562: 556: 552: 537: 524: 508: 502: 499: 498: 447:are treated as 413: 384: 380: 371: 367: 366: 362: 347: 333: 331: 328: 327: 284:Avogadro number 253: 197: 46:arrangement of 42:in a periodic, 32: 17: 12: 11: 5: 9822: 9812: 9811: 9806: 9801: 9799:Quasiparticles 9784: 9783: 9779:Physics portal 9771: 9768: 9767: 9765: 9764: 9759: 9754: 9749: 9744: 9739: 9734: 9733: 9732: 9722: 9717: 9712: 9707: 9702: 9701: 9700: 9693:Standard Model 9690: 9689: 9688: 9677: 9675: 9671: 9670: 9668: 9667: 9662: 9660:Quasiparticles 9657: 9652: 9647: 9641: 9639: 9635: 9634: 9632: 9631: 9626: 9621: 9616: 9611: 9606: 9601: 9596: 9591: 9586: 9581: 9576: 9571: 9566: 9560: 9558: 9556:Quasiparticles 9552: 9551: 9548: 9547: 9544: 9543: 9541: 9540: 9535: 9530: 9525: 9519: 9517: 9513: 9512: 9509: 9508: 9506: 9505: 9500: 9495: 9489: 9487: 9483: 9482: 9480: 9479: 9474: 9469: 9463: 9461: 9450: 9444: 9443: 9441: 9440: 9435: 9430: 9429: 9428: 9423: 9418: 9413: 9408: 9403: 9393: 9388: 9382: 9380: 9376: 9375: 9372: 9371: 9369: 9368: 9363: 9352: 9350: 9348:Exotic hadrons 9344: 9343: 9341: 9340: 9335: 9330: 9325: 9320: 9315: 9310: 9305: 9300: 9295: 9290: 9285: 9279: 9277: 9271: 9270: 9268: 9267: 9262: 9257: 9252: 9247: 9242: 9241: 9240: 9235: 9230: 9225: 9214: 9212: 9203: 9194: 9188: 9187: 9184: 9183: 9180: 9179: 9177: 9176: 9174:X and Y bosons 9171: 9166: 9161: 9156: 9151: 9146: 9141: 9136: 9131: 9126: 9121: 9116: 9111: 9106: 9101: 9096: 9090: 9088: 9084: 9083: 9080: 9079: 9077: 9076: 9066: 9061: 9056: 9051: 9045: 9043: 9039: 9038: 9036: 9035: 9030: 9025: 9019: 9017: 9008: 8999: 8993: 8992: 8990: 8989: 8983: 8981: 8975: 8974: 8971: 8970: 8968: 8967: 8961: 8959: 8953: 8952: 8950: 8949: 8947:W and Z bosons 8944: 8939: 8933: 8931: 8922: 8916: 8915: 8912: 8911: 8909: 8908: 8907: 8906: 8901: 8896: 8891: 8886: 8881: 8871: 8866: 8861: 8856: 8851: 8846: 8840: 8838: 8832: 8831: 8829: 8828: 8823: 8818: 8813: 8808: 8803: 8801:Strange (quark 8798: 8793: 8788: 8783: 8778: 8773: 8767: 8765: 8756: 8747: 8741: 8740: 8733: 8732: 8725: 8718: 8710: 8704: 8703: 8691: 8686: 8680: 8666: 8665:External links 8663: 8660: 8659: 8602: 8544: 8494: 8487: 8467: 8442: 8429: 8416: 8371: 8334:(3): 199–208. 8311: 8304: 8280: 8269:. 7 April 2015 8251: 8244: 8226: 8198: 8191: 8166: 8159: 8141: 8134: 8103: 8096: 8071: 8064: 8039: 8032: 8014: 8008:978-0486432618 8007: 7989: 7982: 7960: 7944: 7937: 7917: 7910: 7892: 7871: 7851: 7844: 7815: 7808: 7789: 7788: 7786: 7783: 7781: 7780: 7775: 7770: 7768:Surface phonon 7765: 7760: 7755: 7750: 7745: 7740: 7735: 7730: 7725: 7720: 7715: 7710: 7705: 7700: 7694: 7693: 7692: 7689:Physics portal 7676: 7673: 7665: 7664:Other research 7662: 7658:isotope effect 7634: 7631: 7617: 7614: 7597: 7594: 7559: 7558: 7547: 7542: 7538: 7532: 7525: 7521: 7513: 7509: 7505: 7502: 7478: 7467: 7462: 7461: 7448: 7443: 7440: 7435: 7432: 7429: 7425: 7421: 7418: 7415: 7412: 7407: 7403: 7399: 7396: 7391: 7388: 7385: 7381: 7377: 7374: 7369: 7365: 7359: 7350: 7346: 7340: 7335: 7330: 7325: 7322: 7319: 7315: 7309: 7305: 7299: 7296: 7293: 7289: 7285: 7280: 7276: 7270: 7263: 7259: 7244: 7243: 7230: 7225: 7220: 7217: 7214: 7210: 7206: 7203: 7200: 7197: 7192: 7188: 7184: 7181: 7176: 7173: 7170: 7166: 7162: 7159: 7154: 7150: 7144: 7137: 7134: 7129: 7125: 7119: 7114: 7109: 7104: 7101: 7098: 7094: 7088: 7084: 7078: 7075: 7072: 7068: 7064: 7059: 7055: 7049: 7042: 7035: 7031: 7013: 7007: 7000: 6994: 6987: 6981: 6973: 6962: 6951: 6943: 6937: 6931: 6910: 6888: 6862: 6832: 6808: 6807: 6794: 6790: 6784: 6777: 6773: 6765: 6761: 6757: 6752: 6748: 6744: 6739: 6719: 6718: 6706: 6700: 6696: 6692: 6689: 6684: 6679: 6675: 6669: 6664: 6660: 6656: 6651: 6646: 6642: 6637: 6631: 6627: 6620: 6617: 6611: 6606: 6589: 6586: 6561: 6558: 6545: 6532: 6526: 6525: 6511: 6508: 6504: 6497: 6491: 6486: 6478: 6475: 6472: 6468: 6464: 6457: 6453: 6450: 6446: 6441: 6437: 6432: 6429: 6426: 6422: 6418: 6414: 6343: 6342:Thermodynamics 6340: 6335:Brillouin zone 6248: 6247: 6234: 6230: 6227: 6224: 6218: 6215: 6201: 6200: 6187: 6184: 6181: 6177: 6168: 6165: 6162: 6156: 6147: 6143: 6138: 6133: 6130: 6127: 6123: 6114: 6111: 6108: 6102: 6093: 6089: 6058:pseudomomentum 6026:By analogy to 6020:Brillouin zone 6008:Main article: 6005: 6002: 5950:Brillouin zone 5913: 5910: 5906: 5905: 5894: 5890: 5883: 5880: 5874: 5869: 5866: 5863: 5859: 5853: 5846: 5843: 5840: 5836: 5829: 5823: 5820: 5817: 5813: 5808: 5803: 5798: 5795: 5792: 5788: 5782: 5778: 5774: 5769: 5737: 5723: 5716: 5709: 5688: 5680: 5674: 5667: 5662: 5661: 5649: 5642: 5639: 5633: 5628: 5624: 5618: 5611: 5607: 5600: 5594: 5590: 5586: 5581: 5577: 5573: 5568: 5541: 5537: 5514: 5510: 5498: 5497: 5485: 5479: 5476: 5472: 5468: 5463: 5456: 5452: 5445: 5438: 5432: 5428: 5424: 5421: 5414: 5411: 5406: 5402: 5380: 5374: 5371: 5367: 5363: 5358: 5351: 5347: 5340: 5330: 5326: 5322: 5319: 5315: 5309: 5304: 5300: 5283: 5276: 5271: 5270: 5259: 5256: 5252: 5246: 5238: 5235: 5230: 5224: 5219: 5212: 5208: 5201: 5197: 