In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, certain systems can achieve
negative temperatures; that is, their
thermodynamic temperatureThermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an “absolute” scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
can be of a
negativeBeing negative or non-negative is a property of a number which is real, or a member of a subset of real numbers such as rational and integer numbers. A negative number is one that is less than zero, such as −, -1.414, -1. A positive number is one that is greater than zero, such as , 1.414, 1...
quantity. Negative temperatures can be expressed as negative numbers on the
kelvinThe kelvin is a unit increment of temperature and is one of the seven SI base units. The Kelvin scale is a thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero kelvin...
scale.
Temperatures that are expressed as negative numbers on the familiar
CelsiusCelsius is a temperature scale that is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...
or
FahrenheitFahrenheit is the temperature scale proposed in 1724 by, and named after, the physicist Daniel Gabriel Fahrenheit . Today, the scale has been replaced by the Celsius scale in most countries; it is still in use for non-scientific purposes in the United States and a few other nations, such as...
scales are simply colder than the zero points of those scales. By contrast, a system with a truly negative temperature is not colder than
absolute zeroAbsolute zero is a temperature marked by a 0 entropy configuration. It is the coldest temperature theoretically possible and cannot be reached by artificial or natural means, because it is impossible to decouple a system fully from the rest of the universe...
; in fact, temperatures colder than absolute zero are impossible.
In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
, certain systems can achieve
negative temperatures; that is, their
thermodynamic temperatureThermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an “absolute” scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
can be of a
negativeBeing negative or non-negative is a property of a number which is real, or a member of a subset of real numbers such as rational and integer numbers. A negative number is one that is less than zero, such as −, -1.414, -1. A positive number is one that is greater than zero, such as , 1.414, 1...
quantity. Negative temperatures can be expressed as negative numbers on the
kelvinThe kelvin is a unit increment of temperature and is one of the seven SI base units. The Kelvin scale is a thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero kelvin...
scale.
Temperatures that are expressed as negative numbers on the familiar
CelsiusCelsius is a temperature scale that is named after the Swedish astronomer Anders Celsius , who developed a similar temperature scale two years before his death...
or
FahrenheitFahrenheit is the temperature scale proposed in 1724 by, and named after, the physicist Daniel Gabriel Fahrenheit . Today, the scale has been replaced by the Celsius scale in most countries; it is still in use for non-scientific purposes in the United States and a few other nations, such as...
scales are simply colder than the zero points of those scales. By contrast, a system with a truly negative temperature is not colder than
absolute zeroAbsolute zero is a temperature marked by a 0 entropy configuration. It is the coldest temperature theoretically possible and cannot be reached by artificial or natural means, because it is impossible to decouple a system fully from the rest of the universe...
; in fact, temperatures colder than absolute zero are impossible. Rather, a system with a negative temperature is hotter than
any system with a positive temperature (in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system).
Heat and molecular energy distribution
Negative temperatures can only exist in a system where there are a limited number of energy states (see below). As the temperature is increased on such a system, particles move into higher and higher energy states, and as the temperature increases, the number of particles in the lower energy states and in the higher energy states approaches equality. (This is a consequence of the definition of temperature in
statistical mechanicsStatistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force...
for systems with limited states.) By injecting energy into these systems in the right fashion, it is possible to create a system in which there are more particles in the higher energy states than in the lower ones. The system can then be characterised as having a negative temperature. A substance with a negative temperature is not colder than
absolute zeroAbsolute zero is a temperature marked by a 0 entropy configuration. It is the coldest temperature theoretically possible and cannot be reached by artificial or natural means, because it is impossible to decouple a system fully from the rest of the universe...
, but rather it is hotter than infinite temperature. As Kittel and Kroemer (p.462) put it, "The temperature scale from cold to hot runs:
+0 K, . . . , +300 K, . . . , +∞ K, −∞ K, . . . , −300 K, . . . , −0 K."
Generally, temperature as it is felt is defined by the kinetic energy of atoms (heat). Since there is no upper bound on momentum of an atom there is no upper bound to the number of energy states available if enough energy is added, and no way to get to a negative temperature. However, temperature is more generally defined by statistical mechanics than just kinetic energy (see below).
Temperature and disorder
The distribution of energy among the various
translationalIn physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
,
vibrationalOscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power...
,
rotationA rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...
al,
electronicIn atomic physics and quantum chemistry, electron configuration is the arrangement of electrons of an atom, a molecule, or other physical structure...
