Cross Sections for Inelastic Scattering of Electrons by Atoms - Selected Topics Related to Electron Microscopy

We begin with a resume of the Bethe theory, which provides a general framework for discussing the inelastic scattering of fast electrons and leads to powerful criteria for judging the reliability of cross-section data. The central notion of the theory is the generalized oscillator strength as a function of both the energy transfer and the momentum transfer, and is the only non-trivial factor in the inelastic-scattering cross section. Although the Bethe theory was initially conceived for free atoms, its basic ideas apply to solids, with suitable generalizations; in this respect, the notion of the dielectric response function is the most fundamental. Topics selected for discussion include the generalized oscillator strengths for the K-shell and L-shell ionization for all atoms with Z ≤ 30, evaluated by use of the Hartree-Slater potential. As a function of the energy transfer, the generalized oscillator strength most often shows a non-monotonic structure near the K-shell and L-shell thresholds, which has been interpreted as manifestations of electron-wave propagation through atomic fields. For molecules and solids, there are additional structures due to the scattering of ejected electrons by the fields of other atoms

The IAEA program on atomic and molecular data for radiotherapy and related research

Radiation measurements and dosimetry usually require reliable values of physical quantities that describe the interactions of radiation with matter. Examples of such quantities are the stopping power and the ionization yield. In any material subjected to ionizing radiation, many energetic particles are present. These may be primary particles, charged or uncharged, or secondary particles, such as electrons ejected in ionizing processes. These particles deliver energy to molecules in the material in various collision processes, and energetic electrons are always the most numerous. Any serious analysis of the energy delivery processes requires knowledge about the collision processes, most importantly the cross sections for all major processes specified by energy transfer values. Such an analysis may be carried out in many ways, depending on the specific purpose (2). Two major classes of approaches are readily recognized: the particle transport theory (3) and Monte Carlo simulations (4), as discussed by Bichsel (5) in the present Symposium. In either approach, it is important to use as input the cross section data that best represent elementary microscopic processes. An analysis based on unrealistic input data must be viewed with caution at best, because results might be misleading. Cross section data found in the literature are often relative rather than absolute, discordant rather than unique, and fragmentary rather than comprehensive (i.e., covering the wide range of variables needed in radiation research and other applications). The purpose of this lecture is to survey major international efforts and activities

