Physics:Quantum Phonons - Revision history

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	'''Acoustic phonons''' correspond to modes in which neighboring atoms move approximately in phase. In the long-wavelength limit this amounts to a uniform displacement of the crystal, which costs no restoring energy, so the frequency tends to zero as the wavelength becomes very large. Acoustic phonons are responsible for ordinary sound propagation in solids. Longitudinal and transverse acoustic phonons are often abbreviated LA and TA.		'''Acoustic phonons''' correspond to modes in which neighboring atoms move approximately in phase. In the long-wavelength limit this amounts to a uniform displacement of the crystal, which costs no restoring energy, so the frequency tends to zero as the wavelength becomes very large. Acoustic phonons are responsible for ordinary sound propagation in solids. Longitudinal and transverse acoustic phonons are often abbreviated LA and TA.

	'''Optical phonons''' correspond to out-of-phase motion of atoms in the basis. In ionic crystals, these oscillations produce an oscillating dipole moment and can couple to electromagnetic radiation, especially in the infrared. For this reason they are called optical phonons.<ref name=girvinYang /> Longitudinal and transverse optical phonons are often labeled LO and TO. Optical phonons have a nonzero frequency at the center of the Brillouin zone.		'''Optical phonons''' correspond to out-of-phase motion of atoms in the basis. In ionic crystals, these oscillations produce an oscillating dipole moment and can couple to electromagnetic radiation, especially in the infrared. For this reason they are called optical phonons.<ref name=girvinYang>{{cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|pages=78–96}}</ref> Longitudinal and transverse optical phonons are often labeled LO and TO. Optical phonons have a nonzero frequency at the center of the Brillouin zone.

	Optical phonons may be infrared active or Raman active, making them important in vibrational spectroscopy and in the optical characterization of crystals.		Optical phonons may be infrared active or Raman active, making them important in vibrational spectroscopy and in the optical characterization of crystals.
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	=References=		= References =
	{{reflist\|3}}		{{reflist\|3}}

	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}
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	{{Short description\|Quantum Collection topic on Quantum Phonons}}~~Book I~~		{{Short description\|Quantum Collection topic on Quantum Phonons}}

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	~~~~{{Short description\|Quantum Collection topic on Quantum Phonons}}		{{Short description\|Quantum Collection topic on Quantum Phonons}}Book I

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	A '''~~phonon~~''' is a quasiparticle or collective excitation associated with mechanical vibrations in a periodic, elastic arrangement of atoms or molecules in condensed matter, especially in solids and some liquids. In quantum mechanics, a phonon is the quantized excitation of a normal mode of vibration in an interacting lattice.~~<ref>{{cite book\|first=Franz \|last=Schwabl \|title=Advanced Quantum Mechanics \|edition=4th \|publisher=Springer \|date=2008 \|page=253 \|isbn=978-3-540-85062-5}}</ref>~~ Phonons can be regarded as quantized sound waves, in close analogy to photons as the quanta of light.<ref name=girvinYang>{{cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|pages=78–96}}</ref>		'''Phonons''' a phonon is a quasiparticle or collective excitation associated with mechanical vibrations in a periodic, elastic arrangement of atoms or molecules in condensed matter, especially in solids and some liquids. In quantum mechanics, a phonon is the quantized excitation of a normal mode of vibration in an interacting lattice. Phonons can be regarded as quantized sound waves, in close analogy to photons as the quanta of light. The study of phonons is a central part of condensed matter physics because phonons play a major role in the thermal, elastic, optical, and electronic properties of materials. They are essential in understanding thermal conductivity, aspects of electrical conductivity, lattice heat capacity, neutron scattering, and important interactions in solids such as electron–phonon coupling. The concept of the phonon was introduced in 1930 by the Soviet physicist Igor Tamm, and the name phonon was suggested by Yakov Frenkel. The word derives from the Greek (), meaning sound or voice, reflecting the fact that long-wavelength phonons correspond to sound propagation in matter. The analogy with photons emphasizes the wave–particle duality of lattice vibrations. Acoustic and optical lattice vibrations in a diatomic chain. In acoustic modes neighboring atoms move largely in phase, while in optical modes they move out of phase. A phonon is the quantum-mechanical description of an elementary vibrational motion in which a crystal lattice oscillates at a single frequency.

	The study of phonons is a central part of condensed matter physics because phonons play a major role in the thermal, elastic, optical, and electronic properties of materials. They are essential in understanding thermal conductivity, aspects of electrical conductivity, lattice heat capacity, neutron scattering, and important interactions in solids such as electron–phonon coupling.

