SOP Report Swastik.docx

8/14/2019 SOP Report Swastik.docx

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A Report

on

the

Quarks In The Dense Core Of

Neutron Stars

BITS Pilani, KK Birla Goa Campus

Swastik Mohapatra

2007B5A8567G


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ACKNOWLEDGEMENT

An activity can be termed as an accomplishment only when the purpose is fulfilled.The accomplishment of any activity involves a continuous unflinching effort,

motivation and support from its mentor. We would like to thank our mentor and guide

for this project, Dr. T. K Jha, for his constant support and guidance, without which our

report would have never been complete. We would also like to forward our heartfelt

thanks to Dr. Chandradew Sharma whose eager help was instrumental in shaping up

of our report.

We would also like to offer our sincere gratitude to our fellow BITSians who were

always very helpful mentally.

We would also like to thank the authors of all the books and websites which provided

us with most of the data used by us in this project.

Last but not the least we would like to thank the God for showing us the right way

throughout the project without whom this report would never have been.

Swastik Mohapatra

2007B5A8567G


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Table of Contents

Topic Page

1. Acknowledgement 2

2. Abstract 5

3. Introduction 6

4. Compact Stars 9

5. Phases of Nuclear Matter 12

6. Aspects of Nuclear Matter 15

7. Relativistic Approach 16

8. Lagrange Formalism 17

9. Theory of Neutron Stars 20

10. Formalism and Methodology 23

11. Runge Kutta (C code) 26


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Topic Page

12. Quark and The Bag Model 28

13. Quarks and Colour 31

14. Conclusion 34

15. References 35


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Abstract

The project aims at studying the quark aspects of the neutron stars. The report covers

the various properties of the nuclear matter, the phase diagrams, Equation of state,

which are approached relativistically along with a study on the Quark bag model

and its implications. It also covers the TOV equations which are solved to obtain the

structural peculiarities of the neutron stars imparted by the dense nuclear mater. We

solved the TOV equations using the Runge Kutta algorithm to find out radius andmass of the neutron stars for varying central densities using C.


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Introduction

The celestial bodies including the stars and planets have wondered the human beings

for many years. For this reason, studies about stars have been a crucial part of humanendeavour. The stars are born in clouds of diffused gases called molecular clouds in

which conglomerations takes place because of the gravitational instability created by

shock waves from supernova, after which the stars undergo different phases to grow

finally into heavy dense star. They also lead an active life span of millions of years

after which they collapse and die. the stars exist in so many forms one of them are the

neutron stars which were first discovered as light emitting highly magnetised rotating

pulsars with a white dwarf companion creating luminous nebula and with a miniature

planetary system comprising of 3 planets and the star.

The neutron stars are composed of neutrons which

follow the Pauli Exclusion principle due to which they are refrained from further

collapsing (follow the quantum laws) and hence form the densest form of matter .

These neutron stars are formed as the remnant of a gravitational collapse with a mass

of 1.35-2 solar mass, radius of 10-12 km and a temperature below 1 MeV. The

density of the stars range from 3 to 10 saturation density (constant) that is the

maximum density we can achieve even if we increase the number of nucleons in it.

For this reason they serve as the natural laboratory for studying the cold dense matter.

these peculiarities have made scientists and researchers to probe into various features

of these stars which are well understood by the notion of Weizsacker, liquid drop

model of nucleus and study of bulk infinite nuclear matter.


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There are several factors that regulate the stellar formation such as the gravity, the

interstellar gas, pressure, rotation, magnetic fields, winds and radiations and shock

waves from nearby stars. As the density increases the temperature of the matter

increases leading to the fusion of hydrogen to form helium and then various

thermonuclear reaction takes place leading to formation of different elements upto

iron. The physical understanding of these features are well accomplished by the study

of the phase diagrams , EOS which are the backbone of the thermodynamics of the

stars , aspects of nuclear matter, TOV equations that gives the structure of the stars.

Given below is a schematic diagram of the thermonuclear fusion leading to the

formation of Neutron Star.


