COLLECTIVE LOSSES IN SOLIDS*

D. Pines

I

This review considers the results achieved in the last several years in understanding the collective nature of the characteristic energy losses that are observed when fast electrons are scattered by thin solid films. We shall begin with what is meant by collective and by individual energy losses. We shall be concerned with portions of energy of the order of 10–25 ev, transferred by the incident electrons to the solid. Energy of this order, apart from certain special cases, is absorbed in a solid by valence electrons (i.e., electrons outside closed atomic shells). If each particle of the system is considered individually, which is permissible only when the Coulomb interaction between valence electrons is negligible, then the excitation spectrum of the valence electrons is described by a set of energy differences \(\hbar\omega_{n0}\) for an individual electron making a transition within one band or from one band to another. When a fast charged particle causes such transitions and, consequently, energy is transferred to an individual valence electron, we are dealing with individual energy losses of the particle.

However, in many cases the Coulomb interaction between valence electrons has a substantial influence on the excitation spectrum of the system. As a result of this interaction, valence electrons in a solid acquire the ability to perform collective oscillations of high frequency, which may differ considerably from most of the frequencies \(\omega_{n0}\) and depends approximately only on the charge, mass, and density of the electrons in the solid. When a fast charged particle excites such collective oscillations, energy is transferred simultaneously to a certain number of electrons moving coherently as a result of their interaction. We call the corresponding energy losses collective. The collective oscillations of valence electrons have much in common with the oscillations of an electron plasma observed in gas discharges. We shall introduce the term “plasmon” to denote the quantum of an elementary excitation corresponding to such high-frequency collective motion. In those cases where we may expect that the plasmon exists as a quite well-defined quantity, its energy will be close to

\[ \hbar\omega_p=\hbar\left(\frac{4\pi n e^2}{m}\right)^{\frac{1}{2}}, \]

* Reviews of Modern Physics 28, 184 (1956). Translated by D. G. Sannikov.

where \(n\) is the density of valence electrons and \(m\) is the mass of the free electron. The study of collective energy losses is then reduced to the study of plasmon excitations in solids.

It is much more difficult to justify the introduction of new elementary excitations in solid-state physics than in the physics of “strange particles.” In this review we shall try to bring together the available experimental and theoretical data indicating the existence of plasmons as quite definite entities in almost all solids. As we shall see, experiments on characteristic losses make it possible to determine the following parameters of plasmons:

1) the plasmon energy, obtained from the values of the observed losses;
2) the plasmon lifetime, obtained from the width of the line of the observed losses;
3) the cross section for plasmon formation by a fast charged particle, obtained from the dependence of the observed losses on the foil thickness and on the energy of the incident particle;
4) the plasmon dispersion (the dependence of the plasmon energy on wavelength), obtained from the dependence of the energy losses on the scattering angle;
5) the minimum wavelength and maximum energy beyond which the plasmon can no longer be regarded as a quite definite kind of excitation of the system, obtained from the value of the maximum scattering angle at which excitation of the plasmon is still observed.

The next two sections of the review are devoted to the theoretical apparatus needed for interpreting the available experimental data. In Section II we shall be concerned chiefly with the dispersion equation for plasmons in solids. In other words, we shall be interested in the possible plasmon energies in a given solid. This dispersion equation can be obtained, in a microscopic approach, from the equations of motion of the electrons or from a Hamiltonian formulation of the problem; it can also be obtained from a phenomenological macroscopic consideration of the solid in terms of an effective dielectric constant. We prefer the microscopic approach, since the use of a macroscopic dielectric permeability leads to certain ambiguities, for the resolution of which it is in any case necessary to resort to a microscopic treatment. In Section III we consider the mechanism of plasmon excitation by a fast charged particle, as well as methods for distinguishing plasmon and individual electron excitations.

In Sections IV and V we compare the theoretical predictions with experimental data on the behavior of plasmons in solids. In Section VI we draw some conclusions and consider possible directions for further research in this field.

II

Let an electron with momentum \(\mathbf{P}_0\) and coordinates \(\mathbf{R}_0\) interact with the valence electrons in a solid. We take the Hamiltonian of such a system in the form\(^*\)

\[ H=\sum_i \frac{p_i^2}{2m}+V(r_i)+2\pi e^2\sum_{i\ne j k}\frac{e^{ik\cdot(x_i-x_j)}}{k^2} +\frac{P_0^2}{2m}+\sum_{ik}\frac{4\pi e^2}{k^2}e^{ik\cdot(R_0-x_i)} . \tag{1} \]

The first term represents the kinetic energy of the electrons, the second the potential created by the ionic core, the third the Coulomb interaction—

\(^*\) We carry out the Fourier expansion in a cube of unit volume:

interaction between valence electrons. We start from the assumption that the influence of the ionic-core electrons on the valence electrons can be described with the aid of some potential. (In what follows we shall return to the consideration of those cases in which this assumption is inadmissible.) The last two terms in (1) give the kinetic energy of the incident particle and its Coulomb interaction with the valence electrons. The interaction of the incident particle with the valence electrons, as is easy to see, depends only on the fluctuation of the density of the valence electrons \(\rho_k\). Since

\[ \rho_k=\int dx\,\rho(x)e^{-ikx} =\int dx\sum_i\delta(x-x_i)e^{-ikx} =\sum_i e^{-ikx_i}, \tag{2} \]

the interaction between the external particle and the electron system can be written in the form

\[ H_{\mathrm{ext}}=-\sum_k \frac{4\pi e^2}{k^2}\rho_k e^{ikR_0}. \tag{3} \]

\(\rho_k\) describes fluctuations of the electron density \(\rho(x)\) about the mean value \(\rho_0=n\), and, by virtue of (3), represents the “natural” variable in the problem of energy losses. \(\rho_k\) also proves to be a natural variable in the study of the collective properties of a dense electron gas\(^{1,2}\). Such a study was first carried out for the model of a free electron gas, when \(V(r_i)\) is taken to be constant. This simplified model of a solid should be considered in more detail, since already in it some of the basic conditions of collective behavior are realized and, consequently, the conditions for exciting plasmons. It was shown in\(^{2}\) that \(\rho_k\) satisfies the following operator equation of motion:

\[ \frac{d^2\rho_k}{dt^2}+\omega_p^2\rho_k = -\sum_i\left(\frac{k\cdot p_i}{m}-\frac{\hbar k^2}{2m}\right)^2 e^{-ikx_i} -\frac{4\pi e^2}{m}\sum_{k'\ne k}\frac{k\cdot k'}{(k')^2}\rho_{k-k'}\rho_{k'}, \tag{4} \]

where \(\omega_p\) is the plasma frequency, equal to

\[ \omega_p=\left(\frac{4\pi n e^2}{m}\right)^{1/2}. \tag{5} \]

To the extent that the terms on the right-hand side of equation (4) may be neglected, the density fluctuations behave as oscillators with frequency \(\omega_p\), and one should expect that it is precisely the collective properties of the system that determine the spectrum of energy losses of the incident charged particle. These oscillations are analogous to the classical longitudinal plasma oscillations discovered in a gas discharge\(^{3}\). In the present case, since the frequencies are very large \((\hbar\omega_p \gg kT)\), it is necessary to take into account the quantum character of the spectrum of the oscillations. As the quantum of elementary excitation of the collective oscillations of the valence electrons we introduce the plasmon with energy \(\hbar\omega_p\). Since the density of valence electrons in solids varies in the range from \(\sim 10^{22}\) to \(\sim 10^{24}\), the range of variation of the plasmon energy \(\hbar\omega_p\) is \(4\)–\(30\) eV. The energy required to excite a plasmon is, of course, considerably greater than the thermal energy. Moreover, in any metal the plasmon energy turns out to be greater than the kinetic energy of an individual conduction electron. Consequently, excitation of plasmons can be observed only in those cases,

when energy is supplied to the system of valence electrons from outside, and in quantities exceeding \(\hbar\omega_p\), which is precisely what occurs when a fast charged particle passes through a solid.

The first term on the right-hand side of equation (4) corresponds to taking into account the influence of the kinetic energy of the electrons on the plasma oscillations. Following \(^{2}\), its role can be estimated by averaging over the electron momenta. We then obtain approximately

\[ \sum_i \left(\frac{\mathbf{k}\mathbf{p}_i}{m}-\frac{\hbar k^2}{2m}\right)^2 e^{-i\mathbf{k}\mathbf{x}_i} \simeq \left\{ \frac{k^2\langle p^2\rangle_{\mathrm{cp}}}{m^2} + \frac{\hbar^2 k^4}{4m^2} \right\}\rho_k . \tag{6} \]

This term becomes comparable in magnitude with the term \(\omega_p^2\rho_k\), due to the Coulomb interaction, for such values of \(k\) that

\[ k^2 \simeq k_c \simeq \frac{\omega_p^2}{\langle v^2\rangle_{\mathrm{cp}}} \simeq \frac{\omega_p^2}{v_0^2}, \tag{7} \]

where \(v_0\) is the electron velocity at the boundary of the Fermi distribution of the electron gas. In our model of a solid, \(k_c^{-1}\) is, in order of magnitude, equal to the mean distance between electrons. For values of \(k\) small in comparison with \(k_c\), the kinetic energy of the electrons will have little influence on the plasmons, and one may then expect collective behavior to appear in the system. On the other hand, for values of \(k\) large in comparison with \(k_c\), the system does not behave as a whole; the concept of a plasmon as an independent formation no longer corresponds to reality, and density fluctuations and elementary excitations are then realized in the form of individual excitations of the aggregate of separate electrons.