	The concept of the phonon was introduced in 1930 by the Soviet physicist Igor Tamm, and the name ''phonon'' was suggested by Yakov Frenkel.<ref>{{Cite book \|last=Kozhevnikov \|first=A. B. \|url=https://archive.org/details/stalinsgreatscie0000kozh/mode/1up?q=tamm+phonon \|title=Stalin's great science: the times and adventures of Soviet physicists \|date=2004 \|publisher=Imperial College Press \|isbn=978-1-86094-419-2 \|location=London \|pages=64–69}}</ref> The word derives from the Greek ~~{{lang\|grc\|φωνή}}~~ (~~{{transliteration\|grc\|phonē}}~~), meaning ''sound'' or ''voice'', reflecting the fact that long-wavelength phonons correspond to sound propagation in matter. The analogy with photons emphasizes the wave–particle duality of lattice vibrations.
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	Acoustic and optical lattice vibrations in a diatomic chain. In acoustic modes neighboring atoms move largely in phase, while in optical modes they move out of phase.
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	~~</div>~~
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	Recent theoretical work has also explored whether phonons and related excitations may be associated with effective mass and even unusual gravitational behavior in appropriate effective descriptions.<ref>Alberto Nicolis and Riccardo Penco. (2017). [https://arxiv.org/abs/1705.08914 Mutual Interactions of Phonons, Rotons, and Gravity], Arxiv.org.</ref><ref>Angelo Esposito, Rafael Krichevsky, and Alberto Nicolis. (2018). [https://arxiv.org/abs/1807.08771 The mass of sound].</ref> Experimental advances have further enabled increasingly direct control and detection of phononic excitations in nanoscale and quantum systems.<ref name="nature">{{Cite journal \|title=Detecting the softest sounds in the Universe \|date=July 1, 2019 \|journal=Nature \|volume=571 \|issue=7763 \|pages=8–9 \|doi=10.1038/d41586-019-02009-5 \|bibcode=2019Natur.571....8. \|s2cid=195774243}}</ref>		Recent theoretical work has also explored whether phonons and related excitations may be associated with effective mass and even unusual gravitational behavior in appropriate effective descriptions.<ref>Alberto Nicolis and Riccardo Penco. (2017). [https://arxiv.org/abs/1705.08914 Mutual Interactions of Phonons, Rotons, and Gravity], Arxiv.org.</ref><ref>Angelo Esposito, Rafael Krichevsky, and Alberto Nicolis. (2018). [https://arxiv.org/abs/1807.08771 The mass of sound].</ref> Experimental advances have further enabled increasingly direct control and detection of phononic excitations in nanoscale and quantum systems.<ref name="nature">{{Cite journal \|title=Detecting the softest sounds in the Universe \|date=July 1, 2019 \|journal=Nature \|volume=571 \|issue=7763 \|pages=8–9 \|doi=10.1038/d41586-019-02009-5 \|bibcode=2019Natur.571....8. \|s2cid=195774243}}</ref>

	~~== See also ==~~
	=See also=		=See also=
	{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

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	{{Short description\|Quantum Collection topic on Quantum Phonons}}		{{Short description\|Quantum Collection topic on Quantum Phonons}}

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	{{Quantum article nav\|previous=Physics:Quantum Semiconductor physics\|previous label=Semiconductor physics\|next=Physics:Quantum Electron-phonon interaction\|next label=Electron-phonon interaction}}		{{Quantum article nav\|previous=Physics:Quantum Semiconductor physics\|previous label=Semiconductor physics\|next=Physics:Quantum Electron-phonon interaction\|next label=Electron-phonon interaction}}

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	A '''phonon''' is a [[quasiparticle]] or [[collective excitation]] associated with mechanical vibrations in a periodic, elastic arrangement of ~~[[atom]]s~~ or ~~[[molecule]]s~~ in [[condensed matter ~~physics\|condensed matter]]~~, especially in ~~[[solid]]s~~ and some ~~[[liquid]]s~~. In quantum mechanics, a phonon is the quantized excitation of a normal mode of vibration in an interacting lattice.<ref>{{cite book\|first=Franz \|last=Schwabl \|title=Advanced Quantum Mechanics \|edition=4th \|publisher=Springer \|date=2008 \|page=253 \|isbn=978-3-540-85062-5}}</ref> Phonons can be regarded as quantized [[sound waves]], in close analogy to ~~[[photon]]s~~ as the quanta of [[light]].<ref name=girvinYang>{{cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|pages=78–96}}</ref>		A '''phonon''' is a quasiparticle or collective excitation associated with mechanical vibrations in a periodic, elastic arrangement of atoms or molecules in condensed matter, especially in solids and some liquids. In quantum mechanics, a phonon is the quantized excitation of a normal mode of vibration in an interacting lattice.<ref>{{cite book\|first=Franz \|last=Schwabl \|title=Advanced Quantum Mechanics \|edition=4th \|publisher=Springer \|date=2008 \|page=253 \|isbn=978-3-540-85062-5}}</ref> Phonons can be regarded as quantized sound waves, in close analogy to photons as the quanta of light.<ref name=girvinYang>{{cite book \|last1=Girvin \|first1=Steven M. \|last2=Yang \|first2=Kun \|title=Modern Condensed Matter Physics \|date=2019 \|publisher=Cambridge University Press \|isbn=978-1-107-13739-4 \|pages=78–96}}</ref>

	The study of phonons is a central part of condensed matter physics because phonons play a major role in the thermal, elastic, optical, and electronic properties of materials. They are essential in understanding [[thermal conductivity]], aspects of [[electrical conductivity]], lattice heat capacity, [[neutron scattering]], and important interactions in solids such as electron–phonon coupling.		The study of phonons is a central part of condensed matter physics because phonons play a major role in the thermal, elastic, optical, and electronic properties of materials. They are essential in understanding thermal conductivity, aspects of electrical conductivity, lattice heat capacity, neutron scattering, and important interactions in solids such as electron–phonon coupling.

