multiplicity distribution of grey particles emitted from hadron-nucleus interactions at high...

8/13/2019 Multiplicity Distribution of Grey Particles Emitted From Hadron-nucleus Interactions at High Energies

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Abstract:

In this project we studied the distribution function of grey particles from

hadron-nucleus interactions at high energies (> 1 GeV) according to a

Geometric model by Andersson, Otterlund and Stenlund . We calculated

the average number of hadron nucleon collisions inside the target nucleus

interactions using the distribution function of the number of hadron-

nucleon collisions according to Glauber theory.

The results are then compared with the experimental data obtained from

proton-Emulsion interactions at high energies (200, 400 and 800 GeV).

Good agreement was obtained with the data, which indicates that the

model is energy independent. It seems also that the production of grey

particles at high energies in hadron-nucleus interactions is insensitive to

the incident energy within the energy used (200-800 GeV).Also the

production of these particles seems to depend strongly on the target size.


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1 Introduction1.1 High energy nuclear detectors1.1.1 Gas filled detectors[1, 2]

Radiation passing through a gas can ionize the gas molecules,

provided the energy delivered by it is higher than the ionization potential

of the gas. The charge pairs thus produced can be made to move in

opposite directions by the application of an external electric field. The

result is an electric pulse that can be measured by an associated

measuring device. A typical gas filled detector would consist of a gas

enclosure and positive and negative electrodes. The electrodes are raised

to a high potential difference that can range from less than 100 volts to a

few thousand volts depending on the design and mode of operation of the

detector. The creation and movement of charge pairs due to passage of

radiation in the gas perturbs the externally applied electric field producinga pulse at the electrodes. The resulting charge, current, or voltage at one

of the electrodes can then be measured, which together with proper

calibration gives information about the energy of the particle beam and/or

its intensity.

It is apparent that such a system would work efficiently if a large

number of charge pairs are not only created but are also readily collected

at the electrodes before they recombine to form neutral molecules. The

choice of gas, the geometry of the detector, and the applied potential give

us controlling power over the production of charge pairs and their

kinematic behavior in the gas.

Examples for filled gas detectors: bubble chambers, cloud chambers,

streamer chambers, spark chambers and proportional tube.


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1.1.2 liquid filled detectors[1, 2]There is no reason why a liquid cannot be used as an ionizing medium

for detection of radiation. When radiation passes through a liquid, itproduces charge pairs, which can be directed towards electrodes for

generation of a pulse. If the liquid assures good proportionality between

the energy deposited and the number of charge pairs generated, the height

of the pulse would give a good measure of the energy deposited. As it

turns out, there are a number of liquids that have fairly good

proportionality and therefore can be used as detection media. Now, one

would expect the charge recombination probability in a liquid to be much

higher than that in a typical gas. This is certainly true but we should also

remember that the higher density ensures production of larger number of

charge pairs as well. We will discuss these two competing factors later in

the chapter, but the point to consider is that, in principle, liquids can be

used as ionizing media to detect and measure radiation. Apart from a

section on bubble chambers, in this chapter we will concentrate on

different types of electronic detectors that use liquids as detection media.

The bubble chambers, as we will see later, do not work like conventional

electronic detectors in which the voltage or current is measured at the

readout electrode. Instead, the particles passing through them produce

bubbles that are photographed and hen visually inspected. There is also a

class of detectors, called liquid scintillation detectors, in which the liquids

produce light when their molecules are excited by incident radiation.

Such devices will be discussed in the chapter on scintillation detectors.

Examples for liquid filled detectors: liquid ionization chambers and

liquid proportional counters.


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1.1.3 Semiconductor detectors[3]In these detectors, radiation is measured by means of the number of

charge carriers set free in the detector, which is arranged between twoelectrodes. Ionizing radiation produces free electrons and holes. The

number of electron-hole pairs is proportional to theenergy transmitted by

the radiation to the semiconductor. As a result, a number of electrons are

transferred from the valence band to the conduction band,and an equal

number of holes are created in the valence band. Under the influence of

anelectric field,electrons and holes travel to the electrodes, where they

result in a pulse that can be measured in an outercircuit,as described by

theShockley-Ramo Theorem.The holes travel in the opposite direction

and can also be measured. As the amount of energy required to create an

electron-hole pair is known, and is independent of the energy of the

incident radiation, measuring the number of electron-hole pairs allows the

energy of the incident radiation to be found.

