Phonons Thermal energy Heat capacity Einstein model Density of states Debye model Anharmonic effects Thermal expansion Thermal conduction by phonons

Transcription

1 3b. Lattice Dynamics Phonons Thermal energy Heat capacity Einstein model Density of states Debye model Anharmonic effects Thermal expansion Thermal conduction by phonons Neutron scattering G. Bracco-Material Science SERP CHEM 1 Phonons So far we discussed a classical approach to the lattice vibrations. Solving the problems of vibrations of atoms in lattice we have found that coupled harmonic oscillations of atoms give rise to non coupled normal modes of oscillation Hamiltonian of a harmonic oscillator: = + α As we know from quantum mechanics the energy levels of the harmonic oscillator are quantized. Similarly any normal mode of the lattice is quantized and the quantum of vibration is called a phonon in analogy with the photon, which is the quantum of the electromagnetic wave travelling wave of lattice vibrations Similarly to photons, phonon number is not conserved: phonons can be created and destroyed during processes G. Bracco-Material Science SERP CHEM 2

2 Phonons Quantum harmonic oscillators have equally spaced energy levels with separation = oscillators can accept or lose energy only in discrete units of energy and allowed energies for the oscillator are ε n =(n+ ) n is the number of phonons and ½ħω is the zero point energy (for n=0) A transition from states with different number of phonons: if n 2 <n 1 phonon annihilation if n 2 >n 1 phonon creation 1 ε = n 2 + hω n Since the phonon moves in the crystal its wave vector is k = but, although it λ interacts with other particles (such as electrons) as it had a momentum ħ, however a phonon does not carry physical momentum because the phonon coordinates are relative coordinates of atoms quasi-momentum or crystal momentum ħ is conserved within ħ ħ = ħ + ħ 1 2 hω G. Bracco-Material Science SERP CHEM 3 Phonons If Phonons are to be considered as particles (quasi-particle to study collisions with other phonons or other particles), a mode with a well defined k is not localized, travelling waves of a single frequency and wavelength imply a vibration of the whole lattice. space From the Heisenberg uncertainty principle, to localize a phonon the k must be spread on a range k around k The velocity of the wave packed (envelope) is the group velocity (in 3D is the gradient ) Green point move with group velocity Red point with the phase velocity G. Bracco-Material Science SERP CHEM 4

3 Phonons Therefore, considering the dispersion relations we have found for a diatomic chain, the phonon will not propagate at the BZ boundary and for optical phonons at the center of the zone =0 As observed, in these points we have stationary waves superposition of two travelling waves with opposite directions (+k and k) due to Bragg reflection G. Bracco-Material Science SERP CHEM 5 Phonons The lattice with s atoms in a unit cell is described by 3s independent oscillators (3 acoustic modes and 3s-3 optical modes) The frequencies of normal modes of these oscillators will be given by the solution of 3s linear equations as we have discussed before. Each mode has a frequency ω p (k), where p denotes a particular mode, i.e. p=1, 3s. The energy of this mode is given by ℇ = ( )ħ () the integer n p (k) is the occupation number of the normal mode p,k. The total energy of the lattice = ( + ) ħ () G. Bracco-Material Science SERP CHEM 6

4 Thermal energy and lattice vibrations Atoms vibrate about their equilibrium position. The coupling produces vibrational waves. This motion is increased as the temperature is raised In a solid, the energy associated with these vibrations is the thermal energy of the lattice and this concept is fundamental to understand many of the basic properties of solids: the vibrational energy changes with temperature a measure of the heat energy which is necessary to raise the temperature of the material (the specific heat or heat capacity is the thermal energy which is required to raise the temperature of unit mass or 1 gmole by one Kelvin). Scattering of a conduction electrons in a metal this scattering gives rise to electrical resistance. The energy related to lattice vibrations is the contribution to the heat capacity present in all the solids. It is the only contribution present in insulators, instead for metal there is also a contribution from the conduction electrons. G. Bracco-Material Science SERP CHEM 7 Thermal energy and lattice vibrations Classically, the thermal energy of an oscillator has one contribution from the kinetic energy and one from the potential energy average energy <K>= <V>= Average energy for 1D oscillator <E>=<K>+<V>= For 3D oscillator <E>=<K>+<V>=3 Specific heat per atom =3 (or per mol =3R about 25 J/mol K, R=gas constant) law of Dulong and Petit This value is observed at room temperature for most of the solids but, in general, for T 0 K where the harmonic approximation should be a good approximation, 0. G. Bracco-Material Science SERP CHEM 8

