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| [[File:CF3I spectrum2.png|thumb|right|400px|Part of the rotational spectrum of [[trifluoroiodomethane]], CF<sub>3</sub>I.<ref group=notes>The spectrum was measured over a couple of hours with the aid of a chirped-pulse Fourier transform microwave spectrometer at the University of Bristol.</ref> Each rotational transition is labeled with the quantum numbers, ''J'', of the final and initial states, and is extensively split by the effects of [[Nuclear quadrupole resonance|nuclear quadrupole coupling]] with the <sup>127</sup>I nucleus.]]
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| '''Rotational [[spectroscopy]]''' is concerned with the measurement of the energies of transitions between quantized rotational states of [[molecule]]s in the [[gas phase]]. The spectra of [[chemical polarity|polar]] molecules can be measured in [[Absorption (optics)|absorption]] or [[Emission (electromagnetic radiation)|emission]] by [[microwave]] spectroscopy<ref>{{cite book|last=Gordy|first=W.|title=Microwave Molecular Spectra in Technique of Organic Chemistry, Vol. IX, Edited by A. Weissberger|year=1970|publisher=Interscience|location=New York}}</ref> or by [[far infrared]] spectroscopy. The rotational spectra of non-polar molecules cannot be observed by those methods, but can be observed and measured by [[Raman spectroscopy]]. Rotational spectroscopy is sometimes referred to as ''pure'' rotational spectroscopy to distinguish it from [[rotational-vibrational spectroscopy]] where changes in rotational energy occur together with changes in vibrational energy, and also from ro-vibronic spectroscopy (or just [[vibronic spectroscopy]]) where rotational, vibrational and electronic energy changes occur simultaneously.
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| For rotational spectroscopy, molecules are classified according to symmetry into spherical top, linear and symmetric top; analytical expressions can be derived for the rotational energy terms of these molecules. Analytical expressions cannot be derived for the fourth category, asymmetric top, but spectra can be fitted using numerical methods. The rotational energies are derived theoretically by considering the molecules to be [[rigid rotor]]s and then applying extra terms to account for centrifugal distortion, fine structure, hyperfine structure and Coriolis coupling. Fitting the spectra to the theoretical expressions gives numerical values of the angular [[moment of inertia|moments of inertia]] from which very precise values of molecular bond lengths and angles can be derived in favorable cases. In the presence of an electrostatic field there is [[Stark effect|Stark splitting]] which allows molecular [[electric dipole moment]]s to be determined.
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| An important application of rotational spectroscopy is in exploration of the chemical composition of the [[interstellar medium]] using [[radio telescope]]s.
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| ==Applications==
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| Rotational spectroscopy has primarily been used to investigate fundamental aspects of molecular physics. It is a uniquely precise tool for the determination of molecular structure in gas phase molecules. It can be used to establish barriers to internal rotation such as that associated with the rotation of the CH<sub>3</sub> group relative to the C<sub>6</sub>H<sub>4</sub>Cl group in [[chlorotoluene]] (C<sub>7</sub>H<sub>7</sub>Cl).<ref>{{cite journal|last=Nair|first=K.P.R.|coauthors=Demaison, J.; Wlodarczak, G.; Merke, I.;|title=Millimeterwave rotational spectrum and internal rotation in o-chlorotoluene|journal=Journal of Molecular Spectroscopy|year=236|volume=237|issue=2|pages=137–142|doi=10.1016/j.jms.2006.03.011|bibcode = 2006JMoSp.237..137N }}</ref> When fine or hyperfine structure can be observed, the technique also provides information on the electronic structures of molecules. Much of current understanding of the nature of weak molecular interactions such as [[van der Waals]], [[Hydrogen bond|hydrogen]] and [[Halogen bond|halogen]] bonds has been established through rotational spectroscopy. In connection with [[radio astronomy]], the technique has a key role in exploration of the chemical composition of the [[interstellar medium]]. Microwave transitions are measured in the laboratory and matched
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| to emissions from the [[interstellar medium]] using a [[radio telescope]]. [[ammonia|NH<sub>3</sub>]] was the first stable [[polyatomic]] molecule to be identified in the [[interstellar medium]].<ref>{{cite journal|last=Cheung|first=A.C.|coauthors=Rank, D.M.; Townes, C.H.; Thornton, D.D. and Welch, W.J.|title=Detection of NH<sub>3</sub> molecules in the interstellar medium by their microwave emission spectra|journal=Physical Review Letters|year=1968|volume=21|pages=1701–1705|doi=10.1103/PhysRevLett.21.1701|bibcode = 1968PhRvL..21.1701C|issue=25 }}</ref> The measurement of [[chlorine monoxide]]<ref>{{cite journal|last=Ricaud|first=P.|coauthors=Baron, P; de La Noë, J.|title=Quality assessment of ground-based microwave measurements of chlorine monoxide, ozone, and nitrogen dioxide from the NDSC radiometer at the Plateau de Bure|journal=Ann. Geophys.|year=2004|volume=22|pages=1903–1915|doi=10.5194/angeo-22-1903-2004|bibcode = 2004AnGeo..22.1903R }}</ref> is important for [[atmospheric chemistry]]. Current projects in astrochemistry involve both laboratory microwave spectroscopy and observations made using modern radiotelescopes such as the [[Atacama Large Millimeter Array|Atacama Large Millimetre Array]] (ALMA).<ref>{{cite web|title=Astrochemistry in Virginia|url=http://www.virginia.edu/ccu/molecspectroscopy.html|accessdate=2 December 2012}}</ref> Unlike [[Nuclear magnetic resonance|NMR]], [[Infrared spectroscopy|Infrared]] and [[UV-visible spectroscopy|UV-Visible]] spectroscopies, microwave spectroscopy has not yet found widespread application in [[analytical chemistry]].
