2 The Coulomb logarithm in classical kinetic theory – a brief review

One definition of  $\unicode[STIX]{x1D6EC}$ is $\unicode[STIX]{x1D6EC}\doteq \unicode[STIX]{x1D706}_{\text{D}}/b_{\text{min}}$ ( $\doteq$ denotes a definition), where $\unicode[STIX]{x1D706}_{\text{D}}$  is the Debye screening length (the effective maximum impact parameter for two-body scattering) and $b_{\text{min}}$  is a characteristic length to be determined. In classical scattering theory,Footnote ¹   $b_{\text{min}}=b_{0}$ , where

(2.1) $$\begin{eqnarray}b_{0}\doteq \frac{q_{1}q_{2}}{\unicode[STIX]{x1D707}u^{2}}\end{eqnarray}$$

is the impact parameter for $90^{\circ }$ scattering between particles 1 and 2. Here $q$  is the charge, $\unicode[STIX]{x1D707}$  is the reduced mass and $u$  is the relative velocity. The distance of closest approach is  $2b_{0}$ . For a weakly coupled plasma, $b_{0}$  is very much smaller than  $\unicode[STIX]{x1D706}_{\text{D}}$ ; specifically, $b_{0}/\unicode[STIX]{x1D706}_{\text{D}}=O(\unicode[STIX]{x1D716}_{\text{p}})$ , where $\unicode[STIX]{x1D716}_{\text{p}}\doteq 1/(\overline{n}\unicode[STIX]{x1D706}_{\text{D}}^{3})$ is the so-called plasma (expansion) parameter ( $\overline{n}$  is the mean density).

The result $b_{\text{min}}=b_{0}$ arises in classical, weakly coupled plasma kinetic theory (Montgomery & Tidman Reference Montgomery and Tidman1964) as follows. One can determine the net scattering cross-section  $\unicode[STIX]{x1D70E}$ , or in more detail the velocity-dependent collision operator, from an approximate calculation of the pair correlation function in the limit of small  $\unicode[STIX]{x1D716}_{\text{p}}$ . That leads to an integration over all possible impact parameters  $b$ or, equivalently, over wavenumber magnitude $k\doteq b^{-1}$ . In particular, with $\unicode[STIX]{x1D750}_{\boldsymbol{k}}\doteq -4\unicode[STIX]{x03C0}\text{i}\boldsymbol{k}/k^{2}$ being the Fourier representation of the bare Coulomb field of a unit point charge, ${\mathcal{D}}(\boldsymbol{k},\unicode[STIX]{x1D714})$ being the Vlasov dielectric function and $f$  being the one-particle distribution function, the Balescu–Lenard collision operator (which captures the effect of dielectric shieldingFootnote ² but not large-angle scattering) for collisions of species  $s$ on species  $\overline{s}$ isFootnote ³

(2.2) $$\begin{eqnarray}\displaystyle C_{s\overline{s}}[f] & \doteq & \displaystyle -\unicode[STIX]{x03C0}(\overline{n}m)_{s}^{-1}(\overline{n}q^{2})_{s}(\overline{n}q^{2})_{\overline{s}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}}\boldsymbol{\cdot }\int \text{d}\overline{\boldsymbol{v}}\int \frac{\text{d}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{3}}\frac{\unicode[STIX]{x1D750}_{\boldsymbol{k}}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}^{\ast }}{|{\mathcal{D}}(\boldsymbol{k},\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v})|^{2}}\unicode[STIX]{x1D6FF}(\boldsymbol{k}\boldsymbol{\cdot }(\boldsymbol{v}-\overline{\boldsymbol{v}}))\nonumber\\ \displaystyle & & \displaystyle \boldsymbol{\cdot }\left(\frac{1}{m_{s}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\boldsymbol{v}}-\frac{1}{m_{\overline{s}}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\overline{\boldsymbol{v}}}\right)f_{s}(\boldsymbol{v})f_{\overline{s}}(\overline{\boldsymbol{v}}).\end{eqnarray}$$

The Vlasov dielectric has the well-known form

(2.3) $$\begin{eqnarray}{\mathcal{D}}(\boldsymbol{k},\unicode[STIX]{x1D714})=1+\mathop{\sum }_{\overline{s}}\frac{\unicode[STIX]{x1D714}_{\text{p}\overline{s}}^{2}}{k^{2}}\int \text{d}\overline{\boldsymbol{v}}\,\frac{\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x2202}_{\overline{\boldsymbol{v}}}f_{\overline{s}}(\overline{\boldsymbol{v}})}{\unicode[STIX]{x1D714}-\boldsymbol{k}\boldsymbol{\cdot }\overline{\boldsymbol{v}}+\text{i}\unicode[STIX]{x1D716}},\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{\text{p}\overline{s}}$ is the plasma frequency. According to (2.2), this is to be evaluated at the characteristic streaming or transit frequency $\unicode[STIX]{x1D714}=\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}$ (not a normal-mode frequency). Thus, the dependence on wavenumber magnitude cancels out under the $\overline{\boldsymbol{v}}$ integral in (2.3), and without further approximation the integration over wavenumber magnitude requires evaluation of

(2.4) $$\begin{eqnarray}I(k)\doteq \int _{0}^{k_{\text{max}}}\frac{\text{d}k}{k}\left(\frac{1}{[1+A(k\unicode[STIX]{x1D706}_{\text{D}})^{-2}]^{2}+[B(k\unicode[STIX]{x1D706}_{\text{D}})^{-2}]^{2}}\right),\end{eqnarray}$$

where $A(\widehat{\boldsymbol{k}},\boldsymbol{v})$ and  $B(\widehat{\boldsymbol{k}},\boldsymbol{v})$ are functions that at least for thermal particles are of order unity (Montgomery & Tidman Reference Montgomery and Tidman1964, Chap. 7). The integral (2.4) can be done exactly, but the details are not important for qualitative discussion; for $k_{\text{max}}\unicode[STIX]{x1D706}_{\text{D}}\gg 1$ , one finds

(2.5) $$\begin{eqnarray}I(k)=\ln (k_{\text{max}}\unicode[STIX]{x1D706}_{\text{D}})+O(1).\end{eqnarray}$$

To understand this result physically, note that the denominator inside the large parentheses in (2.4) represents the mean-square effect of dielectric shielding and makes the integral strongly convergent at $k\rightarrow 0$ , effectively limiting the wavenumber integration to $k\gtrsim k_{\text{min}}\doteq k_{\text{D}}$ , where $k_{\text{D}}\doteq \unicode[STIX]{x1D706}_{\text{D}}^{-1}$ ; this corresponds to a maximum impact parameter $b_{\text{max}}\approx \unicode[STIX]{x1D706}_{\text{D}}$ . Because Debye shielding is ineffective at small scales, equation (2.4) is logarithmically divergent at large  $k$ and must be cut off at some  $k_{\text{max}}$ (or $b_{\text{min}}\doteq k_{\text{max}}^{-1}$ ); the dominant term in (2.5) obviously arises from evaluating $\int _{k_{\text{D}}}^{k_{\text{max}}}\text{d}k\,k^{-1}=\int _{b_{\text{min}}}^{\unicode[STIX]{x1D706}_{\text{D}}}\text{d}b\,b^{-1}$ . However, this large- $k$ or small-scale divergence is not physical. It arises because the Balescu–Lenard derivation is perturbative, based on zeroth-order straight-line trajectories; thus, large-angle scattering (the effect of impact parameters $b\lesssim b_{0}$ ) is misrepresented. There is no solution for this difficulty within the Balescu–Lenard framework. However, one can asymptotically match between an ‘inner’ solution that treats large-angle scattering correctly and an ‘outer’ solution appropriate for small-angle scattering, as was done, for example, by Frieman & Book (Reference Frieman and Book1963) for the classical regime. Such matching removes any apparent divergence at the small scales, in agreement with the result that the Rutherford cross-section is integrable as the scattering angle $\unicode[STIX]{x1D703}\rightarrow \unicode[STIX]{x03C0}$ or $b\rightarrow 0$ . Of course, the scale  $b_{0}$ remains in the asymptotically matched solution: it determines the two-body relationship between scattering angle and impact parameter,