The energy required for production of electron-hole-pairs is very low

compared to the energy required for production of paired ions in a gas

detector. Consequently, in semiconductor detectors the statistical

variation of the pulse height is smaller and the energy resolution is

higher. As the electrons travel fast, the time resolution is also very good,

and is dependent upon rise time. Compared with gaseous ionizationdetectors, the density of a semiconductor detector is very high, and

charged particles of high energy can give off their energy in a

semiconductor of relatively small dimensions.

Examples for semiconductor detectors: PIN diode, diamond detectors,

and Thermo luminescent Detectors.
http://en.wikipedia.org/wiki/Radiationhttp://en.wikipedia.org/wiki/Charge_carrierhttp://en.wikipedia.org/wiki/Electrodehttp://en.wikipedia.org/wiki/Electronhttp://en.wikipedia.org/wiki/Electron_holehttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Valence_bandhttp://en.wikipedia.org/wiki/Conduction_bandhttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Electrical_networkhttp://en.wikipedia.org/wiki/Shockley-Ramo_Theoremhttp://en.wikipedia.org/wiki/Statistical_variabilityhttp://en.wikipedia.org/wiki/Statistical_variabilityhttp://en.wikipedia.org/wiki/Rise_timehttp://en.wikipedia.org/wiki/Gaseous_ionization_detectorshttp://en.wikipedia.org/wiki/Gaseous_ionization_detectorshttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Gaseous_ionization_detectorshttp://en.wikipedia.org/wiki/Gaseous_ionization_detectorshttp://en.wikipedia.org/wiki/Rise_timehttp://en.wikipedia.org/wiki/Statistical_variabilityhttp://en.wikipedia.org/wiki/Statistical_variabilityhttp://en.wikipedia.org/wiki/Shockley-Ramo_Theoremhttp://en.wikipedia.org/wiki/Electrical_networkhttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Conduction_bandhttp://en.wikipedia.org/wiki/Valence_bandhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Electron_holehttp://en.wikipedia.org/wiki/Electronhttp://en.wikipedia.org/wiki/Electrodehttp://en.wikipedia.org/wiki/Charge_carrierhttp://en.wikipedia.org/wiki/Radiation


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1.1.4 Calorimeters[1, 4]Methods of particle energy measurement in modern high energy

physics have to cover a large dynamical range of more than 20 orders ofmagnitude in energy. Detection of extremely small energies (milli-

electron-volts) is of great importance in astrophysics if one searches for

the remnants of the Big Bang. At the other end of the spectrum, one

measures cosmic ray particles with energies of up to 1020 eV, which are

presumably of extragalactic origin. Calorimetric methods imply total

absorption of the particle energy in a bulk of material followed by the

measurement of the deposited energy. Let us take as an example a 10

GeV muon. Passing through material this particle loses its energy mainly

by the ionization of atoms while other contributions are negligible. To

absorb all the energy of the muon one needs about 9m of iron or about

8m of lead. It is quite a big bulk of material! On the other hand, high-

energy photons, electrons and hadrons can interact with media producing

secondary particles which lead to a shower development. Then the

particle energy is deposited in the material much more efficiently. Thus

calorimeters are most widely used in high energy physics to detect the

electromagnetic and hadronic showers. At very high energies (1TeV),

however, also muon calorimetry becomes possible because TeV muons in

iron and lead undergo mainly interaction processes where the energy loss

is proportional to the muon energy, thus allowing muon calorimetry. This

technique will become relevant for very high-energy colliders ( 1TeV

muon energy).

Examples for calorimeters: Electromagnetic calorimeters and Hadron

calorimeters.