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Matter as we know is made up of two types fundamental particles. Hadrons and

leptons. Leptons are known to be the most fundamental particles while hadrons are

known to be further composed of quarks and antiquarks, which are the most

fundamental building blocks of matter on this side of the spectrum.

Quarks as we know never exist naturally as single particles, but in form of baryons

and mesons which are culmination of quarks in groups of threes and twos

respectively. They follow the bag model which states that the quarks exist in groups

inside a hypothetical elastic bag and the bag breaks to form infinite nuclear matter

only under the conditions of extremely high pressure and density. These conditions

are available readily in the neutron stars and other some other compact stars making

them ideal natural laboratories for their study.


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Compact Stars

Stars have different sizes. Some are very small, so much so that they have even

insufficient pressure and temperature inside them to start fusion. Other stars are huge

and deplete their energy rapidly. As the thermonuclear fusion reactions diminish in the

core of stars, the thermal pressure decreases. Eventually, a critical point is reached.

When the force of gravity is no longer balanced by the thermal pressure the star starts

to collapse. What happens after that depends on the mass of the progenitor star. If the

star is a few times more massive than the Sun, the collapse is eventually halted due tothe degeneracy pressure of electrons and a so-called white dwarf forms. If the star is

more massive, around ten solar masses, the collapse continues until the atomic nuclei

start to overlap and the core stabilises as a dense neutron star. For progenitor stars of

still higher mass, the collapse is assumed to lead to the formation of a black hole. The

threshold mass, when a star collapses to a black hole instead of a neutron star is,

however, only approximately known, since the details of the collapse are poorly

understood.

White dwarfs, neutron stars and black holes are extremely dense objects, so-called

compact objects, which are left behind in the debris when normal stars die. That is,

when most of the nuclear fuel has been consumed and they collapse under the pull of

gravity. With the exception of small black holes, which evaporate quickly due to

Hawking radiation, and accreting neutron stars and white dwarfs, all three types of

compact objects are essentially static over the lifetime of the universe and therefore

represent the final stage of stellar evolution.


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White Dwarf

Assume a thought experiment where we keep on adding mass to a cold object ignoring

thermal pressure. Gravitational pull tends to increase. We initially form planet like

structures at the core of which due to high density the free electrons would start

becoming degenerate i.e. they would be forced to be forced to fall into the lowest

energy states available. With increasing mass put into it the core now expands and

more and more of the object becomes degenerate.

Stars made up of such degenerate material are called degenerate dwarfs or more

popularly white dwarfs. White dwarfs arise from the cores ofmain-sequence stars and

are therefore very hot when they are formed. As they cool they redden and dim to

form dark black dwarfs. Now more the mass we add the smaller the object we get,

ultimately forming a mass with radius some bare thousands of kilometres. On adding

further matter it tends to approach the upper mass limit of the white dwarf known as

the Chandrashekhar Limit, which is about 1.4 times the mass of our sun. On further

increase in pressure at the core the phenomena of combination of proton and electron

is observed by inverse beta decay to form more and more neutrons. The equilibrium

would shift towards heavier, more neutron-rich nuclei which are not stable at

everyday densities. As the density increases, these nuclei become still larger and less

well-bound. At a critical density of about 41014

kg/m, called theneutron drip line,

the atomic nucleus would tend to fall apart into protons and neutrons. Eventually we

would reach a point where the matter is on the order of the density (~21017

kg/m) of

an atomic nucleus. Here we encounter the next kind of celestial body which has a

large majority of its matter as neutrons and some traces of protons and electrons. They

are known as neutron stars.

