Astroparticle Physics

46 3 Kinematics and Cross Sections3.4 Cross Sections“Physicists are, as a general rule, highbrows.They think and talk in long, Latinwords, and when they write anythingdown they usually include at least onepartial differential and three Greek letters.”Stephen Whitecross sectionApart from the kinematics of interaction processes the crosssection for a reaction is of particular importance. In the mostsimple case the cross section can be considered as an effectivearea which the target particle represents for the collisionwith a projectile. If the target has an area of πrT 2 and the projectilesize corresponds to πrP 2 , the geometrical cross sectionfor a collision is obtained to beσ = π(r T + r P ) 2 . (3.53)interaction lengthIn most cases the cross section also depends on other parameters,for example, on the energy of the particle. The atomiccross section σ A , measured in cm 2 , is related to the interactionlength λ according toAλ {cm} =N A {g −1 } ϱ {g/cm 3 } σ A {cm 2 }(3.54)absorption coefficientinteraction rates(N A – Avogadro number; A – atomic mass of the target, ϱ– density). Frequently, the interaction length is expressed by(λ ϱ) {g/cm 2 }. Correspondingly, the absorption coefficientis defined to beµ {cm −1 }= N A ϱσ A= 1 A λ ; (3.55)equivalently, the absorption coefficient can also be expressedby (µ/ϱ) {(g/cm 2 ) −1 }.The absorption coefficient also provides a useful relationfor the determination of interaction probabilities or rates,φ {(g/cm 2 ) −1 }= µ ϱ = N AA σ A . (3.56)If σ N is a cross section per nucleon, one hasφ {(g/cm 2 ) −1 }=σ N N A . (3.57)