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.

- 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.

- 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!]

- Note that distributing uniformly in area is
- 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.

- The number in each bin labeled by x_i, \Delta N_sc(x_i),
defines the partial cross section \Delta \sigma(x_i) by

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!

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. - Start with a constant, geometric cross section picture.
- 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.

- Associate the cross section with a radius R_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).

- 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!

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Copyright © 1997,1998 Richard Furnstahl and James Steele.