# On the relationship between ultrasonic attenuation and Brillouin linewidths: for a re-definition of the phonon lifetime

Yvan Simon, Thierry Hocquet

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Yvan Simon, Thierry Hocquet. On the relationship between ultrasonic attenuation and Brillouin linewidths: for a re-definition of the phonon lifetime. 2025. hal-05190403v1

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# On the relationship between ultrasonic attenuation and Brillouin linewidths: for a re-definition of the phonon lifetime

Yvan Simon

Professeur honoraire, Sorbonne Université

Thierry Hocquet*

Laboratoire Matière et Systèmes Complexes, UMR 7057 CNRS,
Université Paris Cité, 10 rue Alice Domon et Léonie Duquet, 75013 PARIS, France
and Sorbonne Université, 4 place Jussieu, 75005 Paris, France

In a crystal, the sound amplitude attenuation $\alpha_{\ell}$ of acoustic waves of the mode $\ell(\mathbf{k}, s)$ of wave vector $\mathbf{k}$ and branch $s$ in a frequency range where the dispersion is linear ($k = \omega/v$, e.g. $f < 1$ THz in the Si[100] direction) is directly related to the relaxation time of a large deviation from equilibrium of the number $n_{\ell}$ of phonons of mode $\ell$. This relaxation time, denoted as $\theta_{\ell}$ in the text ($\theta_{\ell} = 1/2\alpha_{\ell} v$), is commonly referred to in the literature as the phonon lifetime, a term we reserve rather for a lifetime $\tau_{\ell}$ defined as the reciprocal of the probability per unit time of a simple transition $n_{\ell} \rightarrow n_{\ell} \pm 1$ resulting from crystal anharmonicity. We argue that in a weakly anharmonic and transparent crystal the Brillouin backscattered light has a Lorentzian distribution with full width at half maximum (FWHM) $\Delta\omega \sim 1/\tau_{\ell B}$, where $\ell_B$ is the longitudinal acoustic mode of the one phonon emitted or absorbed by a scattered photon, at variance with the accepted equation $\Delta\omega = 1/\theta_{\ell B}$. In usual backscattering experiments at room temperature, $\tau_{\ell B} \approx \theta_{\ell B} x^2$ ($x = h f/k_B T$) is at least three orders of magnitude smaller than $\theta_{\ell B}$. In Sec. III we review in detail the classical theory of Brillouin scattering when dealing with an opaque cubic crystal. To the fluctuations of the phonon occupation number $n_{\ell}$ classically correspond not only amplitude fluctuations, but also random phase jumps of the scattered electric field at the rate $2/\tau_{\ell}$. The lines are shown to be still Lorentzians, whose FWHM $\Delta\omega$ is the sum $\Delta\omega_1 + \Delta\omega_2$ of a phonon lifetime broadening $\Delta\omega_1 = 4/\tau_{\ell B}$ and of a well-determined opacity broadening $\Delta\omega_2$, first studied by Sandercock (1972), in cases where the latter dominates ($\Delta\omega \approx \Delta\omega_2$; e.g. in Ge). The comparison of experimental and theoretical values of the attenuation in Si at 300 K with those derived from the Si Brillouin spectrum recorded by Olsson et al. (2015) corroborates the fact that $\tau_{\ell B}$ is the proper phonon lifetime to interpret the anharmonic broadening of the Brillouin lines.

## I. INTRODUCTION

Insofar as surface effects are not involved, the sound attenuation in a pure and structurally perfect dielectric crystal is entirely due to the anharmonicity of interatomic forces. Anharmonicity also accounts for the finiteness of the thermal conductivity. In general, when dealing with weakly anharmonic crystals, such as those of standard semiconductors crystallizing in the diamond or zincblende structure (Si, GaAs, ZnSe, GaP, ...), or alkali halides, the quantum treatment of both phenomena retains only the cubic term $W$ of the anharmonic Hamiltonian. Then, $W$, viewed as a time-independent small perturbation, causes transitions between the eigenstates $|\dots, n_{\ell}, \dots, n_{\ell'}', \dots, n_{\ell''}', \dots\rangle$ of the harmonic Hamiltonian. Here, $n_{\ell}$ stands for the phonon occupation number of the mode $\ell(\mathbf{k}, s)$ of wave vector $\mathbf{k}$ and branch index $s$. Let $n_{0\ell}$ denote its mean value for thermal equilibrium, $n_{1\ell} = n_{\ell} - n_{0\ell}$ and $n_{1\ell}^*$ the root mean square of the fluctuations. According to Bose statistics and assuming $x = \hbar\omega_{\ell}/k_B T$, where $\omega_{\ell} = \omega_s(\mathbf{k})$ is the angular frequency

of the mode,

$$n_{0\ell} = \frac{1}{e^x - 1} \quad \text{and} \quad n_{1\ell}^* = \sqrt{n_{0\ell}(n_{0\ell} + 1)}. \quad (1)$$

With each mode $\ell$, one may associate a relaxation time $\theta_{\ell}$ defined by

$$\frac{1}{\theta_{\ell}} = - \frac{1}{n_{1\ell}} \frac{dn_{1\ell}}{dt}, \quad (2)$$

when $n_{\ell}$ departs for equilibrium, while all other modes have equilibrium occupation [1, 2].

This definition will be clarified in the Appendix, where we briefly recall how the expression (A12) for $\theta_{\ell}$, which is now universally accepted, can be derived in the framework of the first-order perturbation theory. The purpose of this rather long Appendix is twofold. Firstly, we prefer to present separately some well-known notions and results, whose review is necessary for our discussion in Sec. II and IV, but which might make it too cumbersome. At the same time, we introduce some shortened notations to be used in Sec. II. On the other hand, we want to show that the full expression for $\theta_{\ell}$, or for the so-called amplitude damping constant $\Gamma_{\ell B}$ (see eq. 3 below), was practically available in the first edition of Peierls' book (1955) [3], shortly after Klemens introduced the concept of relaxation time.

<footnote>* thierry.hocquet@sorbonne-universite.fr</footnote>
In the literature, the so-defined relaxation time is usu-
ally denoted by the letter $\tau$, and since the 1960s, referred
to as the *phonon lifetime*. As explained in Sec. II, we
shall reserve the symbol $\tau_{\ell}$ to designate another under-
standing of the concept of phonon lifetime. The relevance
of making such a distinction is the central argument of
the present paper.

In the following, we will deal only with acoustic
phonons (indexed $s = 1, 2, 3$ or labeled $T_s$ (slow), $T_f$
(fast), L), and we will be especially concerned with the
relaxation time $\theta_{\ell}$ of the longitudinal phonons ($s = 3$ or
L) in the low-frequency range where dispersion is linear
($2\pi f_{\ell} = \omega_{\ell} = vk$, $v = v_3(\hat{k})$, $f_{\ell} < 1$ THz). Herring
carefully discussed the relation of these $\theta_{\ell}$ to thermal
conduction, a rather complicated matter [2].

By contrast, there is a simple and direct relation of $\theta_{\ell}$ to the amplitude attenuation coefficient $\alpha_{\ell}$ of the sound waves of the same mode. A direct measurement of $\alpha_{\ell}$ consists in observing the propagation (with the velocity $v = v_3(\hat{x})$, over a distance $d \lesssim 1$ mm, of a quasi-monochromatic wave train centered on the mode $\ell$ [4]. Its amplitude decreases as $e^{-\alpha_{\ell}x}$, and in time as $e^{-\alpha_{\ell}vt}$. We can look at it as a wave packet of $N_{\ell}$ phonons, then treat $N_{\ell}$ as a macroscopic deviation $n_{1\ell} = N_{\ell} \gg n_{0\ell}$, so that $N_{\ell}(t) = N_{\ell}(0) e^{-t/\theta_{\ell}}$. Since a $\Delta n_{\ell}$ represents a variation in energy, the amplitude of the sound signal decreases as $e^{-t/2\theta_{\ell}}$. Hence

$$
\alpha_{\ell} v = \frac{1}{2\theta_{\ell}} = \Gamma_{\ell B}, \qquad (3)
$$

by definition of the *amplitude damping constant* Γ<sub>ℓB</sub>. The
θ<sub>ℓ</sub> (or α<sub>ℓ</sub>) of the modes L [100], (f<sub>ℓ</sub> < 1 THz) in GaAs at
50 K, calculated from Eq. (A12) [5, 6] and measured α<sub>ℓ</sub>'s
[4] are in remarkable agreement, at least in comparison
with other earlier attempts to reconcile experimental and
theoretical α<sub>ℓ</sub> values, especially in Si, in the 30 - 300 K
and 30 - 100 GHz ranges [7, 8]. We will return to this
matter in Sec. IV A.

Another effect of anharmonicity is the broadening of
the one-phonon peaks in neutron or light scattering [9].
We restrict ourselves to the case of *Brillouin lines* ob-
served at room temperature and in *backscattering* in
the direction [100] or [110] of a cubic crystal. If *q*₀ =
(2π/λ₀) *x* = (Ω/c) *x* is the wave vector of the incident
photons and *ν*₁ the real part of the crystal's refractive
index, the laws of energy and crystal momentum con-
servation determine the wave vector **k** = ±K *x* and the
frequency *f*<sub>B</sub> of the longitudinal phonon emitted (+) or
absorbed (−) [9]:

$$
\begin{align}
K &= \frac{2\Omega \nu_1}{c} \frac{1}{1 \pm \frac{v\nu_1}{c}} \approx \frac{2\Omega \nu_1}{c} = \frac{4\pi \nu_1}{\lambda_0}, \nonumber \\
f_B &= \frac{\omega_B}{2\pi} = \frac{vK}{2\pi} = \frac{2v\nu_1}{\lambda_0}. \tag{4}
\end{align}
$$

Typically, $\lambda_0 \sim 450 - 650$ nm, $\nu_1 \sim 4$, $K \sim 10^8$ m$^{-1}$ and $f_B \sim 100$ GHz. If the sample is a weakly anharmonic crystal, anharmonic shifts and linewidths are small compared to the Brillouin shift $\omega_B$. Then, the Stokes and

anti-Stokes lines are practically centered at $\Omega - \omega_B$ and $\Omega + \omega_B$. Under these conditions, the well-known theory of the one-phonon scattering cross section [10] concludes that the shape of a line is well approximated by a Lorentzian with a full width at half maximum (FWHM)

$$
\Delta\omega = 2\Gamma_{\ell B} = 1/\theta_{\ell B}, \quad (5)
$$

where $\theta_{\ell B}$ is the relaxation time of the “Brillouin” mode
$\ell_B = (\frac{\omega_B}{v}\hat{\mathbf{x}}, 3)$. Brillouin scattering thus becomes an in-
direct and rather simple way to measuring $\alpha$ [11, 12].

Irrespective of the fact that our “phonon lifetime”
concept challenges Eq. (5) and the underlying theory, it
should be pointed out that Eq. (5) is in any case no
longer valid if the sample is not perfectly transparent.
A so-called opacity broadening adds to the anharmonic
broadening, sometimes dominating or even masking it
completely, as observed in Ge [13] and GaAs [14].

