In the quantitative treatment of transition rates, both radiative and electronic Auger transitions are considered. Similarly, the muon electron screening effect does not have a significant impact on the results. T 0.1 Some useful properties of muon and electron T 0.2 Comparative properties of muonic and electron atoms.
F 2.9 Modification of the monopole operator due to electronic screening F 2.10 Effects of penetration on electron slug velocities. F 3.1 Comparison of our results with nuclear internal conversion coefficients F 3.2 Comparison of the penetration effect with other calculations. F 3.3 Schematic representation of the photoelectric electron emission F 3.4 Comparison of our results with experimental photoelectric effect data.
Nuclear "._ap~~ This phase depends greatly on the nature of the particle and the nucleus. That program included dipole radiation transitions and the dipole part of the muon-electron interaction.
CHAPTER 1: INITIAL HISTORY OF THE MUON IN THE ATOM
This reduction of the interaction confirms the reduction of the lowest-order perturbation term (one quantum exchange between the muon and the atom). It follows from the slow motion of the muon that the maximum energy loss in each quantum collision is limited by the classical maximum momentum transfer given by [18]. In fact, the most likely energy loss is only about 1/10 of the maximum value given in eq.
This means that the trajectory of the muon is not modified much by the energy loss. These arguments fully justify the use of the classical description of the muon's trajectory and its interaction with electrons. When the corresponding integral is evaluated, it is found that the stopping power is proportional to the muon speed and, therefore, represents a frictional force.
Inelastic scattering cross section for Cl in rubidium chloride in A2/eV as a function of energy loss E for several values of muon energy E, indicating curves. Also b0 (L,!:;L) means the solution of the impact parameter as a function of the moment quantity (from liL = ('lm µ EP b.
Oxides
CHAPTER 2: QUANTAL CASCADE AND TRANSITION RATES
Appendix A contains most of the mathematical details and a compilation of formulas involved in this development. Note that most of the problem is concentrated in the evaluation of the matrix element of the perturbation Hamiltonian. But in such a case the transition rate between states of different n vanishes, due to the orthogonality of the radial parts of the wave functions.
This process is, of course, nothing more than a quantum picture of the muon-electron collisions classically described in Chapter 2. The only possible errors in radiation are the use of simplified wave functions for the muon and the neglect of spin-related terms in the Hamiltonian. The exponential retardation factor, as a direct result of relativity, takes care of the necessary oscillations of the photon field until it is absorbed at another point.
The relative intensity of the different multipoles is found using order-of-magnitude estimates for the multipole matrix elements. Then the ratio between two consecutive multipoles, excluding factors of order of unity, is roughly. The strong dependence of the speeds lies in the transition frequency w, where the speeds are proportional to w 2L +l. All other things being equal, the transition with the smallest n2 is the most intense.
In the next subsection, we will see quite different characteristics of Auger levels. Inspection of these rates and radiation rates confirms the fact that the conversion coefficient increases with increasing order of multipolarity [50]. As a demonstration, in Figure 2.9 we show a modification of the monopoly operator in the presence of an unrealistically strong screen.
Derived by equating the exact values of the wave functions (54) point by point to the hydrogen-like formulas and solving numerically for Z*. This is a wide open question, for which the classical part of the cascade could serve as a guideline. In the final version of the program we have made provisions so that the initial population of the muon can be spread over the entire (n,l) spectrum.
CHAPTER 3: TESTS AND COMPARISON WITH THEORY
Note the existence of a zero in the conversion coefficient without delay near E = l keV. Comparison between our calculation of the penetration effect (solid curves) and the calculation of ref. Nevertheless, the agreement is acceptable in most of the energy range, considering the approaches involved.
Researchers have historically been concerned with calculating penetration; reference (52) is for penetration estimation only. In Figure 3.2, we have reproduced some relevant figures from their paper, with our results overlaid; as we can see, the two families of curves closely follow each other. The conversion coefficient of the Auger process and the rate of electron ejection through the photoelectric effect shown in Figure 3.3.
The final result is that without too many approximations the ratio of the conversion coefficient to the photoelectric cross section is given by. Of course, for high energies we need to include delay for the equation to make any sense. Note that penetration has no meaning in this case and that we have used only dipole transitions.
