The equations in this area either do not use axioms of breakthrough mechanics or use relations for which there exists a absolute accord in classical mechanics.
For example, accede a adamant regular, crystalline, i.e. not amorphous, filigree composed of N particles. We will accredit to these particles as atoms, although in a absolute solid these may be molecules. N is some ample number, say about 1023 (on the adjustment of Avogadro's number) for a archetypal sample of solid. If the filigree is rigid, the atoms accept to be advance armament on one addition to accumulate anniversary atom abreast its calm position. These armament may be Van der Waals forces, covalent bonds, electrostatic attractions, and others, all of which are ultimately due to the electric acreage force. Magnetic and gravitational armament are about negligible. The armament amid anniversary brace of atoms may be characterized by a abeyant activity action V that depends on the ambit of break of the atoms. The abeyant activity of the absolute filigree is the sum of all pairwise abeyant energies:2
\,\sum_{i < j} V(r_i - r_j)
where \, r_i is the position of the \, ith atom, and \, V is the abeyant activity amid two atoms.
It is difficult to break this many-body botheration in abounding generality, in either classical or breakthrough mechanics. In adjustment to abridge the task, we acquaint two important approximations. First, we accomplish the sum over adjoining atoms only. Although the electric armament in absolute debris extend to infinity, this approximation is about accurate because the fields produced by abroad atoms are screened. Secondly, we amusement the potentials \, V as harmonic potentials: this is permissible as continued as the atoms abide abutting to their calm positions. (Formally, this is done by Taylor accretion \, V about its calm bulk to boxlike order, giving \, V proportional to the displacement \, x^2 and the adaptable force artlessly proportional to \, x. The absurdity in blank college adjustment agreement charcoal baby if \, x charcoal abutting to the calm position).
The consistent filigree may be visualized as a arrangement of assurance affiliated by springs. The afterward bulk shows a cubic lattice, which is a acceptable archetypal for abounding types of apparent solid. Added lattices cover a beeline chain, which is a actual simple filigree which we will anon use for modelling phonons. Added accepted lattices may be begin in the commodity on clear structure.
Cubic.svg
The abeyant activity of the filigree may now be accounting as
\sum_{\{ij\} (nn)} {1\over2} m \omega^2 (R_i - R_j)^2.
Here, \,\omega is the accustomed abundance of the harmonic potentials, which we accept to be the aforementioned back the filigree is regular. \, R_i is the position alike of the \, ith atom, which we now admeasurement from its calm position. The sum over abutting neighbors is denoted as "(nn)".
edit Filigree waves
Phonon breeding through a aboveboard filigree (atom displacements abundantly exaggerated)
Due to the access amid atoms, the displacement of one or added atoms from their calm positions will accord acceleration to a set of beating after-effects breeding through the lattice. One such beachcomber is apparent in the bulk to the right. The amplitude of the beachcomber is accustomed by the displacements of the atoms from their calm positions. The amicableness \,\lambda is marked.
There is a minimum accessible wavelength, accustomed by alert the calm break a amid atoms. As we shall see in the afterward sections, any amicableness beneath than this can be mapped assimilate a amicableness best than 2a, due to the aeon of the lattice.
Not every accessible filigree beating has a categorical amicableness and frequency. However, the accustomed modes do acquire categorical wavelengths and frequencies.
edit One dimensional lattice
In adjustment to abridge the assay bare for a 3-dimensional filigree of atoms it is acceptable to archetypal a 1-dimensional filigree or beeline chain. This archetypal is circuitous abundant to affectation the arresting appearance of phonons.
edit Classical treatment
The armament amid the atoms are affected to be beeline and nearest-neighbour, and they are represented by an adaptable spring. Anniversary atom is affected to be a point atom and the basis and electrons move in step.(adiabatic approximation)
n-1 n n+1 ← d →
\cdotso++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o\cdots
→→ → →→→
u_{n-1} \qquad\quad u_n \qquad\quad u_{n+1}
Where n labels the n'th atom, d is the ambit amid atoms if the alternation is in calm and un the displacement of the n'th atom from its calm position.
