The Theory and Computation of Energy Deposition Properties

Remigio Cabrera-Trujillo; John R. Sabin    Quantum Theory Project, Department of Physics, University of Florida, Gainesville, FL, USA

1 Introduction

To set the stage for this volume, let us first consider a swift ion of charge Z1 moving with some laboratory frame velocity v through a medium. By colliding with the nuclei and electrons and thereby transferring energy to them, the projectile is slowed, and the energy loss, ΔE(v), per unit path length, Δx, is known as the stopping power (−ΔE(v)/Δx) of the target material. The quantity is velocity dependent and carries a negative sign so that the stopping power will be positive as the projectile slows. As simple energy and momentum conservation demand, at all but the lowest projectile velocities, the stopping power comes from collisions of the projectile with target electrons, and thus from energy transfer to the electronic structure of the target atoms, molecules, or solid.

The stopping power is generally normalized by the target scatterer density n, to produce the stopping cross-section, S(v). If one removes the primary velocity dependence and constants from the cross-section, one obtains the stopping number, L(v), where interesting physics is concentrated. These quantities are related as

d ⁢ E d ⁢ x = n ⁢ S ⁢ v = 4 ⁢ π ⁢ n ⁢ e 4 ⁢ Z 1 2 ⁢ Z 2 m e ⁢ v 2 ⁢ L ⁢ v

where e and me are the electron charge and mass, respectively.

It is convenient to expand the stopping number in a Born series

⁢ v = ∑ ∞ i = 0 ⁢ Z 1 i ⁢ L i ⁢ v

in the projectile charge, Z1. The leading term, L0, is dominant over most of the range of commonly encountered projectile velocities [1]; however, other terms in the series become large at very low projectile velocity (L1 and L2) or at very high (relativistic) velocities (e.g., L5) [2]. In fact, each term in the Born series can be again expanded in series

i ⁢ v = ∑ ∞ j = 0 ⁢ L i ⁢ j ⁢ v

where the terms beyond Li0 correct for assumptions made in the derivation of Li0.

The best-known example is for the first term in the expansion when the Bethe formulation is employed. In this case, the first, or Bethe–Born term in L0 can be written, including the relativistic terms [3], as

00 = ln ⁢ 2 ⁢ m ⁢ v 2 I 0 - ln ⁢ 1 - β 2 - β 2

where I0 is the mean excitation energy or first moment of the energy weighted dipole oscillator strength distribution for the target and, as usual, β=v/c. However, the Bethe theory is developed assuming that the projectile has a much larger velocity than do the target electrons. To correct for this error, a new term, L01=−C/Z2, called the shell corrections, was introduced [4–7]. For the most part, terms beyond i=2 and j=1 have not been investigated.

It is primarily with the various methods and approaches for calculating the stopping cross-section or properties related to it, as well as the various terms in the series expansion of the stopping cross-section that many of the chapters in this volume will be concerned.

2 Some history

Although there was early work on energy deposition using classical theory and free particle targets by Darwin [8] and Thomson [9] in 1912, the first formulation of the energy deposition problem based on the realization that the binding of electrons in a target atom is important was due to Niels Bohr in 1913 [10] and 1915 [11], when he realized the importance of considering the target electrons to be bound. His treatment was classical, but considered the target electrons to be harmonically bound, and gave a leading term in the Born series that depends on the logarithm of v3/ω, where ω is the characteristic frequency of the target electron. These seminal papers by Niels Bohr can be considered the beginning of energy deposition theory and since then, there has been considerable interest in the interaction and penetration of energetic ions with matter.

Following Bohr, there was significant activity (for an excellent review of the early history and an extensive list of references up to 1980, see Sections I and III of Ref. [12]). An important step was taken by Bethe in 1930 with the quantum mechanical solution of the problem in the first Born approximation, and using the dipole approximation to the oscillator strength distribution [13]. Again, as in Bohr's treatment, the leading term in the Born series was found to be proportional to the logarithm of a power of the projectile velocity over a characteristic energy, but this time it was v2/I0, where I0 is again the mean excitation energy of the target. Bethe extended his treatment to the relativistic regime in 1932 [14].

In 1933, Bloch revisited the problem [15], and found that, in the dipole approximation, the odd powers in the Born series vanish, but that there is a non-negligible third term in the Born series, giving S(v)∝Z14. This term is now referred to as the Bloch correction and corresponds to L2 in the Born series (equation (2)).

In 1963, Barkas noticed the difference in stopping powers when measured with positively and negatively charges projectiles [16], which implied an odd power L1 contribution to the Born series, and the Z13 contribution to the stopping cross-section is now referred to as the Barkas correction.

Higher order terms in the Born series, relevant to relativistic interactions, are seldom, but occasionally (cf. e.g., Refs. [2,17]) discussed.

Details of the forgoing can be found in the classic work by Livingston and Bethe [18], which contains a comprehensive review of experimental data, as well as in the extensive review papers by Fano [3], Fano and Cooper [19], Inokuti [20], Bichsel [4], and by Ahlen [12].

The preceding was developed mostly for atomic or molecular targets, and, indeed, much experimental work has been carried on gaseous samples. If one thinks along the lines of the Bragg rule [21], then one would not expect such considerations to differ much from results obtained on solid samples, and for most molecular or van der Waals solids – indeed they do not [22].

However, metals have a fundamentally different electronic structure than do atoms and molecules: the electrons are described more realistically by bands than by localized orbitals. In the early 1950s, using a density functional based theory, Lindhard [23–26] developed a successful method for treating solids directly that is still in use today.

3 The situation today

During the past decade, there has been a revitalization of theoretical interest in energy deposition. This interest has been supported by the availability of extensive computational resources and stimulated by new and more accurate measurement techniques, so that details of the interactions between swift ions and materials can now be both measured and computed.

Many of the articles in this volume reflect the differences in the plan of attack on the energy deposition problem as it is presently formulated with respect to that of two decades ago. In the most general terms, the treatment of energy deposition in terms of considering each perturbation to the Bethe (or Bohr) leading term in the Born series has given way to what one might call in more avant garde terms, a more holistic approach.

Some of the contributions retain the framework of the first Born approximation, and explore ways to improve L0 and thus gain a better understanding of some of the corrections,...