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1.1.5 Drift Tubes[4]The drift tube (DT) system measures positions of particles that have

high velocity such as muons. Each wide tube contains a stretched wire

within a gas volume. When a muon or any charged particle passes

through the volume it knocks electrons off the atoms of the gas. These

follow the electric field ending up at the positively-charged wire. By

registering where along the wire electrons hit (in the diagram, the wires

are going into the page) as well as by calculating the muon's original

distance away from the wire (shown here as horizontal distance and

calculated by multiplying the speed of an electron in the tube by the time

taken) DTs give two coordinates for the muons position.

Figure [1-1]: Mechanism of detection by Drift tubes.


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1.1.6 Cathode Strip Chambers [4]Cathode strip chambers (CSC) are used in experiments that in which

the magnetic field is uneven and particle rates are high.

CSCs consist of arrays of positively-charged anode wires crossed with

negatively-charged copper cathode strips within a gas volume. When

muons pass through, they knock electrons off the gas atoms, which flock

to the anode wires creating an avalanche of electrons. Positive ions move

away from the wire and towards the copper cathode, also inducing a

charge pulse in the strips, at right angles to the wire direction. Because

the strips and the wires are perpendicular, we get two position coordinates

for each passing particle.

Figure [1-2]: Mechanism of detection by Cathode strip chamber.


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1.1.7 Resistive Plate Chambers [4]The Resistive Plate Chambers (RPC's) consist of two parallel plates

made out of Bakelite with a bulk resistivity of 109- 10

10cm, separated

by a gas gap of 2mm. The whole structure is made of gas tight. The outer

surfaces of resistive material are coated with conductive graphite paint to

form the high voltage and ground electrodes. The readout is performed by

means of aluminum strips separated from the graphite coating by an

insulating Polyethylene (PET) film. Resistive Plate Chambers (RPCs) are

very important detectors, because of excellent time resolution which is

crucial at Large Hadrons Collider (LHC) due to the beam crossing time of

25ns.When a muon passes through the chamber, electrons are knocked

out of gas atoms. These electrons in turn hit other atoms causing an

avalanche of electrons. The electrodes are transparent to the signal (the

electrons), which are instead picked up by external metallic strips after a

small but precise time delay.

Figure [1-3]: Resistive plate chamber


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1.2 Nuclear Emulsion[5]Nuclear emulsion is a 3-D tracking detector for charged particles

having a special resolution of less than 1 m. it is a photographic platewhere it is used to detect the tracks of charged particles. A photographic

emulsion is consists of a large number of small crystals of silver halides,

mostly bromide, where the silver halide has high sensitivity to light.

Nuclear emulsion used as a target and as a detector, where the

emulsion is a heterogeneous target because it contains different nuclei

with different mass numbers (H, C, N, O, Ag, Br).

Nuclear emulsion has been largely used in high energy physics,

leading to the discovery of new particles and to the measurements of their

properties. The high sensitivity and grain uniformity of nuclear emulsions

make them capable of observing tracks of single particles with sub

micrometric space resolution and therefore they are especially suitable for

the observation of short-lived particles

1.2.1 Nuclear emulsion as a detector:-The photographic emulsion is used in recording the nuclear reactions

in the 4-space. It consists of a large number of small crystals of silver

halide (usually AgBr) which are transparent embedded in gelatin

medium. When a charged particle pass through the emulsion , some of the

halide grains are modified but their modifications are invisible, this effect

is described as "the latent image formation", that is an invisible image

produced by the exposure of the film to light. Then by immersing the

nuclear emulsion plate in reducing bath which is called "developer", the

latent images are turned into grains of metallic silver which is black. So,

along the charged particle path, we can see a trail of black grains of


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metallic silver under a microscope. That means we can obtain a 3-

dimension image of particle trajectory.

Processing of the nuclear emulsion includes developing, fixing,

washing and drying. Where after developing emulsion plate placed in a

bath called "fixer" which dissolves the unaffected grains of silver, finally,

the plate is washed and dried.

The interaction with emulsion is shaped as a Star because the particles

can be emitted in all directions and the star will be as in figure [1-4].

Figure [1-4] the shape of star (interaction with emulsion plate).

1.2.2 Advantages of nuclear emulsion:-1. It can be used as a target and as a detector of 4-space

geometry.


