Lorentzian Width Research Articles

The exact expression of the Voigt profile function

We solve in exact and closed-form a classical problem of mathematical physics of great interest in spectroscopy: the convolution of a Gaussian and a Lorentzian distribution that define the Voigt profile function, V ( x ) . The solution is based in three steps: a power series development following the integral expression for V ( x ) , the ordinary differential equation (ODE) satisfied by that expansion, and the corresponding solution of the ODE. This work converts in obsolete all graphical, numerical and semi-analytical approximations published previously. All results are clearly expressed in terms of the complementary error function Φ c ( a ) = 1 - erf ( a ) , where a = ln 2 γ L / γ G is, basically, the relation between Lorentzian and Gaussian widths.

Defect dynamics observed by NMR of quadrupolar nuclei in gallium nitride.

Hexagonal and cubic polytypes of bulk gallium nitride powders are characterized by 69,71Ga and 14N MAS NMR at 11.7 T. The (corrected) 71Ga chemical shifts are 333.0 and 357.5 ppm, respectively; the corresponding 14N chemical shifts are -301.8 and -297.0 ppm (all shifts referenced to 1 M gallium nitrate). The 69,71Ga nuclear quadrupole coupling constants (NQCC) in the hexagonal form are axially symmetric and agree with previous single-crystal determinations. The 71Ga MAS NMR satellite pattern envelope of the cubic form has a large Gaussian half-height width of 297 kHz, due to nonzero NQCC values induced by defects. The 14N MAS NMR spinning sideband pattern of the cubic form has a Lorentzian envelope half-height width of 17.5 kHz for the same reason. A sample containing both phases shows an unexpected marked loss of the 71Ga MAS NMR satellite transition intensity expected for the hexagonal phase. Static 71Ga-selective Hahn spin-echo measurements at the perpendicular edge of the powder pattern for the hexagonal form in this sample show a large reduction in T2, especially at higher temperatures. The partial destruction of both spin-echoes and rotational echoes is due to a chemical-exchange type process involving sites having different NQCC values.

Temperature and Pressure Dependence of High-Resolution Air-Broadened Absorption Cross Sections of NO2(415−525 nm)

Cross sections of air-broadened NO_2 in the 415−525 nm region are reported. These are retrieved from 21 absorption spectra recorded at 0.060 cm^(-1) resolution with the McMath−Pierce Fourier Transform Spectrometer located on Kitt Peak in Arizona. The measurements are obtained for pressures (1−760 Torr) and temperatures (220−298 K) that are representative of typical tropospheric and stratospheric conditions. Two sigma uncertainty (95% confidence interval ≈ 2σ_(mean)) for the absolute absorption cross sections is below ±7% over the reported wavelength range. The average integrated intensity of all our data is 〈σ〉_(400-500 nm) = 4.53 × 10^(-17) cm^2 nm, which is within 0.2% of the averaged value from the recent literature. The wavelength (referred to vacuum) accuracy is 0.011 cm^(-1) (2.8 × 10^(-4) nm at 500 nm) and precision is 0.0022 cm^(-1) throughout the investigated wavelength range. In agreement with previous observations, high-resolution features in the NO_2 absorption spectrum display a strong pressure dependence with an effective pressure broadening parameter of 0.116 ± 0.003 cm^(-1)/atm (the rate of increase of Lorentzian half width at half-maximum with pressure). Temperature has a relatively minor effect on the shapes of individual high-resolution features, but it exerts a complex dependence on the relative line intensities. Absorption cross sections reported here represent the highest resolution data available over a substantial (>100 nm) wavelength range for quantitative analysis of NO_2 atmospheric column absorption spectra.

Frequency comb linewidth of an actively mode-locked fiber laser.

We report what is to our knowledge the first measurement of the linewidth of the frequency comb lines of a mode-locked Er-doped fiber laser. By propagating the output pulses through fiber as long as 1000 km in a modified self-heterodyne arrangement, we have measured the effective linewidth of the comb lines to be less than 12 kHz on a 5-ms time scale; the width is due primarily to frequency jitter from environmental fluctuations. Deconvolution of the spectral line shapes by use of Voigt analysis yields an upper limit of the intrinsic Lorentzian width of 3 kHz.

A study of collision broadening in the O 2A-band with the noble gases using fourier transform spectroscopy

Fourier transform spectroscopy has been used to record the absorption line shapes for rotational transitions in the Δ v=0, b 1 Σ g +←X 3 Σ g − band of O 2 in the presence of the noble gases. We determined angular momentum dependent collision broadening coefficients for each noble gas using the Lorentzian widths that were extracted from Voigt fits to the line shapes. The values were largest at low J ′′ and, on average, decreased by 39% for the highest J ′′ states. Ten “scaling/fitting laws” designed for the analysis of rotational energy transfer data were fit to the broadening cross-section data. Both the angular momentum based ECS-P and the energy based EPGL-0 descriptions accurately represented the data across the entire range of rotational levels probed in this study. Pressure-induced line shifting was also studied in the O 2–Xe collision system. The pressure shifts were all negative.

