Negative temperature

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https://en.wikipedia.org/wiki/Negative_temperature

SI temperature/coldness conversion scale: Temperatures on the Kelvin scale are shown in blue (Celsius scale in green, Fahrenheit scale in red), coldness values in gigabyte per nanojoule are shown in black. Infinite temperature (coldness zero) is shown at the top of the diagram; positive values of coldness/temperature are on the right-hand side, negative values on the left-hand side.

Certain systems can achieve negative thermodynamic temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales. This should be distinguished from temperatures expressed as negative numbers on non-thermodynamic Celsius or Fahrenheit scales, which are nevertheless higher than absolute zero. A system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system. A standard example of such a system is population inversion in laser physics.

Thermodynamic systems with unbounded phase space cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy. This is only possible if the number of high energy states is limited. For a system of ordinary (quantum or classical) particles such as atoms or dust, the number of high energy states is unlimited (particle momenta can in principle be increased indefinitely). Some systems, however (see the examples below), have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.

History

The possibility of negative temperatures was first predicted by Lars Onsager in 1949. Onsager was investigating 2D vortices confined within a finite area, and realized that since their positions are not independent degrees of freedom from their momenta, the resulting phase space must also be bounded by the finite area. Bounded phase space is the essential property that allows for negative temperatures, and can occur in both classical and quantum systems. As shown by Onsager, a system with bounded phase space necessarily has a peak in the entropy as energy is increased. For energies exceeding the value where the peak occurs, the entropy decreases as energy increases, and high-energy states necessarily have negative Boltzmann temperature.

The limited range of states accessible to a system with negative temperature means that negative temperature is associated with emergent ordering of the system at high energies. For example in Onsager's point-vortex analysis negative temperature is associated with the emergence of large-scale clusters of vortices. This spontaneous ordering in equilibrium statistical mechanics goes against common physical intuition that increased energy leads to increased disorder.

Definition of temperature

The absolute temperature (Kelvin) scale can be loosely interpreted as the average kinetic energy of the system's particles. The existence of negative temperature, let alone negative temperature representing "hotter" systems than positive temperature, would seem paradoxical in this interpretation. The paradox is resolved by considering the more rigorous definition of thermodynamic temperature in terms of Boltzmann's entropy formula. This reveals the tradeoff between internal energy and entropy contained in the system, with "coldness", the reciprocal of temperature, being the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system, while systems with a negative temperature will decrease in entropy as one adds energy to the system.

The definition of thermodynamic temperature T is a function of the change in the system's entropy S under reversible heat transfer Q_rev:

${\displaystyle T=\frac{d{Q}_{\mathrm{r}\mathrm{e}\mathrm{v}}}{dS}.}$

Entropy being a state function, the integral of dS over any cyclical process is zero. For a system in which the entropy is purely a function of the system's energy E, the temperature can be defined as:

${\displaystyle T={\left(\frac{dS}{dE}\right)}^{-1}.}$

Equivalently, thermodynamic beta, or "coldness", is defined as

${\displaystyle \beta =\frac{1}{kT}=\frac{1}{k}\frac{dS}{dE},}$

where k is the Boltzmann constant.

Note that in classical thermodynamics, S is defined in terms of temperature. This is reversed here, S is the statistical entropy, a function of the possible microstates of the system, and temperature conveys information on the distribution of energy levels among the possible microstates. For systems with many degrees of freedom, the statistical and thermodynamic definitions of entropy are generally consistent with each other.

Some theorists have proposed using an alternative definition of entropy as a way to resolve perceived inconsistencies between statistical and thermodynamic entropy for small systems and systems where the number of states decreases with energy, and the temperatures derived from these entropies are different. It has been argued that the new definition would create other inconsistencies; its proponents have argued that this is only apparent.

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 mechanics 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 zero, 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 (−273.15 °C), …, +100 K (−173.15 °C), …, +300 K (+26.85 °C), …, +1000 K (+726.85 °C), …, +∞ K (+∞ °C), −∞ K (−∞ °C), …, −1000 K (−1273.15 °C), …, −300 K (−573.15 °C), …, −100 K (−373.15 °C), …, −0 K (−273.15 °C).

The corresponding inverse temperature scale, for the quantity β = 1/kT (where k is the Boltzmann constant), runs continuously from low energy to high as +∞, …, 0, …, −∞. Because it avoids the abrupt jump from +∞ to −∞, β is considered more natural than T. Although a system can have multiple negative temperature regions and thus have −∞ to +∞ discontinuities.

In many familiar physical systems, temperature is associated to the kinetic energy of atoms. Since there is no upper bound on the momentum of an atom, there is no upper bound to the number of energy states available when more energy is added, and therefore no way to get to a negative temperature. However, in statistical mechanics, temperature can correspond to other degrees of freedom than just kinetic energy (see below).

Temperature and disorder

The distribution of energy among the various translational, vibrational, rotational, electronic, and nuclear modes of a system determines the macroscopic temperature. In a "normal" system, thermal energy is constantly being exchanged between the various modes.

However, in some situations, 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 scale of this exchange is much slower than for the exchanges within the isolated mode. One example is the case of nuclear spins in a strong external magnetic field. 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 associated to other modes.

A definition of temperature can be based on the relationship:

${\displaystyle T=\frac{d{q}_{\mathrm{r}\mathrm{e}\mathrm{v}}}{dS}}$

The relationship suggests that a positive temperature corresponds to the condition where entropy, S, increases as thermal energy, q_rev, 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 increases the entropy.

