Discoverer: Davy, Sir H. and Thnard, L.-J. Assume Eg=0.67 eV and room-temperature (298 K) Alinear expansion coefficientis usually employed in describing the expansion of a solid, while a volume expansion coefficient is more useful for a liquid or a gas. So, although they can conduct electricity, they are inefficient at the task. 2023 Springer Nature Switzerland AG. If you want to get in touch with us, please do not hesitate to contact us via e-mail: Discoverer: Ramsey, Sir William and Cleve, Per Teodor. PubMedGoogle Scholar. Weisskopf, Phys. (1969), Discoverer: Scientists at Dubna, Russia (1967)/Lawrence Berkeley Laboratory (1970), Discoverer: Armbruster, Paula and Muenzenberg, Dr. Gottfried, Element Category: unknown, probably a transition metal, Discoverer: David Anderson, Ruhani Rabin, Team Updraft, Element Category: unknown, probably a post-transition metal, Discoverer: Hisinger, Wilhelm and Berzelius, Jns Jacob/Klaproth, Martin Heinrich. Our Website follows all legal requirements to protect your privacy. Fornonmetallic solids,kis determined primarily bykph, which increases as the frequency of interactions between the atoms and the lattice decreases. Therefore higher temperatures implygreater distancebetween atoms. E. Discoverer: De Marignac, Charles Galissard, Discoverer: De Marignac, Jean Charles Galissard, Discoverer: Ghring, Otto and Fajans, Kasimir. Rev. The results are thus of the same form as reported by Lark-Horovitz. 302, 137 (1983). Authors: Yang, Gang; Kooi, Kyler; Wang, Guojian; Mei, Hao; Li, Yangyang; Mei, Dongming Award ID(s): 1738632 Publication Date: 2018-05-01 NSF-PAR ID: 10063257 Journal . On applying an electric field of intensity 10 V/cm across a semiconductor at a certain temperature the average drift velocity of free electrons is measured to be 70 m/s. Higher RRR values indicate higher purity and lower electrical and thermal resistance leading to higher thermal conductivity. A cylindrical sample of solid germanium has length 55.0 mm and diameter 5.0 mm. The vibrations of atoms are not independent of each other, but are rather strongly coupled with neighboring atoms. The key variable in applying Ohm's Law is a metal's resistivity. In general, the thermal conductivity of gases increases with increasing temperature. The standard units of conductivity is Siemens per meter. When, ni at 300K = 2.5 1013 /cm and n and p in germanium are 3800 and 1800 cm2/Vs respectively. NS-23, 81 (1976), W.L. Sodium chloride melts at 801C. : National Defense Research Committee Discoverer: Corson, Dale R. and Mackenzie, K. R. The actinide or actinoid series encompasses the 15 metallic chemical elements with atomic numbers from 89 to 103, actinium through lawrencium. This is commonly measured across the opposite faces of a one-meter cube of material and described as an ohm meter (m). Chem. Theboiling pointof a substance is the temperature at which this phase change (boiling or vaporization) occurs. Anderson, N. Apsley, Semicond. Solution: Given: 7. The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient, and the most relevant for fluids. Different materials have different bonding forces and therefore different expansion coefficients. Phonons play a major role in many of the physical properties of condensed matter, like thermal conductivity and electrical conductivity. Thus, at room temperature in Ge (Si), one atom out of about 109 (1012) contributes one electron-hole pair through breaking of covalent bond. 110 6 S/m. The heat can be removed by channeling the liquid through a heat exchanger. But volume charge density equals the number of electrons (holes) per unit volume multiplied by the charge of each carrier. Forbidden . Accordingly, transport of thermal energy may be due to two effects: The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specificallyelectrons. Department of Physics, The University of South Dakota, Vermillion, SD, 57069, USA, Gang Yang,Kyler Kooi,Guojian Wang,Hao Mei,Yangyang Li&Dongming Mei, You can also search for this author in Electrical conductivity and its converse, electrical resistivity, is a fundamental property of a material that quantifies how Germanium conducts the flow of electric current. It is a measure of a substances ability to transfer heat through a material by conduction. As with boiling points, the melting point of a solid is dependent on the strength of those attractive forces. In general: Most materials are very nearly homogeneous, therefore we can usually writek = k (T). In general, the thermal conductivity of gases increases with increasing temperature. The temperature dependent hopping distance and hopping energy are obtained for a-Ge fabricated as the electrical contact materials for high-purity Ge planar detectors. In this paper, the electrical conductivity of a nondegenerate electron gas with a complex shaped quantum well is calculated. Need an editable periodic table to edit? Both the mobility and are positive quantities and are constant for a given material upto electrical fields of about 105 volts/meter. Metalsin general havehigh electrical conductivity,high thermal conductivity, and high density. Electrical resistance is expressed in Ohms. Notes on the properties of Germanium: Brinell Hardness: converted from Mohs scale. Light gases, such ashydrogenandheliumtypically havehigh thermal conductivity. ThoughtCo, Aug. 27, 2020, thoughtco.com/table-of-electrical-resistivity-conductivity-608499. Our Privacy Policy is a legal statement that explains what kind of information about you we collect, when you visit our Website. Thelinear thermal expansion coefficientis defined as: whereLis a particular length measurement anddL/dTis the rate of change of that linear dimension per unit change in temperature. Using these parameters, we predict that the surface leakage current from a Ge detector with a-Ge contacts can be much smaller than one yocto amp (yA) at helium temperature, suitable for rare-event physics searches. Phys. Note that Fouriers law applies for all matter, regardless of its state (solid, liquid, or gas), therefore, it is also defined for liquids and gases. At theboiling pointthe two phases of a substance, liquid and vapor, have identical free energies and therefore are equally likely to exist. (0 K), the electrical conductivity of a semiconductor has a value of zero (i.e., the conductivity is at its minimum) whereas a metal exhibits its maximum electrical conductivity at absolute zero; furthermore, conductivity increases with increasing temperature in a . In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress. Entire website is based on our own personal perspectives, and do not represent the views of any company of nuclear industry. K). Retrieved from https://www.thoughtco.com/table-of-electrical-resistivity-conductivity-608499. Carbon is an insulator with a very low conductivity. 101/1-B, Chemical and electrical properties of cavities in silicon and germanium. The symbol of resistivity is usually the Greek letter (rho). If a single ball strikes multiple other balls, each of those will carry only a fraction of the energy. https://www.thoughtco.com/table-of-electrical-resistivity-conductivity-608499 (accessed March 4, 2023). Answer (1 of 2): Both silicon and germanium have four valence electrons. Journal of Physics and Chemistry of Solids, Abstract Studies were made of electrical conductivity at low temperatures in germanium crystals grown with indium and antimony EE majority and minority impurities respectively. The measured electrical conductivity of type germanium as a function of temperature. Most recent implementations of low-cost electrical conductivity (EC) sensors intended for water quality measurements are based on simple two-pole designs. In gases, the thermal conduction is caused by diffusion of molecules from higher energy level to the lower level. In liquids, the thermal conduction is caused by atomic or molecular diffusion. Mean energy of hole as a function of electronic field F at different lattice temperatures. Conductivity sensors for water bodies measure from a few micro-siemens per cm (S/cm) to about 200 milli-siemens per cm (mS/cm). To understand the difference betweenmetals,semiconductorsandelectrical insulators, we have to define the following terms from solid-state physics: This website was founded as a non-profit project, build entirely by a group of nuclear engineers. 2) Note! The melting point of ice is 0 C. The lower the resistivity, the more readily the material permits the flow of electric charge. Notes on the Electrical Conductivity of particular elements: Carbon: Value given for graphite. Mobility also varies with applied electrical field. Transport of thermal energy in solids may be generally due to two effects: Copyright 2023 Periodic Table | All Rights Reserved |. The experimental values of several physical properties have been employed to determine . The electrical conductivity vs T of all samples is presented in Fig. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. >> Physics. A liquid in a partial vacuum has a lower boiling point than when that liquid is at atmospheric pressure. Resistivity is often represented by the Greek letter rho (). The table at right give the electrical conductivity of elements with diamond solid state structure. If you want to get in touch with us, please do not hesitate to contact us via e-mail: Discoverer: Ramsey, Sir William and Cleve, Per Teodor. Find step-by-step Engineering solutions and your answer to the following textbook question: A metal alloy is known to have electrical conductivity and electron mobility values of $1.2 \times 10^{7}(\Omega \cdot \mathrm{m})^{-1}$ and $0.0050 \mathrm{m}^{2} / \mathrm{V} \cdot \mathrm{s}$, respectively. - the resistivity depends strongly on the presence of impurities in the material. The electrical conductivity of the CCZ reinforced with 0.1 wt% graphene was comparable to that of CCZ with no graphene content (65 vs. 63% IACS). The thermal conductivity of nonmetallic liquids generally decreases with increasing temperature. Electon Dev. I. Germanium, High Purity Germanium - Low Temperature Hall Analyses, Radial and axial impurity distribution in high-purity germanium crystals, https://doi.org/10.1016/j.jcrysgro.2011.12.042, Investigation of influential factors on the purification of zone-refined germanium ingot: Investigation of influential factors on the purification of zone-refined germanium ingot, Effect of annealing on contact performance and electrical properties of p-type high purity germanium single crystal, https://doi.org/10.1007/s00339-012-7518-x, Electrical properties of lightly doped polycrystalline silicon, Temperature dependence of carrier transport in polycrystalline silicon, https://doi.org/10.1016/S0026-2692(86)80133-1, The dc voltage dependence of semiconductor grainboundary resistance, On the mobility of polycrystalline semiconductors, https://doi.org/10.1016/0038-1101(80)90196-3, The Hall effect in III-V semiconductor assessment, https://doi.org/10.1088/0268-1242/1/3/006, A conduction model for semiconductor-grain-boundary-semiconductor barriers in polycrystalline-silicon films, Characterization of high-purity germanium (Ge) crystals for developing novel Ge detectors, https://doi.org/10.1088/1748-0221/15/10/t10010, Fabrication and characterization of high-purity germanium detectors with amorphous germanium contacts, https://doi.org/10.1088/1748-0221/14/02/p02019, Characterization of high-purity germanium detectors with amorphous germanium contacts in cryogenic liquids, https://doi.org/10.1140/epjc/s10052-020-8235-9, Investigation of the electrical conduction mechanisms in P-type amorphous germanium electrical contacts for germanium detectors in searching for rare-event physics, https://doi.org/10.1140/epjc/s10052-020-08529-z, The electrical properties and distribution of indium in germanium crystals, https://doi.org/10.1016/j.mssp.2017.11.004, Materials Science in Semiconductor Processing, Vol. In a semiconductor, there are two types of charge carriers namely free electrons and holes. Transport of thermal energy in solids may be generally due to two effects: Copyright 2023 Periodic Table | All Rights Reserved |, Zinc Boiling Melting Point Thermal Conductivity Expansion, Arsenic Boiling Melting Point Thermal Conductivity Expansion. 4.510 7 S/m. As a result, we find that the hopping energy in a-Ge increases as temperature increases while the hopping distance in a-Ge decreases as temperature increases. Silicon and germanium are similar to each other and have an intermediate conductivity. Table 1 gives the value of and other important physical quantities for Ge and Si. The law, named afterGerman physicist Georg Ohm, appeared in 1827 in a published paper laying out how current and voltage are measured via electrical circuits. J. Semiconductor metals (or metalloids) have a higher number of valence electrons (usually four or more). Applied Physics A Thethermal conductivityof most liquids and solids varies with temperature. Further . In fact, for crystalline, nonmetallic solids such as diamond, kphcan be quite large, exceeding values of k associated with good conductors, such as aluminum. Hansen, G.S. Conductivity is a measure of how well a material conducts an electric current. Also, isentropic compressibility was calculated. Discoverer: Scientists at Dubna, Russia (1964)/Albert Ghiorso et. Rev. Germanium: 4.610 1: 2.17: Sea water: 210 1: 4.8: Drinking water: 210 1 to 210 3: 510 4 to 510 2: . The electrical conductivity of pure germanium can be increased by (a) increasing the temperature (b) doping