	The concept of the phonon was introduced in 1930 by the Soviet physicist Igor Tamm, and the name ''phonon'' was suggested by Yakov Frenkel.<ref>{{Cite book \|last=Kozhevnikov \|first=A. B. \|url=https://archive.org/details/stalinsgreatscie0000kozh/mode/1up?q=tamm+phonon \|title=Stalin's great science: the times and adventures of Soviet physicists \|date=2004 \|publisher=Imperial College Press \|isbn=978-1-86094-419-2 \|location=London \|pages=64–69}}</ref> The word derives from the Greek {{lang\|grc\|φωνή}} ({{transliteration\|grc\|phonē}}), meaning ''sound'' or ''voice'', reflecting the fact that long-wavelength phonons correspond to sound propagation in matter. The analogy with photons emphasizes the wave–particle duality of lattice vibrations.		The concept of the phonon was introduced in 1930 by the Soviet physicist Igor Tamm, and the name ''phonon'' was suggested by Yakov Frenkel.<ref>{{Cite book \|last=Kozhevnikov \|first=A. B. \|url=https://archive.org/details/stalinsgreatscie0000kozh/mode/1up?q=tamm+phonon \|title=Stalin's great science: the times and adventures of Soviet physicists \|date=2004 \|publisher=Imperial College Press \|isbn=978-1-86094-419-2 \|location=London \|pages=64–69}}</ref> The word derives from the Greek {{lang\|grc\|φωνή}} ({{transliteration\|grc\|phonē}}), meaning ''sound'' or ''voice'', reflecting the fact that long-wavelength phonons correspond to sound propagation in matter. The analogy with photons emphasizes the wave–particle duality of lattice vibrations.
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	== Definition ==		== Definition ==

	A phonon is the quantum-mechanical description of an elementary vibrational motion in which a crystal lattice oscillates at a single frequency.<ref>{{cite book\|last1=Simon\|first1=Steven H.\|title=The Oxford solid state basics\|date=2013\|publisher=Oxford University Press\|location=Oxford\|isbn=978-0-19-968077-1\|page=82\|edition=1st}}</ref> In classical mechanics, this corresponds to a [[normal mode]] of vibration. Any general lattice vibration may be decomposed into a superposition of such normal modes, in much the same way that an arbitrary periodic signal can be decomposed into Fourier components.		A phonon is the quantum-mechanical description of an elementary vibrational motion in which a crystal lattice oscillates at a single frequency.<ref>{{cite book\|last1=Simon\|first1=Steven H.\|title=The Oxford solid state basics\|date=2013\|publisher=Oxford University Press\|location=Oxford\|isbn=978-0-19-968077-1\|page=82\|edition=1st}}</ref> In classical mechanics, this corresponds to a normal mode of vibration. Any general lattice vibration may be decomposed into a superposition of such normal modes, in much the same way that an arbitrary periodic signal can be decomposed into Fourier components.

	In the classical description these modes are wave-like. In the quantum description they also have particle-like properties, and the discrete packet of vibrational energy associated with a given mode is called a phonon.		In the classical description these modes are wave-like. In the quantum description they also have particle-like properties, and the discrete packet of vibrational energy associated with a given mode is called a phonon.
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	== Lattice dynamics ==		== Lattice dynamics ==

	In a crystalline solid, atoms occupy equilibrium positions in a regular lattice. Since the atoms remain near these positions, they must exert forces on one another. These restoring forces may arise from [[covalent ~~bond]]s~~, electrostatic attraction, Van der Waals forcess, and related interactions. The full many-body problem is complicated, so in lattice dynamics two common approximations are often introduced.		In a crystalline solid, atoms occupy equilibrium positions in a regular lattice. Since the atoms remain near these positions, they must exert forces on one another. These restoring forces may arise from covalent bonds, electrostatic attraction, Van der Waals forcess, and related interactions. The full many-body problem is complicated, so in lattice dynamics two common approximations are often introduced.