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| ==Overview==
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| A molecule in the [[gas phase]] is free to rotate relative to a set of mutually [[orthogonal]] axes of fixed orientation in space, centered on the [[center of mass]] of the molecule. Free rotation is not possible for molecules in liquid or solid phases due to the presence of [[intermolecular force]]s. Rotation about each unique axis is associated with a set of quantized energy levels dependent on the moment of inertia about that axis and a quantum number. Thus, for linear molecules the energy levels are described by a single moment of inertia and a single quantum number, J. For symmetric tops there are two moments of inertia and two rotational quantum numbers to consider. Analysis of spectroscopic data with the expressions detailed below results in quantitative determination of the value(s) of the moment(s) of inertia. From these precise values of the molecular structure and dimensions may be obtained.
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| For a linear molecule, analysis of the rotational spectrum provides values for the [[Rigid rotor#Quantum mechanical linear rigid rotor|rotational constant]] and the moment of inertia of the molecule, and, knowing the atomic masses, can be used to determine the [[bond length]] directly. For [[diatomic|diatomic molecules]] this process is straightforward. For linear molecules with more than two atoms it is necessary to measure the spectra of two or more [[isotopologue]]s, such as <sup>16</sup>O<sup>12</sup>C<sup>32</sup>S and <sup>16</sup>O<sup>12</sup>C<sup>34</sup>S. This allows a set of [[simultaneous equations]] to be set up and solved for the [[bond length]]s).<ref group=notes>For a symmetric top, the values of the 2 moments of inertia can be used to derive 2 molecular parameters. Values from each additional isotopologue provide the information for one more molecular parameter. For asymmetric tops a single isotopologue provides information for at most 3 molecular parameters.</ref> It should be noted that a bond length obtained in this way is slightly different from the equilibrium bond length. This is because there is [[zero-point energy]] in the vibrational ground state, to which the rotational states refer, whereas the equilibrium bond length is at the minimum in the potential energy curve. The relation between the rotational constants is given by
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| :<math>B_\nu=B-\alpha(\nu+{1\over 2})</math> | |
| where ν is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated if the B values for two different vibrational states can be found.<ref>Banwell and McCash, p99</ref>
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| For other molecules, if the spectra can be resolved and individual transitions assigned both [[bond length]]s and [[Molecular geometry|bond angles]] can be deduced. When this is not possible, as with most asymmetric tops, all that can be done is to fit the spectra to three moments of inertia calculated from an assumed molecular structure. By varying the molecular structure the fit can be improved, giving a qualitative estimate of the structure. Isotopic substitution is invaluable when using this approach to the determination of molecular structure.
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| === Classification of molecular rotors ===
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| In [[quantum mechanics]] the free rotation of a molecule is [[angular momentum quantization|quantized]], so that the [[rotational energy]] and the [[angular momentum]] can take only certain fixed values, which are related simply to the [[moment of inertia]], <math> I </math>, of the molecule. For any molecule, there are three moments of inertia: <math>I_A</math>, <math>I_B</math> and <math>I_C</math> about three mutually orthogonal axes ''A'', ''B'', and ''C'' with the origin at the [[center of mass]] of the system. The general convention, used in this article, is to define the axes such that <math>I_A \leq I_B \leq I_C</math>, with axis <math>A</math> corresponding to the smallest moment of inertia. Some authors, however, define the <math>A</math> axis as the molecular [[molecular symmetry#Elements|rotation axis]] of highest order.
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| The particular pattern of [[energy level]]s (and, hence, of transitions in the rotational spectrum) for a molecule is determined by its symmetry. A convenient way to look at the molecules is to divide them into four different classes, based on the symmetry of their structure. These are
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| *Spherical tops (spherical rotors) All three moments of inertia are equal to each other: <math>I_A = I_B = I_C</math>. Examples of spherical tops include [[Allotropes of phosphorus#White phosphorus|phosphorus tetramer (P<sub>4</sub>)]], [[Carbon tetrachloride|carbon tetrachloride (CCl<sub>4</sub>)]] and other tetrahalides, [[methane|methane (CH<sub>4</sub>)]], [[silane|silane, (SiH<sub>4</sub>)]], [[Sulfur hexafluoride|sulfur hexafluoride (SF<sub>6</sub>)]] and other hexahalides. The molecules all belong to the cubic [[molecular point group|point group]]s T<sub>d</sub> or O<sub>h</sub>.
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| *Linear molecules. For a linear molecule the moments of inertia are related by <math>I_A << I_B = I_C </math>. For most purposes, <math>I_A</math> can be taken to be zero. Examples of linear molecules include [[Oxygen|dioxygen, O<sub>2</sub>]], [[nitrogen|dinitrogen, N<sub>2</sub>]], [[Carbon monoxide|carbon monoxide, CO]], [[Hydroxyl radical|hydroxy radical, OH]], [[Carbon dioxide|carbon dioxide, CO<sub>2</sub>]], [[Hydrogen cyanide|hydrogen cyanide, HCN]], [[Carbonyl sulfide|carbonyl sulfide, OCS]], [[Acetylene|acetylene (ethyne, HC≡CH)]] and dihaloethynes. These molecules belong to the point groups C<sub>∞v</sub> or D<sub>∞h</sub>
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| *Symmetric tops (symmetric rotors) A symmetric top is a molecule in which two moments of inertia are the same, <math>I_A= I_B</math> or <math>I_B= I_C</math>. By definition a symmetric top must have a 3-fold or higher order [[molecular symmetry#Elements|rotation axis]]. As a matter of convenience, spectroscopists divide molecules into two classes of symmetric tops, ''[[Oblate spheroid|Oblate]] symmetric tops'' (saucer or disc shaped) with <math>I_A = I_B < I_C</math> and ''[[Prolate]] symmetric tops'' (rugby football, or cigar shaped) with <math>I_A < I_B = I_C </math>. The spectra look rather different, and are instantly recognizable. Examples of symmetric tops include
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| : [[Oblate spheroid|Oblate]]: [[Benzene|benzene, C<sub>6</sub>H<sub>6</sub>]], [[Ammonia|ammonia, NH<sub>3</sub>]]
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| : [[Prolate]]: [[Chloromethane|chloromethane, CH<sub>3</sub>Cl]], [[Methylacetylene|propyne, CH<sub>3</sub>C≡CH]]
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| :As a detailed example, ammonia has a moment of inertia I<sub>C</sub> = 4.4128 × 10<sup>−47</sup> kg m<sup>2</sup> about the 3-fold rotation axis, and moments I<sub>A</sub> = I<sub>B</sub> = 2.8059 × 10<sup>−47</sup> kg m<sup>2</sup> about any axis perpendicular to the C<sub>3</sub> axis. Since the unique moment of inertia is larger than the other two, the molecule is an oblate symmetric top.<ref>Moment of inertia values from P. Atkins and J. de Paula, ''Physical Chemistry'', 8th ed. (W.H. Freeman 2006), p. 445</ref>
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| *Asymmetric tops (asymmetric rotors) The three moments of inertia have different values. Examples of small molecules that are asymmetric tops include [[Water (molecule)|water, H<sub>2</sub>O]] and [[Nitrogen dioxide|nitrogen dioxide, NO<sub>2</sub>]] whose symmetry axis of highest order is a 2-fold rotation axis. Most large molecules are asymmetric tops.
