By use of classical mechanics, Bohr developed an equation of stopping power, -dE/dx, given as the product of a kinematic factor and a stopping number.
The kinematic factor includes such terms as the electronic charge and mass, the number of atoms per cubic centimetre of the medium, and the velocity of the incident charged particle. The stopping number includes the atomic number and the natural logarithm of a term that includes the velocity of the incident particle as well as its charge, a typical transition energy in the system (see ; a crude estimate is adequate because the quantity appears within the logarithm), and Planck’s constant, h. Bohr’s stopping-power formula does not require knowledge of the details of atomic binding. In terms of the stopping number, B, the full expression for stopping power is given by -dE/dx = (4πZ12e4N/mv2)B, where Z1 is the atomic number of the penetrating particle and N is the atomic density of the medium (in atoms/volume).
For a heavy incident charged particle in the nonrelativistic range (e.g., an alpha particle, a helium nucleus with two positive charges), the stopping number B, according to the German-born American physicist Hans Bethe, is given by quantum mechanics as equal to the atomic number (Z) of the absorbing medium times the natural logarithm (ln) of two times the electronic mass times the velocity squared of the particle, divided by a mean excitation potential (I) of the atom; i.e., B = Z ln (2mv2/I).
Bethe’s stopping number for a heavy particle may be modified by including corrections for particle speed in the relativistic range (β2 + ln [1 - β2]), in which the Greek letter beta, β, represents the velocity of the particle divided by the velocity of light, and polarization screening (i.e., reduction of interaction force by intervening charges, represented by the symbol δ/2), as well as an atomic-shell correction (represented by the ratio of a constant C to the atomic number of the medium); i.e., B = Z (ln 2mv2/I - β2 - ln[1 - β2] - C/Z - δ/2).
The most important nontrivial quantity in the equation for stopping number is the mean excitation potential, I. Experimental values of this parameter, or quantity, are known for most atoms, but no single theory gives it over the whole range of atomic numbers because the calculation would require knowledge of the ground states and all excited states. Statistical models of the atom, however, come close to providing a theory. Calculations by the American physicist Felix Bloch in 1933 showed that the mean excitation potential in electron volts is about 14 times the atomic number of the element through which the charged particle is passing (I = 14Z). A later calculation gives the ratio of the potential to atomic number as equal to a constant (a) plus another constant (b) times the atomic number raised to the -2/3 power in which a = 9.2 and b = 4.5—i.e., I/Z = a + bZ-2/3. This formula is widely applicable. Other exact quantum-mechanical calculations for hydrogen give its mean excitation potential as equal to 15 eV.
Even though the basic stopping-power theory has been developed for atoms, it is readily applied to molecules by virtue of Bragg’s rule (named for the British physicist William H. Bragg), which states that the stopping number of a molecule is the sum of the stopping numbers of all the atoms composing the molecule. For most molecules Bragg’s rule applies impressively within a few percent, though hydrogen (H2) and nitrous oxide (NO) are notable exceptions. The rule implies: (1) similarity of atomic binding in different molecules having one common atom or more, and (2) that the vacuum ultraviolet transitions, in which most electronic transitions are concentrated under such irradiation, involve energy losses much higher than the strengths of most chemical bonds.
The charge on a heavy positive ion fluctuates during penetration of a medium. In the beginning it captures an electron, which it quickly loses. As it slows down, however, the cross section of electron loss decreases relative to that for capture. Basically, the impinging ion undergoes charge-exchange cycles involving a single capture followed by a single loss. Ultimately, an electron is permanently bound when it becomes energetically impossible for the ion to lose it. A second charge-exchange cycle then occurs. This phenomenon continues repeatedly until the velocity of the heavy ion approximates the orbital velocity of the electron in Bohr’s theory of the atom, when the ion spends part of its time as singly charged and another part as a neutral atom. The kinematic factor in the expression for stopping power is proportional to the square of the nuclear charge of the penetrating particle, and it is modified to account for electron capture as the particle slows down. On slowing down further, the electronic energy-loss mechanism becomes ineffective, and energy loss by elastic scattering dominates. The mathematical expressions presented here apply strictly in the high-velocity, electronic excitation domain.
