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Universal Journal of and Application 10(3): 68-71, 2016 http://www.hrpub.org DOI: 10.13189/ujpa.2016.100302

Origin of Strong Magnetic Fields of

Qiu-he Peng1,2,*, Jie Zhang2, Men-quan Liu2, Chich-gang Chou3

1Department of , Nanjing University, China 2Department of Physics, West-China Normal University, China 3National Astronomical , Chinese Academy of Sciences, China

Copyright©2016 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License

3 Abstract Since there is P2 superfluid in neutron Iwazaki(2005) proposed that the huge of interior, it can be treated as a system of magnetic dipoles. magnetars is some color ferromagnetism of . Under the presence of background magnetic field, the Recently Vink & Kuiper (2006) suggest that magnetars may magnetic dipoles tend to align in the same direction. When originate from the rapid rotating proto-neutron . the temperature is lower than 107K, the strong magnetic In this paper we propose a new idea for the origin of the fields of the magnetars may originate from the induced magnetars. The strong magnetic fields of the magnetars may 3 3 of the P2 neutron Cooper pairs in the originate from the induced magnetic moment of the P2 anisotropic neutron superfluid. And this gives a convenient neutron Cooper pairs in the anisotropic neutron superfluid. explanation of the strong magnetic field of magnetars. Keywords Magnetic Fields, Stars, Neutron, , 3 General 2. Anisotropic P2 Neutron Superfluid There are two relevant regimes of neutron superfluid in 1 the interior: one is the isotropic S0 neutron superfluid with the critical temperature is Tλ 1 10 10 3 14 ( S0)≈1×10 Kina range 1×10 <ρ (g/cm ) <1.6×10 . 3 1. Introduction The other is the anisotropic P2 neutron superfluid with a × 14 ρ 3 × 14 It is generally believed that there is very strong magnetic wide density range1.3 10 < (g/cm ) < 7.2 10 .The 12 3 field, B > 10 , for most of the neutron stars (e.g. critical temperatureTPλ ()2 is Shapiro and Teukolski, 1983). There probably are magnetars 33 8 with strong magnetic field strength exceeding the quantum TPλ (2 )=∆ max ( P 2 ) /1.78 k ≈× 3.25 10 K. (1) critical threshold, H =4.414×1013 gauss (Duncan & cr ∆ 3 Thompson 1992; Paczynski 1992; Usov 1992; Thompson & We note that the gap ( P2) is almost a constant Duncan 1995,1996). Anomalous X-ray (AXPs) and about the maximum with an error less than 3% in a rather × 14 ρ 3 × 14 Soft Gamma Repeaters (SGRs) are classes of candidates to wide density region 3.3 10 < (g/cm ) < 5.2 10 ( see Fig. 8 of Elgagøy et al. 1996, but we neglect the F state of neutron magnetars (e.g. Kouvelliotou et al. 1998, 1999; Hurley et al. Cooper pair here). 1999; Mereghetti& Stella 1995; Wilson etal. 1999; Kaspi et A3P neutronCooper pair has a spin angular al. 1999). The magnetic field of the magnetars may be so 2 with a spin quantum number, σ =1. The magnetic moment of strong as to reach two orders of above the 3 the P2neutronCooper pair is twice that of the abnormal quantum critical threshold Hcr. magnetic moment of a neutron, 2 µn in magnitude, and its It is believed that the strong magnetic field of neutron stars projection on the external magnetic field (z-direction) is originates from the collapse of the core of a with σ ×(2 µ ) , σ =1,0,-1, where the conservation of (e.g. Ostriker & Gunn, z n z 1969). However, it is rather difficult to get the strong −23 µn =−×0.966 10 erg/gauss magnetic field such as those expected for some of the pulsars 13 3 with B > 10 gauss, especially for the strong field strength of It is interesting to note that the behavior of the P2 neutron 3 magnetars by the collapse process alone. superfluid is very similar to that of the liquid He at very low Several models for the origin of magnetars have already temperature (Leggett 1975): been proposed. Ferrario & Wickrammasinghe(2005) suggest 1) The projection distribution for the magnetic moment of 3 that the strong magnetic field of magnetars is descended the P2neutronCooper pairs in the absence of external from their stellar progenitor with high magnetic field core. magnetic field is stochastic, or “Equal Spin Pair” (ESP) Universal Journal of Physics and Application 10(3): 68-71, 2016 69