5192: 5184: 5181: 5176: 5172: 5167: 5163: 5157: 5151: 5145: 5142: 5138: 5135: 5131: 5127: 5123: 5117: 5109: 5106: 5101: 5095: 5090: 5086: 5081: 5062: 5061: 5049: 5043: 5039: 5030: 5026: 5022: 5018: 5013: 5008: 5005: 5001: 4996: 4988: 4985: 4978: 4974: 4970: 4963: 4958: 4951: 4947: 4923: 4917: 4914: 4910: 4901: 4897: 4893: 4889: 4884: 4879: 4875: 4870: 4862: 4859: 4852: 4848: 4844: 4837: 4832: 4828: 4798: 4794: 4771: 4767: 4744: 4702: 4699: 4691:shear stresses 4624:primitive cell 4616:Brillouin zone 4568:primitive cell 4549:group velocity 4530: 4520:speed of sound 4474: 4470: 4467: 4460: 4457: 4442: 4441: 4430: 4420: 4416: 4410: 4406: 4398: 4394: 4391: 4385: 4380: 4376: 4372: 4366: 4361: 4356: 4348: 4344: 4340: 4335: 4328: 4324: 4320: 4314: 4307: 4304: 4300: 4292: 4288: 4284: 4279: 4272: 4268: 4264: 4258: 4254: 4251: 4246: 4241: 4237: 4213: 4206: 4155: 4152: 4113: 4110: 4091:case when the 4044: 4043: 4032: 4026: 4023: 4017: 4014: 4005: 4002: 3999: 3993: 3990: 3981: 3978: 3975: 3969: 3966: 3949: 3948: 3937: 3934: 3931: 3928: 3925: 3922: 3919: 3916: 3913: 3910: 3904: 3900: 3896: 3892: 3888: 3885: 3879: 3876: 3869: 3865: 3860: 3856: 3839: 3823: 3822: 3808: 3803: 3800: 3795: 3792: 3789: 3786: 3783: 3780: 3777: 3774: 3771: 3768: 3765: 3762: 3748: 3745: 3740: 3737: 3734: 3728: 3723: 3719: 3715: 3712: 3671: 3670: 3658: 3652: 3649: 3645: 3639: 3635: 3629: 3624: 3620: 3614: 3610: 3606: 3601: 3598: 3594: 3588: 3584: 3579: 3573: 3569: 3562: 3559: 3555: 3550: 3545: 3529: 3528: 3516: 3510: 3506: 3503: 3497: 3494: 3490: 3486: 3483: 3480: 3474: 3469: 3466: 3462: 3459: 3456: 3453: 3449: 3443: 3439: 3435: 3430: 3425: 3421: 3406: 3405: 3392: 3389: 3385: 3379: 3375: 3369: 3362: 3358: 3352: 3347: 3343: 3336: 3333: 3327: 3324: 3319: 3316: 3313: 3310: 3306: 3302: 3297: 3294: 3291: 3287: 3283: 3280: 3277: 3272: 3269: 3265: 3259: 3255: 3249: 3245: 3239: 3235: 3231: 3225: 3222: 3216: 3211: 3206: 3200: 3197: 3194: 3190: 3186: 3181: 3177: 3172: 3165: 3161: 3155: 3151: 3147: 3141: 3138: 3121: 3120: 3103: 3100: 3096: 3090: 3086: 3080: 3076: 3072: 3069: 3067: 3063: 3056: 3052: 3044: 3040: 3036: 3035: 3030: 3027: 3024: 3021: 3017: 3011: 3008: 3004: 2998: 2994: 2988: 2984: 2980: 2974: 2971: 2967: 2964: 2961: 2957: 2950: 2945: 2942: 2938: 2935: 2931: 2927: 2924: 2921: 2917: 2911: 2907: 2900: 2897: 2892: 2886: 2882: 2875: 2872: 2868: 2864: 2858: 2855: 2850: 2847: 2845: 2841: 2838: 2835: 2831: 2825: 2821: 2815: 2811: 2807: 2806: 2792: 2791: 2776: 2773: 2769: 2762: 2759: 2754: 2750: 2745: 2741: 2736: 2732: 2729: 2727: 2724: 2717: 2714: 2709: 2705: 2700: 2696: 2691: 2687: 2686: 2680: 2677: 2673: 2670: 2666: 2662: 2659: 2656: 2650: 2645: 2642: 2638: 2635: 2631: 2627: 2624: 2621: 2617: 2611: 2607: 2601: 2597: 2594: 2588: 2585: 2583: 2581: 2577: 2571: 2567: 2563: 2558: 2554: 2549: 2543: 2540: 2536: 2533: 2529: 2526: 2522: 2516: 2513: 2510: 2507: 2503: 2497: 2494: 2491: 2487: 2481: 2478: 2473: 2470: 2468: 2465: 2458: 2455: 2450: 2446: 2441: 2437: 2432: 2428: 2427: 2422: 2419: 2416: 2412: 2408: 2405: 2402: 2399: 2397: 2394: 2388: 2384: 2380: 2375: 2371: 2366: 2362: 2361: 2328: 2327: 2312: 2307: 2303: 2297: 2294: 2291: 2288: 2285: 2281: 2275: 2271: 2264: 2260: 2255: 2252: 2250: 2246: 2242: 2238: 2237: 2232: 2228: 2222: 2219: 2216: 2213: 2209: 2203: 2199: 2192: 2188: 2183: 2180: 2178: 2174: 2170: 2166: 2165: 2149: 2142: 2131: 2120: 2078: 2071: 2062: 2061: 2048: 2043: 2037: 2033: 2029: 2024: 2020: 2015: 2008: 2004: 2001: 1997: 1994: 1991: 1988: 1985: 1981: 1975: 1971: 1967: 1962: 1959: 1954: 1948: 1945: 1939: 1934: 1930: 1922: 1917: 1914: 1911: 1907: 1903: 1898: 1867: 1863: 1842: 1822: 1819: 1814: 1810: 1789: 1786: 1781: 1777: 1773: 1768: 1764: 1741: 1737: 1717: 1714: 1697: 1694: 1691: 1688: 1685: 1682: 1679: 1654: 1612: 1593: 1588: 1587: 1576: 1571: 1567: 1564: 1560: 1557: 1554: 1551: 1548: 1543: 1539: 1536: 1528: 1523: 1519: 1514: 1509: 1504: 1500: 1496: 1492: 1486: 1482: 1478: 1473: 1469: 1450: 1449: 1438: 1430: 1426: 1422: 1415: 1411: 1405: 1401: 1394: 1391: 1386: 1382: 1378: 1374: 1371: 1368: 1365: 1361: 1358: 1355: 1352: 1349: 1319: 1316: 1313: 1310: 1307: 1287: 1284: 1281: 1278: 1274: 1270: 1267: 1264: 1261: 1258: 1236: 1233: 1230: 1227: 1223: 1219: 1214: 1210: 1191: 1175: 1174: 1163: 1158: 1155: 1152: 1149: 1145: 1139: 1135: 1129: 1124: 1121: 1118: 1115: 1111: 1107: 1104: 1101: 1097: 1093: 1088: 1084: 1059: 1058: 1047: 1039: 1035: 1031: 1024: 1020: 1014: 1010: 1003: 1000: 996: 990: 987: 984: 980: 976: 971: 968: 965: 961: 956: 952: 949: 944: 940: 936: 933: 930: 894: 872: 871: 870: 869: 868: 867: 866: 865: 864: 863: 862: 861: 860: 859: 858: 857: 851: 841: 830: 825: 796: 795: 794: 793: 792: 791: 790: 789: 788: 787: 786: 785: 784: 783: 782: 781: 724: 721: 706: 703: 649: 646: 629: 616: 615: 604: 599: 594: 588: 584: 580: 575: 571: 566: 559: 555: 551: 545: 542: 534: 530: 527: 523: 520: 517: 514: 511: 507: 492: 491: 436:approximations 411: 406: 405: 393: 387: 383: 379: 374: 370: 