Sec. 4.8] PROTONS, DEUTERONS, AND ALPHA PARTICLES 87 



The evaluation of the integral must be carried out numerically, and when 

 possible the integration is usually started at some well-established experi- 

 mental value of the range and energy rather than that at zero kinetic energy, 

 thus avoiding the error introduced by the poor fit of the stopping formula at 

 very low energies where electron capture and loss become effective. 



Unlike the comparatively indeterminate range of beta particles and the 

 exponential absorption of gamma rays, the ranges of heavy charged particles 

 are well defined in that all particles of an initially monoenergetic beam are 

 brought to rest after traversing the same distance through an absorber, with 

 only a small statistical spread or straggling about the mean range. On the 

 other hand, heavier particles with a higher charge number such as fission 

 fragments do not exhibit well-defined ranges. Their charge varies rapidly 

 with velocity due to electron capture, and elastic collisions with nuclei play 

 a more important part; hence, the stopping formula cannot be applied even 

 for a first approximation. It is probable, however, that at very great 

 energies, in the order of 1,000 mev or more depending upon the mass of the 

 particle, the stopping formula would also be applicable to these particles. 

 At such energies the velocity would be great enough to satisfy the conditions 

 required by the formula, and the charge would then remain constant. How- 

 ever, even at such energies it is doubtful that the stopping formula will prove 

 to be very useful since the range of a highly charged particle is small and the 

 straggling large. 



At the present time the most accurately measured ranges are those of alpha 

 particles from the natural radioactive isotopes. They have been measured 

 repeatedly by numerous investigators over a period of many years and now 

 stand as the final test of the accuracy of range calculations, at least for 

 energies up to 10 mev. 



On the basis of early measurements of the range of alpha particles, three 

 empirical relations were established for the variation of range in air as a 

 function of energy: 



1. Range between to 3 cm, 



R r^ V H ~ E 3/i cm 



2. Range between 3 to 7 cm (Geiger formula), 



R = 9.67 X 10" 2 V cm 



3. Range above 7 cm, 



R ^ y i r^ E 2 cm 



These relations should be regarded only as qualitative, because in the 

 variation of R with v n , the exponent n also depends on the velocity and 

 increases from n = 1.4 for low velocities to n — 4.0 for very high velocities [6]. 



The relative range of a particle in a substance compared with its range in 