The second term on the right-hand side of equation (4) takes into account the influence of nonlinear interactions between density fluctuations on the equation of motion of \(\rho_k\). Since this term is nonlinear and \(\rho_k\) is diminished for long wavelengths owing to Coulomb correlations*), its magnitude will evidently be small. However, with our method of treatment it is difficult to study in detail the influence of this term on the change of \(\rho_k^2\). In \(^{2}\), where the whole problem is studied within the framework of the Hamiltonian formalism, it is shown that for all \(k<k_c\) the nonlinear term is always considerably smaller than the kinetic-energy term. Consequently, the criterion given in the preceding paragraph is sufficient in order to draw a distinction between the collective and individual behavior of the particles of a free electron gas.

The free-electron model is a rather good approximation for a solid, provided that we are dealing with a metal and are interested in transitions of conduction electrons only within a given band. If, however, we consider a nonmetal or the influence of transitions between different bands on the behavior of plasmons, it is necessary to take into account the influence on the electron motion of the term \(V(\mathbf{r}_i)\) in equation (1). This was first done by Mott \(^{4}\), who, using a semiclassical approximation, considered plasma oscillations as waves of polarization in a solid. We carried out a similar treatment \(^{5}\), linking it more closely with the relations given above.

* Thus, for long wavelengths \(\rho_k\) behaves as a set of oscillators if the nonlinear term is absent. In this approximation the mean square fluctuation of \(\rho_k\) is determined by the zero-point energy of the plasmons and corresponds to \((\rho_k^2)_{\mathrm{cp}}\sim(\hbar k^2/4m\omega_p)\rho n\). This should be compared with the value \(n^2\) for free electrons.

As was already noted, we are interested in the matrix elements \(\rho_k\) for transitions between different states of the system of valence electrons. We shall describe these states as eigenstates of the operator

\[ H_0=\sum_i \frac{p_i^2}{2m}+V(r_i). \]

Thus, for the system of valence electrons we have \(H_0\psi_n=E_n\psi_n\). Since the thermal energy is small in comparison with the energies of the transitions under consideration, it may be assumed that, before its interaction with a fast charged particle, the system of valence electrons was in its lowest state. Let us now calculate the matrix element of the transition \((\rho_k)_{n0}\) between the lowest state and all states \(n\) that differ from the lowest one by the momentum \(\hbar k\). Examining the equation of motion for \((\rho_k)_{n0}\), we obtain, by analogy with (4),

\[ \left[\frac{d^2\rho_k}{dt^2}+\omega_p^2\rho_k\right]_{n0} = -\omega_{n0}^2(\rho_k)_{n0} - \sum_{k'\ne k} \frac{4\pi e^2}{m(k')^2}\,\mathbf{k}\cdot\mathbf{k}'\,(\rho_k-k'\rho_{k'})_{n0}, \tag{8} \]

where \(\omega_{n0}=(E_n-E_0)/\hbar\) is the frequency difference between states \(0\) and \(n\) for the system of valence electrons. Thus, in the case of a metal, \(\omega_{n0}\) corresponds to the change in energy of some electron that has made a transition within the same band or from the conduction band into a higher band; and in the case of a semiconductor or an insulator, to a transition from the valence band to the conduction band or to one lying still higher. Let us again suppose that the nonlinear term on the right-hand side of equation (8) may be neglected. Then the criterion for collective behavior will be the fulfillment of the inequality \(\omega_p^2\gg \omega_{n0}^2\) for the most significant matrix elements \((\rho_k)_{n0}\). Under these conditions the Coulomb interaction between electrons, expressed by the quantity \(\omega_p^2\), will prevail over the individual behavior of the electrons, represented by the quantity \(\omega_{n0}^2\). Therefore, perhaps, there is nothing surprising in the fact that \(\rho_k\) will oscillate with a frequency close to \(\omega_p\); in other words, under the given conditions the forces acting on an electron from the other electrons are much more substantial than the forces arising from the periodic field of the ions. In a metal, for nearly free conduction electrons, the criterion \(k\ll k_c\) is a special case of the criterion \(\omega_{n0}\ll\omega_p\), since the model of free electrons considered by us describes well the influence of transitions within the conduction band on the collective behavior of electrons.

The influence of the individual motion of a particle on collective behavior can be understood still better if one decomposes \(\rho_k\) into collective and individual parts, as was done in \({}^{2}\). For a gas of free electrons this can be done by finding an operator which, in the linear approximation, has a purely oscillator equation of motion. Then for the oscillator part of \(\rho_k\) one obtains the expression

\[ q_k=\sum_i \frac{\omega_p^2}{\omega^2-\left[(\mathbf{k}\cdot\mathbf{p}_i/m)-(\hbar k^2/2m)\right]^2} e^{-i\mathbf{k}\cdot\mathbf{x}_i}, \tag{9} \]

and the corresponding dispersion equation takes the form

\[ 1=\frac{4\pi e^2}{m}\sum_i \frac{1}{\left[\omega-(\mathbf{k}\cdot\mathbf{p}_i/m)\right]^2-(\hbar^2 k^4/4m^2)}. \tag{10} \]

For long waves it may be written in the following form:

\[ \omega^2 \simeq \omega_p^2 + k^2 \langle v^2\rangle_{\mathrm{cp}} + (\hbar^2 k^4/4m^2). \tag{11} \]

The correction terms in (10) take into account the change in the dispersion equation of plasmons caused by the coupling between free electrons and free plasmons; this coupling is represented by the first term on the right-hand side of equation (4). From (9) and (11) it is seen that for long waves \((k \ll k_c)\) we have \(\rho_k \simeq q_k\), and, consequently, the density fluctuations are, by their nature, almost completely collective. On the other hand, if \(k \gg k_c\), then \(q_k \ll \rho_k\), and \(\rho_k\) describes an aggregate of individual particles.

An analogous procedure for the case of valence electrons in solids was applied in \(^{5}\). The collective component of the matrix element \((\rho_k)_{n0}\) in this case is equal to

\[ (q_k)_{n0}=-\frac{\omega_p^2}{\omega^2-\omega_{n0}^2}(\rho_k)_{n0}, \tag{12} \]

and the dispersion equation takes the form

\[ 1=\frac{4\pi e^2}{\hbar k^2}\sum_n \frac{2\omega_{n0}|(\rho_k)_{n0}|^2}{\omega^2-\omega_{n0}^2}. \tag{13} \]

The dispersion equation (13) may be written in the form

\[ 1=\frac{4\pi e^2}{m}\sum_n \frac{f_{n0}}{\omega^2-\omega_{n0}^2}, \tag{14} \]

where \(f_{n0}\) is the generalized oscillator strength for the transition of our electronic system from the zero state to the excited state \(n\); it is equal to

\[ f_{n0}=\frac{2m}{\hbar k^2}\,\omega_{n0}|(\rho_k)_{n0}|^2. \tag{15} \]

If it is assumed that the zero and excited states are represented by Slater determinants constructed from one-electron wave functions \(\varphi_k(x_l)\), then the relations written above take the form

\[ 1=\frac{4\pi e^2}{m}\sum_{kK}\frac{f_{kK}}{\omega^2-\omega_{kK}^2}, \tag{16} \]

where \(f_{kK}\) in the case of long waves is equal to

\[ f_{kK}=\frac{2m}{3\hbar}\,\omega_{kK} \left|\int dx\,\varphi^*_{k+K}(x)\,x\varphi_k(x)\right|^2. \tag{17} \]

\(f_{kK}\) is the oscillator strength for the transition of one electron from the state \(k\) to the state \(k+K\), where \(K\) is a reciprocal-lattice vector, and \(\omega_{kK}\) is the corresponding frequency difference for one electron. For \(K=0\) (transitions within a band) we have

\[ f_{k0}=\frac{m}{\hbar^2}\frac{\partial^2 E(k)}{\partial k^2}=\frac{m}{m^*}, \tag{18} \]

where \(E(k)\) is the energy of one electron, and \(m^*\) is defined by (18). The dispersion equation in the form (16) is identical with the equation that was derived by Mott \(^{4}\).