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| === Selection rules ===
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| {{main|selection rules}}
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| Transitions between rotational states can be observed in molecules with a permanent [[electric dipole moment]].<ref>Hollas p95</ref><ref group=notes>Such transitions are called electric dipole-allowed transitions. Other transitions involving quadrupoles, octupoles, hexadecapoles etc. may also be allowed but the spectral intensity is very much smaller, so these transitions are difficult to observe. Magnetic-dipole-allowed transitions can occur in [[paramagnetic]] molecules such as [[dioxygen]], O<sub>2</sub> and [[nitric oxide]], NO</ref> A consequence of this rule is that no microwave spectrum can be observed for centrosymmetric linear molecules such as N<sub>2</sub> ([[dinitrogen]]) or HCCH ([[ethyne]]), which are non-polar. Tetrahedral molecules such as CH<sub>4</sub> ([[methane]]), which have both a zero dipole moment and isotropic polarizability, would not have a pure rotation spectrum but for the effect of centrifugal distortion; when the molecule rotates about a 3-fold symmetry axis a small dipole moment is created, allowing a weak rotation spectrum to be observed by microwave spectroscopy.<ref>Hollas, p.104 shows part of the observed rotational spectrum of [[silane]]</ref>
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| With symmetric tops, the selection rule for electric-dipole-allowed pure rotation transitions is Δ''K''=0, Δ''J'' = ±1. Moreover the quantum number ''K'' is limited to have values between and including +''J'' to -''J''.<ref>Banwell and McCash, p.49</ref>
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| For [[raman spectroscopy|Raman spectra]] the general rule is that the molecular [[polarizability]] must be [[anisotropic]], which means that it is not the same in all directions.<ref>Hollas p111</ref> Polarizability is a 3-dimensional [[tensor]] that can be represented as an ellipsoid. The polarizability ellipsoid of spherical top molecules is in fact spherical so those molecules show no rotational Raman spectrum. For all other molecules both Stokes and anti-Stokes lines<ref group=notes>In Raman spectroscopy the photon energies for Stokes and anti-Stokes scattering are respectively less than and greater than the incident photon energy. See the energy-level diagram at [[Raman spectroscopy]].</ref> can be observed and they have similar intensities due to the fact than many rotational states are thermally populated. The selection rule for linear molecules is ΔJ = 0, ±2. The reason for the value of 2 is that the ellipsoid must rotate twice during a transition.
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| The selection rule for symmetric top molecules is
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| :Δ''K'' = 0
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| :If ''K''=0, then Δ''J'' = ±2
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| :If ''K'' ≠ 0, then Δ''J'' = 0, ±1, ±2
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| Transitions with Δ''J'' = +1 are said to belong the an ''R'' series, whereas transitions with Δ''J'' = +2 belong to an ''S'' series.<ref>Banwell and McCash, section 4.2, p 105, ''Pure Rotational Raman Spectra</ref>
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| === Units===
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| The units used for rotational constants depend on the type of measurement. With infrared spectra, the unit of measurement is usually wavenumbers per cm, written as cm<sup>−1</sup> and shown with the symbol <math>\tilde \nu</math>. Wavenumbers per cm is literally the number of waves in one centimeter, or the reciprocal of wavelength in cm. On the other hand, microwave spectra are usually measured in [[hz|Gigahertz]]. The relationship between the two units is derived from the expression | |
| :<math> \nu \times \lambda = c</math>
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| where ν is a [[frequency]], λ is a [[wavelength]] and ''c'' is the [[velocity of light]]. It follows that
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| :<math>\tilde \nu /cm^{-1}= \frac{1}{\lambda /cm} = \frac{\nu /s^{-1}}{c /cm \ s^{-1}} =
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| \frac{\nu /s^{-1}}{2.99792458 \times 10^{10}}</math>
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| As 1 GHz = 10<sup>9</sup> hz, the numerical conversion can be expressed as
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| :<math> \tilde \nu /cm^{-1}\approx \frac {\nu /Ghz}{30}</math>
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| ===Effect of vibration on rotation===
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| The population of vibrationally excited states follows a Boltzmann distribution, so low frequency vibrational states are appreciably populated even at room temperatures. As the moment of inertia is higher when a vibration is excited, the rotational constants, ''B'' decrease. Consequently, the rotation frequencies in each vibration state are different from each other. This can give rise to "satellite" lines in the rotational spectrum. An example is provided by [[cyanodiacetylene]], H-C≡C−C≡C-C≡N,<ref>{{cite journal|last=Alexander|first=A.J|coauthors=Kroto, H.W; Walton, D.R.M.|title=The microwave spectrum, substitution structure and dipole moment of cyanobutadiyne|journal=J. Mol. Spectrosc.|year=1967|volume=62|pages=175–180|doi=10.1016/0022-2852(76)90347-7|bibcode = 1976JMoSp..62..175A }} Illustrated in Hollas, p97</ref>
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| Further, there is a [[fictitious force]], [[Coriolis effect|Coriolis coupling]], between the vibrational motion of the nuclei in the rotating (non-inertial) frame. However, as long as the vibrational quantum number does not change (i.e., the molecule is in only one state of vibration), the effect of vibration on rotation is not important, because the time for vibration is much shorter than the time required for rotation. The Coriolis coupling is often negligible, too, if one is interested in low vibrational and rotational quantum numbers only.
