CHAPTER 1
Interaction of Millimetre Wavelength Electromagnetic Radiation with Gases
Millimetre wavelength (MMW) radiation forms the 30–300 GHz band of the electromagnetic spectrum and is used to study the mainly rotational spectra of gaseous molecules. Transitions can also be measured in the inversion spectrum of ammonia, between molecular rotamer configurations and between magnetic fine structure components of molecules possessing a magnetic dipole, e.g. O2 and NO. The absorption spectra normally studied in small molecules (<200 Dalton) become more intense at higher frequencies, as illustrated in Figure 1.1 for sulfur dioxide, so for quantitative work it is usually advantageous to work in the MMW band.
The rotational levels lie at low energies and are all populated at ambient temperatures. This gives rise to abundant spectra of narrow lines, width < 1 MHz at Pascal (Pa) pressures, that are spread over this wide spectral region. Spectral interference between species in real mixtures is unusual and regions can be chosen for measurements where the target species are represented without spectral overlap. This is quite different from other spectrometric methods in the ultraviolet-visible to mid-infrared region where the analyst is usually constrained to work at one of a few frequencies. Separation prior to analysis for gas mixtures is not required apart perhaps from gross scrubbing of suspended solids or liquids, making rotational spectrometry unique amongst quantitative analytical methods.
Rotational spectroscopy is essentially a low-pressure technique if one is to exploit its remarkable selectivity, although quantitative measurements can be made at pressures up to atmospheric. Not all gases are rotationally active; with the exception of dioxygen all the homonuclear diatomic molecules and of course monatomic gases are inactive. Molecules with a high degree of symmetry, notably methane, ethane, ethene, benzene and carbon dioxide, are likewise inactive or have weak spectra, e.g. propane and butane. Oxygen and water, possibly the two most-determined molecular species in the modern world, are rotationally active and can be measured readily in those matrix gases.
Modern teaching in analytical spectrometry deals only rarely with quantitative rotational spectrometry in any depth and this lack of attention gives rise to some misunderstandings about the technique. Much of that derives from the unusual and sometimes seemingly mysterious combination of optical and electronic phenomena that characterise this spectral region. In reality of course, MMW spectrometry is quite simple and has been made much easier in recent years with the advent of solid state electronic instrumentation and devices of very high quality.
The theory too is fundamentally simple and has been very well developed although less emphasis has been placed on the quantitative aspects of the subject than was perhaps desirable, with the notable exception of the book devoted to this topic by Varma and Hrubesh.
The origin and details of molecular rotational spectra are explicitly laid out in the comprehensive texts by Townes and Schawlow, and Gordy and Cook, the latter two being the sources for much of the theory outlined below. The present authors have focused only upon those aspects that are directly relevant to quantitative MMW cavity spectrometry, and the reader is referred to the original texts for a more comprehensive treatment. Readers will soon notice that the original texts, and indeed the majority of all published MMW spectrometry papers and texts to this day, use cgs rather than SI units. Consequently, some of the equations differ from those in other texts and translation of units will be necessary to compare them with other work.
1 Basic Spectroscopic Theory
The purpose of this section is to give readers access to the most important relationships between the spectroscopy and the quantitative measurement of molecular species in mixtures. The theory aims to form a relationship between the composition of a gas mixture and the peak intensity αmax of a spectral transition. The value of αmax is influenced by pressure broadening, temperature effects and power saturation, and takes on different forms with respect to the size and structure of the molecule. These factors all influence the spectral line shape that is directly related to the fractional abundance of that absorbing species in the sample being measured. The theory also permits at least semi-quantitative design of experimental parameters such as optimum working pressure, measurement frequencies for optimum sensitivity and location with concomitant gas absorption lines. It will help understanding of some design requirements and limitations of the technique. Most of all it is intended to remove some of the mystique from the subject. The initial goal is therefore to equate the maximum absorption coefficient αmax to the species' spectroscopic parameters and fractional abundance. It will be approached from first principles, with an eye focused always on the quantitative aspects of the theory.
The main difference between optical and MMW spectrometry lies in the fact that energies involved in the rotational transitions are ~ 100-1000 times less than those involved in the usually more familiar vibrational and electronic transitions of molecular and atomic species.
The Boltzmann equation exponent hv/kT at ambient temperatures is approximately 4.8 X 10-3 σ/cm-1 or 1.6 X 10-4 v/GHz. Thus for rotational energy levels lying typically in the tens-hundreds cm-1 region* above the ground state the exponential term is not much less than unity, indicating that most levels will hold a significant fraction of the overall population of species. This has mostly positive consequences through rich well-distributed spectra and a choice of frequency region for observation of particular gas mixtures.
