Lifetime broadening

We next evaluate the lineshape function (8.16) for two concrete situations in gases, (The complexity of molecular motions in liquids precludes computation of their dipole correlation functions in a text of this scope.) In the first situation, we imagine that we are examining lineshapes in the far-infrared spectrum of a collision-free, rotating polar molecule. Its dipole moment /Iq is assumed to rotate classically without interruption with angular frequency cwq about an axis normal to /Iq. In a dilute gas, we would then have 

These two delta functions correspond to absorption and emission of radiation at frequency c o, respectively, with spectral lineshapes exhibiting zero full width at half maximum (fwhm). Such uninterrupted molecular rotation, in which the dipole correlation function (8.17) maintains perfect sinusoidal coherence for an indefinite period of time, produces no broadening in the lineshape function I co). 

Suppose now that we have the more realistic correlation function 

This would represent a molecule rotating classically with frequency (Oq, with a transition moment that decays exponentially with 1/e lifetime x = 2/y following excitation at time t = 0. Such decay may occur via radiationless transition, spontaneous emission (Section 8.4), or collisional deactivation in the excited state. The corresponding (real) lineshape function is 

Like Eq. 8.18, this expression exhibits terms corresponding to absorption and emission, respectively. The normalized absorption lineshape function 

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Equation B2.5.48 introduces the effective average collision cross section Here, the lifetime broadening results from the (coiiisionai) perttirbation of A by collisions with M. 

Moderately slow exchange. The state lifetime is 2t we ask how the absorption band is affected as this becomes smaller. The uncertainty principle argument given earlier is applicable here lifetime broadening will occur as the state lifetime decreases. Thus, we expect resonance absorption at (or near) frequencies Va nnd Vb but the bands will be broader than in the very slow exchange limit. Equation (4-68) is applicable in this regime. 

Fig. 5. Effective g assignment of the low-field S = IEPR signals in D. vulgaris Fepr protein [from 11)]. The spectrum was recorded at the optimEd temperature of 12 K, that is, at which the amplitude is maximal and lifetime broadening is not significEmt. EPR conditions microwave frequency, 9.33 GHz microwave power, 80 mW modulation amplitude, 0.8 mT.

On the experimental side the observed thermally activated transition from Mu to Mu (Holzschuh et al., 1982 Odermatt et al., 1988) is important in two respects. First, it demonstrates conclusively that, at least in diamond, Mu is metastable with Mu being the stable configuration. Figure 10 shows the lifetime broadening of the Mu signal (depolarization... 

Another unusual feature of CuCl and CuBr is the presence of two Mu centers with nearly identical isotropic hyperfine parameters. One of the centers, Mu7, occurs preferentially at low temperatures but is metastable as evidenced by a thermally activated transition to the second center, Mu77 (see Fig. 13). As the temperature increases, the effects of this transition first appear as an increse of the Mu7 depolarization rate (lifetime broadening). At higher temperatures the transition becomes fast enough so that... 

This is known as the inhomogeneous limit (convoluted with Lorentzian lifetime broadening). 

The ultimate (minimum) linewidth of an optical band is due to the natural or lifetime broadening. This broadening arises from the Heisenberg s uncertainty principle, AvAt < U2jt, Av being the full frequency width at half maximum of the transition and the time available to measure the frequency of the transition (basically, the life-... 

Another, less straightforward way to determine the vibrational lifetime is by studies of the infrared absorption peak shape. Consider a single adsorbed molecule at 0 K. The width of the peak is then determined by the lifetime broadening and in the first approximation it has a Lorentaan shape with a full width at half maximum (FWHM) A = (2nct), t then being the lifetime. However, as usual we have to consider an ensemble of molecules at finite temperatures and then there exist other peak broadening mechanisms that must be taken into account. 

Fig. IX The calculated infrared absorption peak, assuming lifetime broadening and vibrational damping via local charge oscillations, (a) Fitted to the experimental spectrum in Fig. 11 for an ordered overlayer incorporating the dipole-dipole interaction, (b) The same calculation for a single adsorb molecule. (Reproduced by poTtassUm from Crljen and Langreth. )...

To conclude, even if there exist several processes that affect the vibrational line shape it seems probable that when most of them have been sorted out and with the good agreement between theory and experiment, the lifetime broadening for a chemisorbed CO molecule is of the order of a few cm, corresponding to a lifetime of a few ps. The main vibrational energy relaxation mechanism is creation of electron-hole pairs caused by the local charge oscillations between the metal and the 2n molecular resonance crossing the Fermi level. 

If the thermal relaxation time Ti is sufficiently short, this time may determine the width of the resonance line. This effect is referred to as lifetime broadening. The Uncertainty Principle expressed in terms of energy and time is as follows... 

Two technical questions naturally arise (i) are all the carbons counted and (ii) what limits resolution In liquids all carbons are represented provided that the repetition period is substantially longer than the longest carbon T c Resolution is generally restricted by static field inhomogeneity or by lifetime broadening. Circumstances in solids are less clearcut. 

We then compare the measured linewidth to that implied by the "lifetime broadening" of at an rf field of 66 kHz and static field of 15 MHz for the piperidine cured DGEBA epoxy at 33°C. 

If highest resolution is required, then the strategy is to use thermal activation to stir away the conformatonal anisotropy operate as close to the glass transition temperature as possible, up to the point at which Ti lifetime broadening predominates. On the other hand, at lower temperatures the lineshape and its temperature dependence may provide useful information (34). 

For metal/semiconductor interfaces the limitations coming from 1) are not very severe since the cross sections for photoemission and the lifetime broadening of such deep state photopeaks along with the poor energy resolution of the photon source... 

The frequency-resolved spectrum, at the right of Figure 2.2, shows that the single ZOBS transition is split into a multiplet, due to the coupling with the ZODS, which is lifetime broadened, exhibiting a linewidth F due to interaction with the bath states. The observation of such a spectrum provides evidence for IVR. The lifetime-imposed linewidth is proportional to the product of the mean-squared... 

As the temperature is lowered, the line width may significantly decrease and then approach a lower limit. This sharpening also increases the apparent intensity of the signal. The increased line width at high temperature is usually attributable to lifetime broadening. If many thermally accessible states are present, lowering... 

L. Asplund, U. Gelius, S. Hedman, K. Helenelund, K. Siegbahn, P.E.M. Siegbahn, Vibrational structure and lifetime broadening in core-ionised methane, J. Phys. B 18 (1985) 1569. 

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