The main question that arises in deriving relations (14) and (16) is that it is difficult to estimate the validity of the linear approximation. It is unclear, for example, whether in some cases it is necessary to take into account the Lorentz polarization correction (which can arise only from the terms that have been discarded). It is also difficult to allow for the damping of plasma oscillations caused by individual electrons (it is assumed to be small). For these reasons it is desirable to consider the dispersion and absorption of plasmons within the framework of the Hamiltonian formalism, where these questions can be investigated.

Such a consideration for the case of free electrons was carried out in ^2. A generalization to the case of valence electrons in solids was first given by Kanazawa ^6 and independently by Adams ^7. We shall briefly discuss an analogous treatment carried out in ^5. The basic Hamiltonian for the system of valence electrons (1) can be rewritten in the following form:

\[ H=\sum_i \frac{p_i^2}{2m}+V(\mathbf r_i)+ \sum_{i,j;\,k>k_c}\frac{2\pi e^2}{k^2}e^{i\mathbf k(\mathbf x_i-\mathbf x_j)}+ \]

\[ +\sum_{k<k_c}\frac{P_kP_k^*}{2} +\frac{\omega_p^2 Q_k^*Q_k}{2} +\sum_{k<k_c}\left(\frac{4\pi e^2}{k^2}\right)^{\frac12} \left(\frac{\mathbf k\cdot\mathbf p_i}{m}-\frac{\hbar k^2}{2m}\right) Q_k e^{i\mathbf k\cdot\mathbf x_i}+ \]

\[ +\sum_{\substack{i,\,k;\,l<k_c\\ k\ne -l}} \frac{2\pi e^2}{m}\, \frac{\mathbf k\cdot\mathbf l}{|k|\,|l|}\, Q_k Q_l e^{i(\mathbf k+\mathbf l)\mathbf x_i}. \tag{19} \]

This is possible if we impose a system of additional conditions on the wave function of the complete system:

\[ \left(P_k-i\left(\frac{4\pi e^2}{k^2}\right)^{\frac12}\rho_k^*\right)\psi=0 \quad (k<k_c). \tag{20} \]

In (19) and (20) we have introduced \(n'=k_c^3/6\pi^2\) degrees of freedom of the plasmon, expressing them in terms of the coordinates \(Q_k\) and momenta \(P_k\). The long-range part of the Coulomb interaction between the valence electrons is now described through quantities characterizing plasmons. The third term in (19) represents the short-range screened interaction of the valence electrons; the fourth and fifth terms are the field energy of plasmons of frequency \(\omega_p\), and the sixth is the linear interaction between electrons and plasmons. The last additional term gives the nonlinear interaction of electrons and plasmons.

The linear part of the interaction of electrons with plasmons gives rise to two effects: a change in the plasmon frequency and an effective interaction between electrons. In addition, it leads to absorption of plasmons by the electron system. The shift in the plasmon spectrum due to this interaction is investigated by means of a canonical transformation that makes it possible to eliminate the first order of the interaction. The desired transformations and the corresponding results can be obtained easily if one works in a mixed representation, in which the plasmon operators are taken in a representation in which

\[ H_0=\sum_i \frac{p_i^2}{2m}+V(\mathbf r_i) \]

is diagonal. One of the consequences of these transformations is that the plasmon variables no longer enter into the transformed additional conditions. Thus, we obtain a system of \(n'\) independentיז

of plasmons with maximum momentum \(k_c\) and an ensemble of \(3n\) electrons subject to the transformed supplementary conditions. The dispersion equation for the plasmons takes the form

\[ \omega^2=\omega_p^2+\frac{4\pi e^2}{\hbar k^2}\sum_n \frac{\left|\left[\sum_i\left(\frac{\mathbf{k}\mathbf{p}_i}{m}-\frac{\hbar k^2}{2m}\right)e^{i\mathbf{k}\mathbf{x}_i}\right]_{n0}\right|^2}{\omega^2-\omega_{n0}^2}\,\omega_{n0}, \tag{21} \]

where \(\omega_{n0}\) is defined above*.

If the lower and excited states of the valence electrons are described with the aid of Slater determinants, then we obtain the results of Kanazawa\(^6\). The equivalence of (21) and (14) can be established if one uses the identity

\[ \omega_{n0}(\rho_k)_{n0} = \left[ \sum_i\left(\frac{\mathbf{k}\mathbf{p}_i}{m}-\frac{\hbar k^2}{2m}\right)e^{i\mathbf{k}\mathbf{x}_i} \right]_{n0} \]

and the generalized sum rule

\[ \sum_n f_{n0}=n, \]

so that there is as yet nothing new here.

However, the Hamiltonian formulation of the problem makes it possible to investigate the terms that we neglected in deriving (14) or (21). Such terms are the nonlinear interaction of the electrons with the plasmons and the short-range part of the interaction between electrons. A detailed investigation shows that, when the nonlinear interaction between electrons and plasmons is taken into account, corrections enter the plasmon dispersion equation that are analogous to the Lorentz correction to the local polarization of the field. Moreover, when the linear interaction between electrons and plasmons can be regarded as a comparatively weak perturbation \((\omega_{n0}<\omega_p)\), it is not difficult to show that the nonlinear interaction, and consequently also the Lorentz correction, are negligibly small. This is not surprising, since use of the Lorentz correction means that the given electron can be regarded as localized in a given region of the crystal and, consequently, that its effective binding frequency \(\omega_{n0}\) is large in comparison with all other frequencies under study. If \(\omega_p<\omega_{n0}\), then such a treatment is of course inapplicable.

The short-range part of the interaction between electrons affects the plasmon spectrum, since the electrons are coupled to the plasmons by the term of the linear interaction and to each other by the term of the screened Coulomb interaction. If the linear interaction of the plasmons with the electrons is excluded, then an interaction plasmon—electron—electron will thereby automatically be introduced, which can cause absorption of plasmons and change their frequency. It was found that in the case \(\omega_{n0}\ll\omega_p\) this change of frequency is approximately the same as for a gas of free electrons and is equal to

\[ (\Delta\omega)^2\sim -\,k^2 E_{\mathrm{exch}}^{\mathrm{c.d.}}, \tag{22} \]

\[ \overline{\phantom{xxxxxxxxxxxxxxxx}} \]

* In deriving (21) we neglected the influence of plasmon damping on the dispersion equation, since this influence is unimportant for those cases in which excitation of plasmons is observed.

where \(E_{\mathrm{exch.}}^{\mathrm{s.r.}}\) is the exchange energy for the short-range part of the interaction

\[ \sum_{k>k_c}\frac{2\pi e^2}{k^2} e^{ik(r_i-r_j)}. \]

This shift is usually considerably smaller than the correction terms taken into account in (11).

There is a close connection between the optical properties of a solid and the influence of valence electrons on plasmons. This is so because the Hamiltonian describing the interaction between the transverse electromagnetic field and the valence electrons is basically of the same form as that part of (19) which describes the plasmon field and its interaction with the electrons. The difference is that the photon is a transverse wave and that for free photons the dispersion equation has the form

\[ \omega_0^2=c^2k^2+\omega_p^2. \]

Thus, as shown in \(^{5}\), the modified frequency \(\omega_{k\mu}\) in the presence of a photon of polarization \(\varepsilon_{k\mu}\) is equal to

\[ \omega_{k\mu}^{2} = c^{2}k^{2}+\omega_{p}^{2} + \frac{4\pi e^{2}}{m^{2}} \sum_{n} \frac{ 2 \left| \left[ \sum_i \varepsilon_{k\mu}\rho_i e^{ikx_i} \right]_{n0} \right|^{2} }{ \omega_{k\mu}^{2}-\omega_{n0}^{2} } \,\omega_{n0}. \tag{23} \]

In the limiting case of long waves, (23) takes the form

\[ \omega_{k\mu}^{2} = c^{2}k^{2}+\omega_{p}^{2} + \frac{4\pi e^{2}}{m} \sum_{kK} \frac{f_{kK}\omega_{kK}^{2}} {\omega_{k\mu}^{2}-\omega_{kK}^{2}}. \tag{24} \]

We obtain the connection with the usual optical constants if we write:

\[ \omega_{k\mu}^{2}\varepsilon(\omega_{k\mu})=c^{2}k^{2}, \tag{25} \]

where \(\varepsilon(\omega)\) is the dielectric permittivity for frequency \(\omega\). Then we obtain the well-known expression for \(\varepsilon(\omega)\):

\[ \varepsilon(\omega) = 1-\frac{4\pi e^{2}}{m} \sum_{kK} \frac{f_{kK}}{\omega^{2}-\omega_{kK}^{2}}. \tag{26} \]

Thus, knowledge of the zeros of \(\varepsilon(\omega)\) makes it possible, from optical data, to obtain the dispersion equation for plasmons. In reality this is feasible only in the case of alkali metals; the plasmon energies obtained in this way are considered in the next section. The fact that the dispersion equation for plasmons is equivalent to the condition \(\varepsilon(\omega)=0\) was apparently first noted by Mott \(^{4}\). The close connection between the optical and plasmon properties of a solid was emphasized by Hubbard \(^{8}\), Fröhlich and Pelzer \(^{9}\), and Fano \(^{10}\).

