Casimir Friction Between a Small Particle and a Plane Surface

Abstract

In this section, the friction force acting on a small neutral particle moving relative to a flat surface of a solid is considered in the framework of the fluctuation electrodynamic in non-relativistic and relativistic cases. The friction force in the particle–plate configuration is deduced from the friction force in the plate–plate configuration, assuming one of the plates to be sufficiently rarefied. The effect of the multiple scattering of the electromagnetic field between a particle and substrate is also studied. These effects can be important for physisorbed molecules. For physisorbed molecules, high-order processes, which are not included in the theory of Casimir friction, can dominate the damping rate of physisorbed molecules. The results of the theoretical calculations are compared with experimental data. The special case of Casimir friction force acting on a small neutral particle moving relative to the black-body radiation is also analyzed.

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In this section, the friction force acting on a small neutral particle moving relative to a flat surface of a solid is considered in the framework of the fluctuation electrodynamic in non-relativistic and relativistic cases. The friction force in the particle–plate configuration is deduced from the friction force in the plate–plate configuration assuming one of the plates to be sufficiently rarefied. The effect of the multiple scattering of the electromagnetic field between a particle and substrate is also studied. These effects can be important for physisorbed molecules. For physisorbed molecules, high-order processes, which are not included in the theory of Casimir friction, can dominate the damping rate of physisorbed molecules. The results of the theoretical calculations are compared with experimental data. The special case of Casimir friction force acting on a small neutral particle moving relative to black-body radiation is also analyzed.

8.1 Friction Force on a Particle Moving Parallel to a Plane Surface: Non-relativistic Theory

The friction force acting on a small particle during motion parallel to a flat surface can be obtained from the friction between two semi-infinite bodies in the limit when one of the bodies is sufficiently rarefied. For $d\ll \lambda _T=c\hbar /k_BT$, in (7.28), it is possible to neglect the first term and, in the second term, the integration can be extended to the whole q-plane, and we can put $k_z\approx iq$. Let us assume that the rarefied body consists of particles with dipole and magnetic moments. The dielectric and magnetic susceptibilities of the rarefied body—let us say body 2 will be close to unity, i.e. $\varepsilon _2-1\rightarrow 4\pi n\alpha _{E2}\ll 1$ and $\mu _2-1\rightarrow 4\pi n\alpha _{H2}\ll 1$, where n is the particle concentration in body 2, and $\alpha _{E2}$ and $\alpha _{H2}$ are their dielectric and magnetic susceptibilities, respectively. In linear order, in the particle concentration n, according to (C.17) from Appendix, C, the reflection amplitudes are given by

$$ R_{2p} = \frac{\varepsilon _2k_z - \sqrt{\varepsilon _2\mu _2 k^2 - q^2}}{\varepsilon _2k_z + \sqrt{\varepsilon _2\mu _2 k^2 - q^2}} \approx \frac{\varepsilon _2-1}{\varepsilon _2+1}\approx 2\pi n\alpha _E, $$

$$ R_{2s} = \frac{\mu _2k_z - \sqrt{\varepsilon _2\mu _2 k^2 - q^2}}{\mu _2k_z + \sqrt{\varepsilon _2\mu _2 k^2 - q^2}} \approx \frac{\mu _2-1}{\mu _2+1}\approx 2\pi n\alpha _H. $$

If the particle–surface separation $d\ll \lambda _T$, then the friction force acting on a particle moving parallel to a plane surface can be obtained from (7.28) as the ratio between the change of the frictional shear stress after the displacement of body 2 by small distance, dz, and the number of the particles in a slab with thickness dz:

$$\begin{aligned} F_x^{part} =\,\frac{d\sigma _{\Vert }(z)}{ndz}\Big |_{z=d}&= \frac{2\hbar }{\pi ^2}\int _0^\infty d\omega \int d^2qq_xqe^{-2qd}\left( n_2(\omega ^{\prime })-n_1(\omega )\right) \times \nonumber \\&\quad \times \left( \mathrm {Im}R_p \mathrm {Im}\alpha _E^{\prime } + \mathrm {Im}R_s \mathrm {Im}\alpha _H^{\prime }\right) , \end{aligned}$$

(8.1)

where $\alpha _{E(H)}^{\prime } = \alpha _{E(H)}(\omega ^{\prime })$. After the transformation, we get:

$$ F_x^{part}= \frac{2\hbar }{\pi ^2}\int _{-\infty }^\infty dq_y\int _0^\infty dq_xq_xqe^{-2qd}\left\{ \int _0^\infty d\omega \big [n(\omega )-n(\omega +q_xv)\big ]\times \right. $$

$$ \times \,\Big [\left( \mathrm {Im}R_{1p}(\omega + q_xv)\mathrm {Im}\alpha _E(\omega )+[\omega +q_xv\leftrightarrow \omega ]\right) + $$

$$ +\,\big (\mathrm {Im}R_{1s}(\omega +q_xv)\mathrm {Im}\alpha _H(\omega )+[\omega +q_xv\leftrightarrow \omega ]\big ) \Big ]- $$

$$ -\,\int _0^{q_xv}d\omega [n(\omega )+1/2]\Big [\left( \mathrm {Im}R_{1p}(\omega -q_xv)\mathrm {Im}\alpha _E(\omega )+[\omega -q_xv\leftrightarrow \omega ]\right) + $$

$$\begin{aligned} +\,\left( \mathrm {Im}R_{1p}(\omega -q_xv)\mathrm {Im}\alpha _H(\omega )+[\omega -q_xv\leftrightarrow \omega ]\right) \Big ] \Bigg \}, \end{aligned}$$

(8.2)

where the symbols $[\omega \,\pm \, q_xv\leftrightarrow \omega ]$ stand for the terms that can be obtained from the preceding terms by interchanging $\omega \,\pm \, q_xv$ and $\omega$. An alternative derivation of (8.2), using the law of energy conservation without taking into account the contribution from magnetic moment, is given in Appendix H. To linear order in the sliding velocity v from (8.2), we get $F_{fric}=-\Gamma _{\Vert }v$, where

$$\begin{aligned} \Gamma _{\Vert }&=\frac{2\hbar }{\pi } \int _0^\infty \mathrm {d}\omega \left( - \frac{\partial n(\omega )}{\partial \omega }\right) \int _0^\infty \mathrm {d} qq^4e^{-2qd}\times \nonumber \\&\quad \times \Big [\mathrm {Im}R_p(q,\omega )\mathrm {Im}\alpha _E(\omega )+ \mathrm {Im}R_s(q,\omega )\mathrm {Im}\alpha _H(\omega )\Big ]. \end{aligned}$$

(8.3)

In the non-retarded limit, without taking into account the contribution from the magnetic moment, this equation reduces to the formula obtained by Tomassone and Widom [145].

For a spherical particle with radius R, the electrical and magnetic susceptibilities are given by (5.58) and (5.59), respectively. In this case, from (8.3) in the limit $d<\mid \varepsilon (\omega =k_BT/\hbar )\mid ^{-1/2}\lambda _T$ for the contribution to the friction coefficient from p-polarized waves, we get [139]:

$$\begin{aligned} \Gamma _{p\Vert }\approx 3\frac{\hbar }{d^5}{\left( \frac{k_BT}{4\pi \hbar } \right) }^2\sigma _1^{-1}\sigma _2^{-1}R^3, \end{aligned}$$

(8.4)

and for the contribution from s-polarized waves, we get

$$\begin{aligned} \Gamma _{s\Vert }\approx \frac{\pi ^3}{45}\frac{\hbar }{\lambda _T^2} \left( \frac{\sigma _1\sigma _2R^2}{c^2}\right) {\left( \frac{R}{d}\right) } ^3, \end{aligned}$$

(8.5)

where $\sigma _1$ and $\sigma _2$ are the conductivities of substrate and particle, respectively. For $d=2R=10$ nm, $\sigma _1=\sigma _2=4\,\times \, 10^{17}$ s$^{-1}$ (which corresponds to gold at room temperature), we get the very small friction coefficients: $\Gamma _p \sim 10^{-29}$ kg$\,\cdot \,\mathrm{{s}}^{-1}$ and $\Gamma _s \sim 10^{-24}$ kg$\,\cdot \,\mathrm{{s}}^{-1}$.  Note that the contribution to the friction between a small particle and a plane metal surface is mainly due to the s-polarized waves, just as in the case of the friction between two plane surfaces, and is many orders of magnitude larger than contribution from the p-polarized waves. This is related to the screening of the electromagnetic field inside the metal volume, which is stronger for the p-polarized waves. As a result, the energy dissipation and, consequently, the friction will be larger for the s-polarized waves. Figure 8.1 shows the velocity-dependence of the friction force acting on a small copper particle with $R=10$ nm, moving above a copper sample at $d=20$ nm. The contributions from the electric dipole and magnetic moments are shown separately. At small velocities, the contribution from the magnetic moment is seven orders of magnitude larger than the contribution from the electric dipole moment.

Fig. 8.1

The velocity dependence of the friction force acting on a small copper particle with radius $R=10$ nm moving above a copper sample at the separation $d=20$ nm. The contributions from the electric dipole moment and the magnetic moment are shown separately. (The base of the logarithm is 10)

The friction can be greatly enhanced for high-resistivity materials. Using (8.3) in the non-retarded limit (which can be formally obtained in the limit $c\rightarrow \infty$), and for high-resistivity material ($4\pi \sigma<<k_BT/\hbar$), we get:

$$\begin{aligned} \Gamma _{p\Vert }=0.9\frac{k_BT}{4\pi \sigma }\frac{R^3}{d^5} \end{aligned}$$

(8.6)

where we have assumed that the particle and the substrate have the same dielectric function $\varepsilon =1+4\pi i\sigma /\omega$. As discussed above (see Sect. 7.3.1), the macroscopic theory (which was used in obtaining (8.6)) is only valid when $\sigma \gg \sigma _{min}\sim e^2\tau /md^3$. For $\sigma \sim \sigma _{min},\, d=2R=10$ nm and $\tau =10^{-15}$ s (8.6) gives $\Gamma _{p\Vert \max }\sim 10^{-18}$ kg s$^{-1}$.

If the particle and the substrate are made from the same material, able to support the surface phonon–polaritons, the friction is given by

$$\begin{aligned} \Gamma _{\Vert }=\frac{9k_BT\eta R^3}{d^5}\left( \frac{1}{\omega _s^2}+\frac{1}{\omega _p^2}\right) , \end{aligned}$$

(8.7)

where $\omega _s$ and $\omega _p$ are the frequencies of the surface phonon–polaritons for the substrate and the particle, respectively. If the substrate and the particle are made from silicon carbide (SiC), $\omega _s=1.79\times 10^{14}$ s$^{-1}$ and $\omega _p=1.76\,\times \, 10^{14}$ s$^{-1}$. Thus, for $d=2R=10$ nm, $\eta =8.9 \times 10^{11}$ s$^{-1}$, and $T=300$ K, we get $\Gamma \sim 10^{-21}$ kg s$^{-1}$. This friction coefficient is three orders of magnitude larger than for the good conductors.

Fig. 8.2

The schemes of the configurations: a plate–plate; b particle–plate in the lab frame; and c particle–plate in the rest frame of a particle. The friction force in the particle–plate configuration in the lab frame can be obtained from the friction force in the plate–plate configuration, assuming that plate 2 is rarefied and in the rest frame of a particle and assuming that plate 1 is rarefied

8.2 Friction Force on a Particle Moving Parallel to Plane Surface: Relativistic Theory

The friction force acting on a particle moving parallel to a dielectric plate can be obtained from the friction force between two plates each sliding relative to other,  assuming one of the plates as sufficiently rarefied (see Fig. 8.2). According to a fully relativistic theory [128] (see also Sect. 7.3), the contributions to the friction force, $F_{1x}$, and the radiation power, $P_1$, absorbed by plate 1 from evanescent waves, which dominate at small separations and low temperatures, are determined by the formulas

$$\begin{aligned} \left( \begin{array}{c} F_{1x}\\ P_1 \end{array} \right) = \int \frac{d^2q}{(2\pi )^2} \int _0^{cq} \frac{d\omega }{2\pi } \left( \begin{array}{c} \hbar q_x\\ \hbar \omega \end{array} \right) \Gamma _{12}(\omega , \mathbf {q})\big [n_2(\omega ^{\prime }) - n_1(\omega )\big ], \end{aligned}$$

(8.8)

where the positive quantity

$$\begin{aligned} \Gamma _{12}(\omega , \mathbf {q})&= \frac{4\mathrm {sgn}(\omega ^{\prime })}{|\Delta |^2} \left[ (q^2 - \beta kq_x)^2 - \beta ^2k_z^2q_y^2\right] \Big \{\mathrm {Im}R_{1p}\big [(q^2 - \beta kq_x)^2\mathrm {Im}R_{2p}^{\prime }|D_{ss}|^2 \nonumber \\&\quad + \beta ^2k_z^2q_y^2 \mathrm {Im}R_{2s}^{\prime }|D_{sp}|^2\big ] +(p\leftrightarrow s)\Big \}e^{-2 k_z d}, \end{aligned}$$

(8.9)

can be identified as a spectrally resolved photon emission rate:

$$ \Delta = \big (q^2 - \beta kq_x\big )^{2}D_{ss}D_{pp} - \beta ^2k_z^2q_y^2D_{ps}D_{sp},\,\, $$

$$ D_{pp} = 1 - e^{-2k_zd}R_{1p}R_{2p}^{\prime },\, D_{sp} = 1 + e^{-2k_zd}R_{1s}R_{2p}^{\prime }, $$

$n_i(\omega )=[\exp (\hbar \omega /k_BT_i)-1]^{-1}$, $k_z=\sqrt{q^2-(\omega /c)^2}$, $R_{1 p(s)}$ is the reflection amplitude for surface 1 in the K frame for a p(s)-polarized electromagnetic wave, $R_{2 p(s)}^{\prime } = R_{2 p(s)}(\omega ^{\prime }, q^{\prime })$ is the reflection amplitude for surface 2 in the $K^{\prime }$ frame for a p(s)-polarized electromagnetic wave, $\omega ^{\prime }=\gamma (\omega -q_xv)$, $q_x^{\prime }=\gamma (q_x- \beta k)$, $D_{ps}=D_{sp}(p\leftrightarrow s)$. The symbol $(p\leftrightarrow s$) denotes the terms that are obtained from the preceding terms by permutation of the p and s indexes.