Spectral line parameters in the (2 ← 0) overtone band and the dipole moment function of the HI molecule

We present here new high-resolution experimental data on the linestrengths and pressure-broadened Lorentzian widths for the P 2(13) to R 2(12) vibration–rotation lines in the first overtone absorption band of hydrogen iodide. By combining the measured linestrengths with our previous results for the fundamental band [J. Mol. Spectrosc. 218 (2003) 75] and with other published data for the higher-overtone bands of this molecule, an improved dipole moment expansion as a function of the dimensionless reduced nuclear displacement x is obtained: μ( x)=0.4471(5)−0.0772(2) x+0.542(3) x 2−1.90(2) x 3. This experimental dipole moment function is compared with the results of a few recent non-relativistic and relativistic ab initio calculations. The agreement between theoretical and experimental Herman–Wallis coefficients for the first two vibrational bands of HI is found best as ever reported before.

Pulsed photoacoustic calibration of a differential absorption water vapor lidar

We have designed and constructed a photoacoustic (PA) cell to be used for wavelength calibration in a water vapor differential absorption lidar (DIAL) system. This system will be used to remotely determine atmospheric H2O16 concentrations. The accuracy of these measurements depends critically on the ability to tune to and detune from lines in the water vapor spectrum. Specifically, the signal-to-noise ratio of the PA signal must be strong enough to locate the center frequency of an absorption line with accuracy and repeatability better than 0.2 times the Lorentzian half width of the absorption (0.1 cm−1). So, although it represents only a peripheral part of the DIAL experiment, the calibration provided by the PA cell is critical to its overall success. In the environment of a mobile lidar system, problems with mechanical and electronic noise are potentially greater than those in a typical laboratory setting, and the cell is designed to address this situation. The PA cell utilizes a “figure-8” beam tunnel geometry which partially separates the tunnel into two subchambers, each with a microphone mounted in it. The laser beam is allowed to pass through only one of these subchambers. The microphone in the side that receives the laser beam records both PA signal and background noise, while the other microphone records only background noise. A differential lowpass filter and microphone amplifier circuit, of a design dedicated to this application, uses the output of the two microphones to subtract the background noise from the PA signal. The PA cell described within is of simple mechanical design and is constructed at very little cost.

Studies of narrow autoionizing resonances in gadolinium

The autoionization (AI) spectrum of gadolinium between the first and second limits has been investigated by triple-resonance excitation with high-resolution cw lasers. A large number of narrow AI resonances have been observed and assigned total angular momentum J values. The resonances are further divided into members of AI Rydberg series converging to the second limit or other ‘interloping’ levels. Fine structure in the Rydberg series has been identified and interpreted in terms of J c j coupling. A number of detailed studies have been performed on the interloping resonances: these include lifetime determination by line-shape analysis, isotope shifts, hyperfine structure, and photoionization saturation parameters. The electronic structure of the interloping levels is discussed in terms of these studies. Line-widths generally decrease with increasing total angular momentum and the J=7 resonances are extremely narrow with Lorentzian widths ranging from <1 MHz up to 157 MHz. The strongest resonances are found to have cross-sections of ∼10 −12 cm 2 and photoionization can be saturated with powers available from cw diode lasers.

Pressure broadening of the lowest rotational transition of [formula omitted] by N 2 and O 2

The nitrogen and oxygen pressure broadening coefficients of the J=1←0 transition of H 35 Cl at 626 GHz ( 21 cm −1 ) have been measured using a high resolution tunable far infrared sideband spectrometer. The profiles of the three 35 Cl hyperfine components of the transition were separately fitted to Voigt line shapes and the Lorentzian widths determined over a range of broadening gas pressures up to 1.1 Torr for nitrogen and 1 Torr for oxygen. The room temperature pressure broadening coefficient for nitrogen was derived as 3.46(6) MHz/Torr and for oxygen as 2.70(5) MHz/Torr.

Absorption line profiles of N 2O measured for the J=25–24 and 26–25 rotational transitions

The self- and foreign (Ar, N 2, and O 2)-pressure effects on the submillimeter-wave absorption line profiles have been investigated for the J=26–25 and 25–24 rotational transitions of N 2O using the AIST terahertz spectrometer. By fitting the observed absorption profiles with the Galatry functions, the integral intensity, line-center position, Lorentzian width and contraction parameter were determined for each absorption line. The obtained pressure-broadening coefficients are consistent with the literature values reported for infrared spectra. The pressure dependence of contraction parameter was determined for the self-effect. The line shifts were found to be small.