Up to now we have assumed that the influence of the electrons of the ionic core on the behavior of plasmons can be described by means of a potential \(V(\mathbf r)\). Such an assumption is not quite correct, since it does not allow one to take into account the polarization of the ionic core by the plasmon field. The influence of the ionic core on the dispersion equation of plasmons is easily taken into account in the semiclassical approximation of Mott, in which the ionic core

is considered as a system of oscillators with frequencies \(\omega_i\) and strengths \(f_i\). This leads to the equation

\[ 1=\frac{4\pi e^2}{m}\sum_{kK}\frac{f_{kK}}{\omega^2-\omega_{kK}^2} +\frac{4\pi e^2}{m}\sum_i\frac{f_i}{\omega^2-\omega_i^2}. \tag{27} \]

In \({}^{5}\), where a Hamiltonian formulation of the problem is given, the ionic core was considered on an equal footing with the valence electrons. It turned out that the method of treatment described above had to be modified somewhat. However, the final result—the dispersion equation for plasmons—coincides completely with (27).

The damping of plasma waves in a solid is connected mainly with two mechanisms. These are the short-range interaction between electrons and the linear interaction of plasmons with electrons. For a gas of free electrons only the first mechanism is possible, since in this case, in transitions caused by the linear interaction of plasmons with electrons, the conservation laws of energy and momentum are not satisfied. The lifetime of a plasmon, determined by collisions of electrons, was calculated in \({}^{5}\) and is approximately equal to

\[ \frac{1}{\tau_1}\simeq \frac{\hbar\omega}{(\hbar^2 k^2/2m)}\,\omega_p . \tag{28} \]

Such a dependence of the plasmon lifetime on its wavelength was to be expected. For long-wavelength plasmons the lifetime is extremely large. It decreases proportionally to the square of the wavelength.

In a real solid a much more effective absorption is produced by the linear interaction of plasmons with electrons. The lifetime determined by this mechanism (absorption of a plasmon by an electron making a transition from one band to another) was first calculated by Wolff \({}^{11}\). It can be expressed very simply in terms of the optical constants \(n\) and \(k\):

\[ \frac{1}{\tau_2}=nk\omega_p . \tag{29} \]

An analogous result was obtained by Kanazawa, and also in \({}^{5}\). Here too optical experiments do not give us the values of \(n\) and \(k\) in the required frequency region. However, since \(1/\tau_2\) is proportional to the oscillator strengths and to the density of final states for transitions with frequencies \(\omega_{0n}\) close to \(\omega_p\), whenever a large shift is observed in the value of the plasmon energy from its value for free electrons, one should also expect a correspondingly large broadening of the energy-loss lines.

III

Let us now consider the formation of plasmons by a fast charged particle. This question was first investigated in \({}^{1}\), where, in a semiclassical approximation based on the method of density fluctuations, the mean free path for the formation of one plasmon was calculated. The energy losses for plasmon formation were calculated by the method previously applied to the Cherenkov effect; these phenomena are very similar in that in both cases processes occur with conservation of energy and momentum, caused by the linear interaction of particles with the field. A quantum formulation of the problem can be found in \({}^{12}\). A macroscopic treatment, in which an effective dielectric constant is used, was given by Hubbard \({}^{8}\) and by Fröhlich and Pelzer \({}^{9}\). We shall present

here there is a somewhat different derivation of the cross section for plasmon formation, similar to that given by Ferrell\(^{13}\). This method makes it possible to investigate the angular distribution of the electrons that have excited plasmons, and gives results coinciding with those found in \(^{1}\) and \(^{12}\).

The interaction between a fast charged particle and the valence electrons is specified by expression (3). Let us consider the long-wavelength part of this interaction:

\[ \sum_{k<k_c}\frac{4\pi e^2}{k^2}\rho_k e^{i\mathbf{k}\mathbf{R}_0}. \tag{30} \]

As a result of the canonical transformation, which eliminates the interactions of plasmons with electrons [and leads to the dispersion equation for plasmons (21)], expression (30) takes the form*)

\[ \sum_{k<k_c} i\left(\frac{4\pi e^3}{k^2}\right)^{1/2} P_k e^{i\mathbf{k}\mathbf{R}_0}, \tag{31} \]

where \(P_k\) is the momentum of a plasmon with energy \(\hbar\omega\) and wave vector \(k\). If we now consider, in first-order perturbation theory, the effect of this interaction on a fast charged particle, then for the probability that the particle, per unit time, forms a plasmon with wave vector \(k\) and energy \(\hbar\omega\), and in so doing is scattered into the element of solid angle \(d\Omega\), we obtain the value

\[ w=\frac{d\Omega}{2\pi a_0}\frac{\omega P_0}{\hbar k^2}, \tag{32} \]

where \(a_0\) is the Bohr radius. In Fig. 1 the conditions imposed by the law of conservation of momentum are shown. It is easy to verify that

\[ \hbar^2 k^2=(\Delta P)^2+P_0^2\theta^2=P_0^2(\theta^2+\theta_E^2), \tag{33} \]

where **)

\[ \theta_E=\frac{\Delta P}{P_0}=\frac{1}{2}\frac{\hbar\omega}{E_0}\ll 1 \tag{34} \]

and \(E_0\) is the energy of the incident fast particle \((E_0\gg\hbar\omega)\).

Using (32) and (33), it is not difficult to show that the differential scattering cross section through an angle \(\theta\), calculated per one valence electron, is equal to:

\[ \sigma(\theta)d\Omega=\frac{d\Omega}{2\pi n a_0}\frac{\theta_E}{\theta^2+\theta_E^2}. \tag{35} \]

The maximum scattering angle is determined by the relation

\[ \theta_c\sim(\hbar k_c/P_0) \tag{36} \]

and under ordinary experimental conditions, when the scattered electrons

Fig. 1. Conservation law of energy and momentum for a fast electron exciting a plasmon.

*) We neglect the term of the screened interaction of the electrons; see equation (42) in \(^{12}\).

) \[ \frac{\Delta P}{P_0} = \frac{\Delta\sqrt{2ME}}{\sqrt{2ME_0}} = \frac{\Delta E}{2E_0} = \frac{\hbar\omega}{2E_0} \]
(\(P_0\) is the momentum of the incident particle, \(\Delta P\) is the change of momentum in the collision). (Translator’s note.)

with an energy of several keV, \(\theta_E \ll \theta_c \ll 1\). Integrating (35) over the solid angle, we obtain the cross section for plasmon formation

\[ \sigma \simeq \frac{\theta_E}{n a_0}\ln\frac{\theta_c}{\theta_E} = \frac{\hbar\omega}{2n a_0 E_0}\ln\frac{k_c P_0}{m\omega}. \tag{37} \]

It corresponds to the mean free path for plasmon formation \(\lambda\), equal to

\[ \lambda = 2a_0\left(\frac{E_0}{\hbar\omega}\right) \frac{1}{\ln(k_c P_0/m\omega)}. \tag{38} \]

The mean free path for excitation of a plasmon with energy \(15\) eV by an electron with energy \(10\) keV, for a value of \(k_c\) typical in metals, will be \(\sim 250\) Å. The maximum scattering angles \(\theta_c\) are, in order of magnitude, equal to several hundredths of a radian.

Fig. 2. Schematic of possible transitions with small momenta from one band to another. In order for a narrow loss line to result, the group of transitions marked by hatching must noticeably predominate over the other transitions.

