Mass absorption coefficient, equations

Equation 1-5 was written for a sample containing a single element upon which monochromatic x-rays are incident. In so far as x-ray absorption is an atomic property, the mass absorption coefficients for other samples are additive functions of the weight-fractions of the elements, free or combined, that are present that is,... 

Since the mass absorption coefficient of hydrogen changes very little with wavelength, its value (0.435) may be taken from the literature. For carbon, 0.567, which was obtained from the measured kf for benzene by allowing for the absorption due to hydrogen, will serve. For chlorine, the rounded value na = 12 was chosen on the basis of the literature, which gives 11.6 at 0.71 A.13 With the numerical values inserted, Equation 3-8 becomes... 

Effective wavelengths have been included in Table 3-1 to show the changes that occur in this important variable when one gas is substituted for another. These wavelengths correspond to mass absorption coefficients calculated from Equation 3-14 and were obtained by interpolation from tabulated values of absorption coefficients for different wavelengths.15... 

At the wavelength (0.71 A) of molybdenum Ka, the three mass absorption coefficients of Equation 5-6 have the values given (0.381) ... 

It is gratifying that no empirical calibrating factor was needed with the Fe-55 source, which means that the results were predictable from Equation 5-6 by insertion of accepted values for the mass absorption coefficients. The deviation corrected by the introduction of this empirical factor (Equation 5-7) was of the kind produced by the filtering of polychromatic beams. About all that can be said about such empirical factors and about background corrections is this Always unwelcome, not to be introduced unless necessary, the need for them does not in itself make a method less desirable, but it does usually indicate that something is incompletely understood. 

Even if c (Equation 5-12) is fairly large, an element cannot be precisely determined—or may even escape detection—if it is present in too small amount relative to the matrix. What amount is too small depends not only upon the relative mass (or weight-fraction) of element sought but also upon the mass absorption coefficient of the matrix, as Equations 5-8 and 5-9 imply. 

In Equation 6-1, Io/I, usually an intensity ratio, is the quotient of the corrected average counting rate for the exposed substrate by that for the substrate covered with d cm of iron foil. The mass absorption coefficients of iron are m, a mean value for the incident (polychromatic) beam and g2, for the characteristic line being counted. The angles d and 02 are those made with the sample by the incident beam (30°) and by the emergent beam (60°), the beams being regarded as rays and... 

The change in the absorption effect with chemical composition can be calculated in simple cases by properly adjusting a (Equation 7-5) to allow for the change in mass absorption coefficient. The mass absorption coefficient of the sample for a given wavelength is given by... 

Equation (8) therefore allows us to use directly tabulated subshell photo-ionization cross sections (p) instead of mass absorption coefficients (a). F is the integrated photoelectron signal from an appropriate subshell of the monolayer adatom Yg the integrated signal from the relevant subshell of the substrate which is not simply the area of the core-level peak p and p. 

X-Ray Absorption Data. The x-ray absorption data shown in Figure 2 were obtained using a computer program based on the mass absorption coefficients, y, total energy, as found in the paper of Storm and Israel (1 ) and using the fundamental absorption equation for x-rays (II). 

The linear and mass absorption coefficients decrease exponentially when the wavelength decreases. Using equation (13.3), it can be calculated that household aluminium foil with a thickness of 12 pm (p — 2.7) absorbs 57% of the TiKa line (pM = 264) but only 1% of the AgKo line (//M = 2.54). 

In the simplest case, this system can be solved using the method of multilinear regression [30,31,40], In some cases, it is necessary to experimentally determine p the mass absorption coefficient of the mixture [39,40], To carry out this procedure, a monochromatic CuKa radiation was used, and with the help of the following equation [39,40]... 

If the phases within the mixture are known or were identified from the powder pattern of the mixture, we can calculate the mass absorption coefficients in equation (26). The most common method of determining Wa is to compare a peak in the mixture with the same peak in the pure phase. For a pure phase we may write from equation (21)... 

Equation 4.14 makes use of the fact that the scattered intensity is proportional to the amount of a particular phase, e.g. see Eqs. 2.17 to 2.19 in Chapter 2, with a correction to account for different absorption of x-rays by two components in the mixture. Since the ratio of intensities fi om a pure phase and a mixture is employed, diffraction patterns from both the pure material and from the analyzed mixture must be measured at identical instrumental settings, in addition to identical sample characteristics such as preparation, shape, amount, packing density, surface roughness, etc. King s equation becomes a simple intensity ratio when two phases have identical absorption coefficients, i.e. when p/pa = p/pb. We note that the composition of the second phase (or a mixture of all other phases) should be known in order to determine its mass absorption coefficient. Otherwise, mass absorption should be determined experimentally. When absorption effects are ignored, the accuracy of quantitative analysis may be lowered drastically. 

Equation 2.2 expresses the exponential in terms of (pp 1), which is called the mass absorption coefficient. It is independent of physical state of solid or liquid. The mass absorption coefficient varies with chemical element an element of higher atomic number has a higher gtp l value than an element of lower atomic number. 

The linear absorption coefficient p is proportional to the density p, which means that the quantity pjp is a constant of the material and independent of its physical state (solid, liquid, or gas). This latter quantity, called the mass absorption coefficient, is the one usually tabulated. Equation (1-10) may then be rewritten in a more usable form ... 

This equation permits quantitative analysis of a two-phase mixture, provided that the mass absorption coefficients of each phase are known. If they are not known, a calibration curve can be prepared by using mixtures of known composition. In each case, a specimen of pure a must be available as a reference material, and the measurements of and I p must be made under identical conditions. 

The use of Equation (22) in QPA once again eliminates the need to measure the instrument calibration constant and the sample mass absorption coefficient. However, in a similar manner to Equation (15), the method normalizes the sum of the analysed weight fractions to 1.0. Thus, if the sample contains amorphous phases, and/or minor amounts of unidentified crystalline phases, the analysed weight fractions will be overestimated. Once again this can be addressed by inclusion of an internal standard and modification of the measured lV s via Equations (16) and (17). 

Example What is the mass absorption coefficient of LiCl for Cu Ka radiation According to equation (27),... 

The meaning of equation (32) is that the intensity of any peak hkl in the mixture / divided by the intensity of the same peak in the pure sample of phase a, la is given by the right hand side of the equation. The mixture and the pure phase measurements are carried out exactly in the same way. It is assumed that the phases have been identified prior to the intensity measurements, so their mass absorption coefficients may be calculated. As an example we will consider a mixture of LiCl and CaS04. The mass absorption coefficient of calcium sulfate is 77.37 cm /g. Let us assume that the mixture contains 10 weight percent CaS04. The mass absorption coefficient of the mixture is then... 

Quantitative calculations when there is more than one absorber are based on the additive absorption of different elements. The mass-absorption coefficient of a sample containing different elements is given by an equation similar to Equation 14.10. 

Strictly speaking, this equation holds only for monochromatic light. The sensitivity for a given element depends markedly on the sample composition. Very small concentrations of elements with large mass-absorption coefficients can be determined in samples where the bulk matrix has a low mass-absorption coefficient. Thus, for example, 10" ° g of phosphorus can be determined in biological tissues. Classical examples of quantitative x-ray absorption analysis are the determination of lead tetraethyl in gasoline and sulfur in petroleum. 

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