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2. It provides us with the possibilities of measuring energies andangles, where it detects and conserves the nuclear events for a

long time (i.e. it Acts as a memory for reaction).

3. Used in studying the characteristics of new elementary particlesand detection of unstable and neutral particles decay.

4. It allows us to study the projectile interactions with differenttargets, where the projectile can interact with free and quasi-free

nuclei (H), light nuclei (C, N and O) and heavy nuclei (Ag and

Br).

5. It is sensitive to study slow-energy particles which giveappreciable information about the thermal excitation of the

target nucleus.

6. Emulsion medium has high stopping power so, a large fractionof short-lived particles are brought to rest in it before decay and

hence their ranges and half-life time can be measured

accurately.

The emulsion is a suitable tool for studying the interactions at high and

ultra-high energies. Our project will concern mainly with this detector

which is available at Dr. M. El-Nadi labat Cairo University.

1.2.3 Specific ionization:-When the charged particle goes through the photographic emulsion, it

will slow down by losing its kinetic energy due to inelastic collisions with

the emulsion atoms.

It loses its energy by the ionization, elastic and inelastic scattering

making a trail of silver grains along its path. If the grain density is I

(number of the developed grains per unit path length of the track and

depend on the velocity and charge of ionizing particle), it is necessary to


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determine where is the minimum ionization in the plateau region atthe same depth at the emulsion.

Specific ionization is the probability that at the passage of an ionizing

particle through a grain of silver halide to produce a developed grain. The

value of this probability depends on the energy dissipated in the grain so;

it is a function of the specific energy loss of the particle.

1.2.4 Classification of emitted particles:-By performing of the grain density measurements of the tracks of the

primary beam at random depths and in different regions, the tracks of the

emitted particles are classified according to into:-1- Shower tracks:These are tracks with low ionization (

) and corresponding to

highly relativistic particles having . Most of them are pionswith energy > 60 Mev admixture with fast protons (E > 400 Mev),

charged K-mesons, antiprotons and hyperons. The number of shower

particles is denoted by, where its value gives a good estimation of thenumber of charged -mesons produced in the interaction.

2- Gray tracks:These are tracks having a specific ionization in the range 1.4 3000. They are mostly recoiling protons with energies (30< E < 400 MeV) with small admixtures of deuterons, tritons and slow

pions (E < 50 MeV). The number of gray particles is denoted by.


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3- Black tracks:These are tracks having corresponding to and their

ranges . The black tracks are due to slow charged targetfragments with energies E per nucleon (protons, deuterons,tritons, , and small fraction of heavier targets fragments. The

black tracks with ranges less than 5 were left uncounted as they wereassumed to be due to recoil nuclei. The number of black particles is

denoted by.Heavily ionizing particles:

They are the summation of grey and black particles with . Thenumber of heavily ionized particles is denoted by.1.2.5 Interactions with light and heavy nuclei from emulsion:-

The projectile nucleus will interact with the individual components of

the emulsion according to a certain cross section. The interaction with

emulsion is divided into three main groups:-

1- Interaction with free nuclei (H).2- Interaction with light nuclei (CNO group).3- Interaction with heavy nuclei (AgBr group).

Separating the interactions with (H), (CNO) and (AgBr) groups is quite

difficult and the important parameter the helps in separating the

interactions is the number of heavily ionized particles where:- Events with ( are classified as collisions with

hydrogen.

Events with

are classified as collisions with light

nuclei (CNO).


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Events with are classified as collisions with heavy nuclei(AgBr).

But in events with

there is an admixture of (CNO) target events

and peripheral collisions with (AgBr) target. We can justify the admixture

by taking into account the following conditions:-

1- (AgBr) events must fulfill the following criteria:-a) At least one track with range in emulsion must be

found.

b)No tracks with should be found.2- The residual events belong to the interactions with (CNO).1.2.6 Grey particles emitted from hadron nucleus interaction:-

In emulsion experiment the slow particles emitted from hadron

nucleus interaction are divided into two categories according to their

energies. The fast part (protons with energies about 30-400 Mev) is

denoted grey particles and the rest part is denoted by black particles. The

term grey and black are originated from emulsion experiments where

visual appearance of the produced tracks is used for the classification.

