Computational Lab in Physics:
Final Project
Monte Carlo Nuclear Collisions: Glauber Model
Nuclear Charge Densities

Charge densities: similar to a hard
sphere.

Edge is “fuzzy”.
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For the Pb nucleus (used at LHC)

Woods-Saxon density: r (r) =




R = 1.07 fm * A 1/3
a =0.54 fm
A = 208 nucleons
Probability :
r0
1+ e
r-R
a
µ r 2 r (r)
3
Nuclei: A bunch of nucleons

Each nucleon is distributed with:
P(r,q , f ) = r(r)dV = r(r)r 2 drd(cosq )df

Angular probabilities:

Flat in f, flat in cos(q).
4
Impact parameter distribution




Like hitting a target:
Rings have more area
Area:
Area proportional to probability
2p bdb
5
Collision:



2 Nuclei colliding
Red: nucleons from nucleus A
Blue: nucleons from nucleus B
6
Interaction Probability vs. Impact
Parameter, b


After 100,000 events
Beyond b~2R Nuclei
miss each other


Largest probability:


Note fuzzy edge
Collision at b~12-14
fm
Head on collisions:

b~0: Small probability
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Binary Collisions, Number of
participants




If two nucleons get closer than d<s/p they collide.
Each colliding nucleon is a “participant” (Dark colors)
Count number of binary collisions.
Count number of participants
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Find Npart, Ncoll, b distributions

Nuclear Collisions
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Final Project: Monte Carlo Model
of Nuclear Collisions
1.
Nuclear Density Function

2.
Make plots of the nuclear density for the Pb
nucleus (see slide 3),
Distribution of nucleons in the nucleus


Using the nuclear density function, write a
function that will randomly distribute A
nucleons in the nucleus (A=208 for Pb).
Make a plots of the x-y, and x-z coordinates
of the nucleons in sample nucleus.

You will need to distribute them in 3D. You can
use spherical polar coordinates (slide 3,4),
then convert to cartesian.
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Final Project: Monte Carlo Model
of Nuclear Collisions
3.
Impact Parameter, b


4.
Make a plot of the impact parameter probability
distribution (slide 5)
For b = 6 fm, make an example collision between two
nuclei. Plot the x-y coordinates of the nucleons in each
nucleus (color coded, see slide 6, left plot).
Number of collisions, Number of participants

For each pair of nucleons (one from nucleus A, one from
nucleus B), check if there is a collision.

Nucleon-Nucleon Collision:




Find the distance d in the x-y plane between each nucleon-nucleon pair
(the z axis is the beam axis, see slide 6)
Collision: when d2<s/p. Use s = 60 mb (where 1 b = 10-28 m2).
Any nucleon that collides is called a “participant”. Color each participant a
darker color (as in slides 6, 8).
Count the number of nucleon-nucleon collisions.
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Final Project: Monte Carlo Model
of Nuclear Collisions
5.
Many collisions!




Simulate 106 nucleus-nucleus collision events.
Draw a random impact parameter from the
distribution in slide 5.
Calculate Npart, Ncoll for each collision.
For those events where there was an
interaction (Ncoll>1), fill histograms of
the impact parameter, b.
 the number of participants
 the number of collisions (see slide 9)


In part II of the project, we will model particle
production, and compare it against CMS data. 12
For compiling with root: Using a
makefile
Enter the following in a file called e.g. make_mcglauber
#ROOTCFLAGS =$(shell root-config --cflags)
#ROOTLIBS =$(shell root-config --libs)
#ROOTGLIBS =$(shell root-config --glibs)
CXXFLAGS =$(shell root-config --cflags)
LIBS
=$(shell root-config --libs)
mcglauber:
g++ $(CXXFLAGS) -o mcglauber mcglauber.cc $(LIBS)
Then, call the makefile using:
make -f make_mcglauber
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Useful code Snippets.
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Text output into many files: ofstream
#include <fstream>
char filename[255];
sprintf(filename,"MandelbrotRunDataAdapIter%i.txt",i);
cout << filename << endl;
ofstream mandelOutput(filename);
for (double c_real=RealPartMin[i];
c_real<RealPartMax[i];c_real+=stepSizeRe) {
for (double c_imag=ImagPartMin[i]; c_imag<ImagPartMax[i];
c_imag+=stepSizeIm) {
// iterate until iter>100 or |z|>4
mandelOutput << c_real << '\t' << c_imag << '\t' << iter << endl;
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Using the complex class

Operators defined for complex numbers


Overloading of operators:




+, -, *, /, -=, +=, /=, *=, =, ==, !=
complex_ob + scalar
scalar + complex_ob
complex_ob + complex_ob
Functions:

Returning a scalar:






T abs(const complex<T> &ob) : absolute value
T arg(const complex<t> &ob) : phase angle
T conj(…) : complex conjugate
cos, cosh, exp, imag, log, log10, pow, real, sin, sinh sqrt, tan, tanh, do the
obvious
T norm(const complex<T> &ob) : magnitude squared
complex<T> polar(const T& v, const T& theta=0) : magnitude specified by
v and phase angle specified by theta
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Demonstrating complex class
#include <iostream>
#include <complex>
using namespace std;
int main() {
complex<double> cmpx1(1,0);
complex<double> cmpx2(1,1);
cout << cmpx1 << “ “ << cmpx2 << endl;
complex<double> cmpx3 = cmpx1*sqrt(cmpx2);
cout << cmpx3 << endl;
cmpx3+=10;
cout << cmpx3 << endl;
return 0;
}
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Reading data from text files
//loop to read many files that have a similar name
// and one number to index them.
for (int i=0; i<nPlots; ++i) {
char filename[255];
sprintf(filename,"MandelbrotRunData%i.txt",i);
ifstream inputFile(filename);
//read the data
// example, read 1 line:
char buffer[255]l;
inputFile.getline(buffer,255);
//example, read 3 numbers from a line
double number1, number2, number3;
inputFile >> number1 >> number2 >> number3;
// note that one must take care of end-of-lines.
// another useful loop: while(!inputFile.eof()) { // keep reading next line }
// do something with data.
}
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Playing with Palettes

gStyle:



SetPalette(1,0);
 RGB colors in the
range 51 to 99
SetPalette(51,0);
 Deep blue to light blue
Look in TColor class:
 SetPalette
 CreateGradientColorTa
ble
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More control over colors:
void colorPalette() {
//example of new colors (greys) and definition of a new palette
const Int_t NRGBs = 5;
const Int_t NCont = 256;
Double_t stops[NRGBs] = { 0.00, 0.30, 0.61, 0.84, 1.00 };
Double_t red[NRGBs] = { 0.00, 0.00, 0.57, 0.90, 0.51 };
Double_t green[NRGBs] = { 0.00, 0.65, 0.95, 0.20, 0.00 };
Double_t blue[NRGBs] = { 0.51, 0.55, 0.15, 0.00, 0.10 };
TColor::CreateGradientColorTable(NRGBs, stops, red, green, blue, NCont);
gStyle->SetNumberContours(NCont);
}
TF2 *f2 = new TF2("f2","exp(-(x^2) - (y^2))",-1.5,1.5,-1.5,1.5);
//f2->SetContour(colNum);
f2->SetNpx(300);
f2->SetNpy(300);
f2->Draw("colz");
return;
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