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M. β , − 6 .TeatosaotdteToa em oe o hi calculatio their for model Fermi Thomas the adopted authors The ]. ea n ec etoiaini hmcleulbimgvnby given equilibrium chemical is neutronization hence and decay β − ∗ .INTRODUCTION I. ea a cu hnteeeg feetosbcmshge tha higher becomes electrons of energy the when occur can decay orsodn Author Corresponding E § edrv h eto rpdniyi ytmwt eve ions. heavier with system a in density drip neutron the derive we 3 e F + E p F = R E γ ∼ & n F − te uhr netgtdnmrclytevariations the numerically investigated authors Other ei ed nncersrcuehsbe tde in studied been has structure nuclear on fields netic R uhacniini xetdt emdfidi the in modified be to expected is condition a Such . 70km, , 10 a us rmnurnsas tsc ihdensity high such At stars. neutron from burst ray awti antr.Bsdo ul self-consistent fully on Based magnetars. within na g este.Ti ilb epu oitrrtthe interpret to helpful be will This densities. igh rnsaswihmgthv infiatimplications significant have might which stars tron ∼ n h ylto nryo neeto exceed electron an of energy cyclotron the and y ergon tt.Hne h opsto fthe of composition the Hence, state. ground lear 17 cfil ed ob ucetyhg,mkn the making high, sufficiently be to needs field ic gei ed[ field agnetic .Teerslsmyhv important have may results These G. n,i eea,ncerastrophysics nuclear general, in and, s 00m,temaximum the 5000km), eas ge ue rsso eto tr a icse in discussed was stars neutron of crusts outer erature. e ale ndffrn otxs o example, For contexts. different in earlier ted lcrn n rtn r ae h generic The same. are protons and electrons f osuyteeeto hg)mgei edon field magnetic (high) of effect the study to B os ntepeec fmagnetic of presence the in rons, int & irin,teeeto magnetic of effect the ions, vier rsneo antcfil.It field. magnetic of presence e ∼ h etoiaindniyi neutron- a in density neutronization the 10 & ∼ 10 17 e ihdniywiedaf having dwarfs white density high sed 10 ∗ 10 re fmgiuehigher magnitude of order ,wihpeual orsod to corresponds presumably which G, 15 17 18 mgea oe)[ model) (magnetar G 8 [ G ,wihw ilrcl in recall will we which ], [ G 02 India 0012, 4 2 .Sc mle ht dwarfs’ white smaller Such ]. , 3 ,weetedniyi also is density the where ], B rpline drip int neutronization. f ol erestricted be could 1 n hence and ] § again 3 sand ns the n (1) 2

F where Ee,p,n are the Fermi energies for electron, proton and neutron respectively given by

F F 2 2 4 2 e,p Ee,p = p c + ∆e,pc with ∆e,p = me,p(1+2νe,pBD ) (2) q and

F F 2 2 2 4 En = pn c + mnc , (3) q e,p where me,p,n are the masses of electron, proton and neutron respectively, BD are the magnetic fields in the units of e 13 p 20 respective critical fields Bc =4.414 × 10 G and Bc =1.364 × 10 G, for electrons and protons respectively, νe,p are F F the occupied Landau levels in the respective energies, p is the Fermi momentum of electrons and protons, pn is the Fermi momentum of neutrons and c is the speed of light. The neutronization density for this system was investigated by previous authors [2]. Nevertheless, here we discuss the results categorically. Let us consider three possible cases, some of them may turn out to be absurd. F 2 2 a. Case I: p /c >> ∆e,p: This happens when both protons and electrons are highly relativistic. At the beginning of formation of neutrons (and hence the corresponding momentum is zero), from eqn. (1) we obtain

F p /c = mn/2, (4) which however violates initial choice as mp ∼ mn. Hence, this situation is unphysical to reveal the neutronization. F 2 2 b. Case II: p /c ∼ ∆p >> ∆e: This happens when electrons are highly relativistic but protons are just becoming relativistic. From eqn. (1) the initiation of neutronization occurs at

m2 − ∆ pF /c = n p , (5) 2mn

2 F 2 which, however, turns out to be negative unless ∆p ∼ mp. However, this corresponds to Ep ∼ mpc which cannot reveal relativistic protons. Hence, this case is unphysical. F 2 2 c. Case III: p /c ∼ ∆e << ∆p: This corresponds to relativistic electrons but nonrelativistic protons, which reads eqn. (1) as

pF 2 ∆1/2 + + ∆ = m (6) p c e n s  at the neutronization, revealing further

2 2 F mp p me e p /c = mn 1 − 1+2νpBD − (1+2νeBD). (7) s mn mn  q    When the Fermi energy and magnetic field of the system are such that the electrons lie in the ground Landau level 2 only (with νe = 0) and the protons are not affected by the magnetic field (with EF pmax ∼ mpc ), the neutronization density 2eBe Be ρ = D c pF (m + m )=0.343Be 107 gm/cc, (8) neut h2c e p D where e is the charge of electrons and h the Planck’s constant. Eq. (8) implies that the neutronization density increases linearly with the magnetic field and for the increment to be at least an order of magnitude compared to the nonmagnetic case, the magnetic field B has to be ∼ 1015G. However, beyond neutronization, the chemical equilibrium eqn. (1) gives

F 2 2 2 F 2 2 2 F 2 2 2 p + mec + p + mpc = pn + mnc , (9) q q q e where we have assumed that all the electrons reside in the ground Landau level and for this to be true, BD > 2.77 (see [2]), and the protons are not affected by the chosen value of magnetic field. 3

16 10 e B D=0 15 Be =100 10 e D B D=1000 1014 13

) 10 3 1012 11

(gm/cm 10 bd ρ 1010 109 108 107 0.0001 0.001 0.01 0.1 1 F x n

FIG. 1: Density beyond neutronization in an n − p − e system as a function of (dimensionless) Fermi momentum of neutrons, for different magnetic fields. All electrons reside in the ground Landau level and the protons are not affected by the chosen magnetic fields.

Therefore, the density beyond neutronization is given by

ǫ 2Be pF ≈ n D ρbd mpNe + 2 , where Ne = 2 3 , (10) c (2π) λe mec where

2 mnc F F F ~ ~ ǫn = 3 χ(xn ), xn = pn /mnc, λe = /mec, λn = /mnc, λn 1 2 χ(xF ) = xF (1+2xF ) 1+ xF 2 − log(xF + 1+ xF 2) . n 8π2 n n n n n  q q  (11)

Here Ne is the number density of electrons (which is also the number density of protons), ǫn is the energy density of neutrons, λe and λn are the Compton wavelengths of the electron and neutron respectively and ~ = h/2π. Now eqn. F F F e (9) expresses p in terms of pn . Hence ρbd varies as a function of xn for a given BD, as shown in Fig. 1.

III. NEUTRON DRIP IN A GAS WITH HEAVY IONS

The results of this section are mostly applicable for neutron stars and maybe for white dwarfs with heavier ion(s). We adopt the Harrison-Wheeler formalism [9] in describing the equilibrium composition of nuclear matter. We do not include the shell effects in the model, as they have a negligible effect on the density of the system. Here we intend to determine the variation in the density at the onset of drip with the variation in magnetic field. Hence, the lattice energy effects can be neglected, as they have a significant influence only on the composition of the system and an even lesser influence on the pressure. They do not alter the density of the mixture [10]. Moreover, the Harrison-Wheeler model has the added advantage of allowing one to obtain a purely analytical relation between the drip density and the field strength, as we shall show below. Previous authors obtained a relation between the field and drip pressure (see, e.g., [8]), however purely in numerical computations. Their results have the disadvantage that the drip pressure versus magnetic field relation is obtained numerically at discreet values of the field with reasonably large intervals. This kills an important physics revealing oscillations of the drip density at the lower values of field, which we address below 4 in detail in our analysis. Moreover, they considered only a limited range of fields while studying the drip pressure variation. Whereas we investigate the density for a wide range of field strengths so as to understand the regimes where there is a significant change in the drip density. We assume that given enough time, following nuclear burning, cold catalyzed material will achieve complete ther- modynamic equilibrium. The matter composition and equation of state will then be determined by the lowest possible energy state of the matter. The constituents of the matter are assumed to be ions, free electrons and free neutrons and the drip density is obtained by finding the density of the equilibrium composition, when the neutron number density is zero (neutrons just start appearing). We further assume the strong magnetic field present in the matter is constant throughout. We then obtain the equilibrium density by minimizing the energy density of the mixture with respect to the number densities of the ions, electrons and neutrons. Due to the presence of the magnetic field, the electrons are Landau quantized but the field of present interest (say, upto the value which a neutron star core could plausibly have) is small enough so that it does not affect the ions. Now we proceed to establish our results. Note that for the convenience of the readers, we follow the well-known textbook by Shapiro & Teukolsky [10] in developing our results, which discusses the non-magnetic results in detail. The total energy density of the system can be written as