If a molecule possessing two rotational energy levels E described by the quantum numbers m for the lower level and n for the upper level, is exposed to radiation of frequency v where
hv = En - Em (1.1)
and h is Planck's constant, a transition will take place with probability
Pm [right arrow] n = p(v)Bm [right arrow] n (1.2)
where pm [right arrow] n/s-1 is the rate of change of the probability that the molecule will be found in the upper state; p(v) is the energy density of the radiation per unit frequency interval/J m-3 s and Bm [right arrow] n/J-1 is the Einstein coefficient of absorption for that transition.
In the excited state n the interaction of the radiation field with the molecule induces emission, and the molecule relaxes to the lower state m. The induced emission is indistinguishable from the field that caused it. Furthermore, it is of no use analytically unless the excitation source radiation is interrupted before the upper state population has had time to relax. That becomes the case in Pulsed Fourier Transform microwave spectroscopy where the excitation source is pulse modulated and the induced radiation is emitted against a very low microwave background.
The Einstein coefficients for induced emission and absorption are identical Bmn and can be expressed as
Bmn = (2π2/3ε0h2) [|(m|µx|n)|2 + |{m\µy\n)|2 + ({m|µz|n)|2] (1.3)
The terms in µmn/C m are the matrix elements of the molecule's dipole moment projected onto axes x, y, z fixed in space rather than onto the molecule itself; ε0/J-1C2 m-1 is the permittivity of free space.
In MMW spectrometry the radiation is commonly plane polarised as it passes through the absorbing gas. This effect can give rise to alterations in the intensity of observed spectral transitions depending upon the extent of that polarisation and its relationship with other electrical and magnetic fields that may be introduced. Indeed, the effect is exploited to great advantage in Zeeman magnetic field modulation for the determination of radicals, and for Stark electric field modulation of the absorption line frequency; see Chapter 5. The Zeeman effect is noticeable in oxygen, where the energy levels are split in the Earth's magnetic field, and the intensity of the split transitions can be altered depending upon the orientation of the cavity in the laboratory, and the polarisation of the MMW radiation. This analytically unwelcome splitting can be easily overcome by magnetic shielding of the cavity (Figure 1.2).
Another route to depopulation of the upper level is spontaneous emission, which is described by an analogous spontaneous emission coefficient Am [left arrow] n/s-1. This mechanism is the basis of atomic emission spectrometry in the ultraviolet-visible spectrum particularly. In the MMW region with its much lower frequency transitions it is not a significant contributor, as will be demonstrated below. Modifying Equation 1.2 to bring in the populations N of the upper and lower states one can write for the absorption and emission processes
d/dt(ΔNm [right arrow] n) = Nm Bmnρ(v) (1.4)
and for the reverse process
d/dt(ΔNm[left arrow]n) = Nn [Bmnρ(v) + Am[left arrow]n] (1.5)
If the two processes are in thermodynamic equilibrium these rates will be equal. The ratio of the populations of the two levels NnNm will equate to the Boltzmann relationship and conform to Planck's radiation law. From this can be derived the relationship
Am[left arrow]n/Bmn = 8πhv3 /c3 /J sm-3 (1.6)
The strong frequency dependence of A/B shows why even though spontaneous emission is analytically useful in the ultraviolet-visible spectrum, it is not at MMW frequencies. At 300 nm the ratio is 6.17 X 10-13 Jsm-3 whereas at 300 GHz it is 1.67 X 1023 Jsm-3, with typical lifetimes 1/Amn of 10-8 s for ultraviolet visible, and 100 s for MMW emitter upper states.
It is normal therefore to work in the absorption mode to make quantitative measurements at MMW frequencies and the absorption coefficient α can be derived from a consideration of the power absorbed by a volume V of gas containing N molecules per unit volume. The number of molecules per second undergoing the transition m [right arrow] n is the product of the total number of molecules and the probability of the transition taking place pm [right arrow] n. As each photon has an energy hv the total power absorbed from the field is, from Equation 1.2:
Pm [right arrow] n = VNmρ(v) Bmnhv (1.7)
The radiation field also stimulates emission from the upper state n and the radiation returned to the field by this process is described by the analogous equation
Pm[left arrow]n = VNnρ{v)Bmnhv (1.8)
and the nett change in power in the radiation field will be given by subtracting Equations 1.8-1.7:
ΔP = V(Nn - Nmρ(v)Bmnhv (1.9)
This equality underlines the important feature that the power absorbed is directly proportional to the difference in populations of the two levels involved in the transition (Nn - Nm). That has some important implications for MMW spectro-metry in the light of both the absolute values of the energy levels and the difference in energy between them, as is discussed below.
Under conditions of local thermodynamic equilibrium (LTE) thermal relaxation processes will be rapid and maintain the population of the lower level. If, however, one were able to depopulate the upper level n compared with its LTE population, the absorption signals would increase. The technique of double-resonance spectrometry exploits this by application of an intense MMW or laser field that excites a transition, e.g. n [right arrow] q, where q is a higher state (Figure 1.3). This depletes the population of n and permits greater absorption of a second MMW field, exciting the transitions m right arrow] or q [right arrow] r.
Freezing the gas to a low temperature increases the population of state m with respect to n both in absolute and relative terms. This is of particular consequence as levels nearer to the ground state are measured. Significant sensitivity enhancement can be obtained by freezing the gas down to a few K rotational temperature as the absorption measurement is carried out in, for example, a supersonic jet. There is a trade-off, however, between population increase of the lower states by depopulation of the higher states, because the intermediate states' populations too will increase as the rotational temperature falls. This is reached in OCS, for example, at j = 5 and j = 6 at 4 K.
The term (Nn - Nm) can be expressed using the Boltzmann equation, assuming LTE:
Nn - Nm = Nm[1 - exp(-hv/kT)] (1.10)
Equation 1.9 becomes
ΔP = VNm[1 - exp(-hv/kT)] ρ(v)Bmnhv (1.11)
Equation 1.11 can be made more practically useful by defining the absorption coefficient α as the fractional change in power per unit path-length Δl in the cell
α = - (ΔP/P)/Δl/m-1 (1.12)
The volume and power density terms can be rationalised also: a volume element in the cell can be expressed as AΔl where A is the cross-sectional area of the element. The energy density ρ(v) can be expressed in terms of A and P by considering the wave-front from a monochromatic source propagating through the cell at velocity c:
ρ = P/cA (1.13)
The term Nm is more usefully expressed as the total concentration of molecules of that species N multiplied by the fraction in the lower state of the MMW transition fm. Combining Equations 1.11, 1.12 and 1.13 yields
αmn = (Nfm/c)[1 - exp (-hv/kT)]Bmnhv (1.14)
The exponential term can be expanded because hv < kT in the longer MMW region:
hv/kT = 0.048v(/GHz)/7(/K) (1.15)
although at higher frequencies and low temperatures, this assumption should not be automatic: at 100 GHz and 10 K, hv/kT = 0.48.
Retaining the second term of the expansion for inspection shows that at low temperature and high frequency measurements, it can have a significant influence on the calculated absorption coefficient:
αmn = Nfm(hv)2/ckT (1 1/2 hv/kT + ...)Bmn (1.16)
but for most practical purposes it can be truncated at the first term of the expansion
αmn ≈ Nfm (hv)2/ckT Bmn (1.17)
Equations 1.16 and 1.17 are based on the assumption that all the molecules undergoing the transition m [right arrow] n do so at the same frequency v. In reality they will have slightly different transition frequencies centred around the centre frequency v0 due predominately to collisional interactions between molecules. Doppler broadening also makes a small contribution giving a Gaussian shape to the line (Figure 1.4), but the overall result is a profile approximated by the Lorentz shape function S(v):
S(v) = Δv/π[(v0 - v)2 + Δ v2] (1.18)
where v is close to v0 and Δv is the half-width of the spectral profile measured at the half-intensity points (HWHM).
Bmn in Equation 1.17 can be replaced in terms of the sum of the squares of the dipole moment matrix elements (Equation 1.3) and the equation modified by the Lorentz function, yielding the general form of the equation for αv the absorption coefficient at any point on the spectral profile:
αv = 2πNfmv2/3 ε0ckT µ2mn Δv/(v0 - v)2 + Δv2 (1.19)
The integrated line absorption intensity I
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20)
is independent of line width. Integrated absorption intensity measurements should and largely do compensate for the varying broadening effects that may occur as the composition of the analyte mixture changes, and are the norm for quantitative MMW spectrometry.
When v = v0 the centre frequency and maximum of the line profile, the peak absorption coefficient αmax is given by
αmax = 2πNfmv2/3 ε0ckT Δv µ2mn (1.21)
The expression above must be summed over all possible transitions between states m and n to obtain a total absorption coefficient. In rotational spectroscopy, every transition originating in state J consists of 2 J + 1 overlapping transitions whose magnetic quantum number M ranges from -J to + J, but whose frequencies are identical. For plane polarised MMW interacting with all molecules, the relative strength of each M component works out as 1 - M2/(J + 1)2 and, after summation, their nett contribution to the transition dipole strength becomes (J + 1)µ2, where ITLµITL is the molecular dipole moment.