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| === Effect of rotation on vibrational spectra ===
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| {{main|Rotational-vibrational spectroscopy}} Historically, the theory of rotational energy levels was developed to account for observations of vibration-rotation spectra of gases in [[infrared spectroscopy]], which was used before microwave spectroscopy had become practical. To a first approximation the energy of rotation is added to, or subtracted from the energy of vibration. The vibration-rotation wavenumbers of transitions for a harmonic oscillator with rigid rotor are given by
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| :<math>\tilde \nu = \tilde \nu_{vib} \pm BJ(J+1) </math>
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| In reality, this expression has to be modified for the effects of anharmonicity of the vibrations, for centrifugal distortion and for Coriolis coupling.<ref>Banwell and McCash, p63</ref> The plus sign implies simultaneous excitation of both vibration and rotation, giving the so-called ''R'' branch in the spectrum, whereas with the minus sign a quantum of rotational energy is lost while a quantum of vibrational energy is gained, giving the ''P'' branch. The pure vibration, Δ''J''=0, gives rise to the ''Q'' branch of the spectrum. Because of the thermal population of the rotational states the ''P'' branch is slightly less intense than the ''R'' branch.
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| Rotational constants obtained from infrared measurements are in good accord with those obtained by microwave spectroscopy while the latter usually offers greater precision.
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| ==Structure of rotational spectra==
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| ===Spherical top===
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| Spherical top molecules have no net dipole moment. A pure rotational spectrum cannot be observed by absorption or emission spectrocopy because there is no permanent dipole moment whose rotation can be accelerated by the electric field of an incident photon. Also the polarizability is isotropic, so that pure rotational transitions cannot be observed by Raman spectroscopy either. Nevertheless, rotational constants can be obtained by [[Rovibrational coupling|ro-vibrational spectroscopy]]. This occurs when a molecule is polar in the vibrationally excited state. For example, the molecule [[methane]] is a symmetric top but the asymmetric C-H stretching band shows rotational fine structure in the infrared spectrum, illustrated in [[rovibrational coupling]]. This spectrum is also interesting because it shows clear evidence of [[Coriolis effect|Coriolis coupling]] in the asymmetric structure of the band.
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| ===Linear molecules===
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| [[File:Rotational spectrum example.png|right|thumb|300px|Energy levels and line positions calculated in the rigid rotor approximation]]
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| The [[rigid rotor]] is a good starting point from which to construct a model of a rotating molecule. It is assumed that component atoms are [[point particle|point masses]] connected by rigid bonds. A linear molecule lies on a single axis and each atom moves on the surface of a sphere around the centre of mass. The two degrees of rotational freedom correspond to the [[spherical coordinates]] θ and φ which describe the direction of the molecular axis, and the quantum state is determined by two quantum numbers J and M. J defines the magnitude of the rotational angular momentum, and M its component about an axis fixed in space, such as an external electric or magnetic field. In the absence of external fields, the energy depends only on J. Under the [[rigid rotor]] model, the rotational energy levels, ''F''(J), of the molecule can be expressed as,
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| :<math> F\left( J \right) = \tilde B J \left( J+1 \right) \qquad J = 0,1,2,...</math>
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| where <math> \tilde B </math> is the rotational constant of the molecule and is related to the moment of inertia of the molecule. In a linear molecule the moment of inertia about an axis perpendicular to the molecular axis is unique, that is, <math> I_B = I_C, I_A=0 </math>, so
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| :<math> \tilde B /cm^{-1} = {h \over{8\pi^2cI_B}}= {h \over{8\pi^2cI_C}}</math>
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| For a diatomic molecule
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| :<math> I=\frac{m_1m_2}{m_1 +m_2}d^2 </math>
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| where ''m''<sub>1</sub> and ''m''<sub>2</sub> are the masses of the atoms and ''d'' is the distance between them.
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| [[Selection rules]] dictate that during emission or absorption the rotational quantum number has to change by unity; i.e., <math> \Delta J = J^{\prime} - J^{\prime\prime} = \pm 1 </math>. Thus, the locations of the lines in a rotational spectrum will be given by
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| :<math> \tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime}} = F\left( J^{\prime} \right) - F\left( J^{\prime\prime} \right) = 2 \tilde B \left( J^{\prime\prime} + 1 \right) \qquad J^{\prime\prime} = 0,1,2,...</math>
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| where <math>J^{\prime\prime}</math> denotes the lower level and <math>J^{\prime}</math> denotes the upper level involved in the transition.
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| The diagram illustrates rotational transitions that obey the <math>\Delta J</math>=1 selection rule. The dashed lines show how these transitions map onto features that can be observed experimentally. Adjacent <math>J^{\prime\prime}{\leftarrow}J^{\prime}</math> transitions are separated by 2''B'' in the observed spectrum. Frequency or wavenumber units can also be used for the ''x'' axis of this plot.
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| ==== Rotational line intensities ====
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| [[File:Populations of rotational states.png|thumb|Rotational level populations with ''Bhc/kT'' = 0.05. J is the quantum number of the lower rotational state]]
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| The probability of a transition taking place is the most important factor influencing the intensity of an observed rotational line. This probability is proportional to the population of the initial state involved in the transition. The population of a rotational state depends on two factors. The number of molecules in an excited state with quantum number J, relative to the number of molecules in the ground state, ''N<sub>J</sub>/N<sub>0</sub>'' is given by the [[Boltzmann distribution]] as
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| :<math>\frac{N_J}{N_0}=e^{-\frac{E_J}{kT}} =e^{-\frac {BhcJ(J+1)}{kT}}</math>,
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| where k is the [[Boltzmann constant]] and T the [[absolute temperature]]. This factor decreases as J increases. The second factor is the [[Degenerate energy levels|degeneracy]] of the rotational state, which is equal to 2J+1. This factor increases as J increases. Combining the two factors<ref>Banwell and McCash, p40</ref>
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| :<math>\mathrm{population} \propto (2J+1)e^{-\frac{E_J}{kT}} </math>
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| The maximum relative intensity occurs at
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| :<math>J=\sqrt{\frac{kT}{2hcB}}</math>
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| The diagram at the right shows an intensity pattern roughly corresponding to the spectrum above it.