5 Energy and heat capacity of a harmonic oscillator Atomic vibrations leads to band of normal mode frequencies from zero up to some maximum value. Calculation of the lattice energy and heat capacity of a solid therefore requires two steps: i) the evaluation of the contribution of a single mode ii) the summation over the frequency distribution of the modes. At temperature T, each energy level of the oscillator is populated with a probability given by the Boltzmann factor 1 P n =exp( ( )) ε n = n + ω 2 h E= Total energy of the harmonic oscillator energy of the lattice mode with angular frequency ω at temperature T G. Bracco-Material Science SERP CHEM 9 = ( / ) ( / ) Let s introduce the partition function = ( + / ) ( ) derivative + ( + / ) E= Performing the calculation Z= ( + / ) = ( / ) ( / ) = ( / ) ( / ) n since ( / )<1 the geometric series for x<1 converges to = = ( / ) partition function of the oscillator ((/ )) whose G. Bracco-Material Science SERP CHEM 10

6 Performing the derivative = E= + This is the mean energy of phonons. The first term in the above equation is the zero-point energy, the minimum energy of the oscillator even at T=0 K. Rearranging the result = + = <n(ω)> + = <n(ω)>= mean excitation number of phonon at frequency ω The average number of phonons is given by the Planck distribution which is a special form of the Bose-Einstein distribution. Phonons are Bose-Einstein particles Phonons are bosons. G. Bracco-Material Science SERP CHEM 11 E k B T Mean energy of a harmonic oscillator as a function of T 1 hω 2 T = + low temperature limit Since exponential term gets bigger = zero point energy high temperature limit 1 = 1 + +! + retaining up to the linear term = + the energy steps are now small compared with the energy of the harmonic oscillator classical limit corrected by the zero point energy (negligible at high temperatures). G. Bracco-Material Science SERP CHEM 12

7 Heat Capacity C Heat capacity C can be found by differentiating the average energy of phonons = + = = ( ) = = () ( ) θ E θ E θ E 1 Where θ E = has the dimensions of T and is the Einstein temperature. G. Bracco-Material Science SERP CHEM 13 Einstein model for the heat capacity of solids The theory developed by Einstein is the first quantum theory of solids. He made the simplifying assumption that all 3N vibrational modes of a 3D solid of N atoms had the same frequency, so that the whole solid had a heat capacity given by the previous results for 1D oscillator multiplied by 3N (3 degrees of freedom for the N oscillators) = 3 = 3 θ E θ E θ E 1 A. Einstein, Ann. Physik, vol. 22, p. 186 (1907) In this model, the atoms are treated as identical independent oscillators with energy. G. Bracco-Material Science SERP CHEM 14