Let us note further that the probability of simultaneous excitation of two plasmons by an incident fast charged particle is so small that, when calculating the scattering of fast electrons by thin films, this probability may be neglected. Only a repeated act of plasmon formation is possible. The probability that an electron will excite \(N\) plasmons while passing through a foil of thickness \(t\) is given by the Poisson distribution

\[ P_N(t)=\frac{1}{N!}\left(\frac{t}{\lambda}\right)^N e^{-t/\lambda}. \]

Before proceeding to the consideration of experimental data on plasmon formation, it is worth dwelling on the alternative possibility of an individual energy loss (i.e., a loss corresponding to the transition of an individual valence electron from one band to another). Let us consider a metal and suppose that the energy loss goes into excitation of an electron from the conduction band into a higher-lying band. If we restrict ourselves to comparatively low-energy transitions (say \(<15\) eV), then only transitions to the nearest band, which generally speaking has a substantially different form, may be taken into account. In the approximation of almost free electrons this is schematically shown in Fig. 2. It is clear from the figure that the energy-loss line will be narrow (\(\Gamma\) small in comparison with the magnitude of the loss \(\Delta E\)) only when the transition probability varies very strongly within the band and has a large value only for a comparatively small number of possible transitions between bands. Neglecting the Coulomb interaction of the electrons, one can easily calculate the probability of transfer of energy \(\Delta E=\hbar\omega_{n0}\) by a fast incident particle to one valence electron per unit time. As a result we obtain

\[ w=\frac{d\Omega}{2\pi a_0}\,\frac{4\pi e^2}{m}\, \frac{t_{n0}\hbar}{\Delta E}\, \frac{P_0}{\hbar k^2}. \tag{39} \]

If we are interested in the energy loss with width \(\Gamma\) for a given value of \(\Delta E\), then in expression (39) it is necessary to include all transitions between

bands that contribute to this energy region. Then we find

\[ w=\frac{d\Omega}{2\pi a_0}\left(\frac{\hbar\omega_p}{\Delta E}\right) \left(\frac{n_{\mathrm{cp}}}{n}f_{\mathrm{cp}}\right) \frac{P_0\omega_p}{\hbar k^2}, \tag{40} \]

where \(f_{\mathrm{cp}}\) is the average oscillator strength for the transition \(\Delta E\), and \(n_{\mathrm{cp}}/n\) is the relative number of electrons participating in the transitions for the given band.

From comparison of (40) and (32) it is seen that, for \(\hbar\omega_p \simeq \Delta E\), the ratio of the probability of energy transfer to one valence electron to the probability of plasmon excitation is equal to:

\[ \frac{n_{\mathrm{cp}}}{n}f_{\mathrm{cp}}. \]

\(f_{\mathrm{cp}}\), generally speaking, is somewhat less than unity, while \(n_{\mathrm{cp}}/n\), in order for the energy-loss line to be narrow, must be substantially less than unity. Consequently, for transitions with momenta smaller than \(k_c\), plasmon excitation predominates over one-electron transitions between bands. In \({}^{5}\) it is shown that this predominance becomes still sharper if the Coulomb correlation between valence electrons is taken into account. For \(\omega_{n0}<\omega_p\), the correlation reduces the part of the matrix element \((\rho_k)_{n0}\) associated with individual electrons by the factor \((\omega_{n0}/\omega_p)^2\). Expression (40) then takes the form

\[ w_{\mathrm{corr}}\sim \frac{d\Omega}{2\pi a_0} \left(\frac{\Delta E}{\hbar\omega_p}\right)^3 \left(\frac{n_{\mathrm{cp}}}{n}f_{\mathrm{cp}}\right) \frac{P_0\omega_p}{\hbar k^2}, \tag{41} \]

so that the cross section for transfer of energy \(\Delta E<\hbar\omega_p\) to an individual electron is smaller than the cross section for plasmon formation by the ratio

\[ \left(\frac{\Delta E}{\hbar\omega_p}\right)^3 \frac{n_{\mathrm{cp}}}{n}f_{\mathrm{cp}}. \tag{42} \]

This conclusion will not seem surprising if one recalls that the stopping power of valence electrons depends only on the sum rule and does not depend on the mechanism of energy transfer*). The effect of Coulomb correlation considerably increases the probability of transferring the energy \(\hbar\omega\) to the ensemble of valence electrons. In order that the sum rule for the stopping power be preserved, this correlation must at the same time reduce the probability of individual particle transitions. Expression (42) reflects simultaneously the increase in the energy of plasmons and the corresponding suppression of low-energy transitions from one band to another.

IV

Let us now turn to the experimental data that testify to the excitation of plasmons in solids**).

* ) Precisely because the stopping power is insensitive to the mechanism of stopping, we shall not concern ourselves here with its calculation. Historically, the plasmon approach as applied to the electron gas in metals was first used by Kronig and Korringa \({}^{14}\), who considered the influence of electron interactions on the stopping power of metals. The relative contribution of individual electrons and plasmons to the stopping power is discussed in \({}^{12}\).

**) The material of this section is borrowed to a considerable extent from \({}^{5}\) and from the paper by D. Pines, included in \({}^{15}\).

Already the first experiments in this area by Ruthemann\(^{16}\) and Lang\(^{17}\) showed that in the spectrum of characteristic losses in solid thin films two classes of lines predominate. For Be and Al these authors found several comparatively narrow lines, multiples of the principal loss quantum, approximately equal to 19 eV for Be and 15 eV for Al, while for Cu and Ag there is a single loss line, considerably broader than the lines for Be and Al. It is located near 20 eV for Cu and near 23 eV for Ag.

Let us compare these values with the magnitude of the plasmon quantum calculated on the assumption that the valence electrons are free. For Be and Al, with plasmon energies respectively equal to 19 eV and 16 eV, good agreement is obtained. On the other hand, for Cu and Ag the plasmon energies in the case of free electrons would be 11 eV and 9 eV. Thus, if we wish to explain both types of lines as the result of excitation of plasmons, it is necessary to clarify why in the case of Be and Al the valence electrons behave as free ones (insofar as this concerns the behavior of plasmons) and several loss lines are observed, whereas in the case of Cu and Ag the plasmon energy is considerably higher than follows from the calculation for free electrons, and only one loss line is observed. The first contradiction was resolved by Mott\(^{4}\), who showed that if the valence electrons are weakly bound, while the electrons of the ionic core are strongly bound (in comparison with the plasmon energy for free electrons \(\hbar\omega_p\)), then the fact that we have a solid, and not free electrons, will not have any particular effect on the plasmon energy. The answer to the second question was given by Herring (private communication) and Wolff\(^{11}\), who showed that the coupling of plasmons to the electrons of the Cu and Ag ionic core must lead to a significant broadening of the plasma resonance and to an increase in the plasmon energy.

With respect to the spectrum of characteristic energy losses, most solids, as we shall see, can be divided into two groups. In those cases where the loss line is comparatively narrow, it is located near the value of the plasmon energy calculated for free electrons, and several loss lines are usually observed. On the other hand, when the loss line is broad, usually only one line is observed, differing strongly from the value of the plasmon energy calculated for free electrons.

Let us consider the case of weakly bound valence electrons and strongly bound electrons of the ionic core, i.e., let us assume that \(\omega_{kK}^{2} \ll \omega^{2} \ll \omega_i^{2}\), for transitions \(\omega_{kK}, \omega_i\), for which the oscillator strengths have a noticeable value. In this case (27) may be written approximately in the form

\[ 1 \simeq \frac{4\pi e^2}{m}\sum_{kK}\frac{f_{kK}}{\omega^2} -\frac{4\pi e^2}{m}\sum_i\frac{f_i}{\omega_i^2}. \tag{43} \]

Using the sum rule \(\sum_{kK} f_{kK}=1\) and introducing the static dielectric constant of the ionic core

\[ \varepsilon_c = 1+\frac{4\pi e^2}{m}\sum_i\frac{f_i}{\omega_i^2}, \tag{44} \]

we obtain the plasmon dispersion equation in the form

\[ \omega^2 \simeq \omega_p^2/\varepsilon_c . \tag{45} \]

Under these conditions \(\varepsilon_c\) will be of order unity, since we have assumed-

assumed that the electrons of the ionic core are strongly bound. Since both Al and Be are metals in which the valence electrons are weakly bound, while the electrons of the ionic core are strongly bound, it is not surprising that the experimentally observed loss line agrees well with the plasmon energy calculated for free electrons. Such agreement prompts one to make a similar comparison for other solids as well.

In comparing our theoretical value of the plasma losses (45) with experiment, it must be kept in mind that for many solids the spectra of energy losses obtained by different experimenters exhibit considerable discrepancies. A review of the experimental material on the spectra of characteristic energy losses was given by Marton et al.^18 and was also discussed in detail by them at the conference on electron physics in Merilend.^19 Therefore we shall not consider here in detail the possible reasons for such discrepancies. Among these reasons one may point to the difficulties of obtaining clean films with thicknesses of several hundred Å, and to the possibility of damage and contamination of the films in the course of their bombardment by electrons.^20 In addition, the discrepancy in the number of observed plasma lines, according to (35) and (38), may be due to differences in the film thickness, in the energy of the bombarding electrons, and in the angular aperture of the spectral analyzer.

In Figs. 3 and 4 an incomplete summary is given of the values of characteristic energy losses obtained to date. In identifying the experimental values of plasma losses we shall be guided by the following criteria:

1) If a solid has been investigated by different experimenters, we shall take into account only those loss lines that were found by all experimenters.

2) Losses appear either as a whole series of equidistant lines or, when the experimental conditions did not permit observation of multiple losses, as one rather broad line.

3) In those cases where the relative intensity of different loss lines is given, the plasma losses are identified with the most distinct lines in the spectrum.

4) If there are several different values for a given loss line, then as the experimental value of the loss we choose the mean of these values. These criteria are to some extent suggested by the considerations of the preceding section, according to which the plasma lines should be the most noticeable in the loss spectrum.