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| ====Centrifugal distortion====
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| When a molecule rotates, the [[centrifugal force]] pulls the atoms apart. As a result, the moment of inertia of the molecule increases, thus decreasing the value of <math> \tilde B </math>, when it is calculated using the expression for the rigid rotor. To account for this a centrifugal distortion correction term is added to the rotational energy levels of the diatomic molecule.
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| :<math> F\left( J \right) = \tilde B J \left( J+1 \right) - \tilde D J^2 \left( J+1 \right)^2 \qquad J = 0,1,2,...</math>
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| where <math> \tilde D</math> is the centrifugal distortion constant.
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| Therefore, the line positions for the rotational mode change to
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| :<math> \tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime}} = 2 \tilde B \left( J^{\prime\prime} + 1 \right) - 4\tilde D \left( J^{\prime\prime} +1 \right)^3 \qquad J^{\prime\prime} = 0,1,2,...</math>
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| In consequence, the spacing between lines is not constant, as in the rigid rotor approximation, but decreases with increasing rotational quantum number.
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| An assumption underlying these expressions is that the molecular vibration follows [[simple harmonic motion]]. In the harmonic approximation the centrifugal constant ''D'' can be derived as
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| :<math> \tilde D = \frac{h^3}{32 \pi^4 I^2 r^2 k c}</math>
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| where ''k'' is the vibrational [[force constant]]. The relationship between B and D
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| :<math>\tilde D=\frac{4\tilde B^3}{\tilde \omega ^2}</math> | |
| where :<math> \tilde \omega </math> is the harmonic vibration frequency, follows. If anharmonicity is to be taken into account, terms in higher powers of J should be added to the expressions for the energy levels and line positions.<ref>Banwell and McCash, p 45.</ref> A striking example concerns the rotational spectrum of [[hydrogen fluoride]] which was fitted to terms up to ''[J(J+1)]<sup>5</sup>''.<ref>{{cite journal|last=Jennings|first=D.A.|coauthors=Evenson, K.M; Zink, L.R.;Demuynck, C.;Destombes, J.L.;Lemoine, B;Johns,J.W.C.|title=High-resolution spectroscopy of HF from 40 to 1100 cm<sup>−1</sup>: Highly accurate rotational constants|journal=Journal of Molecular Spectroscopy|date=April 1987|volume=122|issue=2|pages=477–480|doi=10.1016/0022-2852(87)90021-X|bibcode = 1987JMoSp.122..477J }}[http://tf.nist.gov/general/pdf/632.pdf pdf]</ref>
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| ==== Oxygen ====
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| The electric dipole moment of the dioxygen molecule, O<sub>2</sub> is zero, but the molecule is [[paramagnetic]] with two unpaired electrons so that there are magnetic-dipole allowed transitions which can be observed by microwave spectroscopy. The unit electron spin has three spatial orientations with respect to the given molecular rotational angular momentum vector, K, so that each rotational level is split into three states, J = K + 1, K, and K - 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. The energy difference between successive J terms in any of these triplets is about 2 cm<sup>−1</sup> (60 GHz), with the single exception of J = 1←0 difference which is about 4 cm<sup>−1</sup>. 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 K there are two allowed transitions. The <sup>16</sup>O nucleus has zero nuclear spin angular momentum, so that symmetry considerations demand that K have only odd values.<ref>{{cite journal|last=Strandberg,|first=M. W. P.|coauthors=Meng, C. Y.;Ingersoll, J. G.|title=The Microwave Absorption Spectrum of Oxygen|journal=Phys.Rev.|year=1949|volume=75|issue=10|pages=1524–1528|doi=10.1103/PhysRev.75.1524|bibcode = 1949PhRv...75.1524S }}[http://dspace.mit.edu/bitstream/handle/1721.1/4963/RLE-TR-087-14236979.pdf pdf]</ref><ref>{{cite journal|last=Krupenie|first=Paul H.|title=The Spectrum of Molecular Oxygen|journal=J. Phys. Chem. Ref. Data 1, 423 (1972);|year=1972|volume=1|issue=2|pages=423–534|doi=10.1063/1.3253101|url=http://www.nist.gov/data/PDFfiles/jpcrd8.pdf}}</ref>
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| ===Symmetric top===
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| 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 2''J''+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 [[molecular symmetry|principal rotation axis]] of the molecule. In the absence of an external electrical field, the rotational energy of a symmetric top is a function of only J and K and, in the rigid rotor approximation, the energy of each rotational state is given by
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| :<math> F\left( J,K \right) = \tilde B J \left( J+1 \right) + \left( \tilde A - \tilde B \right) K^2 \qquad
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| J = 0,1,2,... \quad \mbox{and}\quad K = +J, ... 0 ... -J</math>
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| where <math> \tilde B = {h\over{8\pi^2cI_B}} </math> and <math> \tilde A = {h\over{8\pi^2cI_A}} </math> for a ''prolate'' symmetric top molecule or
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| <math> \tilde A = {h\over{8\pi^2cI_C}} </math> for an ''oblate'' molecule.