365: 361: 356: 353: 350: 346: 340: 337: 296:covalent bonds 261:correspondence 252: 249: 196: 193: 15: 9: 6: 4: 3: 2: 9821: 9810: 9807: 9805: 9802: 9800: 9797: 9796: 9794: 9781: 9780: 9775: 9769: 9763: 9760: 9758: 9755: 9753: 9750: 9748: 9745: 9743: 9740: 9738: 9737:Exotic matter 9735: 9731: 9728: 9727: 9726: 9725:Eightfold way 9723: 9721: 9718: 9716: 9715:Antiparticles 9713: 9711: 9708: 9706: 9703: 9699: 9696: 9695: 9694: 9691: 9687: 9684: 9683: 9682: 9679: 9678: 9676: 9672: 9666: 9663: 9661: 9658: 9656: 9653: 9651: 9648: 9646: 9643: 9642: 9640: 9636: 9630: 9627: 9625: 9622: 9620: 9617: 9615: 9612: 9610: 9607: 9605: 9602: 9600: 9597: 9595: 9592: 9590: 9587: 9585: 9582: 9580: 9577: 9575: 9572: 9570: 9567: 9565: 9562: 9561: 9559: 9557: 9553: 9539: 9536: 9534: 9531: 9529: 9526: 9524: 9521: 9520: 9518: 9514: 9504: 9501: 9499: 9496: 9494: 9491: 9490: 9488: 9484: 9478: 9475: 9473: 9470: 9468: 9465: 9464: 9462: 9458: 9454: 9451: 9449: 9445: 9439: 9436: 9434: 9431: 9427: 9424: 9422: 9419: 9417: 9414: 9412: 9409: 9407: 9404: 9402: 9399: 9398: 9397: 9394: 9392: 9389: 9387: 9386:Atomic nuclei 9384: 9383: 9381: 9377: 9367: 9364: 9361: 9357: 9354: 9353: 9351: 9349: 9345: 9339: 9336: 9334: 9331: 9329: 9326: 9324: 9321: 9319: 9318:Upsilon meson 9316: 9314: 9311: 9309: 9306: 9304: 9301: 9299: 9296: 9294: 9291: 9289: 9286: 9284: 9281: 9280: 9278: 9276: 9272: 9266: 9263: 9261: 9258: 9256: 9253: 9251: 9250:Lambda baryon 9248: 9246: 9243: 9239: 9236: 9234: 9231: 9229: 9226: 9224: 9221: 9220: 9219: 9216: 9215: 9213: 9211: 9207: 9204: 9202: 9198: 9195: 9193: 9189: 9175: 9172: 9170: 9167: 9165: 9162: 9160: 9157: 9155: 9152: 9150: 9147: 9145: 9142: 9140: 9137: 9135: 9132: 9130: 9127: 9125: 9122: 9120: 9117: 9115: 9112: 9110: 9109:Dual graviton 9107: 9105: 9102: 9100: 9097: 9095: 9092: 9091: 9089: 9085: 9074: 9070: 9067: 9065: 9062: 9060: 9057: 9055: 9052: 9050: 9047: 9046: 9044: 9040: 9034: 9031: 9029: 9026: 9024: 9021: 9020: 9018: 9016: 9012: 9009: 9007: 9006:Superpartners 9003: 9000: 8998: 8994: 8988: 8985: 8984: 8982: 8980: 8976: 8966: 8963: 8962: 8960: 8958: 8954: 8948: 8945: 8943: 8940: 8938: 8935: 8934: 8932: 8930: 8926: 8923: 8921: 8917: 8905: 8902: 8900: 8897: 8895: 8892: 8890: 8889:Muon neutrino 8887: 8885: 8882: 8880: 8877: 8876: 8875: 8872: 8870: 8867: 8865: 8862: 8860: 8857: 8855: 8852: 8850: 8847: 8845: 8842: 8841: 8839: 8837: 8833: 8827: 8824: 8822: 8821:Bottom (quark 8819: 8817: 8814: 8812: 8809: 8807: 8804: 8802: 8799: 8797: 8794: 8792: 8789: 8787: 8784: 8782: 8779: 8777: 8774: 8772: 8769: 8768: 8766: 8764: 8760: 8757: 8755: 8751: 8748: 8746: 8742: 8738: 8731: 8726: 8724: 8719: 8717: 8712: 8711: 8708: 8701: 8698: 8695: 8692: 8690: 8687: 8684: 8681: 8678: 8673: 8669: 8668: 8655: 8651: 8647: 8643: 8639: 8635: 8630: 8625: 8621: 8617: 8613: 8606: 8598: 8594: 8590: 8586: 8582: 8578: 8573: 8568: 8565:(1): 011020. 8564: 8560: 8556: 8548: 8540: 8536: 8531: 8526: 8522: 8518: 8514: 8510: 8509: 8504: 8498: 8490: 8484: 8480: 8479: 8471: 8456: 8452: 8446: 8439: 8433: 8426: 8420: 8411: 8406: 8402: 8398: 8394: 8390: 8386: 8382: 8375: 8367: 8363: 8359: 8355: 8351: 8347: 8342: 8337: 8333: 8329: 8322: 8315: 8307: 8301: 8297: 8293: 8292: 8284: 8268: 8264: 8258: 8256: 8247: 8241: 8237: 8230: 8215: 8211: 8205: 8203: 8194: 8188: 8184: 8180: 8179:, 8th Edition 8178: 8170: 8162: 8156: 8152: 8145: 8137: 8135:0-03-083993-9 8131: 8127: 8120: 8118: 8116: 8114: 8112: 8110: 8108: 8099: 8093: 8089: 8085: 8078: 8076: 8067: 8061: 8057: 8053: 8046: 8044: 8035: 8029: 8025: 8018: 8010: 8004: 8000: 7993: 7985: 7983:9780070409545 7979: 7974: 7973: 7964: 7957: 7951: 7949: 7940: 7934: 7930: 7929: 7921: 7913: 7907: 7903: 7896: 7888: 7882: 7874: 7868: 7864: 7863: 7855: 7847: 7841: 7837: 7830: 7828: 7826: 7824: 7822: 7820: 7811: 7805: 7801: 7794: 7790: 7779: 7776: 7774: 7771: 7769: 7766: 7764: 7761: 7759: 7756: 7754: 7751: 7749: 7746: 7744: 7741: 7739: 7738:Rayleigh wave 7736: 7734: 7731: 7729: 7726: 7724: 7721: 7719: 7716: 7714: 7711: 7709: 7706: 7704: 7701: 7699: 7696: 7695: 7690: 7684: 7679: 7672: 7669: 7661: 7659: 7655: 7651: 7647: 7644:vanishes and 7643: 7639: 7630: 7627: 7626:negative mass 7623: 7613: 7611: 7607: 7603: 7593: 7591: 7585: 7581: 7576: = 7573: 7569: 7562: 7545: 7540: 7536: 7530: 7523: 7519: 7511: 7507: 7503: 7500: 7493: 7492: 7491: 7489: 7485: 7481: 7474: 7470: 7441: 7438: 7433: 7430: 7427: 7423: 7419: 7413: 7410: 7405: 7401: 7394: 7389: 7386: 7383: 7379: 7375: 7372: 7367: 7363: 7348: 7344: 7338: 7328: 7323: 7320: 7317: 7313: 7307: 7303: 7297: 7294: 7291: 7287: 7283: 7278: 7274: 7261: 7257: 7249: 7248: 7247: 7223: 7218: 7215: 7212: 7208: 7204: 7198: 7195: 7190: 7186: 7179: 7174: 7171: 7168: 7164: 7160: 7157: 7152: 7148: 7135: 7132: 7127: 7123: 7117: 7107: 7102: 7099: 7096: 7092: 7086: 7082: 7076: 7073: 7070: 7066: 7062: 7057: 7053: 7040: 7033: 7029: 7019: 7018: 7017: 7012: 7006: 6999: 6993: 6990: + 6986: 6980: 6976: 6969: 6965: 6958: 6954: 6942: 6936: 6930: 6924: 6919: 6917: 6913: 6891: 6869: 6868:infinite term 6865: 6843: 6839: 6835: 6813: 6792: 6788: 6782: 6775: 6771: 6763: 6759: 6750: 6746: 6742: 6728: 6727: 6726: 6724: 6704: 6698: 6694: 6687: 6682: 6677: 6673: 6667: 6662: 6658: 6654: 6649: 6644: 6640: 6635: 6629: 6625: 6618: 6615: 6609: 6595: 6594: 6593: 6585: 6583: 6582:heat transfer 6579: 6575: 6571: 6568:behavior (or 6567: 6557: 6555: 6551: 6544: 6539: 6535: 6531: 6509: 6506: 6502: 6495: 6484: 6476: 6473: 6470: 6466: 6462: 6455: 6451: 6448: 6444: 6439: 6435: 6430: 6427: 6424: 6420: 6416: 6412: 6405: 6404: 6403: 6401: 6397: 6393: 6389: 6383: 6381: 6377: 6373: 6369: 6365: 6364:absolute zero 6360: 6357: 6356:heat capacity 6353: 6349: 6348:thermodynamic 6339: 6337: 6336: 6330: 6326: 6317: 6313: 6311: 6303: 6287: ± 6286: 6278: 6262: ± 6261: 6257: 6253: 6232: 6228: 6225: 6222: 6216: 6213: 6206: 6205: 6204: 6185: 6182: 6179: 6154: 6145: 6136: 6131: 6128: 6125: 6121: 6100: 6091: 6087: 6079: 6078: 6077: 6075: 6071: 6067: 6063: 6059: 6055: 6051: 6047: 6044: 6040: 6037: 6033: 6029: 6021: 6016: 6011: 6001: 5999: 5995: 5991: 5987: 5983: 5979: 5974: 5972: 5968: 5964: 5960: 5956: 5951: 5946: 5944: 5940: 5936: 5932: 5928: 5925: 5921: 5919: 5909: 5892: 5888: 5881: 5878: 5872: 5867: 5864: 5861: 5857: 5851: 5844: 5841: 5838: 5834: 5827: 5821: 5818: 5815: 5811: 5801: 5796: 5793: 5790: 5786: 5780: 5776: 5772: 5758: 5757: 5756: 5753: 5751: 5750: 5744: 5740: 5733: 5728: 5726: 5719: 5712: 5704: 5702: 5699: 5695: 5691: 5683: 5677: 5670: 5647: 5640: 5637: 5631: 5626: 5622: 5616: 5609: 5605: 5598: 5592: 5588: 5579: 5575: 5571: 5557: 5556: 5555: 5539: 5512: 5508: 5483: 5477: 5474: 5470: 5466: 5461: 5454: 5450: 5443: 5436: 5430: 5426: 5422: 5419: 5412: 5409: 5404: 5378: 5372: 5369: 5365: 5361: 5356: 5349: 5345: 5338: 5328: 5324: 5320: 5317: 5313: 5307: 5302: 5298: 5290: 5289: 5288: 5286: 5279: 5257: 5254: 5250: 5244: 5236: 5233: 5228: 5222: 5217: 5210: 5206: 5199: 5195: 5182: 5179: 5174: 5170: 5165: 5161: 5149: 5143: 5140: 5136: 5133: 5129: 5125: 5121: 5115: 5107: 5104: 5099: 5093: 5088: 5084: 5079: 5071: 5070: 5069: 5067: 5047: 5041: 5028: 5024: 5020: 5016: 5011: 5006: 5003: 4999: 4994: 4983: 4976: 4972: 4968: 4961: 4956: 4949: 4945: 4921: 4915: 4912: 4899: 4895: 4891: 4887: 4882: 4877: 4873: 4868: 4857: 4850: 4846: 4842: 4835: 4830: 4826: 4818: 4817: 4816: 4814: 4796: 4769: 4765: 4732: 4728: 4724: 4720: 4716: 4712: 4708: 4698: 4696: 4692: 4687: 4682: 4680: 4675: 4671: 4668: 4664: 4660: 4655: 4651: 4649: 4645: 4641: 4637: 4633: 4629: 4625: 4621: 4617: 4609: 4589: 4573: 4569: 4564: 4562: 4558: 4554: 4550: 4542: 4533: 4521: 4516: 4514: 4510: 4506: 4502: 4498: 4495: = 4494: 4489: 4472: 4468: 4465: 4458: 4455: 4447: 4428: 4418: 4414: 4408: 4404: 4396: 4392: 4389: 4383: 4378: 4374: 4370: 4364: 4359: 4354: 4346: 4342: 4338: 4333: 4326: 4322: 4318: 4312: 4305: 4302: 4298: 4290: 4286: 4282: 4277: 4270: 4266: 4262: 4256: 4252: 4249: 4244: 4239: 4235: 4227: 4226: 4225: 4223: 4219: 4212: 4205: 4197: 4193: 4190: = 4189: 4184: 4176: 4168: 4160: 4151: 4149: 4145: 4141: 4137: 4132: 4130: 4126: 4123: 4119: 4108: 4103: 4101: 4096: 4094: 4090: 4086: 4083: 4079: 4074: 4072: 4068: 4064: 4030: 4024: 4015: 4012: 4003: 4000: 3991: 3988: 3979: 3976: 3967: 3964: 3954: 3953: 3952: 3935: 3932: 3929: 3926: 3923: 3920: 3917: 3914: 3911: 3908: 3902: 3898: 3890: 3886: 3883: 3877: 3874: 3867: 3863: 3858: 3854: 3846: 3845: 3844: 3842: 3834: 3832: 3828: 3806: 3801: 3798: 3793: 3790: 3787: 3784: 3781: 3778: 3775: 3772: 3769: 3766: 3763: 3760: 3746: 3743: 3738: 3735: 3732: 3726: 3721: 3717: 3713: 3710: 3703: 3702: 3701: 3699: 3695: 3690: 3688: 3684: 3680: 3676: 3656: 3650: 3647: 3643: 3637: 3633: 3627: 3622: 3618: 3612: 3608: 3604: 3599: 3596: 3586: 3577: 3571: 3567: 3560: 3557: 3553: 3548: 3534: 3533: 3532: 3514: 3508: 3504: 3501: 3495: 3492: 3488: 3484: 3481: 3478: 3472: 3467: 3464: 3460: 3457: 3454: 3451: 3447: 3441: 3437: 3433: 3428: 3423: 3419: 3411: 3410: 3409: 3390: 3387: 3383: 3377: 3373: 3367: 3360: 3356: 3350: 3345: 3341: 3334: 3331: 3325: 3317: 3314: 3311: 3308: 3304: 3300: 3295: 3292: 3289: 3285: 3281: 3278: 3270: 3267: 3263: 3257: 3253: 3247: 3243: 3237: 3233: 3229: 3223: 3220: 3214: 3209: 3204: 3198: 3195: 3192: 3188: 3184: 3179: 3175: 3170: 3163: 3159: 3153: 3149: 3145: 3139: 3136: 3126: 3125: 3124: 3101: 3098: 3088: 3078: 3074: 3070: 3068: 3061: 3054: 3050: 3042: 3038: 3028: 3025: 3022: 3019: 3015: 3009: 3006: 3002: 2996: 2992: 2986: 2982: 2978: 2972: 2969: 2965: 2962: 2959: 2955: 2948: 2943: 2940: 2936: 2933: 2929: 2925: 2922: 2919: 2915: 2909: 2905: 2898: 2895: 2890: 2884: 2880: 2873: 2870: 2866: 2862: 2856: 2853: 2848: 2846: 2839: 2836: 2833: 2829: 2823: 2819: 2813: 2809: 2797: 2796: 2795: 2774: 2771: 2767: 2760: 2757: 2748: 2743: 2734: 2730: 2728: 2722: 2715: 2712: 2707: 2703: 2698: 2694: 2689: 2678: 2675: 2671: 2668: 2664: 2657: 2654: 2648: 2643: 2640: 2636: 2633: 2629: 2625: 2622: 2619: 2615: 2609: 2605: 2599: 2592: 2586: 2584: 