Preface

The neutron stars are kind of stellar objects, produced from the gravitational

collapses of massive stars during a Type II or Type Ib or Type Ic supernova

event when their nuclear fuels get exhausted. The mass of the progenitors

are ranging from 8M⊙ to 20M⊙. The mass of a typical neutron star is

about 1.4M⊙ to 3.2M⊙ with a corresponding radius ∼ 10 km. Constituents

of neutron stars are almost entirely neutrons, a small amount (∼ 4%) of

protons and an equal amount of electrons to make the system overall charge

neutral. Neutron stars are supported against further collapse because of its

strong self-gravity, is by the quantum degeneracy pressure of dense neutron

matter. The origin of this degeneracy pressure is Pauli exclusion principle.

The density of a typical neutron star varies from ≤ 109 kg/m3 at the crustal

region to 6 − 8 × 1017 kg/m3 near the central core region. In Fig.(1), we

have shown diagrammatically the schematic picture of internal structure for

a typical neutron star [1, 2, 3].

3

Preface

Figure 1: The schematic diagram of internal structure for a typical neutronstar. Outer core is composed of free electrons, free muons, super-fluid neutronmatter and super-conducting protons. Since the density of inner core is highenough, many exotic type of phases, e.g., Kaon condensed phase, dense quarkmatter phase etc., may exist in this region.

4

Preface

In this Ph.D. thesis work, entitled Some Theoretical Studies on Physics

and Astrophysics of Crustal Matter of Strongly Magnetized Compact Stars,

we have investigated the following important aspects related to dense crustal

matter, which is mainly compact metallic iron crystal, in presence of strong

quantizing magnetic field relevant for strongly magnetized neutron stars or

magnetars (see the references:[4, 5, 6, 7, 8, 9, 10, 11, 12, 13]):

• Developed a formalism for Thomas-Fermi-Dirac model for low density

stellar matter in presence of a strong quantizing magnetic field.

• Investigated some of the properties of crustal matter of magnetars.

• Studied some of the important properties of relatively low density stel-

lar matter in presence of strong quantizing magnetic field.

• Assuming that a first order superconducting phase transition can oc-

cur from the core region of a compact quark matter star, we have

investigated the expulsion of magnetic flux lines from the growing su-

perconducting core of a magnetized quark star.

Each of these chapters are self-explanatory, consisting of contents and intro-

duction, mathematical formalism, results and conclusions, along with the use-

ful references. Throughout this thesis work we have concentrated our studies

on the crustal region of strongly magnetized neutron stars or magnetars and

only the mechanism of flux expulsion from the growing superconducting zone

of bulk quark matter has been studied for quark stars.

5

Preface

The crustal region of a typical neutron star extends down to about one

km below the surface. The outermost layer, known as the outer crust, com-

posed of mainly dense crystalline metallic iron. In this region, because of

high density, the atoms are fully ionized. The free electrons behave like a

relativistic and degenerate Fermi gas. As one moves towards the centre the

density of matter increases, as a consequence the nuclei become more and

more neutron rich owing to electron captures which convert protons into neu-

trons inside the nuclei. In the region with densities ≤ 1011 gm cm−3, due to

the capture of electrons by the nuclei to form nuclei with more neutrons. In

this region, the matter is mainly composed of heavy nuclei with very small

amount of protons. In region with densities reaching 4 × 1011gm cm−3, the

atomic nucleus with the most abundant neutrons, 118Kr, starts to release

neutrons. This process is called neutron drip. Therefore, the matter in these

region is mainly composed of atomic nuclei, electrons, protons and dripped

neutrons.

The neutron drip process becomes more violent with the increase of den-

sity. In region with densities reaching the nuclear density (ρnuc ≈ 2.4 × 1014

gm cm−3), the atomic nuclei dissolve to the matter of mainly degenerate

neutrons and a small amount of electrons and protons. In the inner core re-

gion, it is generally believed that normal neutron matter undergoes a phase

transition to super-fluid phase, whereas very small quantity of proton matter

(≤ 4%) behaves like type-II superconductor. To make the system electrically

6

Preface

charge neutral, there are an equal amount of electrons and if the density per-

mits, negatively charged muons may also appear in dense neutron matter.

In the later case chemical potential of electrons becomes ≥ 100MeV.

In region with ρ > 1014 gm cm−3, the energy/particle of the degenerate

neutrons will exceed the rest masses of baryon resonances. As a consequence,

Λ and∑

hyperons are produced in dense neutron matter along with K-

mesons. Because of extremely high matter density, the central core region

may be in a degenerate exotic phase of quark matter.

