Elementary charge

The elementary charge, usually denoted as e or sometimes q, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron.[1] This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called the "elementary positive charge". This charge has a measured value of approximately 1.602176487(40)×10−19 coulombs.[2] In the cgs system, the value for e is 4.80320427(12)×10−10 statcoulombs.[3] In the system of atomic units as well as some other systems of natural units, e functions as the unit of electric charge, i.e. e = 1 in those unit systems.

The magnitude of the elementary charge was first measured in Robert A. Millikan's noted oil-drop experiment in 1909.[4]

Charge quantization is the statement that every stable and independent object (meaning an object that can exist independently for a prolonged period of time) has a charge which is an integer multiple of the elementary charge e: A charge can be exactly 0, or exactly e, –e, 2e, etc., but not, say, 1⁄2 e, or –3.8e, etc. (This statement must not be interpreted to include quarks or quasiparticles, since neither quarks nor quasiparticles possess the ability to exist on their own for prolonged periods of time. Quarks have charges that are integer multiples of 1⁄3 e.)

This is the reason for the terminology "elementary charge": it is meant to imply that it is an indivisible unit of charge.

Contents

Charges less than an elementary charge

There are two known sorts of exceptions to the indivisibility of the elementary charge: quarks and quasiparticles.

What is the quantum of charge?

All known elementary particles, including quarks, have charges that are integer multiples of 1⁄3e. Therefore, one can say that the "quantum of charge" is 1⁄3e. In this case, one says that the "elementary charge" is three times as large as the "quantum of charge".

On the other hand, all isolatable particles have charges that are integer multiples of e. (Quarks cannot be isolated, except in combinations like protons that have total charges which are integer multiples of e.) Therefore, one can say that the "quantum of charge" is e, with the proviso that quarks are not to be included. In this case, "elementary charge" would be synonymous with the "quantum of charge".

In fact, both terminologies are used.[5] For this reason, phrases like "the quantum of charge" or "the indivisible unit of charge" can be ambiguous, unless further specification is given. On the other hand, the term "elementary charge" is unambiguous: It universally refers to the charge of a proton.

Experimental measurements of the elementary charge

In terms of the Avogadro constant and Faraday constant

If the Avogadro constant NA and the Faraday constant F are independently known, the value of the elementary charge can be deduced, using the formula

e = \frac{F}{N_{\mathrm{A}}}

(In other words, the charge of one mole of electrons, divided by the number of electrons in a mole, equals the charge of a single electron.)

In practice, this method is not how the most accurate values are measured today: nevertheless, it is a legitimate and still quite accurate method, and experimental methodologies are described below:

The value of the Avogadro constant NA was first approximated by Johann Josef Loschmidt who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas.[6] Today the value of NA can be measured at very high accuracy by taking an extremely pure crystal (in practice, often silicon), measuring how far apart the atoms are spaced using X-ray diffraction or another method, and accurately measuring the density of the crystal. From this information, one can deduce the mass (m) of a single atom; and since the molar mass (M) is known, the number of atoms in a mole can be calculated: NA = M/m.[7]

The value of F can be measured directly using Faraday's laws of electrolysis. Faraday's laws of electrolysis are quantitative relationships based on the electrochemical researches published by Michael Faraday in 1834.[8] In an electrolysis experiment, there is a one-to-one correspondence between the electrons passing through the anode-to-cathode wire and the ions that plate onto or off of the anode or cathode. Measuring the mass change of the anode or cathode, and the total charge passing through the wire (which can be measured as the time-integral of electric current), and also taking into account the molar mass of the ions, one can deduce F.[7]

The limit to the precision of the method is the measurement of F: the best experimental value has a relative uncertainty of 1.6 ppm, about thirty times higher than other modern methods of measuring or calculating the elementary charge.[7][9]

Oil-drop experiment

Main article: Oil-drop experiment

A famous method for measuring e is Millikan's oil-drop experiment. A small drop of oil in an electric field would move at a rate that balanced the forces of gravity, viscosity (of traveling through the air), and electric force. The forces due to gravity and viscosity could be calculated based on the size and velocity of the oil drop, so electric force could be deduced. Since electric force, in turn, is the product of the electric charge and the known electric field, the electric charge of the oil drop could be accurately computed. By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e.