	First, one usually considers mainly interactions between neighboring atoms. Second, the interaction potential is approximated as harmonic near equilibrium. This means the potential is expanded to quadratic order, so the restoring force is proportional to displacement. In that approximation, the lattice behaves like a collection of masses connected by springs.		First, one usually considers mainly interactions between neighboring atoms. Second, the interaction potential is approximated as harmonic near equilibrium. This means the potential is expanded to quadratic order, so the restoring force is proportional to displacement. In that approximation, the lattice behaves like a collection of masses connected by springs.
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	:<math>\omega_k=\sqrt{ \frac{2C}{m}(1-\cos{ka})}.</math>		:<math>\omega_k=\sqrt{ \frac{2C}{m}(1-\cos{ka})}.</math>

	Each normal coordinate corresponds to a vibrational normal mode of the lattice. The relation between ''ω'' and ''k'' is called the [[dispersion relation]]. In the long-wavelength limit, the dispersion becomes approximately linear and the resulting wave behaves like an ordinary sound wave.		Each normal coordinate corresponds to a vibrational normal mode of the lattice. The relation between ''ω'' and ''k'' is called the dispersion relation. In the long-wavelength limit, the dispersion becomes approximately linear and the resulting wave behaves like an ordinary sound wave.

	=== Quantum treatment ===		=== Quantum treatment ===
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	For long wavelengths, the acoustic branch is nearly linear. Its slope gives the group velocity and therefore the speed of sound in the crystal. At larger wavevectors, the dispersion deviates from linear behavior because of the underlying lattice structure.		For long wavelengths, the acoustic branch is nearly linear. Its slope gives the group velocity and therefore the speed of sound in the crystal. At larger wavevectors, the dispersion deviates from linear behavior because of the underlying lattice structure.

	Crystals with ''p'' atoms in the primitive cell have 3''p'' phonon branches in three dimensions: three acoustic branches and 3''p'' − 3 optical branches. Measured phonon dispersion curves, often obtained by [[inelastic neutron scattering]], provide detailed information about the dynamics of real materials.<ref name=Cardona>		Crystals with ''p'' atoms in the primitive cell have 3''p'' phonon branches in three dimensions: three acoustic branches and 3''p'' − 3 optical branches. Measured phonon dispersion curves, often obtained by inelastic neutron scattering, provide detailed information about the dynamics of real materials.<ref name=Cardona>
	{{cite book \|title=Fundamentals of Semiconductors \|series=Physics and Materials Properties \|first1=Peter Y. \|last1=Yu \|first2=Manuel \|last2=Cardona \|chapter-url=https://books.google.com/books?id=5aBuKYBT_hsC&pg=PA111 \|page=111 \|chapter=Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes \|isbn=978-3-642-00709-5 \|year=2010 \|edition=4th \|publisher=Springer}}</ref>		{{cite book \|title=Fundamentals of Semiconductors \|series=Physics and Materials Properties \|first1=Peter Y. \|last1=Yu \|first2=Manuel \|last2=Cardona \|chapter-url=https://books.google.com/books?id=5aBuKYBT_hsC&pg=PA111 \|page=111 \|chapter=Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes \|isbn=978-3-642-00709-5 \|year=2010 \|edition=4th \|publisher=Springer}}</ref>

	== Acoustic and optical phonons ==		== Acoustic and optical phonons ==

	Solids with more than one atom in the smallest [[unit cell]] exhibit both '''acoustic phonons''' and '''optical phonons'''.		Solids with more than one atom in the smallest unit cell exhibit both '''acoustic phonons''' and '''optical phonons'''.

	'''Acoustic phonons''' correspond to modes in which neighboring atoms move approximately in phase. In the long-wavelength limit this amounts to a uniform displacement of the crystal, which costs no restoring energy, so the frequency tends to zero as the wavelength becomes very large. Acoustic phonons are responsible for ordinary sound propagation in solids. Longitudinal and transverse acoustic phonons are often abbreviated LA and TA.		'''Acoustic phonons''' correspond to modes in which neighboring atoms move approximately in phase. In the long-wavelength limit this amounts to a uniform displacement of the crystal, which costs no restoring energy, so the frequency tends to zero as the wavelength becomes very large. Acoustic phonons are responsible for ordinary sound propagation in solids. Longitudinal and transverse acoustic phonons are often abbreviated LA and TA.
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	:<math>\mathcal{H} = \sum_\alpha\hbar\omega_\alpha {a_\alpha}^\dagger a_\alpha.</math>		:<math>\mathcal{H} = \sum_\alpha\hbar\omega_\alpha {a_\alpha}^\dagger a_\alpha.</math>

