A recent journal article by Wissel *et al.*^{1} describes a dc glow discharge experiment proposed as part of the curriculum for undergraduate students. As plasma physicists, we wholeheartedly support the idea of including these inexpensive but physically meaningful experiments into our student curriculum, and we are grateful to these authors for bringing such beautiful experiments wider attention. That said, we do have a few comments on this article.

In Sec. II A, there is an important typo in Paschen's law, which is written as

where the product *pd* is the pressure multiplied by the spark gap distance, *B* is a constant that depends on the gas, and *C* is a constant that is (incorrectly) written as $C=lnA\u2212ln(1+1/\gamma )$. This constant should be written $C=lnA\u2212ln[ ln(1+1/\gamma ) ]$, which is consistent with the published literature and can be verified from the two equations given immediately preceding their expression for Paschen's law.

We feel the need, however, to comment that the *status quo* in regard to Eq. (1) (with the correct *C*) is a sorry state of affairs through no fault of the authors. Rather, it is a practice upheld by long tradition (see, for example, their Ref. 13 and references therein). The problem is that the argument of the natural logarithm is never given in dimensionless units. There are obvious problems with this situation which is severely exacerbated by a lack of consensus for the units of pressure; mbar, atm, Torr, mm Hg, and Pa are all common in the literature. Furthermore, since all derivations of Paschen's law begin with exponential functions (e.g., $\alpha =Ape\u2212Bp/E$, where *A* is a second gas constant and *E* is the breakdown electric field) in which the argument of the exponential function is dimensionless, it is somewhat inconsistent to then have dimensions in the argument of the natural logarithm. Thus, we assert that despite current practice, Eq. (1) is better expressed as

where $C=ln[ ln(1+1/\gamma ) ]$. In this equation, the (dimensionless) quantity *Apd* does not depend on the units used, which we believe is more efficient and pedagogically desirable.

Before leaving this point we note that in Fig. 3 of Ref. 1 the authors plot Eq. (1) for the known constants of air and show a minimum breakdown voltage occurring for *pd* near 1 Pa cm. However, in the published literature (see, for example, Lisovskiy *et al.*,^{2} who consider variations with respect to gap length and plate radius ratio) the minimum breakdown voltage for air occurs for $pd\u223c70$–100 Pa cm, about 2 orders of magnitude larger than what is shown in Ref. 1. This is a problem with the units,^{3} which could be eliminated by consistently employing standard SI units, a practice that should be recommended to all students.

Concerning technique, we have two comments. The authors correctly state that the glow of the discharge is generally observed above the actual minimum voltage a discharge occurs (i.e., the measured data in Fig. 3 have a minima above that of the theory curve) because a weak (low density) discharge can have optical emissions too low to be observed, particularly by the naked eye. This error can be substantially eliminated by determining the breakdown voltage by observing the jump in current of the power supply (instead of looking for the glow) as it is the jump in conductivity that is the most immediate result of breakdown. In Sec. IV, the authors also correctly describe that a Langmuir probe's I–V characteristics will be exponential between the floating potential and the plasma potential, and not so beyond the plasma potential, respectively, due to collection of Maxwellian electrons and sheath expansion. The shape of the I–V trace beyond the plasma potential is not strictly linear and largely depends on the geometry of the probe. However, it should be obvious that a student can determine the plasma potential by looking for the transition point where the I–V characteristic changes from exponential to non-exponential, a region of the I–V trace sometimes called the “knee” (see Fig. 2 in Ref. 6 paper of Wissel *et al.*, most easily observed with planar probe tips). This method presents a more direct measurement of the plasma potential than inferring it from the floating potential, and should be learned by students. This method also permits a simple determination of the electron density from the probe current at the plasma potential (from the electron saturation current) from the equation $Isat=AprobeneTe/ne$. For cylindrical Langmuir probe tips, this “knee” of the I–V trace is famously difficult to discern, as it becomes blurred by extra current collection from sheath expansion for probe potentials greater than that of the plasma potential. In this case, a student can find the “knee” and estimate the plasma potential by looking for an inflection point—a peak of the first voltage derivative of the I–V trace—provided that the probe is clean. The references given for electrostatic probes are excellent, but we would add one: Hershkowitz's article^{4} is comprehensive and includes practical suggestions for improving accuracy. For example, the accuracy of the Langmuir probe partially depends on the cleanness of its surface. A dirty probe tends to falsely record higher electron temperature and lower plasma density, which seems consistent with the author's disagreement between the Langmuir probe and optical measurements. Students are typically surprised that probes can get dirty in vacuum chamber! A simple way to clean the probe is to heat it up by ohmic heating by drawing a current from the plasma; this can be done by biasing the probe at a sufficiently high positive voltage while carefully monitoring the current so as not to damage the probe.

Finally, the authors state that each Paschen's-law curve is different. We appreciate the delightful open-endedness that this creates and the model-making that these differences inspire. The authors clearly celebrate these things. We hope though that this is done in a context in which reproducibility is broadly sought, and in which careful attention to experimental detail (e.g., comparisons to published experimental results and their details) is also fostered. Likewise, students should be encouraged to attempt to account for the difference between the Langmuir probe and optical measurements through literary and computational studies on the limitations of each of these measurement techniques, and where they attempt to correct their measurements to see if they can find agreement.

## References

*et al*., private communication (