Shot noise

Any electric current will be associated with noise from a variety of sources. One of these sources, the shot noise, is due to the fact that any current is not a smooth continual flow, but rather, it consists of discrete electrons which pass by one at a time. By carefully analyzing the noise of a current, the charge of an electron can be calculated. This method, first proposed by Walter H. Schottky, can only give a value of e accurate to a few percent.[10] However, it was used in the first direct observation of Laughlin quasiparticles, implicated in the fractional quantum Hall effect.[11]

From the Josephson and von Klitzing constants

Another accurate method for measuring the elementary charge is by inferring it from measurements of two effects in quantum mechanics: The Josephson effect, voltage oscillations that arise in certain superconducting structures; and the quantum Hall effect, a quantum effect of electrons at low temperatures, strong magnetic fields, and confinement into two dimensions. The Josephson constant is

K_\mathrm{J} = \frac{2e}{h}

(where h is the Planck constant). It can be measured directly using the Josephson effect.

The von Klitzing constant is

R_\mathrm{K} = \frac{h}{e^2}

It can be measured directly using the quantum Hall effect.

From these two constants, the elementary charge can be deduced:

e = \frac{2}{R_\mathrm{K} K_\mathrm{J}}

CODATA method

In the most recent CODATA adjustments,[7] the elementary charge is not an independently refined quantity. Instead, a value is derived from the relation

e^2 = \frac{2h\alpha}{\mu_0 c_0}

where h is the Planck constant, α is the fine structure constant, μ0 is the magnetic constant and c0 is the speed of light. The uncertainty in the value of e is currently determined entirely by the uncertainty in the Planck constant.

The most precise values of the Planck constant come from watt balance experiments, which are currently used to measure the product K2JRK. The most precise values of the fine structure constant come from comparisons of the measured and calculated value of the gyromagnetic ratio of the electron.[7]

References

  1. ↑ Note that the symbol e has many other meanings. Somewhat confusingly, in atomic physics, e sometimes denotes the electron charge, i.e. the negative of the elementary charge.
  2. ↑ See the NIST posted CODATA value for e
  3. ↑ This is derived from the NIST value and uncertainty, using the fact that one coulomb is exactly 2,997,924,580 statcoulombs. (The conversion is ten times the numerical speed of light in meters/second.)
  4. ↑ Robert Millikan: The Oil-Drop Experiment
  5. ↑ Q is for Quantum, by John R. Gribbin, Mary Gribbin, Jonathan Gribbin, page 296, Web link
  6. ↑ Loschmidt, J. (1865). "Zur Grösse der Luftmoleküle". Sitzungsberichte der kaiserlichen Akademie der Wissenschaften Wien 52 (2): 395–413.  English translation.
  7. ↑ 7.0 7.1 7.2 7.3 7.4 Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80: 633–730. doi:10.1103/RevModPhys.80.633. http://physics.nist.gov/cuu/Constants/codata.pdf.  Direct link to value..
  8. ↑ Ehl, Rosemary Gene; Ihde, Aaron (1954). "Faraday's Electrochemical Laws and the Determination of Equivalent Weights". Journal of Chemical Education 31 (May): 226–232. doi:10.1021/ed031p226. 
  9. ↑ Mohr, Peter J.; Taylor, Barry N. (1999). "CODATA recommended values of the fundamental physical constants: 1998". J. Phys. Chem. Ref. Data 28 (6): 1713–1852. doi:10.1103/RevModPhys.72.351. .
  10. ↑ Beenakker, Carlo; Schönenberger, Christian, Quantum Shot Noise. Fluctuations in the flow of electrons signal the transition from particle to wave behavior, http://arxiv.org/pdf/cond-mat/0605025 .
  11. ↑ de-Picciotto, R.; Reznikov, M.; Heiblum, M.; Umansky, V.; Bunin, G.; Mahalu, D. (1997), "Direct observation of a fractional charge", Nature 389 (162–164): 162, doi:10.1038/38241 .