Assuming that the dielectric permittivity of the rarefied plate is close to the unity, i.e. $\varepsilon -1\rightarrow 4\pi \alpha N\ll 1$, where N is the concentration of particles in a plate in the co-moving reference frame, then, to linear order in the concentration N, the reflection amplitudes for the rarefied plate in the co-moving frame are:

$$\begin{aligned} R_{p}&=\,\frac{\varepsilon k_z - \sqrt{k_z^2 - (\varepsilon -1)\big (\frac{\omega }{c}\big )^{2}}}{\varepsilon k_z + \sqrt{k_z^2 - (\varepsilon -1) \big (\frac{\omega }{c}\big )^{2}}} \approx N\pi \frac{q^2+k_z^2}{k_z^2} \alpha ,\nonumber \\ R_{s}&=\,\frac{k_z - \sqrt{k_z^2 - (\varepsilon -1) \big (\frac{\omega }{c}\big )^{2}}}{k_z +\sqrt{k_z^2 - (\varepsilon -1) \big (\frac{\omega }{c}\big )^{2}}} \approx N\pi \frac{q^2-k_z^2}{k_z^2} \alpha . \end{aligned}$$

(8.10)

Because $R_{p(s)}\ll 1$ for the rarefied plate, it is possible to neglect multiple scattering of the electromagnetic waves between the surfaces. In this approximation, $\Delta _{pp}\approx \Delta _{ss}\approx \Delta _{sp} \approx \Delta _{sp}\approx 1$ and

$$\begin{aligned} \Delta \approx (q^2 - \beta kq_x)^2 - \beta ^2k_z^2q_y^2= \frac{(qq^{\prime })^2}{\gamma ^2}, \end{aligned}$$

(8.11)

$$\begin{aligned} (q^2 - \beta kq_x)^2\mathrm {Im}R_{2p}^{\prime }|\Delta _{ss}|^2&+\,\beta ^2k_z^2q_y^2\mathrm {Im}R_{2s}^{\prime }|\Delta _{sp}|^2\,\nonumber \\&\quad \approx \,\frac{(qq^{\prime })^2}{\gamma ^2} \mathrm {Im}R_{2p}^{\prime } +\,\beta ^2k_z^2q_y^2\mathrm {Im}(R_{2p}^{\prime }+R_{2s}^{\prime }), \end{aligned}$$

(8.12)

$$\begin{aligned} \Gamma _{12}&=\,-4 \left[ \left( \mathrm {Im}R_{1p}\mathrm {Im}R_{2p}^{\prime }+\mathrm {Im}R_{1s}\mathrm {Im}R_{2s}^{\prime }\right) \left( 1+\gamma ^2\beta ^2\frac{k_z^2q_y^2}{q^2q^{\prime 2}}\right) \right. \nonumber \\&\qquad \qquad \,\left. +\,\gamma ^2\beta ^2\frac{k_z^2q_y^2}{q^2q^{\prime 2}}\left( \mathrm {Im}R_{1p}\mathrm {Im}R_{2s}^{\prime }+\mathrm {Im}R_{1s}\mathrm {Im}R_{2p}^{\prime }\right) \right] . \end{aligned}$$

(8.13)

The friction force $f_x$ acting on a particle, and the radiation power w absorbed by it, can be obtained in the K frame from (8.13), under the assumption that plate 2 is sufficiently rarefied [150] (see Fig. 8.2b). In this case, the friction force acting on the surface 2, $F_{2x}$, and the radiation power absorbed by it, $W_2$, are

$$\begin{aligned} \left( \begin{array}{c} F_{2x}\\ W_2 \end{array} \right) = \left( \begin{array}{c} -F_{1x}\\ -W_1 \end{array} \right) = N^{\prime }\int _d^{\infty } dz \left( \begin{array}{c} f_{x}(z)\\ w(z) \end{array} \right) , \end{aligned}$$

(8.14)

where $N^{\prime }= \gamma N$ is the concentration of particles in plate 2 in the K frame,

$$\begin{aligned} \left( \begin{array}{c} f_{x}(z)\\ w(z) \end{array} \right)&=\,\frac{1}{\gamma \pi ^2}\int d^2q \nonumber \\&\quad \times \int _0^{cq} d\omega \left( \begin{array}{c} \hbar q_x\\ \hbar \omega \end{array} \right) \frac{e^{-2 k_z z}}{k_z} \Big [\mathrm {Im}R_{1p}(\omega )\phi _p + \mathrm {Im}R_{1s}(\omega )\phi _s\Big ]\mathrm {Im}\alpha (\omega ^{\prime })\Big [n_2(\omega ^{\prime }) - n_1(\omega )\Big ], \end{aligned}$$

(8.15)

$$ \phi _p={\left( \frac{\omega ^{\prime }}{c}\right) }^2+2\gamma ^2 \left( q^2-\beta ^2q_x^2\right) \frac{k_z^2}{q^2},\,\, \phi _s={\left( \frac{\omega {^\prime }}{c}\right) }^2+2\gamma ^2\beta ^2q_y^2\frac{k_z^2}{q^2}. $$

In the rest reference frame of an object, the radiation power absorbed by it is equal to the heating power for the object. Thus, $-w$ is equal to the heating power for plate 1. Equation (8.15) agrees with the results obtained in [150, 152, 155]. However, as shown in [61, 128], the acceleration and heating of the particle are determined by the friction force $f_x^{\prime }$ and by the radiation power $w^{\prime }$ absorbed by the particle in the rest reference frame of a particle (the $K^{\prime }$ frame)

$$\begin{aligned} m_0\gamma ^3\frac{dv}{dt}=m_0\frac{dv^{\prime }}{dt^{\prime }} = f_x^{\prime }, \end{aligned}$$

(8.16)

$$\begin{aligned} w^{\prime }=\frac{dm_0}{dt^{\prime }}c^2 \end{aligned}$$

(8.17)

where $m_0$ is the rest mass of the particle, $v^{\prime }\ll v$ and $t^{\prime }$ are the velocity and time in the $K^{\prime }$ frame, respectively. These quantities can be also obtained assuming plate 2 to be sufficiently rarefied (see Fig. 8.2c). In this case, in the $K^{\prime }$ frame, the friction force acting on the surface 2, $F_{2x}^{\prime }$, and the radiation power absorbed by it, $W_2^{\prime }$, are

$$\begin{aligned} \left( \begin{array}{c} F_{2x}^{\prime }\\ W_2^{\prime } \end{array} \right) = \left( \begin{array}{c} -\tilde{F}_{1x}\\ \tilde{W}_1 \end{array} \right) = N\int _d^{\infty } dz \left( \begin{array}{c} f_{x}^{\prime }(z)\\ w^{\prime }(z) \end{array} \right) , \end{aligned}$$

(8.18)

where $\tilde{F}_{1x}$ and $\tilde{W}_1$ are obtained from $F_{1x}$ and $W_1$ after the replacement of the indexes $1\leftrightarrow 2$,

$$\begin{aligned} \left( \begin{array}{c} f_{x}^{\prime }\\ w^{\prime } \end{array} \right)&=\,\frac{1}{\pi ^2}\int _{0}^{\infty } dq_x\int _{-\infty }^{\infty } dq_y \nonumber \\&\quad \times \int _0^{cq} d\omega \left( \begin{array}{c} \hbar q_x\\ -\hbar \omega \end{array} \right) \frac{e^{-2 k_z d}}{k_z} \left[ \mathrm {Im}R_{1p}(\omega ^{\prime })\phi _p^{\prime } + \mathrm {Im}R_{1s}(\omega ^{\prime })\phi _s^{\prime }\right] \mathrm {Im}\alpha (\omega )\Big [n_1(\omega ^{\prime }) - n_2(\omega )\Big ], \end{aligned}$$

(8.19)

$$ \phi _p^{\prime }={\left( \frac{\omega }{c}\right) }^2+2\gamma ^2 \left( q^{\prime 2}-\beta ^2q_x^{\prime 2}\right) \frac{k_z^2}{q^{\prime 2}},\,\, \phi _s^{\prime }={\left( \frac{\omega }{c}\right) }^2+2\gamma ^2\beta ^2q_y^2\frac{k_z^2}{q^{\prime 2}}. $$

The relation between the different quantities in the K and $K^{\prime }$ frames can be found using the Lorentz transformations for the energy-momentum tensor for a plate 2, according to which

$$\begin{aligned} F_{2x}=\gamma \left( F_{2x}^{\prime }+v\frac{W_2^{\prime }}{c^2}\right) ,\,\, W_2=\gamma \left( W_2^{\prime }+vF_{2x}^{\prime }\right) , \end{aligned}$$

(8.20)

Using (8.14) and (8.18) gives

$$\begin{aligned} f_x = f_x^{\prime } + v \frac{w^{\prime }}{c^2},\,\, w = w^{\prime } + vf_x^{\prime }. \end{aligned}$$

(8.21)

These relations also can be found using the Lorentz transformation for the energy-momentum for a particle, according to which

$$\begin{aligned} p_x=\gamma \left( p_x^{\prime }+vm_0\right) ,\,\,\varepsilon =\gamma \left( m_0c^2+vp_x^{\prime }\right) , \end{aligned}$$

(8.22)

where $p_x$ and $\varepsilon$ are the momentum and energy, respectively, of a particle in the K frame and $p_x^{\prime }$ and $m_0c^2$ are the same quantities in the $K^{\prime }$ frame. When the derivative of the 4-momentum is taken with respect to lab time, then the factor $\gamma$ in (8.22) disappears because $dt=\gamma dt^{\prime }$ and the relations (8.21) are obtained [161]. From the inverse transformations

$$\begin{aligned} F_{2x}^{\prime }=\gamma \left( F_{2x}-v\frac{W_2}{c^2}\right) ,\,\, W_2^{\prime }=\gamma (W_2-vF_{2x}), \end{aligned}$$

(8.23)

follows

$$\begin{aligned} f_x^{\prime } = \gamma ^2 \left( f_x - v \frac{w}{c^2}\right) ,\,\, w^{\prime } = \gamma ^2(w - vf_x). \end{aligned}$$

(8.24)

These relations also can be obtained as above using the Lorentz transformation for the energy-momentum for a particle. We note that, in contrast to the relations (8.21) on the right side of the relations (8.24), there is an extra factor $\gamma ^2$. This is because the friction force and the radiation power are not 4-vectors. The kinetic energy of a particle in the lab frame is $\varepsilon _K=\varepsilon -m_0c^2$ where the total energy of a particle in the K frame is $\varepsilon =\gamma (m_0c^2+p_x^{\prime }v)$. The rate of change of the kinetic energy is

$$\begin{aligned} \frac{d\varepsilon _K}{dt}=w^{\prime }+f_x^{\prime }v-\frac{w^{\prime }}{\gamma }=w-\frac{w^{\prime }}{\gamma }=vf_x-\frac{(\gamma -1)w^{\prime }}{\gamma ^2}. \end{aligned}$$

(8.25)

Thus, the rate of change of the kinetic energy in the K frame is equal to the friction force power in this frame only when $w^{\prime }=0$.

8.3 Effect of Multiple Scattering of the Electromagnetic Waves

Equation (8.3) does not take into account the screening effect, which is due to multiple scattering of the electromagnetic waves between particle and substrate. This effect becomes important in the case of the resonant photon tunneling between the localized particle and substrate modes. A theory of friction that takes into account the screening effects to linear order in velocity was developed in [139] using the semi-classical theory of the fluctuating electromagnetic field. In [129] (see also Appendix K), the same results were obtained using quantum field theory. Resonances that can be excited by thermal radiation can exist only on the surfaces of semiconductors and dielectrics. For good conductors, the surface plasmoms have high frequencies, and cannot be excited by thermal radiation. Thus, for a metal particle, moving above a metal surface, multiple scattering effects can be neglected. For a dielectric particle, it is necessary to take into account only the dipole moment of the particle; the magnetic moment can be neglected.

The screening effects are especially important for physisorbed molecules, because they contribute to friction that is proportional to the absolute value of molecule susceptibility. Without taking screening effects into account, the friction for a particles is proportional to the imaginary part of the susceptibility, which is usually very small for those frequencies that can be excited by thermal radiation. However, the real part of susceptibility can be very large at the resonant frequency.

There is fundamental difference between ‘vacuum’ friction for two flat surfaces and for point dipole above a flat surface. In the former case, scattering of electromagnetic waves by the surfaces conserves the parallel momentum. Hence, the only possible process of momentum transfer between two flat surfaces is the emission of electromagnetic waves by one body, and the subsequent absorption by the other body. For the evanescent waves, this process gives a contribution that is proportional to the product of the imaginary part of the reflection factors for both surfaces. In the case of a particle above the flat surface, the component of the momentum parallel to the surface can change during scattering by the particle, resulting in momentum transfer. This process gives a contribution that is proportional to product of the imaginary part of the reflection factor for a metal at a different value of the momentum parallel to surface, and absolute value of the point dipole polarizability. The results obtained below are applied to the problem of the vibrational energy relaxation of a physisorbed molecule, and the friction and the heat transfer between an STM tip and a metal surface.

We consider a semi-infinite metal having with a flat surface, which coincides with the xy-coordinate plane, and with the z-axes pointed along the upward normal. A point dipole is located at $\mathbf {r}_0=(0, 0, d)$, performing small amplitude vibrations with the displacement vector $\mathbf {u}(t)=\mathbf {u}_0e^{-\mathrm {i}\omega _0t}$. To linear order in the vibrational coordinate $\mathbf {u}(t)$, the polarization density corresponding to the point dipole can be written in the form:

$$\begin{aligned} \mathbf {p}(\mathbf {r}, t)= & {} \mathbf {p}_0\delta (\mathbf {r}-\mathbf {r}_0)e^{- \mathrm {i}\omega t}+\mathbf {p}_1(\mathbf {r},\omega )e^{-\mathrm {i}(\omega +\omega _0)t}, \end{aligned}$$

(8.26)

$$\begin{aligned} \mathbf {p}_1(\mathbf {r},\omega )= & {} \mathbf {p}_1\delta (\mathbf {r}-\mathbf {r} _0)-\mathbf {p}_0\mathbf {u}_0\cdot \frac{\partial }{\partial \mathbf {r}}\delta (\mathbf {r}-\mathbf {r}_0), \end{aligned}$$

(8.27)

where $\mathbf {p}_0=\mathbf {p}^f+\alpha (\omega ) \mathbf {E}_0,\mathbf {p} _1=\alpha (\omega +\omega _0)\mathbf {E}_1,\,\mathbf {E}(t)=\mathbf {E}_0e^{- \mathrm {i}\omega t}+\mathbf {E}_1e^{-\mathrm {i}(\omega +\omega _0)t}$ is an external electric field at the position of the dipole, $\alpha (\omega )$ is the dipole polarizability of the particle, $\mathbf {p}^f$ is a fluctuating dipole moment, which, according to the fluctuation–dissipation theorem, is characterized by spectral function of fluctuations [8, 184, 185]:

$$\begin{aligned} {\Big \langle p_i^fp_j^{f*}\Big \rangle _{\omega }}= & {} \frac{\hbar }{\pi }\left( \frac{1}{2}+n(\omega )\right) \mathrm { Im}\alpha (\omega )\delta _{ij}. \end{aligned}$$

(8.28)

Outside the metal, the electric field is given by the sum of the electric field from the point dipole, $\mathbf {E}_d(\mathbf {r}, t)$, the electric field from the metal polarization charge induced by the point dipole, $\mathbf {E}_d^{ ind }(\mathbf {r}, t)$, and the electric field from the metal $\mathbf {E}^s(\mathbf {r}, t)$ in absence of point dipole, and originates from thermal and quantum fluctuation of polarization inside the metal:

$$\begin{aligned} \mathbf {E}^{ total }(\mathbf {r}, t)=\mathbf {E}_d(\mathbf {r}, t)+\mathbf {E}_d^{ ind }(\mathbf {r}, t)+\mathbf {E}^s(\mathbf {r}, t). \end{aligned}$$

(8.29)

The force acting on the dipole is only determined by the last two fields. The electric field $\mathbf {E}_d^{ ind }(\mathbf {r}, t)$ can be written in the form:

$$\begin{aligned} \mathbf {E}_d^{ ind }(\mathbf {r}, t)= & {} \mathbf {E}_{d0}^{ ind }(\mathbf {r},\omega )e^{-\mathrm {i}\omega t}+\mathbf {E}_{d1}^{ ind }(\mathbf {r},\omega +\omega _0)e^{- \mathrm {i}(\omega +\omega _0)t}, \end{aligned}$$

(8.30)

$$\begin{aligned} E_{d0i}^{ ind }(\mathbf {r},\omega )= & {} D_{ij}(\mathbf {r},\mathbf {r}_0,\omega )p_{0j}, \end{aligned}$$

(8.31)

$$\begin{aligned} E_{d1i}^{ ind }(\mathbf {r},\omega +\omega _0)=D_{ij}(\mathbf {r},\mathbf {r}_0,\omega +\omega _0)p_{1j}\,+\,\mathbf {u}_0\cdot \frac{\partial }{\partial \mathbf {r} ^{\prime }}D_{ij}(\mathbf {r},\mathbf {r}^{\prime },\omega +\omega _0)_{ \mathbf {r}^{\prime }=\mathbf {r}_0}p_{0j} \end{aligned}$$