Examples

Noninteracting two-level particles

Entropy, thermodynamic beta, and temperature as a function of the energy for a system of N noninteracting two-level particles.

The simplest example, albeit a rather nonphysical one, is to consider a system of N particles, each of which can take an energy of either +ε or −ε but are otherwise noninteracting. This can be understood as a limit of the Ising model in which the interaction term becomes negligible. The total energy of the system is

${\displaystyle E=\epsilon \sum _{i=1}^N{\sigma }_i=\epsilon j}$

where σ_i is the sign of the ith particle and j is the number of particles with positive energy minus the number of particles with negative energy. From elementary combinatorics, the total number of microstates with this amount of energy is a binomial coefficient:

${\displaystyle {\mathrm{\Omega }}_E=(\genfrac{}{}{0ex}{}{N}{\frac{N+j}{2}})=\frac{N!}{\left(\frac{N+j}{2}\right)!\left(\frac{N-j}{2}\right)!}.}$

By the fundamental assumption of statistical mechanics, the entropy of this microcanonical ensemble is

${\displaystyle S=k_{\text{B}}\ln {\mathrm{\Omega }}_E}$

We can solve for thermodynamic beta (β = 1/k_BT) by considering it as a central difference without taking the continuum limit:

${\displaystyle \begin{array}{rl}\beta & =\frac{1}{k_{\mathrm{B}}}\frac{{\delta }_{2\epsilon }[S]}{2\epsilon }\\ & =\frac{1}{2\epsilon }\left(\ln {\mathrm{\Omega }}_{E+\epsilon }-\ln {\mathrm{\Omega }}_{E-\epsilon }\right)\\ & =\frac{1}{2\epsilon }\ln \left(\frac{\left(\frac{N+j-1}{2}\right)!\left(\frac{N-j+1}{2}\right)!}{\left(\frac{N+j+1}{2}\right)!\left(\frac{N-j-1}{2}\right)!}\right)\\ & =\frac{1}{2\epsilon }\ln \left(\frac{N-j+1}{N+j+1}\right).\end{array}}$

hence the temperature

${\displaystyle T(E)=\frac{2\epsilon }{k_{\text{B}}}{\left[\ln \left(\frac{(N+1)\epsilon -E}{(N+1)\epsilon +E}\right)\right]}^{-1}.}$

This entire proof assumes the microcanonical ensemble with energy fixed and temperature being the emergent property. In the canonical ensemble, the temperature is fixed and energy is the emergent property. This leads to (ε refers to microstates):

${\displaystyle \begin{array}{rl}Z(T)& =\sum _{i=1}^N{e}^{-{\epsilon }_i\beta }\\ E(T)& =\frac{1}{Z}\sum _{i=1}^N{\epsilon }_i{e}^{-{\epsilon }_i\beta }\\ S(T)& =k_{\text{B}}\ln (Z)+\frac{E}{T}\end{array}}$

Following the previous example, we choose a state with two levels and two particles. This leads to microstates ε₁ = 0, ε₂ = 1, ε₃ = 1, and ε₄ = 2.

${\displaystyle \begin{array}{rl}Z(T)& =e^{-0\beta }+2e^{-1\beta }+e^{-2\beta }\\ & =1+2e^{-\beta }+e^{-2\beta }\\ E(T)& =\frac{0e^{-0\beta }+2\times 1e^{-1\beta }+2e^{-2\beta }}{Z}\\ & =\frac{2e^{-\beta }+2e^{-2\beta }}{Z}\\ & =\frac{2e^{-\beta }+2e^{-2\beta }}{1+2e^{-\beta }+e^{-2\beta }}\\ S(T)& =k_{\text{B}}\ln \left(1+2e^{-\beta }+e^{-2\beta }\right)+\frac{2e^{-\beta }+2e^{-2\beta }}{\left(1+2e^{-\beta }+e^{-2\beta }\right)T}\end{array}}$

The resulting values for S, E, and Z all increase with T and never need to enter a negative temperature regime.

Nuclear spins

The previous example is approximately realized by a system of nuclear spins in an external magnetic field. This allows the experiment to be run as a variation of nuclear magnetic resonance spectroscopy. 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, such a two-spin system would have maximum entropy when half the atoms are in the spin-up state and half are in the spin-down state, and so one would expect to find the system with close to an equal distribution of spins. 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 will assume the spin-down state is the lower-energy state). It is possible to add energy to the spin system using radio frequency 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, this initially 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. In NMR spectroscopy, this corresponds to pulses with a pulse width of over 180° (for a given spin). While relaxation is fast in solids, it can take several seconds in solutions and even longer in gases and in ultracold systems; several hours were reported for silver and rhodium at picokelvin temperatures. It is still important to understand that the temperature is negative only with respect to nuclear spins. Other degrees of freedom, such as molecular vibrational, electronic and electron spin levels are at a positive temperature, so the object still has positive sensible heat. Relaxation actually happens by exchange of energy between the nuclear spin states and other states (e.g. through the nuclear Overhauser effect with other spins).

Lasers

This phenomenon can also be observed in many lasing systems, wherein a large fraction of the system's atoms (for chemical and gas lasers) or electrons (in semiconductor lasers) are in excited states. This is referred to as a population inversion.