2575: 2569: 2565: 2561: 2556: 2552: 2547: 2541: 2538: 2534: 2531: 2527: 2524: 2520: 2514: 2511: 2508: 2505: 2501: 2495: 2492: 2489: 2485: 2479: 2476: 2471: 2469: 2463: 2456: 2453: 2444: 2439: 2435: 2430: 2420: 2417: 2414: 2410: 2403: 2400: 2398: 2392: 2386: 2382: 2378: 2373: 2369: 2364: 2352: 2351: 2350: 2347: 2345: 2337: 2333: 2330:The quantity 2310: 2305: 2301: 2295: 2292: 2289: 2286: 2283: 2279: 2273: 2269: 2262: 2258: 2253: 2251: 2244: 2230: 2226: 2220: 2217: 2214: 2211: 2207: 2201: 2197: 2190: 2186: 2181: 2179: 2172: 2168: 2156: 2155: 2154: 2152: 2145: 2138: 2134: 2127: 2123: 2116: 2111: 2109: 2105: 2101: 2097: 2096:Fourier space 2093: 2089: 2085: 2081: 2074: 2067: 2046: 2041: 2035: 2031: 2027: 2022: 2018: 2013: 1989: 1986: 1979: 1973: 1969: 1965: 1960: 1957: 1952: 1946: 1943: 1937: 1932: 1928: 1920: 1915: 1912: 1909: 1905: 1901: 1887: 1886: 1885: 1883: 1865: 1861: 1840: 1820: 1817: 1812: 1808: 1787: 1784: 1779: 1775: 1771: 1766: 1762: 1739: 1735: 1725: 1723: 1713: 1711: 1695: 1692: 1689: 1683: 1677: 1667: 1663: 1657: 1648: 1635: 1630: 1628: 1624: 1620: 1615: 1606: 1604: 1596: 1574: 1565: 1562: 1558: 1555: 1552: 1549: 1541: 1537: 1534: 1526: 1521: 1517: 1512: 1507: 1502: 1498: 1494: 1490: 1484: 1480: 1476: 1471: 1467: 1459: 1458: 1457: 1455: 1436: 1428: 1424: 1420: 1413: 1409: 1403: 1399: 1392: 1389: 1384: 1380: 1372: 1369: 1366: 1363: 1359: 1356: 1350: 1347: 1340: 1339: 1338: 1336: 1331: 1317: 1314: 1311: 1308: 1305: 1282: 1279: 1272: 1268: 1265: 1262: 1259: 1256: 1234: 1231: 1228: 1225: 1221: 1217: 1212: 1208: 1199: 1194: 1181: 1161: 1156: 1153: 1150: 1147: 1143: 1137: 1133: 1127: 1122: 1119: 1116: 1113: 1109: 1105: 1102: 1099: 1095: 1091: 1086: 1082: 1074: 1073: 1072: 1069: 1067: 1062: 1045: 1037: 1033: 1029: 1022: 1018: 1012: 1008: 1001: 998: 994: 988: 985: 982: 978: 974: 969: 966: 963: 959: 954: 950: 947: 942: 938: 934: 931: 928: 921: 920: 919: 909: 904: 897: 854: 850: 844: 833: 829: 826: 815: 814: 813: 812: 811: 810: 809: 808: 807: 806: 805: 804: 803: 802: 801: 800: 799: 776: 765: 759: 752: 749: 748: 747: 746: 745: 744: 743: 742: 741: 740: 739: 738: 737: 736: 735: 734: 733: 731: 720: 711: 702: 700: 696: 691: 689: 685: 681: 676: 674: 671: 667: 663: 654: 648:Lattice waves 645: 644: 638: 636: 632: 625: 621: 602: 597: 592: 586: 582: 578: 573: 569: 564: 557: 553: 549: 543: 540: 515: 512: 505: 497: 496: 495: 489: 485: 484: 483: 481: 475: 473: 469: 465: 461: 457: 454: 450: 446: 442: 437: 433: 428: 426: 423:th atom, and 422: 418: 414: 391: 385: 381: 377: 372: 368: 363: 359: 354: 351: 348: 344: 338: 335: 326: 325: 324: 321: 317: 313: 312:gravitational 309: 305: 301: 297: 293: 289: 285: 281: 277: 273: 269: 264: 262: 258: 248: 246: 242: 241:particle-like 238: 234: 230: 226: 225:superposition 222: 218: 214: 210: 206: 202: 192: 190: 186: 183:exhibit both 182: 178: 174: 170: 166: 162: 158: 154: 149: 138: 134: 133:Yakov Frenkel 130: 126: 122: 117: 115: 111: 107: 102: 100: 97:as quantized 96: 93:, similar to 92: 88: 84: 81: 77: 76:excited state 73: 69: 68:quasiparticle 65: 61: 57: 53: 49: 45: 41: 37: 30: 26: 22: 9777: 9598: 9448:Hypothetical 9396:Exotic atoms 9265:Omega baryon 9255:Sigma baryon 9245:Delta baryon 8997:Hypothetical 8979:Ghost fields 8965:Higgs boson 8899:Tau neutrino 8791:Charm (quark 8679:at Wikiquote 8619: 8615: 8605: 8562: 8558: 8547: 8512: 8506: 8497: 8477: 8470: 8458:. Retrieved 8454: 8445: 8432: 8419: 8388: 8384: 8374: 8331: 8327: 8314: 8290: 8283: 8271:. Retrieved 8267:News.mit.edu 8266: 8235: 8229: 8217:. Retrieved 8213: 8175: 8169: 8150: 8144: 8125: 8087: 8055: 8023: 8017: 7998: 7992: 7971: 7963: 7955: 7927: 7920: 7901: 7895: 7861: 7854: 7835: 7799: 7793: 7758:Second sound 7670: 7667: 7650:Cooper pairs 7636: 7619: 7599: 7596:Nonlinearity 7583: 7579: 7571: 7567: 7563: 7560: 7476: 7472: 7465: 7463: 7245: 7010: 7004: 6997: 6991: 6984: 6978: 6971: 6970:is given by 6967: 6960: 6956: 6949: 6940: 6934: 6928: 6923:vacuum state 6920: 6908: 6886: 6860: 6830: 6809: 6720: 6591: 6569: 6563: 6553: 6542: 6537: 6533: 6529: 6527: 6384: 6368:ground state 6361: 6345: 6332: 6328: 6324: 6322: 6301: 6284: 6276: 6259: 6255: 6251: 6249: 6202: 6073: 6069: 6061: 6057: 6053: 6049: 6045: 6038: 6032:matter waves 6025: 5981: 5975: 5963:Raman active 5962: 5958: 5947: 5934: 5930: 5929: 5923: 5922: 5915: 5907: 5754: 5746: 5735: 5729: 5721: 5714: 5707: 5705: 5686: 5678: 5672: 5665: 5663: 5499: 5281: 5274: 5272: 5063: 4704: 4695:viscoelastic 4683: 4676: 4672: 4666: 4662: 4658: 4656: 4652: 4643: 4635: 4619: 4607: 4587: 4565: 4560: 4556: 4552: 4540: 4528: 4517: 4512: 4508: 4500: 4496: 4492: 4490: 4445: 4443: 4221: 4217: 4210: 4203: 4201: 4195: 4191: 4187: 4140:polarization 4135: 4133: 4128: 4124: 4117: 4115: 4097: 4084: 4077: 4075: 4062: 4045: 3950: 3837: 3835: 3830: 3826: 3824: 3697: 3693: 3691: 3686: 3678: 3674: 3672: 3530: 3407: 3122: 2793: 2348: 2331: 2329: 2147: 2140: 2136: 2129: 2118: 2114: 2112: 2100:normal modes 2087: 2076: 2069: 2065: 2063: 