, and
nuclearThe nucleus is the very dense region consisting of nucleons at the center of an atom. Almost all of the mass in an atom is made up from the protons and neutrons in the nucleus, with a very small contribution from the orbiting electrons....
modes of a system determines the macroscopic temperature. In a "normal" system, thermal energy is constantly being exchanged between the various modes.
However, for some cases it is possible to isolate one or more of the modes. In practice the isolated modes still exchange energy with the other modes, but the
time scaleA time scale specifies divisions of time and possibly an epoch that marks the time when the time scale began .*A time standard is a specification of either the rate at which time passes, or points in time, or both....
of this exchange is much slower than for the exchanges within the isolated mode. One example is the case of
nuclearThe nucleus is the very dense region consisting of nucleons at the center of an atom. Almost all of the mass in an atom is made up from the protons and neutrons in the nucleus, with a very small contribution from the orbiting electrons....
spinIn particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....
s in a strong external
magnetic fieldMagnetic fields surround magnetic materials and electric currents and are detected by the force they exert on other magnetic materials and moving electric charges...
. In this case, energy flows fairly rapidly among the spin states of interacting atoms, but energy transfer between the nuclear spins and other modes is relatively slow. Since the energy flow is predominantly within the spin system, it makes sense to think of a spin temperature that is distinct from the temperature due to other modes.
A definition of temperature can be based on the relationship:
(See here for discussion)
The relationship suggests that a
positive temperature corresponds to the condition where
entropyEntropy is a concept of information maintaining great importance in physics, chemistry, and information theory...
,
S, increases as thermal energy,
qrev, is added to the system. This is the "normal" condition in the macroscopic world and is always the case for the translational, vibrational, rotational, and non-spin related electronic and nuclear modes. The reason for this is that there are an infinite number of these types of modes and adding more heat to the system increases the number of modes that are energetically accessible, and thus the entropy.
Nuclear spins
In the case of electronic and nuclear spin systems there are only a finite number of modes available, often just two, corresponding to spin up and spin down. In the absence of a magnetic field, these spin states are
degenerate, meaning that they correspond to the same energy. When an external magnetic field is applied, the energy levels are split, since those spin states that are aligned with the magnetic field will have a different energy from those that are anti-parallel to it.
In the absence of a magnetic field, one would expect such a two-spin system to have roughly half the atoms in the spin-up state and half in the spin-down state, since this maximizes entropy. Upon application of a magnetic field, some of the atoms will tend to align so as to minimize the energy of the system, thus slightly more atoms should be in the lower-energy state (for the purposes of this example we'll assume the spin-down state is the lower-energy state). It is possible to add energy to the spin system using
radio frequencyRadio frequency is a frequency, or rate of oscillation, of electromagnetic radiation within the range of about 3 Hz to 300 GHz. This range corresponds to the frequency of alternating current electrical signals used to produce and detect radio waves...
(RF) techniques. This causes atoms to
flip from spin-down to spin-up.
Since we started with over half the atoms in the spin-down state, initially this drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point more than half of the spins are in the spin-up position. In this case, adding additional energy reduces the entropy since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature.
Lasers
This phenomenon can also be observed in many
lasingA laser is a device that emits light through a process called stimulated emission. Laser light is usually spatially coherent, which means that the light either is emitted in a narrow, low-divergence beam, or can be converted into one with the help of optical components such as lenses...
systems, wherein a large fraction of the system's
atomThe atom is a basic unit of matter consisting of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s (for chemical and gas lasers) or
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s (in
semiconductorA semiconductor is a material that has an electrical resistivity between that of a conductor and an insulator, that is, generally in the range 10
3 Siemens/cm to 10
−8 S/cm. Devices made from semiconductor materials are the foundation of modern electronics, including radio,...
lasers) are in excited states. This is referred to as a
population inversionIn physics, specifically statistical mechanics, a population inversion occurs when a system exists in state with more members in an excited state than in lower energy states...
.
The
HamiltonianIn quantum mechanics, the Hamiltonian H is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
for a single mode of a luminescent radiation field at frequency ν is .
The density operator in the
grand canonical ensembleIn statistical mechanics, a grand canonical ensemble is an imaginary collection of model systems put together to mirror the calculated probability distribution of microscopic states of a given physical system which is being maintained in a given macroscopic state...
is .
For the system to have a ground state, the trace to converge, and the density operator to be generally meaningful,
βH must be positive semidefinite. So if
hν < μ and
H is negative semidefinite, then β must itself be negative, implying a negative temperature.
Further reading
External links
Recent applications