The Hamiltonian for a single mode of a luminescent radiation field at frequency ν is

${\displaystyle H=(h\nu -\mu )a^{†}a.}$

The density operator in the grand canonical ensemble is

${\displaystyle \rho =\frac{e^{-\beta H}}{\mathrm{Tr}\left(e^{-\beta H}\right)}.}$

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.

Motional degrees of freedom

Negative temperatures have also been achieved in motional degrees of freedom. Using an optical lattice, upper bounds were placed on the kinetic energy, interaction energy and potential energy of cold potassium-39 atoms. This was done by tuning the interactions of the atoms from repulsive to attractive using a Feshbach resonance and changing the overall harmonic potential from trapping to anti-trapping, thus transforming the Bose-Hubbard Hamiltonian from Ĥ → −Ĥ. Performing this transformation adiabatically while keeping the atoms in the Mott insulator regime, it is possible to go from a low entropy positive temperature state to a low entropy negative temperature state. In the negative temperature state, the atoms macroscopically occupy the maximum momentum state of the lattice. The negative temperature ensembles equilibrated and showed long lifetimes in an anti-trapping harmonic potential.

Two-dimensional vortex motion

The two-dimensional systems of vortices confined to a finite area can form thermal equilibrium states at negative temperature, and indeed negative temperature states were first predicted by Onsager in his analysis of classical point vortices. Onsager's prediction was confirmed experimentally for a system of quantum vortices in a Bose-Einstein condensate in 2019.

Rotational–vibrational spectroscopy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Rotational%E2%80%93vibrational_spectroscopy

Rotational–vibrational spectroscopy is a branch of molecular spectroscopy concerned with infrared and Raman spectra of molecules in the gas phase. Transitions involving changes in both vibrational and rotational states can be abbreviated as rovibrational (or ro-vibrational) transitions. When such transitions emit or absorb photons (electromagnetic radiation), the frequency is proportional to the difference in energy levels and can be detected by certain kinds of spectroscopy. Since changes in rotational energy levels are typically much smaller than changes in vibrational energy levels, changes in rotational state are said to give fine structure to the vibrational spectrum. For a given vibrational transition, the same theoretical treatment as for pure rotational spectroscopy gives the rotational quantum numbers, energy levels, and selection rules. In linear and spherical top molecules, rotational lines are found as simple progressions at both higher and lower frequencies relative to the pure vibration frequency. In symmetric top molecules the transitions are classified as parallel when the dipole moment change is parallel to the principal axis of rotation, and perpendicular when the change is perpendicular to that axis. The ro-vibrational spectrum of the asymmetric rotor water is important because of the presence of water vapor in the atmosphere.

Overview

See also: Rotational spectroscopy and Vibronic spectroscopy

Ro-vibrational spectroscopy concerns molecules in the gas phase. There are sequences of quantized rotational levels associated with both the ground and excited vibrational states. The spectra are often resolved into lines due to transitions from one rotational level in the ground vibrational state to one rotational level in the vibrationally excited state. The lines corresponding to a given vibrational transition form a band.

In the simplest cases the part of the infrared spectrum involving vibrational transitions with the same rotational quantum number (ΔJ = 0) in ground and excited states is called the Q-branch. On the high frequency side of the Q-branch the energy of rotational transitions is added to the energy of the vibrational transition. This is known as the R-branch of the spectrum for ΔJ = +1. The P-branch for ΔJ = −1 lies on the low wavenumber side of the Q branch. The appearance of the R-branch is very similar to the appearance of the pure rotation spectrum (but shifted to much higher wavenumbers), and the P-branch appears as a nearly mirror image of the R-branch. The Q branch is sometimes missing because of transitions with no change in J being forbidden.

The appearance of rotational fine structure is determined by the symmetry of the molecular rotors which are classified, in the same way as for pure rotational spectroscopy, into linear molecules, spherical-, symmetric- and asymmetric- rotor classes. The quantum mechanical treatment of rotational fine structure is the same as for pure rotation.

The strength of an absorption line is related to the number of molecules with the initial values of the vibrational quantum number ν and the rotational quantum number ${\displaystyle J}$, and depends on temperature. Since there are actually ${\displaystyle 2J+1}$ states with rotational quantum number ${\displaystyle J}$, the population with value ${\displaystyle J}$ increases with ${\displaystyle J}$ initially, and then decays at higher ${\displaystyle J}$. This gives the characteristic shape of the P and R branches.

A general convention is to label quantities that refer to the vibrational ground and excited states of a transition with double prime and single prime, respectively. For example, the rotational constant for the ground state is written as ${\displaystyle B^{\mathrm{\prime }\mathrm{\prime }},}$ and that of the excited state as ${\displaystyle B^{\mathrm{\prime }}.}$ Also, these constants are expressed in the molecular spectroscopist's units of cm⁻¹. so that ${\displaystyle B}$ in this article corresponds to ${\displaystyle \overline{B}=B/hc}$ in the definition of rotational constant at Rigid rotor.

