# Who invented the term "Coulomb logarithm"?

Who is the author of the term "Coulomb logarithm"? In fact, Coulomb logarithm was computed by Langmuir in his paper of 1928 where the term "plasma" was introduced into physics, but the term "Coulomb logarithm" seemed to appear later. The earliest reference I found is the paper Relativistic kinetic equation by Belyaev & Budker (1956).

UPDATE: Using Google Labs I found a reference dated by 1937 in very rare Georgian journal in Russian.

• Have you checked to make sure that Boltzmann or Maxwell did not have a hand in this? The term comes from impact parameter estimates and their work on gas theory might contain early versions of this. Commented Apr 24, 2015 at 12:25
• I suggest asking here: hsm.stackexchange.com
– user20250
Commented Sep 30, 2016 at 13:07
• Interesting question; according to google books ngrams, it became commonly used in the 1960s.
– Alf
Commented Mar 15, 2018 at 10:38

Interesting question. I believe that the term "Coulomb logarithm" was first used by Landau in a 1936 paper, "Die kinetische Gleichung für den fall Coulombscher wechselwirkung," which was published in the German-language journal Physikalische Zeitschrift der Sowjetunion.

It is difficult to prove a negative, so I cannot claim to be 100% certain that there are no earlier sources. However, as evidence, here is a quote from the classic 1976 monograph "Theory of plasma transport in toroidal confinement systems" by Hinton and Hazeltine (emphasis added):

All of the transport literature under review is based on the assumption that the distribution function $$f_a(\mathbf{\overrightarrow x}, \mathbf{\overrightarrow v}, t)$$ satisfies the Fokker-Planck equation, $$\frac{\partial f_a}{\partial t} + \mathbf{\overrightarrow v} \cdot \mathbf{\overrightarrow \nabla}f_a + \left(\frac{e_a}{m_a}\right)(\mathbf{\overrightarrow E} + c^{-1} \mathbf{\overrightarrow v} \times \mathbf{\overrightarrow B}) \cdot \frac{\partial f_a}{\partial \mathbf{\overrightarrow v}} = C_a(f), \tag{1.1}$$ where $$e_a$$ is the charge and $$m_a$$ the mass of particles of species $$a$$, $$\mathbf{\overrightarrow E}$$ and $$\mathbf{\overrightarrow B}$$ are, respectively, the (macroscopic) electric and magnetic fields, and $$C_a$$ is the Fokker-Planck collision operator: $$C_a = \sum_b C_{ab},$$ $$C_{ab} = -\frac{2\pi e_a^2 e_b^2}{m_a} \ln \Lambda \frac{\partial}{\partial v_\alpha} \int d^3v' \left[ \frac{f_a(v)}{m_b} \frac{\partial f_b(\mathbf{\overrightarrow v}')}{\partial v_\beta'} \\- \frac{f_b(v')}{m_a} \frac{\partial f_a(\mathbf{\overrightarrow v}')}{\partial v_\beta'} \right] U_{\alpha\beta} \left(\mathbf{\overrightarrow v} - \mathbf{\overrightarrow {v'}}\right) \tag{1.2}$$ Here, a sum over repeated Cartesian indices $$(\alpha, \beta)$$ is implied and $$U_{\alpha\beta}(\mathbf{\overrightarrow x}) = x^{-3}(x^2 \delta_{\alpha\beta} - x_\alpha x_\beta) \tag{1.3}$$ The Coulomb logarithm is formally given by $$\ln \Lambda = \ln (9N) \tag{1.4}$$ where $$N \gg 1$$ is the number of particles in a sphere of radius $$\lambda_D$$, the Debye length. Equation (1.2) was first derived by Landau (1936). Chandrasekhar (1943) used a different argument to derive a special case of Eq. (1.2), which was later extended, somewhat, by Cohen, Spitzer and Routly (1950).

Also, here is a quote from a 2013 paper, "Effective Potential Theory for Transport Coefficients across Coupling Regimes":

The bare Coulomb potential neglects screening of the intervening medium and leads to a divergent collision operator. To fix this unphysical divergence, Landau utilized the weak coupling assumption to cut off the impact parameter at the Debye screening length [3], hence, imposing an effective potential. This approximation leads to the traditional Coulomb logarithm, $$\ln \Lambda$$, where $$\Lambda \sim \Gamma^{-3/2}$$ is the plasma parameter. It is valid in the limit that this parameter is asymptotically large.

• It is a pity that this answer came after I passed the proofreading of my textbook. There I write that the term Coulomb logarithm came into use in the early 1950s. In the Russian version of this article by Landau, the term is not used. Unfortunately, I do not have a German version of Landau's article, so I could not verify your information. An important fact: the German version was published before the Russian one.
– Igor
Commented Mar 3, 2021 at 6:15