To study the effect of strong quantizing magnetic field on neutron star/magnetar

crustal matter, we start with a brief overview of magnetic field strength in

naturally available systems. In the table as given below we have shown these

typical values:

7

Preface

Figure 2: Brief Overview of Magnetic Field Strength in Natural Systems

8

Preface

Magnetars are strongly magnetized young neutron stars, of age ∼ 104yrs

and surface magnetic field B ≥ 1015G [14, 15, 16, 17, 18]. They are also called

soft gamma repeaters (SGR). These exotic stellar objects are the pulsars

(rotating magnetized neutron stars) emitting bright and repeating flashes of

soft gamma rays. Since the wave lengths associated with the emitted soft

gamma rays may also lie in the hard X-ray region as well, these exotic objects

are also known by the name anomalous X-ray pulsars (AXP).

Since the magnetic field in the crustal region of a typical magnetar is

large enough, the free electrons from fully ionized dense metallic crystal will

be strongly affected in this region. Since the cyclotron quantum associated

with the moving electrons in strong magnetic field in the crustal region is

greater than the corresponding rest mass energy; the effect of strong magnetic

field will be quantum mechanical in nature [19, 20, 21, 22, 23, 24, 25]. The

quantum mechanical magnetic effect on degenerate crustal electron gas is

called Landau diamagnetism. For electrons of mass me, carrying charge of

magnitude qe, the critical magnetic field strength beyond which Landau levels

will be populated is given by B(e)c = m2

e/qe (throughout this thesis paper we

have used the natural unit with h̄ = c = 1). The numerical value for the

strength of B(e)c for electron is ≈ 4.4× 1013G. In such strange situation if we

assume that the constant magnetic field is along z-axis, then the z component

of electron momentum (pz) varies continuously from −∞ to +∞ at finite

temperature, but from −pF to +pF at zero temperature or if the actual

9

Preface

temperature of the system is less than the corresponding Fermi temperature

of the electron gas, where pF is the electron Fermi momentum. Interestingly

the orthogonal component (p⊥ = (p2x + p2

y)1/2, in the relativistic case) varies

in a discrete manner and is given by (2νqeB)1/2, where ν = 0, 1, 2, ........,

the Landau quantum numbers for the electrons. Therefore, the orthogonal

part of electron momentum gets quantized, the phenomenon is called Landau

quantization, which is a purely quantum mechanical effect of strong magnetic

field. In presence of such strong quantizing magnetic field the momentum

space which was spherical in nature for B < B(e)c becomes cylindrical, with

its symmetry axis along the direction of magnetic field. In figs.(3) and (4) we

have shown schematically the momentum space for degenerate electron gas

at zero temperature for the magnetic field strength below the critical value

and above the critical value respectively. Fig.(3) is the Fermi sphere for the

electrons, when B = 0 or B < B(e)c , whereas Fig.(4) is for B > B(e)

c . In

this case the longitudinal part of electron momentum is continuous along the

direction of magnetic field, which is also the symmetry axis of the cylinders

and −pF ≤ pz ≤ +pF , whereas p⊥ is changing in a discrete manner. Here

each cylinder corresponds a particular value of p⊥.

10

Preface

As a consequence the phase space volume element in momentum space

gets modified and is given by

d3p

(2π)3=d2p⊥dpz

(2π)3=giqiB

4π2

∑

ν

(2 − δν0)dpz (1)

where gi is the electron degeneracy factor (= 2, spin degeneracy), the upper

limit νmax for Landau quantum number will be finite at zero temperature

but infinity for T 6= 0 and the factor (2− δν0) takes care of singly degenerate

zeroth Landau level and doubly degenerate all other levels with ν 6= 0.

The Fermi integral for T = 0 is therefore given by

I0 =∫

d3pf(p) =giqiB

4π2

νmax∑

ν=0

(2 − δν0) ×∫ +pF

−pF

fν(pz)dpz (2)

whereas for T 6= 0 we have

I =∫

d3pf(p, T ) =giqiB

4π2

∞∑

ν=0

(2 − δν0) ×∫ +∞

−∞fν(pz, T )dpz (3)

where fν(pz) is a function of longitudinal momentum pz and also depends on

the Landau quantum number ν. Whereas in the finite temperature case this

function also depends on system temperature. At T = 0, in the relativistic

scenario the upper limit of Landau quantum number is given by νmax =

(µ2i −m2

i )/(2qiB) for the i-th type charged particle, which can be obtained

from the non-negative nature of p2F .