Neutron Stars

Due to extreme pressures created by exorbitant amount of gravity a huge amount of

energy is released in form of neutrinos and very small amount in terms of photons.
http://en.wikipedia.org/wiki/Main-sequence_starhttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Cubic_metrehttp://en.wikipedia.org/wiki/Neutron_drip_linehttp://en.wikipedia.org/wiki/Neutron_drip_linehttp://en.wikipedia.org/wiki/Cubic_metrehttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Main-sequence_star


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Phases of Nuclear Matter

An infinitely large system of nucleons without any well defined nuclear shell is

known as neutron star. It is a hypothetical stuff comprising of neutrons and protons

interacting with one another by nuclear forces and no coulomb forces. Such kind of

matter is usually found in extreme conditions of very high pressure and density e.g.

compact stars, neutron stars etc. The matter inside the compact stars, the early

universe, a nucleon gas, or quark-gluon plasma (QGP) are some of the types of

nuclear matter.

As it is known that water has three distinct phases, similarly nucleonic matter also has

distinct phases. So in order to describe the phase structure of nuclear matter we can

use the analogy of water and its phases. At atmospheric pressure and at temperatures

below the 0C freezing point, water takes the form of ice. Between 0C and 100C at

atmospheric pressure, water is a liquid. Above the boiling point of 100C at

atmospheric pressure, water becomes the gas that called steam. We also know that we

can raise the temperature of water by heating it, which is to say by adding energy.

When the liquid water at 100oC is heated its temperature becomes constant and it

starts turning into vapour. This type of transition is referred to as first order transition.

As the pressure is raised, the boiling temperature of water increases until it reaches a

critical point at a pressure 218 times atmospheric pressure (22.1 MPa) and a

temperature of 374C. There the phase coexistence stops and the phase transition

becomes continuous or second order. Thus with the help of a phase diagram ofwater now it can be shown how its phases are directly dependent on pressure and

temperature.

Exactly the same analogy can be applied here in case of nuclear matter which also

very much like atoms and molecules depends on the factors of pressure and

temperature. In their normal states of lowest energy, nuclei show liquid-like

characteristics and have a density of 0.17 nucleons/fm3. In more conventional units,


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this corresponds to 2.7 X 1017

kg/m3, or 270 trillion times the density of liquid water.

In a laboratory, the only possible way to heat nuclei to significant temperatures is by

colliding them with other nuclei. The temperatures reached during these collisions are

astounding. The temperatures that can be reached in nuclear collisions range up to 100

MeV and abovemore than 200 million times the temperature at the surface of the

Sun (~5,500 K).

On heating a nucleus to a temperature of a few MeV, some of the nuclear liquid will

evaporate. From knowing the general form of the interactions between nucleons, it can

be said that, just like water, the nuclear liquid also has a latent heat of vaporization,

and that nuclei should undergo a first-order phase transition. This liquid-gas

coexistence is also expected to terminate at a critical point, the critical point of nuclear

matter. In order to generate that critical state heavy ion collisions are necessary. But

the time of sustenance of such extreme conditions is very short lived i.e. of the order

of 10-21

seconds roughly. Furthermore other problems like rapid expansion and

cooling causes even more difficulty in making observations. So observables such as:

1. Abundance of isotopes.2. Population of excited nuclear states.3. Shapes of energy spectra from nuclear collision remnants.4. Production of particles from the collisions like pions etc.

Neutron stars act as perfect laboratory conditions for observing such experiment as

they have conditions of sustained high density and pressure. Each nucleon can be

considered as an elastic bag consisting of quarks and gluons. If these bags are

subjected to conditions of extremely high density and pressure they tend to overlap.

The nuclear shell then tends to dissolve and what is obtained is a continuous large

mass of nuclear matter. The overlap of the bags allows the quarks and the gluons of

different nuclei mix freely to form a state of quark-gluon plasma. From theoretical

calculations, we also expect the phase transition to a quark-gluon plasma to be of first

order, with a phase coexistence region. Given below is the phase diagram regarding


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the phase change of nuclear matter with respect to the change in temperature and

density.

The yellow part of figure shows that the phase transition between the nuclear liquid

and a gas of nucleons. The study of this phase diagram helps us understand about the

history of the universe and the properties of the celestial bodies like neutron stars.