Table I

Comparison of $\hbar\omega$ with $\Delta E_{\mathrm{obs}}$ for solids in which the valence electrons are weakly bound and the electrons of the ionic core are strongly bound $(\omega_{kk}^2 < \omega^2 < \omega_i^2)$.

$Z$ is the adopted number of valence electrons per atom participating in plasma oscillations

Table I compares the theoretical and experimental data for solids for which the conditions of weakly

Figs. 3 and 4. Summary of experimental data on characteristic energy losses (borrowed from 15, vol. I, p. 432). The energy losses are given in eV. The numbers correspond to the works of the following investigators: 1) Ruthemann¹⁶, 2) Lang¹⁷, 3) Möllenstedt²¹, 4) Marton and Leder²², 5) Klein²³, 6) Watanabe²⁴, 7) Gabor and Djall²⁰.

binding of the valence electrons and a strong binding of the electrons of the ionic core. \(\hbar \omega\) is the plasmon energy calculated from (45), and \(\Delta E_{\text{obs}}\) is the experimental value of the loss, which, according to the considerations set forth, should correspond to plasmon excitation. We retain only two significant figures in \(\hbar \omega\) and \(\Delta E_{\text{obs}}\), since neither of these quantities can be regarded as determined with high accuracy. The agreement obtained is quite satisfactory. Moreover, the discrepancy where it is largest, for B and C, occurs in a direction that can be understood. In these elements the valence electrons are comparatively strongly bound, so that interband transitions corresponding to a rather high energy should substantially affect the behavior of plasmons. If the energy of such transitions \(\hbar \omega_{gK}\) is greater than \(\hbar \omega_p\), then according to (16) one should expect a decrease of the plasmon energy, which, in all probability, is what occurs. The same thing may be expressed less precisely by saying that not all valence electrons are sufficiently free to take part in plasmon excitation; hence a decrease of the plasmon energy is observed.

One might have thought that the alkali metals also belong to this same group of elements with weakly bound valence electrons and strongly bound electrons of the ionic core. But here it is especially difficult to exclude the formation of an oxide film, and therefore one cannot rely entirely on the experimental data on characteristic energy losses[^22]. However, for the alkali metals, Budd’s optical investigations[^25] of the transition from reflection of light to transmission through a film give us the plasmon energy directly, since according to (26) such a transition must occur when the photon energy becomes equal to the plasmon energy1. In Table II we compare the optically determined plasmon energy2 with the theoretical value calculated by means of (45). For comparison, the value calculated for free electrons is given, as well as that one of the experimental values found by Marton and Leder which is closest to the optical value. Very good agreement is obtained between the value calculated by us, on the one hand, and the optical value, on the other. The slight discrepancy can easily be explained by low-energy interband transitions, whose inclusion leads to a certain increase of the calculated values.

Table II

Comparison of \(\hbar \omega\) with optical data and with observed values of energy losses for alkali metals. All energies are given in eV

For most solids in Table I one should expect a rather narrow line, whose width could in principle be calculated from (29), if the corresponding optical data were available. Since they are absent, we can undertake a qualitative discussion. With a strongly bound ionic core and weakly bound valence electrons, the number of transitions with

frequencies close to \(\omega\), is relatively small, and for such transitions the oscillator strengths are apparently small. Therefore one may expect that, for metals such as Al, the plasmon lifetime will be of the order of \((10/\omega_p)\sim 10^{-15}\) sec or even longer. Other elements with a relatively long lifetime may be Be, Mg, and Ge. For most of them the true line width apparently has not been measured, since the experimentally observed width does not exceed the width obtained for electrons that have, in general, suffered no energy loss in the film. On the other hand, the considerably broader lines for B and C (shorter plasmon lifetime \(\sim 10^{-16}\) sec) are explained if one admits that for these solids the bond of the valence electrons cannot be completely neglected. This seems quite plausible, since a frequency shift is also observed. It is surprising that such good agreement is obtained between \(\hbar\omega\) and \(\Delta E_{\mathrm{obs}}\) for Si, since for it too the loss line is broad.

Let us now turn to the second group of solids, which, according to our assumption, no longer possess weakly bound valence electrons and strongly bound electrons of the ionic residue. Let us assume that in transition metals both the \(s\)- and the \(d\)-electrons are valence electrons. Then transitions between bands can no longer be neglected, since as the total number of valence electrons increases the charge of the ionic residue increases, as a result of which some of the valence electrons prove to be strongly bound and can make interband transitions corresponding to high energy. This means that we are dealing with strongly bound valence electrons. Qualitatively this case can be analyzed in the following way. Consider the dispersion equation (16). If all \(s\)- and \(d\)-electrons are regarded as valence electrons, then transitions in the ionic residue may, of course, be neglected. Further, we see that interband transitions with energy smaller than the energy of the free-electron plasmon increase the plasmon energy, whereas transitions with larger energy decrease it. Thus, if the transition elements are considered in order of increasing valence, one should expect that the value of the plasmon energy calculated by us will at first lie below the experimental one, because low-frequency transitions between bands increase the plasmon energy in comparison with (45). With increasing valence, some of the valence electrons become so strongly bound that the oscillator strengths for high-frequency \((\omega_{kk'}\gg\omega_p)\) transitions between bands become significant. Such transitions lower \(\omega\) in comparison with its value following from formula (45). Consequently, one may suppose that there exists a range of valence values within which the influence of the lower-lying levels is canceled by the influence of the higher-lying levels, as a result of which we obtain a value very close to (45). For larger valences the values obtained with the aid of (45) will, of course, be higher than the experimental ones.

It is seen from Table III that this qualitative picture is in agreement with the experimental results. As before, here \(\hbar\omega\) is the quantity calculated with allowance for the polarization of the ionic residue, and \(\Delta E_{\mathrm{obs}}\) is the experimental value of the plasma loss, chosen on the basis of the criteria considered above. As we see, for Ti, which has four valence electrons, (45) gives a value of \(\hbar\omega\) lying substantially below the experimental one. Cr, with six valence electrons, apparently lies in the region for which the influence of the various transitions between bands mutually cancels. For the elements following Cr, transitions between bands definitely lower the plasmon energy as compared with ...

Table III

Comparison of \(\hbar\omega\) with \(\Delta E_{\text{obs}}\) for solids in which either the valence electrons are strongly bound, or the electrons of the ionic core are weakly bound.
\(Z\) is the accepted number of valence electrons per atom participating in plasma oscillations.

comparison with (45). An additional indication that, for six electrons outside a closed shell, low-frequency transitions between bands tend to neutralize the effect of high-frequency transitions follows from the values of the plasmon energies for Mo, Te, W, and Se, which, as Table III shows, are very close to the values calculated for free electrons.

There is no reason to expect that such a purely qualitative consideration will make it possible to draw definite conclusions about the lifetime of a plasmon. However, one may make an assumption which apparently does not contradict experiment. Namely, one may expect that in the case of six valence electrons the loss peaks will be especially broad, since here the damping of plasma oscillations is caused both by high-frequency and by low-frequency transitions between bands. At the same time, in the case of four or eight valence electrons, considerable damping is produced, respectively, either only by low-frequency transitions or only by high-frequency transitions. This conclusion is in agreement with the experiments of Marton and Leder, who showed that for Cr and Mn the lines are approximately twice as broad as for Ti and Co\(^*\), and is not consistent with the results of Watanabe\(^ {**}\).

Let us now consider the noble metals. Here we encounter the case of a weakly bound ionic core. For such metals it is possible

* It should be noted that our approach in the present case differs substantially from Wolff’s approach\(^{11}\). Wolff described the behavior of plasmons in transition metals by considering the interaction of the plasmon of the \(s\)-electrons with individual \(d\)-electrons. His conclusion that, in the case of transition metals, the loss lines broaden as the valence increases apparently contradicts experiment.

** Cited by Marton\(^{18}\).

two approaches. One of them was proposed by Wolff. If one takes the plasmon formed by the interaction of the \(s\)-electrons, for example in Cu, then its energy will be \(\sim 11\) eV. This energy, however, is large in comparison with the energy required to excite one \(d\)-electron of the ionic core into the \(s\)-band (\(\sim 4\) eV). Therefore it is necessary to take into account the influence of transitions between the ionic core and the valence region on the behavior of the plasmons. This rather strong interaction appreciably broadens the plasmon line and noticeably changes its position. Perturbation theory in this case does not give reliable results, but Wolff made a rough estimate which led to the correct order of magnitude for the broadening and the shift. By means of the same mechanism one can explain the large width and shift observed in Ag and Au.