	In this description, each phonon mode behaves like an independent quantum harmonic oscillator. The operators {{math\|''a''<sup>†</sup>}} and {{math\|''a''}} create and annihilate phonons respectively. The occupation number gives how many phonons are present in a given mode. This shows directly that phonons obey Bose–Einstein statistics and are therefore ~~[[boson]]s~~.<ref name=ashcrroftMermin>{{cite book \|last1=Ashcroft \|first1=Neil W. \|last2=Mermin \|first2=N. David \|title=Solid State Physics \|date=1976 \|publisher=Saunders College Publishing \|isbn=0-03-083993-9 \|pages=780–783}}</ref><ref>{{cite book \|title= Statistical Mechanics, A Set of Lectures \|last= Feynman \|first= Richard P. \|year= 1982 \|publisher= Benjamin-Cummings \|location= Reading, MA \|isbn= 978-0-8053-2508-9 \|page= [https://archive.org/details/statisticalmecha00rich/page/159 159] \|url= https://archive.org/details/statisticalmecha00rich/page/159 }}</ref>		In this description, each phonon mode behaves like an independent quantum harmonic oscillator. The operators {{math\|''a''<sup>†</sup>}} and {{math\|''a''}} create and annihilate phonons respectively. The occupation number gives how many phonons are present in a given mode. This shows directly that phonons obey Bose–Einstein statistics and are therefore bosons.<ref name=ashcrroftMermin>{{cite book \|last1=Ashcroft \|first1=Neil W. \|last2=Mermin \|first2=N. David \|title=Solid State Physics \|date=1976 \|publisher=Saunders College Publishing \|isbn=0-03-083993-9 \|pages=780–783}}</ref><ref>{{cite book \|title= Statistical Mechanics, A Set of Lectures \|last= Feynman \|first= Richard P. \|year= 1982 \|publisher= Benjamin-Cummings \|location= Reading, MA \|isbn= 978-0-8053-2508-9 \|page= [https://archive.org/details/statisticalmecha00rich/page/159 159] \|url= https://archive.org/details/statisticalmecha00rich/page/159 }}</ref>

	The vacuum state is the state with no phonons. Excited states correspond to one or more phonons occupying one or more modes.		The vacuum state is the state with no phonons. Excited states correspond to one or more phonons occupying one or more modes.
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	The thermodynamic properties of a crystal are strongly shaped by its phonon spectrum. The full set of allowed phonon states defines the phonon density of states, which determines how vibrational modes contribute to heat capacity, internal energy, and thermal transport.		The thermodynamic properties of a crystal are strongly shaped by its phonon spectrum. The full set of allowed phonon states defines the phonon density of states, which determines how vibrational modes contribute to heat capacity, internal energy, and thermal transport.

	At [[absolute zero]], a harmonic crystal is in its ground state and contains no thermal phonons. At nonzero temperature, random lattice vibrations populate the available phonon states. These thermal phonons can be described statistically using the Bose–Einstein distribution:		At absolute zero, a harmonic crystal is in its ground state and contains no thermal phonons. At nonzero temperature, random lattice vibrations populate the available phonon states. These thermal phonons can be described statistically using the Bose–Einstein distribution:

	:<math>n\left(\omega_{k,s}\right) = \frac{1}{\exp\left(\dfrac{\hbar\omega_{k,s}}{k_\mathrm{B}T}\right) - 1}</math>		:<math>n\left(\omega_{k,s}\right) = \frac{1}{\exp\left(\dfrac{\hbar\omega_{k,s}}{k_\mathrm{B}T}\right) - 1}</math>
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	== Phonon tunneling ==		== Phonon tunneling ==

	Phonons have also been shown to exhibit [[quantum tunneling]] behavior. Across nanometre-scale gaps, heat can be transferred by phonons that effectively tunnel between materials.<ref name=mitPhononTunneling>{{Cite web\|url=https://news.mit.edu/2015/phonon-tunneling-heat-flow-nanometer-gaps-0407\|title=Tunneling across a tiny gap\|website=News.mit.edu\|date=7 April 2015 \|access-date=13 August 2019}}</ref> This effect occurs over distances too large for normal contact conduction and too small for ordinary thermal radiation, making it a distinctly nanoscale quantum heat-transfer mechanism.		Phonons have also been shown to exhibit quantum tunneling behavior. Across nanometre-scale gaps, heat can be transferred by phonons that effectively tunnel between materials.<ref name=mitPhononTunneling>{{Cite web\|url=https://news.mit.edu/2015/phonon-tunneling-heat-flow-nanometer-gaps-0407\|title=Tunneling across a tiny gap\|website=News.mit.edu\|date=7 April 2015 \|access-date=13 August 2019}}</ref> This effect occurs over distances too large for normal contact conduction and too small for ordinary thermal radiation, making it a distinctly nanoscale quantum heat-transfer mechanism.

	== Superconductivity ==		== Superconductivity ==

	Phonons play a fundamental role in conventional [[superconductivity]]. In such materials, electrons form Cooper pairss due to an effective attractive interaction mediated by phonons. This interaction, treated in the BCS theory framework, explains why superconductors can exhibit zero electrical resistance and expel magnetic fields below a critical temperature. Evidence for the importance of phonons is provided by the isotope effect, in which the superconducting critical temperature depends on ionic mass.<ref>{{cite book \|last1=Tinkham \|first1=Michael \|title=Introduction to Superconductivity \|date=1996 \|publisher=Dover Publications, Inc. \|location=Mineola, New York \|isbn=0-486-43503-2 \|page=9\|url=https://archive.org/details/introductiontosu0000mich/page/8/mode/2up?view=theater}}</ref>		Phonons play a fundamental role in conventional superconductivity. In such materials, electrons form Cooper pairss due to an effective attractive interaction mediated by phonons. This interaction, treated in the BCS theory framework, explains why superconductors can exhibit zero electrical resistance and expel magnetic fields below a critical temperature. Evidence for the importance of phonons is provided by the isotope effect, in which the superconducting critical temperature depends on ionic mass.<ref>{{cite book \|last1=Tinkham \|first1=Michael \|title=Introduction to Superconductivity \|date=1996 \|publisher=Dover Publications, Inc. \|location=Mineola, New York \|isbn=0-486-43503-2 \|page=9\|url=https://archive.org/details/introductiontosu0000mich/page/8/mode/2up?view=theater}}</ref>