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Aspects of Nuclear Matter

The various aspects of nuclear matter are studied for a symmetric nuclear matter at

saturation density and zero temperature. There are five physical parameters called

nuclear saturation properties all satisfy the most general EOS for the matter

namely :- 1. Binding energy per Nucleon: (B/A - mn) = -16 MeV

2. Saturation density: 0 = 0.153 fm-3

3. Nuclear Incompressibility: K = ( 200 - 300 )MeV

4. Nucleon Effective Mass: m* = ( 0.8 - 0.9 ) mn

5. Symmetry energy coefficient: asym = 32 MeV

The first two condition normalizes the equation of state. The nuclear incompressibility

and nucleon effective mass implies that the extrapolation to high density remains

meaningful and valid in the vicinity of saturation density. Similarly, the value of the

symmetry energy coefficient implies that our extrapolation to asymmetric matter is

also reasonable. All analysis and studies on EOS goes along with the validity and

exactness of the above constraints. So, many studies have been done that measures the

values of different physical quantities and ensures the correctness of the nuclear

matter aspects.

The calculated value of the ( B/A mn ) of -16 MeV gives the absolute difference

between the mechanical energy of breaking the nucleus and the mass of the neutrons

which is a classically obtained data.

The saturation density has a constant value of 0.153 and all are done in the vicinity of

this value .

The nuclear incompressibility (K) factor is used while solving the EOS and from

measurements and observations its value ranges from 180-800 MeV .So, no

relativistic or non-relativistic model successfully describes this range exactly but the

non relativistic model gives a range of 210-240 MeV while relativistic model gives a

range of 200-300 MeV.


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The nucleon effective mass is observed as the mass a nucleon experiences inside a

bound nuclei due to interaction with other nucleons and is found to range from (0.8-

0.9) nucleon mass which is less than the bare nucleon mass.

It is a consequence of the Dirac field and forms an essential element in the solution of

EOS and success of relativistic phenomenology.

According to Fermi gas model, the decrease of effective mass induces an increase in

the symmetry energy contribution to the nuclear binding energy

Where symmetry energy coefficient is constant value of 32 MeV calculated for a

symmetric nuclear matter. To address various issues , one needs a model that has the

desired attributes of the relativistic framework and which can be successfully applied

to various nuclear force problem both in the vicinity of 0 as well as at higher

densities.

Relativistic approach

Relativistic approach is naturally favored for studying nuclear structure and nuclear

dynamics. The baryons and mesons, are the actual degrees of freedom observed in

experiments at intermediate energies. The interaction between the hadrons and mesons

is well represented by a Lagrangian, and nuclear force characteristics are studied using

the symmetries and conservation laws. The Dirac equation is gives equation of the

motion of a nucleon inside a nucleus. This Dirac equations successfully reasons the

B/A ratio and the effective nuclear mass of the nucleons which arises due to the scalar

and vector Lorentz fields. There are nucleon-nucleon interaction due to exchanges

facilitated by ,,, mesons. The mesons contributes towards the attractive and the

repulsive potential, thereby leading to saturation in nuclear matter.


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To solve for density effects

we need to solve the

Lagrangian for this we use

the mean field approach . In

such an approach we assume a uniform nuclear

matter, the ground state expectation value of the spatial component of

the fields vanishes leading to the non vanishing scalar field () and vacuum

expectation values of the vector () and the is vector () field.

Lagrange formalism

the fields for different particles satisfy equations analogous to those derived for a

continuous system obtained by a limiting process. The Hamiltons equation of motion

are used derived for a classical action . If the lagrangian is aLorentz scalar then the

equations of motion are Lorentz covariant . so we require that Lagrange can be a

scalar. And we can exploit the symmetry properties of the nuclei like the most

strongly bound condition (N=Z) favours iso spin symmetric configurations to form

the most sensible Lagrangian.

The Lagrangian density is a function of the fields, say (x) and their derivatives

i.e.