(8.32)

where $\tilde{D}_{ik}(\mathbf {r},\mathbf {r}^{\prime },\omega )=D_{ik}^0( \mathbf {r},\mathbf {r}^{\prime },\omega )+D_{ik}(\mathbf {r},\mathbf {r} ^{\prime },\omega )$ obeys (3.33) and (3.34), and the function $D_{ik}^0(\mathbf {r},\mathbf {r}^{\prime },\omega )$ obeys the inhomogeneous equations (3.33) and (3.34) for free space, i.e. in absence of the medium. The function $D_{ik}(\mathbf {r},\mathbf {r}^{\prime },\omega )$ determines the electric field induced by a unit dipole due to polarization of the medium. Outside the medium, this function has no singularities and obeys the homogeneous equations (3.33) and (3.34). The solution of (3.33) and (3.34) is described in detail in Appendix C. The electric field from the metal $\mathbf {E}^s(\mathbf {r}, t)=\mathbf {E}^s(\mathbf {r},\omega )e^{-\mathrm {i}\omega t}$ is characterized by the following spectral function of fluctuations [10, 11, 15, 183, 185]

$$\begin{aligned} \Big \langle E_i^s(\mathbf {r})E_j^{s*}(\mathbf {r}^{\prime })\Big \rangle _{\omega } =\frac{\hbar }{\pi }\left( \frac{1}{2}+n(\omega )\right) \mathrm {Im}D_{ij}(\mathbf {r},\mathbf {r}^{{\prime } },\omega ). \end{aligned}$$

(8.33)

The electric fields $\mathbf {E}_0$ and $\mathbf {E}_1$ at the position of the point dipole can be found from the condition of self-consistency:

$$\begin{aligned} E_{0i}= & {} D_{ii}(\mathbf {r}_0,\mathbf {r}_0,\omega )p_{0i}+E_i^s(\mathbf {r} _0,\omega ), \end{aligned}$$

(8.34)

$$\begin{aligned} E_{1i}= & {} D_{ii}(\mathbf {r_0},\mathbf {r}_0,\omega +\omega _0)\alpha (\omega +\omega _0)E_{1i}+\mathbf {u}_0\cdot \frac{\partial }{\partial \mathbf {r}} \Biggl (E_i^s(\mathbf {r},\omega )+ \nonumber \\&+D_{ij}(\mathbf {r_0},\mathbf {r},\omega +\omega _0)p_{0j}+D_{ij}(\mathbf {r}, \mathbf {r}_0,\omega )p_{0j}\Biggr )_{\mathbf {r}=\mathbf {r}_0}. \end{aligned}$$

(8.35)

In (8.34) and (8.35), it was given that $D_{ik}(\mathbf {r},\mathbf {r} )=\delta _{ik}D_{ii}(\mathbf {r},\mathbf {r})$. From (8.34)–(8.35):

$$\begin{aligned} E_{0i}=\frac{E_i^s(\mathbf {r}_0)+D_{ii}(\mathbf {r}_0,\mathbf {r}_0,\omega )p_i^f}{1-\alpha (\omega )D_{ii}(\mathbf {r}_0,\mathbf {r}_0,\omega )}, \end{aligned}$$

(8.36)

$$\begin{aligned} p_{0i}=\frac{p_i^f+\alpha (\omega )E_i^s(\mathbf {r}_0,\omega )}{1-\alpha (\omega )D_{ii}(\mathbf {r}_0,\mathbf {r}_0,\omega )}, \end{aligned}$$

(8.37)

$$\begin{aligned} E_{1i}=\frac{\mathbf {\ u}_0\cdot \frac{\partial }{\partial \mathbf {r}}\Biggl ( E_i^s(\mathbf {r})+D_{ij}(\mathbf {r_0},\mathbf {r},\omega +\omega _0)p_{0j}+D_{ij}(\mathbf {r},\mathbf {r}_0,\omega )p_{0j}\Biggr )_{\mathbf {r}= \mathbf {r}_0}}{1-\alpha (\omega +\omega _0)D_{ii}(\mathbf {r}_0,\mathbf {r} _0,\omega +\omega _0)}. \end{aligned}$$

(8.38)

The total electromagnetic force acting on a fluctuating dipole is determined by the Lorentz force:

$$\begin{aligned} \mathbf {F}=\int _{-\infty }^\infty \mathrm {d}\omega \int \mathrm {d}^3r\left( \left\langle \rho \mathbf {E}\right\rangle +\frac{1}{c}\left\langle \mathbf {j} \times \mathbf {B}\right\rangle \right) , \end{aligned}$$

(8.39)

where the integration is over the volume of the dipole

$$\begin{aligned} \mathbf {E}(\mathbf {r}, t)= \mathbf {E}_0(\mathbf {r},\omega ) e^{-\mathrm {i}\omega t}+\mathbf {E}_1(\mathbf {r}, \omega +\omega _0)e^{-\mathrm {i}(\omega +\omega _0)t} \end{aligned}$$

(8.40)

and the magnetic induction field, which can be obtained from the electric field using Maxwell’s equations

$$\begin{aligned} \mathbf {B}(\mathbf {r}, t)=-\mathrm {i}c\mathbf {\nabla }\times \Biggl ( \mathbf {E}_0(\mathbf {r},\omega ) \frac{e^{-\mathrm {i}\omega t}}{\omega }+\mathbf {E}_1(\mathbf {r},\omega +\omega _0)\frac{e^{-\mathrm {i}(\omega +\omega _0)t}}{\omega +\omega _0}\Biggr ). \end{aligned}$$

(8.41)

In (8.39), $\rho (\mathbf {r}, t)$ and $\mathbf {j}(\mathbf {r}, t)$ are the electron and current densities of the dipole, which can be expressed through the polarization density $\mathbf {p}(\mathbf {r}, t)$:

$$\begin{aligned} \rho (\mathbf {r}, t)=-\mathbf {\nabla }\cdot \mathbf {p}(\mathbf {r}, t)=-\frac{\partial }{\partial x_l}\left( p_{0l}(\mathbf {r},\omega )e^{-\mathrm {i}\omega t}+p_{1l}(\mathbf {r},\omega )e^{-\mathrm {i}(\omega +\omega _0)t}\right) , \end{aligned}$$

(8.42)

$$\begin{aligned} \mathbf {j}(\mathbf {r}, t)=\frac{\partial }{\partial t}\mathbf {p}(\mathbf {r} , t)=- i \left( \omega \mathbf {p}_0(\mathbf {r},\omega )e^{-\mathrm {i} \omega t}+(\omega +\omega _0)\mathbf {p}_1(\mathbf {r},\omega )e^{-\mathrm {i} (\omega +\omega _0)t}\right) . \end{aligned}$$

(8.43)

To linear order in the vibrational coordinate $\mathbf {u}(t)$ and frequency $\omega _0$, the total force acting on the point dipole can be written in the form

$$\begin{aligned} \mathbf {F}(t)=\mathbf {F}_{st}(\mathbf {r}_0)+\mathbf {F}_{dc}(t)+\mathbf {F}_{ fric }(t). \end{aligned}$$

(8.44)

Here, the first term determines the conservative van der Waals force at the position $\mathbf {r}=\mathbf {r}_0$. The second term is the change of the conservative van der Waals force during vibration, given by

$$\begin{aligned} \mathbf {F}_{dc}(t)=\mathbf {u}(t)\cdot \frac{\mathrm {d}}{\mathrm {d}\mathbf {r} _0}\mathbf {F}_{st}(\mathbf {r}_0). \end{aligned}$$

(8.45)

The last term in (8.44) determines the friction force:

$$\begin{aligned} \mathbf {F}_{ fric }(t)= i \omega _0\mathop {\mathbf {\Gamma }}\limits ^{\leftrightarrow }\cdot \mathbf {u}(t)=-\mathop {\mathbf {\Gamma }}\limits ^{\leftrightarrow }\cdot \dot{\mathbf {u}}(t). \end{aligned}$$

(8.46)

Using results from Appendix I, for a particle moving parallel to the surface we get:

(8.47)

and for the motion normal to the surface we get

$$\begin{aligned} \Gamma _{\perp }= & {} \frac{2\hbar }{\pi }\int _0^\infty \mathrm {d}\omega \left( -\frac{\partial n}{\partial \omega } \right) \sum _{l=x, y, z}\Biggl \{\frac{\partial ^2}{\partial z\partial z^{\prime }}\Biggl [\mathrm {Im}D_{ll}(\mathbf {r},\mathbf {r}^{\prime },\omega )+ \nonumber \\&+\,\mathrm {Im}\Biggl (\frac{\alpha (\omega )D_{ll}(\mathbf {r},\mathbf {r} _0,\omega )D_{ll}(\mathbf {r}^{\prime },\mathbf {r}_0,\omega )}{1-\alpha (\omega )D_{ll}(\mathbf {r}_0,\mathbf {r}_0,\omega )}\Biggr ) \Biggr ]\mathrm {Im} \frac{\alpha (\omega )}{1-\alpha (\omega )D_{ll}(\mathbf {r}_0,\mathbf {r} _0,\omega )}+ \nonumber \\&+\,\Biggl (\frac{\partial }{\partial z}\mathrm {Im}\Biggl ( \frac{\alpha (\omega )D_{ll}(\mathbf {r},\mathbf {r}_0,\omega )}{1-\alpha (\omega )D_{ll}(\mathbf {r} _0,\mathbf {r}_0,\omega )}\Biggr )\Biggr )^2\Biggr \} _{\begin{array}{c} \mathbf {r}= \mathbf {r}_0 \\ \mathbf {r}^{\prime }=\mathbf {r}_0 \end{array}}. \end{aligned}$$

(8.48)

8.4 Friction Force on Physisorbed Molecules

The sliding of lubricated surfaces has been studied for many years but the microscopic origin of the friction force is still not well understood. During sliding at low velocities, the lubrication fluid will be squeezed out from the contact areas between the two solids, but usually one or a few monolayers of lubrication molecules will be trapped between the surfaces (boundary lubrication). If the lateral corrugation of the adsorbate–substrate interaction potential is weak, as is typically the case for saturated hydrocarbons, then, during sliding, the molecules will slip relative to the surfaces. One important problem in sliding friction is to understand the origin and magnitude of the friction force acting on the individual molecules during slip. If the adsorbate velocity, V, is much smaller than the sound velocity and (for a metallic substrate) the Fermi velocity of the substrate, then the friction force acting on a molecule is proportional to the velocity

$$\begin{aligned} \mathbf {F}=-M\eta \mathbf {V,} \end{aligned}$$

(8.49)

where M is the molecule mass and $\eta$ is the friction coefficient. For insulating surfaces (e.g., most metal oxides), this atomic scale friction $\eta$ can only be due to phonon emission, but on metallic surfaces, both phonon and electronic friction occur and the latter is connected with the energy transfer to the metal conduction electrons.

Information about the friction parameter $\eta$ can be deduced from infrared spectroscopy and inelastic helium scattering measurements since $\eta$ determines the linewidth of adsorbate vibrations if inhomogeneous broadening and pure dephasing processes can be neglected [205, 206]. Information about $\eta$ can also be deduced from quartz crystal microbalance (QCM) measurements.

In the measurements by Krim et al. [238], one side of a quartz crystal was covered by a thin silver or gold film. When a voltage is applied to the crystal, it performs in-plane oscillations. If adsorbates are adsorbed on the metal film, the resulting mass load will decrease the resonance frequency of the QC oscillator. However, Krim et al. [238] also observed an increased damping of the QC oscillator that can only result if, due to the inertia force, the adsorbates slide relative to the metal surface. If the pinning by the corrugated substrate potential can be neglected, then, from the adsorbate-induced change in the resonance frequency and damping of the QC oscillator, one can deduce both the adsorbate concentration and the damping $\eta$. Finally, for metals, the electronic contribution to the friction $\eta$ can be deduced from surface resistivity measurements [203, 239, 240]. In these measurements, the adsorbate-induced change $\Delta R$ of the resistivity of a thin metallic film is measured. It is easy to prove that $\Delta R \sim \eta$ by equalizing the ohmic energy dissipation with the frictional energy dissipation calculated in a reference frame moving with the drift velocity of the conductions electrons.

In this section, we estimate the electronic friction for inert adsorbates on metal surfaces. The metal is treated in the semi-infinite jellium model and the adsorbate–substrate interaction is assumed to consist of the long-range attractive van der Waals interaction plus a short-range repulsion, resulting from the overlap of the electron clouds of the adsorbate and the substrate. We also discuss the relative importance of the phononic and electronic friction for noble gas atoms and for saturated hydrocarbons adsorbed on metal surfaces.

In Sect. 8.4.1, the friction force acting on an adsorbed molecules is studied using the theory of Casimir friction. However, this theory considers only the interaction between molecule and substrate which can be described within the framework of the dielectric formalism. At small separations between molecule and substrate, other processes, which are not described by this formalism, become important. In contrast to Casimir friction, these higher-order processes give a non-vanishing, linear in the velocity, contribution to the friction at $T=0$ K. The friction acting on physisorbed molecules, taking into account the high-order processes, was considered in [140, 241, 243]. It is well known that accurate results for the electron response of metal surfaces requires realistic model for the surface, i.e., the Lang–Kohn model. In early calculations [140, 241], very simple models were used, and the screening by the conduction electrons was neglected, and the accuracy of the results is not clear. The theory of friction of physisorbed molecules presented in Sect. 8.4.2 takes into account high-order processes and screening effects. In this theory, the friction is related to the surface response function for which accurate calculations exist; for example, using the time-dependent local density approximation.

8.4.1 Casimir Friction

The friction force acting on the moving molecule usually is written in the form

$$\begin{aligned} \mathbf {F}=-M\eta \mathbf {V,} \end{aligned}$$

(8.50)

where M is the mass and $\eta$ is the coefficient of friction. For a physisorbed molecule, we can neglect retardation effects. Formally, this corresponds to the limit $c\rightarrow \infty$ in the formulae for the Green’s function in (8.47) and (8.48). In Appendix C, we show that in the non-retarded limit

$$\begin{aligned} D_{xx}(\mathbf {r},\mathbf {r}^{\prime })= & {} \int \frac{\mathrm {d}^2q}{2\pi } \frac{q_x^2}{q}R_p(\mathbf {q},\omega )e^{\mathrm {i}\mathbf {q}(\mathbf {x}- \mathbf {x}^{\prime })-q(z+z^{\prime })}, \end{aligned}$$

(8.51)

$$\begin{aligned} D_{yy}(\mathbf {r},\mathbf {r}^{\prime })= & {} \int \frac{\mathrm {d}^2q}{2\pi } \frac{q_y^2}{q}R_p(\mathbf {q},\omega )e^{\mathrm {i}\mathbf {q}(\mathbf {x}- \mathbf {x}^{\prime })-q(z+z^{\prime })}, \end{aligned}$$

(8.52)

$$\begin{aligned} D_{zz}(\mathbf {r},\mathbf {r}^{\prime })= & {} \int \frac{\mathrm {d}^2q}{2\pi } qR_p(\mathbf {q},\omega )e^{\mathrm {i}\mathbf {q}(\mathbf {x}-\mathbf {x} ^{\prime })-q(z+z^{\prime })}, \end{aligned}$$

(8.53)

$$\begin{aligned} D_{xz}(\mathbf {r},\mathbf {r}^{\prime })= & {} -\mathrm {i}\int \frac{\mathrm {d}^2q }{2\pi }q_xR_p(\mathbf {q},\omega )e^{\mathrm {i}\mathbf {q}(\mathbf {x}-\mathbf { x}^{\prime })-q(z+z^{\prime })}, \end{aligned}$$

(8.54)

where $R_p$ is the reflection amplitude for p-polarized electromagnetic waves. For physical adsorption, for most molecules, the imaginary part of molecule polarizability $\alpha (\omega )$ is non-vanishing vanish only for very high frequencies. Thus, for frequencies that can be excited by thermal radiation ($\omega \le \omega _T =k_BT/\hbar$), $\mathrm {Im}\alpha \approx 0$. Then, to linear order in $\alpha (0)z_0^{-3}$, from (8.47), (8.48) we get

$$\begin{aligned} \eta _{\parallel }= & {} \frac{\hbar \alpha ^2(0)}{2\pi M}\int _0^\infty \mathrm { d}\omega \left( -\frac{\partial n}{\partial \omega }\right) \Biggl [3\frac{\partial ^2}{\partial z^2}\mathrm {Im}D_{zz}(z, z_0)\mathrm {Im}D_{zz}(z_0, z_0)- \nonumber \\&-2\bigg ( \frac{\partial }{\partial z}\mathrm {Im}D_{zz}(z, z_0)\bigg ) ^{2} \Biggr ]_{z=z_0}, \end{aligned}$$

(8.55)

$$\begin{aligned} \eta _{\perp }= & {} \frac{3\hbar \alpha ^2(0)}{\pi M}\int _0^\infty \mathrm {d} \omega \left( -\frac{\partial n}{\partial \omega }\right) \Biggl [\frac{\partial ^2}{\partial z^2}\mathrm {Im}D_{zz}(z, z_0)\mathrm {Im}D_{zz}(z_0, z_0)+ \nonumber \\&+\,\bigg ( \frac{\partial }{\partial z}\mathrm {Im}D_{zz}(z, z_0)\bigg ) ^{2}\Biggr ] _{z=z_0}. \end{aligned}$$

(8.56)

For $z_0<\min (l,\hbar v_F/k_BT)$, where l is the electron mean free path and $v_F$ is the Fermi velocity, the reflection amplitude $R_p(\mathbf {q},\omega )$ must be calculated using non-local optics. The non-local surface contribution to $\mathrm {Im}R_p$ is given by (6.25). Using this expression for $\mathrm {Im}R_p$ in (8.55) and (8.56), we get the surface contribution to the friction:

$$\begin{aligned} \eta _{\parallel surf }=1.9\frac{\xi ^2\hbar \alpha ^2(0)}{Mz_0^8} \Bigg ( \frac{k_BT}{\hbar \omega _p}\Bigg ) ^{2}\frac{1}{(k_Fz_0)^2} \end{aligned}$$

(8.57)

and $\eta _{\perp surf }=8.4\eta _{\parallel surf }$. From low energy electron diffraction studies, it is known that for physical adsorption of Xe on Ag(111), the separation between the Xe nucleus and the jellium edge of Ag(111) is $d=2.4$ Å. The static polarizability of Xe is $\alpha (0)=4.0$ Å$^3$ and $k_F=1.9$ Å$^{-1}$. At room temperature, $k_BT/\hbar \omega _p\sim 10^{-3}$. Using (8.57), we get $\eta _{\parallel }\sim 10^2$ s$^{-1}$ and $\eta _{\perp }\sim 10^3$ s$^{-1}$. These values are a factor $\sim$10$^{-6}$ smaller than estimations obtained in [242] in the framework of a model taking into account the (Pauli) repulsion of the conduction electrons from region occupied by molecule. This means that, in the case of physisorption on normal metals, the friction is determined by higher order processes that are not considered in the present theory. The calculation of friction force taking into account these processes is given in Sect. 8.4.2.

In the case of physisorption on high-resistivity material, we can neglect non-local effects and use a local optic expression for reflection factor

$$\begin{aligned} R_p=\frac{\varepsilon (\omega )-1}{\varepsilon (\omega )+1}, \end{aligned}$$

(8.58)

and the Green’s function is given by

$$\begin{aligned} D_{zz}(z, z^{\prime })=\frac{\varepsilon (\omega )-1}{\varepsilon (\omega )+1} \frac{2}{(z+z^{\prime })^3}. \end{aligned}$$

(8.59)

As an example we consider adsorption of a molecule on SiC. For this case, dielectric function $\varepsilon (\omega )$ is determined by (6.34). Using (8.55), (8.56) and (8.59) for adsorption of a molecule on SiC, in the resonance approximation (see Sect. 6.3) we get

$$\begin{aligned} \eta _{\parallel }=\frac{9 }{128}\frac{\hbar \alpha ^2(0)}{Mz_0^8} \frac{\hbar \omega _a}{k_BT}\frac{1}{\sinh ^2(\hbar \omega _0/2k_BT)} \end{aligned}$$

(8.60)

and $\eta _{\perp }=7\eta _{\parallel }$. For parameters, corresponding to Xe (see above) on SiC (see Sect. 6.3) $\eta _{\Vert }\sim 4.6\times 10^5$ s$^{-1}$ and $\eta _{\perp }\sim 3.2 \times 10^{6}$ s$^{-1}$. Thus, for physical adsorption on semiconductors and dielectrics surfaces, Casimir friction can be many orders of magnitude larger than on metal surfaces. Another possible mechanism of friction on dielectrics is related to the emission of substrate phonons (see Chap. 15). In the case of flat surfaces for the parallel motion of a particle, the effectiveness of this mechanism will be low, and the main mechanism will be determined by Casimir friction.

8.4.2 High-Order Processes

Consider a neutral molecule outside a metal surface. Let $\hat{\mu }$ be the dipole moment operator of the molecule. In the dipole approximation, the molecule–metal interaction Hamiltonian has the form

$$\begin{aligned} H^{\prime }=-\hat{\mu }\cdot \nabla \int d^3x^{\prime }\frac{\hat{\rho }(\mathbf {x}^{\prime })}{|\mathbf {x}_a -\mathbf {x}^{\prime }|}, \end{aligned}$$

(8.61)

where $\rho (\mathbf {x}^{\prime })$ is the charge density operator of the metal electrons and $\mathbf {x}_a$ is the ion-core position of the molecule. Instead of calculating the friction force on a uniformly moving adsorbate, it is more convenient to consider an oscillating adsorbate (both treatments give identical results). Let us write $\mathbf {x}_a = \mathbf {x}_0 + Q_0\mathbf {e}(b + b^+)$, where $Q_0 = (\hbar /2M\omega )^{1/2}$, b and $b^+$ are the annihilation and creation operators for the oscillator. The direction $\mathbf {e}$ of the oscillation will be taken to be $\hat{\mathbf {z}}$ or $\hat{\mathbf {x}}$, where (x, y, z) is a coordinate system with the z-axis normal to the surface, and the positive z direction pointing away from the metal. Expanding (8.61) to linear order in $Q_0$ gives

$$ H^{\prime }=-\hat{\mu }\cdot \nabla \int d^3x^{\prime }\frac{\hat{\rho }(\mathbf {x}^{\prime })}{|\mathbf {x}_0 -\mathbf {x}^{\prime }|}- $$

$$\begin{aligned} -\hat{\mu }\cdot \nabla \mathbf {e}\cdot \nabla \int d^3x^{\prime }\frac{\hat{\rho }(\mathbf {x}^{\prime })}{|\mathbf {x}_0 -\mathbf {x}^{\prime }|}Q_0(b + b^+). \end{aligned}$$

(8.62)

Using expansion in 2D integral Fourier at $(z_0 - z)>0$

$$ \frac{1}{|\mathbf {x}_0 -\mathbf {x}|} = \int \frac{d^2q}{2\pi q}e^{i\mathbf {K}\cdot (\mathbf {x} - \mathbf {x}_0)}, $$

where $\mathbf {K}=\mathbf {q} - iq{\hat{\mathbf {z}}}$, and $\mathbf {q}=(q_x, q_y)$, (8.62) can be transformed to

$$\begin{aligned} H^{\prime } = \mu _i\int d^3x \hat{\rho }(\mathbf{{x}}) V_i(\mathbf{{x}})+ \mu _ie_j\int d^3x \hat{\rho }(\mathbf{{x}}) V_{ij}(\mathbf{{x}})Q_0 (b + b^+), \end{aligned}$$

(8.63)

where

$$\begin{aligned} V_i(\mathbf{{x}}) = \int \frac{d^2q}{2\pi q}K_ie^{i\mathbf{{K}}\cdot (\mathbf{{x}} - \mathbf{{x}}_0)}, \end{aligned}$$

(8.64)

$$\begin{aligned} V_{ij}(\mathbf{{x}}) = \int \frac{d^2q}{2\pi q}K_{i}K_{j}e^{i\mathbf{{K}}\cdot (\mathbf{{x}} - \mathbf{{x}}_0)}. \end{aligned}$$

(8.65)

Now, assume that the molecule is in its electronic ground state $\left| A \right. \rangle$, and the oscillator is in its first vibrational excited state $(n=1)$. Consider the decay rate w from $n=1\rightarrow n=0$. Since $\langle A\left| \hat{\mu }\right| A\rangle =0$, we must go to the second order in perturbation theory when calculating w:

$$\begin{aligned} w=\frac{2\pi }{\hbar }\sum _f\Bigg |\Big \langle A, f, n=0\Bigg |H^{\prime }\frac{1}{H_0-E_0}H^{\prime }\Bigg |A, i, n=1\Big \rangle \Bigg |^{2} \delta (E_f-E_i-\hbar \omega ), \end{aligned}$$

(8.66)

where $E_0 = E_A + E_i+\hbar \omega$ is the initial energy and where $\left| f\right. \rangle$ and $\left| i\right. \rangle$ denote the final and initial states of the metal. Now, two final states are possible for the metal, namely with (a) one or (b) two electron-hole pair excitations, as indicated in Fig. 8.3.

Fig. 8.3

Elementary processes involved in the damping of the adsorbate vibration: a A process that results in a one electron-hole pair. This process is taken into account in the theory of Casimir friction. b The process that results in two electron-hole pairs, this process is not taken into account in the theory of Casimir friction

Processes (a) can be described in the framework of the dielectric formalism and are taken into account with the theory of Casimir friction (see Sect. 8.4.1). As shown in Sect. 8.4.1, the contribution to the friction from this process vanishes at zero temperature. Processes (b) are not taken into account in the theory of Casimir friction. However, as will be shown below, at short separations, these processes can give a larger contribution to friction than the Casimir friction. Due to the fact that the transitions from the initial state $\left| i\right. \rangle$ to the final state $\left| f\right. \rangle$ in Fig. 8.3 proceed in two steps, the interaction

$$ \hat{V} =H^{\prime }\frac{1}{H_0-E_0}H^{\prime } $$

is non-local. However, we can formally define an effective local potential $V_{eff}(\mathbf {x})$ in such a way that the matrix element $M_{if} = e\langle |V_{eff}(\mathbf {x})|\rangle$ equals the matrix element associated with the two-step process:

$$\begin{aligned} M_{if} = \Big< f\left| \Big < A, n=0\left| H^{\prime }\frac{1}{H_0-E_0}H^{\prime }\right| A, n=1\Big> \right| i \Big >. \end{aligned}$$

(8.67)

If we assume that $\omega _a \gg \omega _p$, where $\hbar \omega _a = E_B - E_A$ is the energy of the virtual atomic transition and $\omega _p$ is the metal plasma frequency, we can neglect the screening during the electronic transitions $kA \rightarrow k^{\prime \prime }B \rightarrow k^{\prime }A$, in Fig. 8.3. This follows from the fact that when $\omega _a \gg \omega _p$ the electrons in the metal have no time to follow the rapidly changing potential during such a virtual transition. On the other hand, the system stays a long time in the final state and screening in this state is important. Substituting (8.66) in (8.67) and neglecting screening effects during the rapid transitions, and assuming $|\epsilon _F - \epsilon _{k^{\prime \prime }}| \ll \hbar \omega _a$, gives

$$M_{fi}=\frac{1}{2}\alpha _{ij}(0)e^2\int d^3xd^3x^{\prime }\varphi _k{(\mathbf {x})}\varphi _{k^{\prime }}^{*}(\mathbf {x}^{\prime })\Psi (\mathbf {x, x}^{\prime })\times $$

$$\begin{aligned} \times \left[ V_i(\mathbf{{x}}) V_{jl}(\mathbf{{x}}^{\prime })+ V_i(\mathbf{{x}}^{\prime }) V_{jl}(\mathbf{{x}})\right] e_l, \end{aligned}$$

(8.68)

where $\varphi _k$, $\varphi _{k^{\prime }}$, $\varphi _{k^{\prime \prime }}$ are the wave functions of the electron in the initial, final and intermediate states, respectively. The function

$$\begin{aligned} \Psi (\mathbf{{x, x}}^{\prime })=\sum _{k^{\prime \prime }}(1-2n_{k^{\prime \prime }})\varphi _{k^{\prime \prime }}(\mathbf{{x}})\varphi _{k^{\prime \prime }}^{*}(\mathbf{{x}}^{\prime }), \end{aligned}$$

(8.69)

where $n_{k^{\prime \prime }}$ is the Fermi distribution function ($n_k =1$, for $k<k_F$, and zero otherwise). The static polarizability for spherical molecule $\alpha _{ij}(0)=\alpha (0)\delta _{ij}$, where

$$\begin{aligned} \alpha (0)=2\sum _B\frac{\big | \left\langle B|\hat{\mu }_x|A\right\rangle \big | ^{2}}{E_B-E_A}. \end{aligned}$$

(8.70)

Now, let us consider the function $\Psi (\mathbf {x, x}^{\prime })$. The second part of (8.69) includes summation over occupied states and this function is only significant for $z, z'< 0$ where $z=0$ corresponds to the jellium edge. On the other hand, the product $V_i(\mathbf{{x}})V_{jl}(\mathbf{{x'}})$ is rather small in this region of space due to the rapid decay of this function with increasing distance from the the jellium edge. In this region, we can neglect the second term in (8.69) to get

$$\begin{aligned} \Psi (\mathbf{{x, x}}^{\prime })\approx {\sum _{k^{\prime \prime }}\varphi _{k^{\prime \prime }}^{}(\mathbf{{x}})}\varphi _{k^{\prime \prime }}^{*}(\mathbf{{x}}^{\prime }{} \mathbf{{)=}}\,\delta (\mathbf{{x}}-\mathbf{{x}}^{\prime }). \end{aligned}$$

(8.71)

The matrix element (8.68) can now be written as

$$\begin{aligned} M_{fi}=\alpha (0)e^2\int d^3x\varphi _k(\mathbf{{x}})\varphi _{k^{\prime }}^{*}(\mathbf{{x}}) V_i(\mathbf{{x}})V_{il}(\mathbf{{x}}) e_l. \end{aligned}$$

(8.72)

so that

$$\begin{aligned} V_{eff}(\mathbf{{x}}) = \alpha (0)e V_i(\mathbf{{x}})V_{il}(\mathbf{{x}}) e_l. \end{aligned}$$

(8.73)

With this effective potential, according to the Kubo formula, the friction coefficient is given by [205, 206]

$$\begin{aligned} \eta =\frac{e^2}{M\omega }\int d^3xd^3x^{\prime }V_{eff}(\mathbf{{x}})\mathrm{{Im}}\chi (\mathbf{{x, x}}^{\prime },\omega )V_{eff}(\mathbf{{x}}^{\prime }), \end{aligned}$$

(8.74)

where $\chi$ is the density–density correlation function

$$\begin{aligned} \chi =\frac{i}{\hbar }\int _0^\infty dte^{i\omega t}\left\langle \left[ \hat{\rho }(\mathbf{{x}}, t),\hat{\rho }(\mathbf{{x}}^{\prime }, 0)\right] \right\rangle . \end{aligned}$$

(8.75)

In order to relate $\eta$ to the reflection amplitude $R_p(q,\omega )$, we must expand $V_{eff}$ in evanescent plane waves. Such an expansion is exact only in a region of space where the function to be expanded satisfies the Laplace equation. In the present case, $V_{eff}$ does not satisfy the Laplace equation anywhere, but in the surface region of the metal it is nevertheless possible to approximate this function accurately with a sum of evanescent plane waves (see Appendix L). We get

$$\begin{aligned} V_{eff}\approx \int \frac{d^2q}{(2\pi )^2}V_{\mathbf{{q}}}e^{i\mathbf{{q\cdot x}}_{\Vert }+qz}, \end{aligned}$$