In Sec. III, we reconsider the classical theory of Brillouin scattering with the aim of making it consistent with our quantum interpretation of the anharmonic broadening argued in Sec. II, and taking advantage of the fact that the classical approach allows the simultaneous treatment of both broadenings.

In Sec. IV we focus on the case of *silicon* and in particular on the attenuation $\alpha$ at 300 K of low-frequency longitudinal phonons along the [100] or [110] direction. We have three sources of data, namely:

i) Direct measurements of $\alpha_L$[100] in the 30-90 GHz range by Daly *et al.* [8].

ii) Theoretical calculations based on equations (A12) and (3) [6, 15].

iii) The Brillouin spectrum of Si measured in the backscattering geometry in the direction [110] by Olsson *et al.* [16].

From these data, we obtain, for example, at 90 GHz, in
the [100] direction: i) $\alpha_L \simeq 195 \text{ cm}^{-1}$; ii) $\alpha_L \sim 1 \text{ cm}^{-1}$;
iii) and, relying on the anharmonic broadening expres-
sion (5) then taking into account the weak anisotropy of
a Si crystal, $\alpha_L[100] \simeq \alpha_L[110] \simeq 1560 \text{ cm}^{-1}$. Our goal
is to explain and remove such a puzzling discrepancy.

## II. WHAT LIFETIME IS IMPLICIT IN THE BRILLOUIN LINEWIDTHS?

Consider an event in which one photon of energy
E = ħΩ absorbs one phonon of mode ℓ. It is scat-
tered with an energy E' > E and gives rise to a count
in the recording of an anti-Stokes line. In a rigorously
harmonic crystal, whose Hamiltonian is the sum of 3N
independent oscillator Hamiltonians h<sub>ℓ</sub> (eigentates |n<sub>ℓ</sub>⟩,
non-degenerate eigenvalues (n<sub>ℓ</sub> + 1/2)ħω<sub>ℓ</sub>), such a photon
would have induced a simple transition |n<sub>ℓ</sub>⟩ → |n<sub>ℓ</sub> − 1⟩
and emerged with exactly an energy E' = E + ħω<sub>ℓ</sub>. In
fact, the observed Brillouin lines of a transparent crystal are not perfectly sharp because of a fundamental and unavoidable dispersion of the energy $E'$ of the scattered photons. That this is the effect of internal phonon coupling and that the linewidths should be expressed as a function of the coupling constants (A1) is beyond dispute. And yet, as we shall see, different approaches can lead to very different results.

The energy spreading of the anti-Stokes photons can be estimated from the quantum point of view of the time-energy uncertainty relation [17]. Under the action of the anharmonic perturbation $W$ the transition $|n_\ell\rangle \rightarrow |n_\ell - 1\rangle$ involved in the anti-Stokes scattering can arise freely from either of the three-phonon processes $\mathbf{a}_-$ and $\mathbf{b}_-$ described in the Appendix; its probability per unit time, in statistical averaging, is given by Eqs. (A5, A6, A8 and A9) as

$$ \overline{w}_- = \overline{w(\mathbf{a}_-) + w(\mathbf{b}_-)} = \sum_{s', s'', \mathbf{k}'_i} AP + BQ. \quad (6) $$

Its reciprocal defines a “lowering” phonon lifetime $\tau_\ell$. The energy conservation law determining the energy $E'$ of the anti-Stokes photons is then weakened as $|E' - E - \hbar\omega_\ell| \sim \hbar/\tau_\ell$ giving an uncertainty $\Delta E' \sim \hbar/\tau_\ell$. It is readily seen that the mean probability per unit time of a Stokes transition $|n_\ell\rangle \rightarrow |n_\ell + 1\rangle$, $\overline{w}_+ = w(\mathbf{a}_+) + w(\mathbf{b}_+)$, has the same value (6), so that $\tau_\ell$ can also be considered as a “raising” phonon lifetime, and $\tau_\ell/2$ as the lifetime of the quasi-stationary state $|n_\ell\rangle$.

Therefore, we conjecture that the FWHM of the Brillouin lines relative to absorption or emission of a phonon of mode $\ell$ is

$$ \Delta\omega = \frac{\Delta E'}{\hbar} = \frac{1}{\tau_\ell} = \sum_{s', s'', \mathbf{k}'_i} AP + BQ, \quad (7) $$

it being understood that this relation, written in analogy to relation (5), conveys only an order of magnitude and may be corrected by a factor or order unity.

On the other hand, the time constant $\theta_\ell$ calculated in the Appendix is the relaxation time of a macroscopic deviation of the occupation number $n_\ell$ and is clearly related to the amplitude attenuation $\alpha_\ell$ of a sound wave of mode $\ell$. The return to equilibrium of a macroscopic deviation $n_{1\ell}$ occurs as a long sequence of random jumps $\Delta n_\ell = +1$ and $\Delta n_\ell = -1$, the latter being indeed more probable than the former. As a result, $\theta_\ell$ is a relatively long time, and it is not clear a priori what role it could play in the one-phonon contribution to light scattering. For example, in Si at 300 K, the calculated attenuation of the above L [100] mode at 90 GHz, $\alpha \sim 1 \text{ cm}^{-1}$, yields a relaxation time $\theta = 1/(2\alpha v) \sim 0.6 \text{ µs}$ ($v = 8430 \text{ m/s}$).

There is a simple and general relationship between the lifetime $\tau_\ell$ defined above and the relaxation time $\theta_\ell$.

The intermediate equation (A10) may be rewritten as

$$ \frac{1}{\theta_{\ell}} = \sum_{s', s'', \mathbf{k}'_i} A[n'_{0}(n''_{0} + 1) - (n'_{0} + 1)n''_{0}] + B[(n'_{0} + 1)(n'''_{0} + 1) - n'_{0}n'''_{0}]. \quad (8) $$

Expressing the terms between brackets as a function of the products $P$ and $Q$ ($n'_{0}(n''_{0} + 1) = P/n_0$, etc.), Eq. (8) becomes

$$ \frac{1}{\theta_{\ell}} = \left(\frac{1}{n_0} - \frac{1}{n_0 + 1}\right) \sum_{s', s'', \mathbf{k}'_i} AP + BQ = 2(\cosh x - 1) \frac{1}{\tau_{\ell}}, $$

with $x = h f/k_B T$. For Brillouin shifts $f_B \sim 30 - 150 \text{ GHz}$ usually encountered in backscattering experiments at room temperature, $x \lesssim 2.4 \times 10^{-2} \ll 1$, so that

$$ \tau_{\ell} \approx x^2 \theta_{\ell}, \quad (9) $$

to better than 0.005%.

There is such an order of magnitude difference between $\tau_\ell$ and $\theta_\ell$ that the question of whether the Brillouin line FWHM $\Delta\omega$ is $1/\theta_\ell$ or of the order of $1/\tau_\ell$ should be easily resolved by a simple backscattering measurement of the two LA Brillouin lines. However, under two conditions:

i) The crystal is transparent in the range $\lambda_0 = 450 - 650 \text{ nm}$ of the visible spectrum, or at least where non-dominant opacity broadening can be subtracted (a point studied in Sec. III within the framework of classical light scattering theory).

ii) Theoretical calculation of $\theta_\ell$ (and hence $\tau_\ell$), or equivalently, direct and reliable measurements of the intrinsic attenuation $\alpha_\ell$ are available.

GaAs satisfies the second condition [4, 6], but not the first [14]. Crystals of ZnSe, GaP are transparent, but we have no data on $\theta_\ell$ or $\alpha_\ell$. The case of silicon discussed in Sec. IV may well be conclusive.

### III. REVISITING THE CLASSICAL THEORY OF LIGHT SCATTERING

#### A. The backscattering field from an opaque crystal

In many ways, a cubic crystal viewed as a continuous medium, behaves like a fluid. In terms of dielectric and optical properties a cubic crystal is isotropic. Fluctuations $\delta\varepsilon(\mathbf{r}, t)$ of the dielectric constant about its mean value $\bar{\varepsilon} = \varepsilon_0(1 + \bar{\chi}) = \varepsilon_0\nu^2$ are the assumed classical source of light scattering; $\nu = \nu_1 + i\nu_2$ is the refractive index of the crystal. The thermodynamic differential equations relating the local state variables, density $\rho$, temperature $T$, entropy $S$ (J/kg.K) and pressure $p$, defined from the stress tensor $s_{ik}$ as $p = -s_{ii}/3$, are shown to be the same as for a fluid, as are well-known formulas
such as those giving $c_p - c_V$ or the ratio $B_S/B_T = c_p/c_V$ of the adiabatic and isothermal bulk moduli. As in a fluid, $\epsilon$ can be expressed as a function of two state variables, say $\rho$ and $T$, or $p$ and $S$. Pressure and entropy fluctuations are statistically independent ($\overline{\delta p \delta S} = 0$) [18]. It follows that the irradiance $I$ of the scattered wave is the sum of two independent terms $I_S + I_p$, resulting from isobaric entropy fluctuations and isentropic pressure fluctuations, respectively [19]. In weakly anharmonic crystals the spectral densities of $I_S$ and $I_p$ do not overlap, so that the latter, i.e. the Brillouin spectrum, can be considered and calculated separately.

We will argue about a simple backscattering geometry. A monochromatic plane wave $E_0(\mathbf{r}, t) = E_0 \hat{\mathbf{z}} e^{iq_0 x} e^{-i\Omega t}$ ($q_0 = 2\pi/\lambda_0 = \Omega/c$) is incident on the $x=0$ face of a silicon crystal of volume $V = L^3 \sim 1 \text{ mm}^3$ (say $0 < x < L, -L/2 < y, z < L/2$). The $x$-axis is taken along a crystalline direction [100] or [110]. To estimate orders of magnitude, we suppose $T = 300$ K and $\lambda_0 = 532$ nm [16] at which $\nu(\text{Si}) = 4.14 + i0.033$. The transmitted wave is written as $\mathbf{E}(\mathbf{r}, t) = \mathbf{E} \hat{\mathbf{z}} e^{i\Omega\nu x} e^{-i\Omega t} = \mathbf{E} \hat{\mathbf{z}} e^{-x/\delta} e^{i\Omega\nu_1 x/c} e^{-i\Omega t}$, where $\delta = \lambda_0/(2\pi\nu_2) = 2.6 \ \mu\text{m}$ is a small penetration depth. Only a very small part of the sample is irradiated. And the fact that the scattered wave is similarly attenuated means that backscattering is required.