W, F, and I

A topical area of Professor Doke's recurring work concerns physical quantities basic to radiation physics and dosimetry. Therefore, I believe it appropriate to dedicate to him with sincere respect the present essay reporting on the current status of understanding of the most fundamental three listed in the title. Throughout I will focus on the physical meanings of these quantities and their magnitudes, considered from the structure of atoms, molecules, and their aggregates, viz., condensed matter, with minimal reference to technical details. 1. w Background The W value is an index of the mean number of ions produced in a gas subjected to ionizing radiation. Forrnally, it is defined as the radiation energy absorbed (usually expressed in units of eV) "per ion pair of either sign produced", or, in a simpler language, "per electron liberated". The basic knowledge up to 1961 is eloquently articulated in a classic essay by Platzman 1) The W value for a given gas depends weakly on the properties of the radiation such as the mass and charge of particles or initial energies (provided they are sufficiently F The submitted manuscript has been authored by a contractor of the U. S. Government Accordingly, the U. S. Government retains a nonexcluwve, royalty-free hcen?,e to publlsh or reproduce the published form of this contribution, or allow others to do so, for DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Potiions of this document may be illegible in electronic image products. Images are produced from the best available original document. high). This makes the ionization measurement useful as a method of dosimetry, viz., the determination of the absorbed energy. 2) The ratio W/I is always greater than unity because a part of the absorbed energy must be used in nonionizing events such as discrete excitation or molecular dissociation into neutral fragments and also in producing subexcitation electrons, viz., electrons with kinetic energies too low to cause e~ectronic excitation or ionization 3) The ratio Wfl is 1.7-1.8 for rare gases, and 2.1-2.6 for gases of common molecules (depending on the electronic structure, going from "hard" to "soft"). Calculation of the W value is possible from three approaches: i) the energy balance of Platzman, heuristic for general understanding and appropriate for an estimate; ii) the states resulting from the incidence of particles of relatively low energies, while the Spencer-Fano method is good for the incidence of high-energy particles. The Spencer-Fano method is informative in showing the relative importance of ranges of different particle energies, in the form of the yield spectrum, as explained below. Consider a gas consisting of n molecules per unit volume of a single chemical species. Let cTi(T,, be the total ionization cross section of the molecule by an electron of kinetic energy T. During its passage over an infinitesimal path length dx, the electron produces the number of ion pairs (more precisely, of liberated electrons) expressed as dlli = n CV(2") dx. Thus the total number of ion pairs is given by the integral over the total path length. If one may regard x and T as related by a well defined smooth function, then one may write dx = (dT/d.x)-l dT and carry out the above integration, where dT.\dxis the mean energy loss per unit path length, viz., the stopping power. This However, the CSDA is inadequate for electrons in general because the energy 10SSof an electron can be as high as (?'-/)/2, which occurs on a head-on collision, and also because the CSDA cannot account for the production of secondary electrons. Spencer and Fano Then, Eq. (1) becomes In general Tranges from several eV up to keV or even MeV, and therefore it is sensible to consider in T as an independent variable and to rewrite Eq. (2) as Ni = n jTq(T) y(T) d(ln 2). and is also useful in the context of the Bethe theory An example of the yield spectrum is seen in [12], which treats ion incidence. Main findings are as follows. 1) For the same set of cross-section data, the method of Fowler and the method of Spencer and Fano lead to the same result. Indeed, these two methods represent two alternative mathematical pictures of the same physics 2) The systematic of the W value has been explored more broadly. For vapors of atoms from hydrogen to argon, approximate estimates of the W values were made [17] and other data. Outlook As the above summary indicates, we have now a fair understanding about principles that govern W values in general, and also about their numerical magnitudes for common gases. For better understanding of the basics, I wish to see work in the two following directions. First, the quantum yield of ionization, viz., the probability of ionization of a molecule when it has received a fixed excitation energy (exceeding the first ionization threshold 1 ) has been extensively studied Consequently, the statistical fluctuations in the number of ion pairs produced are restricted, leading to a smaller variance, viz., to F less than unity. When the initial kinetic energy of an incident charged particle is vexy high (i. e., much higher than I), F tends to a constant value, which may be regarded as a property of the "material. Ni of the number of ion pairs, viz., Recent Developments Vi= n \pi(T) y(T) dT = n fTpi(T) y(T) d(ln T). (4) Here y(T) is the same degradation spectrum as in Eq. (2), but pi(T) represents a new quantity 9 pi(T) = &~(T) tN(T)k)2, where~(T) is the cross section for a collision of kind k (ionization or discrete excitation, further classified in terms of energy transfer in the collision) of an electron of kinetic energy T and AN(T)k is the increase in the mean number of ion pairs due to that collision The Fano factor F = Vi /Ni can be expressed as F=~pi(T) y(T) dTf j G(T) y(T) dT. (6) There must be a value T* in the same interval of the two integrals, which covers the range of the kinetic energies of all the electrons present in the medium, according to Cauchy's generalized mean-vrdue theorem [50]. The value of T* depends on y(T). However, if the ratio pi(T)/~i(T) varies slowly on~then the precise value of T* is immaterial. This is indeed generally the case, as illustrated in Indeed the plateau value of the ratio attained around T = 500 eV represents the value of 3) By far the most notable discovery in recent years is the close correlation between F and W, which may be called the Krajcar-Bronic relation approximately obeyed by data on many materials, where a and b are constants. From a basic point of view, the Krajcar-Bronic relation is qualitatively understandable in the following way. When W/I is small, many ions are readily produced; this means that many ionizing collisions, as well as many nonionizing collisions, occu~this in turn means that the history of individual collision processes is certainly diverse but is subject to the energy conservation. Therefore, F should be appreciably less than unity. When W/I is large, a modest number of ions are produced as a result of a modest number of collision processes; then, the energy conservation is less relevant, and therefore F should be large (within the general limit F being less than unity). In summary, F and Wfl should go together when one looks at different materials. However, it has been difficult to explain the Fano factor of argon for protons nearly the same as for elecwons of the same speed. Furthermore, it might be fruitful to study closely the Fano factors of polyatomic molecules for protons and a particles, for the diversity of energy transfer processes is greater because of vibrational and rotational degrees of freedom in addition to the translation of a molecule as a whole. The appreciable difference of the Fano factor for different particles is in sharp contrast with the W value, which is insensitive to different particles, as already noted by Platzman [1]. However, I find it mildly puzzling how to reconcile the close comelation Outlook The data surveys I am unaware of any measurements on the statistical fluctuations in the number of excited states rather than ions. The simplest index of the fluctuations is the variance, and the ratio of the variance to the mean maybe called the Fano factor for the yield of excited states. This quantity can be theoretically treated similarly as the Fano factor for ionization yield. More precisely, one should talk about a particular excited state, which may be detected and scored for instance through fluorescence, i. e., light emitted at a specific frequency, or through products of a specific chemical reaction. The cross section for discrete excitation by collisions of an electron or another charged particle shows different dependence on the kinetic energy, depending on the molecular species and on the character of a transition involved, for instance, whether it is dipole-allowed or not The mean excitation energy 1 is the sole nontrivial property of the stopping medium, and is defined in terms of the dipole oscillator-strength spectrum dfld.l?as . where both of the integrals run over the entire range of excitation energies E including discrete and continuous. Most often the spectrum is normalized so that the denominator equals Z. Sometimes, for instance, in Fano [57], the denominator is set at unity. For a medium consisting of molecules, Z means the average atomic number as readily evaluated from the molecular structure, and N means the molecular number density. For condensed matter, it is customary to express the oscillator-strength spectrum in terms of the dielectric-response function s(E), which represents the electric displacement induced by an applied electric field that has unit magnitude, is uniform in space, and oscillates at frequency 20% .The probability for the transfer of energy E from a fast charged particle to matter through glancing collisions is proportional to q(E) = Im [-1/ E(E)], and the product E q(E) maybe viewed as equal to dfldE, apart from a constant of proportionality where The 1 value thus defined plays a key role in the determination of stopping power. Unlike the ionization threshold, the mean excitation energy for stopping power is governed by the entire oscillator-strength spectrum, and is roughly proportional to z a rough estimate is given by UZ = 10 eV, except for Z less than 10, for which 1 is larger that given by the relation. A more accurate estimate is often desirable because the stopping power is often amenable to measurements to a high precision; in general, when the stopping power is known under the condition of applicability of Eq. (8) to a precision of AS/S, then it implies the knowledge of 1 to a precision of about 10 NH. From the point of view of basic physics, the Z value is linked with many other electronic properties through sum rules Another example concerns data on water [59