(2.6) $$\begin{eqnarray}\tan (\unicode[STIX]{x1D703}/2)=b_{0}/b;\end{eqnarray}$$

it sets the basic size of the classical cross-section $\unicode[STIX]{x1D70E}_{\text{cl}}$ , which is finite and proportional to  $b_{0}^{2}$ ; and the asymptotic matching leads to $k_{\text{max}}=b_{0}^{-1}$ (this is clear on dimensional grounds). One is led to the dominant resultFootnote ⁴ $\unicode[STIX]{x1D70E}_{\text{cl}}\sim b_{0}^{2}\ln \unicode[STIX]{x1D6EC}_{\text{cl}}$ , whereFootnote ⁵

(2.7) $$\begin{eqnarray}\ln \unicode[STIX]{x1D6EC}_{\text{cl}}\equiv L_{\text{cl}}\doteq \ln (\unicode[STIX]{x1D706}_{\text{D}}/b_{0})=O(\ln (\unicode[STIX]{x1D716}_{\text{p}}^{-1})).\end{eqnarray}$$

Since this $L_{\text{cl}}$ is independent of  $\widehat{\boldsymbol{k}}$ and  $\boldsymbol{v}$ , one can perform the integration over solid angle in (2.2); one is ultimately led (Lenard Reference Lenard1960) to the Landau collision operator (Landau Reference Landau1936), which is generally used in practice. The Landau operator is discussed in appendix B of Krommes (Reference Krommes2018a ) and in various textbooks.

An alternative way of arriving at $\ln \unicode[STIX]{x1D6EC}_{\text{cl}}$ is to begin with the Boltzmann collision operator (which does not incorporate Debye shielding but does handle large-angle scattering correctly), then to take its small-angle limit. (That amounts to considering a Fokker–Planck operator that does not incorporate shielding effects.) In that limit, one encounters the integral $\int _{k_{\text{min}}}^{k_{\text{max}}}\text{d}k\,k^{-1}=\int _{b_{\text{min}}}^{b_{\text{max}}}\text{d}b\,b^{-1}$ , to be considered in the classical regime. Clearly, the logarithmic divergence at large  $b$ must be rectified by inserting the cutoff $b_{\text{max}}=\unicode[STIX]{x1D706}_{\text{D}}$ , which as I discussed follows systematically from the Balescu–Lenard formalism.

In both of the above two procedures, which are correct only for small-angle scattering, the necessity for a small-scale cutoff remains. In order to deal with the misrepresentation of large-angle scattering in lieu of complicated asymptotic analysis, students are generally taught to merely insert the short-scale cutoff $b_{\text{min}}=b_{0}$ or $k_{\text{max}}=b_{0}^{-1}$ , which describes the cross-over between large- and small-angle scattering. That is the correct result, and it is more than hand waving. It must be stressed that that recipe encapsulates the result of a systematic asymptotic matching between the classical regimes of large and small impact parameters, that the underlying physics is convergent for small impact parameters and that no discontinuity or exponentially rapid variation that would lead to diffraction occurs in the vicinity of  $b_{0}$ . These remarks expand upon some of those in the paper by Mulser et al. (Reference Mulser, Alber and Murakami2014).

It is insightful to transform the impact-parameter integration to an integration over scattering angle  $\unicode[STIX]{x1D703}$ . When that is done, one sees according to (2.6) that the effect of dielectric shielding is to introduce a cutoff at small  $\unicode[STIX]{x1D703}$ , so $\unicode[STIX]{x1D70E}\sim \int _{\unicode[STIX]{x1D703}_{\text{min}}}^{}\text{d}\unicode[STIX]{x1D703}/\unicode[STIX]{x1D703}\sim \ln \unicode[STIX]{x1D703}_{\text{min}}^{-1}$ ; see, for example, Landau & Lifshitz (Reference Landau and Lifshitz1981, (41.9)). Classically, one has $\unicode[STIX]{x1D703}_{\text{min}}\approx \unicode[STIX]{x1D703}_{0}\doteq 2b_{0}/\unicode[STIX]{x1D706}_{\text{D}}$ (which follows from the small-angle limit of (2.6)). In the next section, we shall see that quantum-mechanical diffraction effects on the Debye scale lead to a larger value of  $\unicode[STIX]{x1D703}_{\text{min}}$ for sufficiently high temperatures, and thus to a smaller value of  $\ln \unicode[STIX]{x1D6EC}$ .

3 Quantum-mechanical corrections to $\ln \unicode[STIX]{x1D6EC}$

As noted by Spitzer and more clearly explained in the 149 page review article by Sivukhin (Reference Sivukhin1966), at sufficiently high temperature the classical approximation fails, quantum-mechanical considerations apply, and one finds $b_{\text{min}}=\unicode[STIX]{x1D706}_{\text{B}}\doteq h/\unicode[STIX]{x1D707}u$ , the ‘de Broglie wavelength of the test particle in the coordinate system in which the scattering center …is at rest’ (Sivukhin Reference Sivukhin1966). Interpolation between the classical and quantum-mechanical limits leads to the standard prescription

(3.1) $$\begin{eqnarray}b_{\text{min}}=\max \{\unicode[STIX]{x1D706}_{\text{B}},b_{0}\}.\end{eqnarray}$$

Spitzer (Reference Spitzer1962, p. 128) said it this way:Footnote ⁶

Essentially the same discussion appears in Cohen et al. (Reference Cohen, Spitzer and Routly1950, p. 233). Spitzer referred to the choice of a deflection angle, called $\unicode[STIX]{x1D703}_{\text{min}}$ above. His words correspond to the fact that when $\unicode[STIX]{x1D706}_{\text{B}}>b_{0}$ diffraction of the de Broglie wave by an opaque disc of radius  $\unicode[STIX]{x1D706}_{\text{D}}$ (not the very much smaller radius  $\unicode[STIX]{x1D706}_{\text{B}}$ ) produces a diffraction angle  $\unicode[STIX]{x1D703}_{\text{B}}$ that is larger than the classical deflection angle  $\unicode[STIX]{x1D703}_{0}$ for a particle incident with maximum impact parameter $b_{\text{max}}=\unicode[STIX]{x1D706}_{\text{D}}$ ; thus, $\unicode[STIX]{x1D70E}_{\text{qu}}\sim \ln \unicode[STIX]{x1D703}_{\text{B}}^{-1}<\unicode[STIX]{x1D70E}_{\text{cl}}\sim \ln \unicode[STIX]{x1D703}_{0}^{-1}$ .

A possible source of confusion is that Spitzer did not completely spell out the argument; he did not explicitly state that the diffraction angle should be evaluated with $b_{\text{max}}=\unicode[STIX]{x1D706}_{\text{D}}$ , although this is clearly implied by the fact that $\unicode[STIX]{x1D706}_{\text{D}}$ appears in the classical  $\unicode[STIX]{x1D703}_{0}$ , and by Spitzer’s discussion (on the page preceding the above quotation) of  $\unicode[STIX]{x1D706}_{\text{D}}$ as the maximum impact parameter). Instead, he cited a paper by Marshak (Reference Marshak1941). Tracing back through the historical record is interesting. Marshak stated,Footnote ⁷

Thus far we have not given any explicit form for  $I$ [ $I$  being the $\unicode[STIX]{x1D703}$ integral of the Rutherford differential scattering cross section  $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D703})$ ]. If we look at the integral expression for  $I$ [(3.8) below with $\unicode[STIX]{x1D700}=0$ ] we see that it diverges if we integrate between the limits 0 and  $\unicode[STIX]{x03C0}$ . However, there are physical grounds for extending this integration only to some small angle  $\unicode[STIX]{x1D703}_{\text{min}}$ , in which case:

$$\begin{eqnarray}2I=\log _{e}\frac{2}{(1-\cos \unicode[STIX]{x1D703}_{\text{min}})}.\end{eqnarray}$$

Now it can be shown that where is the de Broglie wavelength of the electron participating in the collision, and ‘ $a$ ’ is the screening radius ….

Marshak referred to the screening radius ‘of the atom’, but it is clear that in the present plasma context one should replace  $a$ by  $\unicode[STIX]{x1D706}_{\text{D}}$ . Thus, we see more clearly that Spitzer was recapitulating and providing a physical interpretation of Marshak’s argument, which was focused on the physics of  $\unicode[STIX]{x1D703}_{\text{min}}$ or, equivalently, the physics associated with maximum impact parameter  $\unicode[STIX]{x1D706}_{\text{D}}$ . In turn, Marshak’s * footnote, which provides the background to his assertion ‘it can be shown that …’, refers to p. 497 of the famous paper of Bethe (Reference Bethe1933) on the quantum mechanics of one- and two-electron atoms. Bethe worked in the first Born approximation (for which he cited earlier literature) and discussed a form factor  $F$ that determines  $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D703})$ . In modern notation, the formula for  $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D703})$ is given by (3.6). (This reduces to the Rutherford cross-section when $\unicode[STIX]{x1D703}_{\text{B}}\rightarrow 0$ .) Bethe emphasized,Footnote ⁸

(The last ratio is not dimensionless; obviously, the intended comparison is between $\sin \frac{1}{2}\unicode[STIX]{x1D703}$ and $\frac{1}{2}\unicode[STIX]{x1D703}_{\text{B}}$ or, for small angle, between  $\unicode[STIX]{x1D703}$ and  $\unicode[STIX]{x1D703}_{\text{B}}$ .) It is the quantum correction to the Rutherford cross-section, which is important at small scattering angle and was well known to the pioneers of quantum mechanics, that underlies Marshak’s conclusion and Spitzer’s interpretation.

The first Born approximation does not, in and of itself, predict the interpolation recipe (3.1). Indeed, in that approximation the integral that defines the total momentum transfer can be done exactly, a result known to Sivukhin (see (3.6) in the excerpt below) and surely earlier workers as well. The inner length  $b_{0}$ enters the resulting formula only multiplicatively (the total scattering cross-section is proportional to  $b_{0}^{2}$ ). However, the first Born approximation is valid (at best; see later discussion) only at high energies. Sivukhin explains the issue clearly:Footnote ⁹

3. The classical-mechanics analysis applies so long as $(2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706})b_{0}\gg 1$ , where $\unicode[STIX]{x1D706}=h/\unicode[STIX]{x1D707}u$ is the de Broglie wavelength of the test particle in the coordinate system in which the scattering center (field particle) is at rest. …We can write this condition in the form

(3.2) $$\begin{eqnarray}u\ll \unicode[STIX]{x1D6FC}c,\end{eqnarray}$$

where

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\left|\frac{ee^{\ast }}{\hbar c}\right|\end{eqnarray}$$

( $\hbar =h/2\unicode[STIX]{x03C0}=1.05\cdot 10^{-27}~\text{erg}~\text{s}$ ). If  $e$ and  $e^{\ast }$ are equal to the elementary charge the constant $\unicode[STIX]{x1D6FC}=e^{2}/\hbar c=1/137$ is the fine-structure constant.

The classical analysis cannot be used if (3.2) is not satisfied. This result might appear strange at first glance since the exact quantum-mechanical solution for scattering of a charged particle in a Coulomb field yields an expression for $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D703},u)$ which is exactly the same as the classical expression …(cf., for example, Davydov (Reference Davydov1976) or any text on quantum mechanics). The essential point here, however, is that the results coincide only when the scattering field is a Coulomb field over all space. In the case of a cutoff Coulomb field the wave properties of the particle are appreciably different from those given by the classical analysis.

When

(3.4) $$\begin{eqnarray}u\gg \unicode[STIX]{x1D6FC}c,\end{eqnarray}$$

the quantum-mechanical scattering problem can be solved relatively easily by means of the Born approximation. The solution of the problem is simplified if the cutoff Coulomb field is replaced by a Debye field with potential

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D711}=\frac{e^{\ast }}{r}\text{e}^{-r/\unicode[STIX]{x1D706}_{\text{D}}}.\end{eqnarray}$$

If (3.4) is satisfied, it is well known that the quantum-mechanical analysis leads to the result

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D703},u)=\left(\frac{ee^{\ast }}{2\unicode[STIX]{x1D707}u^{2}}\right)^{2}\frac{1}{\left(\sin ^{2}\displaystyle \frac{\unicode[STIX]{x1D703}}{2}+\unicode[STIX]{x1D700}^{2}\right)^{2}},\end{eqnarray}$$

where

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D700}=\frac{\unicode[STIX]{x1D706}}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{\text{D}}}=\frac{\hbar }{2\unicode[STIX]{x1D707}u\unicode[STIX]{x1D706}_{\text{D}}}\end{eqnarray}$$

(cf., for example, [any textbook discussion of quantum-mechanical scattering theory]).

By substituting (3.6) [into the formula for mean momentum transfer] we recover [the classical result (2.7)] with the sole difference that the classical value of the Coulomb logarithm is replaced by the quantum-mechanical value

(3.8) $$\begin{eqnarray}\displaystyle L_{\text{qu}} & = & \displaystyle \frac{1}{4}\int _{0}^{\unicode[STIX]{x03C0}}\frac{\sin ^{2}\displaystyle \frac{\unicode[STIX]{x1D703}}{2}\sin \unicode[STIX]{x1D703}}{\left(\sin ^{2}\displaystyle \frac{\unicode[STIX]{x1D703}}{2}+\unicode[STIX]{x1D700}^{2}\right)^{2}}\,\text{d}\unicode[STIX]{x1D703}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\ln \frac{1+\unicode[STIX]{x1D700}^{2}}{\unicode[STIX]{x1D700}^{2}}-\frac{1}{2(1+\unicode[STIX]{x1D700}^{2})}.\end{eqnarray}$$

In all cases of physical interest it is found that

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D700}=\frac{\unicode[STIX]{x1D706}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{\text{D}}}\ll 1,\end{eqnarray}$$

so that the square of  $\unicode[STIX]{x1D700}$ can be neglected compared with unity. In this approximation

(3.10) $$\begin{eqnarray}L_{\text{qu}}=\ln \frac{1}{\unicode[STIX]{x1D700}}-\frac{1}{2}=\ln \frac{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{\text{D}}}{\unicode[STIX]{x1D706}}-\frac{1}{2}.\end{eqnarray}$$

If the term $-1/2$ is neglected this expression differs from the classical value [ $\ln (\unicode[STIX]{x1D706}_{\text{D}}/b_{0})$ ] only in that the lower limit  $b_{0}$ is replaced by $\unicode[STIX]{x1D706}/4\unicode[STIX]{x03C0}$ . This result is easily understood: the De Broglie wave associated with the incident particle is diffracted on the Debye sphere surrounding the scattering center. Diffraction theory or elementary interference considerations shows that to within a factor of order unity the mean value of the diffraction angle is $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D706}/2\unicode[STIX]{x1D706}_{\text{D}}$ . If this value exceeds the classical limit $\unicode[STIX]{x1D703}_{\text{min}}=2b_{0}/\unicode[STIX]{x1D706}_{\text{D}}$ , the classical formula …no longer applies and $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D706}/2\unicode[STIX]{x1D706}_{\text{D}}$ is to be taken as the lower limit in the integral in (3.8). This procedure leads to $L=\ln (4\unicode[STIX]{x1D706}_{\text{D}}/\unicode[STIX]{x1D706})$ , which differs from (3.10) by the unimportant factor of  $\unicode[STIX]{x03C0}$ under the logarithm. It also follows from this qualitative description that scattering on a cutoff Coulomb field is essentially the same as scattering on a Debye field (3.5).

The quantum-mechanical relation (3.10) can be written in the form

(3.11) $$\begin{eqnarray}L_{\text{qu}}=L_{\text{cl}}+\ln \frac{2\unicode[STIX]{x1D6FC}c}{u}-\frac{1}{2}\end{eqnarray}$$

[which, since $\unicode[STIX]{x1D6FC}c/u<1$ , shows that when the diffraction correction is valid the size of the Coulomb logarithm is reduced].

We recall that this formula is derived under the assumption that $u\gg \unicode[STIX]{x1D6FC}c$ , whereas the classical expression (2.7) applies when $u\ll \unicode[STIX]{x1D6FC}c$ .

4. The quantum-mechanical calculation becomes extremely complicated in the intermediate region. It would not be very meaningful to be concerned with this region because we are already using the binary-collision approximation with its artificial and, indeed, somewhat arbitrary truncation of the Coulomb forces so that any improvement in the values of the Coulomb logarithm obtained as a result of more complex calculations would be quite illusory. Instead, it is simpler and more in the spirit of the approximation used here to proceed as follows. The formulas for the limiting cases (2.7) and (3.10) show that the Coulomb logarithm contains the velocity  $u$ under the logarithm so that the latter is a slowly varying function of  $u$ . It is physically obvious that this slow variation also obtains in the intermediate region. Hence, without incurring any serious error we can extrapolate (2.7) and (3.10) into the intermediate region up to the value $u=u_{\text{lim}}$ , at which both expressions coincide. When $u<u_{\text{lim}}$ the classical formula (2.7) is to be used; when $u\gg u_{\text{lim}}$ the quantum-mechanical formulas (3.10) or (3.11) are used.