On the other hand, since the energy required to excite electrons of the ionic core in these metals is very small, the electrons of the ionic core, with respect to their plasmon behavior, should more properly be regarded as valence electrons. For Cu one then obtains nine valence electrons and a plasmon energy \(\sim 36\) eV. One may expect that high-frequency transitions between bands substantially reduce this energy, which also explains the observed value of 20 eV. Thus the noble metals may be considered either as a case of a weakly bound ionic core or as a case of strongly bound valence electrons; the former, apparently, is in better agreement with numerical calculations.

In the divalent metals Zn and Cd the electrons of the ionic core likewise cannot be regarded as strongly bound. The plasmon energy of the \(s\)-electrons of Zn is 13 eV, while the energy required to excite an electron of the ionic core is apparently somewhat less than 10 eV. The same is true for Cd, where the plasmon energy of the \(s\)-electrons is \(\sim 11\) eV, and the excitation energy of an electron of the ionic core is about 10 eV. It is clear that these metals must be considered on the same basis as the noble metals; the coupling between the valence electrons and the ionic core then makes it possible to understand the origin of the large width and shift of the loss lines.

For the metals situated after Cd (In, Sn, Sb, and Te) the situation is much better. Although the electrons of the ionic core in In and Sn are in fact not strongly bound (excitation energies \(\sim 17\) and \(\sim 22\) eV), so that the ionic cores possess considerable polarizability (\(\epsilon_c \sim 1.35\) and 1.23, respectively), nevertheless the plasmon energies are very close to the values obtained from the dispersion equation (45). On the other hand, it is clear that for Te the electrons of the ionic core still play a substantial role in determining the plasmon energy. The good agreement for Sb and Te, Ta and W, and Pb and Bi may be the result of mutual cancellation of the effects of low-frequency and high-frequency transitions between bands, as was already pointed out above. It is necessary to note that, despite the good numerical agreement between \(\hbar\omega\) and \(\Delta E_{\mathrm{obs}}\), the width of the lines for these elements is definitely larger than for Be, Mg, Al, and Ge.

Let us now try to compare the theoretical and experimental data on energy losses in compounds. We shall calculate the plasmon energy neglecting the correction for the polarizability of the ionic core, which should give a very good approximation for the strongly bound ionic cores of the compounds considered by us. In calculating \(\hbar\omega\) we shall take the total number of valence electrons of the compound (i.e., for \(\mathrm{Al}_2\mathrm{O}_3\), six from Al and eighteen from O). In choosing \(\Delta E_{\mathrm{obs}}\) we are guided by the same considerations as in the case of monatomic solids. In Table IV, where \(\Delta E_{\mathrm{obs}}\) and \(\hbar\omega\) are compared, \(\hbar\omega\) has been calculated under the assumption that the coupling of the valence electrons is small in comparison with the plasmon energy \(\hbar\omega\).

Table IV

Comparison of $\hbar\omega$ with $\Delta E_{\text{obs}}$ for compounds. $Z$ is the accepted average number of valence electrons per atom that take part in plasma oscillations. Experimentalists: Marton and Leder (ML), Watanabe (W), and Möllenstedt (M).

As can be seen from the table, this assumption proves to be surprisingly successful. Almost complete agreement is observed between $\hbar\omega$ and $\Delta E_{\text{obs}}$. For the sulfides PbTe, PbSe, mica, BeO, MgO, Li$_2$CO$_3$, C$_2$(OH)$_2$, MoO$_3$, and SiO$_2$, these quantities essentially coincide. Consequently, for these substances the transitions between the valence band and the conduction band correspond to energies considerably smaller than the plasmon energies $\hbar\omega$. In Al$_2$O$_3$, TeO$_2$, and SnO$_2$ the covalent bond is apparently stronger than in the preceding compounds, which affects the plasmon energy in the expected direction. Here, as in the case of C, the strong valence bond gives rise to high-frequency transitions between bands and therefore reduces the plasmon energy in comparison with $\hbar\omega_p$. This effect in the compounds under consideration is, in order of magnitude, precisely what could have been expected from the shift observed for C.

The loss lines in KBr, KCl, and NaCl are rather difficult to identify, since for KBr and KCl many lines of approximately equal intensity are observed, while for NaCl the loss spectrum is affected by the collodion film used as a backing. However, the experimental loss lines cited by us appear to be the most intense, and, indeed, they repeat with decreasing intensity. It is interesting to note that the positions of these lines agree with the calculated plasmon energies.

V

In the preceding section experimental data were considered concerning the energy and lifetime of plasmons in solids. In this section we shall dwell on the results of the experimental study of the mechanism of interaction of an incident fast charged particle with an ensemble of plasmons. In particular, we shall be interested in the mean free path for plasmon formation and the angular distribution of the electrons that have transferred energy to plasmons.

Unfortunately, we do not have reliable data on the mean free path for plasmon excitation. From Lang’s data$^{17}$ on the dependence of plasmon excitation in Al on the foil thickness, one can derive, though with a very low degree of accuracy, a value of the mean

free path, which turns out to be somewhat less than 180 Å. This agrees with the theoretical value of 190 Å calculated from (38) for electrons with an energy of 7 keV, which Lang used. Recently, Blackstock, Ritchie, and Birkhoff^26 carried out a careful study of the excitation of plasmons in Al, Mg, and Cu by electrons with energies ranging from 20 keV to 100 keV. In Fig. 5 their results are reproduced for

Fig. 5. Energy spectrum of electrons with initial energies of 46 keV and 100 keV after passing through an Al film of thickness 15 mg/cm² (after Blackstock, Birkhoff, and Ritchie^26). The right-hand peak represents electrons that have experienced no energy loss. The other peaks, with intervals of \(\sim 15\) eV, correspond to plasmon excitation.

the excitation of plasmons in Al at 45 keV and 100 keV. The relative change of the various lines shows that the mean free path for plasmon excitation decreases with increasing energy of the incident electron, as is to be expected from (38). Blackstock et al. made a detailed comparison of formula (38) with experiment for different Al-film thicknesses and different incident-electron energies and obtained satisfactory agreement. In the case of Mg the situation proved less clear. The authors found several loss lines, multiples of \(\sim 10\) eV; the character of these lines changed with the energy of the incident electrons. However, different methods of determining the thickness of the Mg film gave quite different results. One of them leads to good agreement with (38). On the other hand, for Cu only one loss line at \(\sim 23\) eV was found, independent of the film thickness and of the energy of the incident particle. This line is rather broad, and therefore it is difficult to estimate \(\lambda\) accurately. The value they obtained for the mean free path lies, within the limits of experimental error, in good agreement with the theoretical value.

Marton, Simpson, and McGraw^27 investigated the angular distribution of electrons with an energy of 20 keV scattered by a thin gold film. The dependence they obtained of the intensity on angle for those electrons which had lost 24 eV was analyzed in detail by Ferrell^13 on the basis of formula (35). He showed that the experimental results agree with the assumption that losses of 24 eV correspond to plasmon excitation.

Watanabe^28 investigated the dependence of the energy loss on angle for scattering of 25-keV electrons by Be, Mg, Al, Ge, and graphite. His re-

results provide important information about the dispersion equation of the plasmon and the critical value of the wave vector \(k_c\), beginning from which the plasmon can no longer be regarded as an independent quantity.

If the free-electron model is applicable to the given solid, then the plasmon energy as a function of the wave vector may, according to (11), be written in the form

\[ \Delta E(k)=\hbar\omega_p\left(1+\frac{\hbar k^2}{m\omega_p}\alpha\right), \tag{46} \]

where

\[ \alpha=\frac{3}{5}\frac{E_0}{\hbar\omega_p}, \tag{47} \]

and it is assumed that \((\hbar k^2\alpha/m\omega_p)\) is considerably less than unity. Here \(E_0\) is the electron energy at the boundary of the Fermi distribution. In \(^5\) it is shown that expression (46) should be well satisfied for \(k \lesssim k_c\); \(\alpha\) usually differs from the value (47), calculated for free electrons, owing to the short-range interaction of the electrons and owing to transitions between bands.

Fig. 6. Loss spectrum for Al (after Watanabe \(^ {28}\)). Along the ordinate is plotted the angle at which the electron was scattered; along the abscissa, the energy of the scattered electron (the energy of electrons that did not undergo scattering in the film is taken as zero).

Fig. 7. Diagram of the dependence of the energy loss on the angle, obtained from Fig. 6 (after Watanabe \(^ {28}\)). Along the ordinate is plotted the scattering angle in radians; along the abscissa, the energy loss divided by twice the energy of the incident electron.