acceptor impurities . If you continue to use this site we will assume that you are happy with it. The electrical conductivity of the graphene reinforced composites improves (40-43%) with increasing the sintering temperature and diminishes (3-8%) with increasing the graphene content. https://doi.org/10.1007/s00339-018-1803-2, DOI: https://doi.org/10.1007/s00339-018-1803-2. K). Specific Heat: Value given for solid phase. The mobility decreases many fold due to the influence of scattering by optical modes, By clicking accept or continuing to use the site, you agree to the terms outlined in our. Whereas in the case of conductors, when the temperature increases its resistivity increases as electrons collide more frequently with vibrating atoms. Understanding Electrical, Thermal, and Sound Conductors, Superconductor Definition, Types, and Uses, The Basics: An Introduction to Electricity and Electronics, Rhodium, a Rare Platinum Group Metal, and Its Applications, Find the Anode and Cathode of a Galvanic Cell, Effect of Relative Humidity and Sea Level Pressure on Electrical Conductivity of Air over Indian Ocean, Ph.D., Biomedical Sciences, University of Tennessee at Knoxville, B.A., Physics and Mathematics, Hastings College. Tax calculation will be finalised during checkout. Also discussed are the pros and cons of using thin amorphous semiconductor materials as an alternative contact technology in large-scale germanium experiments, For the first time, electrical conduction mechanisms in the disordered material system is experimentally studied for p-type amorphous germanium (a-Ge) used for high-purity Ge detector contacts. Thermal expansion is common for solids, liquids and for gases. Google Scholar, E.E. In this article we will discuss different electrical properties of germanium and silicon like conductivity, drift current, concentration forbidden energy gap and mobility. Where E is the applied electric field in volt/meter, J is the resulting drift current density in amperes/m2. In contrast, for alloys, the contribution of kph to k is no longer negligible. An increasing temperature means that, on average, there is greater mobility of the atoms that constitute the metal. Hubbard, F.S. Helmenstine, Anne Marie, Ph.D. "Table of Electrical Resistivity and Conductivity." Sci. On substituting these values of Qvn and Qvp in equation (2) we get. Fermi Level "Fermi level" is the term used to describe the top of the collection of electron energy levels at absolute zero temperature. Due to this property one material can conduct electricity. GaAs transistor was first introduced in the year of 1970. Why are semiconductors valuable in modern . Thus by substituting the values we will get the electrical conductivity of germanium. of a-Ge is on the order of 2.13$$^{-0.05}_{+0.07}$$A to 5.07$$^{0.83}_{+2.58}$$A,depending on the density of states near the Fermi energy level within bandgap. While resistivity is a material property, resistance is the property of an object. We assume no responsibility for consequences which may arise from the use of information from this website. Its electrical conductivity is 112 -1 m -1. This is a preview of subscription content, access via your institution. Draw the Covalent Bonding of the semiconductor materials. 74 Ge. -05 ii: SAMPLE OHM-CM 3 5 3 1100 1000 900 800 -K FIG. Which of the following are semiconductor materials? The liquid can be said to be saturated with thermal energy. The Electrical Conductivity of Silicon and Germanium, to be made of the concentration and mobility of light holes and prove that the number of minima in the conduction band of germanium equals four. 965, 1126 (1954), E.G.S. Convert between Electrical Resistivity Units. However, in other classes of materials, known as semiconductors (e.g. Haller, W.L. The temperature dependence of electrical conductivity of single-crystal and polycrystalline high-purity germanium (HPGe) samples has been investigated in the temperature range from 7 to 100 K. Report No. For vapors, it also depends upon pressure. 