The volume $V$ is imagined to be divided into small volume elements [19], e. g. enlarged cubic cells $w_i$ of side $d=m$ times the lattice constant $a$ centered in $M_i(x_i, y_i, z_i)$. The volume $w = d^3$ must contain so many atoms that the local notions of density or pressure make sense, while $d$ remains small compared to the light wavelength $\lambda = \lambda_0/\nu_1 \gtrsim 100$ nm (e. g. Si, $a = 5.43$ Å, $m = 6$, $6^3 \times 8 = 1700$ atoms, $d = 3.3$ nm). An oscillating electric dipole $\mathbf{p}_i(t)$ is attached to each cell $w_i$. The monochromatic transmitted and reflected waves are a solution of the Maxwell's equations obtained in first approximation by assuming $\mathbf{p}_i(t) = \epsilon_0 \bar{\chi} \mathbf{E}(\mathbf{r}_i, t) w$. In the second approximation (and discarding entropy fluctuations) $\mathbf{p}_i$ is corrected by the fluctuating dipole moment

$$ \delta \mathbf{p}_i = \left. \frac{\partial \varepsilon}{\partial p} \right|_S \delta p(\mathbf{r}_i, t) \mathbf{E}(\mathbf{r}_i, t) w \\ = \text{const.} \delta p(\mathbf{r}_i, t) e^{-i\Omega t} e^{i\Omega\nu x_i/c} w \hat{\mathbf{z}}. \quad (10) $$

Let $\mathbf{E}'_i(\mathbf{r}, t)$ be the electric field of the radiated spherical wave emitted in the surrounding space by the dipole (10). As agreed, $\mathbf{E}'(\mathbf{r}, t) = \sum_i E'_i(\mathbf{r}, t)$ is the field of the Brillouin scattered wave.

Consider the scattering pattern thus formed in a very distant plane $\Pi(x = -R)$ in directions less than, say, $4^\circ$ from the $x$-axis, which corresponds inside the crystal ($\nu_1 \simeq 4$) to scattering angles greater than $179^\circ$. At an observation point $P$ of $\Pi$ all $\mathbf{E}'_i(P, t) = E'_i(\hat{\mathbf{\theta}}(P))$ have practically the same direction $\hat{\mathbf{\theta}} = \hat{\mathbf{s}} \times (\hat{\mathbf{s}} \times \hat{\mathbf{z}})/\sin\theta$ ($\mathbf{OP} = r \hat{\mathbf{s}}, \sin\theta = |\hat{\mathbf{s}} \times \hat{\mathbf{z}}|$) within better than $0.1^\circ$; at point $O'(-R, 0, 0) \hat{\mathbf{\theta}} = -\hat{\mathbf{z}}$. Thus, the scalar treatment of the optical field applies, and $E'(P, t) = \sum_i E'_i(P, t)$

can be calculated in the same way as a diffracted pattern under Fraunhofer conditions [20] (e. g. $\lambda_0 \simeq 500$ nm, $R = 20 \text{ m} \gg L^2/\lambda_0 \simeq 2 \text{ m}$). Note that the exact backscattered light cannot be detected because it is lost in the bright monochromatic beam reflected by the crystal. A practical arrangement, as used in [13], is to place a collecting lens and a small opaque disk, e.g. 14 mm and 3.5 mm diameter respectively, about 100 mm in front of the crystal. The disk prevents the reflected light from entering the interferometer. In this way, the scattered wave is collected in directions at $1^\circ$ to $4^\circ$ from optical axis ($\tan^{-1} 0.07 \simeq 4^\circ$).

Wa are only interested in the irradiance of the scattered wave, $I = \epsilon_0 c \overline{E'}^* E'$, or more precisely in the normalized spectral density $\frac{1}{I} \frac{dI}{d\omega}$. This quantity, where we can replace $I$ for short by an “intensity” $\mathcal{I} = \overline{E'}^* E'$, is independent of the point of observation, whether it is a point of a far-field pattern or one of its image in the focal plane of the collecting lens. Finally $\frac{1}{\mathcal{I}} \frac{d\mathcal{I}}{d\omega}$ describes the profile of the Brillouin lines. The invisibility of the scattered wave at point $O'$ does not prevent the formal derivation of $\frac{1}{\mathcal{I}} \frac{d\mathcal{I}}{d\omega}$ from the field $E'(O', t)$. The ray going from a point source $M_i$ to $O'$ is refracted at the sample surface quite close to the point $H_i(0, y_i, z_i)$. Let $\mathcal{L}_i$ denote its optical path calculated with the complex index $\nu: \mathcal{L}_i = \nu M_i H_i + H_i O' \simeq \nu x_i + R$ in the Fraunhofer approximation. The interfering fields $E'_i$ at $O'$ differ in both amplitude and phase, first by those of the source strength (10), and then by the propagation factor $e^{i\Omega\mathcal{L}_i/c}$. Thus we obtain

$$ \begin{aligned} E'(O', t) &= D \sum_i \left( \delta p(\mathbf{r}_i, t) e^{-i\Omega t} e^{i\Omega\nu x_i/c} w \right) e^{i\Omega\nu x_i/c} \\ &= D e^{-i\Omega t} \sum_i \delta p(\mathbf{r}_i, t) e^{iKx_i} e^{-\beta x_i} w \end{aligned} \quad (11) $$

where the constant $D$ includes the factor $e^{i\Omega R/c}/R$, $K = 2\Omega\nu_1/c = 4\pi\nu_1/\lambda_0 \simeq 10^8 \text{ m}^{-1}$ and $\beta = 2\Omega\nu_2/c = 2/\delta \simeq 8 \times 10^5 \text{ m}^{-1}$. By the way, one recognizes the magnitude $K$ of the wave vector of the phonon emitted or absorbed in a backscattering quantum process (cf. Eq. (4)) and $\beta$ coincides with the so-called absorption coefficient.

Fluctuations of the local state variables obviously result from the perpetual random motion of the atoms. In classical mechanics, and in the harmonic approximation, the most general motion can be expressed as a sum $\sum_\ell$ of the $3N$ normal modes $\ell(\mathbf{k}, s)$, according to the nomenclature reused from the phonons. Usually, and more conveniently, anharmonic effects are treated in quantum mechanics, as seen for example in the Appendix. However, since we are returning to the classical point of view, whether we are dealing with light or lattice vibrations, we will account for anharmonicity by simply assuming that the amplitudes $A_\ell$ and phases $\varphi_\ell$ of the normal modes are themselves fluctuating variables (see § III B). Moreover, the classical interpretation of light scattering by a continuous medium ignores the microscopic structure of a crystal and considers only a continuous thermal displacement field $\mathbf{u}(\mathbf{r}, t)$ slowly varying on the scale $d$ of
the cells $w_i$. This is expressed as a sum, denoted below as $\sum_{\ell}$, restricted to acoustic modes of wavelength $\Lambda \gg d \sim 3$ nm; their wave vectors ($k \lesssim 2 \times 10^8$ m$^{-1}$) belong to the central region of the Brillouin zone, so that they can read as non-dispersive elastic or sound waves

$$ \mathbf{u}(\mathbf{r}, t) = \sum_{\ell} \tilde{A}_{\ell}(t) e^{i\varphi_{\ell}(t)} e^{i\mathbf{k}\cdot\mathbf{r}} e^{-i\omega_{\ell}t} \mathbf{e}_s(\hat{\mathbf{k}}), \quad (12) $$

where $\mathbf{e}_s(\hat{\mathbf{k}})$ ($s = 1, 2, 3$) are the normalized polarization vectors. In the first approximation, thermal sound waves (12) are isentropic, and thus

$$ \delta p(\mathbf{r}, t) = B_S \frac{\delta \rho}{\rho} = - B_S \operatorname{div} \mathbf{u} \\ = - B_S \sum_{\ell} i \mathbf{k} \cdot \mathbf{e}_s A_{\ell}(t) e^{i\varphi_{\ell}(t)} e^{i\mathbf{k} \cdot \mathbf{r}} e^{-i\omega_{\ell}t} $$

with $\omega_{\ell} = v_s(\hat{\mathbf{k}})k$. In substituting $\delta p(\mathbf{r}, t)$ in Eq. (11) we must use its real part [21], Re($\delta p$) = $-\frac{1}{2}B_S(\sum_{\ell} + \sum_{\ell}^*)$. $E'(O', t)$ thus appears as the sum of two terms, $E'_+ + E'_-$. Putting $G = -DB_S/2$ we have

$$ E'_{\pm}(O', t) = G e^{-i\Omega t} \sum_{\ell} \pm i \mathbf{k} \cdot \mathbf{e}_s A_{\ell}(t) e^{\pm i\varphi_{\ell}} e^{\mp i\omega_{\ell}t} \\ \times \sum_i e^{\pm i\mathbf{k} \cdot \mathbf{r}} e^{iKx_i} e^{-\beta x_i} w. \quad (13) $$

$E'_\pm(O', t)$ are stationary random functions; in related quantities such as $\mathcal{I}_+ = \overline{E'_+ E'_+}$ or the correlation function $C(s) = \overline{E'_+ (t) E'_+ (t+s)}$, the bars mean statistical or time averages indistinctly. $\mathcal{I}(O')$ reduces to the sum $\mathcal{I}_+ + \mathcal{I}_- = \overline{E'_+ E'_+ + E'_- E'_-}$; in averaging cross terms disappear. The spectrum of $E'_+$ (resp. $E'_-$) is mainly governed by the factors $e^{-(\Omega+\omega_\ell)t}$ (resp. $e^{-(\Omega-\omega_\ell)t}$) and is clearly shifted towards high (resp. low) frequencies. In Eq. (10) the effect of a mode $\ell$ consists in an amplitude modulation at the frequency $\omega_\ell$ of the local polarization $\mathbf{P} \propto e^{-i\Omega t}$, from which a double shift $\Omega \pm \omega_\ell$ was indeed expected. Spectral densities $d\mathcal{I}_+/d\omega$ and $d\mathcal{I}_-/d\omega$ characterize the anti-Stokes and Stokes scattering, respectively. We can limit ourselves to the calculation of the former.

To perform the summation $\sum_s \sum_k \sum_i$ of (13) (with the upper sign) we use the $\mathbf{k}$-point cubic mesh $\mathbf{k} = (-K + m_1 \frac{2\pi}{L}, m_2 \frac{2\pi}{L}, m_3 \frac{2\pi}{L})$ which has the full density $V/8\pi^3$ in the reciprocal space, and we replace the sum $\sum_i \cdots w$ by an integration $J = \int_0^L \int_{-L/2}^{L/2} \int_{-L/2}^{L/2} \cdots dx dy dz$. It is seen immediately that $J \neq 0$ if and only if $(m_2, m_3) = (0, 0)$. Only mode $\ell(\mathbf{k} // \hat{\mathbf{x}}, s)$ are involved. Moreover, in the assumed orientation of the crystal, these are either strictly transverse ($s = 1, 2$) or strictly longitudinal ($s = 3$), so that

$\mathbf{k} \cdot \mathbf{e}_1 = \mathbf{k} \cdot \mathbf{e}_2 = 0$ and $\mathbf{k} \cdot \mathbf{e}_3 = k_x$. Thus we first obtain

$$ J = L^2 \int_0^L e^{i m_1 \frac{2\pi}{L} x} e^{-\beta x} dx = L^2 \frac{e^{-\beta L} - 1}{i m_1 \frac{2\pi}{L} - \beta} \\ \approx \frac{L^2}{\beta - i(k_x + K)} \quad (e^{-\beta L} \approx 0), $$

and then Eq. (13) reduces to

$$ E'_{+}(O', t) = F(t) \\ = H e^{-i\Omega t} \sum_{k} A_{\ell}(t) e^{i\varphi_{\ell}(t)} \frac{k}{\beta - i(k + K)}, \quad (14) $$

where, for brevity, $F(t)$ stands for $E'_+(O', t)$ and $k$ stands for $k_x$. If $v$ denotes $v_3(\hat{\mathbf{k}})$, $\omega_\ell = v|k|$. The unimportant constant $H = iGL^2$ should disappear in the final result. As assumed above, the sum $\sum_k$ is limited to small values of $|k|$ ($k = -K + m_1 \frac{2\pi}{L}$, $K \sim 10^8$ m$^{-1}$, $|k| < 2 \times 10^8$ m$^{-1}$, $v[100] < v[110] = 9150$ m/s, $\omega_\ell/2\pi < 300$ GHz, $x < 0.05$).