The unique behavior of strong magnetic field of magnetar is the splitting

or coagulation of X-ray photons traveling through such strong magnetic field.

X-ray photons readily split into two or more, or merge together. Such exotic

11

Preface

Figure 3: B < B(e)c

Figure 4: B > B(e)c

12

Preface

effects of magnetic photon splitting or merging cannot be observed in human

laboratories, but may be detected in this type of strange cosmic laboratories.

Some of the other important effects of strong quantizing magnetic field on

dense neutron star matter, which have already been studied thoroughly are:

a) No phase transition to quark matter from dense neutron star matter [26].

b) Equation of state changes significantly. As a consequence the gross prop-

erties of neutron stars, e.g.; mass, radius, moment of inertia etc. will

be severely affected [27, 28, 29, 30, 31, 32].

c) The strong quantizing magnetic field affects the fundamental interactions,

in particular the electro-magnetic and weak interaction process. As a

consequence the thermal evolution of neutron stars will be modified

[28].

d) Electron gas behaves like a neutral super-fluid [33].

e) The chiral symmetry of the quark matter system breaks down. Strong

quantizing magnetic field behaves like a catalyst to generate mass and

breaks the chiral symmetry [34, 35, 36, 37].

f) The strong quantizing magnetic field deforms the geometrical structure of

neutron stars. In the extreme scenario the strongly magnetized neutron

star becomes either a black string or a black disc [39, 40].

13

Preface

g) The kinetic pressure of the system becomes anisotropic in presence of

strong quantizing magnetic field [41, 42].

h) The presence of strong quantizing magnetic field in the crustal region of

neutron stars or magnetars breaks the spherical symmetry of electron

distribution around the metallic nuclei [43, 44].

i) The effect of strong quantizing magnetic field on the Fowler-Nordheim

type electron field emission from the poles of strongly magnetized neu-

tron stars [45].

j) The effect of strong quantizing magnetic field on the work function of

metal, which was needed to investigate the Fowler-Nordheim field emis-

sion of electrons [46].

Based on the effect of strong quantizing magnetic field as mentioned above,

in this Ph.D. thesis work we have investigated the following properties of

crustal matter of strongly magnetized neutron stars or magnetars.

1: In chapter-1 we have investigated some of the gross properties of

crustal matter of strongly magnetized neutron stars using non-relativistic

version of Thomas-Fermi-Dirac model. The crustal matter of neutron star

may be assumed to be a regular array of fully ionized metallic ions of iron nu-

clei. We have assumed that the nuclei are at rest at the centre of Wigner-Seitz

cells and the distribution of electrons around the nuclei of metallic atoms.

The electrons are assumed to be non-relativistic in nature and are moving in

14

Preface

presence of strong quantizing magnetic field along a particular direction, say

z-axis. The motion of the electrons are therefore quantized in the x-y plane

and continuous along z-axis. To investigate the gross properties of such mag-

netically distorted matter, we have assumed that the electrons are confined

within the spherically symmetric Wigner-Seitz (WS) cells. Further, We have

obtained the electron-electron exchange interaction within the Wigner-Seitz

cells and incorporated the exchange term in Thomas-Fermi condition and

obtained Thomas-Fermi-Dirac equation. the electrons inside the cells are as-

sumed to behave semi-classically and the potential changes slowly with the

radial parameter, so that Thomas-Fermi model [47, 48] can be used to study

some of the important gross properties of the crustal matter. In this chapter,

the important findings we have noticed are:

(a) The outer crust of a neutron star, particularly in the case of a strong

magnetic field (magnetars) plays a crucial role in the evolution of pulsar

magnetic field. It is really a great challenge to explain field evolution in

these strongly magnetized objects using existing models of field evolution.

(b)These objects require a very rapid field evolution.

(c) Now the TFD model for low density matter in presence of strong mag-

netic fields shows an over all contraction of the outer crust. Since the Ohmic

decay of magnetic field in a conducting material depends on the thickness of

the region, a decrease in width of the outer crust by an order of magnitude

will cause a rapid decay of magnetic field (at least two orders of magnitude

15

Preface

decrease in decay time scale). The equation of state indicates that elec-

trons within the Wigner-Seitz cells are more strongly bound to the positively

charged nuclei in presence of strong quantizing magnetic fields than the non-

magnetic (or non-quantizing) case. Such strong binding of electrons within

the cells may decrease the electrical conductivity of the matter. Which will

further reduce the time scale for Ohmic decay of magnetic field in the outer

crust of these strongly magnetized stellar objects.