(8.76)

where

$$ V_{\mathbf{{q}}}=2q\int d^3xV_{eff}e^{-i\mathbf{{q\cdot x}}_{\Vert }+qz}. $$

Using (8.64), (8.65) and (8.73), we get

$$\begin{aligned} V_{\mathbf{{q}}}=2i\alpha (0)e\int d^2q^{\prime }\frac{qK_i^{\prime }K_i^{\prime \prime }K_j^{\prime \prime }e_j}{q^{\prime }q^{\prime \prime }(q+q^{\prime }+q^{\prime \prime })}e^{-i(q^{\prime }+q^{\prime \prime })z_0}, \end{aligned}$$

(8.77)

where $\mathbf{{q}}^{\prime \prime }=\mathbf{{q}} - \mathbf{{q}}^{\prime }$, $\mathbf{{K}}^{\prime }= \mathbf{{q}}^{\prime } - iq^{\prime }\hat{z}$ $\mathbf{{K}}^{\prime \prime }= \mathbf{{q}}^{\prime \prime } - iq^{\prime \prime }\hat{z}$. Using (8.76) in (8.74), we get

$$\begin{aligned} \eta =\frac{e^2}{M\omega }\int \frac{d^2q}{(2\pi )^2}\int dzdz^{\prime }\big | V_{\mathbf{{q}}}\big | ^{2}e^{qz+qz^{\prime }}\mathrm{{Im}}\chi (\mathbf{{q}}, z, z^{\prime }, \omega ). \end{aligned}$$

(8.78)

According to the linear response theory, in the non-retarded limit, reflection amplitudes for p-polarized evanescent waves can be written in the form [244]

$$\begin{aligned} R_p(\mathbf{{q}},\omega ) =\frac{2\pi }{q}\int dzdz^{\prime }e^{qz+qz^{\prime }}\chi (\mathbf{{q}}, z, z^{\prime }, \omega ). \end{aligned}$$

(8.79)

From (8.78) and (8.79) we get

$$ \eta =\frac{e^2}{2\pi M\omega }\int \frac{d^2q}{(2\pi )^2}\big | V_{\mathbf{{q}}}\big | ^{2}q \mathrm{{Im}}R_p ({q},\omega )= $$

$$\begin{aligned} =\frac{e^2\alpha ^2(0)}{\pi ^2M\omega }\int d^2qq^3{\left| \int d^2q^{\prime }\frac{\mathbf{{K^{\prime }\cdot K^{\prime \prime }K^{\prime \prime }\cdot e }}}{q^{\prime }q^{\prime \prime }(q+q^{\prime }+q^{\prime \prime })}e^{-(q^{\prime }+q^{\prime \prime })z_0}\right| } ^2\mathrm{{Im}}R_p(q,\omega ). \end{aligned}$$

(8.80)

At low frequencies, we have (see [245])

$$ \mathrm{{Im}}R_p(q,\omega )=2\omega q\xi (q)/k_F\omega _p, $$

and (8.80) becomes

$$\begin{aligned} \eta =\frac{2e^2\alpha ^2(0)}{\pi ^2Mk_F\omega _p}\int d^2qq^4\xi (q) {\left| \int d^2q^{\prime }\frac{\mathbf{{K^{\prime }\cdot K^{\prime \prime }K^{\prime \prime }\cdot e }}}{q^{\prime }q^{\prime \prime }(q+q^{\prime }+q^{\prime \prime })}e^{-(q^{\prime }+q^{\prime \prime })z_0}\right| }^2. \end{aligned}$$

(8.81)

For $\mathbf{{e}}=\hat{x}$

$$\begin{aligned} \eta _{\Vert }=\frac{e^2}{\hbar a_0}\frac{\big [ k_F^3\alpha (0)\big ] ^{2}}{(k_Fz_0)^{10}}\frac{m}{M}\frac{\omega _F}{\omega _p}k_Fa_0I_{\Vert }, \end{aligned}$$

(8.82)

where

$$ I_{\Vert }=z_0^{10}\frac{4}{\pi ^2}\int d^2qq^4\xi (q){\left| \int d^2q^{\prime }\frac{(\mathbf{{q}}^{\prime }\cdot \mathbf{{q}}^{\prime \prime }-q^{\prime }q^{\prime \prime })(q_x-q_x^{\prime })}{q^{\prime }|\mathbf{{q-q}}^{\prime }|(q+q^{\prime }+ |\mathbf{{q-q}}^{\prime }|)}e^{-\{q^{\prime }+|\mathbf{{q-q}}^{\prime }|)z_0}\right| }^2, $$

and for $\mathbf{{e}}=\hat{z}$

$$\begin{aligned} \eta _{\Vert }=\frac{e^2}{\hbar a_0}\frac{\big [ k_F^3\alpha (0)\big ] ^{2}}{(k_Fz_0)^{10}}\frac{m}{M}\frac{\omega _F}{\omega _p}k_Fa_0I_{\perp }, \end{aligned}$$

(8.83)

where

$$ I_{\perp }=z_0^{10}\frac{4}{\pi ^2}\int d^2qq^4\xi (q) {\left| \int d^2q^{\prime }\frac{(\mathbf{{q}}^{\prime }\cdot \mathbf{{q}}^{\prime \prime }-q^{\prime }q^{\prime \prime })}{q^{\prime }(q+q^{\prime }+ |\mathbf{{q-q}}^{\prime }|)}e^{-\{q^{\prime }+|\mathbf{{q-q}}^{\prime }|)z_0}\right| }^2. $$

In (8.82) and (8.83), $a_{0}$ is the Bohr radius and $k_F$ the Fermi wave vector. The function $\xi (q)$ has been calculated using the time-dependent local density approximation (TDLDA). The numerical results [246] are accurately approximated by

$$\begin{aligned} \xi \approx \xi _0/\left[ 1+a(q/k_F)^3\right] , \end{aligned}$$

(8.84)

where ($xi_0, a$) = (0.89, 6.25) and (0.43, 2.49 for $r_s=2$ and 3, respectively. Using (8.82) and (8.83), the integrals $I_{\Vert }$ and $I_{\perp }$ have been calculated numerically [243] and the results are displayed in Fig. 8.4 as a function of $k_Fz_0$ where $z_0$ is the distance between the nucleus.

Fig. 8.4

Frictional integrals $I_{\Vert }$ and $I_{\perp }$ (defined in the text) as a function of $k_Fz_0$, where $z_0$ is the distance of the center of the adsorbate from the jellium edge of the substrate. $r_s$ is Wigner–Zeitz radius in atomic units

8.4.3 Comparison of the Theory with Experiment

For inert atoms and molecules adsorbed on metal surfaces, one can (approximately) distinguish between two contributions to the electronic friction associated with (a) the long-ranged attractive van der Waals interaction and (b) the short-ranged Pauli repulsion associated with the overlap of the electron clouds of the adsorbate and the substrate. The latter contribution to $\eta _{\Vert }$ has, for Xe on Ag(111), been estimated [242] to be $\approx 6 \times 10^7$ s$^{-1}$, and will now be compared with the contribution from the van der Waals friction. From low-energy electron diffraction studies, it is known that the separation between the Xe nucleus and the jellium edge of Ag(111) is $d=2.4$ Å. The static polarizability of Xe is $\alpha (0)=4.0$ Å$^3$. From (8.82), the contribution to $\eta _{\Vert }$ from the van der Waals interaction is estimated to be $\eta _{\Vert }\approx 4\times 10^7$ s$^{-1}$, i.e., only 30 % smaller than that associated with the Pauli repulsion. The fact that the two contributions are of a similar magnitude is probably related to the fact that at the equilibrium separation, the attractive and repulsive adsorbate–substrate interactions are of identical magnitude, which should result in dissipative forces of similar magnitudes.

The electronic friction for Xe on Ag(111) can be deduced from surface resistivity [247] data, $\eta _{\Vert } \sim 3\times 10^8$ s$^{-1}$. This value is only a factor of three larger than estimated above, but it is likely that a non-negligible contribution to the electronic friction comes from ‘chemical’ effects, namely from the fact that the Xe-6s electronic resonance state is located around the vacuum energy [248] with a tail extending down to the Fermi energy. In [242], the chemical contribution to $\eta _{\Vert }$ was estimated to be $\sim$1.5$\,\times 10^8$ s$^{-1}$. For the lighter noble gas atoms and for saturated hydrocarbonates, chemical effects should be negligible since no electronic resonance states occur close to the Fermi energy. An example is $C_2H_6$, for which surface resistivity data gives [239, 240] $\eta \sim 3\times 10^8$ s$^{-1}$. Since $C_2H_6$ has almost the same binding energy as Xe, and since the effective Lennard-Jones radius of $C_2H_6$ is practically identical to that of Xe, the theoretical $\eta _{\Vert }$, derived above for Xe should also be valid for $C_2H_6$ when scaled by the mass ratio $M(Xe)/M(C_2H_4)=4.4$. This gives the electronic friction $\eta _{\Vert } \sim 4\times 10^8$ s$^{-1}$, which is in good agreement with surface resistivity data.

Table 8.1 Resonance frequencies and damping (full width at half-maximum) of the vibrational modes (frustrated translations) normal to the surface for hydrocarbons physisorbed on Cu(100). The calculated damping rates are obtained from (8.85) using the observed resonance frequencies. Experimental data are from [249]

Many lubrication fluids consist of long-chain hydrocarbons but very little experimental data relevant for sliding friction is available for such molecules adsorbed on metallic surfaces. However, Witte and Wöll [249] have performed inelastic He-atom scattering from the saturated hydrocarbons n-hexane, n-octane, n-decane, and cyclohexane adsorbed on Cu(100). The perpendicular vibrations of all these molecules was found to be $\hbar \omega _{\perp }\approx 7$ meV. In the most simple interaction model between a physisorbed molecule and a metal surface, the interaction strength, and the force constant, are proportional to $\alpha (0)$. Therefore the perpendicular vibrational energies should be proportional to $\sqrt{\alpha (0)/M}$. Witte and Wöll have shown that $\sqrt{\alpha (0)/M}$ is almost equal for all the saturated hydrocarbons given above, indicating that, as expected, all these molecules are physisorbed on Cu(100). In Table 8.1, we have summarized the observed linewidth $\gamma$ of the perpendicular adsorbate vibrations. Assuming that the linewidth is due to energy relaxation, one can deduce the friction $\hbar \eta _{\perp }=\gamma$. This gives $\eta _{\perp }\sim 2\times 10^{12}$ s$^{-1}$ for the systems quoted in Table 8.1. The large magnitudes of the frictions quoted in Table 8.1 cannot be explained as resulting from the electronic contribution which gives $(\eta _{\perp })_{el}\sim 1\times 10^9$ s$^{-1}$, i.e., roughly three orders of magnitude smaller than the observed friction. When $\omega _{\Vert }\ll \omega _D$ (where $\omega _D \approx 30$ is the Debye frequency), the phononic friction is accurately given by [250]

$$\begin{aligned} \gamma _{\perp }=\frac{3 }{8\pi }\frac{M}{\rho } {\left( \frac{\omega _{\perp }}{c_t}\right) }^3\omega _{\perp }, \end{aligned}$$

(8.85)

where $c_t$ is the transverse sound velocity and $\rho$ the mass density. The friction values calculated from this formula are in close agreement with experiment, see Table 8.1. The reason for the importance of the phonon friction in these cases is the relative high frequency of the perpendicular vibrational modes (note that: $\eta _{ph}\sim \omega _{\perp }^4$) while the electronic friction $\eta _{el}$ is independent of the resonance frequency $\omega _{\perp }$ of the adsorbate vibration). Witte and Wöll could not detect any adsorbate vibrations parallel to the surfaces, which indicates that these modes have a frequency that is too low frequency ($\hbar \omega _{\Vert }<0.3$ meV), to be detected with the resolution of the He-atom equipment. Using the formula for the phononic friction,

$$\begin{aligned} \gamma _{\Vert }=\frac{3 }{8\pi }\frac{M}{\rho }{\left( \frac{\omega _{\Vert }}{ c_t}\right) }^3\omega _{\Vert }, \end{aligned}$$

(8.86)

gives $\eta _{\Vert }< 8\times 10^6$ s$^{-1}$. This should be compared with the calculated electronic friction $(\eta _{\Vert })_{el} \sim 3\times 10^8$ s$^{-1}$. Hence, the parallel friction is mainly of electronic origin.

8.5 Force on a Particle in a Thermal Field

8.5.1 The Case of Small Velocities

Casimir friction also occurs when a particle moves relative to black-body radiation; for example, relative to the walls of an oven, or relative to the cosmic microwave background. This kind of friction has no position dependence, i.e. it is spatially homogeneous. The consequence is a universal dissipative drag acting on all matter in relative motion with respect to a thermalized photon gas. To calculate this universal drag to linear order in velocity, we can use the same approach as in Sect. 8.3. Assuming that the size of the particle is smaller than $\lambda _T$, we get the friction coefficient for a particle moving relative to black-body radiation.

$$\begin{aligned} \Gamma ^{BB}=\frac{2\hbar }{\pi }\int _0^\infty d\omega \left( -\frac{\partial n }{\partial \omega }\right) \sum _{k=x, y, z}\mathrm {Im}\alpha _{kk}\frac{\partial ^2}{ \partial x\partial x^{\prime }}\mathrm {Im}D_{kk}^{BB}(\mathbf {r},\mathbf {r}^{\prime },\omega )\Big |_{\mathbf {r}=\mathbf {r}^{\prime }}, \end{aligned}$$

(8.87)

where $\alpha _{kk}$ is the polarizability of the particle, and $D_{kk}^{BB}(\mathbf {r},\mathbf {r}^{\prime },\omega )$ is the Green’s function of the black-body radiation. For a spherical particle $\alpha _{kk}=\alpha$, and using the formula (see [107] and also Sect. 3.1.2)

$$\begin{aligned} \sum _{k=x, y, z}D_{kk}^{BB}(\mathbf {r},\mathbf {r}^{\prime },\omega )=2\left\{ \frac{\omega ^2}{c^2}\frac{1}{|\mathbf {r}-\mathbf {r}^{\prime }|}\exp \left( \frac{i\omega }{c}|\mathbf {r}-\mathbf {r}^{\prime }|\right) -2\pi \delta (\mathbf {r}-\mathbf {r}^{\prime })\right\} , \end{aligned}$$

(8.88)

we get

$$\begin{aligned} \Gamma ^{BB}=\frac{\beta \hbar ^2}{3\pi c^5}\int _0^\infty d\omega \frac{ \omega ^5\mathrm {Im}\alpha (\omega )}{\sinh ^2\big (\frac{1}{2}\beta \hbar \omega \big )}, \end{aligned}$$

(8.89)

where $\beta ^{-1}=k_BT$. Equation (8.124) was first obtained in [157] using a different approach. The photon gas exerts a drag on any polarizable particle that moves with respect to the reference frame in which the photon gas is thermalized, and this drag is proportional to the relative velocity.

Tungsten ovens can operate at temperatures as high as 3000 K. If a beam of atoms, ions or molecules passes through such an oven, it will be subject to drag due to the Casimir friction mediated by the thermal radiation. For an atom or molecule, the polarizability $\alpha$ can be characterized by a single absorption line at $\omega =\omega _0$. In this case, $\mathrm {Im}\alpha (\omega )=\alpha _0\delta (\omega /\omega _0-1)$, where $\alpha _0$ is the static polarizability at $\omega =0$.

Setting $m/\tau =\Gamma ^{BB}$, where m is the mass of the molecule and $\tau$ is the relaxation time, gives

$$\begin{aligned} \tau =\frac{3\pi mc^5\hbar ^4}{2^6\alpha _0(k_BT)^5}\frac{\sinh ^2(x)}{x^6}, \end{aligned}$$

(8.90)

where $x=\beta \hbar \omega _0/2$. The relaxation time has a minimum at a temperature-dependent frequency that coincides with the minimum of the function $f(x)=\sinh ^2(x)/x^6$, at $x_m=2.98$, where $f(x_m)=0.137$. Ba$^{+}$ has a resonance near 2 eV, which is approximately six times the thermal energy associated with a 3000 K oven. For this resonance, the relaxation time would be near the minimum, and for the ion polarizability $\alpha _0\approx 1.0\times 10^{-30}$ m$^3$, one obtains the relaxation time $\approx 10^5$ s, i.e. 1 day. This relaxation time can be measured using ion traps.