8 = 3 Einstein model for the heat capacity of solids θ E θ E θ E 1 This crude model gave the correct limit at high temperatures heat capacity given by Dulong-Petit law 3R (R is the gas constant) It tend to zero for T 0 K, but its exponential decreasing to zero is too fast with respect values measured for solids. atomic oscillators in a solid are not Temperature dependence of according to the Einstein model with θ E =1320 K in comparison with experimental values for diamond (A. Einstein, Ann. Physik, vol. 22, p. 186, 1907 ) isolated therefore they vibrate with frequencies distributed in optical and acoustic branches. For acoustic modes 0 therefore always exists a mode fulfilling 1 and the decay to zero is not exponential. G. Bracco-Material Science SERP CHEM 15 Density of States (DOS) The Einstein model correspond to the first step (contribution of a single mode), for the second step we need to count how many energy level are present in a interval E E+ E N=g(E) E where g(e) is the function density of states According to Quantum Mechanics if a particle is constrained the energy of the particle can only have discrete energy values. These energy differences can be so small depending on the system that the energy can be considered as continuous This is the case of classical mechanics. But on atomic scale the energy can only jump by a discrete amount from one value to another. Definite energy levels differences get small Almost a continuum This discussion on DOS is valid also for electrons (, boson E(k), fermion). G. Bracco-Material Science SERP CHEM 16

9 Density of States In some cases, each particular energy level can be associated with more than one different state (mode for phonons or wavefunction for electrons) This energy level is said to be degenerate. The density of states is the number of discrete states per unit energy interval, and so that the number of states between ε and ε+ ε is g(ε) and =N number of stated between and. The dispersion relations (ω(k) or E(k)) provide the way to calculate g. In 1D for monoatomic chain: ω(k ) ω(k ) Both blue regions (stripes) contribute to the counting of states (factor 2 because they are equal) The number of states in the red region (=g(ω(k ))dω) is twice the number of state between k and k k k k k G. Bracco-Material Science SERP CHEM 17 Density of States (1D) k traveling waves: 4π 2π 0 2π 4π 6π L L L L L Periodic boundary condition (Born-von Karman) Distinct k wavevectors corresponding to traveling waves ( () ) all positive and negative values of n in the 1 st BZ are allowed Length of the 1D chain L=Na (N cells) k= n -N/2<n N/2 These allowed wavenumbers are uniformly distibuted in k, any k occupy a volume density in k between k and k+dk ρ(k)= and in this volume there is a single value dn=1 = = (constant density) Number of states between ω(k) ω(k)+dω g(ω)dω = 2 ρ(k)dk= 2 dk= dk g(ω)dω = dk G. Bracco-Material Science SERP CHEM 18

10 Density of States (1D) It is worth noting that the density of states is independent from the specific boundary condition, in fact choosing the vanishing of the vibration at the boundaries to get standing waves: Distinct k wavevectors corresponding to standing waves (sin(kx-ωt)) only positive values of n in the 1 st BZ are allowed Length of the 1D chain L=Na (N cells) sin(kl)=0 k= n 0<n N These allowed wavenumbers are uniformly distibuted in k, any k occupy a volume and in this volume there is a single value dn=1 density in k between k and k+dk ρ(k)= (constant density) = = Number of states between ω(k) ω(k)+dω g(ω)dω = ρ(k)dk= dk g(ω)dω = dk density equal to the case of Born-von Karman BC. G. Bracco-Material Science SERP CHEM 19 g(ω) = or in term of the group velocity g(ω) = Density of States (1D) The density of state is greater where the group velocity is smaller and is divergent where = 0 ω(k)= sin ( ) = cos ( ) 1 ( ) 2 ω ω Multiplying by ( ) 2 = ω2 ω 2 g(ω) = ω ω = ω ω G. Bracco-Material Science SERP CHEM 20

11 The energy of lattice vibrations will then be found by integrating the energy of single oscillator over the distribution of vibration frequencies. This has a typical structure: sum over all the states, each state at ω populates in average by <n> phonons 1 hω ε = hω + g / kt ( ω) dω h 2 e ω 1 0 for 1D g(ω)= 2N ω π Density of States (1D) 2 2 ( ) 1/ 2 max ω Mean energy of a harmonic oscillator G. Bracco-Material Science SERP CHEM 21 For a 2D rectangular crystal, the Born-von Karman boundary conditions are given by u l,m (t) = u l+nx,m (t) L x =N x a x u l,m (t) = u l,m+ny (t) L y =N y a y, k x = x n x -N x /2<n x N x /2 k y = y n y -N y /2<n y N y /2 any k occupy a volume x point dn=1 y = and in this volume there is a single for an isotropic oscillator (same dispersion along x and y) and a square crystal (L x =L y =L) density in k between k and k+dk (the annular region of area da=2πkdk) ρ(k)dk= 2πkdk = 2πk ρ(k) = 2πk and g(ω)= 2πk / Density of States (2D) G. Bracco-Material Science SERP CHEM 22