If C is the adaptable connected of the bounce and m the accumulation of the atom again the blueprint of motion of the n'th atom is :
-2Cu_n + C(u_{n+1} + u_{n-1}) = m{\operatorname{d^2}u_n\over\operatorname{d}t^2}
This is a set of accompanying equations and back we apprehend the solutions to be oscillatory, new coordinates can be authentic by a detached Fourier transform, in adjustment to de-couple them.3
Put
u_n = \sum_{k=1}^N U_k e^{iknd}
Here nd replaces the accepted connected capricious x. The Uk are accepted as the accustomed coordinates. Substitution into the blueprint of motion produces the afterward decoupled equations.(This requires a cogent abetment appliance the orthonormality and abyss relations of the detached fourier transform 4)
2C(\cos\,kd-1)U_k = m{\operatorname{d^2}U_k\over\operatorname{d}t^2}
These are the equations for harmonic oscillators which accept the solution:
U_k=A_ke^{i\omega_kt};\qquad\quad \omega_k=\sqrt{ {2C \over m}(1-\cos{kd})}
Each accustomed alike Uk represents an absolute vibrational approach of the filigree with wavenumber k which is accepted as a accustomed mode. The added blueprint for ωk is accepted as the burning affiliation amid the angular abundance and the wavenumber.5
edit Breakthrough treatment
Consider a apparent breakthrough automated harmonic alternation of N identical atoms. This is the simplest breakthrough automated archetypal of a lattice, and we will see how phonons appear from it. The ceremonial that we will advance for this archetypal is readily generalizable to two and three dimensions. The Hamiltonian for this arrangement is
\mathbf{H} = \sum_{i=1}^N {p_i^2 \over 2m} + {1\over 2} m \omega^2 \sum_{\{ij\} (nn)} (x_i - x_j)^2\
where \, m is the accumulation of anniversary atom, and \, x_i and \, p_i are the position and drive operators for the \, ith atom. A altercation of agnate Hamiltonians may be begin in the commodity on the breakthrough harmonic oscillator.
We acquaint a set of \, N "normal coordinates" \, Q_k, authentic as the detached Fourier transforms of the \, x's and \, N "conjugate momenta" \,\Pi authentic as the Fourier transforms of the \, p's:
x_j = {1\over\sqrt{N}} \sum_{n=-N/2}^{N/2} Q_{k_n} e^{ik_nja}
p_j = {1\over\sqrt{N}} \sum_{n=-N/2}^{N/2} \Pi_{k_n} e^{-ik_nja}.
The abundance \, k_n will about-face out to be the beachcomber bulk of the phonon, i.e. \, 2\,\pi disconnected by the wavelength. It takes on quantized values, because the bulk of atoms is finite. The anatomy of the quantization depends on the best of abuttals conditions; for simplicity, we appoint alternate abuttals conditions, defining the \, (N+1)th atom as agnate to the aboriginal atom. Physically, this corresponds to abutting the alternation at its ends. The consistent quantization is
k_n = {2n\pi \over Na} \quad \hbox{for}\ n = 0, \pm1, \pm2, ... , \pm {N \over 2}.\
The top apprenticed to \, n comes from the minimum wavelength, which is alert the filigree agreement \, a, as discussed above.
By inverting the detached Fourier transforms to accurate the \, Q's in agreement of the \, x's and the \,\Pi's in agreement of the \, p's, and appliance the approved barter relations amid the \, x's and \, p's, we can appearance that
\left Q_k , \Pi_{k'} \right = i \hbar \delta_{k k'} \quad ;\quad \left Q_k , Q_{k'} \right = \left \Pi_k , \Pi_{k'} \right = 0.\
In added words, the accustomed coordinates and their conjugate momenta obey the aforementioned barter relations as position and drive operators! Writing the Hamiltonian in agreement of these quantities,
\mathbf{H} = \sum_k \left( { \Pi_k\Pi_{-k} \over 2m } + {1\over2} m \omega_k^2 Q_k Q_{-k} \right)
where
\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))}.\
Notice that the couplings amid the position variables accept been adapted away; if the \, Q's and \,\Pi's were Hermitian (which they are not), the adapted Hamiltonian would call \, N bachelor harmonic oscillators.