(d) Because of strong magnetic field along z-direction, the isotropic nature

of electron kinetic pressure will break and becomes anisotropic inside the

cells. Which has already been studied thoroughly and reported in published

articles from our group and others [41, 42, 49].

(e) It is also found that the upper limit of Landau quantum number

for the electrons within the cells depends on the position of the particular

electron; i.e., is a function of radial coordinate r.

(f) It is further noticed that the electrons at the periphery of WS cells

are fully polarized. Therefore the electron gas at the boundary region of

WS cells behaves like ferromagnetic material instead of having diamagnetic

nature. For magnetic field strength ≥ 1015G the electron gas throughout the

WS cells show ferromagnetism.

(g) One of the most important observation is that the singularity of

Thomas-Fermi equation at the origin is removed by the presence of Landau

levels, which are populated by the presence of strong quantizing magnetic

16

Preface

field.

Some version of the work presented in chapter-1 has already been pub-

lished in IJMPD D11, (2002) 123.

2: In this chapter we have presented our investigation on the effect of

strong magnetic field on the crustal matter of magnetars. The work is divided

into two parts: in the first part, based on one of our very recent work [49],

we have investigated the effect of strong quantizing magnetic field on the

outer crust matter. In the second part, we have studied the properties of

compact sub-nuclear matter at the inner crust region in presence of such

strong quantizing magnetic field [50].

In This chapter we have thoroughly investigated the effect of strong quan-

tizing magnetic field on both the outer and inner crust matter of magnetars.

In the outer crust region, the matter with dense crystalline structure of metal-

lic iron at sub-nuclear density are replaced by an array of spherically sym-

metric WS cells with positively charged nuclei at the centre surrounded by

non-uniform dense electron gas with over all charge neutrality. In the inner

crust we have used the conventional Harrison-Wheeler (HW) and Bethe-

Baym-Pethick (BBP) equation of states for the nuclear matter, consisting

of iron along with some more heavier neutron rich nuclei. Here we have

presented with detailed numerical computation the effect of strong magnetic

field on the inner crust matter, which as mentioned above is assumed to be

a mixture of iron, some heavier neutron rich nuclei, which we found to be

17

Preface

specially true in presence of strong quantizing magnetic field, electrons and

free neutrons. The presence of free neutrons are considered beyond neutron

drip density. The new findings of this investigation are as mentioned below:

(a)spherically symmetric WS cells. It has been observed that the radius

of each cell decreases with the magnetic field strength.

(b) It has also been noticed that the upper limit of Landau quantum

number is a function of positional coordinate of the electron with which it

is associated within the WS cells. We have observed that at the surface

region, for all the values of magnetic field strength, this upper limit becomes

identically zero. Which actually means that the electrons near the WS cell

surface are strongly polarized in the opposite direction of external magnetic

field. Whereas, for B > 1015G, they are polarized at every points within the

cells.

(c) It has been observed that the electron density within the cells increases

with the increase in magnetic field strength. Further, for all the values of

magnetic field strength, the electron density is maximum near the nuclear

surface (r = rn) and minimum at the WS cell boundary (r = rs).

(d) The electron kinetic pressure is found to be positive near the central

portion of WS cell but is negative near the surface region. There is a position

within the cell at which the kinetic pressure vanishes. The position of this

point changes with the strength of magnetic field. We have also studied

the variations of kinetic energy, electron-nucleus interaction energy, electron-

18

Preface

electron direct potential energy and electron-electron exchange interaction

part within the cells. We have shown the variation of these quantities within

the WS cells for a number of magnetic field strengths.

(e) We have used HW and BBP equation of states for the nuclear mass

formula. In the second part of this work, we have investigated some of the

properties of inner crust matter of magnetars. We have used HW and BBP

equation of states for the nuclear mass formula. It has been observed from

both the models, that for a stable inner crust matter, the nuclei present must

be heavier than iron and much more neutron rich. The heaviness is more in

the case of BBP equation of state. We have also noticed that for low and

moderate values of magnetic field strength, the variation of mass number and

the corresponding atomic number with magnetic field is not so significant.

Whereas, for B ≥ 1015G, when electrons occupy only the zeroth Landau

level, then much more heavier neutron rich nuclei are formed in the inner

crust region. It is found that high magnetic field behaves like a catalyst to

generates heavy neutron rich nuclei.