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. The relativistic action is therefore dimensionless, which implies,

The variation of this action leads to the Euler-Lagrangian equations, which

is generally given as

Euler - Lagrangian is solved for different fields and each equation of motion is

obtained for each field . Such a formalism guarantees the covariance and therefore, the

Lorentz invariance of the theory in particular, the invariance of this action

against translational symmetry. Consequently it leads to the conservation of

the energy-momentum tensor, which is given by following equation

After obtaining the equation of state we can obtain the energy density and pressure of

the many baryon system for different fields.

Therefore to define the relativistic approach for the nuclear matter following

characteristics are taken into consideration:

Degrees of freedom: The Baryons and the mesons.


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Lorentz covariance and Relativity: The theory must be Lorentzcovariant and in principles of relativity.

Interpretation of quantum mechanical vacuum and antiparticles:Relativistic approach can account for the quantum mechanical

vacuum and antiparticles, which must be incorporated from the onset.

Chiral Symmetry: The theory should be in accordance with QCD andchiral symmetry must be realized.

Nuclear saturation properties: The theory must satisfy the Nuclearsaturation properties so that the extrapolation to high density or temperature

remains valid.


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Theory of neutron stars

The neutron stars are bound by the gravitational force , not by the nuclear force.

Nuclear force is short ranged therefore it acts in the nearest neighbours butgravitational force is long ranged and acts on all mas energy . The gravity

compresses the matter well upto the saturation density but the nucleons feel the

repulsions at the close distances. Hence the nuclear force contributes negatively to the

binding of the neutron stars but shapes the EOS and structure of the star.

Chemical equilibrium in a star :

Stars evolve from a proto neutron star which consist of only photons and neotrinos so

they have high temperature . but after neutrino emission by X rays and photon

emission the star cool to lower temperature to 100000 K . By the end of this pahse

strong reactions takes place due to nucleon interactions-

N + N N + + K (1)

The associated kaon decay takes place in different ways like-

K0

2 (2)

K-

-+

- + K+

-+

+ + 2 +

But in all these reactions the chemical potential is conserved for all species like in

reaction (1) coupled with decay (2)

= n

Iso-spin and charge favoured baryon species :

The baryon species those have the same sign of electric charge are referred as charge

unfavoured proton because it must appear together with a particle of opposite sign to

maintain charge neutrality . the equation for the iso spin energy condition is

n - qB e > g 03I3B+ mB - gB


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the iso spin term g 03 I3B determines whether a species is iso spin favoured or un

favoured. The sign of 03 will be deduced in the neutron star matter and from this we

know that neutron is the dominant baryon species . therefore the neutron is unfavoured

for iso-spin restoring term . baryons with opposite sign of the iso spin projection are

favoured because their presence will reduce the value of 03 and the energy

density term. These condition decides the existence of favourable hadronic species

inside the neutron star on the basis of energy density and charge.

Development of neutron star sequences:

The resulting neutron sequences are obtained from the Oppenheimer-Volkoff

equations which provides a one parameter family of stellar models corresponding to

any particular equation of state . the differential equations are simple and first order

and are solved for boundary condition imposed at centre for mass = 0 and energy

density c . the equations are integrated from centre by increasing radius upto

boundary where pressure drops to zero and this effectively gives the mass and radius.

Mass as function of central density :


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The above plot gives the mass variation for central core density. It is clear from the

graph that gravity exploits the softening of the EOS at higher density introduced by

hyperionization. Stability of the star is lost at both ends of the sequence . below the

minimum mass the configurations are unstable to radial oscillations while beyond the

maximum mass gravity overwhelms all resistance to collapse . Stars beyond this point

are unstable to an acoustical mode of vibration . The stars central density varies from

half of nuclear energy density to 10 times the nuclear energy density. And at the point

in between the graph where derivative of mass with respect to energy density is zero

gives the maximum plausible mass or the Mmax of the neutron star.

Radius

mass relationship:

It is uniquely related to the underlying EOS . it gives the variations of radius for

changing total mass of the star.

For low mass stars the gravitational attraction is weak the corresponding stars are

large and diffuse . For the limiting star the gravitational attraction is strongest of all

those in stable hydrostatic equilibrium and the corresponding radius is the smallest.


