# Course Outline for Physics 880.05

## II. D. Nucleus-Nucleus Collisions (Chs. 12,13) Part 2

In this period we look at a Glauber approach to nucleus-nucleus collisions in detail. The material corresponds to sections 12.2 and 12.3 in Wong's text.

### Assumptions in Glauber Approach

• High energy, so small wavelength and straight trajectories are appropriate => geometric picture of collision.
• Neglect elastic collisions and diffractive dissociation, for which the energy loss is small.
• Treat multiple collisions as independent (non-coherent) => we can add cross sections.
• Treat the NN cross section \sigma_in (inelastic!) as constant
• How can you estimate a typical value? ("Geometric" cross section)
• Neglect slight energy dependence for problems of interest
• Ignore differences in cross section if incident nucleon becomes an excited baryon after a collision. Treat subsequent collisions still using \sigma_in.

### Review: Definition of Cross Section

• We have an incident beam of projectiles (protons, lead nuclei, etc.), distributed uniformly over the transverse area.
• Note that distributing uniformly in area is not the same as distributing uniformly in the impact parameter b. [This is relevent for the problem set!]
• Let us run our experiment for a fixed time, during which N_in particles are incident over an area "Area".
• We record how many total scattering events occur, call it N_sc.
• It seems reasonable that N_sc will be proportional to N_in; if we run longer so that N_in is three times as large, we expect N_sc to be three times as large.
• It also follows that N_sc is proportional to N_in/Area, the "flux" (we have still fixed the time); if we increased Area with the flux fixed, we'd still expect the same number of scattering events.
• So we define the proportionality constant, which has dimensions of an area (because of Area in the denominator), to be the total cross section \sigma_tot:
N_sc = \sigma_tot N_in/Area
• This means that the probability for one of the incident particles scattering is N_sc/N_in = \sigma_tot/Area (that is, the ratio of scattered events to incident particles in Area is the fraction cross section to Area).
• To define a differential cross section, we assign each of the N_sc scattering events to a bin in some variable(s), such as the x value of the leading protons (as in Fig. 12.1).
• The number in each bin labeled by x_i, \Delta N_sc(x_i), defines the partial cross section \Delta \sigma(x_i) by
\Delta N_sc = \Delta\sigma N_in/Area
just as for the total cross section.
• For small enough bins (and plenty of counts), \Delta N_sc and \Delta\sigma will be directly proportional to the bin width \delta x (double the size of the bins, we'll get twice as many counts per bin). So we introduce the differential cross section:
\Delta\sigma = d\sigma/dx \Delta x
• One can think of \sigma(x) as the accumulated total of the \Delta \sigma(x)'s up to x (in this case, from 0 to x). Then this is a monotonically increasing function, whose slope is the differential cross section d\sigma/dx. [Note that the total cross section is given by \sigma(x=1).]
• We can bin according to any set of kinematic variables, like the three components of momentum, which then defines d\sigma/d^3p and the invariant version E d\sigma/d^3p.

### Basic Setup of the Glauber Approach

The general idea behind the Glauber approach can be stated in terms of figure 12.2 in Wong.

• Projectile nucleus B collides with target nucleus A, at well-defined impact parameter \vec b (which is the vector in the transverse plane [no z component!] from the centerline of A to the centerline of B).
• Any of the B ("B" is a number here) nucleons in B ("B" is the name of the nucleus here) can collide with any of the A nucleons in A. So a total of A*B collisions are possible.
• Our goal (for now) is to find the probability that there will be n nucleon-nucleon (NN) collisions during the big collisions, where n is between 0 and A*B.
• Our inputs are the NN inelastic cross section \sigma_in and the density distributions in the nuclei.
• The idea is to sum over all little volumes in the two nuclei (as in the figure), calculating the probability that there is a nucleon in each and then the probability that two nucleons at such a separation will collide. Let's do it!

### Thickness Function for Nucleon-Nucleon Scattering

Let's start with the last piece of information: The probability that two nucleons at a given separation, the impact parameter b_NN, will collide. [This is a vector in general, but only the magnitude matters for us.] We define the probability in terms of the "thickness function" t(b_NN).

• Use our given knowledge of the total cross section \sigma_in; split it into contributions from different b_NN's.
• Associate the cross section with a radius R_in:
\sigma_in = \pi R_in^2
• If b_NN < R_in, then no collision (probability=0).
If b_NN > R_in, then there is a collision (probability=1).
• So we can define a probability per unit area t(b_NN) by:
t(b_NN) = 1/\sigma_in if b_NN < R_in => probability=1
t(b_NN) = 0 if b_NN > R_in => probability=0
• Note that the probability per unit area integrated over all impact parameters is 1 (as expected):
\int t(b_NN) d\vec b = 1/\sigma_in * \pi R_in^2 = 1
• The probability of a collision in this model is t(b_NN)*\sigma_in.
• Define t(b_NN) so that the probability of a collision in general is t(b_NN)*\sigma_in. So the geometric interpretation of t(b_NN) is the probability/unit area of finding the nucleon at b_NN.
• A more realistic model is t(b_NN) = (1/2\pi\beta_p^2) e^{-b_NN^2/2\beta_p^2}
where \beta_p = 0.68 fm. This number comes from the analysis of NN elastic scattering; think of it as the matter distribution inside a proton or neutron.
• Again, t(b_NN) is normalized.
• The simplest model to use, reasonable for most purposes, is that t(b_NN) is just a delta function. That is, the nucleons collide when they line up (b_NN=0).