(f) We have observed that initially the electron density increases with

the increase in mass number, but as soon as free neutrons appear in the

system, the electron density decreases and saturates to some constant value

which depends on the magnetic field strength. In the case of HW equation

of state, free neutron density does not depend on the strength of magnetic

field, whereas, for BBP case, because of chemical equilibrium condition, the

19

Preface

free neutron density depends very weakly on the magnetic field strength. We

have noticed that in the case of BBP equation of state the overall qualitative

difference is because of chemical equilibrium among the constituents.

(g) The total baryon density rises sharply like an avalanche for the value

of A at which free neutrons appear in the system. However, for BBP equation

of state, because of chemical equilibrium condition, the rise is not so sharply

visible for a given magnetic field B.

(h) The qualitative nature of equation of states are almost identical. It

is found that in presence of strong magnetic field, the inner crust matter

becomes mechanically stable (with the positive value of kinetic pressure)

only at very high density.

Some version of the work presented in chapter-2 is published in Euro.

Phys. Jour. A 45, 99 (2010).

3: In chapter-3 we have studied the effect of strong quantizing magnetic

field on low density stellar matter at the crustal region using Thomas-Fermi

and Thomas-Fermi-Dirac (TFD) models. The Wigner-Seitz cell structure is

assumed for the low density matter. The significant changes in the prop-

erties of such low density matter in presence of strong magnetic fields are

discussed. The effect of strong quantizing magnetic field on the equation of

state of matter at the outer crust region of magnetars is studied. The density

of such matter is low enough compared to the matter density at the inner

crust or outer core region. Based on the relativistic version of semi-classical

20

Preface

Thomas-Fermi-Dirac model in presence of strong quantizing magnetic field

a formalism is developed to investigate this specific problem. The equation

of state of such low density crustal matter is obtained by replacing the com-

pressed atoms/ions by Wigner-Seitz cells with nonuniform electron density.

The results are compared with other possible scenarios. The appearance of

Thomas-Fermi induced electric charge within each Wigner-Seitz cell is also

discussed. The important findings of this chapter are:

(a) We have noticed that in this formalism, to solve the Poisson equation

numerically it is necessary to include a few more conditions, which were

absent in the usual field free non-relativistic model or in presence of ultra-

strong magnetic field (νmax = 0).

(b) To remove singularity at the origin, we suggest to use finite dimension

for the nuclei. It has also been noticed that unlike other scenario, one extra

condition appears in the non-relativistic regime with B 6= 0 and νmax 6= 0.

(c) We have also given an approximate method to get an estimate of the

induced charge within each cell and thereby obtain the variation of screening

length with magnetic field strength.

(d) In our model, the Wigner-Seitz cells are assumed to be spherical in

nature and found that the radius of each cell decreases with the increase of

magnetic field strength. The variation is given by ∼ B−1/2.

(e) The formalism is of course not applicable to the inner crust region,

where the matter density is close to the neutron drip point, some of the

21

Preface

neutrons may come out from the cells. We have presented a modified version

of this formalism appropriate for the inner crust region in chapter-2.

(f) We have assumed that all the electrons within the cells are moving

freely, i.e., they are not bound in any one of the atomic orbitals. In reality,

it may happen that the electrons at the vicinity of the nucleus in a cell have

negative energy. These electrons, therefore can not be treated as free. It

is therefore absolutely necessary to get the total energy of an electron as a

function of its position (r or x) within the cell from the numerical solution

of the Poisson’s equation and the expressions for kinetic and various form of

interaction energies. We expect that very close to the nucleus, the electron

energy will be negative and for a particular value of x (= r/µ) (which may be

a function of B) it will become zero (quasi-free electrons) and then becomes

positive. If it is found so, then we can not assume that all the Z-electrons

in the cell are participating in statistical processes. On the other hand, if we

consider the expression for electron energy as given in eqn.(3.86), then from

the physics point of view all the electrons will become free (energy is always

positive). Whereas, if we consider

µ = kinetic energy − eφ = constant,

then we may have bound, quasi-free and free electrons within the cells. The

presence of free electrons in the compressed cells in a dense medium is pop-

ularly known as statistical ionization.

Some version of chapter-3 has been published in Ann. of Phys. 324, 499

22

Preface

(2009).