The above comparisons show that our formulation is generally correct; Indeed, issues surrounding thresholds exist and perfect agreement with results that are simply more accurate cannot be expected. One method could be to empirically discover the necessary corrections needed to minimize the discrepancies and use them in calculating the transition rates.
CHAPTER 4: APPLICATIONS ON THE EXPERIMENTAL RESULTS OF IRON AND THALLIUM
If our predictions were perfect, this would provide an unambiguous measure of the quality of the measurements and the associated errors. Rather than blaming the experiment for the discrepancies, we can directly say that at least part of the discrepancy is due to an incomplete calculation of the matrix elements (in particular, the incompleteness of the 2s wavefunction and the key position of the high-oscillating high-n, low-Z wavefunction cancellation node). To illustrate the sensitivity of the scheme to small variations, we plotted three similar distributions that nevertheless yield significantly higher x2.
Although the best x2 /DF of 4.3 cannot be considered satisfactory, it determines the exact shape of the Z distribution (assuming a model) with rather tight bounds (cf. The dot-and-dash curve shows the beam width and the dashed curve shows the total width of the conditions Although our two standard deviations of the measured values are on the mean, we correctly reproduce all qualitative features of the data.
This phenomenon accounts for most of the highs and lows of the intensities within each range. This mainly affects the intensity of the transitions arising from the n1 = level for the following reason. Note that the inaccuracies of the theory are not included in the calculation of the x2, since they are not randomly distributed and cannot be reliably estimated.
In addition, high Z makes the Auger rates less important and this increases the reliability of the calculation. The rest of the conditions along with the actual numbers are in Table 4.J, along with the data and errors. X2/DF is about 1.2, which means that we can fit the results with reasonable accuracy and the reliability of the calculation is comparable to (or better than) typical experimental accuracies.
Under these circumstances, reliable fits of the initial l-distribution and extrapolation to high levels are almost impossible. Another side use of the program is polarization calculation; there are several experimental results (cf. More iron quality experiments [60] would be desirable to facilitate theory development.
CHAPTER 5: DESCRIPTION OF THE CASCADE PROGRAM
BLOCK DATA
MUON00 ( MAIN)
RMON RDIP
ROCT
MAT EL
R DIPU RQUAU
MATELU
BETA P OPJ
CHAPTER 6: CONCLUSIONS
During our research for the last few years, we have studied the field of muonic atoms from their formation to their dissolution and the interaction of the muon with the surrounding electrons and the nucleus. Since the radial parts of the spin-orbit doublet wavefunctions are, in our non-relativistic treatment, identical, the matrix element A.2 depends only on l. The case L = 0 is given in anticipation of the penetration, which allows the otherwise forbidden monopole transitions.
The electronic part is more complicated, but it comes down to performing integrations of the general form. The muonic part of the transition matrix element is unchanged, while the operator in the electronic part has been modified. Pen~tra~ion ~ffe~ ~~ ~uge~ :r:_~ansiti~~ To precisely evaluate the radial part of the matrix element (A.9), we have the choice to first perform the electronic or muonic integration to feed .
Muon integrals ~one ~irsr:_,_ The first (inner) integral is always incomplete, since the functional form of the integrand changes at r1 = r2• In this case, we expand the Laguerre polynomials that appear in muon wave functions in the power series (finite) . As a result, four interleaved summations need to be performed to evaluate one matrix element, two due to Laguerre polynomials and one each due to the resulting incomplete gamma functions and to expand the final value. If we rewrite the integrals (A.9) in a way that illustrates the penetration contribution, and using the orthogonal properties of the radial parts of the electronic wave functions, we obtain the following formula for the radial double integral from (A.9).
Therefore, it is enough to study a term of the type of (A.19), the general one is formed by their linear combinations. Noting the functional form (A.11) of the continuous wave function and using the definition of the parameter y. Since the muonic integral involves powers and exponentials, a candidate would be a combination of terms of the form.
After evaluating g(r1), the outer integral of the penetration in (A.18) is a modified multipole matrix element, where the operator is a combination of an integer power and an exponential power. The numerical methods for evaluating such a matrix element are very analogous to those of the real multi-pole matrix element. Only the part of the rate formula that is affected is stated; the rest of the factors remain intact.
B: SAMPLE INPUT AND OUTPUT OF THE MUONIC ATOM CASCADE PROGRAM