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| This gives the transition wavenumbers as
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| :<math> \tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K \right) - F\left( J^{\prime\prime},K \right)
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| = 2 \tilde B \left( J^{\prime\prime} + 1 \right)
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| \qquad J^{\prime\prime} = 0,1,2,...</math>
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| which is the same as in the case of a linear molecule.<ref>Hollas, p101</ref> With a first order correction for centrifugal distortion the transition wavenumbers become | |
| :<math> \tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K \right) - F\left( J^{\prime\prime},K \right)
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| = 2 \left( \tilde B - 2D_{JK}K^2 \right)
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| \left( J^{\prime\prime} + 1 \right)
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| -4D_J\left(J^{\prime\prime}+1\right)^3 \qquad J^{\prime\prime} = 0,1,2,...</math>
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| The term in ''D<sub>JK</sub>'' has the effect of removing degeneracy present in the rigid rotor approximation, with different ''K'' values.<ref>Hollas, p102, shows the effect on the microwave spectrum of H<sub>3</sub>SiNCS.</ref>
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| ===Asymmetric top===
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| [[File:Atmospheric terahertz transmittance at Mauna Kea (simulated).png|300 px|thumb|Pure rotation spectrum of atmospheric water vapour measured at Mauna Kea (33 cm<sup>-1</sup> to 100 cm<sup>-1</sup>)]]
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| The quantum number ''J'' refers to the total angular momentum, as before. Since there are three independent moments of inertia, there are two other independent quantum numbers to consider, but the term values for an asymmetric rotor cannot be derived in closed form. They are obtained by individual [[Diagonalizable matrix#Diagonalization|matrix diagonalization]] for each ''J'' value. Formulae are available for molecules whose shape approximates to that of a symmetric top.<ref>Hollas, p103</ref>
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| The water molecule is an important example of an asymmetric top. It has an intense pure rotation spectrum in the far infrared region, below about 200 cm<sup>−1</sup>. For this reason far infrared spectrometers have to be freed of atmospheric water vapour either by purging with a dry gas or by evacuation. The spectrum has been analyzed in detail.<ref>{{cite journal|last=Hall|first=Richard T.|coauthors=Dowling, Jerome M.|title=Pure Rotational Spectrum of Water Vapor|journal=J. Chem. Phys.|year=1967|volume=47|issue=7|pages=2454–2461|doi=10.1063/1.1703330|bibcode = 1967JChPh..47.2454H }}
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| {{cite journal|last=Hall|first=Richard T.|coauthors=Dowling, Jerome M.|title=Erratum: Pure Rotational Spectrum of Water Vapor|journal=J. Chem. Phys.|year=1971|volume=54|issue=11|page=4968|doi=10.1063/1.1674785|bibcode = 1971JChPh..54.4968H }}</ref>
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| == Quadrupole splitting ==
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| When a nucleus has a spin quantum number, ''I'', greater than 1/2 it has a [[quadrupole]] moment. In that case, coupling of nuclear spin angular momentum with rotational angular momentum causes splitting of the rotational energy levels. If the quantum number ''J'' of a rotational level is greater than ''I'', 2''I''+1 levels are produced; but if ''J'' is less than ''I'', 2''J''+1 levels result. The effect is known as [[hyperfine splitting]]. For example, with <sup>14</sup></sup>N (''I'' = 1) in HCN, all levels with J > 0 are split into 3. The energy of the sub-levels are proportional to the [[quadrupole moment|nuclear quadrupole moment]] and a function of ''F'' and ''J''. where ''F'' = ''J''+''I'', ''J''+''I''-1, ..., 0, ... |''J''-''I''|. Thus, observation of nuclear quadrupole splitting permits the magnitude of the nuclear quadrupole moment to be determined.<ref>{{cite journal|last=Simmons|first=James W.|coauthors=Anderson, Wallace E.; Gordy,Walter|title=Microwave Spectrum and Molecular Constants of Hydrogen Cyanide|journal=Phys. Rev.|year=1950|volume=77|pages=77–79|doi=10.1103/PhysRev.77.77|bibcode = 1950PhRv...77...77S }}</ref>
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| This is an alternative method to the use of [[nuclear quadrupole resonance]] spectroscopy. The selection rule for rotational transitions becomes<ref>{{cite book |last=Chang| first=Raymond| title=Basic Principles of Spectroscopy| year=1971| publisher=McGraw-Hill}} p139</ref>
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| :<math>\Delta J= \pm 1, \Delta F = 0, \pm 1 </math>
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| ==Stark and Zeeman effects==
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| In the presence of a static external [[electric field]] the 2''J''+1 degeneracy of each rotational state is partly removed, an instance of a [[Stark effect]]. For example in linear molecules each energy level is split into ''J''+1 components. The extent of splitting depends on the square of the electric field strength and the square of the dipole moment of the molecule.<ref>Hollas, p102 gives the equations for diatomic molecules and symmetric tops</ref> In principle this provides a means to determine the value of the molecular dipole moment with high precision. Examples include [[carbonyl sulfide]], OCS, with μ = 0.71521 ± 0.00020 [[Debye (unit)|Debye]]. However, because the splitting depends on μ<sup>2</sup>, the orientation of the dipole must be deduced from quantum mechanical considerations.<ref>Hollas, p102</ref>
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| A similar removal of degeneracy will occur when a [[paramagnetic]] molecule is placed in a magnetic field, an instance of the [[Zeeman effect]]. Most species which can be observed in the gaseous state are [[diamagnetic]] . Exceptions, known as [[odd molecule]]s, include [[nitric oxide]], NO, [[nitrogen dioxide]], NO<sub>2</sub>, some [[chlorine oxide]]s and the [[hydroxyl radical]]. The Zeeman effect has been observed with [[dioxygen]], O<sub>2</sub><ref>.{{cite journal|last=Burkhalter|first=James H.|coauthors=Roy S. Anderson, William V. Smith, and Walter Gordy|title=The Fine Structure of the Microwave Absorption Spectrum of Oxygen|journal=Phys. Rev.|year=1950|volume=79|issue=4|pages=651–655|doi=10.1103/PhysRev.79.651|bibcode = 1950PhRv...79..651B }}</ref>
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| ==Rotational Raman spectroscopy==
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| Molecular rotational transitions can also be observed by [[Raman spectroscopy]]. Rotational transitions are Raman-allowed for any molecule with an [[anisotropic]] [[polarizability]] which includes all molecules except for spherical tops. This means that rotational transitions of molecules with no permanent dipole moment, which cannot be observed in absorption or emission, can be observed, by scattering, in Raman spectroscopy. Very high resolution Raman spectra can be obtained by adapting a [[Fourier transform infrared spectroscopy|Fourier Transform Infrared Spectrometer]]. An example is the spectrum of <sup>15</sup>N<sub>2</sub>. It shows the effect of nuclear spin, resulting in intensities variation of 3:1 in adjacent lines. A bond length of 109.9985 ± 0.0010 pm was deduced from the data.<ref>Hollas, p113, illustrates the spectrum of <sup>15</sup>N<sub>2</sub> obtained using 476.5 nm radiation from an [[argon ion laser]].</ref>
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| ==Instruments and Methods==
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| The great majority of contemporary spectrometers use a mixture of commercially available and bespoke components which users integrate according to their particular needs. Instruments can be broadly categorised according to their general operating principals. Although rotational transitions can be found across a very broad region of the [[electromagnetic spectrum]], fundamental physical constraints exist on the operational bandwidth of instrument components. It is often impractical and costly to switch to measurements within an entirely different frequency region. The instruments and operating principals described below are generally appropriate to microwave spectroscopy experiments conducted at frequencies between 6 and 24 GHz.