From (33), expressing the law of conservation of energy and momentum, it follows that \(\theta \sim \hbar k/P_0\), where \(\theta\) is the angle through which an electron with momentum \(P_0\) is scattered as a result of the excitation of a plasmon with momentum \(k\). Then from (46), for the relation between the angle through which the electron is scattered and the corresponding plasmon energy, one obtains the expression

\[ \Delta E=\hbar\omega_p+\frac{P_0^2}{m}\alpha\theta^2. \tag{48} \]

In Figs. 6 and 7 the results obtained by Watanabe for Al are reproduced. The lines \(B\) and \(B'\) correspond to losses of 15 eV and 30 eV and represent

electrons which excite, respectively, one or two plasmons. The dependence of the energy loss on angle for these electrons agrees with (48). On the other hand, the straight diffuse line \(D\) corresponds to a loss line at 23 eV. Lines similar to \(B\) were also found by Watanabe in Be (19 eV), Mg (105 eV), Ge (16.5 eV), and graphite (7.5 eV). Lines similar to \(D\) were found in MgO (11.4 eV), Ag (25 eV), and Au (25 eV).

Table V compares the experimental values of \(\alpha\) with the values calculated from (47) for free electrons. The comparison is made for electrons for which loss lines of type \(B\) were observed. For Be and Al the agreement is quite good; for Mg and Ge it is somewhat poorer. In a certain sense the agreement is more surprising than its absence, since, in addition to the kinetic energy of the free electrons, there is a whole series of other factors affecting the value of \(\alpha\). The loss lines for Be, Mg, Al, and Ge have already been identified by us as lines corresponding to the excitation of plasmons. The loss line at 7 eV in graphite clearly does not correspond to a plasmon excited by all the valence electrons \((\hbar\omega \sim 25\ \text{eV})\); it may correspond to plasma oscillations of the \(\pi\)-electrons in graphite (one electron per C atom), which, apparently, are capable of undergoing independent oscillations with a much lower frequency. If one assumes that these electrons are weakly bound, then \(\hbar\omega \sim 12\ \text{eV}\) is obtained, which agrees reasonably well with the observed 7 eV. (The agreement may be improved if the “static” polarization of the remaining valence electrons is taken into account.) It is therefore natural that in the case of graphite the value of \(\alpha\) for free electrons does not agree with the value of \(\alpha\) observed experimentally.

Table V

Comparison of experimental and theoretical data on the dispersion equation for plasmons

All \(D\)-lines are rather broad, as a result of which a reliable determination of the dependence of the energy loss on angle is hardly possible. The line at 23 eV observed for Al may correspond to the excitation of a plasmon in \(\mathrm{Al_2O_3}\); the lines at 25 eV for Ag and Au likewise do not contradict the assumption that excitation of plasmons takes place here. On the other hand, the 11.4 eV line for MgO apparently corresponds to an interband transition.

From the maximum value of the energy loss in cases of type \(B\), one can derive the maximum value of the wave vector \(k_c\), beginning with which the plasmon can no longer be regarded as a well-defined type of excitation of the system. An estimate of this quantity was given by us in another paper\({}^{15}\) on the basis of a variational determination of the minimum energy of the ground state of a gas of free electrons. The value obtained in this way is

\[ k_c \sim 0.353 r_s^{-1/2} k_0, \tag{49} \]

where \(k_0\) is the wave vector of an electron at the Fermi distribution boundary, and \(r_s\) is the mean distance between electrons, measured in units of the Bohr radius. In the case of Al one should expect that the maximum scattering angle \(\theta_c \sim \hbar k_c/P_0\) will be of the order of \(1.1 \cdot 10^{-2}\) radians for electrons with energy 25 keV, used by Watanabe. Watanabe obtained experimentally a somewhat larger value, equal to \(1.5\text{–}1.8 \cdot 10^{-2}\) radians. This over-

the value \(k_c\) can be explained on the basis of recent work by Ferrell and Quinn \({}^{29}\). They found that even a comparatively short-range interaction between electrons can lead to collective energy losses, so that the cutoff of the collective excitation is in fact not very sharp. It turns out that for values of \(k\) larger than \(k_c\), determined by relation (49), when the plasmon model is no longer applicable, there nevertheless still exist collective excitations with an energy close to the plasmon energy. Ferrell and Quinn found that the boundary of the region of collective excitations is effectively pushed back to a value that coincides, in order of magnitude, with that observed experimentally for Al.

VI

In conclusion, let us return once more to the question of whether all the energy losses considered should properly be interpreted as excitation of plasmons. In the case of the narrow loss lines observed for Be, Mg, Al, and Ge, no doubt arises in this respect. The energy-loss lines appear where we expect them according to the theory, and the dependence of the energy losses on the scattering angle is also in good agreement with theoretical predictions. In the alkali metals, the existence of plasmons is confirmed by the agreement between theory and optical experiments.

As for the conclusions drawn on the basis of Table III, the author is convinced that they may be regarded as equally reliable. The general course of the change in plasmon energy across the series of elements given in this table is in complete agreement with experiment. In a certain sense what is surprising is not the existence of a discrepancy, but such good agreement. And this, perhaps, applies to an even greater degree to the compounds considered in Table IV.

When we are dealing with a very broad loss line, as was the case for many of the elements considered above, the question naturally arises whether it is necessary, in order to explain it, to invoke a new type of elementary excitation. Indeed, in this case the large width of the line will simply mean that the excitation has an extremely short lifetime because there exists, in its immediate vicinity, a large number of possible electronic transitions between bands. Then, perhaps, this line would be better explained as a superposition of narrow loss lines associated with individual electrons, and, in more refined experiments, the fine structure of the loss lines would be revealed. (Of course, it is not excluded that such a fine structure may also result from the solution of a complicated dispersion equation.) In a certain sense such an assumption is always valid, since we are dealing with a collective of individual electrons, and any excitation, in principle, can be described through their motion separately. But in the case when the correlation between electrons becomes of great importance and determines the character of the given excited state, a description in terms of individual electrons becomes very complicated and, possibly, even unsuitable. It is precisely under such conditions that it is convenient to introduce a simple collective method of describing the excited state, for which purpose we have resorted to plasmons. In the case of narrow lines this method undoubtedly justifies itself. The author is convinced that it is equally useful in the case of the broad lines considered by us. The position and intensity of these lines indicate that the Coulomb correlation causes a substantial enhancement and displacement of the energy lines-

...losses, as a result of which the plasmon description becomes more adequate.

Of course, individual electrons undergo transitions between bands, and to this mechanism we are inclined to attribute many of the low-lying loss lines that we have excluded from consideration. It is clear that much more work must be done to reconcile such an interpretation with the available data on the band structure and on the transition probabilities for a given solid*). The agreement between the fine structure of X-rays and the characteristic loss lines discussed by Leder, Mendlowitz, and Marton ³¹, in our opinion, does not prove the individual character of the losses. Above all, these processes are entirely different. In one case an electron of an ionic core makes a transition into the upper part of the valence band or into one of still higher bands. In the other case we are dealing with a certain average over all transitions of electrons from the valence band into higher bands.

It would be very strange if the change in the density of initial states in the latter case did not lead to a significant difference between these two processes. The coincidence of the energy differences could be explained, for example, as follows: the correlation between valence electrons that gives rise to plasmon excitations is just such that it increases the density of those states of the system which are situated by \(\hbar\omega\) above the ground state of the valence electrons. This increases the probability of transition of an electron of an ionic core into such a state. Such an increase in probability also leads to the appearance of fine structure in the X-ray absorption spectrum.

Undoubtedly, much remains to be done, both experimentally and theoretically, before we can regard our understanding of the spectrum of energy losses as fully satisfactory. It is necessary to eliminate the discrepancies that exist between the work of various investigators carried out on one and the same substance. It would be desirable to extend our information concerning energy-loss spectra by adding as many as possible of the elements and compounds not yet investigated. It is interesting to verify whether the good agreement between the most prominent loss lines and the predicted behavior of plasmons will then be preserved. In addition, it would be fruitful to concentrate attention on some one substance, in order to understand as well as possible the origin of the various energy losses (in other words, to determine whether they are single or multiple elementary acts, whether they correspond to a plasmon or to an interband transition), to investigate their angular dependence, the corresponding cross sections, and so on.

Another interesting question, deserving special study, is the dependence, considered by Tabor ³², of the cross section for plasmon formation on film thickness for very thin films.

I should like to conclude this review with words of greeting addressed to the experimentalists working in this field. In dealing with plasmons, the experimental solid-state physicist, at least in one important respect, can surpass his colleagues working in the field of elementary particles. The shortest-lived of the elementary particles now known is the \(\pi^{0}\)-meson, whose lifetime is \(\sim 10^{-14}\) sec; in the case of plasmons we can observe elementary excitations with lifetimes of \(\sim 10^{-16}\) sec.

* In this connection it is necessary to note the work of Rudberg and Slater ³⁰.

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1. The experiments in question are those in which it was shown that, as the frequency of light increases in the far-ultraviolet region, thin metallic films cease to reflect light. The frequency at which this occurs satisfies the condition \(\varepsilon(\omega)=0\), coinciding with the condition for the existence of plasmons. (Translator’s note.) ↩

2. A similar table was first given by Ferrell[^13]. ↩