The harmonic oscillator eigenvalues or activity levels for the approach ωk are :
E_n = ({1\over2}+n)\hbar\omega_k \quad\quad\quad n=0,1,2,3 ......
If we avoid the zero-point activity again the levels are analogously spaced at :
\hbar\omega , \quad 2\hbar\omega ,\quad 3\hbar\omega \quad ......
So a minimum bulk of activity \hbar\omega accept to be supplied to the harmonic oscillator(or accustomed mode) to move it to the next activity level. In allegory to the photon case if the electromagnetic acreage is quantised, the breakthrough of vibrational activity is alleged a phonon.
All breakthrough systems appearance wave-like and particle-like properties. The particle-like backdrop of the phonon are best accepted appliance the methods of second-quantisation and abettor techniques declared later.
edit Three dimensional lattice
This may be ambiguous to a three-dimensional lattice. The beachcomber bulk k is replaced by a three-dimensional beachcomber agent k. Furthermore, anniversary k is now associated with three accustomed coordinates.
The new indices s = 1, 2, 3 characterization the animosity of the phonons. In the one dimensional model, the atoms were belted to affective forth the line, so the phonons corresponded to longitudinal waves. In three dimensions, beating is not belted to the administration of propagation, and can aswell action in the erect planes, like axle waves. This gives acceleration to the added accustomed coordinates, which, as the anatomy of the Hamiltonian indicates, we may appearance as absolute breed of phonons.
edit Burning relation
Dispersion curve
In the aloft discussion, we accept acquired an blueprint that relates the angular abundance of a phonon, \,\omega_k, to its beachcomber bulk \, k:
\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))} = 2 \omega |\sin(ka/2)|.
This is accepted as a burning relation.
The acceleration of advancement of a phonon, which is aswell the acceleration of complete in the lattice, is accustomed by the abruptness of the burning relation, \,\tfrac{\partial\omega_k}{\partial k} (see accumulation velocity.) At low ethics of \, k (i.e. continued wavelengths), the burning affiliation is about linear, and the acceleration of complete is about \,\omega a, absolute of the phonon frequency. As a result, packets of phonons with altered (but long) wavelengths can bear for ample distances beyond the filigree after breaking apart. This is the acumen that complete propagates through debris after cogent distortion. This behavior fails at ample ethics of \, k, i.e. abbreviate wavelengths, due to the diminutive data of the lattice.
For a clear that has at atomic two atoms in its archaic corpuscle (which may or may not be different), the burning relations display two types of phonons, namely, optical and acoustic modes agnate to the top and lower sets of curves in the diagram, respectively. The vertical arbor is the activity or abundance of phonon, while the accumbent arbor is the wave-vector. The boundaries at -km and km are those of the aboriginal Brillouin zone. The blue, violet, and amber curves are those of longitudinal acoustic, axle acoustic 1, and axle acoustic 2 modes, respectively.
In some crystals the two axle acoustic modes accept absolutely the aforementioned burning curve. It is aswell absorbing that for a clear with N ( > 2) altered atoms in a archaic cell, there are consistently three acoustic modes. The bulk of optical modes is 3N - 3. Abounding phonon burning curves accept been abstinent by neutron scattering.
The physics of complete in fluids differs from the physics of complete in solids, although both are body waves: complete after-effects in fluids alone accept longitudinal components, admitting complete after-effects in debris accept longitudinal and axle components. This is because fluids can't abutment microburst stresses. (but see viscoelastic fluids, which alone administer to top frequencies, though).
edit Interpretation of phonons appliance added quantisation techniques
In fact, the above-derived Hamiltonian looks like the classical Hamiltonian function, but if it is interpreted as an operator, again it describes a breakthrough acreage approach of non-interacting bosons. This leads to new physics.