4: The expulsion of magnetic flux lines from a growing superconduct-

ing core of a quark star has been investigated in chapter-4. The idea of

impurity diffusion in molten alloys and an identical mechanism of baryon

number transport from hot quark-gluon-plasma phase to hadronic phase

during quark-hadron phase transition in the early universe, micro-second

after big bang have been used. The possibility of Mullins-Sekerka normal-

superconducting interface instability has also been studied [51].

In the present chapter we have assumed a type-I superconducting phase

transition in quark matter at the core region of a quark star and investi-

gated the mechanism by which the magnetic flux lines are expelled from the

superconducting zone.

In a very recent work by Konenkov and Geppert have investigated the

expulsion of magnetic flux lines from superconducting core region of neutron

stars. They have considered a type-II superconducting transition at the core

region and studied the movement of quantized fluxoids. They have also given

a mechanism by which the flux lines expelled from the core into the crustal

region undergo ohmic decay.

As has been discussed in chapter-4, we can have only uu, dd, ud and ss

Cooper pairs in the system.

Since it is expected that the magnetic field strengths at the core region

of a quark star are much less than the corresponding critical value for the

23

Preface

destruction of superconducting property and the temperature is also low

enough, then during such a type-I superconducting phase transition, the

magnetic flux lines from the superconducting quark sector of the quark star

will be pushed out towards the normal crustal region. Now for a small type-

I superconducting laboratory sample placed in an external magnetic field

less than the corresponding critical value, the expulsion of magnetic field

takes place instantaneously. Whereas in the quark star scenario, the picture

may be completely different. It may take several thousands of years for the

magnetic flux lines to get expelled from the superconducting core region.

Which further means, that the growth of superconducting phase in quark

stars will not be instantaneous. Therefore in a quark star / hybrid star, with

type-I superconducting quark matter at the core, the magnetic flux lines will

be completely expelled by Meissner effect not instantaneously, it takes several

thousand years of time.

The aim of the present chapter is to investigate the expulsion of mag-

netic flux lines from the growing superconducting core of a quark star. We

have used the idea of impurity diffusion in molten alloys or the transport

of baryon numbers from hot quark matter soup to hadronic matter during

quark-hadron phase transition in the early universe, micro-second after big

bang (the first mechanism is used by the material scientists and metallur-

gists whereas the later one is used by cosmologists working in the field of

big bang nucleo-synthesis, We have also studied the possibility of Mullins-

24

Preface

Sekerka normal-superconducting interface instability in quark matter. This

is generally observed in the case of solidification of pure molten metals at

the solid-liquid interface, if there is a temperature gradient. The interface

will always be stable if the temperature gradient is positive and otherwise it

will be unstable. In alloys, the criteria for stable / unstable behavior is more

complicated. It is seen that, during the solidification of an alloy, there is a

substantial change in the concentration ahead of the interface. Here solute

diffusion as well as the heat flow effects must be considered simultaneously.

The particular problem we are going to investigate here is analogous to solute

diffusion during solidification of an alloy.

In chapter-4, the new findings are as given below:

(a) If a superconducting transition occurs in a quark star, the magnetic

properties of such bulk object are entirely different from that of a small

laboratory superconducting sample.

(b) The expulsion of magnetic flux lines from the superconducting zone

is not at all instantaneous. The typical time scale is 105 − 106 yrs.

(c) We have noticed that this time scale is very close to the magnetic field

decay time scale in a neutron star.

(d) Due to the presence of excess magnetic flux lines at the interface,

which is actually true if the diffusion rate of magnetic lines of forces in the

normal phase is less than the rate of growth of the superconducting zone, the

topological structure of normal-superconducting boundary layer may change

25

Preface

significantly.

(e) It may take dendritic shape instead of planer structure. The stability

of planer interface also depends on the strength of interface magnetic field at

the boundary layer.

(f) Since the expulsion time scale is very high, we expect that there will

be no instability at the interface between normal and superconducting quark

matter phase. The superconducting phase will grow steadily.

Some version of this work has been published in Astrphys. Space Science

323, 123 (2009).

26

Bibliography

[1] see the review work by P.Haensel, A.Y.Potekhin and D.G.Yakovlev,

Neutron Stars 1: Equation of State and Structure, Springer, Vol. 326

(2007).

[2] S.L. Shapiro and S.A. Teukolsky, Black Holes, White Dwarfs and Neu-

tron Stars, John Wiley and Sons, New York, (1983).

