Inversion spectra

For many molecules the potential energy surface comprises two symmetrically located wells separated by a barrier. The most vivid example is the molecule of ammonia, NH3. Two equilibrium configurations are possible for this molecule which are obtained from each other by means of the inversion of the N atom with respect to the plane of H3. The potential energy surface of the ammonia molecule as a function of the distance between the nitrogen atom and the plane of H3, is presented in Fig. 31. In this figure the top of the potential barrier corresponds to the location of the N atom within the plane of H3. 

In this formula, y is the vibration frequency in a separate potential well and p is the momentum within the region where classic motion is forbidden. The physical meaning of this formula is simple. The splitting energy is determined chiefly by the coordinate region between the potential wells and, consequently, is proportional to exp( - j p dx). The magnitude of the preexponential factor can be further determined (with an accuracy of up to n) on the basis of the dimensional consideration. 

Numerous works have been dedicated to the experimental study of the vibrational and rotational spectra of ammonia, starting from the studies by Cleeton and Williams [84] who were pioneers of microwave spectroscopy. It has been determined that the splitting of energies is the greatest for the fully symmetric deformational vibrations with the frequency of 950 cm 1. Evidently, these vibrations (Fig. 32) provide the most direct way from one potential well to another. The splitting of the fully symmetric deformational vibra- 

Inversion splitting of the vibrational spectrum of ammonia has been used to create the first molecular microwave amplifier (maser) [86, 87]. The inversion population in the ammonia maser is achieved by transmission of the molecular beam through a non-homogeneous electric field. Ammonia molecules in symmetric and antisymmetric states interact with the electric field in different ways and they are therefore separated in this field. They are then directed to the resonator. 

Besides ammonia, the inversion splitting occurs in other molecules of similar type (e.g. PH3, AsH3, etc.). The splitting, however, is substantially less in the latter case. For example, it has been estimated [88] that, in the case of the AsH3 molecule, the inversion caused by tunneling occurs approximately once in two years which, is, of course, too rare to observe by spectroscopic measurements. 

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Finally, we have shown that the inversion spectra of polyatomic molecules, such as NH3 and ND3, are potentially even more sensitive to the time variation of p. This has already been used in astrophysical measurements to put the most stringent limit. Equation 16.36, on the time variation of p on a cosmological timescale. The corresponding laboratory experiments would require slow molecules, and molecular fountains and traps hold the promise of providing them [46]. 

Of special interest to chemists is the tunneling of H in NHj, which yields the characteristic inversion spectra. The appearance of these spectra can be only explained by tunneling. This is regarded as a great achievement of quantum mechanics. " ... 

Secondly, although stable solutions covering the entire temporal range of interest are attainable, the spectra may not be well resolved that is, for a given dataset and noise, a limit exists on the smallest resolvable structure (or separation of structures) in the Laplace inversion spectrum [54]. Estimates can be made on this resolution parameter based on a singular-value decomposition analysis of K and the signal-to-noise ratio of the data [56], It is important to keep in mind the concept of the spectral resolution in order to interpret the LI results, such as DDIF, properly. 

Fig. 17. Energy levels of the rotation-inversion spectrum of ammonia. The quantum numbers (J,K) are given for each level. The heavy arrows indicate the inversion transitions detected in interstellar space and their frequencies in MHz. Thin arrows indicate the rotation-inversion transitions located in the submillimeter wave region. Dashed arrows indicate some collision induced transitions...

From its inception, microwave rotational spectroscopy has contributed greatly to our knowledge about classical inorganic compounds. It all began with a low resolution recording of the ammonia inversion spectrum in 1934. The first high resolution microwave spectra were recorded... 

The high order one-dimensional approach was first used to consider the inversion spectrum of ammonia along the v2 inversion axis [45]. A similar calculation has been used to examine the influence of solvent on the splitting due to tunneling when a hydrogen atom, for example, can move between two positions of equilibrium [46],... 

The micro-wave spectrum is the pure inversion spectrum between the two ground states. 

The inversion spectrum of the J,K) = 1,1) level of ND3 consists of 72 hyperfine transitions in a frequency interval of about 300 kHz. Due to this spectral congestion, the hyperfine structure could not be resolved in an earlier molecular beam experiment [71]. We have therefore carried out this experiment with a Stark-decelerated molecular beam, using the setup shown schematically in Figure 14.19a. A beam of ammonia molecules is decelerated from 280 m/sec to either 100 or 50 m/sec, and focused into a microwave zone. The microwave zone provides a nearly rectangular... 

LIMIT ON THE TIME VARIATION OF Ji FROM THE INVERSION SPECTRUM OF AMMONIA... 

On the other hand, an only slightly smaller enhancement exists for the inversion spectrum of NH3, often observed in astrophysics even for high z objects. This spectrum was used in Ref. [3] to obtain limit 16.1, which we will now discuss in more detail. 

In addition to the rotational structure 16.20, the inversion spectrum has a hyper-fine structure. For the main nitrogen isotope thehyperfine structure is dominated by the electric quadrupole interaction ( 1 MHz) [69]. Because of the dipole selection rule, AAl = 0 and the levels with 7 = AT are metastable. In beam experiments, the width of the corresponding inversion lines is usually determined by collisional broadening. In astrophysical observations, the lines with 7 = AT are also narrower and stronger than others, but the hyperfine structure of the spectra with high redshifts is unresolved. 

The inversion spectrum of Equation 16.20 can be approximately described by the Hamiltonian... 

We see from Equations 16.25 and 16.30 that the inversion frequency and the rotational intervals (Oinv(7i,ATi) — (Oinv(-/2. 2) have different dependencies on p. In principle, this allows one to study time variation of p by comparing different intervals in the inversion spectrum of ammonia. For example, if we compare the rotational interval to the inversion frequency, then Equations 16.25 and 16.30 give... 

Again, as in the case of A-doubling in the OH molecule, it is more promising to compare the inversion spectrum of NH3 with the rotational spectra of other molecules, where... 

Inversion chem Change of a compound into an isomeric form, j in var zhan ) Inversion spectrum spect Lines in the microwave spectra of certain molecules (such... 

In a coupled spin system, the number of observed lines in a spectrum does not match the number of independent z magnetizations and, fiirthennore, the spectra depend on the flip angle of the pulse used to observe them. Because of the complicated spectroscopy of homonuclear coupled spins, it is only recently that selective inversions in simple coupled spin systems [23] have been studied. This means that slow chemical exchange can be studied using proton spectra without the requirement of single characteristic peaks, such as methyl groups. 

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