The decomposition of the atomic thermal motion as a sum of elastic waves as in Eq. (12) is the starting idea of Brillouin in its original theory of light scattering in a transparent medium. In his 1922 paper [22], Brillouin suggested that the thermal sound plane wave of a longitudinal mode $\ell(\mathbf{k}, 3)$ acts as a reflecting three-dimensional grating of period $\Lambda = 2\pi/|k|$, like a set of atomic planes in the Von Laue experiment (1912). Its scattering contribution occurs only in the direction of a mirror reflection and under the further condition, if $i$ is the angle of incidence, that the Bragg equation is satisfied, i.e. $2\Lambda \cos i = \lambda$. Thus, only longitudinal modes of wave vector $\mathbf{k} // \hat{\mathbf{x}}$ can give rise to backscattering waves, provided that $2\Lambda = \lambda = \lambda_0/\nu_1$ or $|k| = 4\pi\nu_1/\lambda_0 = K$. This is indeed the result we would have obtained above for a transparent crystal: assuming $\beta = 0$, $J \neq 0$ if and only if $m_1 = m_2 = m_3 = 0$, so that $k = -K$ is determined (anti-Stokes case). When $\beta \neq 0$, the transmitted wave covers only about fifty periods $\Lambda$ of the “sound” grating over a depth $\delta \sim 3$ µm, so the Bragg condition becomes less restrictive and some constructive interference can occur for $k$ values slightly different from $K$. This explains the opacity broadening of the Brillouin lines.

## B. Correlation function and spectral density of the backscattered field

The values and fluctuations of the amplitude $A_\ell$ and phase $\varphi_\ell$ of a mode $\ell$ are the classical expression of the energy level and evolution of the quantum state $|n_\ell\rangle$ due to anharmonicity. By equating the classical and quantum expressions of its energy contribution (potentiel + kinetic) to the internal energy of a crystal of mass $M$, one obtains a relationship between $A_\ell$ and $n_\ell$, as follows:

$$ \frac{1}{2} M \omega_{\ell}^2 A_{\ell}^2 = \left(n_{\ell} + \frac{1}{2}\right) \hbar \omega_{\ell} \rightarrow A_{\ell}^2 = \frac{2\hbar}{M\omega_{\ell}} \left(n_{\ell} + \frac{1}{2}\right). $$
On the average, $\bar{n}_{\ell} + \frac{1}{2} = \frac{1}{e^{x-1}} + \frac{1}{2} = \frac{1}{x}(1 + \frac{x^2}{12} + \dots)$ and

$$ \overline{A}_{\ell}^2 = \frac{2k_B T}{M\omega_{\ell}^2} \left(1 + \frac{x^2}{12} + \dots \right) \approx \frac{2k_B T}{M\omega_{\ell}^2} \quad (\text{accuracy} \sim 10^{-4}). \qquad (15) $$

As noted and justified in the Appendix, the occupation numbers of different modes are uncorrelated. Furthermore, to quote Peierls [3]: “The occupation numbers should contain all the information that is obtainable about the system, and there should be no phase relations between

states with different phonon numbers”. Thus, by logical correspondance, we assume that all variables $A_{\ell}$ and $\varphi_{\ell}$ should be statistically independent of each other. Consequently, in the autocorrelation function of $F(t)$,

$$ C(s) = \overline{F^*(t)F(t+s)} = C^*(-s) \quad (C(0) = \overline{F^*F} = \mathcal{I}_+), $$

calculated from (14) as a double sum $\sum_{k_1} \sum_{k_2}$, the cross terms disappear in averaging. Thus, $C(s)$ reduces to the simple sum

$$ C(s) = |H|^2 e^{-i\Omega s} \sum_k \overline{A_{\ell}(t)A_{\ell}(t+s)} \frac{k^2}{\beta^2 + (k+K)^2} $$

recalling that the index $\ell$ here denotes the longitudinal mode of wave vector $\mathbf{k} = k\hat{\mathbf{x}}$ ($k = -K + m_1 2\pi/L$), and $\omega_{\ell} = v|k|$. For brevity, we rewrite

$$ C(s) = |H|^2 e^{-i\Omega s} \sum_k C_{AA}(s) C_{\varphi\varphi}(s) e^{-i v|k|s} f(k) \quad (16) $$

where $C_{AA}$ and $C_{\varphi\varphi}$ are the correlation functions of $A_{\ell}(t)$ and $e^{-i\varphi_{\ell}(t)}$, and we will drop the index $\ell$ when unnecessary.

The occupation number $n_{\ell}$ undergo jumps $\Delta n_{\ell} = \pm 1$ at the mean rate $2/\tau_{\ell}$, i.e. the sum of the probabilities per unit times $\overline{w}_{-} + \overline{w}_{+} = 2\overline{w}_{-}$ (cf. Eq. (6)); $\tau_{\ell}/2$ can be referred to as the lifetime of the state $|n_{\ell}\rangle$, or as the “collision time” of the phonons $\ell$. Since $n_{0\ell} \gg 1$ (or $x \ll 1$), these jumps will end in large fluctuations (r.m.s $n_{1\ell}^* \simeq n_{0\ell}$). Thus, $n$ and $A$ can be regarded as continuous variables. Let $a$ denote the deviation $A - \overline{A}$ (so that $\overline{a} = 0$, $\overline{a^2} = \overline{A^2} - \overline{A^2}$). According to a common assumption [18, 19, 23] the evolution of the fluctuation $a(t)$, or of any physical quantity behaving classically, should be governed by the same damping constant as a macroscopic deviation from equilibrium, in the following sense. Assume $a$ has a given value $a(0)$ at time $t=0$, then $\overline{a}(t>0)$ relaxes to zero as $\overline{a}(t) = a(0)e^{-\Gamma t}$, where $\Gamma = \alpha v = 1/2\theta$ is the amplitude damping constant of a usual acoustic wave of the considered mode (cf. Eq. (3)). Hence, the correlation function $C_{aa}$ of $a(t)$ can be derived, as shown in Ref. [18] (see Eqs. 118.6 to 8 therein), and then that of $A(t)$:

$$ C_{AA}(s) = \overline{A(t)A(t+s)} = C_{aa} + \overline{A^2} = \overline{a^2}e^{-\Gamma|s|} + \overline{A^2}. $$

We have ignored relatively small jumps $\Delta A$ associated with those of $n$ as long as $\Delta A/\overline{A} \sim x \ll 1$. But we cannot ignore the concomitant phase changes, which, as Peierls’ remark above suggests, are undetermined. We then assume that in the classical description the phase $\varphi_{\ell}$ suffers sudden changes at random time intervals, after

which it is equally likely to take any value in the range 0 to $2\pi$. Between jumps the phase remains constant. $C_{\varphi\varphi}$ is written as

$$ C_{\varphi\varphi}(s) = \frac{\overline{e^{-i\varphi(t)}e^{i\varphi(t+s)}} - \overline{e^{i(\varphi(t+s)-\varphi(t))}}}{\overline{e^{i(\varphi(s)-\varphi(0))}} - \overline{X} + i\overline{Y}} $$

where the last averages are understood in a statistical sense. $\overline{X} = \cos(\varphi(s) - \varphi(0))$ and $\overline{Y}$ can be estimated using a standard collision time model as follows. $P = e^{-2t/\tau}$ is the probability that no “collision” occurs between $t=0$ and $t=s > 0$ (event A), then $1-P$ that one or more “collisions” occurs (event B). So the probability density $p(X) = P p(X/A) + (1-P) p(X/B)$; hence, $\overline{X} = P \times 1 + (1-P) \times 0 = P$, then $C_{\varphi\varphi}(s > 0) = e^{-2s/\tau}$. Since $C_{\varphi\varphi}(-s) = C_{\varphi\varphi}^*(s)$ (a general property), we finally obtain

$$ C_{\varphi\varphi}(s) = e^{-2|s|/\tau}, \quad \text{or} \quad e^{-\mu|s|} $$

as a short working notation.

Before substituting the above expressions of $C_{AA}$ and $C_{\varphi\varphi}$ into Eq. (16) it is worth noting that their product simplifies considerably. Seeing from (9) that $\Gamma = 1/2\theta = x^2/2\tau = \mu x^2/4 \ll \mu$ ($f_{\ell} < 300$ GHz, $x^2/4 < 6 \times 10^{-4}$), we get

$$ \begin{aligned} C_{AA}C_{\varphi\varphi} &= (\overline{a^2} e^{-\Gamma|s|} + \overline{A^2}) e^{-\mu|s|} \\ &= \overline{a^2} e^{-\mu|s|(1+\frac{x^2}{4})} + \overline{A^2} e^{-\mu|s|} \\ &\approx (\overline{a^2} + \overline{A^2}) e^{-\mu|s|} = \overline{A^2} e^{-\mu|s|}. \end{aligned} $$

On the other hand, calculations of the attenuation of $\alpha(\omega)$ of longitudinal phonons in the [100] direction in Si at 300 K show that the well-known classical law $\alpha \propto \omega^2$ at very low frequencies holds in the ultrasonic range 0-300 GHz considered here. We return to this question in Sec. IV. If $\alpha = 1/2\nu\theta \propto \omega^2$, then $\tau = \theta x^2$ and $\mu = 1/2\tau$ are constants in the same frequency range.
<figure><img src="image_2.png" /><figcaption>FIG. 1. The Lorentzian $L(k)$ shows how the wave vectors $k\hat{x}$ of the absorbed phonons in an anti-Stokes backscattering are dispersed around $-K\hat{x}$ due to the opacity of the crystal (here in Si at 300 K and $\lambda_0 = 532$ nm).</figcaption></figure>

Noting also that from Eq. (15) $\overline{A_{\ell}^2} k^2 = 2k_B T/M v^2 = b = \text{const}$, Eq. (16) becomes

$$C(s) = |H|^2 e^{-i\Omega s} \sum_k \frac{\overline{A_{\ell}^2} e^{-\mu|s|} e^{-iv|k|s}}{\beta^2 + (k+K)^2} \frac{k^2}{\beta^2 + (k+K)^2} \\ = |H|^2 b \sum_k e^{-\mu|s|} e^{-i\Omega s} e^{-iv|k|s} L(k),$$

where $L(k) = 1/(\beta^2 + (k+K)^2)$ is a Lorentzian of half width at half maximum $\beta$, centered at $-K$. At $\lambda_0 = 532$ nm, where $K+i\beta = 4\pi\nu/\lambda_0 \approx 10^8(1+0.008i)\text{ m}^{-1}$, $L(k)$ takes significant values only in the interval $-K \pm 10\beta$ (see Fig. 1), so that we can formally sum over $k$ from $-\infty$ to $+\infty$, assuming $k < 0$ and $|k| = -k$. Letting $\Delta k = k+K$ and $\omega_+ = \Omega+vK$ (anti-Stokes line center), $C(s)$ can be written as

$$C(s) = |H|^2 b \sum_{\Delta k} e^{-\mu|s|} \frac{e^{-i\omega_+ s} e^{i v \Delta k s}}{\beta^2 + (k+K)^2} \quad (\Delta k = m_1 \frac{2\pi}{L}). \qquad (17)$$