[3] N. K. Glendenning, Phys. Rev. C37, 2733 (1988); J. D. Walecka,

Ann. Phys., 83, 491, (1974); N. K. Glendenning, Compact Stars, Nu-

clear physics, Particle Physics and General Relativity (Second Edison),

Springer, 162 (2000).

[4] A.Y.Potekhin, D.A.Baiko, P.Haensel and D.G.Yakovlev, Astron. Astro-

phys. 346, 345 (1999).

[5] A.Y.Potekhin, Astron. Astrophys. 306, 999 (1996).

[6] A.Y.Potekhin and D.G. Yakovlev, Astron. Astrophys. 314, 341 (1997).

[7] Bailin D. and Love A., 1984, Phys. Rep. 107, 325.

27

Bibliography

[8] Proceedings of the NATO advanced research workshop on ”Superdense

QCD matter and compact stars”, Yerevan, Armenia, September 27- Oc-

tober 4, 2003. Nato Science Series II:Mathematics, Physics and Chem-

istry, Vol. 197, Springer 2006, D. Blaschke and D. Sedrakian (eds.).

[9] Alford M., Annual Review of Nuclear and Particle Science, 2001, 51,

131.

[10] Blaschke D., Voskresensky D.N. and Grigorian H., 2005, hep-

ph/0510368.

[11] Rajagopal K. and Wilczek F., 2000, hep-ph/0011333.

[12] Alford M.G., Rajagopal K. and Wilczek F., 1999 Nucl. Phys. B537, 443.

[13] Alford M.G, Kouvaris C. and Rajagopal K., 2004, Phys. Rev. Lett. 92,

222001.

[14] R.C. Duncan, astro-ph/0002442v1; R. C. Duncan and C. Thompson,

APJ Lett. 392, L9 (1992); C. Thompson and R. C. Duncan, MNRAS

275, 255 (1995); C.Thompson and R.C. Duncan, APJ, 473, 322 (1996).

[15] P. M. Woods, et al. APJ Lett. 519, L139 (1999).

[16] K. Hurley, et al. APJ Lett. 442, L111 (1999).

[17] S. Mereghetti and L. Stella, APJ 442, L17 (1999).

[18] S. Mereghetti, arXiv:0904.4880v1 [astro-ph] and references therein.

28

Bibliography

[19] S. Chakrabarty, Astrophys. and Space Science, 213,121 (1994).

[20] S. Chakrabarty and A. Goyal, Mod. Phys. Lett. A9, 3611 (1994).

[21] S. Chakrabarty, Phys. Rev. D54, 1306 (1996); S. Chakrabarty, D. Ban-

dopadhyay and S. Pal, Phys. Rev. Lett. 78, 2898 (1997).

[22] D. Bandopadhyay, S. Chakrabarty and S. Pal, Phys. Rev. Lett. 79, 2176

(1997).

[23] S. Chakrabarty, D. Bandopadhyay and S. Pal, Int. Jour. Mod. Phys.

A13, 295 (1998).

[24] D. Bandopadhyay, S. Pal and S. Chakrabarty, Jour. Phys. G24, 1647

(1998).

[25] D. Bandopadhyay, S. Chakrabarty, P. Dey and S. Pal, Phys. Rev. D58,

121301 (1998) (Rapid Communication); S. Pal, D. Bandopadhyaya and

S. Chakrabarty, Jour. Phys. G25, L117 (1999).

[26] S. Chakrabarty, Phys. Rev. D54 (1996) 1306; T. Ghosh and S.

Chakrabarty, Phys. Rev. D63 (2001) 043006; T. Ghosh and S.

Chakrabarty, Int. Jour. Mod. Phys. D10 (2001) 89; S. Chakrabarty,

Phys. Rev. D51 (1995) 4591.

[27] S. Chakrabarty and P.K. Sahu, Phys. Rev. D53, (1996) 4687.

[28] P.K. Sahu and S.K. Patra, Int. Jour. Mod. Phys. 16, (2001) 2435.

29

Bibliography

[29] A.A. Isayev and J. Yang, JETP Lett 92, (2010) 783; A.A. Isayev, Phys.

Rev. C85, (2002) 031302.

[30] A.K. Harding and D. Lai, Rep. Prog. Phys. 69, 2631 (2006).

[31] M. Ruderman, Phys. Rev. Letts. 27, 1306 (1971).

[32] Z.Medin and D.Lai, MNRAS 382, 1833 (2007); Z.Medin and D.Lai,

Phys. Rev. A74, 062507 (2006); A.Harding and D.Lai, astro-

ph/0606674.