The activity spectrum of this Hamiltonian is calmly acquired by the adjustment of ladder operators, agnate to the breakthrough harmonic oscillator problem. We acquaint a set of ladder operators authentic by
\begin{matrix} a_k &=& \sqrt{m\omega_k \over 2\hbar} (Q_k + {i\over m\omega_k} \Pi_{-k}) \\ a_k^\dagger &=& \sqrt{m\omega_k \over 2\hbar} (Q_{-k} - {i\over m\omega_k} \Pi_k). \end{matrix}
The ladder operators amuse the afterward identities:
\mathbf{H} = \sum_k \hbar \omega_k \left(a_k^{\dagger}a_k + 1/2\right)
a_k , a_{k'}^{\dagger} = \delta_{kk'}
a_k , a_{k'} = a_k^{\dagger} , a_{k'}^{\dagger} = 0.
As with the breakthrough harmonic oscillator, we can again appearance that \, a_k^\dagger and \, a_k appropriately actualize and abort one action of activity \,\hbar\omega_k. These excitations are phonons.
We can anon deduce two important backdrop of phonons. Firstly, phonons are bosons, back any bulk of identical excitations can be created by again appliance of the conception abettor \, a_k^\dagger. Secondly, anniversary phonon is a "collective mode" acquired by the motion of every atom in the lattice. This may be apparent from the actuality that the ladder operators accommodate sums over the position and drive operators of every atom.
It is not a priori accessible that these excitations generated by the \, a operators are actually after-effects of filigree displacement, but one may argue oneself of this by artful the position-position alternation function. Let \, |k\rangle denote a accompaniment with a individual breakthrough of approach \, k excited, i.e.
\begin{matrix} | k \rangle = a_k^\dagger | 0 \rangle. \end{matrix}
One can appearance that, for any two atoms \, j and \,\ell,
\langle k | x_j(t) x_{\ell}(0) | k \rangle = \frac{\hbar}{Nm\omega_k} \cos \left k(j-\ell)a - \omega_k t \right + \langle 0 | x_j(t) x_\ell(0) |0 \rangle
which is absolutely what we would apprehend for a filigree beachcomber with abundance \,\omega_k and beachcomber bulk \, k.
In three ambit the Hamiltonian has the form
\mathbf{H} = \sum_k \sum_{s = 1}^3 \hbar \, \omega_{k,s} \left( a_{k,s}^{\dagger}a_{k,s} + 1/2 \right).
For example, accede a adamant regular, crystalline, i.e. not amorphous, filigree composed of N particles. We will accredit to these particles as atoms, although in a absolute solid these may be molecules. N is some ample number, say about 1023 (on the adjustment of Avogadro's number) for a archetypal sample of solid. If the filigree is rigid, the atoms accept to be advance armament on one addition to accumulate anniversary atom abreast its calm position. These armament may be Van der Waals forces, covalent bonds, electrostatic attractions, and others, all of which are ultimately due to the electric acreage force. Magnetic and gravitational armament are about negligible. The armament amid anniversary brace of atoms may be characterized by a abeyant activity action V that depends on the ambit of break of the atoms. The abeyant activity of the absolute filigree is the sum of all pairwise abeyant energies:2
\,\sum_{i < j} V(r_i - r_j)
where \, r_i is the position of the \, ith atom, and \, V is the abeyant activity amid two atoms.
It is difficult to break this many-body botheration in abounding generality, in either classical or breakthrough mechanics. In adjustment to abridge the task, we acquaint two important approximations. First, we accomplish the sum over adjoining atoms only. Although the electric armament in absolute debris extend to infinity, this approximation is about accurate because the fields produced by abroad atoms are screened. Secondly, we amusement the potentials \, V as harmonic potentials: this is permissible as continued as the atoms abide abutting to their calm positions. (Formally, this is done by Taylor accretion \, V about its calm bulk to boxlike order, giving \, V proportional to the displacement \, x^2 and the adaptable force artlessly proportional to \, x. The absurdity in blank college adjustment agreement charcoal baby if \, x charcoal abutting to the calm position).
The consistent filigree may be visualized as a arrangement of assurance affiliated by springs. The afterward bulk shows a cubic lattice, which is a acceptable archetypal for abounding types of apparent solid. Added lattices cover a beeline chain, which is a actual simple filigree which we will anon use for modelling phonons. Added accepted lattices may be begin in the commodity on clear structure.