1833:if particle 1726: 1721: 1719: 1665: 1661: 1652: 1650:held fixed, 1646: 1631: 1621:between the 1610: 1607: 1591: 1589: 1451: 1334: 1332: 1197: 1189: 1179: 1176: 1070: 1063: 1060: 907: 905: 892: 873: 852: 848: 839: 831: 827: 797: 774: 763: 757: 750: 726: 717: 695:normal modes 692: 683: 679: 677: 672: 659: 639: 634: 627: 619: 617: 493: 476: 471: 467: 463: 459: 455: 444: 429: 424: 420: 409: 407: 319: 279: 275: 265: 254: 228: 198: 164: 156: 152: 128: 118: 103: 83:quantization 66:. A type of 35: 33: 9730:Quark model 9498:Theta meson 9401:Positronium 9313:Omega meson 9308:J/psi meson 9238:Antineutron 9149:Dark photon 9114:Graviphoton 9073:Stop squark 8781:Down (quark 7600:As well as 6916:Hamiltonian 6812:Hamiltonian 6378:about some 6372:temperature 4731:eigenvalues 2108:periodicity 2098:which uses 1882:Hamiltonian 1603:normal mode 918:th atom is 878:labels the 699:frequencies 675:is marked. 268:crystalline 221:normal mode 205:vibrational 177:light waves 173:sound waves 127:. The name 99:light waves 91:sound waves 9793:Categories 9472:Heptaquark 9433:Superatoms 9366:Pentaquark 9356:Tetraquark 9338:Quarkonium 9228:Antiproton 9129:Leptoquark 9064:Neutralino 8826:antiquark) 8816:antiquark) 8811:Top (quark 8806:antiquark) 8796:antiquark) 8786:antiquark) 8776:antiquark) 8745:Elementary 8629:2205.05037 8572:2205.05037 8488:0486435032 8395:: 012121. 7785:References 6574:conduction 6388:photon gas 6380:mean value 6036:wavevector 5994:wavelength 5978:wavenumber 4572:wavevector 4122:wavevector 4105:See also: 4080:amount of 2344:wavelength 2336:wavenumber 2104:wavevector 1627:wavenumber 670:wavelength 229:elementary 195:Definition 123:physicist 9710:Particles 9655:Particles 9614:Polariton 9604:Plasmaron 9574:Dropleton 9467:Hexaquark 9438:Molecules 9426:Protonium 9303:Phi meson 9288:Rho meson 9260:Xi baryon 9192:Composite 9028:Gravitino 8771:Up (quark 8654:248665478 8622:(1): 26. 8597:248665478 8539:195774243 8460:13 August 8273:13 August 8219:15 August 7881:cite book 7778:Vibration 7608:and form 7541:α 7531:† 7524:α 7512:α 7508:∑ 7442:… 7428:α 7411:− 7406:α 7387:− 7384:α 7373:… 7349:α 7329:… 7318:α 7308:α 7295:− 7292:α 7284:… 7262:α 7224:… 7213:α 7191:α 7172:− 7169:α 7158:… 7128:α 7108:… 7097:α 7087:α 7074:− 7071:α 7063:… 7041:† 7034:α 6946:…⟩ 6838:continuum 6793:α 6783:† 6776:α 6764:α 6760:ω 6756:ℏ 6751:α 6747:∑ 6699:α 6695:ω 6691:ℏ 6688:− 6678:α 6663:α 6659:ω 6645:α 6630:α 6626:∑ 6578:radiation 6507:− 6467:ω 6463:ℏ 6452:⁡ 6421:ω 6229:π 6176:Π 6142:Π 5918:unit cell 5852:† 5812:ω 5807:ℏ 5787:∑ 5777:∑ 5694:hermitian 5617:† 5589:ω 5585:ℏ 5576:∑ 5536:Π 5475:− 5467:− 5462:† 5427:ω 5420:ℏ 5401:Π 5370:− 5357:† 5325:ω 5314:ℏ 5245:† 5218:† 5130:δ 5116:† 5038:Π 5025:ω 5012:− 5004:− 4987:ℏ 4973:ω 4957:† 4913:− 4909:Π 4896:ω 4861:ℏ 4847:ω 4793:Π 4693:(but see 4473:λ 4469:π 4384:⁡ 4365:− 4303:± 4240:± 4236:ω 4031:⋯ 4025:ω 4022:ℏ 4001:ω 3998:ℏ 3977:ω 3974:ℏ 3936:… 3899:ω 3895:ℏ 3794:± 3791:… 3782:± 3773:± 3756:for 3736:π 3683:Hermitian 3648:− 3619:ω 3597:− 3593:Π 3583:Π 3568:∑ 3496:⁡ 3485:ω 3461:⁡ 3455:− 3438:ω 3420:ω 3388:− 3357:ω 3342:∑ 3309:− 3301:− 3282:− 3268:− 3244:∑ 3234:ω 3185:− 3160:∑ 3150:ω 3099:− 3095:Π 3085:Π 3075:∑ 3039:∑ 3007:− 2983:∑ 2906:∑ 2863:∑ 2810:∑ 2753:Π 2740:Π 2665:δ 2661:ℏ 2637:− 2606:∑ 2596:ℏ 2525:− 2486:∑ 2449:Π 2411:δ 2407:ℏ 2284:− 2270:∑ 2241:Π 2198:∑ 2113:A set of 2028:− 1980:∑ 1970:ω 1906:∑ 1788:… 1690:∝ 1678:ω 1644:→∞, with 1559:⁡ 1553:− 1518:ω 1499:ω 1370:− 1360:⁡ 1315:… 1266:π 1209:ϕ 1117:π 1096:∑ 986:− 929:− 666:amplitude 579:− 554:ω 506:∑ 378:− 352:≠ 345:∑ 318:function 272:amorphous 237:wave-like 227:of these 213:frequency 191:phonons. 181:unit cell 125:Igor Tamm 62:and some 52:molecules 9686:timeline 9538:R-hadron 9493:Glueball 9477:Skyrmion 9411:Tauonium 9124:Inflaton 9119:Graviton 9099:Curvaton 9069:Sfermion 9059:Higgsino 9054:Chargino 9015:Gauginos 8874:Neutrino 8859:Antimuon 8849:Positron 8844:Electron 8754:Fermions 8455:Phys.org 8366:17019967 7675:See also 7654:exchange 7586:⟩ 7574:⟩ 7447:⟩ 7334:⟩ 7229:⟩ 7113:⟩ 6959:, where 6584:models. 6376:randomly 6043:momentum 5701:spectrum 5237:′ 5183:′ 5144:′ 5108:′ 4707:operator 4630:and two 4513:acoustic 3694:periodic 2973:′ 2944:′ 2899:′ 2874:′ 2761:′ 2716:′ 2679:′ 2644:′ 2535:′ 2457:′ 2084:momentum 1625:and the 441:screened 417:position 308:Magnetic 304:electric 185:acoustic 9674:Related 9645:Baryons 9619:Polaron 9609:Plasmon 9584:Fracton 9579:Exciton 9533:Diquark 9528:Pomeron 9503:T meson 9460:Baryons 9421:Pionium 9406:Muonium 9333:D meson 9328:B meson 9233:Neutron 9218:Nucleon 9210:Baryons 9201:Hadrons 9164:Tachyon 9139:Majoron 9104:Dilaton 9033:Photino 8869:Antitau 8836:Leptons 8634:Bibcode 8616:Physics 8577:Bibcode 8517:Bibcode 8397:Bibcode 8346:Bibcode 7708:Fracton 7602:photons 6906:⁠ 6894:⁠ 6884:⁠ 6872:⁠ 6858:⁠ 6846:⁠ 6828:⁠ 6816:⁠ 6548:is the 6306:⁠ 6289:⁠ 6281:⁠ 6264:⁠ 6028:photons 5935:optical 4646:in the 4612:⁠ 4596:⁠ 4592:⁠ 4576:⁠ 4545:⁠ 4524:⁠ 4509:optical 4065:is the 4060:⁠ 4048:⁠ 2128:of the 2102:of the 1632:In the 622:is the 419:of the 415:is the 306:force. 