Polarization of the KαL1x-ray satellite in copper after 2 MeV proton impact

The KαL1satellite spectrum of 29Cuwas measured with a 5 inch Johansson crystal spectrometer after 2 MeVproton impact. The satellite spectrum was simulated by using Voigt profiles witha Lorentzian width which is obtained from the multiconfiguration Dirac–Fock(MCDF) calculations and convoluted with the Gaussian instrumental responsefunction. A unique assignment of the strongest satellite transitions to visiblestructures in the experimental spectrum was based on a comparison of thepredicted linear polarization of the transitions (using in calculations theintermediate coupling scheme) to their measured values. The simulated spectrumof the satellite transitions was fitted to the experiment. The fitted valuesof energies and relative intensities of the transitions are close to thoseobtained from the MCDF calculations. The alignment of the (1s2p)−2double-vacancy state obtained from the measured polarization ofthe satellite spectrum is given by \U0001d49c20(1P1) = −0.77(21)and \U0001d49c20 (3P) = −0.53(10)(values are not corrected for the Coster–Kronig transition). This isin agreement with the value reported for the KαL1peak of Al.

Observation of new electronic states of the Al–H 2/D 2 complex

A study of the electronic spectra of the weakly bound Al–H 2 and Al–D 2 complexes is presented. The complexes were generated in a pulsed, free-jet supersonic beam and detected with laser fluorescence excitation and depletion spectroscopy. All observed excited vibronic levels predissociate, and the transitions were detected by observation of lower-energy, emitting Al atoms. A long excited-state progression in the Al–H 2/D 2 stretch vibrational mode associated with the strongly bound 2 B 2 electronic state correlating with the Al(3d) atomic asymptote was observed. The Lorentzian widths of these bands are large, indicative of strong coupling to the repulsive Al(4s)–H 2 state. A Franck–Condon analysis was carried out to derive an effective one-dimensional potential energy curve for the van der Waals stretch coordinate in the excited 2 B 2 state. This potential energy curve is compared with the previously computed one-dimensional C 2v cut for this state [Faraday Discuss. 118 (2001) 387]. Bands associated with the mixed 5d, 6s←3p electronic transition are also reported. A complex of pattern of vibronic energies was observed, and the Lorentzian widths of these bands vary greatly. The binding energies of these and other Al–H 2 Rydberg states are compared with the computed binding energy of the ionic Al +–H 2 complex.

Simulation of the effect of random noise on Gaussian and Lorentzian widths in a Voigt profile

Experimental line profiles can be altered by so-called spectral interference that appears due to different causes. One of the most commonly appearing sources of interference is random noise due to the detector used for measuring the spectra in the laboratory. In this work, the noise is studied using a simulation procedure. This study helps to pinpoint the changes that the noise produces in the parameters of the experimental profiles and, consequently, in the information provided by them.

General Expression for the Voigt Function That is of Special Interest for Applied Spectroscopy

The purpose of this work is twofold. First we obtain a series expansion for the Voigt function that is valid for all values of the dimensionless parameter a (a measure of the ratio between the Lorentzian and Gaussian widths). Furthermore, the resulting coefficients are independent of the generalized coordinate b (the wavelength measured in units of the Gaussian width). In the second place, we fit an experimental “shaped bell” curve to a Voigt profile using certain theoretical restrictions that relate the maximum height and the full width at half-maximum.

Experimental and theoretical investigation of the rotational structure of the Al–H2/D2 complex

The rotational structure of the Al–H2/D2 complex is investigated in a collaborative experimental and theoretical study. The isotopomeric complexes were prepared in a pulsed supersonic beam, and their 3d←3p and 4p←3d electronic transitions recorded through laser fluorescence excitation spectroscopy. Transitions to quasibound excited vibronic levels were observed by monitoring emission from lower excited Al atomic levels, formed by nonradiative decay of the excited complex. In some bands, the Lorentzian width was sufficiently narrow that resolved rotational lines were observed. Rotational analysis of several bands which typify the two different patterns of observed rotational structure is presented. The derived rotational constants and parity splitting parameters for the ground Al(3p)–oH2/pD2 bend-stretch levels were compared with constants computed from fits to J- and parity-dependent energies calculated with new Al(3p)–H2 potential energy surfaces (PESs), which extend those recently reported by Williams and Alexander [J. Chem. Phys. 112, 5722 (2000)] by inclusion of the dependence on the H2 bond distance. The experimental and computed rotational constants were found to be in very good agreement. This provides strong support for the reliability of the calculated PESs.

Electronic spectroscopy and excited state dynamics of the Al-H2/D2 complex.