Evaluation of discomfort in manual activities using hand mapping: the influence of age

This study analyzes the method of hand mapping to assess the perceived discomfort during the use of hand tools by different age groups. The aim was to determine the influence of age and how it affects the perception of discomfort. Considering that, this was a study that involves subjective aspects, it seeks to analyze results by means of a comparative assessment among different areas of the hand. Hence, the hand palm was divided by both anatomical and interface criteria for the task. The manual activity consisted in a simulation of opening procedure for PET packaging of soft drinks. The results show differences in perception of discomfort among the ages analyzed. The adopted method contributes to the ergonomic design, as the perception of discomfort is an important parameter in the product design, providing more efficient and accessible products

Radially restricted linear energy transfer for high-energy protons: A new analytical approach

Radially restricted linear energy transfer (LET) is a basic physical parameter relevant to radiation biology and radiation protection. In this report a convenient method is presented for the analytical computation of this quantity without the need for complicated simulation. The method uses the energy-re-stricted LETL, as recently redefined in a 1993 ICRU draft document and supplements it by a relatively simple term that represents the energy of fast rays lost within distancer from the track core. The method provides a better fit than other models and is valid over the entire range of radial distance from track center to the maximum radial distance traveled by the most energetic secondary electrons.L r computed by this approach differs only a few percent from the values Contribution to the international symposium on heavy ions research: space, radiation protection and therapy, 21–24 March 1994, Sophia-Antipolis, Franc

Electronic Stopping and Momentum Density of Diamond Obtained from First-Principles Calculations

We calculate the "head" element or the (0,0)-element of the wave-vector and frequency-dependent dielectric matrix of bulk crystals via first-principles, all-electron Kohn-Sham states in the integral of the irreducible polarizability in the random phase approximation. We approximate the macroscopic "head" element of the inverse matrix by its reciprocal value, and integrate over frequencies and momenta to obtain the electronic energy loss of protons at low velocities. Numerical evaluation for diamond targets predicts that the band gap causes a strong non-linear reduction of the electronic stopping power at ion velocities below 0.2 atomic units.Comment: 8 pages, 6 figures, REVTeX

Aspects of Relativistic Sum Rules

The status of our understanding of relativistic sum rules is reviewed. The recent development of new theoretical methods for the evaluation of these sum rules offers hope for further advances in this challenging field. These new techniques are described, along with a discussion of the source of difficulties inherent in such relativistic calculations. A connection is pointed out between certain sum rules for atomic interactions with charged particles and those for interactions with photons.Comment: 32 pages, accepted for publication in Advances in Quantum Chemistr