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| ===Absorption cells and Stark modulation===
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| A microwave spectrometer can be most simply constructed using a source of microwave radiation, an absorption cell into which sample gas can be introduced and a detector such as a [[superheterodyne receiver]]. A spectrum can be obtained by sweeping the frequency of the source while detecting the intensity of transmitted radiation. A simple section of [[waveguide]] can serve as an absorption cell. An important variation of the technique in which an alternating current is applied across electrodes within the absorption cell results in a modulation of the frequencies of rotational transitions. This is referred to as Stark modulation and allows the use of [[Lock-in amplifier|phase-sensitive detection]] methods offering improved sensitivity. Absorption spectroscopy allows the study of samples that are thermodynamically stable at room temperature.
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| The first study of the [[microwave]] spectrum of a molecule (NH<sub>3</sub>) was performed by Cleeton & Williams in 1934.<ref name="Cleeton">{{cite journal|last=Cleeton|first=C.E.|coauthors=Williams, N.H.|title=Electromagnetic waves of 1.1 cm wave-length and the absorption spectrum of ammonia|journal=Physical Reviews|year=1934|volume=45|pages=234–237|doi=10.1103/PhysRev.45.234|bibcode = 1934PhRv...45..234C|issue=4 }}</ref> Subsequent experiments exploited powerful sources of [[microwave]]s such as the [[klystron]], many of which were developed for radio detection and ranging ([[RADAR]]) during the [[World War II|Second World War]]. The number of experiments in microwave spectroscopy surged immediately after the war. By 1948, [[Walter Gordy]] was able to prepare a review of the results contained in approximately 100 research papers.<ref>{{cite journal|last=Gordy|first=W.|title=Microwave spectroscopy|journal=Reviews of Modern Physics|year=1948|volume=20|pages=668–717|doi=10.1103/RevModPhys.20.668|bibcode = 1948RvMP...20..668G|issue=4 }}</ref> Commercial versions<ref>{{cite web|title=June 1971, Hewlett Packard Journal|url=http://www.hpl.hp.com/hpjournal/pdfs/IssuePDFs/1971-06.pdf|accessdate=November 2012}}</ref> of microwave absorption spectrometer were developed by [[Hewlett Packard]] in the 1970s and were once widely used for fundamental research. Most research laboratories now exploit either Balle-[[Willis H. Flygare|Flygare]] or chirped-pulse Fourier transform microwave (FTMW) spectrometers.
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| ===Fourier transform microwave (FTMW) spectroscopy===
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| The theoretical framework<ref>{{cite journal|last=Schwendemann|first=R.H.|journal=Ann. Rev. Phys. Chem.|year=1978|volume=29|pages=537–558|doi= 10.1146/annurev.pc.29.100178.002541
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| |bibcode = 1978ARPC...29..537S }}</ref> underpinning FTMW spectroscopy is analogous to that used to describe [[NMR spectroscopy|FT-NMR spectroscopy]]. The behaviour of the evolving system is described by optical [[Bloch equations]]. First, a short (typically 0-3 microsecond duration) microwave pulse is introduced on resonance with a rotational transition. Those molecules that absorb the energy from this pulse are induced to rotate coherently in phase with the incident radiation. De-activation of the polarisation pulse is followed by microwave emission that accompanies decoherence of the molecular ensemble. This [[free induction decay]] occurs on a timescale of 1-100 microseconds depending on instrument settings. Following pioneering work by Dicke and co-workers in the 1950s,<ref>{{cite journal|last=Dicke|first=R.H.|coauthors=Romer, R.H.|journal=Rev. Sci. Inst.|year=1955|volume=26|issue=10|pages=915–928|doi=10.1063/1.1715156 |bibcode = 1955RScI...26..915D }}</ref> the first FTMW spectrometer was constructed by Ekkers and [[Willis H. Flygare|Flygare]] in 1975.<ref>{{cite journal|last=Ekkers|first=J.|coauthors=Flygare, W.H.|journal=Rev. Sci. Inst.|year=1976|volume=47|issue=4|pages=448–454|doi=10.1063/1.1134647|bibcode = 1976RScI...47..448E }}</ref>
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| ====The Balle-Flygare FTMW spectrometer ====
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| Balle, Campbell, Keenan and Flygare demonstrated that the FTMW technique can be applied within a "free space cell" comprising an evacuated chamber containing a [[Optical cavity|Fabry-Perot cavity]].<ref>{{cite journal|last=Balle|first=T.J.|coauthors=Campbell, E.J.; Keenan, M.R.; Flygare, W.H.|journal=J. Chem. Phys.|year=1980|volume=72|issue=2|pages=922–932|doi=10.1063/1.439210|bibcode = 1980JChPh..72..922B }}</ref> This technique allows a sample to be probed only milliseconds after it undergoes rapid cooling to only a few degrees Kelvin in the throat of an expanding gas jet. This was a revolutionary development because (i) cooling molecules to low temperatures concentrates the available population in the lowest rotational energy levels. Coupled with benefits conferred by the use of a Fabry-Perot cavity, this brought a great enhancement in the sensitivity and resolution of spectrometers along with a reduction in the complexity of observed spectra; (ii) it became possible to isolate and study molecules that are very weakly bound because there is insufficient energy available for them to undergo fragmentation or chemical reaction at such low temperatures. [[William Klemperer]] was a pioneer in using this instrument for the exploration of weakly bound interactions. While the Fabry-Perot cavity of a Balle-Flygare FTMW spectrometer can typically be tuned into resonance at any frequency between 6 and 18 GHz, the bandwidth of individual measurements is restricted to about 1 MHz. An animation illustrates the operation of this instrument which is currently the most widely used tool for microwave spectroscopy.<ref>{{cite web|last=Jager|first=W.|title=Balle-Flygare FTMW spectrometer animation|url=http://www.chem.ualberta.ca/~jaeger/misc/ftmw.swf}}</ref>
