7.1.3 The Bragg Curve

Created October 19,1995

7.1.3.1 The Bragg Curve

THE BRAGG CURVE

Heavy Particles

The plot of specifc energy loss (which can be related to specifc ionization) along the track of a charged particle is called a Bragg Curve. A typical Bragg curve is depicted in the following graphic for an alpha particle of several MeV of initial energy:

As the energy falls, the specific energy loss increases according to the Bethe-Bloch formula. As the energy falls below a threshold, however, an electron will attach to the alpha, dramatically lowering the specific energy loss.

ELECTRONS

The energy deposition of the electron increases more slowly with penetration depth due to the fact that its direction is changed so much more drastically (Johns and Cunningham, p. 213). In fact, these authors state: "There is no increase in energy deposited near the end of the tract and the Bragg peak for electrons is never observed."

The explanation of Sorenson and Phelps, however, helps shed light on this topic. They state (p. 170) : "A similar increase in ionization density is seen at the end of an electron track; however, the peak occurs when the electron energy has been reduced to less than about 1 keV, and it accounts for only a small fraction of its total energy." Therefore it seems that for electrons and their tortuous paths the energy deposition is spread in the transverse direction as it progresses forward in the initial direction, until the last 1 keV which is deposited pretty much along a straight line.

7.1.3.2 Energy Straggling

Energy loss in a material is a statistical or stochastic process. Therefore, a spread of energies always results when an initially monoenergetic beam of particles encounters an absorber. Many particles lose the "average energy," although some will lose not so much and some will lose more than the average. This results in a finite width to the energy distribution curve known as "energy straggling"

Recall our barefoot soldiers in a field of broken glass analogy. With photons, soldiers in a mine field, some can make it through uneffected, i.e. unattenuated. With charged particles, ALL PARTICLES LOSE ENERGY just as all soldiers will suffer cut feet and slow down in the field of glass.

The straggling peak is approximately Gaussian shaped, with a width that increases with the ratio Z/A. In other words, at lower atomic numbers (where the Z/A ratio is higher) we have wider widths. This is due to the fact that there is less shielding of inner electrons in the lower Z elements - there is more stopping influence per electron.

7.1.3.3 Mass Stopping Power

As with the mass attenuation coeficient, which is defined as mu/rho, where mu is the linear attenuation coefficient and rho is the absorber density, we can define a "mass stopping power." The mass stopping power is given by :

(1/rho) * (dT/dx) in units of keV/(gm/cm**2)

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Douglas J. Wagenaar, Ph.D., wagenaar@nucmed.bih.harvard.edu