	== Other properties and research ==		== Other properties and research ==

← Older revision		Revision as of 08:17, 20 May 2026
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	{{Short description\|Quantum Collection topic on Quantum Phonons}}		{{Short description\|Quantum Collection topic on Quantum Phonons}}

	{{Quantum book backlink\|Condensed matter and solid-state physics}}		{{Quantum book backlink\|Condensed matter and solid-state physics}}
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	The study of phonons is a central part of condensed matter physics because phonons play a major role in the thermal, elastic, optical, and electronic properties of materials. They are essential in understanding [[thermal conductivity]], aspects of [[electrical conductivity]], lattice heat capacity, [[neutron scattering]], and important interactions in solids such as electron–phonon coupling.		The study of phonons is a central part of condensed matter physics because phonons play a major role in the thermal, elastic, optical, and electronic properties of materials. They are essential in understanding [[thermal conductivity]], aspects of [[electrical conductivity]], lattice heat capacity, [[neutron scattering]], and important interactions in solids such as electron–phonon coupling.

	The concept of the phonon was introduced in 1930 by the Soviet physicist [[Igor Tamm]], and the name ''phonon'' was suggested by [[Yakov Frenkel]].<ref>{{Cite book \|last=Kozhevnikov \|first=A. B. \|url=https://archive.org/details/stalinsgreatscie0000kozh/mode/1up?q=tamm+phonon \|title=Stalin's great science: the times and adventures of Soviet physicists \|date=2004 \|publisher=Imperial College Press \|isbn=978-1-86094-419-2 \|location=London \|pages=64–69}}</ref> The word derives from the Greek {{lang\|grc\|φωνή}} ({{transliteration\|grc\|phonē}}), meaning ''sound'' or ''voice'', reflecting the fact that long-wavelength phonons correspond to sound propagation in matter. The analogy with photons emphasizes the wave–particle duality of lattice vibrations.		The concept of the phonon was introduced in 1930 by the Soviet physicist Igor Tamm, and the name ''phonon'' was suggested by Yakov Frenkel.<ref>{{Cite book \|last=Kozhevnikov \|first=A. B. \|url=https://archive.org/details/stalinsgreatscie0000kozh/mode/1up?q=tamm+phonon \|title=Stalin's great science: the times and adventures of Soviet physicists \|date=2004 \|publisher=Imperial College Press \|isbn=978-1-86094-419-2 \|location=London \|pages=64–69}}</ref> The word derives from the Greek {{lang\|grc\|φωνή}} ({{transliteration\|grc\|phonē}}), meaning ''sound'' or ''voice'', reflecting the fact that long-wavelength phonons correspond to sound propagation in matter. The analogy with photons emphasizes the wave–particle duality of lattice vibrations.
	<div style="float:right; width:420px; background:#fff8dc; border:1px solid #e0d890; padding:6px; margin:0 0 1em 1em;">		<div style="float:right; width:420px; background:#fff8dc; border:1px solid #e0d890; padding:6px; margin:0 0 1em 1em;">
	<div style="font-size:90%; text-align:center;">		<div style="font-size:90%; text-align:center;">
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	== Lattice dynamics ==		== Lattice dynamics ==

	In a crystalline solid, atoms occupy equilibrium positions in a regular lattice. Since the atoms remain near these positions, they must exert forces on one another. These restoring forces may arise from [[covalent bond]]s, electrostatic attraction, [[Van der Waals ~~force]]s~~, and related interactions. The full many-body problem is complicated, so in lattice dynamics two common approximations are often introduced.		In a crystalline solid, atoms occupy equilibrium positions in a regular lattice. Since the atoms remain near these positions, they must exert forces on one another. These restoring forces may arise from [[covalent bond]]s, electrostatic attraction, Van der Waals forcess, and related interactions. The full many-body problem is complicated, so in lattice dynamics two common approximations are often introduced.

	First, one usually considers mainly interactions between neighboring atoms. Second, the interaction potential is approximated as harmonic near equilibrium. This means the potential is expanded to quadratic order, so the restoring force is proportional to displacement. In that approximation, the lattice behaves like a collection of masses connected by springs.		First, one usually considers mainly interactions between neighboring atoms. Second, the interaction potential is approximated as harmonic near equilibrium. This means the potential is expanded to quadratic order, so the restoring force is proportional to displacement. In that approximation, the lattice behaves like a collection of masses connected by springs.
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	Because atoms in a lattice are coupled, displacing one atom from equilibrium affects its neighbors and gives rise to propagating vibrational waves through the crystal. These are lattice waves. Their amplitudes are given by the atomic displacements, and for periodic normal modes the waves have definite wavelengths and frequencies.		Because atoms in a lattice are coupled, displacing one atom from equilibrium affects its neighbors and gives rise to propagating vibrational waves through the crystal. These are lattice waves. Their amplitudes are given by the atomic displacements, and for periodic normal modes the waves have definite wavelengths and frequencies.