209:lattice 189:optical 95:photons 85:of the 78:in the 72:physics 64:liquids 44:elastic 9804:Bosons 9650:Mesons 9599:Phonon 9594:Magnon 9516:Others 9486:Mesons 9379:Others 9275:Mesons 9223:Proton 9087:Others 9042:Others 9023:Gluino 8957:Scalar 8937:Photon 8920:Bosons 8763:Quarks 8677:Phonon 8652: 8595: 8537: 8508:Nature 8485: 8364: 8302: 8242: 8189: 8157: 8132: 8094: 8062: 8030: 8005: 7980: 7935: 7908: 7869: 7842: 7806: 7622:rotons 7590:bosons 7578:| 7566:| 7475:while 6927:| 6844:, the 6552:, and 6528:where 6333:first 6203:where 5998:photon 5732:bosons 5698:energy 4717:. The 4715:bosons 4686:fluids 4665:and k 4444:where 4089:photon 4082:energy 4046:where 4028: 4007: 3983: 3810: 3408:where 2064:where 1177:Here, 874:where 846: 837: 822: 818: 778: 772: 768: 761: 755: 618:Here, 408:where 288:forces 257:axioms 165:photon 129:phonon 121:Soviet 60:solids 36:phonon 21:photon 9638:Lists 9629:Trion 9624:Roton 9564:Anyon 9391:Atoms 9154:Preon 9094:Axion 9049:Axino 8942:Gluon 8929:Gauge 8650:S2CID 8624:arXiv 8593:S2CID 8567:arXiv 8535:S2CID 8391:(1). 8362:S2CID 8336:arXiv 8324:(PDF) 7753:SASER 7698:Boson 7246:and, 5996:of a 4547:(see 4078:exact 4069:of a 3843:are: 3681:were 2092:waves 1640:→0, 1249:with 714:seen. 662:waves 270:(not 215:. In 161:sound 157:voice 153:sound 148:phonē 139:word 137:Greek 48:atoms 38:is a 27:, or 9589:Hole 9416:Onia 9323:Kaon 9283:Pion 8854:Muon 8696:and 8483:ISBN 8462:2019 8300:ISBN 8275:2019 8240:ISBN 8221:2020 8187:ISBN 8155:ISBN 8130:ISBN 8092:ISBN 8060:ISBN 8028:ISBN 8003:ISBN 7978:ISBN 7933:ISBN 7906:ISBN 7887:link 7867:ISBN 7840:ISBN 7804:ISBN 6346:The 6072:and 6030:and 5713:and 5527:and 5280:and 4640:GaAs 4594:and 3677:and 2135:and 2075:and 1298:for 1071:Put 766:+ 1 753:− 1 310:and 187:and 171:for 142:φωνή 108:and 8864:Tau 8642:doi 8585:doi 8525:doi 8513:571 8405:doi 8389:193 8354:doi 8296:159 8183:100 7490:as 6840:or 6449:exp 6362:At 6056:or 5692:is 4659:ħω. 4375:sin 4076:An 3493:sin 3458:cos 2094:in 1556:cos 1357:cos 906:If 855:+ 1 834:− 1 824:→→→ 732:): 482:.) 155:or 101:. 70:in 54:in 50:or 9795:: 8648:. 8640:. 8632:. 8620:16 8618:. 8614:. 8591:. 8583:. 8575:. 8563:13 8561:. 8557:. 8533:. 8523:. 8511:. 8505:. 8453:. 8403:. 8387:. 8383:. 8360:. 8352:. 8344:. 8332:76 8330:. 8326:. 8298:. 8265:. 8254:^ 8212:. 8201:^ 8185:. 8106:^ 8086:. 8074:^ 8054:. 8042:^ 7947:^ 7883:}} 7879:{{ 7818:^ 7612:. 7592:. 7011:ħω 6998:ħω 6985:ħω 6972:ħω 6918:. 6909:ħω 6887:ħω 6861:ħω 6831:ħω 6283:, 6050:ħk 6046:ħk 5986:cm 5973:. 5727:. 5722:ħω 5671:= 5068:: 4815:: 4681:. 4557:ωa 4488:. 4209:, 4085:ħω 4073:. 4063:ħω 2346:. 2153:: 1659:→ 1647:Na 1636:, 1629:. 1605:. 1330:. 1180:na 886:, 816:→→ 770:← 701:. 298:, 294:, 34:A 23:, 9362:) 9358:( 9075:) 9071:( 8729:e 8722:t 8715:v 8702:. 8656:. 8644:: 8636:: 8626:: 8599:. 8587:: 8579:: 8569:: 8541:. 8527:: 8519:: 8491:. 8464:. 8413:. 8407:: 8399:: 8368:. 8356:: 8348:: 8338:: 8308:. 8277:. 8248:. 8223:. 8195:. 8163:. 8138:. 8100:. 8068:. 8036:. 8011:. 7986:. 7941:. 7914:. 7889:) 7875:. 7848:. 7812:. 7584:α 7582:, 7580:β 7572:β 7570:, 7568:α 7546:. 7537:a 7520:a 7504:= 7501:N 7479:α 7477:a 7473:α 7468:α 7466:a 7439:, 7434:1 7431:+ 7424:n 7420:, 7417:) 7414:1 7402:n 7398:( 7395:, 7390:1 7380:n 7376:, 7368:1 7364:n 7358:| 7345:n 7339:= 7324:1 7321:+ 7314:n 7304:n 7298:1 7288:n 7279:1 7275:n 7269:| 7258:a 7219:1 7216:+ 7209:n 7205:, 7202:) 7199:1 7196:+ 7187:n 7183:( 7180:, 7175:1 7165:n 7161:, 7153:1 7149:n 7143:| 7136:1 7133:+ 7124:n 7118:= 7103:1 7100:+ 7093:n 7083:n 7077:1 7067:n 7058:1 7054:n 7048:| 7030:a 7014:2 7008:1 7005:n 7001:2 6995:2 6992:n 6988:1 6982:1 6979:n 6974:q 6968:α 6963:α 6961:n 6957:α 6952:α 6950:n 6944:3 6941:n 6938:2 6935:n 6932:1 6929:n 6911:q 6903:2 6900:/ 6897:1 6889:q 6881:2 6878:/ 6875:1 6863:q 6855:2 6852:/ 6849:1 6833:q 6825:2 6822:/ 6819:1 6789:a 6772:a 6743:= 6738:H 6705:) 6683:2 6674:q 6668:2 6655:+ 6650:2 6641:p 6636:( 6619:2 6616:1 6610:= 6605:H 6554:T 6546:B 6543:k 6538:s 6536:, 6534:k 6530:ω 6510:1 6503:) 6496:T 6490:B 6485:k 6477:s 6474:, 6471:k 6456:( 6445:1 6440:= 6436:) 6431:s 6428:, 6425:k 6417:( 6413:n 6329:k 6325:k 6302:a 6298:/ 6294:π 6292:4 6285:k 6277:a 6273:/ 6269:π 6267:2 6260:k 6256:k 6252:n 6233:a 6226:n 6223:2 6217:= 6214:K 6186:K 6183:+ 6180:k 6167:f 6164:e 6161:d 6155:= 6146:k 6137:; 6132:K 6129:+ 6126:k 6122:Q 6113:f 6110:e 6107:d 6101:= 6092:k 6088:Q 6074:Π 6070:Q 6062:k 6039:k 5982:ω 5893:. 