[33] Sutapa Ghosh and Somenath Chakrabarty, Mod. Phys. Lett. A17,

(2002) 2147.

[34] Sutapa Ghosh, Soma Mandal and Somenath Chakrabarty, Phys. Rev.

C75, (2007) 015805.

[35] V.P. Gusynin, V.A. Miransky and I.A. Shovkovy, Phys. Rev. D52, 4747

(1995); V.P. Gusynin, V.A. Miransky and I.A. Shovkovy, Phys. Lett.

B349, 477 (1995); V.P. Gusynin and I.A. Shovkovy, Phys. Rev. D56,

5251 (1995); V.P. Gusynin, V.A. Miransky and I.A. Shovkovy, Phys.

Rev. Lett. 83, 1291 (1999).

[36] D.M. Gitman, S.D. Odintsov and Yu.I. Shil’nov, Phys. Rev. D54, 2968

(1996); D.S. Lee, C.N. Leung and Y.J. Ng, Phys. Rev. D55, 6504 (1997);

D.S. Lee, C.N. Leung and Y.J. Ng, Phys. Rev. D57, 5224 (1998); E.V.

Gorbar, Phys. Lett. B491, 305 (2000).

30

Bibliography

[37] T. Inagaki, T. Muta and S.D. Odintsov, Prog. Theo. Phys. 127, 93

(1997); T. Inagaki, S.D. Odintsov and Yu.I. Shil’nov, Int. J. Mod. Phys.

A14, 481 (1999); T. Inagaki, D. Kimura and T. Murata, Prog. Theo.

Phys. 111, 371 (2004).

[38] A. Singh, S. Puri and H. Mishra, arXiv:1106.5260; B. Chatterjee, H.

Mishra and A. Mishra, NPA 862-863:312-315, 2011 (ICPAQGP-2010,

DEcember 2010); ibid Phys. Rev. D84, (2011) 014016.

[39] A. A. Isayev and J. Yang, arXiv:1301.0544; arXiv:1210.3322;

arXiv:1112.5420

[40] D.P. Menezes, M. Benghi Pinto, S.S. Avancini, A. Perez Martinez and

C. Provindencia, Phys. Rev. C79, (2009) 035807; . Gonzalez Felipe, H.J.

Mosquera Cuesta, A. Perez Martinez and H. Perez Rojas, Chin. Jour.

Astron. Astrophys. 5, (2005) 1959.

[41] A. Perez Martinez, H. Perez Rojas, H.J. Mosquera Cuesta, Int. Jour.

Mod. Phys. D17, (2008) 2107.

[42] M. Chaichian, S.S. Masood, C. Montonen, A. Perez Martinez and H.

Perez Rojas, Phys. Rev. Letts. 84, (2000) 5261.

[43] Jon Eilif Skjervold and Erlend Ostgaard, Phys. Scr. 29, 484 (1984).

[44] D. Lai, Rev. Mod. phys. 73, 629 (2001).

31

Bibliography

[45] A. Ghosh and S. Chakrabarty, MNRAS, 425, 1239 (2012).

[46] A. Ghosh and S. Chakrabarty, A & A, 32, 377 (2011).

[47] B. K. Shivamoggi and P. Mulser, EPL 22, 657 (1993).

[48] E.H. Lieb and B. Simon, Phys. Rev. Lett. 31, 681 (1973); E.H. Lieb,

J.P. Solovej and J. Yngvason, Phys. Rev. Lett. 69, 749 (1992); E.H.

Lieb, Bull. Amer. Math. Soc. 22, 1 (1990); R. Ruffini, ”Exploring the

Universe”, a Festschrift in honour of Riccardo Giacconi, Advance Series

in Astrophysics and Cosmology, World Scientific, Eds. H. Gursky, R.

Rufini and L. Stella, vol. 13, 383 (2000); Int. Jour. of Mod. Phys. 5, 507

(1996).

[49] Nandini Nag, Sutapa Ghosh and Somenath Chakrabarty, Ann. of Phys.

324, 499 (2009);

[50] Nandini Nag, Sutapa Ghosh and Somenath Chakrabarty, Euro. Phys.

Jour. A 45, 99 (2010).

[51] Mullins W.W. and Sekerka R.F., Jour. Appl. Phys. 34, 323 (1963);

Mullins W.W. and Sekerka R.F., Jour. Appl. Phys. 1964; 35, 444 (1964);

see also Langer J.S., 1980, Rev. Mod. Phys. 52, 1.

32