	There is a shortest distinct wavelength, on the order of twice the lattice spacing. Wavelengths shorter than this are not independent because of the periodicity of the lattice. This restriction is reflected in the structure of reciprocal space and the first [[Brillouin zone]].		There is a shortest distinct wavelength, on the order of twice the lattice spacing. Wavelengths shorter than this are not independent because of the periodicity of the lattice. This restriction is reflected in the structure of reciprocal space and the first Brillouin zone.

	Not every arbitrary lattice motion has a single wavelength and frequency, but the normal modes do. These normal modes are the basis from which phonons arise upon quantization.		Not every arbitrary lattice motion has a single wavelength and frequency, but the normal modes do. These normal modes are the basis from which phonons arise upon quantization.
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	=== Quantum treatment ===		=== Quantum treatment ===

	In the quantum version of the harmonic chain, the positions and momenta of the atoms become operators, and the lattice is described by a [[Hamiltonian ~~(quantum mechanics)\|Hamiltonian]]~~ of coupled harmonic oscillators:		In the quantum version of the harmonic chain, the positions and momenta of the atoms become operators, and the lattice is described by a Hamiltonian of coupled harmonic oscillators:

	:<math>\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.</math>		:<math>\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.</math>
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	== Crystal momentum ==		== Crystal momentum ==

	Phonons are commonly assigned a quantity ''ħk'' analogous to momentum, but this is not ordinary mechanical momentum. It is more precisely called '''crystal momentum''' or pseudomomentum.<ref>{{Cite book\|title=[[Introduction to Solid State Physics]], 8th Edition\|last=Kittel\|first=Charles\|publisher=Wiley\|year=2004\|isbn=978-0-471-41526-8\|pages=[https://archive.org/details/isbn_9780471415268/page/100 100]}}</ref>		Phonons are commonly assigned a quantity ''ħk'' analogous to momentum, but this is not ordinary mechanical momentum. It is more precisely called '''crystal momentum''' or pseudomomentum.<ref>{{Cite book\|title=Introduction to Solid State Physics, 8th Edition\|last=Kittel\|first=Charles\|publisher=Wiley\|year=2004\|isbn=978-0-471-41526-8\|pages=[https://archive.org/details/isbn_9780471415268/page/100 100]}}</ref>

	Because of lattice periodicity, wavevectors that differ by a reciprocal lattice vector describe equivalent states. This is why phonon wavevectors are usually taken inside the first [[Brillouin zone]]. The concept of crystal momentum is essential for analyzing scattering processes involving phonons, electrons, and photons in solids.		Because of lattice periodicity, wavevectors that differ by a reciprocal lattice vector describe equivalent states. This is why phonon wavevectors are usually taken inside the first Brillouin zone. The concept of crystal momentum is essential for analyzing scattering processes involving phonons, electrons, and photons in solids.

	== Operator formalism and second quantization ==		== Operator formalism and second quantization ==
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	:<math>\mathcal{H} = \sum_\alpha\hbar\omega_\alpha {a_\alpha}^\dagger a_\alpha.</math>		:<math>\mathcal{H} = \sum_\alpha\hbar\omega_\alpha {a_\alpha}^\dagger a_\alpha.</math>

	In this description, each phonon mode behaves like an independent quantum harmonic oscillator. The operators {{math\|''a''<sup>†</sup>}} and {{math\|''a''}} create and annihilate phonons respectively. The occupation number gives how many phonons are present in a given mode. This shows directly that phonons obey [[Bose–Einstein statistics]] and are therefore [[boson]]s.<ref name=ashcrroftMermin>{{cite book \|last1=Ashcroft \|first1=Neil W. \|last2=Mermin \|first2=N. David \|title=Solid State Physics \|date=1976 \|publisher=Saunders College Publishing \|isbn=0-03-083993-9 \|pages=780–783}}</ref><ref>{{cite book \|title= Statistical Mechanics, A Set of Lectures \|last= Feynman \|first= Richard P. \|year= 1982 \|publisher= Benjamin-Cummings \|location= Reading, MA \|isbn= 978-0-8053-2508-9 \|page= [https://archive.org/details/statisticalmecha00rich/page/159 159] \|url= https://archive.org/details/statisticalmecha00rich/page/159 }}</ref>		In this description, each phonon mode behaves like an independent quantum harmonic oscillator. The operators {{math\|''a''<sup>†</sup>}} and {{math\|''a''}} create and annihilate phonons respectively. The occupation number gives how many phonons are present in a given mode. This shows directly that phonons obey Bose–Einstein statistics and are therefore [[boson]]s.<ref name=ashcrroftMermin>{{cite book \|last1=Ashcroft \|first1=Neil W. \|last2=Mermin \|first2=N. David \|title=Solid State Physics \|date=1976 \|publisher=Saunders College Publishing \|isbn=0-03-083993-9 \|pages=780–783}}</ref><ref>{{cite book \|title= Statistical Mechanics, A Set of Lectures \|last= Feynman \|first= Richard P. \|year= 1982 \|publisher= Benjamin-Cummings \|location= Reading, MA \|isbn= 978-0-8053-2508-9 \|page= [https://archive.org/details/statisticalmecha00rich/page/159 159] \|url= https://archive.org/details/statisticalmecha00rich/page/159 }}</ref>