Method of combination differences

Numerical analysis of ro-vibrational spectral data would appear to be complicated by the fact that the wavenumber for each transition depends on two rotational constants, ${\displaystyle B^{\mathrm{\prime }\mathrm{\prime }}}$ and ${\displaystyle B^{\mathrm{\prime }}}$. However combinations which depend on only one rotational constant are found by subtracting wavenumbers of pairs of lines (one in the P-branch and one in the R-branch) which have either the same lower level or the same upper level. For example, in a diatomic molecule the line denoted P(J + 1) is due to the transition (v = 0, J + 1) → (v = 1, J) (meaning a transition from the state with vibrational quantum number ν going from 0 to 1 and the rotational quantum number going from some value J + 1 to J, with J > 0), and the line R(J − 1) is due to the transition (v = 0, J − 1) → (v = 1, J). The difference between the two wavenumbers corresponds to the energy difference between the (J + 1) and (J − 1) levels of the lower vibrational state and is denoted by ${\displaystyle {\mathrm{\Delta }}_2}$ since it is the difference between levels differing by two units of J. If centrifugal distortion is included, it is given by

${\displaystyle {\mathrm{\Delta }}_2^{\mathrm{\prime }\mathrm{\prime }}F(J)=\overline{\nu }(R(J-1))-\overline{\nu }(P(J+1))=(2B^{\mathrm{\prime }\mathrm{\prime }}-3D^{\mathrm{\prime }\mathrm{\prime }})\left(2J+1\right)-D^{\mathrm{\prime }\mathrm{\prime }}{\left(2J+1\right)}^3}$

where ${\displaystyle \overline{\nu }()}$ means the frequency (or wavenumber) of the given line. The main term, ${\displaystyle 2B^{″}(2J+1),}$ comes from the difference in the energy of the ${\displaystyle J+1}$ rotational state, ${\displaystyle B^{″}((J+1)(J+2)),}$ and that of the ${\displaystyle J-1}$ state, ${\displaystyle B^{″}((J-1)J).}$

The rotational constant of the ground vibrational state B′′ and centrifugal distortion constant, D′′ can be found by least-squares fitting this difference as a function of J. The constant B′′ is used to determine the internuclear distance in the ground state as in pure rotational spectroscopy. (See Appendix)

Similarly the difference R(J) − P(J) depends only on the constants B′ and D′ for the excited vibrational state (v = 1), and B′ can be used to determine the internuclear distance in that state (which is inaccessible to pure rotational spectroscopy).

${\displaystyle {\mathrm{\Delta }}_2^{\mathrm{\prime }}F(J)=\overline{\nu }(R(J))-\overline{\nu }(P(J))=(2B^{\mathrm{\prime }}-3D^{\mathrm{\prime }})\left(2J+1\right)-D^{\mathrm{\prime }}{\left(2J+1\right)}^3}$

Linear molecules

Heteronuclear diatomic molecules

Simulated vibration-rotation line spectrum of carbon monoxide, ¹²C¹⁶O. The P-branch is to the left of the gap near 2140 cm⁻¹, the R-branch on the right.

Schematic ro-vibrational energy level diagram for a linear molecule

Diatomic molecules with the general formula AB have one normal mode of vibration involving stretching of the A-B bond. The vibrational term values ${\displaystyle G(v)}$, for an anharmonic oscillator are given, to a first approximation, by

${\displaystyle G(v)={\omega }_e\left(v+\frac{1}{2}\right)-{\omega }_e{\chi }_e{\left(v+\frac{1}{2}\right)}^2}$

where v is a vibrational quantum number, ω_e is the harmonic wavenumber and χ_e is an anharmonicity constant.

When the molecule is in the gas phase, it can rotate about an axis, perpendicular to the molecular axis, passing through the centre of mass of the molecule. The rotational energy is also quantized, with term values to a first approximation given by

${\displaystyle F_v(J)=B_{v}J\left(J+1\right)-D{J}^2{\left(J+1\right)}^2}$

where J is a rotational quantum number and D is a centrifugal distortion constant. The rotational constant, B_v depends on the moment of inertia of the molecule, I_v, which varies with the vibrational quantum number, v

${\displaystyle B_v=\frac{h}{8{\pi }^{2}c{I}_v};I_v=\frac{m_A{m}_B}{m_A+m_B}{d}_v^2}$

where m_A and m_B are the masses of the atoms A and B, and d represents the distance between the atoms. The term values of the ro-vibrational states are found (in the Born–Oppenheimer approximation) by combining the expressions for vibration and rotation.

${\displaystyle G(v)+F_v(J)=\left[{\omega }_e\left(v+\frac{1}{2}\right)+B_{v}J(J+1)\right]-\left[{\omega }_e{\chi }_e{\left(v+\frac{1}{2}\right)}^2+D{J}^2(J+1{)}^2\right]}$

The first two terms in this expression correspond to a harmonic oscillator and a rigid rotor, the second pair of terms make a correction for anharmonicity and centrifugal distortion. A more general expression was given by Dunham.

The selection rule for electric dipole allowed ro-vibrational transitions, in the case of a diamagnetic diatomic molecule is

${\displaystyle \mathrm{\Delta }v=\pm 1(\pm 2,\pm 3,\mathit{\text{etc.}}}$${\displaystyle ),\mathrm{\Delta }J=\pm 1}$

The transition with Δv=±1 is known as the fundamental transition. The selection rule has two consequences.

1. Both the vibrational and rotational quantum numbers must change. The transition :${\displaystyle \mathrm{\Delta }v=\pm 1,\mathrm{\Delta }J=0}$ (Q-branch) is forbidden
2. The energy change of rotation can be either subtracted from or added to the energy change of vibration, giving the P- and R- branches of the spectrum, respectively.

The calculation of the transition wavenumbers is more complicated than for pure rotation because the rotational constant B_ν is different in the ground and excited vibrational states. A simplified expression for the wavenumbers is obtained when the centrifugal distortion constants ${\displaystyle D^{\mathrm{\prime }}}$ and ${\displaystyle D^{\mathrm{\prime }\mathrm{\prime }}}$ are approximately equal to each other.

${\displaystyle \overline{\nu }={\omega }_0+(B^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }})m+(B^{\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})m^2-2(D^{\mathrm{\prime }}+D^{\mathrm{\prime }\mathrm{\prime }})m^3,{\omega }_0={\omega }_e(1-2{\chi }_e)m=\pm 1,\pm 2etc.}$

Spectrum of R-branch of nitric oxide, NO, simulated with Spectralcalc, showing λ-doubling caused by the presence of an unpaired electron in the molecule

where positive m values refer to the R-branch and negative values refer to the P-branch. The term ω₀ gives the position of the (missing) Q-branch, the term ${\displaystyle (B^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }})m}$ implies an progression of equally spaced lines in the P- and R- branches, but the third term, ${\displaystyle (B^{\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})m^2}$shows that the separation between adjacent lines changes with changing rotational quantum number. When ${\displaystyle B^{\mathrm{\prime }\mathrm{\prime }}}$ is greater than ${\displaystyle B^{\mathrm{\prime }}}$, as is usually the case, as J increases the separation between lines decreases in the R-branch and increases in the P-branch. Analysis of data from the infrared spectrum of carbon monoxide, gives value of ${\displaystyle B^{\mathrm{\prime }\mathrm{\prime }}}$ of 1.915 cm⁻¹ and ${\displaystyle B^{\mathrm{\prime }}}$ of 1.898 cm⁻¹. The bond lengths are easily obtained from these constants as r₀ = 113.3 pm, r₁ = 113.6 pm. These bond lengths are slightly different from the equilibrium bond length. This is because there is zero-point energy in the vibrational ground state, whereas the equilibrium bond length is at the minimum in the potential energy curve. The relation between the rotational constants is given by

${\displaystyle B_v=B_{eq}-\alpha \left(v+\frac{1}{2}\right)}$

where ν is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated when the B values for two different vibrational states can be found. For carbon monoxide r_eq = 113.0 pm.

Nitric oxide, NO, is a special case as the molecule is paramagnetic, with one unpaired electron. Coupling of the electron spin angular momentum with the molecular vibration causes lambda-doubling with calculated harmonic frequencies of 1904.03 and 1903.68 cm⁻¹. Rotational levels are also split.

Homonuclear diatomic molecules

The quantum mechanics for homonuclear diatomic molecules such as dinitrogen, N₂, and fluorine, F₂, is qualitatively the same as for heteronuclear diatomic molecules, but the selection rules governing transitions are different. Since the electric dipole moment of the homonuclear diatomics is zero, the fundamental vibrational transition is electric-dipole-forbidden and the molecules are infrared inactive. However, a weak quadrupole-allowed spectrum of N₂ can be observed when using long path-lengths both in the laboratory and in the atmosphere. The spectra of these molecules can be observed by Raman spectroscopy because the molecular vibration is Raman-allowed.

Dioxygen is a special case as the molecule is paramagnetic so magnetic-dipole-allowed transitions can be observed in the infrared. The unit electron spin has three spatial orientations with respect to the molecular rotational angular momentum vector, N, so that each rotational level is split into three states with total angular momentum (molecular rotation plus electron spin) ${\displaystyle \mathrm{J}\hslash }$, J = N + 1, N, and N - 1, each J state of this so-called p-type triplet arising from a different orientation of the spin with respect to the rotational motion of the molecule. Selection rules for magnetic dipole transitions allow transitions between successive members of the triplet (ΔJ = ±1) so that for each value of the rotational angular momentum quantum number N there are two allowed transitions. The ¹⁶O nucleus has zero nuclear spins angular momentum, so that symmetry considerations demand that N may only have odd values.

Raman spectra of diatomic molecules

The selection rule is

${\displaystyle \mathrm{\Delta }J=0,\pm 2}$

so that the spectrum has an O-branch (∆J = −2), a Q-branch (∆J = 0) and an S-branch (∆J=+2). In the approximation that B′′ = B′ = B the wavenumbers are given by

${\displaystyle \overline{\nu }(S(J))={\omega }_0+4BJ+6B={\omega }_0+6B,{\omega }_0+10B,{\omega }_0+14B,...}$

${\displaystyle \overline{\nu }(O(J))={\omega }_0-4BJ+2B={\omega }_0-6B,{\omega }_0-10B,{\omega }_0-14B,...}$

since the S-branch starts at J=0 and the O-branch at J=2. So, to a first approximation, the separation between S(0) and O(2) is 12B and the separation between adjacent lines in both O- and S- branches is 4B. The most obvious effect of the fact that B′′ ≠ B′ is that the Q-branch has a series of closely spaced side lines on the low-frequency side due to transitions in which ΔJ=0 for J=1,2 etc. Useful difference formulae, neglecting centrifugal distortion are as follows.

${\displaystyle {\mathrm{\Delta }}_4^{\mathrm{\prime }\mathrm{\prime }}F(J)=\overline{\nu }[S(J-2)]-\overline{\nu }[O(J+2)]=4B^{\mathrm{\prime }\mathrm{\prime }}(2J+1)}$

${\displaystyle {\mathrm{\Delta }}_4^{\mathrm{\prime }}F(J)=\overline{\nu }[S(J)]-\overline{\nu }[O(J)]=4B^{\mathrm{\prime }}(2J+1)}$

Molecular oxygen is a special case as the molecule is paramagnetic, with two unpaired electrons.

For homonuclear diatomics, nuclear spin statistical weights lead to alternating line intensities between even-${\displaystyle J^{\mathrm{\prime }\mathrm{\prime }}}$ and odd-${\displaystyle J^{\mathrm{\prime }\mathrm{\prime }}}$ levels. For nuclear spin I = 1/2 as in ¹H₂ and ¹⁹F₂ the intensity alternation is 1:3. For ²H₂ and ¹⁴N₂, I=1 and the statistical weights are 6 and 3 so that the even-${\displaystyle J^{\mathrm{\prime }\mathrm{\prime }}}$ levels are twice as intense. For ¹⁶O₂ (I=0) all transitions with even values of ${\displaystyle J^{\mathrm{\prime }\mathrm{\prime }}}$ are forbidden.

Polyatomic linear molecules

Spectrum of bending mode in 14N14N16O simulated with Spectralcalc. The weak superimposed spectrum is due to species containing 15N at natural abundance of 0.3%

Spectrum of a perpendicular band from acetylene, C2H2, simulated with Spectralcalc showing 1,3 intensity alternation in both P- and R- branches. See also Hollas p157

Spectrum of the asymmetric stretching (parallel) band of carbon dioxide, 12C16O2 simulated with Spectralcalc.[6] The weak superimposed spectrum is due to absorption of the first vibrationally excited level (0 11 0), which due to its low energy is populated at room temperature

These molecules fall into two classes, according to symmetry: centrosymmetric molecules with point group D_∞h, such as carbon dioxide, CO₂, and ethyne or acetylene, HCCH; and non-centrosymmetric molecules with point group C_∞v such as hydrogen cyanide, HCN, and nitrous oxide, NNO. Centrosymmetric linear molecules have a dipole moment of zero, so do not show a pure rotation spectrum in the infrared or microwave regions. On the other hand, in certain vibrational excited states the molecules do have a dipole moment so that a ro-vibrational spectrum can be observed in the infrared.

The spectra of these molecules are classified according to the direction of the dipole moment change vector. When the vibration induces a dipole moment change pointing along the molecular axis the term parallel is applied, with the symbol ${\displaystyle \parallel }$. When the vibration induces a dipole moment pointing perpendicular to the molecular axis the term perpendicular is applied, with the symbol ${\displaystyle \perp }$. In both cases the P- and R- branch wavenumbers follow the same trend as in diatomic molecules. The two classes differ in the selection rules that apply to ro-vibrational transitions. For parallel transitions the selection rule is the same as for diatomic molecules, namely, the transition corresponding to the Q-branch is forbidden. An example is the C-H stretching mode of hydrogen cyanide.

For a perpendicular vibration the transition ΔJ=0 is allowed. This means that the transition is allowed for the molecule with the same rotational quantum number in the ground and excited vibrational state, for all the populated rotational states. This makes for an intense, relatively broad, Q-branch consisting of overlapping lines due to each rotational state. The N-N-O bending mode of nitrous oxide, at ca. 590 cm⁻¹ is an example.

The spectra of centrosymmetric molecules exhibit alternating line intensities due to quantum state symmetry effects, since rotation of the molecule by 180° about a 2-fold rotation axis is equivalent to exchanging identical nuclei. In carbon dioxide, the oxygen atoms of the predominant isotopic species ¹²C¹⁶O₂ have spin zero and are bosons, so that the total wavefunction must be symmetric when the two ¹⁶O nuclei are exchanged. The nuclear spin factor is always symmetric for two spin-zero nuclei, so that the rotational factor must also be symmetric which is true only for even-J levels. The odd-J rotational levels cannot exist and the allowed vibrational bands consist of only absorption lines from even-J initial levels. The separation between adjacent lines in the P- and R- branches is close to 4B rather than 2B as alternate lines are missing. For acetylene the hydrogens of ¹H¹²C¹²C¹H have spin ½ and are fermions, so the total wavefunction is antisymmetric when two ¹H nuclei are exchanged. As is true for ortho and para hydrogen the nuclear spin function of the two hydrogens has three symmetric ortho states and one antisymmetric para states. For the three ortho states, the rotational wave function must be antisymmetric corresponding to odd J, and for the one para state it is symmetric corresponding to even J. The population of the odd J levels are therefore three times higher than the even J levels, and alternate line intensities are in the ratio 3:1.

Spherical top molecules

Upper: low-resolution infrared spectrum of asymmetric stretching band of methane (CH₄); lower: PGOPHER simulated line spectrum, ignoring Coriolis coupling

These molecules have equal moments of inertia about any axis, and belong to the point groups T_d (tetrahedral AX₄) and O_h (octahedral AX₆). Molecules with these symmetries have a dipole moment of zero, so do not have a pure rotation spectrum in the infrared or microwave regions.

Tetrahedral molecules such as methane, CH₄, have infrared-active stretching and bending vibrations, belonging to the T₂ (sometimes written as F₂) representation. These vibrations are triply degenerate and the rotational energy levels have three components separated by the Coriolis interaction. The rotational term values are given, to a first order approximation, by

${\displaystyle F^{+}=B_{\nu }J(J+1)+2B_{\nu }{\zeta }_r(J+1)}$

${\displaystyle F^0=B_{\nu }J(J+1)}$

${\displaystyle F^{-}=B_{\nu }J(J+1)-2B_{\nu }{\zeta }_r(J+1)}$

where ${\displaystyle {\zeta }_r}$ is a constant for Coriolis coupling. The selection rule for a fundamental vibration is

${\displaystyle \mathrm{\Delta }J=0,\pm 1}$

Thus, the spectrum is very much like the spectrum from a perpendicular vibration of a linear molecule, with a strong Q-branch composed of many transitions in which the rotational quantum number is the same in the vibrational ground and excited states, ${\displaystyle J^{\mathrm{\prime }}=J^{\mathrm{\prime }\mathrm{\prime }}=1,\mathrm{2...}}$ The effect of Coriolis coupling is clearly visible in the C-H stretching vibration of methane, though detailed study has shown that the first-order formula for Coriolis coupling, given above, is not adequate for methane.

Symmetric top molecules

These molecules have a unique principal rotation axis of order 3 or higher. There are two distinct moments of inertia and therefore two rotational constants. For rotation about any axis perpendicular to the unique axis, the moment of inertia is ${\displaystyle I_{\perp }}$ and the rotational constant is ${\displaystyle B=\frac{h}{8{\pi }^{2}c{I}_{\perp }}}$, as for linear molecules. For rotation about the unique axis, however, the moment of inertia is ${\displaystyle I_{\parallel }}$ and the rotational constant is ${\displaystyle A=\frac{h}{8{\pi }^{2}c{I}_{\parallel }}}$. Examples include ammonia, NH₃ and methyl chloride, CH₃Cl (both of molecular symmetry described by point group C_3v), boron trifluoride, BF₃ and phosphorus pentachloride, PCl₅ (both of point group D_3h), and benzene, C₆H₆ (point group D_6h).

For symmetric rotors a quantum number J is associated with the total angular momentum of the molecule. For a given value of J, there is a 2J+1- fold degeneracy with the quantum number, M taking the values +J ...0 ... -J. The third quantum number, K is associated with rotation about the principal rotation axis of the molecule. As with linear molecules, transitions are classified as parallel, ${\displaystyle \parallel }$ or perpendicular,${\displaystyle \perp }$, in this case according to the direction of the dipole moment change with respect to the principal rotation axis. A third category involves certain overtones and combination bands which share the properties of both parallel and perpendicular transitions. The selection rules are

${\displaystyle \parallel }$ If K ≠ 0, then ΔJ = 0, ±1 and ΔK = 0

If K = 0, then ΔJ = ±1 and ΔK = 0

${\displaystyle \perp }$ ΔJ = 0, ±1 and ΔK = ±1

The fact that the selection rules are different is the justification for the classification and it means that the spectra have a different appearance which can often be immediately recognized. An expression for the calculated wavenumbers of the P- and R- branches may be given as

${\displaystyle \overline{\nu }={\overline{\nu }}_0+(B^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }})m+(B^{\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }}-D_J^{\mathrm{\prime }}+D_J^{\mathrm{\prime }\mathrm{\prime }})m^2}$

${\displaystyle -2(D_J^{\mathrm{\prime }}+D_J^{\mathrm{\prime }\mathrm{\prime }})m^3-(D_J^{\mathrm{\prime }}-D_J^{\mathrm{\prime }\mathrm{\prime }})m^4}$

${\displaystyle +\left\{\left[(A^{\mathrm{\prime }}-B^{\mathrm{\prime }})-(A^{\mathrm{\prime }\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})\right]-\left[D_{JK}^{\mathrm{\prime }}+D_{JK}^{\mathrm{\prime }\mathrm{\prime }}\right]m-\left[D_{JK}^{\mathrm{\prime }}-D_{JK}^{\mathrm{\prime }\mathrm{\prime }}\right]m^2\right\}{K}^2-(D_K^{\mathrm{\prime }}-D_K^{\mathrm{\prime }\mathrm{\prime }})K^4}$

in which m = J+1 for the R-branch and -J for the P-branch. The three centrifugal distortion constants ${\displaystyle D_J,D_{JK}}$, and ${\displaystyle D_K}$ are needed to fit the term values of each level. The wavenumbers of the sub-structure corresponding to each band are given by

${\displaystyle \overline{\nu }={\overline{\nu }}_{sub}+(B^{\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})J(J+1)-(D_J^{\mathrm{\prime }}-D_J^{\mathrm{\prime }\mathrm{\prime }})J^2(J+1{)}^2-(D_{JK}^{\mathrm{\prime }}-D_{JK}^{\mathrm{\prime }\mathrm{\prime }})J(J+1)K^2}$

${\displaystyle {\overline{\nu }}_{sub}}$ represents the Q-branch of the sub-structure, whose position is given by

${\displaystyle {\overline{\nu }}_{sub}={\overline{\nu }}_0+\left[(A^{\mathrm{\prime }}-B^{\mathrm{\prime }})-(A^{\mathrm{\prime }\mathrm{\prime }}-B^{\mathrm{\prime }\mathrm{\prime }})\right]K^2-(D_K^{\mathrm{\prime }}-D_K^{\mathrm{\prime }\mathrm{\prime }})K^4}$.

Spectrum of the C-Cl stretching band in CH3Cl (parallel band) simulated with Spectralcalc.