phase similar to the A- phase of the liquid 3He at very low µ B 3 ∆= − = n , (4) temperature (Leggett 1975). The P2neutron superfluid is n n−+11 n nfn () basically isotropic without significant magnetic moment in kT the absence of the external magnetic field. We name it as the 2sinhx (2 ) 3 = A- phase of the P2neutron superfluid similar to the A phase fx() . (5) 3 + of the liquid He at very low temperature (Leggett 1975). 1 2coshx (2 ) 2) However, the projection distribution for the magnetic The Brillouin function, f(µ B/kT), is introduced to take 3 n moment of the P2 neutron Cooper pairs in the presence of into account the effect of thermal We note that f It is 3 external magnetic field is not stochastic. The number of P2 an increasing function, in neutron Cooper pair with paramagnetic moment is more than particular, fx( )≈ 4 x / 3, for x << 1 and the ones with diamagnetic moment. Therefore, the 3 fx( )→ 1, when x >> 1 . f(µnB/kT) increase with P2neutron superfluid has a whole induced paramagnetic moment and its behavior is anisotropic in the presence of decreasing temperature. And this is the mathematical 3 formula for the B-phase of the 3P superfluid. external magnetic field. We name it as the B- phase of the P2 2 3 A relevant question is how many have been neutron superfluid similar to the B phase of the liquid He at 3 very low temperature (Leggett 1975). combined into the P2 Cooper pairs? The total number of 3 kF 3 The full story of anisotropic P2 neutron superfluid is more dk neutrons is given by NV= (Here k is a wave complicated when magnetic interaction is introduced. We ∫ (2π )3 consider this problem from the effective field theory point of 0 view (Feng & Jin 2005). And for simplicity we only consider vector, pk=  , p is the momentum).The neutrons the zero order approximation. It is possible that the induced combined into the 3P Cooper pairs are just in a thin layer in 3 2 magnetic field of the P2superfluid will boost the neutron star 3 magnetic field to that of magnetars, whenever the stellar the Fermi surface with thickness k∆ , k∆ =2 mPnn ∆ ()2 . internal temperature is below the V k temperature. We will discuss this in see section V. δ ≈ 3 ∆ Then we have Nk2 F (k∆<

3 The induced paramagnetic moment of the P2neutron Here, we would like to emphasize that the energy gap, ∆, superfluid in the B-phase may be simply estimated as is the of the Cooper pair rather than a 3 follows: the system of the P2neutronCooper pairs with spin variation of the Fermi energy due to a disturbance or due to quantum number , s = 1, may be treated as a Bose-Einstein the variation of number density (In the latter case, it system. It obeys the Bose-Einstein statistics. is easy to misunderstand to get A magnetic dipole tends to align in the direction of the δδkE33∆ 3 ∆ 3 q ≈=3 = or q = ). Therefore, external magnetic field. The P2 neutron Cooper pair has 3 kEEFFF22 8π EF energy σZ×2µnB (σZ =1,0,-1) in the applied magnetic field due to the abnormal magnetic moment of the neutrons. Here we think the eq.(6) is a proper estimation. 22 µn is the absolute value of the magnetic moment of a neutron, For the non–relativistic neutron gas ( EF=  km Fn/2 ), we use this convention from now on. B is the background 3 the fraction of the neutrons that combined into the P2 magnetic field if there is no magnetic interaction. While in Cooper pairs is the effective field frame, B is the total magnetic field. 2 3 1/2 3 Detailed discussion will be presented in section V. We 4p pmFn [2∆ ( P2 )] ∆()P2 1/2 3 q = = 3[ ] (7) denote the number density of P2neutronCooper pairs with 3 (4p / 3) pEFF spin projection σz =1,0,-1 by n1, n0 and n-1 respectively. Their relative ratios are The Fermi energy of the neutron system may be calculated by the formula nn−+112/µµB kT −2/B kT = eenn, = , (2) 132/3 2 2/3 2/3 ρ 2/3 nn00 EF= ( )h NAn ( Y ρ )≈ 60( ) MeV .(8) 28mn πρnuc 3 ++ = (3) n−+10 nn 1 nPn () 2 The energy gap of the anisotropic neutron superfluid is 3 3 ∆( P2) ~ 0.045 MeV (Elgagøy et al. (1996)), q ~ 8.7%. Thus, The difference of the number density of P2 neutron 3 Cooper pairs with paramagnetic and diamagnetic moment is the total number of the P2 Cooper pairs is 70 Origin of Strong Magnetic Fields of Magnetars