	The vacuum state is the state with no phonons. Excited states correspond to one or more phonons occupying one or more modes.		The vacuum state is the state with no phonons. Excited states correspond to one or more phonons occupying one or more modes.
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	:<math>n\left(\omega_{k,s}\right) = \frac{1}{\exp\left(\dfrac{\hbar\omega_{k,s}}{k_\mathrm{B}T}\right) - 1}</math>		:<math>n\left(\omega_{k,s}\right) = \frac{1}{\exp\left(\dfrac{\hbar\omega_{k,s}}{k_\mathrm{B}T}\right) - 1}</math>

	where ''ω''<sub>''k'',''s''</sub> is the mode frequency, ''k''<sub>B</sub> is the [[Boltzmann constant]], and ''T'' is the temperature.<ref>{{cite book \|last1=Pathria \|last2=Beale \|title=Statistical Mechanics \|date=2011 \|publisher=Elsevier \|location=India \|isbn=978-93-80931-89-0 \|page=201 \|edition=3}}</ref>		where ''ω''<sub>''k'',''s''</sub> is the mode frequency, ''k''<sub>B</sub> is the Boltzmann constant, and ''T'' is the temperature.<ref>{{cite book \|last1=Pathria \|last2=Beale \|title=Statistical Mechanics \|date=2011 \|publisher=Elsevier \|location=India \|isbn=978-93-80931-89-0 \|page=201 \|edition=3}}</ref>

	High-frequency phonons contribute strongly to heat capacity, while low-frequency phonons are especially important in determining thermal conductivity. The resulting thermal phonon gas is analogous in many ways to a photon gas in black-body radiation.<ref name=nonMetalsThermalPhonons>{{cite web \|title=Non-metals: thermal phonons \|url=https://www.doitpoms.ac.uk/tlplib/thermal_electrical/nonmetal_thermal.php#:~:text=When%20the%20lattice%20is%20heated,as%20a%20gas%20of%20phonons. \|website=University of Cambridge Teaching and Learning Packages Library \|access-date=15 August 2020}}</ref>		High-frequency phonons contribute strongly to heat capacity, while low-frequency phonons are especially important in determining thermal conductivity. The resulting thermal phonon gas is analogous in many ways to a photon gas in black-body radiation.<ref name=nonMetalsThermalPhonons>{{cite web \|title=Non-metals: thermal phonons \|url=https://www.doitpoms.ac.uk/tlplib/thermal_electrical/nonmetal_thermal.php#:~:text=When%20the%20lattice%20is%20heated,as%20a%20gas%20of%20phonons. \|website=University of Cambridge Teaching and Learning Packages Library \|access-date=15 August 2020}}</ref>
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	== Superconductivity ==		== Superconductivity ==

	Phonons play a fundamental role in conventional [[superconductivity]]. In such materials, electrons form [[Cooper ~~pair]]s~~ due to an effective attractive interaction mediated by phonons. This interaction, treated in the [[BCS theory]] framework, explains why superconductors can exhibit zero electrical resistance and expel magnetic fields below a critical temperature. Evidence for the importance of phonons is provided by the isotope effect, in which the superconducting critical temperature depends on ionic mass.<ref>{{cite book \|last1=Tinkham \|first1=Michael \|title=Introduction to Superconductivity \|date=1996 \|publisher=Dover Publications, Inc. \|location=Mineola, New York \|isbn=0-486-43503-2 \|page=9\|url=https://archive.org/details/introductiontosu0000mich/page/8/mode/2up?view=theater}}</ref>		Phonons play a fundamental role in conventional [[superconductivity]]. In such materials, electrons form Cooper pairss due to an effective attractive interaction mediated by phonons. This interaction, treated in the BCS theory framework, explains why superconductors can exhibit zero electrical resistance and expel magnetic fields below a critical temperature. Evidence for the importance of phonons is provided by the isotope effect, in which the superconducting critical temperature depends on ionic mass.<ref>{{cite book \|last1=Tinkham \|first1=Michael \|title=Introduction to Superconductivity \|date=1996 \|publisher=Dover Publications, Inc. \|location=Mineola, New York \|isbn=0-486-43503-2 \|page=9\|url=https://archive.org/details/introductiontosu0000mich/page/8/mode/2up?view=theater}}</ref>

	== Other properties and research ==		== Other properties and research ==