12 Density of States (2D) For anisotropic oscillators the curve at constant frequency (energy) in the k x, k y plane is different from a circle = () To calculate g() we have to sum all the modes between the two lines = and = + Isotropic Anisotropic as in the isotropic case, g() is the area between the two lines divided by the density of k points in the k x, k y plane g()= = dl but dω= ω dω= v g ( v g group velocity) dl g()= dk v g ω+dω dl ω This calculation has to be done for any dispersion curve and the results summed up to get the total g(). 1 st BZ Simple square lattice Curves = G. Bracco-Material Science SERP CHEM 23 For a 3D crystal, the Born-von Karman boundary conditions are given by u l,m,n (t) = u l+nx,m,n (t) L x =N x a x u l,m,n (t) = u l,m+ny,n (t) L y =N y a y, u l,m,n (t) = u l,m,n+nz (t) L z =N z a z k x = n x -N x /2<n x N x /2 k y = n x y -N y /2<n y N y /2 y k z = n z -N z /2<n z N/2 any k occupy a volume x y Density of States (3D) =, in this volume dn=1 z z for an isotropic oscillator (same dispersion along x, y and z) and a cubic crystal (L x =L y =L z =L) density in k between k and k+dk (the spherical shell of volume dv=4πk 2 dk) ρ(k)dk= 4πk2 dk = 4πk2 ρ(k) = 4πk 2 and g(ω)= 4πk 2 / G. Bracco-Material Science SERP CHEM 24

13 Density of States (3D) For anisotropic oscillators the surface at constant frequency (energy) in the k x, k y, k z space is different from a sphere = () To calculate g() we have to sum all the modes between the two surfaces = and = + as in the isotropic case, g() is the area between the two surfaces divided by the density of k points in the k x, k y, k z space dk ω+dω g()= d = ds ω but dω= ω dω= v g ( v g group velocity) d g()= v g Of course, the calculation is performed on any dispersion curve and the results are summed up to get the total g() a non trivial task. G. Bracco-Material Science SERP CHEM 25 Ex. GaAs Curves = () for a section of 3D 1 st BZ (FCC lattice) On the left: results of calculations. On the right the curves determined experimentally. Density of States (3D) For 3D, the calculation of () is a very heavy task and is performed numerically. To understand the thermal properties of a solid Debye proposed a model to approximate the real dispersion curves. In this model only the acoustic branches are considered and the true dispersion curves are approximated by linear dispersion curves. This model gives better results than the Einstein model. G. Bracco-Material Science SERP CHEM 26

14 The Debye model Debye obtained a good approximation to the resulting heat capacity by neglecting optical phonons and the dispersion of the acoustic waves, i.e. assuming ω=v (v the sound velocity) for arbitrary wavenumber v = ω = =v. In a one dimensional crystal this is equivalent to taking ω(k) as given by the dotted lines rather than full curve. In 2D as the conical dispersion curves. Einstein approximation ω=const. Debye approximation ω=v k G. Bracco-Material Science SERP CHEM 27 The DOS is g(ω)= 4πk 2 v 2 s v s = 2 v 3 s The Debye model /vs = k2 v s = If the crystal of volume V contains N cells, there is the following constraint on the maximum frequency for each dispersion curve (polarization) Density of state approximate by the Debye curve: both curves enclose the same area g(ω)ω this cutoff is the Debye frequency 6 2 v 3 sn/v and in turn also a cutoff in k space Debye wave vector 6 2 N/V vs = for a cubic sample with periodicity radius of a sphere in k space, for the real crystal k within the 1 st BZ G. Bracco-Material Science SERP CHEM 28

