4.7.1 Introduction
The equations presented in Secs. 4.3-4.6 for energy loss and range of charged particles were derived with the assumption that the charge of the particle does not change as the particle traverses the medium. This assumption is certainly
valid for electrons, positrons, protons, and deuterons ( Z = 1). It holds well for alphas too ( Z = 2). However, for Z > 2, the charge of the particle cannot be assumed constant, and for this reason the energy loss and range calculations require special treatment.
Consider an atom or an ion with speed greater than the orbital velocity of its own electrons. If this particle enters a certain medium, the atomic electrons will be quickly removed from the atom or ion, leaving behind a bare nucleus.
The nucleus will keep moving through the medium, continuously losing energy in collisions with the electrons of the medium.+ It is probable that the ion will capture an electron in one of these collisions. It is also probable that the electron will be lost in another collision. As the ion slows down and its speed becomes of the same order of magnitude as the orbital speeds of the atomic electrons, the probability for electron capture increases, while the probability for electron loss decreases. When the ion slows down even farther and is slower than the orbiting electrons, the probability of losing an electron becomes essentially zero, while the probability of capturing one becomes significant. As the speed of the ion continues to decrease, a third electron is captured, then a fourth, and so on. At the end, the ion is slower than the least bound electron. By that time, it is a neutral atom. What is left of its kinetic energy is exchanged through nuclear and not electronic collisions. The neutral atom is considered as stopped when it either combines chemically with one of the atoms of the material or is in thermal equilibrium with the medium.
4.7.2 The dE / dr Calculation
The qualitative discussion of Sec. 4.7.1 showed how the charge of a heavy ion changes as the ion slows down in the medium. It is this variation of the charge that makes the energy loss calculation very difficult. There is no single equation given d E / & for all heavy ions and for all stopping materials. Instead, d E / & is calculated differently, depending on the speed of the ion relative to the speed of the orbital electrons.
The stopping power is written, in general, as the sum of two terms:
where (dE/cix), = electronic energy loss ( d E / d x ) , = nuclear energy loss
An excellent review of the subject is presented by Northcliffe" and Lindhard, Scharff, and Schiott." The results are usually presented as universal curves in terms of two dimensionless quantities, the distance s and the energy E , first
t~ollisions with nuclei are not important if the particle moves much faster than the atomic electrons.
146 MEASUREMENT AND DETECTION OF RADIATION
introduced by Lindhard et al.1° and defined as follows:
wheret a = 0.8853a,(~:/~ + ~ , 2 / ~ ) - ' / ~ x = actual distance traveled
a, = h2/me2 = Bohr radius = 5.29 X lo-'' m Z,, MI = charge and mass of incident particle Z2, M2 = charge and mass of stopping material
The parameters N, r,, and mc2 have been defined in Sec. 4.3.
At high ion velocities, u * V , Z ? / ~ , where v, = e2/fi = orbital velocity of the electron in the hydrogen atom, the nuclear energy loss is negligible. The particle has an effective charge equal to Z,, and the energy loss is given by an equation of the form
which is similar to Eq. 4.2.
At velocities of the order of u = u,z:/~, the ion starts picking up electrons and its charge keeps decreasing. The energy loss through nuclear collisions is still negligible.
In the velocity region v < U,Z?/~, the electronic energy loss equation takes the formlo
where
1 12.13
and n has a value very close to T . The constant k depends on Z and A only, not on energy, and its value is less than 1. Some typical values are given in Table 4.2.
Table 4.3 shows the kinetic energy per unit atomic mass, as well as the kinetic energy, of several ions for u = v,Z:/~.
The electronic stopping power for different ions and stopping materials is obtained by using the following semiempirical approach.
se he number 0.8853 = (9.rr2)1/3/27/3 is called the Thomas-Fermi constant.
Table 4.2 Values of k Used in Eq. 4.34
The ratio of stopping power for two ions having the same velocity and traveling in the same medium is given by (using Eq. 4.33):
The application of Eq. 4.35 to heavy ions should take into account the change of the charge Z, as the ion slows down. This is accomplished by replacing Z, with an effective charge,
Zeff = 7721
where 77 is a parameter that depends on energy. The second particle in Eq. 4.35 is taken to be the proton (Z, = A , = I), thus leading to the form14-l6
where the effective proton charge 77, is given by Eq. 4.37, reported by Booth &
rant'^, and T, is the proton kinetic energy in MeV:
77; = [I - exp ( - 150~,)] exp (-0.835e-'~.~'~) (4.37)
Table 4.3 The Kinetic Energy of Heavy Ions for Several Values of v = U , , Z ~ / ~
~-
voZ:'=
(X lo-') P
Ion z1 ( 4 s ) (X 10') T/A T (MeV)
148 MEASUREMENT AND DETECTION OF RADIATION
Equations giving the value of q have been reported by many investigator^.'^-'^
The most recent equation reported by Forster et al.17 valid for 8 I Z , I 20 and for v/v, > 2 is
q = 1 - A ( Z l ) exp with
The proton stopping power is known.lg Brown19 has developed an equation of the form
by least squares fitting the data of Northcliffe and chilling." The most recent data are those of Janni4
The experimental determination of d E / h is achieved by passing ions of known initial energy through a thin layer of a stopping material and measuring the energy loss. The thickness Ax of the material should be small enough that d E / h = AE/Ax. Unfortunately, such a value of Ax is so small, especially for very heavy ions, that the precision of measuring Ax is questionable and the uniformity of the layer has an effect on the measurement. Typical experimental results of stopping power are presented in Fig. 4.12. The data of Fig. 4.12 come from Ref. 13. The solid line is based on the following empirical equation proposed by Bridwell and B U C ~ " and Bridwell and Moak2':
where T is the kinetic energy of the ion in MeV.
For a compound or mixture, d E / h can be obtained by using Eq. 4.12 with ( d E / d ~ ) ~ obtained from Eq. 4.36 or Eq. 4.40.
At velocities v < v , z , ~ / ~ , the energy loss through nuclear elastic collisions becomes important. The so-called nuclear stoppingpower is given by the follow- ing approximate expressionlo :
While the electronic stopping power ( d ~ / d p ) , continuously decreases as the ion speed v decreases, the nuclear stopping power increases as v decreases, goes through a maximum, and then decreases again (Fig. 4.13).
F
lo2 - I I I I 1
20 40 60 80 100 (MeV)
Energy
Figure 4.12 Energy loss of iodine ions in several absorbers (Ref. 13). The curves are based on Eq.
4.40.
4.7.3 Range of Heavy Ions
The range of heavy ions has been measured and calculated for many ions and for different absorbers. But there is no single equation-either theoretical or empirical-giving the range in all cases. Heavy ions are hardly deflected along their path, except very close to the end of their track, where nuclear collisions become important. Thus the range R, which is defined as the depth of penetra- tion along the direction of incidence, will be almost equal to the pathlength, the actual distance traveled by the ion. With this observation in mind, the range is given by the equation
150 MEASUREMENT AND DETECTION OF RADIATION
0.6 r /
Figure 4.13 The electronic and nuclear energy loss as a func-
.- I I I &1 tion of the dimensionless en-
0 1 2 3 4 ergy (Ref. 12).
Results of calculations based on Eq. 4.42 are given by many authors. Based on calculations described in Ref. 12, Siffert and ~ o c h e " present universal graphs for several heavy ions in silicon (Figs. 4.14 and 4.15).
The range of a heavy ion in a compound or mixture is calculated from the range in pure elements by using the e q u a t i ~ n ~ ~ , ~ ~
where R, = range, in kg/m2, in element i w i = weight fraction of ith element