For the cosmos, it is believed that hydrogen atoms condensed from protons and electrons when the radiation cooled to approximately 3000 K, and that the coupling of the cosmic radiation to matter due to Compton scattering becomes ineffective below this condensation temperature [251]. However atoms, ions, and molecules with absorption in the appropriate frequency range should remain coupled to the cosmic radiation as its temperature drops from 3000 K to perhaps 300 K. This coupling could influence the structure and anisotropies observed in recent experiments on the cosmic microwave background [252]. It could also influence the behavior of molecules formed from the residue of novas and supernovas, and then subject to drag from a still hot cosmic microwave (i.e. electromagnetic) background. At much lower temperatures, macroscopic bodies can coalesce, in which geometrically determined resonances may become relevant.

8.5.2 Relativistic Case

We introduce two reference frames, K and $K^{\prime }$. K is the frame of black-body radiation, and $K^{\prime }$ moves with velocity V along the x axis. In the $K^{\prime }$ frame a particle moves with time-dependent velocity $v^{\prime }\ll V$ along the x-axis. In the $K^{\prime }$ frame, we assume the velocity vanishes at $t=t_0$, $v^{\prime }(t_0)=0$. However, since $v^{\prime }\ll V$, the difference between the friction forces in the $K^{\prime }$ frame and the particle rest reference frame can be neglected, and in this section the $K^{\prime }$ frame is denoted as the particle rest reference frame. The relation between the x components of the momentum in the different reference frames is given by

$$\begin{aligned} p_x=(p_x^{\prime }+\beta E^{\prime }/c)\gamma , \end{aligned}$$

(8.91)

where $\beta =V/c$, $\gamma = 1/\sqrt{1-\beta ^2}$, $E^{\prime }=E_0/\sqrt{1-(v^{\prime }/c)^2}$ is the total energy of a particle in the $K^{\prime }$ frame, and $E_0=m_0c^2$ is the rest energy of a particle. The rest energy can change due to the absorption and emission of thermal radiation by a particle. The connection between forces in the K and $K^{\prime }$ frames follows from (8.91)

$$\begin{aligned} \frac{dp_x}{dt}=\frac{1}{1+(Vv^{\prime }/c^2)}\left[ \frac{dp_x^{\prime }}{dt^{\prime }}+V \frac{dm_0}{dt^{\prime }}\frac{1}{\sqrt{1-(v^{\prime }/c)^2}}+m_0V\frac{d}{dt^{\prime }}\left( \frac{1}{\sqrt{1-(v^{\prime }/c)^2}}\right) \right] . \end{aligned}$$

(8.92)

For $v^{\prime }\ll V$ from (8.92) we get

$$\begin{aligned} F_x=F_x^{\prime }+V \frac{dm_0}{dt^{\prime }}, \end{aligned}$$

(8.93)

where $F_x$ and $F_x^{\prime }$ are the forces in the K and $K^{\prime }$ frames, respectively. The last term in (8.93) determines the rate of change of the momentum of a particle in the K frame due to the change of its rest mass as a result of the absorption and emission of radiation by the particle. Taking into account that at $v^{\prime }\ll V$

$$\begin{aligned} \frac{dp_x}{dt} = \frac{d}{dt}\left( \frac{m_0v}{\sqrt{1-(v/c)^2}}\right) =\frac{dm_0}{dt^{\prime }}V + m_0\gamma ^3\frac{dv}{dt}, \end{aligned}$$

(8.94)

from (8.93) we get

$$\begin{aligned} m_0\gamma ^3\frac{dv}{dt}=\frac{dp_x^{\prime }}{dt^{\prime }}=F_x^{\prime }, \end{aligned}$$

(8.95)

where v is the velocity of a particle in the K frame. From (8.95) it follows that acceleration in the K frame is determined by the friction force in the $K^{\prime }$ frame.

A particle will move with a constant velocity ($v=$ const) if an external force $f_x$ is applied to it such that

$$\begin{aligned} \gamma V\frac{dm_0}{dt}=F_x+f_x. \end{aligned}$$

(8.96)

If the force $f_x$ does not change the rest mass of a particle, then its value is the same in the K and $K^{\prime }$ frames, i.e., $f_x=f_x^{\prime }$. In this case

$$\begin{aligned} f_x=-F_x+\gamma V\frac{dm_0}{dt}=-F_x+\frac{dm_0}{dt^{\prime }}V=-F_x^{\prime }. \end{aligned}$$

(8.97)

Thus, for uniform motion of a particle, an external force must be applied which is equal, but opposite in sign, to the friction force acting on the particle in the $K^{\prime }$ frame.

In the $K^{\prime }$-reference frame, the Lorenz force on the particle is determined by the expression

$$\begin{aligned} F_x^{\prime }=\int d^3\mathbf {r}^{\prime }\Big \langle \rho ^{\prime } E_x^{\prime *}\Big \rangle +\frac{1}{c}\int d^3\mathbf {r}^{\prime }\Big \langle \big [\mathbf {j^{\prime }\times B}^{\prime *}\big ]_x\Big \rangle , \end{aligned}$$

(8.98)

where $\mathbf {E}^{\prime }$ and $\mathbf {B}^{\prime }$ are the electric and induction field, respectively, and $\rho ^{\prime }$ and $\mathbf {j}^{\prime }$ are the charge and current densities of the particle, respectively, which can be written in the form

$$\begin{aligned} \rho ^{\prime }=-\mathbf {\nabla ^{\prime } \cdot P^{\prime }}, \end{aligned}$$

(8.99)

$\mathbf {j}^{\prime }= \mathbf {j}_d^{\prime } + \mathbf {j}_m^{\prime }$, where

$$\begin{aligned} \mathbf {j}_d^{\prime } = \frac{\partial \mathbf {P}^{\prime }}{\partial t^{\prime }} \end{aligned}$$

(8.100)

$$\begin{aligned} \mathbf {j}_m^{\prime } = c [\mathbf {\nabla ^{\prime } \times M^{\prime }}], \end{aligned}$$

(8.101)

where $\mathbf {P}^{\prime }$ and $\mathbf {M}^{\prime }$ are the vectors of polarization and magnetization, respectively. For the point particle

$$\begin{aligned} \mathbf {P}^{\prime }=\mathbf {p}_d^{\prime }\delta (\mathbf {r}^{\prime }-\mathbf {r}^{\prime }_0), \end{aligned}$$

(8.102)

$$\begin{aligned} \mathbf {M}=\mathbf {p}_m^{\prime }\delta (\mathbf {r}^{\prime }-\mathbf {r}^{\prime }_0), \end{aligned}$$

(8.103)

where $\mathbf {p}_d^{\prime }$ and $\mathbf {p}_m^{\prime }$ are the dipole and magnetic moments, respectively. The force that acts on the particle, can be written in the form $F_x^{\prime } = F_{xd}^{\prime } + F_{xm}^{\prime }$, where $F_{xd}^{\prime }$ and $F_{xm}^{\prime }$ are the forces, which act on the dipole and magnetic moments, respectively. Taking into account (8.99) and (8.100) the force, acting on the dipole, can be written in the form

$$\begin{aligned} F_{xd}^{\prime }=-\int d^3\mathbf {r}^{\prime }\left\langle \mathbf {\nabla \cdot P}^{\prime } E_x^{\prime *}\right\rangle +\frac{1}{c}\int d^3\mathbf {r}^{\prime }\left\langle \bigg [\frac{\partial \mathbf {P}^{\prime }}{\partial t}\times \mathbf {B}^{\prime *}\bigg ]_{x}\right\rangle . \end{aligned}$$

(8.104)

The integrand in the second term in (8.104) can be written as

$$\begin{aligned} \left\langle \bigg [\frac{\partial \mathbf {P}^{\prime }}{\partial t}\times \mathbf {B}^{\prime *}\bigg ]_{x} \right\rangle = \frac{\partial }{\partial t^{\prime }} \left\langle \bigg [\mathbf {P}^{\prime } \times \mathbf {B}^{\prime *}\bigg ]_{x} \right\rangle - \left\langle \bigg [\mathbf {P}^{\prime } \times \frac{\partial \mathbf {B}^{\prime *}}{\partial t^{\prime }}\bigg ]_{x} \right\rangle . \end{aligned}$$

(8.105)

Due to homogeneity of time, the first term in (8.105) is equal to zero, and taking into account the Maxwell’s equation

$$\begin{aligned} \mathbf {\nabla ^{\prime } \times E^{\prime }} = - \frac{1}{c}\frac{\partial \mathbf {B}^{\prime }}{\partial t^{\prime }}, \end{aligned}$$

(8.106)

the second term can be written in the form

$$\begin{aligned} - \left\langle \bigg [\mathbf {P}^{\prime } \times \frac{\partial \mathbf {B}^{\prime *}}{\partial t^{\prime }}\bigg ]_{x} \right\rangle = \left\langle \bigg [\mathbf {P}^{\prime } \times [\mathbf {\nabla \times E^{\prime }}] \bigg ]_{x} \right\rangle . \end{aligned}$$

(8.107)

After substitution of (8.107) in (8.104) and integration we get

$$\begin{aligned} F_{xd}^{\prime }=\left\langle \mathbf { p_d^{\prime }\cdot \nabla }^{\prime } E_x^{\prime *}\right\rangle _{\mathbf {r^{\prime } =r^{\prime }_0}} + \left\langle \big [ \mathbf {p}^{\prime }_d \times [\mathbf {\nabla ^{\prime } \times E^{\prime *}}]\big ]_x\right\rangle _{\mathbf {r^{\prime } =r^{\prime }_0}}. \end{aligned}$$

(8.108)

Taking into account the vector equality

$$\begin{aligned} \Big [ \mathbf {p}^{\prime }_d \times \big [\mathbf {\nabla ^{\prime } \times E^{\prime *}}\big ]\Big ] = \mathbf {\nabla ^{\prime }}(\mathbf {p^{\prime }_d \cdot E^{\prime *}}) - \mathbf { (p_d^{\prime }\cdot \nabla ^{\prime }) E^{\prime *}} \end{aligned}$$

(8.109)

we get

$$\begin{aligned} F_{xd}^{\prime }= \frac{\partial }{\partial x^{\prime }}\big \langle \mathbf { p_d^{\prime }\cdot E^{\prime *}}\big \rangle _{\mathbf {r^{\prime } =r^{\prime }_0}}. \end{aligned}$$

(8.110)

To calculate $F_{xm}$ we use vector equality

$$\begin{aligned} \mathbf {\bigg [\big [\nabla \times M^{\prime }\big ]\times B^{\prime *}\bigg ]}_x = -\left( \frac{\partial \mathbf {M}^{\prime }}{\partial x^{\prime }}\cdot \mathbf {B}^{\prime *}\right) + (\mathbf {B^{\prime *}\cdot \nabla ^{\prime }})M_x^{\prime }. \end{aligned}$$

(8.111)

Integrating (8.111) over the particle volume and taking into account that $\mathbf {\nabla ^{\prime } \cdot B^{\prime }}=0$, we get

$$\begin{aligned} F_{xm}=\int d^3x^{\prime }\Bigg \langle \mathbf {\bigg [\big [\nabla ^{\prime } \times M^{\prime }\big ]\times B^{\prime *}\bigg ]}_x\Bigg \rangle = \frac{\partial }{\partial x^{\prime }}\Bigg \langle \big (\mathbf {p_m^{\prime }\cdot B^{\prime *}}\big )\Bigg \rangle _{\mathbf {r^{\prime } =r^{\prime }_0}}. \end{aligned}$$

(8.112)

After summation of the contributions due to interaction of the electromagnetic field with the dipole and magnetic moments of the particle, we get

$$\begin{aligned} F_x^{\prime } = \frac{\partial }{\partial x ^{\prime }}\Big \langle \mathbf {p_e^{\prime }\cdot E^{\prime *}\left( r^{\prime }\right) }\Big \rangle _{\mathbf {r^{\prime } =r^{\prime }_0}} + \frac{\partial }{\partial x ^{\prime }}\Big \langle \mathbf {p_m^{\prime }\cdot B^{\prime *}\left( r^{\prime }\right) } \Big \rangle _{\mathbf {r^{\prime } =r^{\prime }_0}}, \end{aligned}$$

(8.113)

where, according to the fluctuation electrodynamics, $\mathbf {d}_{e(m)}^{\prime }=\mathbf {d}_{e(m)}^{f\prime }+\mathbf {d}_{e(m)}^{in\prime }$, $\mathbf {E}^{\prime }= \mathbf {E}^{f\prime }+\mathbf {E}^{in\prime }$, $\mathbf {B}^{\prime }= \mathbf {B}^{f\prime }+\mathbf {B}^{in\prime }$, where $\mathbf {d}_{e(m)}^{f\prime }$ and $\mathbf {E}^{f\prime }$($\mathbf {B}^{f\prime }$) are the fluctuating dipole (magnetic) moment of a particle and the electric (induction) field of the black-body radiation, and $\mathbf {d}_{e(m)}^{in\prime }$ and $\mathbf {E}^{in\prime }$($\mathbf {E}^{in\prime }$) are the dipole (magnetic) moment of a particle induced by the black-body radiation and the electric (induction) field induced by the fluctuating dipole (magnetic) moment of a particle, respectively. Because the calculations of the contributions to the friction force from the dipole and magnetic moments are very similar, below we will consider the contribution only from the dipole moment.

Taking into account the statistical independence of the fluctuating quantities, the Lorentz force can be written in the form

$$\begin{aligned} F_x^{\prime } = F_{1x}^{\prime }+ F_{2x}^{\prime }, \end{aligned}$$

(8.114)

where

$$\begin{aligned} F_{1x}^{\prime } =\frac{\partial }{\partial x ^{\prime }}\Big \langle \mathbf {d_e^{in\prime }\cdot E^{f\prime *}\left( r^{\prime }\right) }\Big \rangle _{\mathbf {r^{\prime } =r^{\prime }_0}}, \end{aligned}$$

(8.115)

$$\begin{aligned} F_{2x}^{\prime } =\frac{\partial }{\partial x ^{\prime }}\Big \langle \mathbf {d_e^{f\prime }\cdot E^{in\prime *}\left( r^{\prime }\right) }\Big \rangle _{\mathbf {r^{\prime } =r^{\prime }_0}}. \end{aligned}$$

(8.116)

To calculate $F_{1x}^{\prime }$ we write the electric field in the $K^{\prime }$ frame as a Fourier integral,

$$ \mathbf {E}^{f\prime }(\mathbf {r}^{\prime }, t^{\prime }) = \int _{-\infty }^{\infty } \frac{d\omega ^{\prime }}{2\pi }\int \frac{d^3k^{\prime }}{(2\pi )^3}e^{i\mathbf {k}^{\prime }\cdot \mathbf {r}^{\prime }-i\omega ^{\prime }t^{\prime }}\mathbf {E}^{f\prime }(\mathbf {k}^{\prime }, \omega ^{\prime }), $$

Using that

$$ \mathbf {d}_e^{in\prime } = \int _{-\infty }^{\infty } \frac{d\omega ^{\prime }}{2\pi }\int \frac{d^3k^{\prime }}{(2\pi )^3}\alpha (\omega ^{\prime })e^{i\mathbf {k}^{\prime }\cdot \mathbf {r}^{\prime }-i\omega ^{\prime }t^{\prime }}\mathbf {E}^{f\prime }(\mathbf {k}^{\prime }, \omega ^{\prime }). $$

where $\alpha (\omega ^{\prime })$ is the particle polarizability, we get

$$\begin{aligned} F_{1x}^{\prime } =-i\int _{\infty }^{\infty } \frac{d\omega ^{\prime }}{2\pi }\int \frac{d^3k^{\prime }}{(2\pi )^3}k_x^{\prime } \alpha (\omega ^{\prime })\Big \langle \mathbf {E^{f\prime }\cdot E^{f\prime *}}\Big \rangle _{\omega ^{\prime }\mathbf {k}^{\prime }}. \end{aligned}$$

(8.117)

When we change from the $K^{\prime }$ frame to the K frame, $\langle \mathbf {E^{\prime }\cdot E^{\prime *}}\rangle _{\omega ^{\prime }\mathbf {k}^{\prime }}$ is transformed as the energy density of a plane electromagnetic field. From the law of transformation of the energy density of a plane electromagnetic field [253] we get

$$\begin{aligned} \bigg \langle \mathbf {E^{f\prime }\cdot E^{f\prime *}}\bigg \rangle _{\omega ^{\prime }\mathbf {k}^{\prime }} = \bigg \langle \mathbf {E^f\cdot E^{f *}}\bigg \rangle _{\omega \mathbf {k}}{\left( \frac{\omega ^{\prime }}{\omega }\right) }^2. \end{aligned}$$

(8.118)

According to the theory of the fluctuating electromagnetic field, the spectral density of the fluctuations of the electric field is determined by [107]

$$\begin{aligned} \bigg \langle E_i^f(\mathbf {r})E_j^{f*}(\mathbf {r^{\prime }})\bigg \rangle _{\omega \mathbf {k}} =\hbar \mathrm {Im}D_{ij}(\mathbf {k}, \omega )\coth \left( \frac{\hbar \omega }{2k_BT_1}\right) , \end{aligned}$$

(8.119)

where the Green’s function of the electromagnetic field in the free space is determined by

$$\begin{aligned} D_{ik}(\omega , \mathbf {k}) = - \frac{\frac{4\pi \omega ^2}{c^2}}{{\frac{\omega ^2}{c^2}} - k^2 + i0\cdot \mathrm {sgn} \,\omega }\left[ \delta _{ik} - \frac{c^2k_ik_k}{\omega ^2 }\right] , \end{aligned}$$

(8.120)

$T_1$ is the temperature of the black-body radiation. Taking into account that

$$ \mathrm {Im}\frac{1}{\frac{\omega ^2}{c^2} - k^2 + i0\cdot \mathrm {sign} \,\omega }= \mathrm {Im}\frac{1}{\frac{\omega ^{\prime 2}}{c^2} - k^{\prime 2} + i0\cdot \mathrm {sgn} \,\omega ^{\prime } }, $$

we get

$$\begin{aligned} \bigg \langle \mathbf {E^{f\prime }\cdot E^{f\prime *}}\bigg \rangle _{\omega ^{\prime } \mathbf {k}^{\prime }} = 4\pi ^2 \hbar k^{\prime } \Bigg \{\delta \bigg (\frac{\omega ^{\prime }}{c}-k^{\prime }\bigg ) - \delta \bigg (\frac{\omega ^{\prime }}{c}+k^{\prime }\bigg )\Bigg \}\coth \left( \frac{\hbar \omega }{2k_BT_1}\right) . \end{aligned}$$

(8.121)

Substitution of (8.121) in (8.117) and integration over $\omega ^{\prime }$ gives

$$\begin{aligned} F_{1x}^{\prime } = \frac{\hbar c}{2\pi ^2}\int d^3k^{\prime } k^{\prime } k_x^{\prime } \mathrm {Im}\alpha (ck^{\prime }) \coth \left( \frac{\hbar \gamma (ck^{\prime }+Vk_x^{\prime })}{2k_BT_1}\right) , \end{aligned}$$

(8.122)

where it was taken into account, that $\omega = (\omega ^{\prime } + k^{\prime }_xV)\gamma$. Introducing the new variable $\omega ^{\prime } = ck^{\prime }$, (8.122) can be written in the form

$$\begin{aligned} F_{1x}^{\prime } = \frac{\hbar }{\pi c^2}\int _0^{\infty } d\omega ^{\prime } \omega ^{\prime 2} \int _{\frac{-\omega ^{\prime }}{c}}^{\frac{\omega ^{\prime }}{c}} dk_x^{\prime } k_x^{\prime } \mathrm {Im}\alpha (\omega ^{\prime }) \coth \left( \frac{\hbar \gamma (\omega ^{\prime }+Vk_x^{\prime })}{2k_BT_1}\right) . \end{aligned}$$

(8.123)

At small velocities ($V\ll c$)$\, F_x = -\Gamma V$, where

$$\begin{aligned} \Gamma =\frac{ \hbar ^2}{3\pi c^5k_BT_1}\int _0^\infty d\omega \frac{ \omega ^5\mathrm {Im}\alpha (\omega )}{\sinh ^2\bigg (\frac{ \hbar \omega }{2k_BT_1} \bigg )}, \end{aligned}$$

(8.124)

Equation (8.124) was first derived in [157] using a different approach. The rate of change of the rest energy of a particle in the $K^{\prime }$ frame due to the absorption of black-body radiation is determined by the equation

$$\begin{aligned} P_1^{\prime }=\frac{dm_0}{dt^{\prime }}c^2=\left\langle \mathbf {j}_e^{in\prime }\cdot \mathbf {E}^{f\prime *}\right\rangle = \frac{\partial }{\partial t ^{\prime }}\left\langle \mathbf {d}_e^{in\prime }(t^{\prime })\cdot \mathbf {E}^{f\prime *}(t^{\prime }_0) \right\rangle _{t^{\prime } =t^{\prime }_0} . \end{aligned}$$

(8.125)

After the calculations, which are similar to those used when calculating $F_{1x}^{\prime }$, we get

$$\begin{aligned} P_1^{\prime } = \frac{\hbar }{\pi c^2}\int _0^{\infty } d\omega ^{\prime } \omega ^{\prime 2} \int _{\frac{-\omega ^{\prime }}{c}}^{\frac{\omega ^{\prime }}{c}} dk_x^{\prime } \omega ^{\prime } \mathrm {Im}\alpha (\omega ^{\prime }) \coth \left( \frac{\hbar \gamma (\omega ^{\prime }+Vk_x^{\prime })}{2k_BT_1}\right) . \end{aligned}$$

(8.126)

From (8.93), the friction force acting on a particle in the K-frame due to the interaction with the black-body radiation is given by

$$\begin{aligned} F_{1x}&=F_{1x}^{\prime }+\beta \frac{P_1^{\prime }}{c} \nonumber \\&= \frac{\hbar }{\pi c^2}\int _0^{\infty } d\omega ^{\prime } \omega ^{\prime 2} \int _{\frac{-\omega ^{\prime }}{c}}^{\frac{\omega ^{\prime }}{c}} dk_x^{\prime } \left( k_x^{\prime }+\beta \frac{\omega ^{\prime }}{c}\right) \mathrm {Im}\alpha (\omega ^{\prime }) \coth \left( \frac{\hbar \gamma (\omega ^{\prime }+Vk_x^{\prime })}{2k_BT_1}\right) . \end{aligned}$$

(8.127)

Introducing the new variables: $k_x^{\prime }=\gamma (q_x-\beta \omega /c)$, $\omega ^{\prime }=\gamma (\omega -Vk_x)$ in the integral (8.127) we get

$$\begin{aligned} F_{1x} = \frac{\hbar \gamma }{\pi c^2}\int _0^{\infty } d\omega \int _{\frac{-\omega }{c}}^{\frac{\omega }{c}} dk_x k_x (\omega -Vk_x)^2\mathrm {Im}\alpha [\gamma (\omega -Vk_x)] \coth \left( \frac{\hbar \omega }{2k_BT_1}\right) , \end{aligned}$$

(8.128)

where we have taken into account that $d\omega ^{\prime }dq_x^{\prime }=d\omega dq_x$. To calculate $F_{2x}^{\prime }$ in the $K^{\prime }$ frame we use the representation of the fluctuating dipole moment of a particle as a Fourier integral

$$\begin{aligned} \mathbf {d}^{f}( t^{\prime }) = \int _{-\infty }^{\infty } \frac{d\omega ^{\prime }}{2\pi }e^{-i\omega ^{\prime }t^{\prime }}\mathbf {d}^{f }( \omega ^{\prime }). \end{aligned}$$

(8.129)

The electric field created in the $K^{\prime }$ frame by the fluctuating dipole moment of a particle is determined by the equation

$$\begin{aligned} E_i^{in\prime }(\mathbf {r}^{\prime }, t^{\prime }) = \int _{-\infty }^{\infty } \frac{d\omega ^{\prime }}{2\pi }\int \frac{d^3k^{\prime }}{(2\pi )^3}e^{i\mathbf {k}^{\prime }\cdot (\mathbf {r}^{\prime }-\mathbf {r}_0^{\prime })-i\omega ^{\prime }t^{\prime }} D_{ik}(\omega ^{\prime }, \mathbf {k}^{\prime })d_k^{f }( \omega ^{\prime }). \end{aligned}$$

(8.130)

According to the fluctuation–dissipation theorem, the spectral density of the fluctuations of the fluctuating dipole moment is determined by the equation [107]

$$\begin{aligned} \bigg \langle d^f_id_k^{f*}\bigg \rangle _{\omega ^{\prime }} =\hbar \mathrm {Im}\alpha ( \omega ^{\prime })\coth \left( \frac{\hbar \omega ^{\prime }}{2k_BT_2}\right) \delta _{ik}, \end{aligned}$$

(8.131)

where $T_2$ is the temperature of a particle. Substituting (8.152) and (8.153) in (8.116) and taking into account (8.154), we get

$$\begin{aligned} F_{2x}^{\prime } = -\frac{\hbar }{\pi c^2}\int _0^{\infty } d\omega ^{\prime } \omega ^{\prime 2} \int _{\frac{-\omega ^{\prime }}{c}}^{\frac{\omega ^{\prime }}{c}} dk_x^{\prime } k_x^{\prime } \mathrm {Im}\alpha (\omega ^{\prime }) \coth \left( \frac{\hbar \gamma \omega ^{\prime }}{2k_BT_2}\right) =0. \end{aligned}$$

(8.132)

Thus, in the rest reference frame of a particle, the friction force due to its own thermal radiation is zero. This result is due to the fact that, in this frame, due to the symmetry, the total radiated momentum from the dipole radiation is identically zero. Thus, the change in momentum of a particle in the rest reference frame is determined by the Lorentz force $F_x^{\prime }$ acting on a particle from the external electromagnetic field associated with the black-body radiation observed in this reference frame. The rate of change of the rest energy of a particle in the $K^{\prime }$ frame, due to its thermal radiation, can be obtained with similar calculations

$$\begin{aligned} P_2^{\prime }&= \bigg \langle \mathbf {j}_e^{f\prime }\cdot \mathbf {E}^{in\prime *}\bigg \rangle = \frac{\partial }{\partial t ^{\prime }}\bigg \langle \mathbf {d}_e^{f\prime }(t^{\prime })\cdot \mathbf {E}^{in\prime *}(t^{\prime }_0)\bigg \rangle _{t^{\prime } =t^{\prime }_0}\nonumber \\&= -\frac{\hbar }{\pi c^2}\int _0^{\infty } d\omega ^{\prime } \omega ^{\prime 2} \int _{\frac{-\omega ^{\prime }}{c}}^{\frac{\omega ^{\prime }}{c}} dk_x^{\prime } \omega ^{\prime } \mathrm {Im}\alpha (\omega ^{\prime }) \coth \left( \frac{\hbar \gamma \omega ^{\prime }}{2k_BT_2}\right) , \end{aligned}$$

(8.133)

and the friction force in the K frame associated with thermal radiation of a particle is given by

$$\begin{aligned} F_{2x}&= F_{2x}^{\prime } + \beta \frac{P_2^{\prime }}{c}\nonumber \\&=\,-\frac{\hbar \gamma }{\pi c^2}\int _0^{\infty } d\omega \times \int _{\frac{-\omega }{c}}^{\frac{\omega }{c}} dk_x k_x (\omega -Vk_x)^2\mathrm {Im}\alpha [\gamma (\omega -Vk_x)] \coth \left( \frac{\hbar \gamma (\omega -Vk_x)}{2k_BT_2}\right) . \end{aligned}$$

(8.134)

The total friction force in the K frame is given by

$$\begin{aligned} F_x=F_{1x}+F_{2x} = \frac{2\hbar \gamma }{\pi c^2}\int _0^{\infty } d\omega \int _{\frac{-\omega }{c}}^{\frac{\omega }{c}} dk_x k_x (\omega -Vk_x)^2\mathrm {Im}\alpha (\gamma (\omega -Vk_x))(n_1(\omega )-n_2(\omega ^{\prime })), \end{aligned}$$

(8.135)

where $n_i(\omega )=[\exp (\hbar \omega /k_BT_i)-1]^{-1}$. Equation (8.135) was first derived in [157]. Note that the friction force $F_x$ can be either positive or negative. However, the acceleration, which is determined by the friction force $F_x^{\prime }$, is always negative. The total heat absorbed by a particle in the $K^{\prime }$-frame is determined by the equation

$$\begin{aligned} P^{\prime }=P^{\prime }_1+P^{\prime }_2=\frac{2\hbar \gamma ^2 }{\pi c^2}\int _0^{\infty } d\omega ^{\prime } \int _{-\omega ^{\prime }/{c}}^{\omega ^{\prime }/{c}} dk_x^{\prime }\omega ^{\prime } (\omega -Vq_x)^2\mathrm {Im}\alpha [\gamma (\omega -Vq_x)](n_1(\omega )-n_2(\omega ^{\prime })). \end{aligned}$$

(8.136)

In the K frame, the total change in energy of a particle due to the interaction with the radiation field can be calculated from the law of the transformation of energy of a particle: $E=\gamma (E^{\prime }+p^{\prime }_xV)$, where E and $E^{\prime }$ are the total energy of a particle in the K and $K^{\prime }$ frames, respectively. From this relation, we get the equation for the rate of change of the energy of a particle in the K frame

$$\begin{aligned} \frac{dE}{dt}=P=P^{\prime }+F_x^{\prime }V=\frac{2\hbar \gamma }{\pi c^2}\int _0^{\infty } d\omega \int _{{-\omega }/{c}}^{{\omega }/{c}} dk_x\omega (\omega -Vk_x)^2\mathrm {Im}\alpha [\gamma (\omega -Vk_x)][n_1(\omega )-n_2(\omega ^{\prime })]. \end{aligned}$$

(8.137)

The rate of change of the energy of the black-body radiation in the K frame is determined by the equation $dW_{BB}/dt=-P$. The steady-state temperature of a particle is determined by the condition $P^{\prime }(T_1, T_2)=0$, and for this state $F_x=F_x^{\prime }$ and $P=F_x^{\prime }V$.