15 The Debye model The thermal energy of the crystal can be calculated as = + This for a single polarization = 2 v 3 s + If we neglect differences between L and T modes the result must be multiplied by 3 (g(ω) 3 g(ω) since the degrees of freedom of the crystal are 3N A better approximation is considering that there is a longitudinal mode and two transverse modes = V ( + ) / = V ( + ) g(ω)ω = 3 V ( + ) = 3 V ( + )= = G. Bracco-Material Science SERP CHEM 29 As we did for the Einstein model, for the Debye model it is useful to introduce the Debye Temperature θ D = D a parameter that provides a way to say if the temperature is high (T>θ D ) or low (T<θ D ). Neglecting the zero point energy which does not contribute to the heat capacity C v = 9 x 3 θ D e x 1 x and the dependence on T is outside the integral and in = θ D /T For T>>θ D (high temperature limit) e x 1+x 3 x 2 x = x 3 = (θ D/T) 3, C v = 3 (Dulong-Petit law) The Debye model G. Bracco-Material Science SERP CHEM 30

16 For T 0 (low temperature limit) x 3 e x x = θ D The Debye model The energy depends on the fourth power of T, therefore the heat capacity C v = = θ D The heat capacity vanishes more slowly than the exponential behaviour of a single harmonic oscillator because the vibration spectrum extends down to zero frequency. G. Bracco-Material Science SERP CHEM 31 The Debye model The figure shows the excellent agreement of this prediction with experiment for an insulator: the lattice heat capacity of solids thus varies as at low temperatures: this is referred to as the Debye law. θ D The heat capacity C v = = is in agreement with experiments in both high and low T limit. Even for intermediate temperature the heat capacity calculated by means of the Debye model shows a good agreement with experiments. G. Bracco-Material Science SERP CHEM 32

17 Debye interpolation scheme θ D The heat capacity C v = = can be used as an interpolation formula for all temperatures even though the actual phonon-density of states curve may differ appreciably from the Debye assumption. Debye frequency and Debye temperature scale with the velocity of sound in the solid. So solids with low densities and large elastic moduli have high. Debye energy ħ can be used to estimate the maximum phonon energy in a solid. Solid Ar Cs Pb G. Bracco-Material Science SERP CHEM 33 The Debye model Left panel, experimental results of specific heats plotted as a function of temperature very different. Right panel, experimental results scaled very similar and very close to the Debye theory. For metals there is another contribution due to electrons which is generally negligible and the lattice heat capacity of solids varies as for not too low temperature. At very low temperatures, the lattice contribution vanishes faster than the electron contribution and the latter can be measurable. At high temperature the measured heat capacity can be also greater than the Dulong-Petit value due to anharmonic effects. G. Bracco-Material Science SERP CHEM 34

18 Einstein & Debye models vs. real crystals The two models can be applied to approximate acoustic and optical dispersion curves as schematically shown in the figure. The different DOS are shown in the figure below. For real crystals the DOS is given by g s ()= d v g for any mode of polarization s g()= g s () Density of state for Al. Singular points where the group velocity is zero (van Hove singularities). G. Bracco-Material Science SERP CHEM 35 Anharmonic Effects Up to now we have considered only vibrations with small amplitude and with this assumption the interaction potential between atoms has been approximated by an harmonic potential retaining only term up to second order. The harmonic potential allows us to calculate crystal properties for T not too high (below the melting point) to fulfill the requirements of small vibration amplitude. One could argue that anharmonic terms can added only to increase the precision of the calculation. Instead some of the phenomena cannot be explained even at low temperature without anharmonicity the specific heat should reach the Dulong-Petit value at high temperature, but experimentally the value is not always reached and inclusion of anharmonic terms is necessary. Thermal expansion: A perfect harmonic crystal does not change size increasing the temperature Thermal conductivity: experimentally thermal conduction shows a complex behavior, but in harmonic approximation is infinite. G. Bracco-Material Science SERP CHEM 36