Range
The total path length traversed by a charged particle before it is stopped is called its range. Range is considered to be taken as the sum of the distance traversed over the crooked path (track), whereas the net projection measured along the initial direction of motion is known as the penetration. The difference between range and penetration distances results from scattering encountered by the particle along its path. For heavy charged particles with high initial velocities (those that are appreciable fractions of the speed of light), large-angle scatterings are rare. The corresponding trajectories are straight, and the difference between range and penetration distance is, for most purposes, negligible.
Particle ranges may be obtained by (numerical) integration of a suitable stopping-power formula. Experimentally, range is more easily measured than is stopping power. For heavy particles a critical incident energy in low-atomic-number mediums is 1,000,000 eV divided by the mass of the particle in atomic mass units (amu)—i.e., 1 MeV/amu. For incident energies higher than this critical value, range is usually well-known, and computation agrees with experiment within about 5 percent. In the case of aluminum, which is the best studied material, the accuracy is within about 0.5 percent. For incident energies less than the critical value, however, range calculations are usually uncertain, and agreement with experiment is poor. The range–energy relation is often given adequately as a power law, that range (R) is proportional to energy (E ) raised to some power (n); that is, R ∝ En. Protons in the energy interval of a few hundred MeV conform to this kind of relation quite well with the exponent n equal to 1.75. Similar situations exist for other heavy particles. Measurements of range and stopping power are of great importance in particle identification and measurement of their energies. Many experimental data and computations are available for ranges of heavy particles as well as of electrons. The theory by which Bethe derived a stopping number is generally accepted as providing the framework for understanding the variation of range with energy, though in practice the mean excitation potential, I, must be obtained in many cases by experimental curve fitting.
Both stopping power and range should be understood as mean (or average) values over an ensemble of atoms or molecules, because energy loss is a statistical phenomenon. Fluctuations are to be expected. In general, these fluctuations are called straggling, and there are several kinds. Most important among them is the range straggling, which suggests that, for statistical reasons, particles in the same medium have varying path lengths between the same initial and final energies. Bohr showed that for long path lengths the range distribution is approximately Gaussian (a type of relationship between number of occurrences and some other variable). For short path lengths, such as those encountered in penetration of thin films, the emergent particles show a kind of energy straggling called Landau type (for the Soviet physicist Lev Landau). This energy straggling means that the distribution of energy losses is asymmetric when a plot is drawn, with a long tail on the high-energy-loss side. The intermediate case is given by a distribution according to Sergey Ivanovich Vavilov, a Soviet physicist, that must be evaluated numerically. There is evidence in support of all three distributions in their respective regions of validity.
The ionization density (number of ions per unit of path length) produced by a fast charged particle along its track increases as the particle slows down. It eventually reaches a maximum called the Bragg peak close to the end of its trajectory. After that, the ionization density dwindles quickly to insignificance. In fact, the ionization density follows closely the LET. With slowing, the LET at first continues to increase because of the strong velocity denominator in the kinematic factor of the stopping-power formula. At low speeds, however, LET goes through a maximum because of: (1) progressive lowering of charge by electron capture, and (2) the effect of the logarithmic term in the stopping-power formula. In general, the maximum occurs at a few times the Bohr orbital velocity. A curve of ionization density (also called specific ionization or number of ion pairs—negative electron and associated positive ion—formed per unit path length) versus distance in a given medium is called a Bragg curve. The Bragg curve includes straggling within a beam of particles; thus, it differs somewhat from the specific ionization curve for an individual particle in that it has a long tail of low ionization density beyond the mean range. The mean range of radium-C′ alpha particles in air at normal temperature and pressure (NTP), for example, is 7.1 centimetres; the Bragg peak occurs at about 6.3 centimetres from the source with a specific ionization of about 60,000 ion pairs per centimetre.