2) You may not distribute or commercially exploit the content, especially on another website. Thequantaof the crystal vibrational field are referred to as phonons. A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, like solids and some liquids. Substances in which electricity can flow are called conductors. Thereafter Iogm increases very rapidly along a straight line which is the same for all specimens. Electrical conductivity is a basic property of material. The mean free path also depends on the diameter of the molecule, with larger molecules more likely to experience collisions than small molecules, which is the average distance traveled by an energy carrier (a molecule) before experiencing a collision. The number of atoms/cm2 in Ge or Si is of the the order of 1022. When considered as the temperature of the reverse change from vapor to liquid, it is referred to as the condensation point. Fluidsare a subset of the phases of matter and includeliquids,gases, plasmas and, to some extent, plastic solids. (auth). Thethermal conductivityof gases and liquids is therefore generally smaller than that of solids. For example, in a sample of germanium at room temperature, the intrinsic carrier concentration is 2.5 x 1019 per m3. (a) The room-temperature electrical conductivity of a silicon specimen is $500(\Omega-\mathrm{m})^{-1}$ The hole concentration is known to be $2.0 \times 10^{22} \mathrm{m}^{-3}$. . ELECTRICAL PROPERTIES OF NEUTRON-IRRADIATED SILICON AT 76$sup 0$K: HALL EFFECT AND ELECTRICAL CONDUCTIVITY. Solid line show the results of calculation in the case where non-parabolic effect are taken into account (Reggiani et al. In gases, the thermal conduction is caused by diffusion of molecules from higher energy level to the lower level. The influence of surfactant concentration and temperature upon these physical properties has also been studied. (The temperature coefficient of resistance of many pure metals is close to 0.004 K-1, so the resistance-temperature graph will extrapolate back to 1/0.004 = 250 K.) You could link this to the idea that the resistance of a pure metal at room temperature is dominated by the vibration of ions, and this will reduce to zero close to 0 K. Sci. Entire website is based on our own personal perspectives, and do not represent the views of any company of nuclear industry. Density, surface tension, electric conductivity, and speed of sound of sodium dodecanoate (sodium laurate) aqueous solutions have been determined. Calculate its charge carrier density 'n e 'and electron mobility at room temperature. Fluidsare a subset of the phases of matter and includeliquids,gases, plasmas and, to some extent, plastic solids. The electrical and thermal conductivities of metalsoriginate fromthe fact that theirouter electrons are delocalized. Abstract. 9, 437 (2006), Article Electrical conductivity of high-purity germanium crystals at low temperature. al. This changes the carrier concentration and transforms n-type germanium into p-type. Electrical resistivity, represented by the Greek letter (rho), is a measure of how strongly a material opposes the flow of electric current. In fact, lattice thermal conduction is the dominant thermal conduction mechanism in nonmetals, if not the only one. An intrinsic semiconductor is just a pure semiconductor without any significant defects or external impurities.The electrical conductivity of intrinsic semiconductors depends strongly on temperature.At absolute zero (T = 0 K T = 0\ \rm K T = 0 K), semiconductors behave like insulators.However, as we increase the temperature, the electrons in the conduction band gain thermal energy and jump . Solids are similar to liquids in that both are condensed states, with particles that are far closer together than those of a gas. Step 1: Classification of elements. Provided by the Springer Nature SharedIt content-sharing initiative, Over 10 million scientific documents at your fingertips, Not logged in Haller, IEEE Trans. Lattice defects produced by fast neutron bombardment introduce localized energy states into the forbidden band. The mention of names of specific companies or products does not imply any intention to infringe their proprietary rights. The temperature dependence of electrical conductivity of single-crystal and polycrystalline high-purity germanium (HPGe) samples has been investigated in the temperature range from 7 to 100 K. At some point the amplitude of vibration becomes so large that the atoms start to invade the space of their nearest neighbors and disturb them and the melting process initiates. Elements Electrical Conductivity Reference Table. Conductivity is a measure of how well a material conducts an electric current. Accordingly, transport of thermal energy may be due to two effects: The unique feature of metals as far as their structure is concerned is the presence of charge carriers, specificallyelectrons. One siemens is equal to the reciprocal of one ohm. Resistivity is the reciprocal of conductivity and its temperature dependence is given by (4) Where Eg - band gap of the material T - Temperature in kelvin K - Boltzmann constant, K - 8.6x10-5 eV/K The resistivity of a semiconductor rises exponentially on decreasing the temperature. A. I B. III C. IV D. V. 3. Liquids tend to have better thermal conductivity than gases, and the ability to flow makes a liquid suitable for removing excess heat from mechanical components. Electrical conductivity in metals is a result of the movement of electrically charged particles. 