It was mentioned earlier that most of the interactions of a charged particle involve the particle and atomic electrons. If the mass of the electron is taken as 1, then the masses of the other common heavyt charged particles are the
'1n this discussion, "heavy" particles are all charged particles except electrons and positrons.
following:
Electron mass = 1 Proton mass = 1840 Deuteron mass = 2(1840)
Alpha mass = 4(1840)
If the incoming charged particle is an electron or a positron, it may collide with an atomic electron and lose all its energy in a single collision because the collision involves two particles of the same mass. Hence, incident electrons or positrons may lose a large fraction of their kinetic energy in one collision. They may also be easily scattered to large angles, as a result of which their trajectory is zig-zag (Fig. 4.2). Heavy charged particles, on the other hand, behave differ- ently. On the average, they lose smaller amounts of energy per collision. They are hardly deflected by atomic electrons, and their trajectory is almost a straight line.
Assuming that all the atoms and their atomic electrons act independently, and considering only energy lost to excitation and ionization, the average energy losst per unit distance traveled by the particle is given by Eqs. 4.2, 4.3, and 4.4.
(For their derivation, see the chapter bibliography: Evans, Segr6, and Roy and Reed.)
Stopping power due to ionization-excitation for p, d, t , a.
Stopping power due to ionization-excitation for electrons.
Stopping power due to ionization-excitation for positrons.
'since E = T + Mc2 and Mc2 = constant, dE/dx = dT/dx; thus, Eqs. 4.2 to 4.4 express the kinetic as well as the total energy loss per unit distance.
*1n SI units, the result would be J/m; 1 MeV = 1.602 X lo-" J.
126 MEASUREMENT AND DETECTION OF RADIATION
Electron or positron
trajectory Figure 4.2 Possible electron and -I-Heavy particle trajectory -heavy particle trajectories.
where r, =
4.rrr; =
mc2 =
Y = M =
P =
N = N =
z =
Z =
I =
e2/mc2 = 2.818 X 10-l5 m = classical electron radius
9.98 x m2 .= m2 = cm2
rest mass energy of the electron = 0.511 MeV (T + M C ~ ) / M C ~ = I / - 4
T = kinetic energy = ( y - 1)Mc2 rest mass of the particle
v/c c = speed of light in vacuum = 2.997930 X lo8 m/s = 3 x lo8 m/s
number of atoms/m3 in the material through which the particle moves
p(NA/A) NA = Avogadro's number = 6.022 X atoms/mol A = atomic weight
atomic number of the material
charge of the incident particle ( z = 1 for e-, e + , p, d; z = 2 for a) mean excitation potential of the material
An approximate equation fo; I , which gives good results for Z > 12,' is
Table 4.1 gives values of I for many common elements.
Many different names have been used for the quantity dE/&: names like energy loss, specific energy loss, differential energy loss, or stopping power. In Table 4.1 Values of Mean Excitation Potentials for
Common Elements and Compoundst
Element I (eV) Element I (eV)
H 20.4 Fe 281*
He 38.5 Ni 303*
Li 57.2 Cu 321*
Be 65.2 Ge 280.6
B 70.3 Zr 380.9
C 73.8 I 491
N 97.8 Cs 488
0 115.7 Ag 469*
Na 149 Au 771 *
A1 160* Pb 818.8
Si 174.5 U 839*
'values of I with * are from experimental results of refs. 2 and 3. Others are from refs. 4 and 5.
this text, the term stoppingpower will be used for d E / h given by Eq. 4.2 to 4.4, as well as for a similar equation for heavier charged particles presented in Sec.
4.7.2.
It should be noted that the stopping power 1. Is independent of the mass of the particle 2. Is proportional to z 2 [(chargeI2] of particle 3. Depends on the speed u of particle
4. Is proportional to the density of the material ( N )
For low kinetic energies, d E / h is almost proportional to l / v 2 . For relativistic energies, the term in brackets predominates and d E / h increases with kinetic energy. Figure 4.3 shows the general behavior of d E / h as a function of kinetic energy. For all particles, d E / h exhibits a minimum that occurs approximately at y = 3. For electrons, y = 3 corresponds to T = 1 MeV; for alphas, y = 3 corresponds to T - 7452 MeV; for protons, y = 3 corresponds to T z 1876 MeV. Therefore, for the energies considered here (see Table 1.1), the d E / h for protons and alphas will always increase, as the kinetic energy of the particle decreases (Fig. 4.3, always on the left of the curve minimum); for electrons, depending on the initial kinetic energy, d E / h may increase or decrease as the electron slows down.
Equations 4.3 and 4.4, giving the stogping power for electrons and positrons, respectively, are essentially the same. Their difference is due to the second term in the bracket, which is always much smaller than the logarithmic term. For an electron and positron with the same kinetic energy, Eqs. 4.3 and 4.4 provide results that are different by about 10 percent or less. For low kinetic energies, d E / h for positrons is larger than that for electrons; at about 2000 keV, the energy loss is the same; for higher kinetic energies, d E / h for positron is less than that for electrons.
As stated earlier, Eqs. 4.2 to 4.4 disregard the effect of forces between atoms and atomic electrons of the attenuating medium. A correction for this density effect6,' has been made, but it is small and it will be neglected here. The density effect reduces the stopping power slightly.
Figure 4.3 Change of stopping power with the kinetic energy of the particle.
128 MEASUREMENT AND DETECTION OF RADIATION
Equations 4.2-4.4 are not valid for very low energies. In the case of Eq. 4.2, a nuclear shell correction is applied (see Ziegler), which appears in the brackets as a negative term and becomes important at low energies ( T I 100 keV). Even without this correction, the value in brackets takes a negative value when (2mc2p2y2)/11 1. The value of this term depends on the medium because of the presence of the ionization potential I. As an example, for oxygen ( I = 89 eV) this term becomes less than 1 for T < 40 keV.
For electrons of very low kinetic energy, Eq. 4.3, takes the form (see Roy &
Reed)
Again for oxygen, the argument of the logarithm becomes less than 1 for electron kinetic energy T < 76 eV. For positrons, the low-energy limit of the validity of Eq. 4.4 is equal to the positron energy for which the whole value within brackets is less than zero.
Example 4.1 What is the stopping power for a 5-MeV alpha particle moving in silicon?
Answer For silicon, A = 28, Z = 14, p = 2.33 kg/m3,
Or, in terms of ~ e v / ( ~ / c m ' ) ,
Example 4.2 What is the stopping power for a 5-MeV electron moving in silicon?
Answer For an electron,
In Ex. 4.2, the stopping power for the 5-MeV electron is, in terms of MeV/
(g/cm2),
Notice the huge difference in the value of stopping power for an alpha versus an electron of the same kinetic energy traversing the same material.
Tables of dE/& values are usually given in units of MeV/(g/cm2) [or in SI units of ~ / ( k ~ / m ' ) ] . The advantage of giving the stopping power in these units is the elimination of the need to define the density of the stopping medium that is necessary, particularly for gases. The following simple equation gives the relationship between the two types of units: