
The mass-to-charge ratio (''m''/''Q'') is a
physical quantity
A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
relating the ''
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
'' (quantity of matter) and the ''
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 ...
'' of a given particle, expressed in
units of
kilograms
The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially ...
per
coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary ch ...
(kg/C). It is most widely used in the
electrodynamics of
charged particles
In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary pa ...
, e.g. in electron optics and
ion optics.
It appears in the scientific fields of
electron microscopy,
cathode ray tubes,
accelerator physics,
nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...
,
Auger electron spectroscopy
A Hanford scientist uses an Auger electron spectrometer to determine the elemental composition of surfaces.
Auger electron spectroscopy (AES; pronounced in French) is a common analytical technique used specifically in the study of surfaces and, ...
,
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosophe ...
and
mass spectrometry
Mass spectrometry (MS) is an analytical technique that is used to measure the mass-to-charge ratio of ions. The results are presented as a '' mass spectrum'', a plot of intensity as a function of the mass-to-charge ratio. Mass spectrometry is u ...
.
The importance of the mass-to-charge ratio, according to classical electrodynamics, is that two particles with the same mass-to-charge ratio move in the same path in a vacuum, when subjected to the same electric and magnetic fields. On rare occasions, the
thomson has been used as its unit in the field of mass spectrometry.
Some disciplines use the charge-to-mass ratio (''Q''/''m'') instead, which is the
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
of the mass-to-charge ratio. 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 International ...
recommended value for an
electron
The electron (, or in nuclear reactions) 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 partic ...
is
Origin
When charged particles move in electric and magnetic fields the following two laws apply:
*
Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an ele ...
law:
*
Newton's second law
Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at re ...
of motion:
where F is the
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
applied to the ion, ''m'' is the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
of the particle, a is the
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
, ''Q'' is the
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 ...
, E is the
electric field, and v × B is the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
of the ion's
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
and the
magnetic flux density.
This differential equation is the classic equation of motion for charged particles. Together with the particle's initial conditions, it completely determines the particle's motion in space and time in terms of ''m''/''Q''. Thus
mass spectrometers could be thought of as "mass-to-charge spectrometers". When presenting data in a
mass spectrum, it is common to use the dimensionless ''m''/''z'', which denotes the dimensionless quantity formed by dividing the mass number of the ion by its charge number.
Combining the two previous equations yields:
This differential equation is the classic equation of motion of a charged particle in vacuum. Together with the particle's initial conditions it determines the particle's motion in space and time. It immediately reveals that two particles with the same ''m''/''Q'' ratio behave in the same way. This is why the mass-to-charge ratio is an important physical quantity in those scientific fields where charged particles interact with magnetic or electric fields.
Exceptions
There are non-classical effects that derive from
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
, such as the
Stern–Gerlach effect that can diverge the path of ions of identical ''m''/''Q''.
Symbols and units
The IUPAC recommended symbol for mass and charge are ''m'' and ''Q'', respectively, however using a lowercase ''q'' for charge is also very common. Charge is a scalar property, meaning that it can be either
positive (+) or
negative (−). The
Coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary ch ...
(C) is the SI unit of charge; however, other units can be used, such as expressing charge in terms of the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a funda ...
(''e''). The
SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
of the physical quantity ''m''/''Q'' is kilogram per coulomb.
Mass spectrometry and ''m''/''z''
The units and notation above are used when dealing with the physics of mass spectrometry; however, the ''m''/''z'' notation is used for the independent variable in a
mass spectrum.
This notation eases data interpretation since it is numerically more related to the
unified atomic mass unit
The dalton or unified atomic mass unit (symbols: Da or u) is a non-SI unit of mass widely used in physics and chemistry. It is defined as of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at r ...
.
For example, if an ion carries one charge the ''m''/''z'' is numerically equivalent to the molecular or atomic mass of the ion in unified atomic mass units (u), where the numerical value of ''m''/''Q'' is abstruse. The ''m'' refers to the molecular or atomic mass number and ''z'' to the
charge number of the
ion; however, the quantity of ''m''/''z'' is dimensionless by definition.
An ion with a mass of 100 u (unified atomic mass units) () carrying two charges () will be observed at . However, the empirical observation is one equation with two unknowns and could have arisen from other ions, such as an ion of mass 50 u carrying one charge. Thus, the ''m''/''z'' of an ion alone neither infers mass nor the number of charges. Additional information, such as the mass spacing between mass isotopomers or the relationship between multiple charge states, is required to assign the charge state and infer the mass of the ion from the ''m''/''z''. This additional information is often but not always available. Thus, the ''m''/''z'' is primarily used to report an empirical observation in mass spectrometry. This observation may be used in conjunction with other lines of evidence to subsequently infer the physical attributes of the ion, such as mass and charge.