505, 3414 (1979), J. Martinez, J. Piqueras, Solid State Electron. 76 Ge. The detectors were fabricated at the Lawrence Berkeley National Laboratory and the University of South Dakota, using crystals grown at the University of South Dakota. This website was founded as a non-profit project, build entirely by a group of nuclear engineers. "Electrical Conductivity of Metals." Such weak intermolecular bonds give organic molecular substances, such as waxes and oils, their soft bulk character, and their low melting points (in liquids, molecules must cease most structured or oriented contact with each other). Goulding, IEEE Trans. 10XX,52,11XX,17,12XX,7,13XX,4,15XX,16,3XXX,2,40XX,10,41XX,12,43XX,5,44XX,4,46XX,5,47XX,3,48XX,3,5XXX,23,6XXX,3,71XX,1,8XXX,22,92XX,5,93XX,1,94XX,4,98XX,2,AISI,66,ASTM,172,Atomic-Mass,327,Atomic-Number,436,Atomic-Radius,86,Atomic-Symbol,329,Atomic-Volume,94,Austenitic,56,Boiling-Point,94,bp1,97,CBS,6,Chemical-Elements,100,Chemical-Symbol,217,CMDS,13,Coefficient-of-Thermal-Expansion,85,Covalent-Radius,87,Crystal-Structure,109,CS,17,CVS,3,Density,297,Duplex,6,el1,100,Elastic-Modulus,30,Electrical-Conductivity,79,Electro-Affinity,87,Electron-Configuration,109,Electronegativity,102,Electrons-per-Shell,111,Enthalpy-of-Fusion,93,Enthalpy-of-Vaporization,95,Ferritic,12,fp1,38,fs1,45,Group-Number,218,HCS,14,Heat-of-Fusion,87,Heat-of-Vaporization,85,HMCS,16,Ionic-Radius,78,Ionization-Energy,102,Ionization-Potential,101,l1,438,LCS,21,List,281,lp1,69,Martensitic,6,MCS,17,MDS,14,Melting-Point,96,mm1,2,mp1,99,MS,4,NCMDBS,6,NCMDS,31,NCS,2,NMDS,8,Oxidation-States,104,p1,44,Period-Number,107,pr1,53,Properties,40,RCLS,1,RCS,16,ref1,5,RRCLS,3,RRCS,4,SAE,201,Site,2,SMS,5,Specific-Gravity,83,Specific-Heat,92,Specific-Weight,1,SS,80,Tests,2,Thermal-Conductivity,105,tm1,274,Valence-Electrons,98,wt1,26, Materials: Electrical Conductivity of Germanium (Ge) [& Color, Uses, Discovery 2022, Electrical Conductivity of Germanium (Ge) [& Color, Uses, Discovery 2022, https://1.bp.blogspot.com/-2TIKNCu3qMA/XV1OOmzQM0I/AAAAAAAAnZs/Fa_hUj_fBe4m6yLSM_ld8_MWsztTDf3OQCLcBGAs/s320/electrical-conductivity-of-germanium.webp, https://1.bp.blogspot.com/-2TIKNCu3qMA/XV1OOmzQM0I/AAAAAAAAnZs/Fa_hUj_fBe4m6yLSM_ld8_MWsztTDf3OQCLcBGAs/s72-c/electrical-conductivity-of-germanium.webp, https://materials.gelsonluz.com/2019/08/electrical-conductivity-of-germanium-ge.html, Not found any post match with your request, STEP 2: Click the link on your social network, Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy, How to calculate density when mass and volume are given, How a small number of atoms can be joined and form completely different substances. At the melting point the two phases of a substance, liquid and vapor, have identical free energies and therefore are equally likely to exist. Measurement and modeling of cavity electrical properties elucidated surface electronic states and indicated a potential for controlled electrical isolation in devices. The inner layers of non-valence electrons "screen off" all of the nuclear charge except for t. germanium), conductivity increases with temperature over a wide range. The dashed line has been computed from equation (1). We use cookies to ensure that we give you the best experience on our website. Dense gases such as xenon and dichlorodifluoromethane have low thermal conductivity. Q9. 531, 490 (1982), J.Y. - 209.239.116.57. At temperatures near room temperature, the temperature dependence of the conductivity fo intrinsic germanium is found to equal: -E o =C (T 3/2' )exp 2kT where C is a temperature-independent constant and T is in Kelvins. The regularity of the lattice arrangement has an important effect onkph, with crystalline (well-ordered) materials likequartzhaving a higher thermal conductivity than amorphous materials like glass. In fact, in pure metals such as gold, silver, copper, and aluminum, the heat current associated with the flow of electrons by far exceeds a small contribution due to the flow of phonons.
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