19 Anharmonic Effects The harmonic potential is symmetric but the real potential is asymmetric. For large deformation, any real crystal resists compression to a smaller volume than its equilibrium value more strongly than expansion due to a larger volume. h = + 2 This asymmetric shape is an anharmonic effect due to the higher order terms in potential which are ignored in harmonic approximation. Thermal expansion is an example of the anharmonic effect. Increasing the energy of the oscillator in harmonic approximation the average position is always centered around. For the real potential the average position R increases with temperature. G. Bracco-Material Science SERP CHEM 37 Anharmonic Effects: Thermal expansion Considering the potential energy in terms of the displacement x between two atoms from their equilibrium position: = + = with c, g, and f>0 Average displacement < > = For small displacement / 1 exp ( ) exp ][1 + ][1 + exp ][ + + β = G. Bracco-Material Science SERP CHEM 38 exp ]exp[ ]exp[ In a similar way the denominator exp ( ) exp ]exp[ ]exp[ exp ][1 + ][1 + exp ][1 + + exp So <x> = The thermal expansion does not involve the symmetric term but only the asymmetric cubic term.

20 Anharmonic Effects: Thermal expansion The linear expansion coefficient at constant pressure α= generally depends on the crystal directions, for cubic crystal only a single value Related to thermal expansion, there is the change in the phonon frequency and this effect is described by the Grüneisen parameter () γ sk = for each mode and a total value where each γ sk is summed with a weight given by the contribution of the mode to the specific heat γ= γ sk c vsk c vsk In this way thermodynamic quantities depend on the volume of the crystal. G. Bracco-Material Science SERP CHEM 39 Anharmonic Effects: Thermal conductivity In harmonic approximation, phonons do not interact with each other, in the absence of boundaries, lattice defects and impurities (which also scatter the phonons), the thermal conductivity is infinite. In fact without collisions there will be no thermal equilibrium. Treating high order term as perturbations, phonons interact (collide) with each other and these collisions limit thermal conductivity which is due to the flow of phonons. The most important terms are: Phonon with wavevector k and branch s decay in 2 phonons due to 3 rd order term, or merging of two phonons in one Decay & Scattering &merging involving 4 Phonons due to 4 th order term All the processes fulfill,, ħ () =,, ħ () energy cons.,, =,, +G crystal momentum conservation G. Bracco-Material Science SERP CHEM 40

21 Anharmonic Effects: Thermal conductivity Consider a rod with the two ends maintained at different temperatures. Thermal conductivity κ is defined as the energy j transmitted per unit time across unit area per unit temperature gradient = in 1D = The thermal energy transfer is a random process involving scattering introduces mean free path of phonons in the problem. Kinetic theory: Let s assume that the energy contributed by a phonon at a point depends on the position of its last collision with an average collision time τ phonons coming from the high temperature end bring more energy than those coming from the low temperature end, thus, although there is no net number flux, there can be energy flux travelling from the high T end to the low T end. Temperature at point x is T(x) and the energy at that point is E(T[x]). Half of the phonons arriving at a point x from the high temperature side carry an energy E(T[x- v x τ]) E(T[x]) v xτ, the other half are from the low-t side and carry an energy E(T[x+v x τ]) E(T[x])+ v xτ. G. Bracco-Material Science SERP CHEM 41 Anharmonic Effects: Thermal conductivity Number of phonons arriving at x per unit time per unit area of cross section is 1/2 n v x, where v x is the phonon speed in x direction. Net energy flux: = v {E(T[x- v x τ])-e(t[x+v x τ])} v 2 τ ( ) Averaging and assuming isotropy < v 2 >= < v 2 >= < v 2 >= < v2 > = = v 2 τ = v l with l the phonon mean free path. In the Debye model the phonon velocity is constant (sound velocity) therefore the temperature dependence of κ= l is that of and l G. Bracco-Material Science SERP CHEM 42