The grey tracks are strongly correlated to the number of hadron-

nucleon interactions inside the nucleus. Many models had investigatedthe correlation between and the number of grey particles

[6-8]

.One of

them is the "Geometric Model" by Andersson,Otterlund and

Stenlund[9,10]

.


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2 Geometric model [9,10]According to Andersson, Otterlund and Stenlund, grey particles

production can be described in the following way: each collision insidethe nucleus gives an independent contribution to the grey particles and

the distribution from each of these collisions is well described by ageometric distribution:

() , where ,So that is the average contribution to the grey particles from

each of the collisions inside the nucleus, A. for collisions the formulabecomes

()

And if this formula is summed over we derive the total distribution for a given target, A, and a given incident hadron, h,()

Since is independent of the impinging hadron, the distributions for a fixed

will not depend on the hadron, so that the only

differences is the distributions originate from summing over different-values in the last equation. When nuclear emulsion is used as a target, afinal step has to be added, namely to sum over the different target

emulsion constituents, H, CNO and AgBr.


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And for Em. And for (CNO) group. And for (AgBr) group.3.2The average number of hadron nucleon collisions in hadron

nucleus interactions:-

One of the main problems arising when hadron nucleus interactions

are studied is how to distinguish between interactions where only one or

few nucleons from the target nucleus have participated in the reaction

with the incident hadron and interactions where several nucleons have

taken part. This problem is equivalent to the problem of separating the

peripheral from the more central interactions, since interactions where

several nucleons are hit by the projectile imply that the projectile has

penetrated the nucleus at rather small impact parameters. It has been

suggested that slow particle production could be used to measure thenumber of collisions,, inside the nucleus.

Most models dealing with hadron nucleus interactions use the

interpretation that the incoming hadron interacts primarily with a certain

number of nucleons,, in its way through the nucleus.From Glauber theory

[13, 14], one can obtain the following relation for

collisions:


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Where T (b) is the thickness function and defined as:

; Where the effective hadron nucleon cross section and its value is istaken as,

{ Is the nuclear density normalized to the mass number A of the

target nucleus andit is taken as follows:

a) For light nuclei (A40):The nuclear density is given by Fermi distribution that given by


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Where:

is the normalization constant.Thus one can deduce the average number of collisions,as;

Where is the inelastic hadron nucleus cross section and given by;

[ ]We calculate the average number of collisions according to eq. (3-3) for

the interactions of pions and protons with some target nuclei. The results

are summarized in table (3-1).

Table [3-1]:The average number of collisions in h-A collisions

hadron

target proton Pion

C 1.755 1.591

Cu63 2.956154 2.163525

Em70 2.994075 2.189037

AgBr94 3.360027 2.400573

Pb108 3.517130 2.492983


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The table shows clearly that the average number of collisions increases

sharply with increasing the target size A.

Also, the

distribution function is given by;

Calculations according to eq.(3-4) is displayed in fig.(3-1) for the

interactions of protons with light (CNO) and heavy (AgBr) emulsion

target nuclei.

Figure.[3-1]: distribution functions for P-CNO (as a light target nuclei) andP-AgBr (as a heavy target nuclei) according to Eq.(3-4).

As shown in the fig., as the target size increases, the extends to values

greater than 10.


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3.3 The distribution of grey particles according to Geometric model:-

In order to calculate the distribution of grey particles according to

Geometric model [eq. (2-2)], we have to differentiate in our calculations

between light nuclei (A40).

a) For light nuclei (A


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Fig. (3-2) displays the Ng-distribution for p-CNO interactions.

Figure [3-3]: Calculated Ng-distribution for p-AgBr according to Eq.(2-2)

b)For heavy nuclei (A>40);-In this case the density distribution is given by Fermi distribution Eq. (3-

2).The normalization constantis determined numerically using:

The thickness function is calculated as:

Where;


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and Rm is taken as the maximum limiting value forthe target radius.

The inelastic hA cross section is thus:

[ ]

[ ] Hence, the

distribution function is;

The Ng-distribution, eq. (2-2) in hadron-nucleus interaction is finally

calculated.