### Extension to Nucleus-Nucleus Scattering

• Return to figure 12.2. With the nuclei at relative impact parameter \vec b, we'll first find the probability of a particular nucleon in B colliding with a particular nucleon in A.
• Sum over all places the nucleons can be, and for each case multiply by the probability of a collision t(b_NN).
• So divide A and B into small volumes, as shown. Their positions are (\vec b_A, z_A) and (\vec b_B, z_B).
• The impact parameter for nucleons in those volumes is
\vec b_NN = \vec b - \vec b_A + \vec b_B
(The sign of the last term is different in the book, but we can take \vec b_B -> -\vec b_B since we integrate over all \vec b_B.)
• We take the nucleons to be distributed according to the matter density obtained from electron scattering from nuclei: \rho_A(\vec r_A) = \rho(\vec b_A, z_A). This is typically parameterized as a Fermi shape. \rho is normalized to one (the integral over all space is one), since the nucleon must be somewhere!
• The probability to find a given nucleon in the "tube" of cross section d\vec b_B is
P_B(\vec b_B) = d\vec b_B * \int dz_B \rho(b_B,z_B) \equiv T(b_B) d\vec b_B
which defines T(b_B). There is an analogous definition for T(b_A).
• If the nucleus is moving, the element dz_B -> 1/\gamma dz_B but the density also changes: \rho -> \gamma \rho, so the \gamma's cancel and the answer is the same. We are just counting how many are in a region defined by a tranverse area, so the result must be invariant under boosts along the z-direction.
• The thickness functions T(b_A) and T(b_B) are well-modeled as gaussians:
T(\vec b_A) = (1/2\pi\beta_A^2) e^{-b^2/2\beta_A)
T(\vec b_B) = (1/2\pi\beta_B^2) e^{-b^2/2\beta_B)
with \beta_i = r'_0 A_i^{1/3} / sqrt{3} where r'_0 = 1.05 fm.
• To find the probability of a collision, we sum (integrate) over all places the nucleons could be (all b_A and b_B):
P = \int d\vec b_A d\vec b_B T_A(b_A) T_B(b_B) t(\vec b - \vec b_A + \vec b_B) \sigma_in \equiv T(\vec b)\sigma_in
which defines the thickness function T(\vec b).
• When gaussians are used for the T's and t, T(\vec b) is:
T(\vec b) = T(b) = (1/2\pi\beta^2) e^{-b^2/2\beta^2}
where \beta^2 \equiv \beta_A^2 + \beta_B^2 + \beta_p^2. [To prove this, write each of the gaussians as its fourier transform and interchange the order of integration.]
• Now that we know the probability of a given collision, what is the probability of n collisions (still at \vec b).
• A total of A*B are possible. We want the probability of exactly n pairs colliding.
• So first we pick the n pairs. There are (A*B n) = (A*B)!/n!(A*B-n)! ways to do this.
• For each pair, the probability of a collision is T(b)\sigma_in and the probability of a miss is (1 - T(b)\sigma_in).
• The collisions must happen n times and the misses the rest (A*B-n), so the probability we seek is:
P(n,\vec b) = (A*B n) (T(b)\sigma_in)^n (1 - T(b)\sigma_in)^(A*B-n)
• Note that \sum_n=0^{A*B} P(n,\vec B) = 1 (either no collisions or at least one!).
• The probability of any collision just excludes n=0:
\sum_n=1^{A*B} P(n,\vec B) = 1 - P(0,b) = 1 - (1 - T(b)\sigma_in)^{A*B}
=> d\sigma^{AB}_in/d\vec b (the ratio of areas)
• The total inelastic cross section is the sum over all impact parameters:
\sigma^{AB}_in = \int d\vec b [1 - (1 - T(b)\sigma_in)^{AB}]
(gaussians) => = -2\pi \beta^2 \sum_1^{A*B} (AB n) (-f)^n/n
where f = \sigma_in/2\pi\beta^2. [For gaussians, we can do all the integrals after expanding (1 - T(b)\sigma_in)^{AB}.]
• We can also compute the average number of collisions at impact parameter b:
<n(b)> = \sum_n=1^{A*B) n P(n,b) = A*B*T(b)*\sigma_in
as well as the average number when there is at least one.
• To get the total probability of n collisions, we integrate over \vec b to get P_tot(n) and <n'> (the ' means at least one collision). The result is
P_tot(n) = [ \int d\vec b P(n,\vec b) ] / \sigma^(AB)_in
where we've normalized the distribution to add up to 1 summed over n=1 (not n=0) to n=A*B.
• Analytic results for gaussians and the case B=1 are given in the book.
• That's it!