# How accurately can you count electrons?

I read that with modern technology they can now shoot one electron at a time.

Can you tell me how accurately it is possible to count charges, how it is made and how they did this in the past?

How do you determine a statcoulomb ($2 \times 10^8$) static charges or a Coulomb ($6.24 \times 10^{18}$) moving charges?

You are probably referring to the Feynman-style double-slit experiment, done with individual electrons. A recent experimental realization of this was reported in

Controlled double-slit electron diffraction. R Bach et al. N. J. Phys. 15 033018 (2013) (open access).

and it is described in more readable detail in

Feynman's double-slit experiment gets a makeover. H Johnston, Physics World, 14 March 2013.

The experimental data looks like this, which you might recognize:

Image source: Bach et al., CC-BY license.

The experiment is in essence really simple: simply take an electron gun (i.e. a cathode-ray tube like you find in old TVs, simply a heated piece of metal that's negatively charged) and dial it down to bring the current to levels where there is only one electron in the system at a time. If your detector is sensitive enough, then you will be able to detect individual electron hits.

The detector in this experiment was a microchannel plate (a metal plate with lots of tiny holes in it), to discretize the electron positions, with a phosphorescent screen behind it to change the electron hits into light, which is then imaged with a (fancy, low-light, but otherwise perfectly normal) camera.

Note, however, that this electron-counting technique only tells you "here's one electron" and "there's another one" and so on. Counting the number of electrons in a given charged object is a lot more difficult. If you want to do that then you can probably achieve single-electron accuracy in objects with tens or possibly hundreds of electrons, depending on the situation, but beyond that you really need to ask exactly what it is you're doing and why.

In particular, it is neither feasible nor reasonable to put together one Coulomb's worth of electrons to single-electron accuracy.

This is partly because, in the end, measurements of charge are not that useful - it is measurements of current that really matter. It is possible, though, to construct experiments that will measure current by observing single electrons pass in single file through a series of Josephson junctions, like the one in

Current measurement by real-time counting of single electrons. J Bylander et al. Nature 434, 361 (2005), arXiv:cond-mat/0411420.

The authors claim to be able to measure currents as large as 1 pA by counting single electrons, which corresponds to 6 million electrons per second.

The current state of the art is that these experiments are approaching the same level of accuracy, precision, stability, reliability, and dynamic range that one can get with traditional current calibration (which is based on the force between two parallel wires, and therefore depends on the SI standards for time, length and mass).

This means that we can turn the relation around, and user experiments like Bylander's to define the ampere as the current caused by the passage of 6.241509×1018 elementary charges per second. (This now depends on the SI second but nothing else.) And indeed, the plan is to do just that, as part of the "new SI" as proposed by the BIPM. This is part of a larger makeover of the SI (including a redefinition of the kilogram in terms of a set value for $\hbar$ by means of watt balances, and re-workings of the kelvin and the mole), so it's coming along but slowly. (In particular, it is not quite clear what the proposed realizations actually are, but it will happen.)

• Thanks, Emilio, that is really interesting. I actually was referring to the experiment they did to measure the esu, which should have 208 194 271 electrons, presumably on a metal plate, how do you count that?. I'll ask a separate question and hope you can answer in detail. – user86008 Sep 21 '15 at 14:28
• @user86008 where did you see that number? It is likely to simply be a calculation of the number of electrons you'd expect an esu to have if you manage to have exactly 1 esu in your system (very hard, but potentially feasible - I don't know), based on precision measurements of the electron charge with respect to the SI force-based ampere standard. – Emilio Pisanty Sep 21 '15 at 14:40
• Measuring that an electron (or two!) has hit a detector is very straightforward, using a channeltron, multichannel plate, or surface barrier detector (this assumes a reasonable electron energy suitable to the detector). An ever-present concern is to avoid having two hit and look like one, but dialing down the current and adding chopping/blanking electron optics takes care of that pretty well (or monitoring your pulse pileup). Note that a standard TEM has, on average, only a few electrons in the entire column at any one time, and only one (or none) in the sample itself. – Jon Custer Sep 21 '15 at 15:00
• 1 esu is 3.3564*6.241509 *10^18/10^10 electrons. 1 esu^2 produces 1 dyne which produces 1 erg = 1J/10^7: 1.50916*12^26 Hz. Any idea about how they managed to determine that? – user86008 Sep 21 '15 at 15:11
• @user86008 That's impossible to tell conclusively because the reference does not include your quotation. However, it looks like it's not a determination of electron numbers - it's simply a charge measurement with an uncertainty larger than $\pm e$. If you provide a specific citation then we can comment on the details. – Emilio Pisanty Sep 21 '15 at 16:12

Addendum as a separate answer because otherwise it's too long and unwieldy. CW to avoid double rep.