Cubic.svg
The abeyant activity of the filigree may now be accounting as
\sum_{\{ij\} (nn)} {1\over2} m \omega^2 (R_i - R_j)^2.
Here, \,\omega is the accustomed abundance of the harmonic potentials, which we accept to be the aforementioned back the filigree is regular. \, R_i is the position alike of the \, ith atom, which we now admeasurement from its calm position. The sum over abutting neighbors is denoted as "(nn)".
edit Filigree waves
Phonon breeding through a aboveboard filigree (atom displacements abundantly exaggerated)
Due to the access amid atoms, the displacement of one or added atoms from their calm positions will accord acceleration to a set of beating after-effects breeding through the lattice. One such beachcomber is apparent in the bulk to the right. The amplitude of the beachcomber is accustomed by the displacements of the atoms from their calm positions. The amicableness \,\lambda is marked.
There is a minimum accessible wavelength, accustomed by alert the calm break a amid atoms. As we shall see in the afterward sections, any amicableness beneath than this can be mapped assimilate a amicableness best than 2a, due to the aeon of the lattice.
Not every accessible filigree beating has a categorical amicableness and frequency. However, the accustomed modes do acquire categorical wavelengths and frequencies.
edit One dimensional lattice
In adjustment to abridge the assay bare for a 3-dimensional filigree of atoms it is acceptable to archetypal a 1-dimensional filigree or beeline chain. This archetypal is circuitous abundant to affectation the arresting appearance of phonons.
edit Classical treatment
The armament amid the atoms are affected to be beeline and nearest-neighbour, and they are represented by an adaptable spring. Anniversary atom is affected to be a point atom and the basis and electrons move in step.(adiabatic approximation)
n-1 n n+1 ← d →
\cdotso++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o\cdots
→→ → →→→
u_{n-1} \qquad\quad u_n \qquad\quad u_{n+1}
Where n labels the n'th atom, d is the ambit amid atoms if the alternation is in calm and un the displacement of the n'th atom from its calm position.
If C is the adaptable connected of the bounce and m the accumulation of the atom again the blueprint of motion of the n'th atom is :
-2Cu_n + C(u_{n+1} + u_{n-1}) = m{\operatorname{d^2}u_n\over\operatorname{d}t^2}
This is a set of accompanying equations and back we apprehend the solutions to be oscillatory, new coordinates can be authentic by a detached Fourier transform, in adjustment to de-couple them.3
Put
u_n = \sum_{k=1}^N U_k e^{iknd}
Here nd replaces the accepted connected capricious x. The Uk are accepted as the accustomed coordinates. Substitution into the blueprint of motion produces the afterward decoupled equations.(This requires a cogent abetment appliance the orthonormality and abyss relations of the detached fourier transform 4)
2C(\cos\,kd-1)U_k = m{\operatorname{d^2}U_k\over\operatorname{d}t^2}
These are the equations for harmonic oscillators which accept the solution:
U_k=A_ke^{i\omega_kt};\qquad\quad \omega_k=\sqrt{ {2C \over m}(1-\cos{kd})}
Each accustomed alike Uk represents an absolute vibrational approach of the filigree with wavenumber k which is accepted as a accustomed mode. The added blueprint for ωk is accepted as the burning affiliation amid the angular abundance and the wavenumber.5
edit Breakthrough treatment
Consider a apparent breakthrough automated harmonic alternation of N identical atoms. This is the simplest breakthrough automated archetypal of a lattice, and we will see how phonons appear from it. The ceremonial that we will advance for this archetypal is readily generalizable to two and three dimensions. The Hamiltonian for this arrangement is
\mathbf{H} = \sum_{i=1}^N {p_i^2 \over 2m} + {1\over 2} m \omega^2 \sum_{\{ij\} (nn)} (x_i - x_j)^2\
where \, m is the accumulation of anniversary atom, and \, x_i and \, p_i are the position and drive operators for the \, ith atom. A altercation of agnate Hamiltonians may be begin in the commodity on the breakthrough harmonic oscillator.