The Fourier transform of $C(s)$ is equal to the spectral density of the stationary random function $F(t)$ (Wiener-Khintchine relations) [23]:

$$S(\omega) = \mathcal{F}C(s) = \frac{1}{2\pi} \int C(s) e^{i\omega s} \, ds \qquad (\text{real} > 0, \text{ another general property}), \text{ and conversely},$$

$$C(s) = \int S(\omega) e^{-i\omega s} \, d\omega \\ \mathcal{I}_+ = \overline{F^*(t)F(t)} = C(0) = \int S(\omega) \, d\omega.$$

To calculate the spectral density $\frac{d\mathcal{I}_+}{d\omega} = S(\omega)$ we use the remarkable properties of the normalized Lorentzians $\ell_a(u)$ (HWHM a, $\int \ell_a du = 1$) viz.

$$\ell_a(u) = \frac{a}{\pi} \frac{1}{u^2 + a^2}, \quad \ell_a(\omega - \omega_0) = \mathcal{F} e^{-a|s|} e^{-i\omega_0 s},$$

and the fact that $\ell_a$ and $\ell_b$ convolute to give $\ell_{a+b} (\ell_a \otimes \ell_b = \ell_{a+b})$. Then the summation $\sum_{\Delta k}$ can be replaced by an integration $\int \frac{L}{2\pi} \cdots d(\Delta k)$, or by the substitution $u = v \Delta k$, $\int \frac{L}{2\pi v} \cdots du$. Thus, Eq. (17) can be rewritten as

$$C(s) = |H|^2 b \frac{L}{2\pi v} \int e^{-\mu|s|} e^{-i(\omega_+ - u)s} \frac{\pi v}{\beta} \ell_{\beta v}(u) \, du.$$

In particular, $\mathcal{I}_+ = C(0) = |H|^2 b L/2\beta$, and [24]

$$\begin{aligned} \frac{d\mathcal{I}_+}{d\omega} &= S(\omega) = C(0) \int \ell_{\mu}(\omega_{+} - \omega - u) \ell_{\beta v}(u) \, du \\ &= \mathcal{I}_+ (\ell_{\mu} \otimes \ell_{\beta v})(\omega_{+} - \omega) \\ &= \mathcal{I}_+ \ell_{\mu+\beta v}(\omega - \omega_{+}). \end{aligned} \qquad (18)$$

Similarly, $\frac{1}{\mathcal{I}_-} \frac{d\mathcal{I}_-}{d\omega} = \ell_{\mu+\beta v}(\omega - \omega_-)$ is the profile of the Stokes line centered at $\omega_- = \Omega - vK$. We are led to the simple conclusion that the Brillouin lines are Lorentzians of HWHM $\mu + \beta v$. The FWHM $\Delta f = \Delta\omega/2\pi = (\mu + \beta v)/\pi$ is the sum $\Delta f_1 + \Delta f_2$ of two independent terms. The first measures the effect of the “phonon lifetime” as understood above:

$$\Delta f_1 = \frac{\mu}{\pi} = \frac{2}{\pi \tau} = \frac{2}{\pi \theta x^2} = \frac{4 \alpha v}{\pi x^2}. \qquad (19)$$

The second measures the effect of the absorption coefficient $\beta$:

$$\Delta f_2 = \frac{\beta v}{\pi} = \frac{4 \nu_2 v}{\lambda_0} = \frac{2 \nu_2}{\nu_1} f_B. \qquad (20)$$

In the last expression, as a function of the Brillouin frequency shift $f_B$ (cf. Eq. (4)), we retrieve the formula proposed by Sandercock [13] and well verified by Kuok et al. in GaAs [14], where $\Delta f_1 \ll \Delta f_2 \approx 30$ GHz at $\lambda_0 = 458$ nm. As will be seen in Sec. IV, both effects are on the same order of magnitude in Si at 532 nm.

# IV. SOUND ATTENUATION AND BRILLOUIN LINES BROADENING IN SILICON

## A. Sound attenuation in cubic crystals; theories and experiments

In the literature dealing with anharmonicity and phonon interactions, most authors introduce in their discussions an *average lifetime of the thermal phonons*, which we will denote as $\theta_0$, roughly estimated from the thermal conductivity $\kappa$ using the familiar argument of the kinetic theory of gases:

$$\kappa = \frac{1}{3} C v_D^2 \theta_0, \qquad (21)$$

where $C$ is the specific heat per unit volume, and $v_D$ the constant Debye velocity of sound. When writing Eq. (21) one is clearly thinking of the lifetimes $\theta_\ell$ of localized wave
packets traveling at the group velocity $v_ℓ$ and having a mean free path $v_ℓθ_ℓ$, summarily averaged as $v_Dθ_0$. In Si at 300 K, $θ_0 \sim 10^{-11}$ s. While we find $θ_0 \simeq 9.0 \times 10^{-12}$ s, Daly et al. take $θ_0 \simeq 9.7 \times 10^{-12}$ s [8] and Stoner and Maris [25] using an improved kinetic formula find $θ_0 \simeq 5 \times 10^{-11}$ s.

The quantum treatment of the sound attenuation in dielectric solids was initiated by Landau and Rumer (LR, 1937) [26]. Their line of argument is that described in the Appendix: the cubic part of the anharmonic potential, considered as a small perturbation, causes transitions between the states of the unperturbed system, and transition probabilities are calculated in the first-order perturbation theory. However, LR restricted themselves to the case of an isotropic solid with linear dispersion relations ($\omega = v_L k$, or $\omega = v_T k$; $v_L > v_T$). They considered “collisions” $a_±$ ($k$ (sound) + $k' = k''$, $\omega + \omega' = \omega''$) for which the wave number $k$ of the sound phonon is very small compared to those, $k'$, $k''$ of the thermal phonons (LR’s Debye quanta), so that they could put approximately $k' \simeq k''$. It is easy to see that under such conditions the only working process is $T$ (sound) + $L \rightleftharpoons L$. The LR model then was faced with two difficulties. According to the authors, to account for the attenuation of long longitudinal waves the quartic terms should be included in the potential energy allowing four-phonon processes to come into play. On the other hand, LR led to a literal expression of $\alpha v_T$ proportional to the first power of the pulsation instead of the expected classical $\omega^2$ law for which either long transverse or longitudinal waves must hold. In the first paragraph of their paper, LR attempt to explain in advance why their calculations will not hold for too long waves. They argue that the energy $\hbar\omega$ of the absorbed sound phonon must exceed a supposed uncertainty in the thermal phonon energy $\Delta E' = \hbar\omega' \sim \hbar/\theta_0$, i.e. $\hbar\omega \gg \hbar/\theta_0$, or $\omega\theta_0 \gg 1$, or else $\Lambda$ (sound) $\ll v_D\theta_0$ (“the free path of the Debye quanta”). Besides the fact that, as argued in Sec. II, neither $\theta_ℓ$ nor $\theta_0$ are the proper lifetimes to express a time-energy uncertainty relation, we do not believe in the relevance of such a restrictive condition; the basic principle of the LR procedure is to calculate transition probabilities between eigenstates $|i\rangle = |\dots, n_ℓ, \dots\rangle$ of the harmonic Hamiltonian, whose energy levels are perfectly defined, and there is no reason to invoke any quantum uncertainty here.

Moreover, Herring has shown that taking the elastic anisotropy of a crystal into account removed the above difficulties [2], and made the LR preamble unnecessary. Three-phonon “collisions” $a_±$, the so-called Herring process $L$ (sound) + $T_s \rightleftharpoons T_f$ and $L + T_s/f \rightleftharpoons L$, become possible and contribute efficiently to the attenuation $\alpha_L$ of long longitudinal waves. In cubic crystals with the diamond, zincblende or sodium chloride structure (crystallographic point groups $O_h$ and $T_d$), the Herring process alone accounts for the attenuation of longitudinal waves in some low-frequency, and $\alpha_L$ or $1/\theta_L \propto \omega^2$ as $\omega \to 0$ [2]. On the other hand, there is indeed a weak contribution to the transverse wave attenuation $\alpha_T$ of the processes

$T_{s/f} + L \rightleftharpoons L$ which vary as $\omega$ in the low frequency limit, as shown by LR. However, processes $T_{s/f} + T \rightleftharpoons T_f$, analog to the Herring process, which are efficient and dominant at low frequencies [5] favor the expected and observed $\omega^2$ dependence of $\alpha_T$.

The estimation of $\alpha_ℓ$ by means of Eq. (A12), which is the achieved formulation of the LR theory, can differ depending on how the coupling coefficients $V(ℓ, ℓ', ℓ'')$ defined in (A1) and the phonon dispersion relation are determined, either using a model based on elastic constants [5], a f.c.c. model with nearest-neighbor central forces [15], or ab initio calculations [6]. The theoretical given in Ref. [6] for Si at 300 K do not allow us to estimate values of $\alpha_L$ at frequencies $f \sim 100$ GHz. However, from Tamura-Maris calculation [15] we find that $\alpha_L[100]$ in Si at 300 K up to about 600 GHz is due to the exclusive contribution of the Herring process, and that [27]

$$ \alpha_L[100] (\text{cm}^{-1}) = 6.7 \times 10^{-5} f^2 (\text{GHz}) \\ (\text{Si, 300 K, } f < 300 \text{ GHz}). \quad (22) $$

Relying on the LR argument, Akhieser (1939) [28] developed a semiclassical approach to deal with the case $\omega\theta_0 \ll 1$. The sound wave is treated macroscopically as an applied external field in which thermal phonons are moving and interacting. The periodic strain of the sound modulates the frequencies of the thermal phonons that are no longer in thermal equilibrium. A Boltzmann equation describes the perturbed distribution of the thermal phonons and their relaxation towards local equilibrium, and finally allows to calculate the rate at which energy is removed from the sound wave, thus damping it. This approach requires only that $x$ (sound, $T$) $\ll 1$. Woodruff and Ehrenreich (WE, 1961) [29], then Maris (1971) [30] resumed the basic idea of Akhieser while extending it to all values of $\omega\theta_0$. It leads to rather complicated expressions for the attenuation. So, simplifying assumptions, more or less justified, reduce them to more convenient forms for comparison with experimental results. In particular,

$$ (a) \quad \alpha = \frac{CT \gamma^{*2}}{2 \rho v^3} \frac{\omega^2 \theta_0}{\omega^2 \theta_0^2 + 1}, \quad (23) $$

$$ (b) \quad \alpha_L = \frac{CT \gamma^2 \omega}{4 \rho v_D^3} \tan^{-1} 2\omega\theta_0, $$

where $\rho$ is the density, $v$ is the velocity of the sound wave ($C$ and $v_D$ have been defined above). The quantity $\gamma^{*2} = \langle \gamma_ℓ^2 \rangle - \langle \gamma_ℓ \rangle^2$ measures the dispersion of the Grüneisen parameters; $\langle \dots \rangle$ are weighted averages over the phonon spectrum, and $\langle \gamma_ℓ \rangle = \gamma(T)$ is an overall Grüneisen parameter [9]. As explained by Daly et al. [8], Eq. (23a) is a simplified form of Maris’ equation (130) [30]. WE gave Eq. (23b) at the end of their paper (where $\Gamma$ denotes $2\alpha_L$) and compared it with the experimental data available for quartz at 1 and 3.9 GHz. When $\omega\theta_0 \gg 1$, Eqs. (23) reduce to the very contrasting ex-
pressions

$$ (a) \quad \alpha = \frac{CT}{2\rho v^3} \gamma^{*2} \frac{1}{\theta_0}, \quad (b) \quad \alpha_L = \frac{\pi CT \gamma^2 \omega}{8 \rho v_D^3}, \quad (24) $$

The former is independent of frequency, the latter is in- dependent of $\theta_0$. Such divergent conclusions affect the reliability of the Boltzmann equation method, at least as far as sound attenuation is concerned.

