Excitation spectrum of valence electrons

Colliex, C.

doi:10.1107/97809553602060000593

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 397-404

Section 4.3.4.3. Excitation spectrum of valence electrons

C. Colliex^a

4.3.4.3. Excitation spectrum of valence electrons

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Most inelastic interaction of fast incident electrons is with outer atomic shells in atoms, or in solids with valence electrons (referred to as conduction electrons in metals). These involve excitations in the 0–50 eV range, but, in a few cases, interband transitions from low-binding-energy shells may also contribute.

4.3.4.3.1. Volume plasmons

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The basic concept introduced by the many-body theory in the interacting free electron gas is the volume plasmon. In a condensed material, the assembly of loosely bound electrons behaves as a plasma in which collective oscillations can be induced by a fast external charged particle. These eigenmodes, known as volume plasmons, are longitudinal charge-density fluctuations around the average bulk density in the plasma n $[\simeq]$ 10²⁸ e⁻/m³). Their eigen frequency is given, in the free electron gas, as $[\omega_p=\left({n\,e^2\over m\varepsilon_0}\right)^{1/2}. \eqno (4.3.4.8)]$ The corresponding $[\hbar\omega_p]$ energy, measured in an energy-loss spectrum (see the famous example of the plasmon in aluminium in Fig. 4.3.4.3), is the plasmon energy, for which typical values in a selection of pure solid elements are gathered in Table 4.3.4.2. The accuracies of the measured values depend on several instrumental parameters. Moreover, they are sensitive to the specimen crystalline state and to its degree of purity. Consequently, there exist slight discrepancies between published values. Numbers listed in Table 4.3.4.2 must therefore be accepted with a 0.1 eV confidence. Some specific cases require comments: amorphous boron, when prepared by vacuum evaporation, is not a well defined specimen. Carbon exists in several allotropic varieties. The selection of the diamond type in the table is made for direct comparison with the other tetravalent specimens (Si, Ge, Sn). The results for lead (Pb) are still subject to confirmation. The volumic mass density is an important factor (through n) in governing the value of the plasmon energy. It varies with temperature and may be different in the crystal, in the amorphous, and in the liquid phases. In simple metals, the amorphous state is generally less dense than the crystalline one, so that its plasmon energy shifts to lower energies.

Table 4.3.4.2| top | pdf |
Plasmon energies measured (and calculated) for a few simple metals; most data have been extracted from Raether (1980)

Monovalent			Divalent			Trivalent			Tetravalent
$[\hbar\omega_p]$ (eV)			$[\hbar\omega_p]$ (eV)			$[\hbar\omega_p]$ (eV)			$[\hbar\omega_p]$ (eV)
	Meas.	Calc.		Meas.	Calc.		Meas.	Calc.		Meas.	Calc.
Li	7.1	(8.0)	Be	18.7	(18.4)	B	22.7	(?)	C	34.0	(31)
Na	5.7	(5.9)	Mg	10.4	(10.9)	Al	14.95	(15.8)	Si	16.5	(16.6)
K	3.7	(4.3)	Ca	8.8	(8.0)	Ga	13.8	(14.5)	Ge	16.0	(15.6)
Rb	3.4	(3.9)	Sr	8.0	(7.0)	In	11.4	(12.5)	Sn	13.7	(14.3)
Cs	2.9	(3.4)	Ba	7.2	(6.7)	Sc	14.0	(12.9)	Pb	(13)	(13.5)

The above description applies only to very small scattering vectors q. In fact, the plasmon energy increases with scattering angle (and with momentum transfer $[\hbar{\bf q}]$ ). This dependence is known as the dispersion relation, in which two distinct behaviours can be described:

(a) For small momentum transfers $[(q\,\lesssim\, q_c)]$ , the dispersion curve is parabolic: $[\hbar\omega_p(q)=\hbar\omega_p(0)+{\alpha\hbar^2 \over m_0}q^2. \eqno (4.3.4.9)]$ The coefficient α has been measured in a number of substances and calculated for the free-electron case in the random phase approximation (Lindhard, 1954); see Table 4.3.4.3 for some data. A simple expression for α is $[\alpha=\textstyle{3\over5} \displaystyle{E_F \over\hbar \omega_p(0)}, \eqno (4.3.4.10)]$ where [E_F] is the Fermi energy of the electron gas. More detailed observations indicated that it is not possible to describe the dispersion curve over a large momentum range with a single [q^2] law. In fact, one has to fit the experiment data with different linear or quadratic slopes as a function of q [see values indicated for Al and In in Table 4.3.4.3, and Höhberger, Otto & Petri (1975)]. Moreover, anisotropy has been found along different q directions in monocrystals (Manzke, 1980). In parallel, refinements have been brought into the calculations by including band-structure effects to deal with the anisotropy of the dispersion relation and with the bending of the experimental curves. Electron–electron correlations have also been considered, which has slightly improved the agreement between calculated and measured values of α (Bross, 1978a,b).

Table 4.3.4.3| top | pdf |
Experimental and theoretical values for the coefficient α in the plasmon dispersion curve together with estimates of the cut-off wavevector (from Raether, 1980)

	Measured α	Calculated α	(Å⁻¹)
Li	0.24	0.35	0.9
Na	0.24	0.32	0.8
K	0.14	0.29	0.8
Mg	0.35	0.39	1.0
Al	0.2 (< 0.5 Å⁻¹)
Al	0.45 (> 0.5 Å⁻¹)	0.43	1.3
In	0.40 (< 0.5 Å⁻¹)
In	0.66 (> 0.5 Å⁻¹)
Si	0.41
Si	0.3	0.45	1.1

(b) For large momentum transfers, there exists a critical wavevector [q_c] , which corresponds to a strong decay of the plasmon mode into single electron–hole pair excitations. This can be calculated using conservation rules in energy and momentum, giving $[\hbar\omega_p(0)+\alpha{\hbar^2\over m_0}\,q^2_c = {\hbar^2\over 2m_0}\,(q^2_c+2q_c q_F), \eqno (4.3.4.11)]$ where [q_F] is the Fermi wavevector. A simple approximation is $[q_c\simeq\omega_p/v_F]$ , [v_F] being the Fermi velocity. Single pair excitations can be created by fast incoming electrons in the domain of scattering conditions contained between the two curves: $[\left.\matrix{\Delta E_{\max}=\displaystyle{\hbar^2\over 2m_0} (q^2+2qq_F) \cr \Delta E_{\min}=\displaystyle{\hbar^2 \over 2m_0}\,(q^2 - 2qq_F)} \right\}\eqno(4.3.4.12)]$ shown in Fig. 4.3.4.13 . They bracket the curve $[\Delta E=\hbar^2q^2/2m_0]$ corresponding to the transfer of energy and momentum to an isolated free electron. For momentum transfers such as $[q\gt q_c]$ , the plasmon mode is heavily damped and it is difficult to distinguish its own specific behaviour from the electron–hole continuum. A few studies, e.g. Batson & Silcox (1983), indicate that the plasmon dispersion curve flattens as it enters the quasiparticle domain and approaches the centre of the continuum close to the free-electron curve. However, not only is the scatter between measurements fairly high, but a satisfactory theory is not yet available [see Schattschneider (1989) for a compilation of data on the subject].

Figure 4.3.4.13| top | pdf |

The dispersion curve for the excitation of a plasmon (curve 1) merges into the continuum of individual electron–hole excitations (between curves 2 and 4) for a critical wavevector q_c. The intermediate curve (3) corresponds to Compton scattering on a free electron.

Plasmon lifetime is inversely proportional to the energy width of the plasmon peak $[\Delta E_{1/2}]$ . Even for Al, with one of the smallest plasmon energy widths ( $[\simeq0.5]$ eV), the lifetime is very short: after about five oscillations, their amplitude is reduced to 1/e. Such a damping demonstrates the strength of the coupling of the collective modes with other processes. Several mechanisms compete for plasmon decay:

(a) For small momentum transfer, it is generally attributed to vertical interband transitions. Table 4.3.4.4, extracted from Raether (1980), compares a few measured values of $[\Delta E_{1/2}(0)]$ , with values calculated using band-structure descriptions.