5889:) 5882:2 5879:1 5873:+ 5868:s 5865:, 5862:k 5858:b 5845:s 5842:, 5839:k 5835:b 5828:( 5822:s 5819:, 5816:k 5802:3 5797:1 5794:= 5791:s 5781:k 5773:= 5768:H 5738:k 5736:b 5724:k 5717:k 5715:b 5710:k 5708:b 5689:k 5687:n 5681:k 5679:b 5675:k 5673:b 5668:k 5666:n 5648:) 5641:2 5638:1 5632:+ 5627:k 5623:b 5610:k 5606:b 5599:( 5593:k 5580:k 5572:= 5567:H 5540:k 5513:k 5509:Q 5484:) 5478:k 5471:b 5455:k 5451:b 5444:( 5437:2 5431:k 5423:m 5413:i 5410:= 5405:k 5379:) 5373:k 5366:b 5362:+ 5350:k 5346:b 5339:( 5329:k 5321:m 5318:2 5308:= 5303:k 5299:Q 5284:k 5282:b 5277:k 5275:b 5258:0 5255:= 5251:] 5234:k 5229:b 5223:, 5211:k 5207:b 5200:[ 5196:= 5191:] 5180:k 5175:b 5171:, 5166:k 5162:b 5156:[ 5150:, 5141:k 5137:, 5134:k 5126:= 5122:] 5105:k 5100:b 5094:, 5089:k 5085:b 5080:[ 5048:) 5042:k 5029:k 5021:m 5017:i 5007:k 5000:Q 4995:( 4984:2 4977:k 4969:m 4962:= 4950:k 4946:b 4922:) 4916:k 4900:k 4892:m 4888:i 4883:+ 4878:k 4874:Q 4869:( 4858:2 4851:k 4843:m 4836:= 4831:k 4827:b 4797:k 4770:k 4766:Q 4743:H 4667:. 4663:ω 4644:k 4636:N 4620:N 4608:a 4604:/ 4600:π 4588:a 4584:/ 4580:π 4561:k 4553:k 4541:k 4539:∂ 4536:/ 4531:k 4529:ω 4527:∂ 4501:k 4499:( 4497:ω 4493:ω 4466:2 4459:= 4456:k 4446:k 4429:, 4419:2 4415:m 4409:1 4405:m 4397:2 4393:a 4390:k 4379:2 4371:4 4360:2 4355:) 4347:2 4343:m 4339:1 4334:+ 4327:1 4323:m 4319:1 4313:( 4306:K 4299:) 4291:2 4287:m 4283:1 4278:+ 4271:1 4267:m 4263:1 4257:( 4253:K 4250:= 4245:2 4222:K 4218:a 4214:2 4211:m 4207:1 4204:m 4196:k 4194:( 4192:ω 4188:ω 4136:s 4129:k 4125:k 4118:k 4057:2 4054:/ 4051:1 4016:2 4013:5 4004:, 3992:2 3989:3 3980:, 3968:2 3965:1 3933:3 3930:, 3927:2 3924:, 3921:1 3918:, 3915:0 3912:= 3909:n 3903:k 3891:) 3887:n 3884:+ 3878:2 3875:1 3868:( 3864:= 3859:n 3855:E 3840:k 3838:ω 3831:a 3827:n 3807:. 3802:2 3799:N 3788:, 3785:2 3779:, 3776:1 3770:, 3767:0 3764:= 3761:n 3747:a 3744:N 3739:n 3733:2 3727:= 3722:n 3718:k 3714:= 3711:k 3698:N 3687:N 3679:Π 3675:Q 3657:) 3651:k 3644:Q 3638:k 3634:Q 3628:2 3623:k 3613:2 3609:m 3605:+ 3600:k 3587:k 3578:( 3572:k 3561:m 3558:2 3554:1 3549:= 3544:H 3515:| 3509:2 3505:a 3502:k 3489:| 3482:2 3479:= 3473:) 3468:a 3465:k 3452:1 3448:( 3442:2 3434:2 3429:= 3424:k 3391:k 3384:Q 3378:k 3374:Q 3368:2 3361:k 3351:m 3346:k 3335:2 3332:1 3326:= 3323:) 3318:a 3315:k 3312:i 3305:e 3296:a 3293:k 3290:i 3286:e 3279:2 3276:( 3271:k 3264:Q 3258:k 3254:Q 3248:k 3238:2 3230:m 3224:2 3221:1 3215:= 3210:2 3205:) 3199:1 3196:+ 3193:j 3189:x 3180:j 3176:x 3171:( 3164:j 3154:2 3146:m 3140:2 3137:1 3102:k 3089:k 3079:k 3071:= 3062:2 3055:l 3051:p 3043:l 3029:k 3026:m 3023:a 3020:i 3016:e 3010:k 3003:Q 2997:k 2993:Q 2987:k 2979:= 2970:k 2966:m 2963:a 2960:i 2956:e 2949:) 2941:k 2937:+ 2934:k 2930:( 2926:l 2923:a 2920:i 2916:e 2910:l 2896:k 2891:Q 2885:k 2881:Q 2871:k 2867:k 2857:N 2854:1 2849:= 2840:m 2837:+ 2834:l 2830:x 2824:l 2820:x 2814:l 2775:0 2772:= 2768:] 2758:k 2749:, 2744:k 2735:[ 2731:= 2723:] 2713:k 2708:Q 2704:, 2699:k 2695:Q 2690:[ 2676:k 2672:, 2669:k 2658:i 2655:= 2649:) 2641:k 2634:k 2630:( 2626:l 2623:a 2620:i 2616:e 2610:l 2600:N 2593:i 2587:= 2576:] 2570:m 2566:p 2562:, 2557:l 2553:x 2548:[ 2542:m 2539:a 2532:k 2528:i 2521:e 2515:l 2512:a 2509:k 2506:i 2502:e 2496:m 2493:, 2490:l 2480:N 2477:1 2472:= 2464:] 2454:k 2445:, 2440:k 2436:Q 2431:[ 2421:m 2418:, 2415:l 2404:i 2401:= 2393:] 2387:m 2383:p 2379:, 2374:l 2370:x 2365:[ 2340:π 2332:k 2311:. 2306:l 2302:p 2296:l 2293:a 2290:k 2287:i 2280:e 2274:l 2263:N 2259:1 2254:= 2245:k 2231:l 2227:x 2221:l 2218:a 2215:k 2212:i 2208:e 2202:l 2191:N 2187:1 2182:= 2173:k 2169:Q 2150:k 2148:p 2143:k 2141:Π 2137:N 2132:k 2130:x 2121:k 2119:Q 2115:N 2088:i 2079:i 2077:p 2072:i 2070:x 2066:m 2047:2 2042:) 2036:j 2032:x 2023:i 2019:x 2014:( 2007:) 2003:n 2000:n 1996:( 1993:} 1990:j 1987:i 1984:{ 1974:2 1966:m 1961:2 1958:1 1953:+ 1947:m 1944:2 1938:2 1933:i 1929:p 1921:N 1916:1 1913:= 1910:i 1902:= 1897:H 1866:i 1862:x 1841:i 1821:0 1818:= 1813:i 1809:x 1785:, 1780:2 1776:x 1772:, 1767:1 1763:x 1740:i 1736:u 1722:N 1696:a 1693:k 1687:) 1684:k 1681:( 1668:) 1666:x 1664:( 1662:φ 1655:n 1653:u 1642:N 1638:a 1613:k 1611:ω 1599:k 1594:k 1592:Q 1575:. 1570:) 1566:a 1563:k 1550:1 1547:( 1542:m 1538:C 1535:2 1527:= 1522:k 1513:; 1508:t 1503:k 1495:i 1491:e 1485:k 1481:A 1477:= 1472:k 1468:Q 1437:. 1429:2 1425:t 1421:d 1414:k 1410:Q 1404:2 1400:d 1393:m 1390:= 1385:k 1381:Q 1377:) 1373:1 1367:a 1364:k 1354:( 1351:C 1348:2 1318:N 1312:1 1309:= 1306:j 1286:) 1283:a 1280:N 1277:( 1273:/ 1269:j 1263:2 1260:= 1257:k 1235:a 1232:n 1229:k 1226:i 1222:e 1218:= 1213:k 1192:k 1190:Q 1185:x 1162:. 1157:a 1154:n 1151:k 1148:i 1144:e 1138:k 1134:Q 1128:N 1123:1 1120:= 1114:2 1110:/ 1106:k 1103:a 1100:N 1092:= 1087:n 1083:u 1046:. 1038:2 1034:t 1030:d 1023:n 1019:u 1013:2 1009:d 1002:m 999:= 995:) 989:1 983:n 979:u 975:+ 970:1 967:+ 964:n 960:u 955:( 951:C 948:+ 943:n 939:u 935:C 932:2 916:n 912:m 908:C 901:n 895:n 893:u 888:a 884:N 880:n 876:n 853:n 849:u 842:n 840:u 832:n 828:u 820:→ 780:→ 775:a 764:n 758:n 751:n 728:( 684:a 680:a 673:λ 635:i 630:i 628:R 620:ω 603:. 598:2 593:) 587:j 583:R 574:i 570:R 565:( 558:2 550:m 544:2 541:1 533:) 529:n 526:n 522:( 519:} 516:j 513:i 510:{ 472:x 468:x 464:x 460:V 456:V 445:V 425:V 421:i 412:i 410:r 392:) 386:j 382:r 373:i 369:r 364:( 360:V 355:j 349:i 339:2 336:1 320:V 280:N 276:N 145:( 31:.

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Index