We acquaint a set of \, N "normal coordinates" \, Q_k, authentic as the detached Fourier transforms of the \, x's and \, N "conjugate momenta" \,\Pi authentic as the Fourier transforms of the \, p's:
x_j = {1\over\sqrt{N}} \sum_{n=-N/2}^{N/2} Q_{k_n} e^{ik_nja}
p_j = {1\over\sqrt{N}} \sum_{n=-N/2}^{N/2} \Pi_{k_n} e^{-ik_nja}.
The abundance \, k_n will about-face out to be the beachcomber bulk of the phonon, i.e. \, 2\,\pi disconnected by the wavelength. It takes on quantized values, because the bulk of atoms is finite. The anatomy of the quantization depends on the best of abuttals conditions; for simplicity, we appoint alternate abuttals conditions, defining the \, (N+1)th atom as agnate to the aboriginal atom. Physically, this corresponds to abutting the alternation at its ends. The consistent quantization is
k_n = {2n\pi \over Na} \quad \hbox{for}\ n = 0, \pm1, \pm2, ... , \pm {N \over 2}.\
The top apprenticed to \, n comes from the minimum wavelength, which is alert the filigree agreement \, a, as discussed above.
By inverting the detached Fourier transforms to accurate the \, Q's in agreement of the \, x's and the \,\Pi's in agreement of the \, p's, and appliance the approved barter relations amid the \, x's and \, p's, we can appearance that
\left Q_k , \Pi_{k'} \right = i \hbar \delta_{k k'} \quad ;\quad \left Q_k , Q_{k'} \right = \left \Pi_k , \Pi_{k'} \right = 0.\
In added words, the accustomed coordinates and their conjugate momenta obey the aforementioned barter relations as position and drive operators! Writing the Hamiltonian in agreement of these quantities,
\mathbf{H} = \sum_k \left( { \Pi_k\Pi_{-k} \over 2m } + {1\over2} m \omega_k^2 Q_k Q_{-k} \right)
where
\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))}.\
Notice that the couplings amid the position variables accept been adapted away; if the \, Q's and \,\Pi's were Hermitian (which they are not), the adapted Hamiltonian would call \, N bachelor harmonic oscillators.
The harmonic oscillator eigenvalues or activity levels for the approach ωk are :
E_n = ({1\over2}+n)\hbar\omega_k \quad\quad\quad n=0,1,2,3 ......
If we avoid the zero-point activity again the levels are analogously spaced at :
\hbar\omega , \quad 2\hbar\omega ,\quad 3\hbar\omega \quad ......
So a minimum bulk of activity \hbar\omega accept to be supplied to the harmonic oscillator(or accustomed mode) to move it to the next activity level. In allegory to the photon case if the electromagnetic acreage is quantised, the breakthrough of vibrational activity is alleged a phonon.
All breakthrough systems appearance wave-like and particle-like properties. The particle-like backdrop of the phonon are best accepted appliance the methods of second-quantisation and abettor techniques declared later.
edit Three dimensional lattice
This may be ambiguous to a three-dimensional lattice. The beachcomber bulk k is replaced by a three-dimensional beachcomber agent k. Furthermore, anniversary k is now associated with three accustomed coordinates.
The new indices s = 1, 2, 3 characterization the animosity of the phonons. In the one dimensional model, the atoms were belted to affective forth the line, so the phonons corresponded to longitudinal waves. In three dimensions, beating is not belted to the administration of propagation, and can aswell action in the erect planes, like axle waves. This gives acceleration to the added accustomed coordinates, which, as the anatomy of the Hamiltonian indicates, we may appearance as absolute breed of phonons.
edit Burning relation
Dispersion curve
In the aloft discussion, we accept acquired an blueprint that relates the angular abundance of a phonon, \,\omega_k, to its beachcomber bulk \, k:
\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))} = 2 \omega |\sin(ka/2)|.
This is accepted as a burning relation.
The acceleration of advancement of a phonon, which is aswell the acceleration of complete in the lattice, is accustomed by the abruptness of the burning relation, \,\tfrac{\partial\omega_k}{\partial k} (see accumulation velocity.) At low ethics of \, k (i.e. continued wavelengths), the burning affiliation is about linear, and the acceleration of complete is about \,\omega a, absolute of the phonon frequency. As a result, packets of phonons with altered (but long) wavelengths can bear for ample distances beyond the filigree after breaking apart. This is the acumen that complete propagates through debris after cogent distortion. This behavior fails at ample ethics of \, k, i.e. abbreviate wavelengths, due to the diminutive data of the lattice.