Table 4.3.4.4| top | pdf |
Comparison of measured and calculated values for the halfwidth ΔE_1/2(0) of the plasmon line (from Raether, 1980)

	Experimental (eV)	Theory (eV)
Li	2.2	2.55
Na	0.3	0.12
K	0.25	0.15
Rb	0.6	0.64
Cs	1.2	0.96
Al	0.53	0.43
Mg	0.7	0.7
Si	3.2	5.4
Ge	3.1	3.9

(b) For moderate momentum transfer q, a variation law such as $[\Delta E_{1/2} (q) = \Delta E_{1/2}(0)+Bq^2+O(q^4) \eqno (4.3.4.13)]$ has been measured. The q dependence of $[\Delta E_{1/2}]$ is mainly accounted for by non-vertical transitions compatible with the band structure, the number of these transitions increasing with q (Sturm, 1982). Other mechanisms have also been suggested, such as phonons, umklapp processes, scattering on surfaces, etc.
(c) For large momentum transfer (i.e. of the order of the critical wavevector ), the collective modes decay into the strong electron–hole-pair channels already described giving rise to a clear increase of the damping for values of $[q\gt q_c]$ .

Within this free-electron-gas description, the differential cross section for the excitation of bulk plasmons by incident electrons of velocity v is given by $[{{\rm d} \sigma_p \over{\rm d} \Omega} (\theta) = {\Delta E_p \over2\pi Na_0 m_0 v^2}\ {1\over \theta^2+\theta^2_E}, \eqno (4.3.4.14)]$ where N is the density of atoms per volume unit and $[\theta_E]$ is the characteristic inelastic angle defined as $[\Delta E_p/2E_0]$ in the non-relativistic description and as $[\Delta E_p/\gamma m_0v^2]$ {with $[\gamma=[1-(v^{2}/c^{2})]^{-1/2}]$ } in the relativistic case. The angular dependence of the differential cross section for plasmon scattering is shown in Fig. 4.3.4.14 . The integral cross section up to an angle $[\beta_0]$ is $[\sigma_p(\beta_0)=\int\limits^{\beta_0}_0\,\left(\displaystyle{{\rm d}\sigma_p \over {\rm d}\Omega}\right){\rm d}\Omega= \,\displaystyle{\Delta E_p\log(\beta_0/\theta_E) \over Na_0 m_0 v^2}. \eqno (4.3.4.15)]$ The total plasmon cross section is calculated for $[\beta_0=\theta_c=q_c/k_0]$ . Converted into mean free path, this becomes $[\Lambda_p={1\over N\sigma_p} = {a_0\over \theta_E} \, \left(\log{\theta_c\over \theta_E}\right)^{-1} \quad \hbox {(non-relativistic formula)}\semi \eqno (4.3.4.16)]$ and $[\Lambda_p= \,{a_0\gamma m_0v^2 \over \Delta E_p}\, \left(\log{\hbar q_c v \over 1.132\,\hbar\omega_p}\right)^{-1} \quad \hbox{(relativistic formula)}. \eqno (4.3.4.17)]$

Figure 4.3.4.14| top | pdf |

Measured angular dependence of the differential cross section dσ/dΩ for the 15 eV plasmon loss in Al (dots) compared with a calculated curve by Ferrell (solid curve) and with a sharp cut-off approximation at θ_c (dashed curved). Also shown along the scattering angle axis, θ_E = characteristic inelastic angle defined as ΔE/2E₀, $[\tilde\theta]$ = median inelastic angle defined by $[\int^{\tilde\theta}_{0}({\rm d}\sigma/{\rm d}\Omega)\,{\rm d}\Omega=1/2\int^{\theta_{c}}_{0}({\rm d}\sigma/{\rm d}\Omega)\,{\rm d}\Omega]$ , and $[\bar\theta]$ = average inelastic angle defined by $[\bar\theta = \int {\theta}({\rm d}\sigma/{\rm d}\Omega)\,{\rm d}\Omega/\int({\rm d}\sigma/{\rm d}\Omega)\,{\rm d}\Omega]$ [courtesy of Egerton (1986)].

The behaviour of $[\Lambda_p]$ as a function of the primary electron energy is shown in Fig. 4.3.4.15 .

Figure 4.3.4.15| top | pdf |

Variation of plasmon excitation mean free path Λ_p as a function of accelerating voltage V in the case of carbon and aluminium [courtesy of Sevely (1985)].

