Reciprocal spectral width

X is the wavelength, and q is the refractive index. The correlation time Tc has been defined as the reciprocal spectral width so as to be equal to the FWHM width in the case of transform-limited pulse. 

T2 transvers relaxation time) The value Tz/Tj- = 5 has been assumed, where T(- is the correlation time (reciprocal spectral width) of light sources. The coherence parameter P represents the extent of random phase distribution, and the dispersion parameter W represents the degree of regular phase-modulation due to material dispersion (see text for definition). The cross relaxation effect has been neglected. 

Phase-modulated coherent pulses are promising for observing phase relaxation effects in view of the possibility of having very broad-band in a controlled manner, though ultimate time-resolution does not strictly reach to the correlation time (reciprocal spectral width). This type of light source is, however, ineffective for the purpose of observing energy relaxation effects. 

The laser used to generate the pump and probe pulses must have appropriate characteristics in both the time and the frequency domains as well as suitable pulse power and repetition rates. The time and frequency domains are related through the Fourier transform relationship that hmits the shortness of the laser pulse time duration and the spectral resolution in reciprocal centimeters. The limitation has its basis in the Heisenberg uncertainty principle. The shorter pulse that has better time resolution has a broader band of wavelengths associated with it, and therefore a poorer spectral resolution. For a 1-ps, sech -shaped pulse, the minimum spectral width is 10.5 cm. The pulse width cannot be <10 ps for a spectral resolution of 1 cm . An optimal choice of time duration and spectral bandwidth are 3.2 ps and 3.5 cm. The pump pulse typically is in the UV region. The probe pulse may also be in the UV region if the signal/noise enhancements of resonance Raman... 

Here <( t ) f(t")> is the autocorrelation function of the electromagnetic field. For the case of excitation by a conventional light source, where the amplitudes and the phases of the field are subject to random fluctuations, the field autocorrelation function differs from zero for time intervals shorter than the reciprocal width of the exciting source. In the limit 8v A, that is when the spectral width, 8v, of the source exceeds the inhomogenously broadened line width, the field autocorrelation function can be represented as a delta function... 

Precisely controllable rf pulse generation is another essential component of the spectrometer. A short, high power radio frequency pulse, referred to as the B field, is used to simultaneously excite all nuclei at the T,arm or frequencies. The B field should ideally be uniform throughout the sample region and be on the order of 10 ]ls or less for the 90° pulse. The width, in Hertz, of the irradiated spectral window is equal to the reciprocal of the 360° pulse duration. This can be used to determine the limitations of the sweep width (SW) irradiated. For example, with a 90° hard pulse of 5 ]ls, one can observe a 50-kHz window a soft pulse of 50 ms irradiates a 5-Hz window. The primary requirements for rf transmitters are high power, fast switching, sharp pulses, variable power output, and accurate control of the phase. 

To consider gas molecules as isolated from interactions with their neighbors is often a useless approximation. When the gas has finite pressure, the molecules do in fact collide. When natural and collision broadening effects are combined, the line shape that results is also a lorentzian, but with a modified half-width at half maximum (HWHM). Twice the reciprocal of the mean time between collisions must be added to the sum of the natural lifetime reciprocals to obtain the new half-width. We may summarize by writing the probability per unit frequency of a transition at a frequency v for the combined natural and collision broadening of spectral lines for a gas under pressure ... 

At constant temperature, the observed widths of the spectral functions decrease with increasing mass of the collisional pair. This fact is a simple consequence of the mean translational energy of a pair, jm v = kT, which is the same for all pairs. The interaction time is roughly proportional to the reciprocal root mean square speed, and thus to the square root of the reduced mass. 

Spatial resolution is one channel which is typically 25 ym wide. Spectral resolution is the product of the channel width and the reciprocal dispersion of the spectrometer. For example, a spectrometer with a focal length of 0.25 m and grating of 152.5 grooves/mm typically produces a reciprocal linear dispersion of 25 nm/mm. Therefore a 25 ym channel will cover 0.64 nm. A 305 g/mm grating used with the same spectrometer would produce a resolution of 0.32 nm/channel. 

The spectral resolution of a detector is defined here as equal to its spatial resolution (in urn ) times the reciprocal linear dispersion of the spectrometer (in nm/um ). It was measured to be 1.5 - 2.5 and 2-4 times poorer for the SPD and SIT, respectively, compared to that of a PMT. All measurements were performed with the same spectometer, utilizing 20 urn slit widths. Because, the proximity focused, microchannel plate (MCP) intensi-fier broadens the line images, the spectral resolution of the ISPD was found to be significantly worse than that of the SPD. Peak widths measured at half maximum intensity were four diodes wide even when only a single diode width was illuminated. 

Linear dispersion defines the extent to which a spectral interval is spread out across the focal field and is expressed in nanometre per millimetre (nm/mm) -(or its inverse, the reciprocal dispersion in mm/nm). Linear dispersion is associated with an instrument s ability to resolve fine spectral detail. It depends of several parameters such as the focal distances and the widths of entrance and exit slits of the instrument. In general the better the dispersion the greater is the physical separation distance between two given wavelengths (Figure 14.11). 

Nmr relaxation measurements allow the molecular motions of the segments of a polymer molecule to be investigated. The spectral line width, which is proportional to I/T2 (i.e. the reciprocal of the spin-spin relaxation time of proton nmr) is a direct measure of segment mobility. It would be anticipated that the anchor segments in the trains attached to the surface of the adsorbent particle would display low mobility. On the other hand, those segments in loops and tails that project into the continuous phase should possess much higher mobility. Two distinct linewidths would therefore be expected for the two different types of segments. 

The reciprocal linear dispersion dX/dx) is the function of the geometric slit width S) and spectral bandpass (AA ,) of the monochromator ... 

The reciprocal linear dispersion for a grating is nearly constant over the entire wavelength region and it is dependent on the number of grooves per unit width, spectral order, and the focal length of the collimator. The resolution of a grating is a function of spectral order (m) and the total number of grooves N) ... 

Figure 12 Graph allowing calculation of the error In measured peak height for a given monochromator spectral bandwidth and natural bandwidth of an absorption band. Spectral bandwidth can be obtained from manufacturers Information or by knowing the physical slit width of the monochromator and the reciprocal dispersion (nm/mm). Since natural bandwidths of, for example, cytochrome absorption bands are about 10 nm at room temperature a spectral bandwidth of 2.5 nm (ratio on the abscissa of 0.25) will Introduce no more than a 3% error in measured peak height. However, for low temperature spectra, spectral bandwidths of about 0.5 nm are required.

Figure 7 A trace of the monochromator spectral bandpass, with an emission slit width of 2 mm, and a reciprocal dispersion, r of 4 nm mm". ...

The effective bandwidth (also called the spectral bandpass or spectral slit width, which is one half the bandwidth when the two slit widths arc identical, is seen to be the range of wavelengths that exit the monochromator at a given wavelength setting. The effective bandwidth can be related to the reciprocal linear dispersion by writing Equation 7-9 in the form... 

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