N(33 P _ pair )≈ qN m ( P )/2. (9) will first discuss the upper limit of the magnetic field of nA22magnetars in our theoretical frame. Since this can be tested 3 by observation directly. Therefore, the total difference of the P2 neutron Cooper pair number with paramagnetic and diamagnetic moment is m B 1 mB ∆=N N(33 P _ pair ) f (n )= N m ( P ) q f ( ) 4. The Upper Limit of Magnetic Field of _ nA22kT 2 kT Magnetars (10) We note that the temperature factor f(µnB/kT) tends to 1 The total induced magnetic moment, of the anisotropic when the temperature decreases low enough. Actually, neutron superfluid is f(µnB/kT) ~1 as long as µnB/kT>>1. For example, this is true (tot )3 3 7 15 µpair ()2P22= µµn ×∆ N = n N A m() P q f(/) µn B kT . whenTK 10 if B= 10 gauss. There is an upper limit for the induced magnetic field of (11) 3 the P2 superfluid according to eq.(12). It corresponds to the 3 Where mP()2 is the of the anisotropic neutron maximum value unity of the temperature factor f(µnB/kT). This upper limit can be realized when all the magnetic superfluid in the neutron star, NA is the A’vgadro constant. 3 The magnetic moment with the dipolar magnetic field is moments of the P2 neutron Cooper pairs are arranged with 3 µNS= BR p NS /2 (Shapiro and Teukolski, 1984).Here Bp the paramagnetic direction as the temperature become low enough. The upper limit of the magnetic field for the is the magnetic field strength and R is the radius of NS magnetars in our point of view is the neutron star. The induced magnetic field is then 3 (tot )3 2m Nm () P m 3 ()3in = nA 2 ()in 2pair ()P2 2mnANm () P2 BPmax() 2 3 q gauss B = = q f(m B /) kT . R . (14) RR33n NS NS NS ≈×2.02 1014η gauss (12) 3 3 Or mP()− ∆ ()P Here η = 2 R 3 []n 2 1/2 is the 0.1m NS ,6 0.05MeV (13) dimensionless factor describes both the macroscopic and The induced magnetic field for the anisotropic neutron microscopic properties of neutron stars. superfluid increases with decreasing temperature (as shown The maximum magnetic field for magnetars depends on the total mass of the anisotropic neutron superfluid of the in the Fig.1). neutron star. It is well known that the upper limit of the mass

for the neutron stars is more than 2.5M Sun . It is therefore possible that the mass of the anisotropic neutron superfluid of the heaviest neutron star may be about. Hence, the maximum of the magnetic field for the heaviest may be estimated to be (3.0-4.0)×1015 gauss. This can be tested directly by magnetar observations.

5. Conclusions and Discussion The strong magnetic field of magnetars may originate from the induced magnetic field due to the paramagnetic 3 moment of the P2 superfluid with significant mass more 7 than 0.1M  at temperature about10 K . The evolutionary scenario of the magnetic field of neutron Figure 1. Variation of the induced magnetic field with the internal stars may be depicted as follows: temperature. Different curves stands for different background magnetic 1) First, the fossil magnetic field of the collapsed core due field. to the conservation of magnetic flux during supernova Fig 1 only shows the case of paramagnetism, that is no would be greatly boosted up more than 90 to magnetic interaction. Since the problem of is still B(0) by the Pauli paramagnetization of the highly degenerate an open question in condensed matter physics (Feng & Jin, relativistic gas just after the formation of the neutron 2005), we will try to discuss this problem in section V. We stars (Peng and Tong 2007). Universal Journal of Physics and Application 10(3): 68-71, 2016 71

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