The electronic spectra of the Al-H2 and Al-D2 complexes are investigated in a collaborative experimental and theoretical study. The complexes were prepared in a pulsed supersonic beam and detected with laser fluorescence excitation spectroscopy. Transitions to bound vibrational levels in electronic states correlating with the excited-state Al(3d, 4p, 4d) + H2/D2 asymptotes were observed by monitoring emission from lower excited Al atomic levels, formed in the non-radiative decay of the excited complex. Fluorescence depletion has also been used to verify that the observed Al-H2 bands all involve the same molecular carrier. The bands have been assigned to the more strongly bound Al-oH2 and Al-pD2 nuclear spin modifications. In contrast to our previous observations for Al(5s)-H2 [X. Yang and P. J. Dagdigian, J. Chem. Phys., 1998, 109, 8920], for which only one potential energy surface (PES) emanates from the dissociation asymptote, the Lorentzian widths of the different vibrational bands in the 3d, 4p, 4d<--3p transitions vary widely, in some cases allowing resolution of the rotational structure of the bands. With the help of the calculated Al(3p)-oH2/pD2 dissociation energies, binding energies of the observed excited vibronic levels are reported. The mechanism of predissociation is investigated theoretically through ab initio calculation of C2 nu cuts of the excited PESs. It is concluded that predissociation occurs through coupling with the repulsive Al(4s)-H2 PES. With these calculations, a qualitative interpretation of the observed bands could be made.

Influence of Excitation Processes on the Shape of Argon and Neon Lines

5-2p 10 ), 703.0 nm (3s 5 -2p 9 ), 750.3 nm (2p1-1S2) as well as the neon line 748.8 nm (3d3-2p10) emitted from glow discharges at low pressures were analysed using a Fabry- Perot interferometer. We showed that the dissociative recombination of Are (or N ) molecular ions which gives rise to the density of non-thermalized atoms, may be regarded as the main process responsible for the distortion of the profiles of the above lines. The distorted profiles were analysed as super- position of the Ballik profile for thermalized atoms and the modified Ballik profile for non-thermalized atoms. The Lorentzian and Gaussian widths as well as relative densities of non-thermalized and thermalized atoms were . determined.

Surface Core-Level Shift as Assessed by Using the Voigt Function.

XPS spectra associated with surface core-level shifts (SCLS) for the InSb{111}A, B-(2×2) surfaces are fitted with the curves represented by the Voigt function. While testing as to how the Voigt function changes with detection angle, the Lorentzian width included in the function remains almost unchanged, whereas the Gaussian width becomes broadened with an increase in detection angle. This Voigt function allows to decompose a spectrum of interest into bulk and SCLS component with sufficient intensity, realizing quantitative estimation for the SCLS intensity.

Core-level broadening mechanisms at silicon surfaces

This article deals with the different mechanisms responsible for core-level broadening as observed by high-resolution photoelectron spectroscopy. Based on a comparative study for three different Si(111) surfaces the fundamental questions of symmetric or asymmetric broadening and scattered electron background are discussed together with the relative contribution of Lorentzian and Gaussian widths to the final core-level broadening. The variation of core-level broadening with temperature is studied at very low temperatures T∼30–300 K in order to evaluate the contribution of phonon broadening for the silicon bulk and the surface components.

Carbon Recombination Lines between 34.5 and 770 MHz toward Cassiopeia A

We present observations of low-frequency recombination lines of carbon toward Cas A near 34.5 MHz (n similar to 575) using the Gauribidanur radio telescope and near 560 MHz (n similar to 225) and 770 MHz (n similar to 205) using the NRAO 140 foot (43 m) telescope in Greenbank. We also present high angular resolution (1') observations of the C270 alpha line near 332 MHz using the Very Large Array in B-configuration. A high signal-to-noise ratio spectrum is obtained at 34.5 MHz, which clearly shows a Voigt profile with distinct Lorentzian wings, resulting from significant pressure and radiation broadening at such high quantum numbers. The emission lines detected near 332, 550, and 770 MHz, on the other hand, are narrow and essentially Doppler-broadened. The measured Lorentzian width at 34.5 MHz constrains the allowed combinations of radiation temperature, electron density, and electron temperature in the line-forming region. Radiation broadening at 34.5 MHz places a lower limit of 115 pc on the separation between Cas A and the line-forming clouds. Modeling the variation in the integrated line-to-continuum ratio with frequency indicates that the region is likely to be associated with the cold atomic hydrogen component of the interstellar medium, and the physical properties of this region are likely to be T-e = 75 K, n(e) = 0.02 cm(-3), T-R100 = 3200 K, and n(H) T-e = 10,000 cm(-3) K. Comparison of the distribution of the C270 alpha recombination line emission across Cas A with that of (CO)-C-12 and H I also supports the above conclusion.