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The energy per nucleon is E/A = /, where = n +b and the isospin

degeneracy factor = 4 for a symmetric nuclear matter composition.

The parameters of the model are used to calculate the physical quantities

obtained by fitting at the saturation point: the binding energy/nucleon B=A = 16:3

MeV,

baryon density = 0:153 fm-3, incompressibility K = (210-380) MeV

and effective (Landau) mass m* = (0:8 - 0:9) m.

Similarly after obtaining the EOS structural properties of the stars are studied by using

the TOV equations that manifests the nuclear matter properties of the stars.

The TOV equation along with the equation of state is the governing equation

to find the mass and radius of a neutron star. This EOS is manifested in the TOV

equation to give the limiting mass and radius that describes the hydrostatic

equilibrium for a non-rotating degenerate star in relativistic frame.

The TOV equation named after the scientists (Tolmann, Openheimer-

Volkoff) contains the structure of a spherically symmetric body of isotropic material

which is in static gravitational equilibrium. The equation is derived by solving the

Einsteins equation for general time invariant symmetric star. The equations are:

( )

( )

( )

Where P, m, r, and G mean pressure, mass, distance from the centre, energy density

and gravity constant respectively.


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These two equations are solved with boundary condition (M(r=0)) =0 and (r=0) =

( ).

The equations gives the dependency of pressure and mass on the radius and central

density of the star which are solved to give the maximum mass of 1.9 solar mass and

corresponding radius to be 9-10 km. It is supplemented by the EOS which has been

solved numerically to give the values of pressure, baryon density and energy density.

Graph shows the radius (km) vs. mass of the star for different EOS. Ref Numerical

survey of neutron crystal structure by B. Datta


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Runge Kutta Method (Code in C)

#include

#include

void main()

{ double f1(double x, double y, double z);// describes the dP/dr function

double f2(double x, double y, double z);// describes the dm/dr function

double ep[2451],t=50.0;// value of energy density epsilon

for(int i=1; i


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Quarks & The Bag Model

The universe is made up of two types of fundamental particles: hadrons and leptons.

Leptons are the particles having very negligible mass compared to the hadrons. The

leptons cannot be further sub-divided. Particles that form the lepton family are

electrons, electron-neutrino, muon, muon-neutrino, tau and tau-neutrino. On the other

hand hadrons can be further divided into baryons and mesons. Baryons and mesons

are made up of still smaller particles called Quarks. Thus we can say that Quarks and

Leptons are the most fundamental building blocks of matter. Like Leptons, there aresix types of Quarks. They are up, down, strange, charm, top and bottom.

Various theories have been formulated throughout the later part of the previous

century to describe the properties and behaviors of these particles. Some of these

theories were significantly successful in this regard. One such successful theory is that

of the Quark Bag Model popularly known as the MIT Bag Model. It was named so

because it was developed in the Massachussets Institute of Technology, Cambridge,

Massachussets, USA.

This model was invented to account for the properties of the elementary particles. It

describes particles as composite systems with their internal structure being associatedwith quark and gluon field variables. This theory doesnt restrict the treatment of


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internal structure of hadrons i.e, quarks as particles only. Its is so because it would

present a quite non-relativistic approach to the description of the whole framework.

rather the internal structure has been described by the usage of the fields. The same

fields used in the conventional relativistic field theories are used in this model.

However the field with which the substructure of the hadrons are described belongs to

the substrucure of a single particle the field variables are not hung on all points of

space as in ordinary field theory. Instead they are hung on a set of points which are

supposed to be in the inside of another extended particle. These sets of points are

called bag, and thus the name Bag Model. All the hadrons comply to the Bag

Model.

Structure of a Hadron

A Hadron is a particle made up of quarks held together tightly by strong forces.

Hadrons are subdivided into two categories namely Baryons and Mesons. Baryons are

made up of three quarks while mesons are made up of a quark and an anti-quark. The

two lightest types of baryons are protons and neutrons. Protons which have a net

charge of unity (+1) has 3 quarks viz. 2 up quarks and one down quark. On the other

hand neutron has two down quarks and one up quark. Up and down quarks carry +2/3

and -1/3 charge. That is why protons have net +1 charge(+2/3 +2/3 -1/3) and neutron

has 0 charge(+2/3 -1/3 -1/3).