′ ǫtotal = NionM(A, Z)+ ǫe(Ne)+ ǫn(Nn). (12) Here M(A, Z) is the energy of a single ion (A, Z), including the rest mass energy, where A and Z are atomic mass and atomic number respectively. It is conventional to include the rest mass′ energy of the electrons in the energy of the ions (see, e.g., [10] for details of non-magnetic results). Therefore, ǫe is the energy density of the electrons after 2 subtracting Nemec from the total energy density ǫe and ǫn is the total energy density of neutrons. The system is neutral in charge. Hence, the number densities Nion, Ne and Nn, of ions, electrons and neutrons respectively, and the total baryon number density N are related as

N = NionA + Nn, Ne = NionZ. (13)

These can be written in terms of mean numbers per baryon as

YionA + Yn = 1, YionZ = Ye, (14) where Yi = Ni/N. Hence at T = 0, we can write ǫtotal as a function of either (N,A,Z,Yn) or (N, Yion, Ye, Yn). The equilibrium composition is obtained by minimizing ǫtotal with respect to A, Z and Yn for a constant N. We use the semi-empirical mass formula of Green [11] based on the liquid drop nuclear model

b A 2 b Z2 M(A, Z)= m c2 b A + b A2/3 − b Z + 4 − Z + 5 , (15) u 1 2 3 A 2 A1/3 "   # where

b1 =0.991749, b2 =0.01911, b3 =0.000840, b4 =0.10175, b5 =0.000763

−24 and mu =1.66 × 10 gm (1 amu). Hence eqn. (12) becomes

N(1 − Y ) ′ ǫ = n M(A, Z)+ ǫ (N )+ ǫ (N ). (16) total A e e n n Moreover,

′ dǫe F 2 dǫn F = Ee − mec , = En . (17) dNe dNn Now we approximate A and Z as continuous variables and equations for equilibrium compositions can be found from eqns. (16) and (17) as ∂ǫ ∂M total =0, and hence = −(EF − m c2), (18) ∂Z ∂Z e e

∂ǫ ∂ M total =0, and hence A2 = Z EF − m c2 , (19) ∂A ∂A A e e  
5 and finally

∂ǫtotal ∂M F =0, and hence using eqn. (19) = En . (20) ∂Yn ∂A Now combining eqns. (18) and (19), we obtain

b Z = 2 A1/2. (21) 2b  5  Eqns. (18) and (19) also give

2Z Z b + b 1 − − 2b m c2 = EF − m c2, (22) 3 4 A 5 A1/3 u e e    

2 2 2 − 1 Z b Z b + b A 1/3 + b − − 5 m c2 = EF , (23) 1 3 2 4 4 A2 3A4/3 u n     F F 2 2 where En = (1 + xn )mnc , and at the onset of neutron drip, xn = 0 when the number density of neutron is just zero. Therefore, at the onset of drip, eqn. (23) can be written as p 2 2 2 − 1 Z b Z m b + b A 1/3 + b − − 5 = n . (24) 1 3 2 4 4 A2 3A4/3 m   u Solving eqns. (21) and (24), we obtain (A, Z) ≈ (122, 39.1) for the onset of drip. Using these values for A and Z, we can obtain the electron Fermi energy at the drip from eqn. (22) as

F Eed 2 = 47.2367. (25) mec

F Figure 2 shows, combining eqns. (21) and (22), that beyond drip, Ee first increases with the increase of A and subsequently saturates to a value ∼ 128.3 at a large A. However, a large A at a high magnetic field corresponds to a F value of Ee and then density well within the maximum possible central density of a neutron star. For example, at e 17 12 BD = 20000 (and hence the magnetic field 8.83 × 10 G), A = 122 corresponds to a density ρbd =4.39 × 10 gm/cc, 11 whereas the corresponding nonmagnetic ρbd =3.25×10 gm/cc. This clearly indicates the huge effects of magnetic field 13 in the matter. Higher values of A = 600 and 4000, at the same magnetic field, correspond to ρbd =1.2 × 10 gm/cc and 7 × 1013gm/cc respectively — all are within the typical nuclear matter density ∼ 1014gm/cc. Hence, highly magnetized neutron stars may exhibit atomic structures with large A. The Fermi energy of a Landau quantized electron with magnetic field strength B confined in νe levels can be rewritten as

2ν B 1/2 EF = c2pF (ν )2 + m2c4 1+ e , e ze e e Be   c  F where pze is z-component of the Fermi momentum of the electron. Therefore, from eqn. (25)

pF 2ν B 1/2 xF (ν )= ze = 2230.31 − e . (26) e e m c Be e  c  Let us now determine the number of states occupied by the Landau quantized electrons [2]. This is given by

νm e νm eB 2BD F N = dN = g e dp = g e x (ν ), (27) e e h2c ν z (2π)2λ3 ν e e νe=0 e νe=0 Z X Z X where gνe the degeneracy factor which is 2−δνe,0. The upper limit νm in the summation is obtained from the condition F 2 that pze(νe) ≥ 0, which gives

F 2 2 F 2 2 (Ee /mec ) − 1 (Eemax/mec ) − 1 νe ≤ e and hence νm = e . (28) 2BD 2BD 6

140

120 )

2 100 c e (m e F

E 80

60

40 0 500 1000 1500 2000 2500 3000 3500 4000 A

FIG. 2: Electron Fermi energy beyond drip as a function of atomic mass.