4.3.4.3.2. Dielectric description

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The description of the bulk plasmon in the free-electron gas can be extended to any type of condensed material by introducing the dielectric response function $[\varepsilon({\bf q},\omega)]$ , which describes the frequency and wavevector-dependent polarizability of the medium; cf. Daniels et al. (1970). One associates, respectively, the $[\varepsilon_T]$ and $[\varepsilon_L]$ functions with the propagation of transverse and longitudinal EM modes through matter. In the small-q limit, these tend towards the same value: $[\lim_{q\rightarrow 0}\, \varepsilon_T({\bf q},\omega) = \lim_{q\rightarrow0}\, \varepsilon_L({\bf q},\omega) = \varepsilon(0,\omega).]$ As transverse dielectric functions are only used for wavevectors close to zero, the T and L indices can be omitted so that: $[\varepsilon_L({\bf q},\omega) = \varepsilon({\bf q},\omega)\quad \hbox{and}\quad \varepsilon_T({\bf q},\omega)=\varepsilon(0,\omega).]$ The transverse solution corresponds to the normal propagation of EM waves in a medium of dielectric coefficient $[\varepsilon(0,\omega)]$ , i.e. to $[{q^2c^2 \over \omega^2}- \varepsilon(0,\omega)=0. \eqno (4.3.4.18)]$ For longitudinal fields, the only solution is $[\varepsilon({\bf q},\omega)=0]$ , which is basically the dispersion relation for the bulk plasmon.

In the framework of the Maxwell description of wave propagation in matter, it has been shown by several authors [see, for instance, Ritchie (1957)] that the transfer of energy between the beam electron and the electrons in the solid is governed by the magnitude of the energy-loss function $[-{\rm Im}[1/\varepsilon({\bf q}, \omega)]]$ , so that $[{{\rm d}^2\sigma \over {\rm d}(\Delta E)\,{\rm d}\Omega} = {1\over N(e\pi a_0)^2}\ {1 \over q^2}\,{\rm Im}\, \left(-{1\over \varepsilon({\bf q},\omega)}\right). \eqno (4.3.4.19)]$ One can deduce (4.3.4.14) by introducing a δ function at energy loss $[\omega_p]$ for the energy-loss function: $[{\rm Im}\left(-{1 \over \varepsilon({\bf q}, \omega)}\right) = {\pi \over 2}\,\omega_p\delta(\omega-\omega_p). \eqno (4.3.4.20)]$ As a consequence of the causality principle, a knowledge of the energy-loss function $[-{\rm Im}[1/\varepsilon(\omega)]]$ over the complete frequency (or energy-loss) range enables one to calculate $[{\rm Re}[1/\varepsilon(\omega)]]$ by Kramers–Kronig analysis: $[{\rm Re}{1\over \varepsilon(\omega)}=1-{2\over \pi}{\rm PP}\int\limits^\infty_0{\rm Im} \left(-\displaystyle{1\over \varepsilon(\omega')}\right) \displaystyle{\omega'\over \omega'^2-\omega^2}\,{\rm d}\omega', \eqno (4.3.4.21)]$ where PP denotes the principal part of the integral. For details of efficient practical evaluation of the above equation, see Johnson (1975).

The dielectric functions can be easily calculated for simple descriptions of the electron gas. In the Drude model , i.e. for a free-electron plasma with a relaxation time τ, the dielectric function at long wavelengths $[(q\rightarrow0)]$ is $[\varepsilon(\omega)=\varepsilon_1(\omega) + i \varepsilon_2(\omega)=1-{\omega^2_p\over \omega^2} {1\over (1-{1/i\omega\tau})}, \eqno (4.3.4.22)]$ with $[\omega^2_p=ne^2/m\varepsilon_0]$ , as above. The behaviour of the different functions, the real and imaginary terms in $[\varepsilon]$ , and the energy-loss function are shown in Fig. 4.3.4.16 . The energy-loss term exhibits a sharp Lorentzian profile centred at $[\omega=\omega_p]$ and of width 1/τ. The narrower and more intense this plasmon peak, the more the involved valence electrons behave like free electrons.

Figure 4.3.4.16| top | pdf |

Dielectric and optical functions calculated in the Drude model of a free-electron gas with ħω_p = 16 eV and τ = 1.64 × 10⁻¹⁶ s. R is the optical reflection coefficient in normal incidence, i.e. R = [(n − 1)² + k²]/(n + 1)² + k²] with n and k the real and imaginary parts of $[\sqrt{\epsilon}]$ . The effective numbers $[n_{\rm eff}(\varepsilon_{2})]$ and $[n_{\rm eff}[{\rm Im}(-1/\varepsilon)]]$ are defined in Subsection 4.3.4.5 [courtesy of Daniels et al. (1970)].