Hao and Maris (2001) [7] and Daly et al. [8] reported measurements of $\alpha_L$[100] in Si in the 30-100 GHz fre- quency range. To our knowledge, there have been no other direct measurements of $\alpha_L$ in Si since then, and none above 100 GHz. Both teams used the so-called picosecond ultrasonic technique [31], in which acoustic pulses are generated by short light pulses incident on one side of a 315 µm [7] or 56 µm [8] thick Si wafer. Two acoustic echoes are detected after the pulses have made one and then two round trips through the wafer. By analyzing the Fourier transforms of these echoes, which cover the 30-100 GHz range, one can estimate the in- trinsic attenuation $\alpha_L$[100]($\omega$) in this range, as far as ex- trinsic losses are unimportant. Incidentally, we remark that these authors and others [5, 6] seem to consider that the anharmonic phonon decay LR model (or the Her- ring process) and the Akhieser's relaxation damping are two physical mechanisms that independently contribute to sound attenuation, or correspond to two regimes as $\omega\theta_0 \gg 1$ and $\omega\theta_0 \ll 1$, respectively. It should be noted that Akhieser himself considered that the thermal phonons relax via collisions caused by the anharmonic in- teraction, and wrote the collision term of its Boltzmann equation exclusively in terms of three-phonon transition probabilities calculated according to the LR procedure; the relaxation time $\theta_0$ does not intervene in his calcula- tion. We emphasize again that, in our view, the LR and Akhieser methods are two alternative ways of analyzing sound attenuation, for whatever values of $\omega\theta_0$.

Under experimental conditions (50-100 GHz, 30- 130 K) for which $\omega\theta_0 > 90$ ($\theta_0 > 3 \times 10^{-10}$ s) Hao and Maris found an attenuation $\alpha_L$[100] independent of fre- quency in agreement with Eq. (24a) [7]. However, the fit of Eq. (24a) to their experimental data at 90, 110 and 130 K assumed $\gamma^{*2} = 12$ instead of an expected value $\gamma^{*2} \sim 1$. Measurements of $\alpha_L$[100] at 300 K from 30 to 90 GHz by Daly et al. [8] are in order of magnitude agreement with law (23a) using $\theta_0 = 9.0 \times 10^{-12}$ s and $\gamma^{*2} = 1$, as shown in Fig. 2a, while Herring's law (22) gives values of $\alpha_L$ at $f \sim 100$ GHz that are more than two orders of magnitude smaller; however, these results are hardly conclusive, because they ignore significant losses that occur when acoustic pulses are reflected at the sur- faces. For example, at 90 GHz and 100 K, where Hao and Maris measured $\alpha_L$[100] = 4 cm$^{-1}$ [7], Daly et al. found an attenuation of 60 cm$^{-1}$ [8], which is essentially and necessarily extrinsic.

In this context, it is worth noting that in the well- studied case of GaAs at 50 K [4–6] we arrive at the op- posite conclusion: $\alpha_L$[100] measured in the 200-400 GHz

<figure><img src="image_3.png" /><figcaption>FIG. 2. The attenuation of longitudinal waves along the [100] direction as a function of frequency. Solid circles are experi- mental data reported a) by Daly et al. [8] in Si at 300 K and b) by Legrand et al [4] in GaAs at 50 K, respectively. Solid lines are theoretical curves. The dashed line is a frequency square fit.</figcaption></figure>

range ($\omega\theta_0 \gg 1$, Fig. 2b) by Legrand et al. [4] increases with $\omega^2$ from 7.5 to 30 cm$^{-1}$, in close agreement with Eq. (A12) using coupling coefficients calculated ab initio [6]. Whereas Eq. (24a) predicts a weak constant attenu- ation of 0.4 cm$^{-1}$.

In order to complete and possibly clarify this contro- versial matter, the question now arises which sound at- tenuation (or which phonon lifetime) can be inferred from the Brillouin line broadening in a transparent or moder- ately opaque crystal such as silicon.

## B. The Brillouin spectrum of silicon

Olsson et al. recorded the Brillouin spectra of sili- con at different temperatures using a multi-pass tandem Fabry-Perot interferometer and a 532 nm laser source [16]. The scattered light was collected in backscatter- ing in the [110] direction. The complex refractive in- dex of Si at room temperature and $\lambda_0 = 532$ nm is $\nu_1 + i\nu_2 = 4.14 + i3.31 \times 10^{-2}$ [32]. The velocity of longitudinal waves in the [110] direction, derived from
the density $\rho = 2329 \text{ kg/m}^3$ and tabulated elastic constants [9], $c_{11} = 166$, $c_{12} = 64$ and $c_{44} = 80 \text{ GPa}$, is $v = [(c_{11} + c_{12} + 2c_{44})/2\rho]^{1/2} = 9150 \text{ m/s}$.

At 298 K, the measured Brillouin lines are Lorentzians with full width at half maximum (FWHM) $\Delta f = 3.4 \text{ GHz}$, centered at $\pm f_B = \pm 142 \text{ GHz}$; the above values of $\lambda_0$, $\nu_1$ and $v$, are in exact agreement with Eq. (4), $f_B = 2\nu_1 v/\lambda_0$. Subtracting the opacity broadening $\Delta f_2 = (2\nu_2/\nu_1)f_B = 2.27 \text{ GHz}$, we obtain the phonon lifetime broadening $\Delta f_1 = 1.13 \text{ GHz}$. Note that a precise an fiable value of the extinction coefficient $\nu_2$ was required [32]; $\nu_2 = 0.05$ would imply that $\Delta f \approx \Delta f_2$ as in Ge [13] and GaAs [14].

Linewidths $\Delta f \sim 1 \text{ GHz}$ are commonly observed under similar conditions (backscattering, 300 K) in transparent cubic crystals, for example in NaCl [12] ($\lambda_0 = 488 \text{ nm}$, L[100], $f_B = 30 \text{ GHz}$, $\Delta f \approx 0.66 \text{ GHz}$), or in ZnSe [33] ($\lambda_0 = 476.5 \text{ nm}$, L[110], $f_B = 54 \text{ GHz}$, $\Delta f \approx 2 \text{ GHz}$). As generally stated in the literature [10, 11, 19] their FWHM $\Delta\omega$ should be equal to $1/\theta_{\ell_B}$. In the classical framework of Sec. III, this statement amounts to ignoring the random phase jumps of the scattered signal and substituting $e^{-\Gamma_{\ell_B}|s|}$ for $e^{-\mu|s|}$ in the correlation function (17). The effect of light absorption is again a simple additive and independent broadening $\Delta f_1 = (2\nu_2/\nu_1)f_B$, while Eq. (19) is replaced by

$$ \Delta f_1 = \frac{\Gamma_{\ell_B}}{\pi} = \frac{1}{2\pi \theta_{\ell_B}} = \frac{\alpha_{\ell_B} v}{\pi} \rightsquigarrow \alpha_{\ell_B} = \frac{\pi \Delta f_1}{v}. \quad (25) $$

Accordingly, the component $\Delta f_1$ of a linewidth is assumed to measure the reciprocal of the relaxation time, then indirectly the intrinsic attenuation $\alpha_{\ell_B}$ at frequency $f_B$. In particular, the Brillouin spectrum of Si at room temperature reported by Olsson et al. should yield

$$ \alpha_L[110](142 \text{ GHz}) = \frac{\pi \times 1.13 \times 10^9}{9150} = 3.88 \times 10^5 \text{ m}^{-1} \\ = 3880 \text{ cm}^{-1}. \quad (26) $$

Such an attenuation is considerably larger than that calculated in one way [6] or another [15] from the LR equation (A12), and even much larger than that predicted by either equation (23a, b). Bonnet et al. [12] used Brillouin spectroscopy and Eq. (25) to study the temperature dependence of $\alpha_L$ in NaCl and KCl crystals; they found equally strong attenuations, e.g. in NaCl at 300 K and 30 GHz, $\alpha_L[100] \approx 4400 \text{ cm}^{-1}$ and $\alpha_L[110] \approx 2250 \text{ cm}^{-1}$. There were no direct measurements of $\alpha_L$ in NaCl in the GHz range for comparison, whereas for silicon, we can compare (26) with experimental data from Daly et al. at 300 K [8]; at 90 GHz they measured $\alpha_L[100] = 195 \text{ cm}^{-1}$ (Fig. 2a), an attenuation that is believed to be largely extrinsic. On the other hand, assuming that $\alpha_L$ varies at most as $\omega^2$, we expect from (26) that at 90 GHz $\alpha_L[110] \ge 1560 \text{ cm}^{-1}$, and presumably $\alpha_L[100] \gtrsim \alpha_L[110]$. Irrespective of the extrinsic or intrinsic sources of losses, the result obtained by Daly et al. means that after two round trips through a Si

wafer of thickness $d = 56 \text{ µm}$, the amplitude of the acoustic pulse, or at least its Fourier component at frequency 90 GHz, was reduced by a factor $e^{-4\alpha d} \sim 10^{-2}$. This experimental fact alone excludes the existence of an intrinsic attenuation of, say, only $1000 \text{ cm}^{-1}$ at 90 GHz, for which one would have $e^{-4\alpha d} \sim 10^{-10}$ and thus no perceptible echoes. Since we do not question the well-founded relation between $\alpha_{\ell}$ and the relaxation time $\theta_{\ell}$, we must conclude that the conventional equation $\Delta\omega = 1/\theta_{\ell_B}$ is refuted by experiment.