…If the relative velocity  $u$ is replaced by the equivalent temperature according to the relation $3T=\unicode[STIX]{x1D707}u^{2}$ …[and upon] substituting the appropriate values of the reduced mass for a deuterium plasma, we obtain the following limiting temperatures for electron–electron, electron–ion, and ion–ion collisions, respectively:

(3.12) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}T_{\text{lim}}^{ee}=6.65~\text{eV},\quad \\ T_{\text{lim}}^{ei}=13.3~\text{eV},\quad \\ T_{\text{lim}}^{ii}=2.45\cdot 10^{4}~\text{eV}=24.5~\text{keV}.\quad \end{array}\right.\end{eqnarray}$$

A similar discussion can be found on p. 239 of the book by Kulsrud (Reference Kulsrud2005), who cites Sivukhin. (Kulsrud arrives at somewhat different but qualitatively similar transition temperatures by using a different interpolation scheme.)

All of the above discussions are consistent in their physical interpretations. Interestingly, however, Mulser et al. (Reference Mulser, Alber and Murakami2014) challenged Spitzer’s heuristic description. They stated,

This characterization of Spitzer’s discussion is incorrect and appears to be a misunderstanding; nowhere in his argument did Spitzer mention ‘a diaphragm of diameter $2\unicode[STIX]{x1D706}_{\text{B}}$ ’ or discuss impact parameters smaller than  $\unicode[STIX]{x1D706}_{\text{B}}$ . In fact, in agreement with the various authors cited above, Mulser et al. (Reference Mulser, Alber and Murakami2014) also concluded that the de Broglie correction was associated with large impact parameters. But although they asserted that this was a new and surprising result, we see that it has been understood for more than a half-century.

Although the basic ideas and results are clear, some discussions in the literature are incomplete; for example, Mulser et al. (Reference Mulser, Alber and Murakami2014) cited a number of references in which apparently the classical cutoff was used. However, closer inspection shows that in several of the papers cited by Mulser et al. (Reference Mulser, Alber and Murakami2014) the authors were, in fact, aware of the quantum correction. For example, Rosenbluth, Macdonald & Judd (Reference Rosenbluth, Macdonald and Judd1957) refer in their footnotes 6 and 3 to the discussion of cutoffs by Cohen et al. (Reference Cohen, Spitzer and Routly1950), which, as we have seen, contains the original version of Spitzer’s argument. And although Balescu (Reference Balescu1988) described the classical situation, he noted (p. 130),

One could quibble with Balescu’s characterization of the issue as ‘minor’, and the first Born approximation does not deserve to be called ‘semi-empirical’. In any event, it is instructive to consider the explanation of Braginskii (Reference Braginskii and Leontovich1965, p. 238), who said,Footnote ¹⁰

Here it is asserted that instead of using the classical formula $\unicode[STIX]{x1D703}_{\text{min}}\sim b_{0}/b_{\text{max}}$ with $b_{\text{max}}=\unicode[STIX]{x1D706}_{\text{D}}$ , one should use $b_{\text{max}}=\unicode[STIX]{x1D706}_{\text{D}}(b_{0}/\unicode[STIX]{x1D706}_{\text{B}})$ or $\unicode[STIX]{x1D703}_{\text{min}}\sim \unicode[STIX]{x1D706}_{\text{B}}/\unicode[STIX]{x1D706}_{\text{D}}$ , the latter ‘quantum uncertainty’ being the diffraction angle of a wave with wavelength  $\unicode[STIX]{x1D706}_{\text{B}}$ encountering an object of radius  $\unicode[STIX]{x1D706}_{\text{D}}$ . But although Braginskii’s argument leads to the correct  $\unicode[STIX]{x1D703}_{\text{min}}$ , his heuristic introduction of a modified  $b_{\text{max}}$ is incorrect; apparently, it was devised in order to obtain agreement with the proper  $\unicode[STIX]{x1D703}_{\text{min}}$ and the earlier discussion of Spitzer. However, Braginskii did appreciate the role of the quantum-mechanical uncertainty due to diffraction by Debye screening clouds of radius  $\unicode[STIX]{x1D706}_{\text{D}}$ , and he understood that the classical cutoff was not to be used for large velocities.

The proper value of  $\unicode[STIX]{x1D703}_{\text{min}}$ must follow from a systematic kinetic theory. Clearly, $\unicode[STIX]{x1D706}_{\text{B}}$  can enter the problem only when quantum-mechanical effects are included. An asymptotic matching between the Born approximation and the quasi-classical regime is described by Landau & Lifshitz (Reference Landau and Lifshitz1981, § 46), where original references are cited. The physics content of that calculation agrees with the heuristic arguments given above. A recent and more refined (as yet unpublished) analysis by R. Kulsrud (private communication, 2018) corrects those estimates by a relatively small amount in the five per cent range.

4 Discussion and summary

Are the quantum-mechanical corrections to $\ln \unicode[STIX]{x1D6EC}$ significant? For definiteness, consider electron–ion collisions. For cold ions, Huba (Reference Huba2016) quotes the easy-to-remember formulaFootnote ¹¹

(4.1) $$\begin{eqnarray}\ln \unicode[STIX]{x1D6EC}_{ei}=\left\{\begin{array}{@{}ll@{}}c_{\text{cl}}-\ln (n_{e}^{1/2}ZT_{e}^{-3/2})\quad & \text{for}~T_{e}<10\,Z^{2}~\text{eV},\\ c_{\text{qu}}-\ln (n_{e}^{1/2}T_{e}^{-1})\quad & \text{for}~10\,Z^{2}~\text{eV}<T_{e},\end{array}\right.\end{eqnarray}$$

where $c_{\text{cl}}\approx 23$ , $c_{\text{qu}}\approx 24$ , $Z$  is the atomic number, $n_{e}$  is in units of $\text{cm}^{-3}$ and $T_{e}$  is in eV. For the ITER-like parameters (FusionWiki 2012) $n_{e}\approx 10^{14}~\text{cm}^{-3}$ , $T_{e}\approx 8.8~\text{KeV}$ and $Z=1$ , one finds that the relative quantum-mechanical correction is approximately 17 %. Although this is not entirely negligible, one might question the quantitative significance of the quantum effects given that there are already subdominant corrections to $\ln \unicode[STIX]{x1D6EC}$ (recall footnote 4 on p. 3). Of course, this is just one numerical example; there are other possibilities best left for the discussion of specific applications. In any event, such a concern misses the point. The quantum effects are conceptually interesting and introduce a new dimension to the physics of the collision processes in plasmas. Clarifying one’s understanding of those processes is a worthy goal in itself.

In summary, excerpts from the literature provide historical perspective and explain the correct interpretation of the quantum-mechanical correction to $\text{ln}\,\unicode[STIX]{x1D6EC}$ . The  $\unicode[STIX]{x1D706}_{\text{B}}$ in the ratio $\unicode[STIX]{x1D6EC}_{\text{qu}}=\unicode[STIX]{x1D706}_{\text{D}}/\unicode[STIX]{x1D706}_{\text{B}}$ that appears in the first Born approximation (which is, roughly speaking, valid for sufficiently high temperatures) ‘is the result of the charge distributions at large impact parameters’ (Mulser et al. Reference Mulser, Alber and Murakami2014), not quantum uncertainty related to the impact parameter  $b_{0}$ for $90^{\circ }$ scattering or to the distance of closest approach ( $2b_{0}$ ). This conclusion was obtained early on by Cohen et al. (Reference Cohen, Spitzer and Routly1950), Spitzer (Reference Spitzer1962) and Sivukhin (Reference Sivukhin1966), and it has been usefully repeated in more modern texts such as those of Kulsrud (Reference Kulsrud2005) and Wesson (Reference Wesson2011, chap. 14.5). Proper understanding of the physics of the Coulomb logarithm is crucial, as that basic quantity figures in a multitude of important plasma-physics applications.