In the Lorentz model , i.e. for a gas of bound electrons with one or several excitation eigenfrequencies $[\omega_i]$ , the dielectric function is $[\varepsilon(\omega)=1+\sum_i \displaystyle{n_i e^2\over m\varepsilon_0}\ {\displaystyle{1\over \omega^2_i-\omega^2+i\omega/\tau_i}}, \eqno (4.3.4.23)]$ where [n_i] denotes the density of electrons oscillating with the frequency $[\omega_i]$ and $[\tau_i]$ is the associated relaxation time. The characteristic $[\varepsilon_1]$ , $[\varepsilon_2]$ , and $[-{\rm Im}(1/\varepsilon)]$ behaviours are displayed in Fig. 4.3.4.17 : a typical `interband' transition (in solid-state terminology) can be revealed as a maximum in the $[\varepsilon_2]$ function, simultaneous with a `plasmon' mode associated with a maximum in the energy-loss function and slightly shifted to higher energies with respect to the annulation conditions of the $[\varepsilon_1]$ function.

Figure 4.3.4.17| top | pdf |

Same as previous figure, but for a Lorentz model with an oscillator of eigenfrequency ħω₀ = 10 eV and relaxation time τ₀ = 6.6 × 10⁻¹⁶ s superposed on the free-electron term [courtesy of Daniels et al. (1970)].

In most practical situations, there coexist a family of [n_f] free electrons (with plasma frequency $[\omega^2_p=n{_f}e^2/m\varepsilon_0)]$ and one or several families of [n_i] bound electrons (with eigenfrequencies $[\omega_i)]$ . The influence of bound electrons is to shift the plasma frequency towards lower values if $[\omega_i\gt\omega_p]$ and to higher values if $[\omega_i\lt\omega_p]$ . As a special case, in an insulator, [n_f=0] and all the electrons [(n_i=n)] have a binding energy at least equal to the band gap $[E_g\simeq\hbar\omega_i]$ , giving $[\omega^2_p=(E_g/\hbar){^2}+ne^2/m\varepsilon_0]$ .

This description constitutes a satisfactory first step into the world of real solids with a complex system of valence and conduction bands between which there is a strong transition rate of individual electrons under the influence of photon or electron beams. In optical spectroscopy, for instance, this transition rate, which governs the absorption coefficient, can be deduced from the calculation of the factor $[\varepsilon_2]$ as $[\varepsilon_2(\omega)={A\over \omega^2}|M_{jj'}|^2J_{jj'}(\omega), \eqno (4.3.4.24)]$ where $[M_{jj'}]$ is the matrix element for the transition from the occupied level j in the valence band to the unoccupied level [j'] in the conduction band, both with the same k value (which means for a vertical transition). $[J_{jj'}(\omega)]$ is the joint density of states (JDOS) with the energy difference $[\hbar\omega]$ . This formula is also valid for small-angle-scattering electron inelastic processes. When parabolic bands are used to represent, respectively, the upper part of the valence band and the lower part of the conduction band in a semiconductor, the dominant JDOS term close to the onset of the interband transitions takes the form $[{\rm JDOS}\propto(E-E_g)^{1/2}, \eqno (4.3.4.25)]$ where [E_g] is the band-gap energy. This concept has been successfully used by Batson (1987) for the detection of gap energy variations between the bulk and the vicinity of a single misfit dislocation in a GaAs specimen. The case of non-vertical transitions involving integration over k-space has also been considered (Fink et al., 1984; Fink & Leising, 1986).

4.3.4.3.3. Real solids

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The dielectric constants of many solids have been deduced from a number of methods involving either primary photon or electron beams. In optical measurements, one obtains the values of $[\varepsilon_1]$ and $[\varepsilon_2]$ from a Krakers–Kronig analysis of the optical absorption and reflection curves, while in electron energy-loss measurements they are deduced from Kramers–Kronig analysis of energy-loss functions.

Fig. 4.3.4.18 shows typical behaviours of the dielectric and energy-loss functions.

(a) For a free-electron metal (Al), the Drude model is a satisfactory description with a well defined narrow and intense maximum of $[{\rm Im}(-1/\varepsilon)]$ corresponding to the collective plasmon excitation together with typical conditions $[\varepsilon_1\simeq\varepsilon_2\simeq 0]$ for this energy $[\hbar\omega_p]$ . One also notices a weak interband transition below 2 eV.