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| ====The Chirped-Pulse FTMW spectrometer ====
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| Noting that digitisers and related electronics technology had significantly progressed since the inception of FTMW spectroscopy, [[Brooks Pate|B.H. Pate]] at the University of Virginia<ref>{{cite web|title=Web page of B.H. Pate Research Group, Department of Chemistry, University of Virginia.|url=http://faculty.virginia.edu/bpate-lab/|accessdate=November 2012}}</ref> designed a spectrometer<ref>{{cite journal|last=Brown|first=G.G.|coauthors=Dian, B.C.; Douglass, K.O.; Geyer, S.M.; Pate, B.H.;|journal=J. Mol. Spectrosc.|year=2006|volume=238|pages=200–212|doi=10.1016/j.jms.2006.05.003|bibcode = 2006JMoSp.238..200B }}</ref> which retains many advantages of the Balle-Flygare FT-MW spectrometer while innovating in (i) the use of a high speed (>4 GS/s) arbitrary waveform generator to generate a "chirped" microwave polarisation pulse that sweeps up to 12 GHz in frequency in less than a microsecond and (ii) the use of a high speed (>40 GS/s) oscilloscope to digitise and Fourier transform the molecular free induction decay. The result is an instrument that allows the study of weakly bound molecules but which is able to exploit a measurement bandwidth (12 GHz) that is greatly enhanced compared with the Balle-Flygare FTMW spectrometer. Modified versions of the original CP-FTMW spectrometer have been constructed by a number of groups in the United States, Canada and Europe.<ref>{{cite journal|last=Grubbs|first=G.S.|coauthors=Dewberry, C.T.; Etchison, K.C.; Kerr, K.E.; Cooke, S.A.|journal=Rev. Sci. Inst.|year=2007|volume=78|issue=9|page=096106|doi=10.1063/1.2786022|bibcode = 2007RScI...78i6106G }}</ref><ref>{{cite journal|last=Wilcox|first=D.S.|coauthors=Hotopp, K.M.; Dian, B.C.;|journal=J. Phys. Chem. A|year=2011|volume=115|issue=32|pages=8895–8905|doi=10.1021/jp2043202
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| }}</ref> The instrument offers a broadband capability that is highly complementary to the high sensitivity and resolution offered by the Balle-Flygare design.
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| == Notes ==
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| {{reflist|group=notes}}
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| ==References==
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| {{reflist}}
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| == Bibliography ==
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| *{{cite book |last1=Banwell |first1=Colin N. |last2=McCash |first2=Elaine M. |title=Fundamentals of molecular spectroscopy |edition= 4th |year=1994 |publisher=McGraw-Hill|isbn=0-07-707976-0 |page=40 }}
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| *{{cite book|last=Hollas|first=M.J.|title=Modern Spectroscopy|edition=3rd|year=1996|publisher=Wiey|isbn=0471965227}}
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| *{{cite book|last=Atkins|first=P.W.|title=Physical Chemistry|year=2006|publisher=Oxford University Press|edition= 8th|isbn=0198700725|pages=431–469|coauthors=de Paula, J.}} Chapter (Molecular Spectroscopy), Section (Pure rotation spectra) and page numbers may be differenct in different editions.
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| *{{Cite book | last1 = [[Charles Hard Townes|Townes]] | first1 = Charles H. | last2 = [[Arthur Leonard Schawlow|Schawlow]] | first2 = Arthur L. | title = Microwave spectroscop | year = 1975 | publisher = Dover Publications | location = New York | isbn = 978-0-486-61798-5 | pages = }}
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| *{{Cite book | last1 = [[Harry Kroto|Kroto]] | first1 = H. W. | title = Molecular rotation spectr | year = 2003 | publisher = Dover Publications | location = Mineola, N.Y. | isbn = 0-486-49540-X | pages = }}
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| *{{Cite book | last1 = Brown | first1 = John M. | last2 = Carrington | first2 = Alan. | title = Rotational spectroscopy of diatomic molecule | year = 2003 | publisher = Cambridge University Press | location = Cambridge ; New York | isbn = 0-521-53078-4 | pages = }}
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| *{{Cite book | last1 = McQuarrie | first1 = Donald A. (Donald Allan) | title = Quantum chemistry | year = 2008 | publisher = University Science Books | location = Sausalito, Calif. | isbn = 1-891389-50-5 | pages = }}
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| ==External links==
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| *[http://www.spectralcalc.com/ infrared gas spectra simulator]
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| *[http://hyperphysics.phy-astr.gsu.edu/HBASE/molecule/rotrig.html Hyperphysics article on Rotational Spectrum]
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| *[http://www.ifpan.edu.pl/~kisiel/rotlinks.htm A list of microwave spectroscopy research groups around the world]
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| {{BranchesofSpectroscopy}}
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| {{DEFAULTSORT:Rotational Spectroscopy}}
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| [[Category:Spectroscopy]]
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| [[Category:Rotation]]
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