Figure (3-3) displays the Ng-distribution for p-AgBr interactions.

Figure [3-3]: Calculated Ng-distribution for p-AgBr according to Eq. (2-2)


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The calculated multiplicities for p-AgBr and p-CNO interactions have

to be summed together to get the corresponding multiplicities for p-Em

interactions according to the formula:

( ) ( ) ( )p Em p CNO p AgBrP Ng uP Ng vP Ng

uand vare determined by[17]:

p CNO

CNO

p CNO p AgBr

CNO A gBr

p A gBr

A gBr

p CNO p AgBr

CNO AgBr

Nu

N N

Nv

N N

is the concentration of a given nucleus i (/cm3

) and is determinedby [12], * +

The Ng distributions for p-Em are shown in figs. (3-4 to 3-6)


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Figure [3-4]:Multiplicity distribution of grey particles for P-Emulsion

interaction: at 200 GeV[18]

as compared with the model, eq. (2-2).

Figure [3-4]:Multiplicity distribution of grey particles for P-Emulsioninteraction: at 400 GeV

[18]as compared with the model, eq. (2-2).


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Figure [3-5]:Multiplicity distribution of grey particles for P-Emulsion

interaction: at 800 GeV[18]

as compared with the model, eq. (2-2).

These figures show the good agreement between the model and the data.

The average number of grey particles emitted from high energy p-A

collisions are also calculated according to the geometric model and

compared with the experiment in table (3-2)

Table [3-2]: The average Ng for p-A interactions

Model Experimentally

CNO AgBr Em 200

(GeV)

400

(GeV)

800

(GeV)

0.652 2.865 2.23937 2.97256 3.1643 3.1871

The table reflects that is nearly energy independent.


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4.1ConclusionIn this project the production of grey particles are studied theoretically

using the geometric model and the calculations are compared with

proton-Em data at incident energies 200, 400 and 800 GeV. The

conclusions are summarized as follow:

1) The independent number of collisions inside the target nucleusdepends mainly on the target size-A.

2) The production of grey particles seems to be independent of theincident energy within the energy range studied values.

3) The increases with increasing the target size-A.


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Appendix

FORTRAN code for calculating the distribution functions for grey

particles according to Geometric model:

This code is designed to obtain the distribution function of grey particles

emitted from hadron-nucleus interactions at high energies (> 1 GeV)

according to Geometric model using FORTRAN programming language

as follow:

!MULTIPLICITY DISTRIBUTION OF FAST PROTONS ACCORDING TO ANDRESON

DIMENSION

W(20),Z(20),T(20),B(20),PJ(30),YM(30,0:25),PY(30,0:25),C(0:30,30)

OPEN(UNIT=2,FILE='INPUTLIGHT.TXT')

OPEN(UNIT=4,FILE='INPUTHEAVY.TXT')

OPEN(UNIT=3,FILE='MULTIPLICITY.TXT')

!============================================================= WRITE (*,*)'Enter the target size?'

WRITE(*,*)'Light press "2" Heavy press "4"'

READ(*,*)JT

READ(JT,*)(W(I),I=1,10)

READ(JT,*)(Z(I),I=1,10)

DO K=1,10

W(21-K)=W(K)

Z(21-K)=-Z(K)

ENDDO

PI=3.141592654

WRITE(3,*)

WRITE(3,*)'**************************************************'


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WRITE(3,*)

READ(JT,*)A,Q1,SHN

!Q1 IS THE TARGET ATOMIC NUMBER Z

IF(A.GT.40.) GOTO 700

!===================================================== !LIGHT TARGETS

!===================================================== WRITE(*,*) 'rms= ?'

READ(*,*)RSA

RS2=RSA*RSA

A2=(2./3.)*(RS2-.81*.81)/(1.-(1./A))

T0=A*SHN/(PI*A2)

STT=0.

DO J=1,20

T(J)=(T0/2.)*(1.+Z(J))

ET=(1.-EXP(-T(J)))/T(J)

STT=STT+W(J)*ET

ENDDO

STT=(A*SHN/2.)*STT

WRITE(3,*)'Inelastic hA cross section',STT,'','fm**2'

X=1.

