Inverse Abel transforms

Inverse Abel transformed images are shown in Fig. 2 for various dissociation wavelengths using the 205.4 nm REMPI scheme. The images that... 

Fig. 2. The inverse Abel transform of the 0(1D2) photofragment images with both laser polarizations vertical and in the plane of the paper. The dissociation wavelength is shown for each image, but each image is arbitrarily scaled in size. The atoms were ionized via the 205.4 nm (2 + 1) REMPI process.

Figure 18. (a) Inverse Abel transformed photoelectron image showing the lab frame PAD for... 

In the CMI measurements, the momentum distribution of the fragment ions is determined in three-dimensional momentum space, so that their angular distribution, 1(9), with respect to the laser polarization direction can be derived in a straightforward manner without the need for a complicated mathematical procedure such as an inverse Abel transformation. The angular distribution can be derived from a thin slice of the three-dimensional momentum distribution at pz 0, as shown in Fig. 1.11 for the three explosion pathways (n = 0-2) of CH3CN. 

After substitution of variables X = and R = r the transformation (4.4.8a) can be written as a convolution (4.2.1), which is readily evaluated in the Fourier transform domain by use of the convolution theorem (cf. Section 4.2.3) [Bral]. The inverse Abel transformation is given by [Majl]... 

For evaluation of radial NMR images Frir) of circular objects, processing of the FID in two steps by Fourier transformation and subsequent inverse Abel transformation is preferred over straight forward Hankel transformation, because established phase correction, baseline correction, and filter routines can be used in calculation of the projections P(jc) as intermediate results [Majl]. As an alternative to Hankel and Abel transformations, the back-projection technique (cf. Section 6.1) can be applied for radial evaluation of circular objects, using copies of just one projection for input. As opposed to the inverse Abel transformation, however, this provides the radial information with nonuniform spatial resolution. 

Different approaches can be taken to obtain radial images. Radial field gradients can be applied by the use of dedicated hardware [Hakl, Leel, Lee2]. Alternatively, a 2D image can be reconstructed from one projection by the backprojection technique, and a radial cross-section can be taken through it. The most direct way to access the radial image from a projection consists in computing the inverse Hankel transformation (cf. Section 4.4.2) of the FID measured in Cartesian k space (cf. Fig. 4.4.1) [Majl]. But in practice, the equivalent route via Fourier transformation of the FID and subsequent inverse Abel transformation (cf. Section 4.4.3) is preferred because established phase and baseline correction routines can be used in the calculation of the projection as an intermediate result. 

Figure 9.10 Exampleofion-velodtymappingofproducts in a photofragmentation experiment. Top photofragment recoil for molecular transitions with fi parallel or perpendicular to the laser field polarization E, and subsequent extraction of ionized fragments. Middle inverse Abel-transformed image of the velocity distribution of ionized D atoms produced in the photolysis of DI at A = 205 nm. Bottom angular and velodty distributions extracted from the ion image map for the D -EI and D -E I fragmentation channels. Data adapted from McDonnell and Heck J. Mass Spectrom., 1998, 33 415, with permission of John Wiley Sons Ltd...

Equation 16.2 is the definition of the Abel transform [33]. In X-ray scattering, Eq. 16.2 is established textbook knowledge [4,34-39]. There it describes the slit smearing. Even the inverse Abel transform... 

The line integral fL(x) can be recognized as a form of the Abel integral equation [63], and is referred to as the Abel transform of f(r). Of more interest to us is f(r) which can be obtained from fL(x) by the inverse Abel transform... 

Figure 6 (a) Three concentric rings whose intensity per pixel is such that the total intensity per ring is constant, (b) Intensity profile of the projection of the rings onto an arbitrary row, convoluted with a Gaussian function 5 pixels wide 3 pixels FWHM. (c) Inverse Abel transform of the projection. The intensity profile along a diameter of the rings is indicated by the stars. 

More quantitative information can be obtained from the images when the full three-dimensional speed and angular distributions are reconstructed using mathematical transformations of the crushed two-dimensional images, or alternatively by using forward convolution simulation techniques. If the initial three-dimensional distribution has cylindrical symmetry, a unique transformation - the inverse Abel transform - can be used to reconstruct the initial three-dimensional velocity distribution. As the photolysis laser vector defines automatically an axis of cylindrical symmetry, the inverse Abel transformation can usually be used, as long as the plane of the position-sensitive detector is placed parallel to the laser polarization vector. 

Figure 3 shows a series of reconstructed ion images of nascent D (deuterium) atoms generated via the photolysis of DI, at three different photolysis wavelengths. These images can be used to illustrate many of the points made above. The upper panels show the inverse Abel-transformed ion images, the... 

Figure 6 Raw images of D atoms formed in the reaction of H+Dj. Intensities are shown in an inverse linear grey scale (darker corresponds to lower intensities). The left and right columns show results obtained for this reaction at centre-of-mass collision energies of 0.54 eV and 1.29 eV, respectively. In the images, the centre-of-mass (CM) of the reaction and the direction of the H atoms are indicated. The circles indicate the calculated positions of the fastest D atoms corresponding to formation of HD in the v= 0, J = 0 state. At the bottom of the figure the differential cross-sections, obtained from the inverse Abel-transformed images are displayed. Reproduced with permission from Heck AJR and Chandler DW (1995) Imaging techniques for the study of chemical reactions dynamics. Annual Review of Physical Chemistry 46 335-378. 1995 Annual Reviews Inc.

Figure 1 Inverse Abel transforms of the SfDz) images observed for photodissociation at 223, 235, and 248 nm. The center-of-mass translational energy releases calculated from these images are also shown. The fast component gradually changes from an isotropic distribution at 223 nm to an anisotropic distribution at 248 nm. The photoabsorption at 248 nm is dominated by a perpendicular transition to A A bimodal speed distribution occurs only in dissociation from the A state.

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