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The rms charge radius is a measure of the size of an
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutro ...
, particularly the
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mas ...
distribution. It can be measured by the scattering of
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s by the nucleus. Relative changes in the mean squared nuclear charge distribution can be precisely measured with
atomic spectroscopy Atomic spectroscopy is the study of the electromagnetic radiation absorbed and emitted by atoms. Since unique elements have characteristic (signature) spectra, atomic spectroscopy, specifically the electromagnetic spectrum or mass spectrum, is appl ...
.


Definition

The problem of defining a radius for the atomic nucleus has some similarity to that of defining a radius for the entire atom; neither have well defined boundaries. However, basic liquid drop models of the nucleus imagine a fairly uniform density of nucleons, theoretically giving a more recognizable surface to a nucleus than an atom, the latter being composed of highly diffuse electron clouds with density gradually reducing away from the centre. For individual protons and neutrons or small nuclei, the concepts of size and boundary can be less clear. A single nucleon needs to be regarded as a " color confined" bag of three valence quarks, binding
gluons A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind qua ...
and so called "sea" of quark-antiquark pairs. Additionally, the nucleon is surrounded by its Yukawa pion field responsible for the
strong nuclear force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
. It could be difficult to decide whether to include the surrounding Yukawa meson field as part of the proton or nucleon size or to regard it as a separate entity. Fundamentally important are realizable experimental procedures to measure some aspect of size, whatever that may mean in the quantum realm of atoms and nuclei. Foremost, the nucleus can be modeled as a sphere of positive charge for the interpretation of
electron scattering Electron scattering occurs when electrons are deviated from their original trajectory. This is due to the electrostatic forces within matter interaction or, if an external magnetic field is present, the electron may be deflected by the Lorentz fo ...
experiments: the electrons "see" a range of cross-sections, for which a mean can be taken. The qualification of "rms" (for "
root mean square In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
") arises because it is the nuclear cross-section, proportional to the square of the radius, which is determining for electron scattering. This definition of charge radius is often applied to composite
hadron In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...
s such as a
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mas ...
,
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beha ...
,
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more ge ...
, or
kaon KAON (Karlsruhe ontology) is an ontology infrastructure developed by the University of Karlsruhe and the Research Center for Information Technologies in Karlsruhe. Its first incarnation was developed in 2002 and supported an enhanced version of ...
, that are made up of more than one
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
. In the case of an anti-matter baryon (e.g. an anti-proton), and some particles with a net zero
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respecti ...
, the composite particle must be modeled as a sphere of negative rather than positive electric charge for the interpretation of electron scattering experiments. In these cases, the square of the charge radius of the particle is defined to be negative, with the same absolute value with units of length squared equal to the positive squared charge radius that it would have had if it was identical in all other respects but each quark in the particle had the opposite electric charge (with the charge radius itself having a value that is an imaginary number with units of length). It is customary when charge radius takes an imaginary numbered value to report the negative valued square of the charge radius, rather than the charge radius itself, for a particle. The best known particle with a negative squared charge radius is the
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beha ...
. The heuristic explanation for why the squared charge radius of a neutron is negative, despite its overall neutral electric charge, is that this is the case because its negatively charged down quarks are, on average, located in the outer part of the neutron, while its positively charged up quark is, on average, located towards the center of the neutron. This asymmetric distribution of charge within the particle gives rise to a small negative squared charge radius for the particle as a whole. But, this is only the simplest of a variety of theoretical models, some of which are more elaborate, that are used to explain this property of a neutron. For
deuteron Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one n ...
s and higher nuclei, it is conventional to distinguish between the scattering charge radius, ''r''d (obtained from scattering data), and the bound-state charge radius, ''R''d, which includes the Darwin–Foldy term to account for the behaviour of the
anomalous magnetic moment In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. (The ''magnetic moment'', also called '' ...
in an electromagnetic field and which is appropriate for treating spectroscopic data. The two radii are related by :R_ = \sqrt, where ''m''e and ''m''d are the masses of the electron and the deuteron respectively while ''λ''C is the
Compton wavelength The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It wa ...
of the electron. For the proton, the two radii are the same.


History

The first estimate of a nuclear charge radius was made by
Hans Geiger Johannes Wilhelm "Hans" Geiger (; ; 30 September 1882 – 24 September 1945) was a German physicist. He is best known as the co-inventor of the detector component of the Geiger counter and for the Geiger–Marsden experiment which discover ...
and
Ernest Marsden Sir Ernest Marsden (19 February 1889 – 15 December 1970) was an English- New Zealand physicist. He is recognised internationally for his contributions to science while working under Ernest Rutherford, which led to the discovery of new theori ...
in 1909, under the direction of
Ernest Rutherford Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics. ''Encyclopædia Britannica'' considers him to be the greatest ...
at the Physical Laboratories of the
University of Manchester , mottoeng = Knowledge, Wisdom, Humanity , established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Unive ...
, UK. The famous experiment involved the scattering of α-particles by
gold Gold is a chemical element with the Symbol (chemistry), symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a Brightness, bright, slightly orange-yellow, dense, s ...
foil, with some of the particles being scattered through angles of more than 90°, that is coming back to the same side of the foil as the α-source. Rutherford was able to put an upper limit on the radius of the gold nucleus of 34 femtometres. Later studies found an empirical relation between the charge radius and the mass number, ''A'', for heavier nuclei (''A'' > 20): :''R'' ≈ ''r''0''A'' where the empirical constant ''r''0 of 1.2–1.5 fm can be interpreted as the
Compton wavelength The Compton wavelength is a quantum mechanical property of a particle. The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It wa ...
of the proton. This gives a charge radius for the gold nucleus (''A'' = 197) of about 7.69 fm.


Modern measurements

Modern direct measurements are based on precision measurements of the atomic
energy level A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The ...
s in hydrogen and deuterium, and measurements of scattering of electrons by nuclei.. There is most interest in knowing the charge radii of
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mas ...
s and
deuteron Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one n ...
s, as these can be compared with the spectrum of atomic
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxi ...
/
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one ...
: the nonzero size of the nucleus causes a shift in the electronic energy levels which shows up as a change in the frequency of the
spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to identi ...
s. Such comparisons are a
test Test(s), testing, or TEST may refer to: * Test (assessment), an educational assessment intended to measure the respondents' knowledge or other abilities Arts and entertainment * ''Test'' (2013 film), an American film * ''Test'' (2014 film), ...
of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED). Since 2002, the proton and deuteron charge radii have been independently refined parameters in the
CODATA The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the Internationa ...
set of recommended values for physical constants, that is both scattering data and spectroscopic data are used to determine the recommended values. The 2014 CODATA recommended values are: :proton: ''R''p = 0.8751(61)×10−15 m :deuteron: ''R''d = 2.1413(25)×10−15 m Recent measurement of the
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
in muonic hydrogen (an
exotic atom An exotic atom is an otherwise normal atom in which one or more sub-atomic particles have been replaced by other particles of the same charge. For example, electrons may be replaced by other negatively charged particles such as muons (muonic atoms) ...
consisting of a proton and a negative muon) indicates a significantly lower value for the proton charge radius, : the reason for this discrepancy is not clear.


References

{{Authority control Physical constants Nuclear chemistry Nuclear physics hu:Atommag#Atommagok tulajdonságai