Part of the spectrum of the asymmetric H-C-H bending vibration in CH3Cl (perpendicular band), simulated with Spectralcalc

Parallel bands

The C-Cl stretching vibration of methyl chloride, CH₃Cl, gives a parallel band since the dipole moment change is aligned with the 3-fold rotation axis. The line spectrum shows the sub-structure of this band rather clearly; in reality, very high resolution spectroscopy would be needed to resolve the fine structure fully. Allen and Cross show parts of the spectrum of CH₃D and give a detailed description of the numerical analysis of the experimental data.

Perpendicular bands

The selection rule for perpendicular bands give rise to more transitions than with parallel bands. A band can be viewed as a series of sub-structures, each with P, Q and R branches. The Q-branches are separated by approximately 2(A′-B′). The asymmetric HCH bending vibration of methyl chloride is typical. It shows a series of intense Q-branches with weak rotational fine structure. Analysis of the spectra is made more complicated by the fact that the ground-state vibration is bound, by symmetry, to be a degenerate vibration, which means that Coriolis coupling also affects the spectrum.

Hybrid bands

Overtones of a degenerate fundamental vibration have components of more than one symmetry type. For example, the first overtone of a vibration belonging to the E representation in a molecule like ammonia, NH₃, will have components belonging to A₁ and E representations. A transition to the A₁ component will give a parallel band and a transition to the E component will give perpendicular bands; the result is a hybrid band.

Inversion in ammonia

Spectrum of central region of the symmetric bending vibration in ammonia simulated with Spectralcalc, illustrating inversion doubling.

Nitrogen inversion in ammonia

For ammonia, NH₃, the symmetric bending vibration is observed as two branches near 930 cm⁻¹ and 965 cm⁻¹. This so-called inversion doubling arises because the symmetric bending vibration is actually a large-amplitude motion known as inversion, in which the nitrogen atom passes through the plane of the three hydrogen atoms, similar to the inversion of an umbrella. The potential energy curve for such a vibration has a double minimum for the two pyramidal geometries, so that the vibrational energy levels occur in pairs which correspond to combinations of the vibrational states in the two potential minima. The two v = 1 states combine to form a symmetric state (1⁺) at 932.5 cm⁻¹ above the ground (0⁺) state and an antisymmetric state (1⁻) at 968.3 cm⁻¹.

The vibrational ground state (v = 0) is also doubled although the energy difference is much smaller, and the transition between the two levels can be measured directly in the microwave region, at ca. 24 Ghz (0.8 cm⁻¹). This transition is historically significant and was used in the ammonia maser, the fore-runner of the laser.

Asymmetric top molecules

Absorption spectrum (attenuation coefficient vs. wavelength) of liquid water (red)  atmospheric water vapor (green) and ice (blue line) between 667 nm and 200 μm. The plot for vapor is a transformation of data Synthetic spectrum for gas mixture "Pure H₂O" (296K, 1 atm) retrieved from Hitran on the Web Information System.

Asymmetric top molecules have at most one or more 2-fold rotation axes. There are three unequal moments of inertia about three mutually perpendicular principal axes. The spectra are very complex. The transition wavenumbers cannot be expressed in terms of an analytical formula but can be calculated using numerical methods.

The water molecule is an important example of this class of molecule, particularly because of the presence of water vapor in the atmosphere. The low-resolution spectrum shown in green illustrates the complexity of the spectrum. At wavelengths greater than 10 μm (or wavenumbers less than 1000 cm⁻¹) the absorption is due to pure rotation. The band around 6.3 μm (1590 cm⁻¹) is due to the HOH bending vibration; the considerable breadth of this band is due to the presence of extensive rotational fine structure. High-resolution spectra of this band are shown in Allen and Cross, p 221. The symmetric and asymmetric stretching vibrations are close to each other, so the rotational fine structures of these bands overlap. The bands at shorter wavelength are overtones and combination bands, all of which show rotational fine structure. Medium resolution spectra of the bands around 1600 cm⁻¹ and 3700 cm⁻¹ are shown in Banwell and McCash, p91.

Ro-vibrational bands of asymmetric top molecules are classed as A-, B- or C- type for transitions in which the dipole moment change is along the axis of smallest moment of inertia to the highest.

Experimental methods

Ro-vibrational spectra are usually measured at high spectral resolution. In the past, this was achieved by using an echelle grating as the spectral dispersion element in a grating spectrometer. This is a type of diffraction grating optimized to use higher diffraction orders. Today at all resolutions the preferred method is FTIR. The primary reason for this is that infrared detectors are inherently noisy, and FTIR detects summed signals at multiple wavelengths simultaneously achieving a higher signal to noise by virtue of Fellgett's advantage for multiplexed methods. The resolving power of an FTIR spectrometer depends on the maximum retardation of the moving mirror. For example, to achieve a resolution of 0.1 cm⁻¹, the moving mirror must have a maximum displacement of 10 cm from its position at zero path difference. Connes measured the vibration-rotation spectrum of Venusian CO₂ at this resolution. A spectrometer with 0.001 cm⁻¹ resolution is now available commercially. The throughput advantage of FTIR is important for high-resolution spectroscopy as the monochromator in a dispersive instrument with the same resolution would have very narrow entrance and exit slits.

When measuring the spectra of gases it is relatively easy to obtain very long path-lengths by using a multiple reflection cell. This is important because it allows the pressure to be reduced so as to minimize pressure broadening of the spectral lines, which may degrade resolution. Path lengths up to 20m are commercially available.