While dealing with the issue of quark confinement using bag model the visualization

is that of an elastic bag which allows thequarksto move freely around, as long as one

doesnt try to pull them further apart. Since it acts like an elastic bag as one tries to

pull any quark apart, the bag stretches, resists and exerts a strong force of attraction.
http://hyperphysics.phy-astr.gsu.edu/hbase/particles/quark.html#c1http://hyperphysics.phy-astr.gsu.edu/hbase/particles/quark.html#c1http://hyperphysics.phy-astr.gsu.edu/hbase/particles/quark.html#c1http://hyperphysics.phy-astr.gsu.edu/hbase/particles/quark.html#c1


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Asymptotic Freedom

Quarks have hardly been ever observed existing alone in nature except in case of

extremely high density states. They are usually observed in pairs (as in mesons) or in

groups of three (as in baryons) enclosed in a hypothetical elastic bag. As one tries to

drag the quarks apart they tend to resist strongly. But as one brings them very close to

one another i.e. inside the bag the strong force of attraction drops down remarkably.

the force of containment gets weaker so that it asymptotically approaches zero for

close confinement. The implication is that the quarks in close confinement are

completely free to move about. Part of the nature of quark confinement is that the

further you try to force the quarks apart, the greater the force of containment.

The potential function that can be used to explain this behavior of the quarks is

V = (-k1/r) + k2 r

Here k1is the strength of the coulomb attraction of the quarks and k2is the strength of

the colour force of attraction (of the order of 1 GeV/fm). The quark-quark coupling

strength decreases for small values of r, as a result of the penetration of the gluon

cloud surrounding the quarks. The gluons carry "color charge" and therefore the

penetration of the cloud would reduce the effective color charge of the quark. The

quark chromodynamics can easily demonstrate the nature and the reason of the

interactions between the quarks via the gluons.
http://hyperphysics.phy-astr.gsu.edu/hbase/particles/quark.html#c6http://hyperphysics.phy-astr.gsu.edu/hbase/particles/quark.html#c6http://hyperphysics.phy-astr.gsu.edu/hbase/particles/expar.html#c1http://hyperphysics.phy-astr.gsu.edu/hbase/particles/expar.html#c1http://hyperphysics.phy-astr.gsu.edu/hbase/particles/expar.html#c1http://hyperphysics.phy-astr.gsu.edu/hbase/particles/quark.html#c6


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Quarks and Colour

The baryons are bound states of three quarks and mesons are bound states of two i.e.

quark and antiquark. This can actually somewhat explain why nucleons have net spin

J= while mesons have J value as 0. This also quite amicably describes the structure

of the proton, neutron and the ++

. But the quark scheme fails to explain why we dont

observe the mesonic states of qq or qq states. No uu states have ever been observed

with a charge of 4/3. Both these problems can be solved by introducing a new

property or a quantum state of colour.

According to this theory it is supposed that quarks come in three primary colours: red,

green, blue denoted by R, G, B. The colour denoted here is not real colour but just a

quantum number denotation. It has been so chosen because quarks behave in perfect

analogy with the colours in real life. Writing the quark structure now becomes easier.

For instance the baryon can now be written as uRuGuB. This eliminates the problem

of identical quarks. Similarly for a proton we can have a structure as uRuGdB, uGuBdR

and so on. But actually only state exists for a proton. So the colour quantum number

has to be introduced without proliferating the number of states. This is complied with

by asserting that all the particles in nature are colourless and unchanged in rotations in

the RGB space. The anti-quarks are given complementary colour quantum numbers

i.e. cyan/anti red (R), magenta/anti green(G) and yellow/anti blue (B). Now the

white colour or the colourless state can be be produced only by a certain set of ways.

