Power spectrum Lorentzian

An important consequence of the lineshape theory discussed above concerns the effect of the bath dynamics on the linewidths of spectral lines. We have already seen this in the discussion of Section 7.5.4, where a Gaussian power spectrum has evolved into a Lorentzian when the timescale associated with random frequency modulations became fast. Let us see how this effect appears in the context of our present discussion based on the Bloch-Redfield theory. 

Therefore, the power spectrum is a sum of two Lorentzians which are shifted in frequency, but their half-width remains determined by the translational diffusion coefficient. The autocorrelahon function in this case is... 

Fig. 5.8.1. Power spectrum of laser light scattered from silver chloride colloids in water. The circles are for the actual experimental data. The solid line is the sum of two Lorentzians, one centered at zero frequency, the second one at 1.97 kHz. The half-width of the first Loren-tzian is 0.35 kHz and that of the second, 0.29 kHz. The instrumental width is 50 Hz. (From Ben-Yosef et al., 1972.)...

When there are more than one species present in a sample, each contributes to give a correlation function which is a sum of exponentials and a power spectrum which is a sum of Lorentzians. The intensity of each component in the composite function is proportional to the product of the molecular weight and the concentration in (w/v) units, assuming the refractive index increments of each component are identical. Two basic approaches are available to extract the particle distribution from the QLLS data. 

Equation (7.68) reveals that the power spectrum has a peak at co = 0 [5(ct) = 0) = 1], giving the dc part 2(/). The first term e/7i) i) represents the shot-noise term, and the third term describes a Lorentzian frequency distribution peaked at co = 0 with the total power (2l7ty) i). This represents the light-beating spectrum, which gives information on the intensity profile I (co) of the incident light wave. 

In a typical laser, it is the rate of the phase fluctuations that determines the laser linewidth. In this case, the field power spectrum is Lorentzian with a FWHM linewidth given in terms of SL experimental parameters by... 

The variance of the recorded fluctuations is equal to the integral of the power spectrum over the recording bandwidth and it is measured with much greater accuracy than the spectral distribution However, undesired noise contributions cannot be easily subtracted from the total variance without the help of the spectral analysis. Furthermore, the measured variance is an underestimation of the actual variance, due to bandwidth limitations (the integral of Lorentzian spectrum up to its cut-off frequency yields only 50% of the variance). 

This equation holds only for an optical electric field which is Gaussian and which possesses an exponential autocorrelatitxi function, i.e., Eq. (66). If the field is non-Gaussian there is no simple relation between the optical spectrum lio)) and the power spectrum. Eq. (69) has three components (1) a shot noise term e(S ln which is independoit of the frequency (Le., white noise), (2) a d.a photocurrent 6(m) which is essentially infinite at extremely low fiequendes (i.e., d.c.) and a light beating spectrum which for an exponential autocorrelation function and Gaussian optical field is a Lorentzian of half width, IF. Fig. 4 shows the experimental data of Benedek et al with calculated points and observed line shape. What is not shown is the infinite d.c. photocurrent at ft)=0. These measurements were obtained by use of a spectrum analyzer which measures directly the power scattered at each frequency. 

In ideal situations, there are two types of motion for particles in an electric field the jostling motion of random Brownian diffusion and an oriented electrophoretic motion. In the power spectrum, the Brownian motion is characterized by a Lorentzian peak centered about the frequency shift produced by electrophoretic motion. Analogously, in the ACF, electrophoretic motion is a... 

In an ELS experiment, there are two types of motions that produce fight intensity fluctuations and frequency shifts in the scattered light i.e., the random Brownian motion and the oriented electrophoretic motion of suspended particles, if the scattering from medium is discounted. The effect of Brownian motion is characterized by a Lorentzian peak in the power spectrum (frequency... 

Electrophoretic applications of this method depend on the known inverse relationship between the first-order correlation function and the power spectrum. (They are a Fourier transform pair.) With random motion the frequencies of the scattered light spread about the incident lighL producing a Lorentzian distribution (the center being at the frequency of the incident light the spectrum is known as a Rayleigh line to distinguish it from other spectral... 

Force calibration is commonly not directly done, but trap stiffness is determined. Then, the force is calculated by multiplying trap stiffness by particle displacement from the trap center. The first method involves again measuring the power spectrum of thermal motion of a bead in the harmonic potential, which is described by a Lorentzian as given by Eq. (3.20). By fitting the power spectrum and using the relation... 

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