The friction force acting on a particle moving relative to the black-body radiation is determined by the imaginary part of the particle polarizability. For an atom, the imaginary part of the polarizability is determined by the atom electronic linewidth broadening due to the radiation mechanism, which can be calculated considering the interaction of an atom with its own radiation. Taking into account this interaction, the dipole moment of an atom induced by an external electric field $E_x^{ext}(\omega ,\mathbf {r}_0)$ can be written in the form [129, 139]

$$\begin{aligned} p_x^{ind}=\alpha _0(\omega )(\omega )\big [E_x^{ind}(\omega ,\mathbf {r}_0)+E_x^{ext}(\omega ,\mathbf {r}_0)\big ], \end{aligned}$$

(8.138)

where, in the single-oscillator model without the radiation linewidth broadening, the atomic polarizability is given by the equation

$$\begin{aligned} \alpha _0(\omega )=\frac{\alpha (0)\omega _0^2}{\omega _0^2-\omega ^2}, \end{aligned}$$

(8.139)

where $\alpha (0)$ is the static polarizability of an atom, and $E_x^{ind}(\omega ,\mathbf {r}_0)$ is the radiation electric field created by the induced dipole moment of an atom. In the Coulomb gauge, which is used in this article, the Green’s function of the electromagnetic field determines the electric field created by the unit point dipole, so $E_x^{ind}(\omega ,\mathbf {r}_0)= \tilde{D}_{xx}(\omega ,\mathbf {r}_0,\mathbf {r}_0)p_x^{ind}$, where $\tilde{D}_{xx}(\omega ,\mathbf {r}_0,\mathbf {r}_0)$ is the reduced part of the Green’s function of the electromagnetic field in the vacuum, which takes into account only the contribution from the propagating electromagnetic waves and determines the radiation in the far field. The Green’s function of the electromagnetic field in the vacuum $D_{xx}(\omega ,\mathbf {r},\mathbf {r}_0)$ diverges at $\mathbf {r}=\mathbf {r}_0$. However, the contribution from the propagating waves remains finite and purely imaginary at $\mathbf {r}=\mathbf {r}_0$, and the divergent contribution from the evanescent waves is real. Therefore, $\tilde{D}_{xx}(\omega ,\mathbf {r}_0,\mathbf {r}_0)=i\mathrm {Im}D_{xx}(\mathbf {r_0},\mathbf {r_0})$. From (8.138) and (8.139) we get

$$\begin{aligned} \mathrm {Im}\alpha (\omega ) = \mathrm {Im}\frac{p_x^{ind}}{E_x^{ext}(\omega ,\mathbf {r}_0)}&=\,\mathrm {Im}\frac{\alpha (0)\omega _0^2}{\omega _0^2-\omega ^2-i\alpha (0)\omega _0^2\mathrm {Im}D_{xx}(\mathbf {r_0},\mathbf {r_0})} \nonumber \\&=\,\frac{\alpha ^2(0)\omega _0^4\mathrm {Im}D_{xx}(\mathbf {r_0},\mathbf {r_0})}{(\omega _0^2-\omega ^2)^2+\big [\alpha (0)\omega _0^2\mathrm {Im}D_{xx}\big ]^2}, \end{aligned}$$

(8.140)

where

$$\begin{aligned} \mathrm {Im}D_{xx}(\mathbf {r_0},\mathbf {r_0})=\mathrm {Im}D_{yy}=\mathrm {Im}D_{zz} = \int \frac{d^3k}{(2\pi )^3} \mathrm {Im}D_{xx}(\omega , \mathbf {k})=\frac{2}{3}{\left( \frac{\omega }{c}\right) }^3\mathrm {sgn}\,\omega . \end{aligned}$$

(8.141)

At resonance ($\omega ^2\approx \omega _0^2$) usually $\alpha (0)\mathrm {Im}D_{xx}\ll 1$ (for example, for a hydrogen atom it is $\sim 10^{-6}$). Thus, the limit $\alpha (0)D_{xx}\rightarrow i0$ can be taken. In this case, the resonant contribution is given by

$$\begin{aligned} \mathrm {Im}\alpha (\omega )\approx \frac{\pi \alpha (0)\omega _0}{2}[\delta (\omega -\omega _0)-\delta (\omega +\omega _0)], \end{aligned}$$

(8.142)

and the off-resonant contribution, which is far from resonance ($\omega ^2\ll \omega _0^2$), can be given by

$$\begin{aligned} \mathrm {Im}\alpha (\omega )\approx \frac{2}{3}{\left( \frac{\omega }{c}\right) }^3\alpha ^2(0)\mathrm {sign}\,\omega . \end{aligned}$$

(8.143)

The result (8.143) was also obtained in [160] using quantum electrodynamics. However, the analysis presented above is much simpler, and it clarifies the physical meaning of the terms in the quantum electrodynamics perturbational theory. Using (8.142) and (8.143) in (8.123) we get the resonant and off-resonant contributions to the friction force

$$\begin{aligned} F_{1x}^{res}=\frac{ \hbar \omega ^5_0\alpha (0)}{ c^4}\int _{-1}^1dxx\bigg [\exp {\left( \frac{\hbar \gamma \omega _0(1+\beta x)}{k_BT_1}\right) }-1\bigg ]^{-1}, \end{aligned}$$

(8.144)

$$\begin{aligned} F_{1x}^{nonres}=-\frac{ 512\pi ^7\hbar \alpha (0)^2\gamma ^6}{945 c^7 }\bigg (\frac{k_BT_1}{\hbar }\bigg )^{8}(7\beta +14\beta ^3+3\beta ^5), \end{aligned}$$

(8.145)

For $\beta \ll 1$ the friction force $F_{1x}=-\Gamma V$, where the resonant and off-resonant contributions to the friction coefficient are

$$\begin{aligned} \Gamma _{res}=\frac{ \hbar ^2\alpha (0)\omega ^6_0}{6 c^5k_BT_1} \frac{ 1}{\sinh ^2\Big (\frac{ \hbar \omega _0}{2k_BT_1} \Big )}, \end{aligned}$$

(8.146)

$$\begin{aligned} \Gamma _{nonres}=\frac{ 512\pi ^7\hbar \alpha (0)^2}{135 c^8 }{\left( \frac{k_BT_1}{\hbar }\right) }^8, \end{aligned}$$

(8.147)

The results (8.146) and (8.147) were obtained in [157, 160]. In the ultrarelativistic case, ($1-\beta \ll k_BT_1/\hbar \omega _0\ll 1$)

$$\begin{aligned} F_{1x}^{res}=\frac{ \omega ^4_0\alpha (0)}{ c^4}\sqrt{2}k_BT_1\sqrt{1-\beta }\ln \frac{\hbar \omega _0\sqrt{1-\beta }}{\sqrt{2}k_BT_1}, \end{aligned}$$

(8.148)

$$\begin{aligned} F_{1x}^{nonres}=\frac{ 216\pi ^7\hbar \alpha (0)^2}{135 c^7 (1-\beta )^3}{\left( \frac{k_BT_1}{\hbar }\right) }^8 \end{aligned}$$

(8.149)

Thus, in this case $F_{1x}^{res}\sim \sqrt{1-\beta }\ln \sqrt{1-\beta }\rightarrow 0$ and $F_{1x}^{nonres}\sim (1-\beta )^{-3}\rightarrow \infty$ at $1-\beta \rightarrow 0$.

For small velocities and typical temperatures, the infrared thermal peak of the black-body radiation is far below the resonance frequency $\omega _0$, so it dominates the far-off-resonant contribution (8.146), which has already been mentioned in [159]. However, in the ultrarelativistic case, the friction is dominated by the far-off-resonant contribution for all temperatures.

According to (8.136), the total heat absorbed by an atom in the $K^{\prime }$ frame is determined by the equation

$$\begin{aligned} P^{\prime }=\frac{ 128\pi ^7 k_B^8\alpha (0)^2}{315 c^6\hbar ^7 }\left[ \gamma ^6(7+35\beta ^2+21\beta ^4+\beta ^6)T_1^8-7T_2^8\right] . \end{aligned}$$

(8.150)

For small velocities, the steady-state temperature of a particle is $T_2=T_1$, and, in the ultrarelativistic case, $T_2\approx (1-\beta )^{-3/8}T_1$. The problem in calculating the imaginary part of the atom’s polarizability in the far off-resonant field was considered in [159]. It was noted that in the literature that questions still remain regarding the gauge invariance of the imaginary part of the polarizability. In this Section, the imaginary part of the atomic polarizability in the field far from resonance is determined by the imaginary part of the electric field of the unit point dipole, which is a gauge-invariant quantity. In the Coulomb gauge, which is used in this article, the electric field of the unit point dipole is the same as the Green’s function of the electromagnetic field. In another gauge, the expression for the Green’s function will change; but, the electric field determined with this Green’s function will remain unchanged. Therefore, despite the fact that the Green’s function for the electromagnetic field is a gauge-dependent quantity, the imaginary part of the atomic polarizability calculated in this Section is a gauge-invariant quantity. The gauge invariance of the obtained results are also confirmed by the direct calculation using quantum electrodynamics [160]. The gauge-invariant formulation presented in this Section confirms that the polarizability of the atom, for small frequencies, is a non-resonant effect, which is proportional to $\omega ^3$ for small driving frequency $\omega$. This is consistent with the gauge-invariant analysis conducted in [160].

For a spherical particle with radius R, the electric and magnetic susceptibilities are given by (5.58) and (5.59), respectively. For metals with $4\pi \sigma \gg k_BT/\hbar$ and for $c\sqrt{2\pi \sigma k_BT}\gg R$, where $\sigma$ is the conductivity, from (5.58) and (5.59) we get [191]

$$\begin{aligned} \mathrm {Im}\alpha _E(\omega ) \approx R^3\frac{3\omega }{4\pi \sigma }, \end{aligned}$$

(8.151)

$$\begin{aligned} \mathrm {Im}\alpha _H(\omega ) = \frac{4\pi \sigma \omega R^5}{30c^2}. \end{aligned}$$

(8.152)

Writing the friction coefficient $\Gamma$ as $m_0/\tau$, where $\tau$ is a relaxation time, and using $m_0 = 4\pi R^3\rho /3$, from (8.127)–(8.152), we get

$$\begin{aligned} \tau ^{-1}_e \approx 10^2\frac{\hbar }{\rho \lambda _T^5}\frac{k_BT}{\hbar \sigma }, \end{aligned}$$

(8.153)

$$\begin{aligned} \tau ^{-1}_m \approx 10^2\frac{\hbar R}{\rho \lambda _T^6}\frac{\sigma R}{c}. \end{aligned}$$

(8.154)

where $\tau _e^{-1}$ and $\tau _m^{-1}$ are the contributions to $1/\tau$ from the electric dipole and magnetic moments, respectively. For $T= 300$ K, $\rho \approx 10^4$ kg/m$^3$, $\sigma \approx 10^{18}$ s$^{-1}$ from (8.151 and 8.154) we get $\tau _e \sim 10^{16}$ s and $\tau _m \sim 10^{12}$ s. When the conductivity decreases, $\tau _e$ also decreases and reaches minimum at $2\pi \sigma \approx k_BT/\hbar$. At $T=3000$ K, this minimum corresponds to approximately a day ($\tau _e^{min} \approx 10^5$ s). In Sect. 8.5.1 the same relaxation time was obtained for $Ba^+$.

8.5.3 Einstein’s Formula

In this section, we give an alternative derivation of the formula for the force on a particle in thermal field. This derivation establishes a link with Einstein and Hopf derivation [254]. According to the linear response theory, the energy absorbed by a particle per unit time can be written in the form [184]

$$\begin{aligned} \frac{dE}{dt}= 2\int _0^{\infty } \frac{d\omega }{2\pi }\int \frac{d^3k}{(2\pi )^3}\omega \big [\mathrm {Im}\alpha _E(\omega ) + \mathrm {Im}\alpha _H(\omega )\big ]\big \langle \mathbf {E\cdot E^{ *}}\big \rangle _{\omega \mathbf {k}} \end{aligned}$$

(8.155)

The force acting on the particle is determined by the momentum absorbed by the particle in the rest reference frame

$$\begin{aligned} F_x=\frac{dP^{\prime }_x}{dt}=\frac{2}{c}\int _0^{\infty } \frac{d\omega ^{\prime }}{2\pi }\int \frac{d^3k^{\prime }}{(2\pi )^3}\omega ^{\prime } \cos {\theta ^{\prime }} \big [\mathrm {Im}\alpha _E(\omega ^{\prime }) + \mathrm {Im}\alpha _H(\omega ^{\prime })\big ]\big \langle \mathbf {E^{\prime }\cdot E^{\prime *}}\big \rangle _{\omega ^{\prime }\mathbf {k}^{\prime }} \end{aligned}$$

(8.156)

Using (8.118), and taking into account that the elementary volume in the ($\omega , \mathbf {k}$) space is invariant under the Lorentz transformation, (8.156) can be rewritten in the form

$$\begin{aligned} F_x=\frac{dP^{\prime }_x}{dt}&=\,\frac{1}{c}\int _0^{\infty } \frac{d\omega }{2\pi }\int _0^{\infty } \frac{4\pi k^2dk}{(2\pi )^3} \nonumber \\&\quad \times \int _0^{\pi }d\theta \sin {\theta }\omega ^{\prime } \cos {\theta ^{\prime }} \big [\mathrm {Im}\alpha _E(\omega ^{\prime }) + \mathrm {Im}\alpha _H(\omega ^{\prime })\big ]\big \langle \mathbf {E\cdot E^{ *}}\big \rangle _{\omega \mathbf {k}}{\left( \frac{\omega ^{\prime }}{\omega }\right) }^2 \end{aligned}$$

(8.157)

Using (3.54) we get

$$\begin{aligned} \int \frac{4\pi k^2dk}{(2\pi )^3}\big \langle \mathbf {E\cdot E^{ *}}\big \rangle _{\omega \mathbf {k}} = \big \langle \mathbf {E}^{2}\big \rangle _{\omega } = 4\pi ^2 u(\omega ), \end{aligned}$$

(8.158)

where the spectral energy density of the thermal electromagnetic field $u(\omega )$ is given by (3.55). Substituting (8.158) in (8.157) we get

$$\begin{aligned} F_x=\frac{dP^{\prime }_x}{dt}=\frac{2\pi }{c}\int _0^{\infty } d\omega \int _0^{\pi } d\theta \sin {\theta } \omega ^{\prime } \cos {\theta ^{\prime }} [\mathrm {Im}\alpha _E(\omega ^{\prime }) + \mathrm {Im}\alpha _H(\omega ^{\prime })]u(\omega ){\left( \frac{\omega ^{\prime }}{\omega }\right) }^2 \end{aligned}$$

(8.159)

From the Lorenz transformation for a plane electromagnetic wave, we get

$$\begin{aligned} \omega = \omega ^{\prime }\gamma \left( 1 + \frac{V}{c}\cos {\theta }\right) . \end{aligned}$$

(8.160)

$$\begin{aligned} \cos {\theta } = \frac{\cos {\theta ^{\prime }} + \frac{V}{c}}{1 + \frac{V}{c}\cos {\theta ^{\prime }}}. \end{aligned}$$

(8.161)

Inserting (8.160) and (8.161) in (8.159) we get

$$\begin{aligned} F_x=\frac{dP^{\prime }_x}{dt}=\frac{2\pi }{c}\int _0^{\infty } d\omega \omega \int _0^{\pi } \frac{d\theta \sin {\theta } \cos {\theta }}{\left( 1 + \frac{V}{c}\cos {\theta }\right) ^3} \Bigg [\mathrm {Im}\alpha _E(\omega ) + \mathrm {Im}\alpha _H(\omega )]u[\gamma \omega \bigg (1 + \frac{V}{c}\cos {\theta }\bigg )\Bigg ], \end{aligned}$$

(8.162)

where we have omitted the index prime. For a two-level atom we can neglect $\mathrm {Im}\alpha _H(\omega )$, and $\mathrm {Im}\alpha _E(\omega )$ can be written in the form

$$\begin{aligned} \mathrm {Im}\alpha _E(\omega ) = \pi M_{12}(N_1 - N_2) \delta (\omega - \omega _0), \end{aligned}$$

(8.163)

where $M_{12} = e^2|\langle 0|\mathbf {r}|1 \rangle |^2/3\hbar$, and where $N_1$ and $N_2$ are the probabilities of the ground and excited states, respectively; $\hbar \omega _0 = E_1 - E_2$, where $E_1$ and $E_2$ are the energies of the ground and excited states, respectively. Substituting (8.163) in (8.162) in the case of small velocities ($V/c \ll 1$) we get

$$\begin{aligned} F_x= -\left( \frac{\hbar \omega _0}{c^2}\right) B_{12}(N_1 - N_2) \left[ u(\omega _0)-\frac{\omega _0}{3}\frac{du(\omega _0)}{d\omega _0}\right] V, \end{aligned}$$

(8.164)

where $B_{12} = 4\pi ^2e^2 |\langle 0|\mathbf {r}|1\rangle |^2/3 \hbar ^2$. Equation (8.164) is Einstein’s equation [254] for the force on an atom moving in a thermal electromagnetic field.