SM=0.

DO MU=1,10

X=X*MU

PJ(MU)=0.

DO J=1,20


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ET=(T(J)**(MU-1)/X)*EXP(-T(J))

PJ(MU)=PJ(MU)+W(J)*ET

ENDDO

PJ(MU)=(A*SHN/2.)*PJ(MU)/STT

SM=SM+MU*PJ(MU)

WRITE(3,*)MU,PJ(MU)

ENDDO

WRITE(3,*)'',SM

GOTO 555

!======================================================= !HEAVY TARGETS

!======================================================= 700 WRITE(*,*)'ro = ?'

READ(*,*)R0

RU=R0*A**(1./3.)

RM=2.*RU

CR=RM/2.

RM2=RM*RM

!CC=1.12*A**(1./3.)

WRITE(*,*)'CC=?'

READ(*,*)CC

WRITE(*,*)'D=?'

READ(*,*)D

SR=0.

DO K=1,20

R=CR*(1.+Z(K))


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R2=R*R

EF=(R-CC)/D

SR=SR+W(K)*R2/(1.+EXP(EF))

ENDDO

RW0=A/(4.*PI*CR*SR)

WRITE(3,*)

!WRITE(3,*)'**************************************************'

WRITE(3,*)

WRITE(3,*)'RW0=',RW0

SB=0.

DO J=1,20

B(J)=(RM/2.)*(1.+Z(J))

B2=B(J)*B(J)

SS=(SQRT(RM2-B2))/2.

SZ=0.

DO K=1,20

XK=(1.+Z(K))*SS

XK2=XK*XK

EZ=EXP((SQRT(XK2+B2)-CC)/D)

SZ=SZ+W(K)/(1.+EZ)

ENDDO

T(J)=2.0*RW0*SHN*SS*SZ

EB=1.-EXP(-T(J))

SB=SB+B(J)*EB*W(J)

ENDDO

STT=PI*RM*SB


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WRITE(3,*)

WRITE(3,*)'INELASTIC hA CROSS SECTION=',STT,'fm^2'

X=1.

SM=0.

DO MU=1,20

X=X*MU

PJ(MU)=0.

DO J=1,20

PJ(MU)=PJ(MU)+W(J)*B(J)*(T(J)**MU)*EXP(-T(J))

ENDDO

PJ(MU)=PJ(MU)*PI*RM/(X*STT)

SM=SM+MU*PJ(MU)

WRITE(3,*)MU,PJ(MU)

ENDDO

WRITE(3,*)'',SM

!***************************************************************** 555 CONTINUE

WRITE(*,*) 'exp= ?'

READ(*,*)GX

AL=GX/SM

XF=AL/(1.+AL)

IF(A.LE.20.)THEN

MUM=10

NFM=10

ELSE

MUM=20


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NFM=20

ENDIF!calculation of Ng distribution

WRITE(3,*)'*********************************************************'

DO NG=0,NFM

C(NG,1)=1.0

GS=0.

DO M=1,MUM

C(0,M)=1.0

C(NG,M+1)=(NG+M)*C(NG,M)/M

PY(M,NG)=C(NG,M)*(XF)**NG*(1.-XF)**(M)

GS=GS+PY(M,NG)*PJ(M)

ENDDO

WRITE(3,*)NG,GS

ENDDO

STOP

END


35/35

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(clarendon, oxford, 1977).

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17.O.M.Osman, PhD. Thesis, Cairo University (1983).18.S.Fredriksson, G.Eilaan and G. Berlad,Phys. Rev. 144, 187 (1987).19.S.Frandrevic, D.Kripic and O.Addamovic, phys. Rev C52, 331

(1995).

20 S Dhamija M Kaur and S Dahia Phys Rev C63 53201 (2001)
http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://cms.web.cern.ch/cms/Detector/index.htmlhttp://cms.web.cern.ch/cms/Detector/index.htmlhttp://en.wikipedia.org/wiki/International_Standard_Book_Number

multiplicity distribution of grey particles emitted from hadron-nucleus interactions at high...

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