Figure 4.3.4.18| top | pdf |

Dielectric coefficients $[\varepsilon_1]$ , $[\varepsilon_2]$ and $[{\rm Im}(-1/\varepsilon)]$ from a collection of typical real solids: (a) aluminium [courtesy of Raether (1965)]; (b) gold [courtesy of Wehenkel (1975)]; (c) InSb [courtesy of Zimmermann (1976)]; (d) solid xenon at ca 5 K [courtesy of Keil (1968)].

(b) For transition and noble metals (such as Au), the results strongly deviate from the free-electron gas function as a consequence of intense interband transitions originating mostly from the partially or fully filled d band lying in the vicinity of, or just below, the Fermi level. There is no clear condition for satisfying the criterion of plasma excitation $[(\varepsilon=0)]$ so that the collective modes are strongly damped. However, the higher-lying peak is more generally of a collective nature because it coincides with the exhaustion of all oscillator strengths for interband transitions.
(c) Similar arguments can be developed for a semiconductor (InSb) or an insulator (Xe solid). In the first case, one detects a few interband transitions at small energies that do not prevent the occurrence of a pronounced volume plasmon peak rather similar to the free-electron case. The difference between the gap and the plasma energy is so great that the valence electrons behave collectively as an assembly of free particles. In contrast, for wide gap insulators (alkali halides, oxides, solid rare gases), a number of peaks are seen, owing to different interband transitions and exciton peaks. Excitons are quasi-particles consisting of a conduction-band electron and a valence-band hole bound to each other by Coloumb interaction. One observes the existence of a band gap [no excitation either in $[\varepsilon_2]$ or in $[{\rm Im}(-1/\varepsilon)]$ below a given critical value ] and again the higher-lying peak is generally of a rather collective nature.

Čerenkov radiation is emitted when the velocity v of an electron travelling through a medium exceeds the speed of light for a particular frequency in this medium. The criterion for Čerenkov emission is $[\varepsilon_1(\omega)\gt {c^2\over v^2}=\beta^{-2}. \eqno (4.3.4.26)]$

In an insulator, $[\varepsilon_1]$ is positive at low energies and can considerably exceed unity, so that a `radiation peak' can be detected in the corresponding energy-loss range (between 2 and 4 eV in Si, Ge, III–V compounds, diamond, $[\ldots]$ ); see Von Festenberg (1968), Kröger (1970), and Chen & Silcox (1971). The associated scattering angle, $[\theta\simeq\lambda_{\rm el}/\lambda_{\rm ph}\simeq10^{-5}\,{\rm rad}]$ for high-energy electrons, is very small and this contribution can only be detected using a limited forward-scattering angular acceptance.

In an anisotropic crystal, the dielectric function has the character of a tensor, so that the energy-loss function is expressed as $[{\rm Im}\left(-{1 \over \sum\limits_i\sum\limits_j \varepsilon_{ij} q_i q_j}\right) . \eqno (4.3.4.27)]$

If it is transformed to its orthogonal principal axes $[(\varepsilon_{11}, \varepsilon_{22}, \varepsilon_{33})]$ , and if the q components in this system are [q_1,q_2,q_3] , the above expression simplifies to $[{\rm Im}\left(-{1 \over \sum\limits_i \varepsilon_{ii} q^2_i}\right) . \eqno (4.3.4.28)]$

In a uniaxial crystal, such as a graphite, $[\varepsilon_{11}=\varepsilon_{22}=\varepsilon_\perp]$ and $[\varepsilon_{33}=\varepsilon_\|]$ (i.e. parallel to the c axis): $[\varepsilon({\bf q}, \omega)=\varepsilon_\perp\sin^2\theta+\varepsilon_\|\cos^2\theta, \eqno (4.3.4.29)]$ where $[\theta]$ is the angle between q and the c axis. The spectrum depends on the direction of q, either parallel or perpendicular to the c axis, as shown in Fig. 4.3.4.19 from Venghaus (1975). These experimental conditions may be achieved by tilting the graphite layer at 45° with respect to the incident axis, and recording spectra in two directions at $[\pm\theta_E]$ with respect to it (see Fig. 4.3.4.20 ).

Figure 4.3.4.19| top | pdf |

Dielectric functions in graphite derived from energy losses for E ⊥ c (i.e. the electric field vector being in the layer plane) and for E||c [from Daniels et al. (1970)]. The dashed line represents data extracted from optical reflectivity measurements [from Taft & Philipp (1965)].

Figure 4.3.4.20| top | pdf |

Geometric conditions for investigating the anisotropic energy-loss function.

4.3.4.3.4. Surface plasmons

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Volume plasmons are longitudinal waves of charge density propagating through the bulk of the solid. Similarly, three exist longitudinal waves of charge density travelling along the surface between two media A and B (one may be a vacuum): these are the surface plasmons (Kliewer & Fuchs, 1974). Boundary conditions imply that $[\varepsilon_A(\omega)+\varepsilon_B(\omega)=0. \eqno (4.3.4.30)]$ The corresponding charge-density fluctuation is contained within the (x) boundary plane, z being normal to the surface: $[\rho({\bf x}, z)\simeq\cos({\bf q}\cdot{\bf x}-\omega t)\delta(z), \eqno (4.3.4.31)]$ and the associated electrostatic potential oscillates in space and time as $[\varphi({\bf x},z)\,\alpha\cos({\bf q}\cdot{\bf x}-\omega t)\exp(-q|z|). \eqno (4.3.4.32)]$ The characteristic energy $[\omega_s]$ of this surface mode is estimated in the free electron case as:

In the planar interface case: $[\left.\matrix{\omega_s = \displaystyle{\omega_p\over \sqrt 2}\hfill \cr \quad\raise2ex\hbox{(interface metal--vacuum)\semi}\hfill \cr \omega_s = \displaystyle{\omega_p \over (1+\varepsilon_d)^{1/2}}\hfill \cr \quad\hbox{(interface metal--dielectric of constant}\,\, \varepsilon_d); \hfill\cr \omega_s =\displaystyle\left({\omega^2_{p_A}\,+\,\omega^2_{p_B} \over2}\right)^{1/2}\hfill \cr \quad\raise2ex\hbox{(interface metal {\it A}--metal {\it B}).}\hfill} \right\} \eqno (4.3.4.33)]$

In the spherical interface case: $[(\omega_s)_l= {\omega_p\over[(2l+1)/l]^{1/2}} \eqno(4.3.4.34a)]$ (metal sphere in vacuum – the modes are now quantified following the l quantum number in spherical geometry); $[(\omega_s)_l={\omega_p\over [(2l+1)/(l+1)]^{1/2}} \eqno (4.3.4.34b)]$ (spherical void within metal).

Thin-film geometry: $[(\omega_s)^\pm={\omega_p\left[{1\pm\exp(-qt) \over 1+\varepsilon_d}\right]^{1/2}} \eqno (4.3.4.35)]$ (metal layer of thickness t embedded in dielectric films of constant $[\varepsilon_d]$ ). The two solutions result from the coupling of the oscillations on the two surfaces, the electric field being symmetric for the $[(\omega_s)^-]$ mode and antisymmetric for the $[(\omega_s)^+]$ .

In a real solid, the surface plasmon modes are determined by the roots of the equation $[\varepsilon(\omega_s)=-1]$ for vacuum coating [or $[\varepsilon(\omega_s)=-\varepsilon_d]$ for dielectric coating].

The probability of surface-loss excitation [P_s] is mostly governed by the $[{\rm Im}\{-1/[1+\varepsilon(\omega)]\}]$ energy-loss function, which is analogous for surface modes to the bulk $[{\rm Im}\{-1/[\varepsilon(\omega)]\}]$ energy-loss function. In normal incidence, the differential scattering cross section $[{\rm d}P_s/\!{\rm d}\Omega]$ is zero in the forward direction, reaches a maximum for $[\theta=\pm\theta_E/3^{1/2}]$ , and decreases as $[\theta^{-3}]$ at large angles. In non-normal incidence, the angular distribution is asymmetrical, goes through a zero value for momentum transfer $[\hbar{\bf q}]$ in a direction perpendicular to the interface, and the total probability increases as $[P_s(\varphi)=\,{P_s(O)\over\cos\varphi}, \eqno (4.3.4.36)]$ where $[\varphi]$ is the incidence angle between the primary beam and the normal to the surface. As a consequence, the probability of producing one (and several) surface losses increases rapidly for grazing incidences.

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