In Sec. II we have argued that the proper phonon lifetime to express the quantum uncertainty in the energy of the scattered photon in a one-phonon process is the time $\tau_{\ell}$ defined in (7) and calculated for the mode $\pm \ell_B$ of the emitted or absorbed phonon. The corresponding line broadening should be $\Delta\omega = 2\pi\Delta f_1 \sim 1/\tau_{\ell_B}$. In room temperature Brillouin backscattering experiments such as that of Olsson et al., $x = hf_B/k_B T \ll 1$, we simply have $\tau_{\ell_B} = \theta_{\ell_B} x^2$ and

$$ \Delta f_1 \sim \frac{1}{2\pi \tau_{\ell_B}} = \frac{1}{2\pi \theta_{\ell_B} x^2} \rightsquigarrow \alpha_{\ell_B} \sim \frac{\pi \Delta f_1}{v} x^2. \quad (27) $$

Hence, the real intrinsic attenuation $\alpha_L[110]$ in Si at 300 K and $f_B = 142 \text{ GHz}$ ($x = 2.27 \times 10^{-2}$) should be $\sim 3880 \text{ cm}^{-2}$ instead of (26). The classical interpretation of Sec. III, according to Eq. (19), gives $\alpha_L[110] = \frac{\pi \Delta f_1 x^2}{v} = 0.5 \text{ cm}^{-1}$.

The remarkable result is that these values of $\alpha_L$ now agree with the predictions of the LR equation (A12). For example, Eq. (22) at 142 GHz gives $\alpha_L[110] \lesssim \alpha_L[100] = 1.35 \text{ cm}^{-1}$. On the other hand, the inset of Fig. 6b in Ref. [6] shows the calculated frequency dependence of $\alpha_L[100]$ in Si at 300 K up to 700 GHz, where $\alpha \approx 100 \text{ cm}^{-1}$; this curve (black line) summarily assimilated to an $\omega^2$ law gives $\alpha_L[100] = 4 \text{ cm}^{-1}$ at 142 GHz.

V. CONCLUSION

Several authors [6, 34] have emphasized the importance in optoacoustics (phonon imaging, optomechanical microcavities) of having a full understanding of the attenuation $\alpha$ of longitudinal acoustic waves in the 100 GHz range, which is just that of the Brillouin frequency shift $f_B$. We have argued in this context that the analysis of the Brillouin spectrum in transparent or moderately opaque crystals provides, if not an exact measurement, at least the correct order of magnitude of the intrinsic attenuation $\alpha_{\ell_B}$, provided that the linewidths are related to the phonon lifetime $\tau$ of the mode $\ell_B$ as defined in Sec. II, and not to the much larger relaxation time $\theta_{\ell_B}$. Thus, we found that in Si at 300 K and 100 GHz, $\alpha_L[100] \sim 1 \text{ cm}^{-1}$, in agreement with LR theory (cf. Eq. (22)). This result and the discussions in Secs. II and IV A have led us to believe that the LR expression (A12) of a relaxation time $\theta_{\ell}$ holds without any restrictive condition or the need to devise another damping mechanism
and to introduce a roughly defined average relaxation time $\theta_0$ of the thermal phonons.

It must be noted, however, that the observations of Daly et al. [34] on a demonstrative imaging experiment are inconsistent with our above estimate of $\alpha_L$, and thus with our interpretation of the Brillouin linewidths. In this experiment, generated longitudinal phonon pulses diffract and propagate in a [111] Si substrate a distance ~ 0.5 mm to reach an Al film used as a transducer; since they have a peak frequency and bandwidth ~ 100 GHz, their absorption lengths $1/\alpha_\ell \sim 1/\alpha_L$ [111] at 100 GHz must be comparable to the 0.5 mm diffraction length. Now the authors claim that this condition is far from being satisfied at room temperature, forcing them to keep the substrate at a temperature below 20 K to minimize phonon-phonon interactions. This report contradicts our conclusions. For even if $\alpha_L$ [111] exceeds $\alpha_L$ [100] by a factor of at most 3, we found that $\alpha_L$ [111] at 100 GHz and 300 K $\lesssim$ 3 cm$^{-1}$, and hence absorption lengths $1/\alpha_\ell \gtrsim$ 3 mm.

To confirm (or refute) our argument, further numerical calculations of $\tau_\ell$, or equivalently of $\theta_\ell$, and/or direct measurements of $\alpha_\ell$ of strictly longitudinal modes (e.g. [100]) could be performed at room temperature in the restricted low-frequency range where the Herring process is expected to dominate phonon-phonon interactions and where, moreover, the limiting law $\alpha_\ell \propto \omega^2$ holds. Then, by observing how the broadening of the Brillouin lines varies with the wavelength $\lambda_0$ of the light source in a [100] backscattering experiment at 300 K, the relation between $\Delta f_1$ and $\tau_\ell$ could be confirmed and quantified. More explicitly:

i) In silicon. According to the Tamura-Maris model [15], $\alpha_\ell$ and $1/\theta_\ell$ vary as $\omega^2$ up to 300 GHz at 300 K; in this range, $x^2 \ll 1$, and $\tau_\ell = \theta_\ell x^2$ is constant. A rather exact value of $\tau_\ell$ can be calculated from the simplified equation (7), $1/\tau_\ell = \sum_{k'} A P(s' = 1, s'' = 2)$, using phonon dispersion relations and coupling coefficients calculated ab initio as in Ref. [6], but with a much finer mesh of the Brillouin zone.

Then, for example, using a Kr+ laser as the light source in a [100] backscattering experiment, $\Delta f_1$ could be measured from the 531 nm green line to the 752 nm near-infrared line, $f_B = 2\nu_1 v/\lambda_0$ decreasing from 131 GHz to 83 GHz, while $\Delta f_2$ decreases from 2.1 to 0.34 GHz. We expect both the calculated $\tau_{\ell_B}$ and the experimental $\Delta f_1$ to remain constant, while the accepted FWHM $\Delta f_1 = 1/2\pi\theta_{\ell_B}$ would decrease by a factor of 2.5.

ii) In a transparent crystal such as zinc selenide. Because of its small extinction coefficient ($\nu_2 < 3.5 \times 10^{-6}$ beyond 530 nm), the same study as for Si of the $\lambda_0$ dependence of the Brillouin linewidths can ignore any opacity broadening. From 531 to 752 nm, $f_B$ decreases from 42 to 28 GHz, $\Delta f_1 \sim 1$ GHz and $\Delta f_2 < 0.1$ MHz.

On the other hand, $\tau_\ell$ can be derived from direct accurate attenuation measurements ($\tau_\ell = x^2/2\alpha_\ell v$). As shown in Fig. 2b, measurements of $\alpha_\ell$ [100] in GaAs at 50 K by Legrand et al. [4] remarkably exhibit the expected $\omega^2$ law, which is a proof of their accuracy. These authors used GaAs/AlAs multilayers as emitters or detectors of longitudinal acoustic waves. This procedure avoids reflection losses and can be applied to ZnSe, which has the same zincblende structure as GaAs and a neighbor lattice constant ($a = 5.67$ and $5.65$ Å, respectively), so that epitaxial ZnSe/GaAs multilayers can be grown on a ZnSe substrate [35].

## Appendix A

In this appendix P(2.x) or Pp.x refers to an equation or a page in Peierls' book [3] (since 1955). It would have been a simple task to derive from Eq. P(2.69) the expression for the relaxation time $\theta_\ell$ of a phonon mode in its accepted present form, as written below as the last equation (A12). But Peierls was primarily concerned with establishing a Boltzmann equation to describe heat conduction. And, in this respect, he said (p.52) that he was not convinced of the usefulness of the concept of relaxation time first introduced by Klemens in 1951 [1].

The cubic part of the anharmonic Hamiltonian can be written as

$$W = \sum_{\ell_1} \sum_{\ell_2} \sum_{\ell_3} V(\ell_1, \ell_2, \ell_3) C_{\ell_1} C_{\ell_2} C_{\ell_3}, \quad (A1)$$

where $C_\ell = a_\ell^\dagger + a_\ell$ are the phonon field operators introduced by Maradudin and Fein [10]. In a crystal of volume V, $N$ unit cells and $N = N r$ atoms, there are $N$ allowed wave vectors $k_i$ uniformly distributed over the Brillouin zone (BZ) with density $V/8\pi^3$, then $3r$ branches $s$, and $3N$ modes $\ell(\mathbf{k}, s)$.

The coupling coefficients $V(\ell_1, \ell_2, \ell_3)$, the analogous coefficients $b(\ell_1, \ell_2, \ell_3)$ of Peierls (Pp.36), and the sometimes used $\Phi(\ell_1, \ell_2, \ell_3)$ [10, 15] defined by Born and Huang [36] are related by

$$ \begin{aligned} b(\ell_1, \ell_2, \ell_3) &= 6 \left(\frac{2M}{\hbar}\right)^{3/2} (\omega_{\ell_1} \omega_{\ell_2} \omega_{\ell_3})^{1/2} V(\ell_1, \ell_2, \ell_3) \\ &= \frac{M^{3/2}}{\sqrt{N}} \Phi(\ell_1, \ell_2, \ell_3), \end{aligned} \quad (A2) $$

where $M$ is the mass of the crystal. $V(\ell_1, \ell_2, \ell_3) = 0$ unless

$$\mathbf{k}_1 + \mathbf{k}_2 + \mathbf{k}_3 = \mathbf{G}, \quad (A3)$$

a vector of the reciprocal lattice (cf. P(2.42)). Given $\mathbf{k}_1$ and $\mathbf{k}_2$, $\mathbf{k}_3$ is the only vector within the BZ that satisfies the condition (A3). The $N$ vectors $\mathbf{k}_i$, or the $3N$ modes, can be paired, as $\ell = (\mathbf{k}, s)$ and $-\ell =$
<figure><img src="image_4.png" /><figcaption>FIG. 3. The four processes likely to cause a change $\Delta n_{\ell} = \pm 1$ in the number of phonons of the sound mode $\ell(\mathbf{k}, s)$. Given two branches $s'$ and $s''$ and a wave vector $\mathbf{k}'$ of the BZ, the vector $\mathbf{k}''$ and thus the processes $\mathbf{a}_-$ and $\mathbf{a}_+$ are well defined by the first equation (A4a). For any $\mathbf{k}'$, however, the second equation (A4a), i.e. the conservation of energy, is not satisfied. Then both processes are either forbidden or only possible for $\mathbf{k}'$ ending on a surface of the BZ. Idem for the processes $\mathbf{b}_-$ and $\mathbf{b}_+$.</figcaption></figure>

($-\mathbf{k}, s$), so that $\sum_i F(-\mathbf{k}_i) = \sum_i F(\mathbf{k}_i)$, or $\sum_{\ell} F(-\ell) = \sum_{\ell} F(\ell)$. For example, knowing that $V^*(\ell_1, \ell_2, \ell_3) = V(-\ell_1, -\ell_2, -\ell_3)$ (cf. P(1.36) and (2.40)), and $C_{-\ell} = C_{\ell}^{\dagger}$, it follows that $W^{\dagger} = W$ is hermitian, as it should be.

As mentioned in the Introduction, a proper description of the effects of weak anharmonicity relies on the study of the transitions that can be induced by the perturbation $W$ between the stationary states of the unperturbed system. When acting on an initial harmonic eigenstate $|i\rangle$ characterized by a definite set of $3N$ phonon occupation numbers, $W$ can produce a state $|f\rangle$ in which all occupation numbers are unchanged, except for three for which $\Delta n = \pm 1$. It should also be noted that the energy conservation prevents the $\Delta n$'s from having the same sign.