Adding equal amounts of red, blue and green. (RGB) Adding equal amounts of cyan, magenta and yellow. (RGB) Adding equal amounts of a primary colour and a complementary colour. (RR,

GG, BB).


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The three qq states i.e. RR, BB, GG are colourless but it is only the combination

RR+BB+GG which is unchanged by rotations in the R,G,B space to form an

observed meson. Now it can be easily explained that why mesons of the form RG,

GB etc. dont exist. It lies in good agreement with the fact that mesons are colourless

(quantum state wise). The colour quantum concept has many more implications the

most important being the charge of nuclear interactions.

Phase Transition of Nucleons

In their normal states of lowest energy, nuclei show liquid-like characteristics and

have a density of 0.17 nucleons/fm3. In more conventional units, this corresponds to

2.7 X 1017

kg/m3, or 270 trillion times the density of liquid water. In a laboratory, the

only possible way to heat nuclei to significant temperatures is by colliding them with

other nuclei. The temperatures reached during these collisions are astounding. The

temperatures that can be reached in nuclear collisions range up to 100 MeV and


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above more than 200 million times the temperature at the surface of the Sun

(~5,500 K).

On heating a nucleus to a temperature of a few MeV, some of the nuclear liquid will

evaporate. From knowing the general form of the interactions between nucleons, it can

be said that, just like water, the nuclear liquid also has a latent heat of vaporization,

and that nuclei should undergo a first-order phase transition. This liquid-gas

coexistence is also expected to terminate at a critical point, the critical point of nuclear

matter. Now what we obtain is a phase of individual nucleons which when further

subjected to higher pressures or temperature tends to break down furthermore. The

nuclear shell then tends to dissolve and what is obtained is a continuous large mass of

nuclear matter. The overlap of the bags allows the quarks and the gluons of different

nuclei mix freely to form a state of quark-gluon plasma.


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Conclusion

The exotics in the dense core of neutron star, in this case Quarks, was done. Quarks

are the most fundamental particles alongside the leptons and obey the bag model and

asymptotic degrees of freedom. The various aspects of quarks in the neutron star,

nucleon to quark phase change process were studied. But a study regarding the

hyperons need to be done in order to get obtain more accurate results regarding the

calculations of mass and radius of the neutron star. In this project a detailed study

about the nuclear matter and its various properties was also done. Formulations of the

Equations of State of the nuclear matter in a neutron star, was done which gave us the

values of the parameters of energy density and pressure P. From these parameters

were formulated the TOV equations alongside the EoS which was solved using the

Runge-Kutta method algorithm. The TOV was also solved numerically using

programming language C. Thus the algorithm produced the results of the radius of the

neutron star and its mass which agrees closely with the experimentally observed

values for neutron star.


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References

Relativistic Nuclear Equation of State in an Effective Model (PhDThesis) by Dr. T. K. Jha.

Isospin Asymmetry in Nuclei and Nuclear Symmetry Energy by TapanMukhopadhyay and D. N. Basu.

Nuclear Physics by V. Devanathan.

An Introduction to Nuclear Physics (2ndEd.) by W. N. Cottingham and D.A. Greenwood.

Nuclear Matter Properties from Nuclear Masses by K. C. Chung, C. S.Wang and A. J. Santiago.

Quarks, Leptons and The Big Bang (2nded) by Jonathan Allday

Quarks and Leptons An Introductory Course in Modern Particle Physicsby F.Halzem,A.Martin.

Exotic Phases of Matter in Compact Stars by Fredrik Sandin

http://en.wikipedia.org/wiki/Nuclear_Matter http://hyperphysics.phy-astr.gsu.edu/hbase/particles/bag model.html
http://en.wikipedia.org/wiki/Nuclear_Matterhttp://en.wikipedia.org/wiki/Nuclear_Matterhttp://hyperphysics.phy-astr.gsu.edu/hbase/particles/bag_model.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/particles/bag_model.htmlhttp://hyperphysics.phy-astr.gsu.edu/hbase/particles/bag_model.htmlhttp://en.wikipedia.org/wiki/Nuclear_Matter

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