For a clear that has at atomic two atoms in its archaic corpuscle (which may or may not be different), the burning relations display two types of phonons, namely, optical and acoustic modes agnate to the top and lower sets of curves in the diagram, respectively. The vertical arbor is the activity or abundance of phonon, while the accumbent arbor is the wave-vector. The boundaries at -km and km are those of the aboriginal Brillouin zone. The blue, violet, and amber curves are those of longitudinal acoustic, axle acoustic 1, and axle acoustic 2 modes, respectively.
In some crystals the two axle acoustic modes accept absolutely the aforementioned burning curve. It is aswell absorbing that for a clear with N ( > 2) altered atoms in a archaic cell, there are consistently three acoustic modes. The bulk of optical modes is 3N - 3. Abounding phonon burning curves accept been abstinent by neutron scattering.
The physics of complete in fluids differs from the physics of complete in solids, although both are body waves: complete after-effects in fluids alone accept longitudinal components, admitting complete after-effects in debris accept longitudinal and axle components. This is because fluids can't abutment microburst stresses. (but see viscoelastic fluids, which alone administer to top frequencies, though).
edit Interpretation of phonons appliance added quantisation techniques
In fact, the above-derived Hamiltonian looks like the classical Hamiltonian function, but if it is interpreted as an operator, again it describes a breakthrough acreage approach of non-interacting bosons. This leads to new physics.
The activity spectrum of this Hamiltonian is calmly acquired by the adjustment of ladder operators, agnate to the breakthrough harmonic oscillator problem. We acquaint a set of ladder operators authentic by
\begin{matrix} a_k &=& \sqrt{m\omega_k \over 2\hbar} (Q_k + {i\over m\omega_k} \Pi_{-k}) \\ a_k^\dagger &=& \sqrt{m\omega_k \over 2\hbar} (Q_{-k} - {i\over m\omega_k} \Pi_k). \end{matrix}
The ladder operators amuse the afterward identities:
\mathbf{H} = \sum_k \hbar \omega_k \left(a_k^{\dagger}a_k + 1/2\right)
a_k , a_{k'}^{\dagger} = \delta_{kk'}
a_k , a_{k'} = a_k^{\dagger} , a_{k'}^{\dagger} = 0.
As with the breakthrough harmonic oscillator, we can again appearance that \, a_k^\dagger and \, a_k appropriately actualize and abort one action of activity \,\hbar\omega_k. These excitations are phonons.
We can anon deduce two important backdrop of phonons. Firstly, phonons are bosons, back any bulk of identical excitations can be created by again appliance of the conception abettor \, a_k^\dagger. Secondly, anniversary phonon is a "collective mode" acquired by the motion of every atom in the lattice. This may be apparent from the actuality that the ladder operators accommodate sums over the position and drive operators of every atom.
It is not a priori accessible that these excitations generated by the \, a operators are actually after-effects of filigree displacement, but one may argue oneself of this by artful the position-position alternation function. Let \, |k\rangle denote a accompaniment with a individual breakthrough of approach \, k excited, i.e.
\begin{matrix} | k \rangle = a_k^\dagger | 0 \rangle. \end{matrix}
One can appearance that, for any two atoms \, j and \,\ell,
\langle k | x_j(t) x_{\ell}(0) | k \rangle = \frac{\hbar}{Nm\omega_k} \cos \left k(j-\ell)a - \omega_k t \right + \langle 0 | x_j(t) x_\ell(0) |0 \rangle
which is absolutely what we would apprehend for a filigree beachcomber with abundance \,\omega_k and beachcomber bulk \, k.
In three ambit the Hamiltonian has the form
\mathbf{H} = \sum_k \sum_{s = 1}^3 \hbar \, \omega_{k,s} \left( a_{k,s}^{\dagger}a_{k,s} + 1/2 \right).
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