Consequently, four kinds of possible transitions $|i\rangle \rightarrow |f\rangle$, which we label $\mathbf{a}_-$, $\mathbf{a}_+$, $\mathbf{b}_-$ and $\mathbf{b}_+$, can reduce or increase the occupation number $n_{\ell}$ of a given mode $\ell$ by only $\pm 1$. As shown schematically in Fig. 3, a transition $\mathbf{a}_-$ can be viewed as a process in which a phonon $\ell$ “collides” with a phonon $\ell'$ to form a single phonon $\ell''$. In a process $\mathbf{b}_-$, a phonon $\ell$ “decays” into two others. The processes $\mathbf{a}_+$ and $\mathbf{b}_+$ are the inverse ones. Given a mode $\ell(\mathbf{k}, s)$, the foreseen final states $|f\rangle$ resulting from

one or another of these three-phonon processes can be indexed $(\ell', s'')$ or $(s', s'', \mathbf{k}')$, since a well-defined $\mathbf{k}''$ always follows from the conservation of the crystal momentum. But the conservation of energy is not necessarily satisfied for any $\mathbf{k}'$.

According to the first-order treatment of a small constant perturbation, the transition probability per unit time $w(\mathbf{a}_-)$ from some initial state $|i\rangle = |\dots, n_{\ell}, \dots, n_{\ell}', \dots, n_{\ell''}, \dots\rangle$ to any final state $|f\rangle = |\dots, n_{\ell-1}, \dots, n_{\ell}', \dots, n_{\ell''} + 1, \dots\rangle$ arising from a “collision” $\mathbf{a}_-$, is independent of time and can be written as (cf. P(2.60))

$$w(\mathbf{a}_-) = \sum_{s', s'', \mathbf{k}'_i} A(\ell, \ell', s'') n n'(n'' + 1),$$

where

$$A(\ell, \ell', s'') = \frac{72\pi}{\hbar^2} |V(\ell, \ell', -\ell'')|^2 \delta(\omega + \omega' - \omega''). \quad (A5)$$

For short, $\omega, \omega', \omega''$ stand for $\omega_{\ell}, \omega_{\ell'}, \omega_{\ell''}$ and $n, n', n''$ stand for $n_{\ell}, n_{\ell'}, n_{\ell''}$. Here, $\mathbf{k} + \mathbf{k}' - \mathbf{k}'' = \mathbf{G}$ determines the mode $-\ell''(-\mathbf{k}'', s'')$, and the general property (A3) ensures that $V(\ell, \ell', -\ell'')$ is nonzero. The $\delta$-function vanishes unless $\omega + \omega' - \omega'' = 0$, i.e. energy is conserved, and allows only a surface within the BZ on which the vector $\mathbf{k}'$ must end.

In the same way we can write

$$w(\mathbf{a}_+) = \sum_{s', s'', \mathbf{k}'_i} A(\ell, \ell', s'') (n+1)(n'+1)n'',$$

$$w(\mathbf{b}_-) = \sum_{s', s'', \mathbf{k}'_i} B(\ell, \ell', s'') n(n'+1)(n''' + 1),$$

$$w(\mathbf{b}_+) = \sum_{s', s'', \mathbf{k}'_i} B(\ell, \ell', s'') (n+1)n'n'''',$$

where

$$B(\ell, \ell', s'') = \frac{36\pi}{\hbar^2} |V(\ell, -\ell', -\ell'')|^2 \delta(\omega - \omega' - \omega'''). \quad (A6)$$

Here $\mathbf{k} - \mathbf{k}' - \mathbf{k}''' = \mathbf{G}$, $\ell''' = (\mathbf{k}'', s'')$. Same remarks as for $A$, except that $B$ includes a factor $1/2$. Otherwise, on summing $\sum_{\ell', s''}$ a given possible process $\ell \rightarrow \ell_1 + \ell_2$ (or $\ell_1 + \ell_2 \rightarrow \ell$) would be counted twice, as $(\ell', s'') = (\ell_1, s_2)$ and then as $(\ell', s'') = (\ell_2, s_1)$.

In quantum averaging,

$$\begin{aligned} \frac{dn_{\ell}}{dt} &= w(\mathbf{a}_-)(-1) + w(\mathbf{a}_+)(+1) + w(\mathbf{b}_-)(-1) + w(\mathbf{b}_+)(+1) \\ &= \sum_{s', s'', \mathbf{k}'_i} A[(n+1)(n'+1)n'' - n n'(n''+1)] + B[(n+1)n'n''' - n(n'+1)(n'''+1)]. \end{aligned} \quad (A7)$$

Equation (A7) measures the effect of phonon interactions on the occupation number of a mode $\ell$ (as well as
on the number $N_\ell$ of phonons in a localized wave packet centered on the mode $\ell$) when the crystal is in any state $|i\rangle$. We can verify that the statistical mean of this effect is zero at thermal equilibrium if we accept with Peierls (cf. pp. 46-49) that the occupation numbers on different modes are statistically independent random variables. Thus, the mean value of the products of three occupation numbers may be replaced by the products of their mean values. Recalling that on average and for each mode

$$ \bar{n} = n_0 = \frac{1}{e^x - 1} \quad (x = \frac{\hbar \omega}{k_B T}), \quad n_0 + 1 = \frac{e^x}{e^x - 1}, $$

it is easy to see that the two products in the first brackets of (A7) have the same mean, which we will refer to as P for further reference. Still dropping the indices $\ell$,

$$ \begin{aligned} P &= n_0 n'_0 (n''_0 + 1) = \frac{e^{x''}}{D} \\ &= (n_0 + 1) (n'_0 + 1) n''_0 = \frac{e^x e^{x'}}{D}, \end{aligned} \quad (A8) $$

where $D = (e^x - 1)(e^{x'} - 1)(e^{x''} - 1)$, and $x + x' = x''$ in the $a_\pm$ processes. Similarly, in the second brackets of (A7),

$$ \begin{aligned} Q &= n_0 (n'_0 + 1) (n'''_0 + 1) = \frac{e^{x'} e^{x''}}{E} \\ &= (n_0 + 1) n'_0 n'''_0 = \frac{e^x}{E}, \end{aligned} \quad (A9) $$

where $E = (e^x - 1)(e^{x'} - 1)(e^{x''} - 1)$, and $x = x' + x'''$ in the $b_\pm$ processes. Therefore, in statistical averaging, $\overline{dn_\ell/dt} = 0$ for each mode, as expected.

As we have pointed out, after substituting expressions (A5) and (A6) for A and B, (A7) coincides with the second term of Eq. P(2.69) [3]; just transform the coupling

coefficients according to P(2.49) and (A2), then replace $\int_{BZ} d^3k' V/8\pi^3 \dots$ by $\sum_{k'_i} \dots$. Now we come to the problem of ultrasound attenuation, which Peierls did not address. Curiously, Peierls was not interested in sound attenuation, and did not mention it in his list (p. 31) of important problems concerning anharmonicity.

Suppose that at some time $t$ the occupation number $n$ of a single mode $\ell$ is artificially raised to a high value $n = n_0 + n_1(t) \gg n_0$, very far from equilibrium. The relaxation of such a large deviation can illustrate the damping of a quasi-monochromatic sound wave train centered on the mode $\ell$. By substituting $n = n_0 + n_1(t)$ in Eq. (A7), while all other modes have their equilibrium occupation, $n' = n'_0$, $n'' = n''_0$ etc., we obtain the statistical mean of $dn_\ell/dt = dn_1/dt$ at time $t$. Taking into account the identities (A8) and (A9), the quantities between brackets are simplified and $n_1$ is factorized, so that

$$ \begin{aligned} \frac{\overline{dn_1}}{\overline{dt}}(t) = n_1(t) & \sum_{s', s'', \mathbf{k}'_i} A[(n'_0 + 1) n''_0 - n'_0 (n''_0 + 1)] \\ & + B[n'_0 n'''_0 - (n'_0 + 1) (n'''_0 + 1)], \end{aligned} \quad (A10) $$

whence, after simplification and by definition of $\theta_\ell$,

$$ \begin{aligned} \frac{1}{\theta_{\ell}} &= - \frac{1}{n_1(t)} \frac{\overline{dn_1}}{\overline{dt}}(t) \\ &= \sum_{s', s'', \mathbf{k}'_i} A(n'_0 - n''_0) + B(n'_0 + n'''_0 - 1). \end{aligned} \quad (A11) $$

Note that $A, B > 0$, and $n'_0 > n''_0$ ($x' < x + x' = x''$), so $\theta_\ell$ is a time-independent positive constant characterizing the mode $\ell$. As long as $n_1$ remains a macroscopic deviation, $n_1(t)$ can be viewed as a smoothed decreasing function of time, whose rate of change $dn_1/dt = -n_1/\theta_\ell$, so that $n_1(t) = n_1(0) e^{-t/\theta_\ell}$.

If A and B are written in extenso according to (A5) and (A6), then (A11) becomes:

$$ \frac{1}{\theta_{\ell}} = \frac{36\pi}{\hbar^2} \left\{ \sum_{s, s', \mathbf{k}'_i} |V(\ell, \ell', -\ell'')|^2 2(n'_0 - n''_0) \delta(\omega + \omega' - \omega'') + \sum_{s', s'', \mathbf{k}'_i} |V(\ell, -\ell', -\ell''')|^2 (n'_0 + n'''_0 - 1) \delta(\omega - \omega' - \omega''') \right\}. $$

Finally, we apply to the second $\sum$ the pairing rule given above, $\sum_i F(\mathbf{k}'_i) = \sum_i F(-\mathbf{k}'_i)$. If $\mathbf{k}'$ is changed into $-\mathbf{k}'$, then $(-\ell', \mathbf{k}''' = \mathbf{k} - \mathbf{k}' + \mathbf{G}, -\ell'''$, $n''', \omega''')$ respectively become $(\ell', \mathbf{k}'', -\ell'', n''', \omega'')$ while $n'_0$ and $\omega'$ remain unchanged. As a result, and by definition of the amplitude damping constant $\Gamma_\ell$,

$$ 2\Gamma_{\ell} = \frac{1}{\theta_{\ell}} = \frac{36\pi}{\hbar^2} \sum_{s', s'', \mathbf{k}'_i} |V(\ell, \ell', -\ell'')|^2 [2(n'_0 - n''_0) \delta(\omega + \omega' - \omega'') + (n'_0 + n''_0 + 1) \delta(\omega - \omega' - \omega'')], \quad (A12) $$

where $\mathbf{k}'' = \mathbf{k} + \mathbf{k}' + \mathbf{G}$. If $\ell(\mathbf{k}, s)$ is a sound mode,

and $v = v_s(\hat{\mathbf{k}})$ is the sound velocity, Eq. (A12) gives the
amplitude attenuation per unit time $\Gamma_{\ell} = \alpha_{\ell}v$ of the sound wave; it represents the complete expression of the Landau-Rumer theory of sound absorption [26], when the elastic anisotropy of the crystal and the dispersion of all

other modes are taken into account, and then all three-phonon processes $a_{\pm}$ and $b_{\pm}$ are considered a priori as possibly effective.

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