Sum-combination frequency

The lack of combination frequencies in three-pulse ESEEM may sometimes be detrimental. If the isotropic and dipolar hyperfine couplings are much smaller than V/, the maximum of the sum combination frequency v+ is given below. ... 

Since V/ is known, one can determine Vdd, and hence the distance between the electron and nuclear spin, even in the presence of small, unknown isotropic hyperfine couplings. The second-order shift with respect to twice the nuclear Zeeman frequency is small. Hence, two-pulse ESEEM with its inferior resolution is not well suited for measuring this shift. The sum combination frequency can be introduced into stimulated-echo ESEEM by inserting an mw tz pulse halfway through the evolution period of length T (sequence in Fig. 11 with fi = 2 = T/2). 

For sufficiently strong m.w. fields the terms containing the nonsecular hyperfine coupling B can be neglected and the sum-combination frequencies become twice the completely decoupled frequencies. 

Theory predicts that for a harmonic oscillator only a change from one vibrational energy level to the next higher is allowed, but for anharmonic oscillators weaker transitions to higher vibrational energy levels can occur. The resulting "overtones" are found at approximate multiples of the frequency of the fundamental. Combination frequencies representing sums... 

Equation (4) shows that modulations of the two-pulse echo amplitude occur at the fundamental hyperfine frequencies and at their sum and difference combination frequencies. The amplitude of the modulations is given by the product of the transition probabilities for the two different transitions associated with branching , Mp wp, while the nonmodulated portion of the echo envelope depends only on the product of the transition probabilities for the nonbranching spins, m " or vf. Substituting the expressions for u and w given by equations (2) and (3) into equation (4) yields... 

The spectra observed in this region involve mainly hydrogen stretching vibrations in, for example, C—H, N—H, and O—H bonds. A fundamental absorption band at a given frequency may be accompanied by bands at all multiples of this frequency. These additional bands are called overtones. The first overtone is much weaker than the fundamental, and successive overtones are progressively weaker still. Absorption bands may also occur at a frequency which is the sum or difference of two fundamental frequencies, or the sum or difference of an overtone and a fundamental frequency. These are called combination frequencies. Overtone and combination bands are most readily observed on comparatively thick samples in the region between the visible and 2.5 pi, where there are no fundamental absorption bands, and all the bands arise from this cause. 

They naturally are much weaker in intensity than the fundamental frequencies. In molecules with a centre of symmetry a number of harmonic and combination frequencies, just as some of the fundamental frequencies, will be inactive as may be seen from a Fourier analysis of the motion. More precisely, in the vibration spectrum those frequencies are inactive for which the sum of the coefficients k dn equation (35 1) belonging... 

For purely harmonic vibrations, overtone or combination frequencies would be exact sums of the constituent fundamental frequencies, but the effect of anharmonicity is to change (usually to reduce) the frequency of an overtone or combination (Section 8.2.2). Difference bands (v,- — v ) behave exactly like the corresponding combination bands (v, + v ) so far as symmetry selection rules, band contours and polarization are concerned, but the energy levels involved are the same as those involved in fundamental transitions, so the difference frequency is precisely equal to the difference between the frequencies of the corresponding fundamentals. For hot bands, one of the effects of anharmonicity is that they do not have exactly the same frequency as the parent transition. 

Equation 14 consists of an unmodulated part with amplitude 1 - U2, the basic frequencies and cop with amplitudes kJl, and the combination frequencies < and w+ with amplitudes k 4, and inverted phase. To compute the frequency-domain spectram, first the unmodulated part is subtracted, as it gives a dominant peak at zero frequency for the usual case of small k values. A cosine Fourier transform (FT) of the time trace results in a spectrum that contains the two nuclear frequencies, w and cop, with positive intensity, and their sum and difference frequencies, a>+ and m, with negative intensity. If the initial part of the time-domain trace is missing, then the spectrum can be severely distorted by frequency-dependent phase shifts and it may be best to FT the time-domain trace and compute the magnitude spectrum. 

When T is varied the echo envelope is modulated only by the two basic frequencies CDa and (Up, the sum and difference frequencies do not appear, in contrast to the two-pulse ESEEM experiment. This is usually advantageous, as it simplifies spectra, but it may also be a disadvantage for disordered systems where the sum-combination line is often the only narrow feature in the ESEEM spectrum. Another important difference is the dependence of the three-pulse ESEEM amplitudes on r, as is apparent from Eq. (17) by the factors 1 - cos(copr) and 1 - cos(cOcir). Due to this suppression effect, individual peaks in the spectrum can disappear completely. These blind spots occur for the a(P) peak when r = 2n /(Up(a) (k = 1, 2,. ..). In principle they can be avoided by using r < Inlco, where (Umax is the maximum nuclear frequency however, this is usually precluded by the spectrometer deadtime. Consequently, the three-pulse ESEEM experiment has to be performed at several r values to avoid misinterpretation of the spectra due to blind-spot artifacts. 

Nondegenerate optical wave mixing, where the frequencies of the incident and the generated waves are different, are due to nonlinear polarization of the form given by Equation (11.35). For example, the incident frequencies cdi, CO2, and CO3 can be combined to create new frequencies 0)4 = cdi CO2 0)3 involving sums or differences. These wave mixing processes are sometimes termed sum-difference frequency generations. 

The ESEEM spectrum is substantially better resolved for stimulated echoes than for the two-pulse echo, because it does not contain combination frequencies (sums and differences of the basic resonance frequencies), and the lines are not broadened by fast transverse spin relaxation. Thus stimulated ESEEM is the more preferable for studying electron-nuclear interactions. 

Figure Bl.5.15 SFG spectrum for the water/air interface at 40 °C using the ssp polarization combination (s-, s- and p-polarized sum-frequency signal, visible input and infrared input beams, respectively). The peaks correspond to OH stretching modes. (After [ ].)...

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