History
In the 19th century, the mass-to-charge ratios of some ions were measured by electrochemical methods. In 1897, the mass-to-charge ratio of the
electron
The electron (, or in nuclear reactions) 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 partic ...
was first measured by
J. J. Thomson. By doing this, he showed that the electron was in fact a particle with a mass and a charge, and that its mass-to-charge ratio was much smaller than that of the hydrogen ion H
+. In 1898,
Wilhelm Wien separated ions (
canal rays) according to their mass-to-charge ratio with an ion optical device with superimposed electric and magnetic fields (
Wien filter). In 1901
Walter Kaufman measured the increase of
electromagnetic mass Electromagnetic mass was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy, is contributing to the mass of charged particles. It was first derived by J. J. Thomson in 1881 and was for ...
of fast electrons (
Kaufmann–Bucherer–Neumann experiments), or
relativistic mass increase in modern terms. In 1913, Thomson measured the mass-to-charge ratio of
ions with an instrument he called a parabola spectrograph. Today, an instrument that measures the mass-to-charge ratio of charged particles is called a
mass spectrometer
Mass spectrometry (MS) is an analytical technique that is used to measure the mass-to-charge ratio of ions. The results are presented as a '' mass spectrum'', a plot of intensity as a function of the mass-to-charge ratio. Mass spectrometry is u ...
.
Charge-to-mass ratio

The charge-to-mass ratio (''Q''/''m'') of an object is, as its name implies, the
charge of an object divided by the mass of the same object. This quantity is generally useful only for objects that may be treated as particles. For extended objects, total charge, charge density, total mass, and mass density are often more useful.
Derivation:
or
Since
,
or
Equations () and () yield
Significance
In some experiments, the charge-to-mass ratio is the only quantity that can be measured directly. Often, the charge can be inferred from theoretical considerations, so that the charge-to-mass ratio provides a way to calculate the mass of a particle.
Often, the charge-to-mass ratio can be determined from observing the deflection of a charged particle in an external
magnetic
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
field. The
cyclotron
A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: J ...
equation, combined with other information such as the
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its a ...
of the particle, will give the charge-to-mass ratio. One application of this principle is the mass spectrometer. The same principle can be used to extract information in experiments involving the
cloud chamber.
The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. It turns out that gravitational forces are negligible on the subatomic level, due to the extremely small masses of subatomic particles.
Electron
The electron charge-to-mass quotient,
, is a quantity that may be measured in experimental physics. It bears significance because the electron mass ''m''
e is difficult to measure directly, and is instead derived from measurements of the elementary charge ''e'' and
. It also has historical significance; the ''Q''/''m'' ratio of the electron was successfully calculated by
J. J. Thomson in 1897—and more successfully by Dunnington, which involves the
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...
and deflection due to a perpendicular
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
. Thomson's measurement convinced him that
cathode rays were particles, which were later identified as
electron
The electron (, or in nuclear reactions) 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 partic ...
s, and he is generally credited with their discovery.
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 International ...
recommended value is CODATA refers to this as the electron charge-to-mass quotient, but ratio is still commonly used.
There are two other common ways of measuring the charge-to-mass ratio of an electron, apart from Thomson and Dunnington's methods.
#The magnetron method: Using a GRD7 Valve (Ferranti valve), electrons are expelled from a hot tungsten-wire filament towards an anode. The electron is then deflected using a solenoid. From the current in the solenoid and the current in the Ferranti Valve, e/m can be calculated.
#Fine beam tube method: A heater heats a cathode, which emits electrons. The electrons are accelerated through a known potential, so the velocity of the electrons is known. The beam path can be seen when the electrons are accelerated through a helium (He) gas. The collisions between the electrons and the helium gas produce a visible trail. A pair of
Helmholtz coils produces a uniform and measurable magnetic field at right angles to the electron beam. This magnetic field deflects the electron beam in a circular path. By measuring the accelerating potential (volts), the current (amps) to the Helmholtz coils, and the radius of the electron beam, e/m can be calculated.
[PASCO scientific, Instruction Manual and Experimental guide for the PASCO scientific Model SE-9638, pg. 1.]
Zeeman Effect
The charge-to-mass ratio of an electron may also be measured with the
Zeeman effect
The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel priz ...
, which gives rise to energy splittings in the presence of a
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
''B'':
Here ''m''
''j'' are quantum integer values ranging from −''j'' to ''j'', with ''j'' as the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
of the
total angular momentum operator
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
J, with
:
where S is the
spin operator with eigenvalue ''s'' and L is the
angular momentum operator with eigenvalue ''l''. ''g''
''J'' is the
Landé g-factor, calculated as
The shift in energy is also given in terms of
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
''υ'' and
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
''λ'' as
Measurements of the Zeeman effect commonly involve the use of a
Fabry–Pérot interferometer, with light from a source (placed in a magnetic field) being passed between two mirrors of the interferometer. If ''δD'' is the change in mirror separation required to bring the ''m''th-order ring of wavelength into coincidence with that of wavelength ''λ'', and Δ''D'' brings the ring of wavelength ''λ'' into coincidence with the ''m''th-order ring, then
It follows then that
Rearranging, it is possible to solve for the charge-to-mass ratio of an electron as
See also
*
Gyromagnetic ratio
*
Thomson (unit)
References
Bibliography
*
*
* CC.
* IUPAP Red Book SUNAMCO 87-1 "Symbols, Units, Nomenclature and Fundamental Constants in Physics" (does not have an online version).
* Symbols Units and Nomenclature in Physics IUPAP-25 IUPAP-25, E.R. Cohen & P. Giacomo, Physics 146A (1987) 1–68.
External links
BIPM SI brochureAIP style manual* NIST o
an
* Physics Today'
instructions on quantities and units
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