The electron energy density at zero temperature is then [2]

m F e ν xe (νe) 2BD F pze ǫe = 2 3 gνe Ee d (2π) λ mec e νe=0 0 X Z   e νm F 2 2BD e xe (νe) = mec 2 3 gνe (1+2νeBD)Ψ e 1/2 , (29) (2π) λ (1+2νeB ) e νe=0 D X   where 1 1 Ψ(z)= z 1+ z2 + ln(z + 1+ z2). 2 2 The density of the mixture of ions and electronsp at the onset of drip pis therefore ǫ N M(A, Z)/Z + ǫ − N m c2 ρ = total = e e e e . (30) drip c2 c2 F e Now eqn. (26) gives xe as a function of BD. Therefore, the above equation gives us a relationship between ρdrip e e F and BD. Note that for B >> Bc , νe = 0 and hence xe becomes independent of B, which renders Ne to be linear e 16 in BD in eqn. (27). Indeed, Fig. 3 shows that for B & 10 G, drip density increases linearly with magnetic field. e This linearity arises due to the fact that beyond a certain BD, the Landau quantized electrons reside in the ground state energy level. We also notice that before the onset of linearity, the drip density oscillates about the non-magnetic result, with small amplitudes. A significant change in the drip density arises only in the linear regime. For instance, the drip density increases by an order, compared to that of the nonmagnetic gas, for a field ∼ 6.4 × 1017G. For completeness, we also provide in Fig. 4 the equation of states (EoSs) of magnetized nuclear matter for different values of (fixed) magnetic field and compare them with the nonmagnetic EoS. All the EoSs are plotted starting from e the density corresponding to A = 56. However, for a very high B (here BD = 20000), the corresponding density increases noticeably compared to that of a lower B and hence the corresponding EoS begins from a larger density, as shown in Fig. 4.

IV. SUMMARY

We have studied the effects of magnetic field on the onsets of neutronization for n − p − e and neutron drip for ion-electron gas systems. We have found that beyond a certain value of magnetic field, the neutronization/drip density 1013 7 ) 3

1012 (gm/cm drip ρ

1011 100 1000 10000 e B D

FIG. 3: Drip density in a gas of electrons and ions as a function of magnetic field.

Be =0 1032 e D B D=440 Be =4400 e D B D=20000 1031 ) 3 1030

P (erg/cm 1029

1028

1027 1011 1012 1013 ρ (gm/cm3)

e FIG. 4: Equation of state for BD = 440, 4400, 20000 and nonmagnetic matter. of either of the systems increases, compared to the nonmagnetic cases, linearly with the magnetic field. This value of e field for the former is BD ∼ 2.7 and for the latter ∼ 1115. The significant change in the drip density arises only in the linear regime. For instance, the drip density for the ionic system increases by an order, compared to that of the nonmagnetic gas, for a field ∼ 6.4 × 1017G. Apart from the various applications to neutron stars, the present findings may have interesting consequences to the recently proposed high density (& 1011gm/cc), high magnetic field (& 1015G) white dwarfs [4, 12, 13, 15, 16]. The 17 inner region of white dwarfs, in particular for Bint & 10 G, would have been neutronized if the drip density had not been changed with the field. Note that some authors have questioned the existence of such white dwarfs [14]. For 8 example, the authors have argued that for the white dwarfs to be stable against neutronization, the magnetic field at the center should be less than few times 1016G for typical matter compositions. However, as discussed in detail in [15] (also see [12]), for the central field ∼ 1016G, the mass of the said highly magnetized white dwarfs already becomes significantly super-Chandrasekhar with the value 2.44 solar mass. Therefore, such super-Chandrasekhar white dwarfs remain stable according to the neutronization limit given by the same authors only. Even if the white dwarfs with very large field consist of central region with density larger than the modified drip density, the present finding will help in putting a constraint on such models of white dwarfs with pure electron degenerate matter.

Acknowledgment

The work has been supported by the project with Grant No. ISRO/RES/2/367/10-11. The authors would like to thank the referee for useful comments and suggestions to improve the quality of the paper.

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