Abstract
This chapter prevents a number of theories associated with soft collisions, that is, those interactions between a charged projectile and the entire atom (which Bohr referred to as nuclear collisions). We begin with applying our earlier analysis of projectile momentum and atomic electron screening in elastic scatter to a new analysis of the conditions of soft collisions. Then, the theory of soft-collision energy loss is developed using classical mechanics. First, the Rutherford formula is developed. This is followed by a thorough analysis of the Bohr theory from both classical and semi-classical points of view. As an aside, the Fermi model of soft-collision energy loss, which is based upon classical electrodynamic theory, is investigated not only on its own right but also for the foundations that it provides for the derivation of the Bethe quantum-mechanical theory of soft-collision energy loss and the effects of a condensed medium upon collision energy loss as discussed in Chap. 12. Bethe’s theory is then developed. Initially, this is for the nonrelativistic regime and is then extended to relativistic energies.
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- 1.
Feynman noted the interesting quandary in orbital velocity that would arise for any elements that exist with Z > 137.
- 2.
And not to be confused with the Rutherford differential cross section we derived in Chap. 4.
- 3.
That the units of the right-hand side of (8.115) should be equal to that of energy may perhaps seem peculiar to the reader, but may be verified using dimensional analysis. To confirm that this is so, first, using SI units, note that the unit for the Fourier transform of the electric field is V•s/m and that that for e 2∣E(ω 0)∣2 is equal to kg2•m2•s−2. Hence, the unit for the right-hand side of (8.99) is kg•m2•s−2 = N•m = J.
- 4.
For the astute reader applying dimensional analysis mentally throughout these derivations, the SI units of this result in (r,t) space are indeed C/m3 as the units of the Dirac δ-function are the reciprocal of its argument.
- 5.
This derivation emphasises the fact that the unit of the linear stopping power is that of force.
- 6.
These ‘shell’ correction factors are discussed in Chap. 12.
- 7.
The energy of the ground state is considered to be equal to zero.
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McParland, B.J. (2014). Soft Collisions. In: Medical Radiation Dosimetry. Springer, London. https://doi.org/10.1007/978-1-4471-5403-7_8
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