OK, so you have clarified your question to the point where I can answer your specific concerns.

Provided you know 8 figures of the coulomb (and you do: 6.24150934), you know the esu, (which is 1C/$c$): 1 esu = 208 194 341 $e$. Can you put/ count as many charges on 2 tiny plates at 1 cm distance and measure the force of repulsion/attraction?

This is a misunderstanding on how the Coulomb is defined and therefore measured in metrological labs. In the SI the Coulomb is defined as

• an ammount of charge such that, if one Coulomb per second flows through a pair of long, thin wires, which are laid parallel and separated by one meter, the magnetic force between the wires is 2×10−7 newtons per metre of length.

The base electrical dimension is the current and not the charge, as you have no doubt read already. This is fundamentally important, because metrological measurements follow from current standards and not charge standards. Measurements of current (which only ever mean comparing two different currents!) can be arbitrarily precise, and they fundamentally rely only on measurements of mass, length and time.

Perhaps more importantly, current is not quantized, because charge comes in lumps but you can take one lump every 1.2 seconds or every 1.20000001231000001 seconds if you're so inclined. (Current does suffer from shot noise, which is a statistical uncertainty caused by an unevenness in the flow of individual charges, but that can be dealt with by appropriate metrological methods.) This means that if your force-measuring apparatus is accurate enough (and your statistical methods are robust enough) then you can determine the current of any source (as a multiple of the ampere standard) to arbitrary precision.

Finally, because it is defined in terms of other dynamical standards, this means that the ratio $1\:\mathrm {statC}/e$ - the number of electrons in one statoulomb - need not be an integer. This has a number of weird consequences, which are nevertheless OK. For example, the statcoulomb is formally the amount of charge such that two 1 statC small charges placed 1 cm apart will experience a force of exactly 1 dyne, but it may not be possible to find such a charge in nature.

Currently, we don't know. The electron charge is known in terms of the Coulomb as $e=1.602 176 6208(98) \times 10^{-19}\:\mathrm C$, where the statistical uncertainty means ±98 on the last two digits; CODATA value. One statcoulomb of charge equals exactly $(2997924580)^{-1}\mathrm C$. Putting in these two we obtain that one statcoulomb is $$\frac{1\:\mathrm{statC}}{1e}=2081943344\pm12$$ electrons. That is, the measurements of the electron charge (which in the case of CODATA are part of a much larger, interconnected measurement campaign; cf. the Wikipedia entry) do not currently permit us to establish exactly how many electrons fit into a single statcoulomb. (For the Coulomb we're much farther away, by about 10 orders of magnitude.)

It is entirely reasonable, however, to suppose that in the near future finer measurements will yield a more dependable value of $e$ in terms of the SI coulomb. In that case, the ratio $1\:\mathrm{statC}/e$ will most likely not be an integer - it would be highly surprising if it is, given that the Coulomb currently depends on things like the mass of the standard kilogram.

And that's OK! Neither the coulomb or the statcoulomb are meant to exist as physical amounts of charges put together on a metal plate to act as a standard. They're set amounts with respect to which we measure other charges: they are metrological constructs that aid in measuring other charges, but in the end they're little more than that.

• You are saying that the esu is an abstract unit and maybe an integer squared cannever produce 1-dyne force. OK, but my question is concrete: if it has not been determined experimentally, can you anyway, with current technology, put 2 billion charges on 2 tiny plates? how do you do that, and to what approximation? and can you measure the repulsive force? Is there any better way you can exactly determine the electrostatic force? If you know some of the answers I'll post a specific question. Thanks! – user86008 Sep 22 '15 at 15:58
• As in, can you put 2,081,943,344 electrons to the unit, on a plate, and measure the resulting force? No, that level of precision is not currently achievable. It's also of very limited scientific value, to be honest. (The current push to define the ampere in terms of a fixed value of $e$ and the SI second has little to do with the Coulomb being a fixed number of electrons, and much more to do with having more reliable and stable standards for electrical units, and with detaching the SI from its dependence on the standard (artefact) kilogram.) – Emilio Pisanty Sep 22 '15 at 20:00
• Re: 'Is there any better way you can exactly determine the electrostatic force?' - all measurements have their uncertainty, and it's always about getting an uncertainty that's small enough for your purposes. Nowadays electrical units are not designed so we can measure forces between charged spheres, they're built to have robust metrological procedures for all our technical endeavours which involve electricity. "Determine electrical forces" nowadays means things like building precise watt balances. – Emilio Pisanty Sep 22 '15 at 20:06