22 Anharmonic Effects: Thermal conductivity For T>> the Dulong-Petit law =3n constant And the average number of phonons n(ω)= but the scattering rate is proportional to the number of phonons which are present therefore the phonon mean free path l The experiment is close to this results with a dependence α 1 2 Pure LiF =732 K For T<< only phonon states with energies up to are populated and for a crystal with few defects, a phonon does not scatter frequently with other phonons and defects. The mean free path is limited mainly by the boundary of the sample l (size effect) and the dependence is that of. law intermediate T G. Bracco-Material Science SERP CHEM 43 Anharmonic Effects: Thermal conductivity In the intermediate region T the conductivity exceeds the trend giving a maximum before to decrease as. Energy is always conserved. Let s analyze the wavevectors involved in the scattering process. At low T, only phonons with low energy and in turn small k vectors are involved: in 3-phonon processes the final k 3 vector is within the 1 st BZ and the crystal momentum is strictly conserved 3 = (normal process) The forward phonon motion is not disturbed T T ( K) In the intermediate region, phonon with energy ω D (and k k D ) are in number and for them the crystal momentum is conserved within a G vector 3 = (umklapp process) G. Bracco-Material Science SERP CHEM 44

23 Anharmonic Effects: Thermal conductivity (umklapp process) the forward motion is perturbed since 3 exceed the 1st BZ ( 3 >/) and with a suitable choice of G it is folded back in the zone the phonon velocity is backward. The process due to anharmonicity provides a backward motion that allows the phonon system to relax to equilibrium. We have treated the collection of phonons as an ideal gas but there are differences G. Bracco-Material Science SERP CHEM 45 Boundary-scattering limited thermal conductivity Pure NaCl =321 K Worlock, Phys Rev 1966 G. Bracco-Material Science SERP CHEM 46

24 Debye temp Melting temp Glassbrenner and Slack, Phys Rev 1964 G. Bracco-Material Science SERP CHEM 47 Neutron scattering Phonon dispersion curves can be determined by neutron scattering. A beam of neutron is produced by a nuclear reactor and interacting with a solid the beam may reemerge with a different energy (inelastic scattering). X-ray are scattered by the electron density, instead neutrons are scattered by the interaction with nuclei. The interaction is strong but is very localized (nucleus size m). As in the case of X-ray, the interaction can be elastic (diffraction) and the Laue relations must be fulfilled. The neutron momentum and energy are = ħ (neutron wavevector) and = energy conservation = = and the Laue condition = +. On the other hand the interaction with the solid can be inelastic and phonons can be created or annihilated. In case of single phonon exchange Energy cons. = ± ħ () (+ for annihilation for creation) Crystal momentum cons. = + +. In this way the dispersion curve () can be measured. G. Bracco-Material Science SERP CHEM 48

25 Neutron scattering The neutron beam produced by a reactor has a very broad energy distribution and the elastic scattering with a crystal (monochromator) can be employed to select an energy in a small range (quasi-monochromatic beam). This energy selected beam can be employed for inelastic scattering. Neutrons have an energy similar to phonon energies ( mev) therefore during an inelastic scattering there is a relatively huge energy exchange, that can easily be measured. The same technique can be used for X-rays to get monochromatic beams but the energy of phonons is in the kev range and, for X-rays, phonon energy exchange is negligible. G. Bracco-Material Science SERP CHEM 49 Neutron scattering Typical experimental results of neutron inelastic scattering for copper and for germanium. The phonon peaks are broadened by anharmonic effects that limit the lifetime τ of the measured phonons E ħ G. Bracco-Material Science SERP CHEM 50