tokens: mass with a ratio coefficient f : and the accretion rate will be where f may be of the order of 0.5, as argued by van den Heuvel Bitzaraki ( 1995 ). Because the mass statistics of NSs and MSPs illustrate that the accreted mass at least is about for MSP formation (Zhang et al. 2011 ; Kiziltan et al. 2013 ; Ozel Freire 2016 ; Antoniadis et al. 2016 ), we prefer f to be about 0.1−0.2. A NS/LMXB with m = 1.4 and R 6 = 1 for possesses an accretion rate higher than . However, it cannot accrete at a rate over the Eddington limit, namely . With the limit of the accretion rate , it is possible to derive an upper limit companion mass in the main sequence according to equation ( 11 ): With 0.1 f 0.2, will be about to . Hence a possible mass distribution of the companion in the main sequence in a LMXB is , corresponding to an accretion rate of . In such a LXMB, the NS will evolve to be a MSP with a magnetic field of . For , however, the accretion time might be too short for the binary system to produce a MSP. Table 2. The 16 radio MSPs with a low magnetic fields of (data from the ATNF pulsar catalogue). 2.4 On the minimum magnetic fields of AMXPs AMXPs exhibit an X-ray outburst during the accreting phase, with a luminosity a few magnitudes higher than that in the quiescent phase (Wijnands van der Klis 1998 ; van der Klis 2000 , 2006 ; Mukherjee et al. 2015 ). The variation of the luminosity leads to a change in the accretion rate, and thus the magnetic field of AMXPs can be derived by equating the magnetosphere radius with the co-rotation radius R co of the NS (Burderi et al. 1996 ; Zhang Kojima 2006 ): where P −3 is the spin period P of AMXPs in units of milliseconds. When R co equals the NS radius, the AMXP will have the minimum magnetic field. For 19 AMXPs with spin periods in the range ~1.7–6.1 ms (Patruno, Haskell Andersson 2017b ), their R co are in the range of ~24–56 km, which is larger than the radii of NSs of ~10–15 km. Thus, it can be concluded that the magnetic fields of 19 AMXPs have not achieved their lowest values. Another method for judging the minimum magnetic field of AMXPs is comparing their spin frequencies to the kilohertz quasi-periodic oscillation (kHz QPO) frequencies. The upper-limit kHz QPO frequency of a AMXP is believed to be the Keplerian frequency with orbital accreting matter (van der Klis 2000 ; Zhang 2004 ): where G is the gravitational constant. For a NS with m = 1.4 and R 6 = 1.5, the orbital frequency ν k is . Some AMXPs have an upper kHz QPO frequency over (van der Klis 2000 , 2006 ; Wang et al. 2017 ), which is comparable to ν k . However, the spin frequency of 19 AMXPs ranges from 164 to (Patruno et al. 2017a ), which is much lower than the upper kHz QPO frequency or ν k , and hence we conclude that the minimum magnetic fields of 19 AMXPs have not yet been achieved. If the co-rotation radius approaches the stellar radius, AMXPs will obtain the minimum magnetic field through accretion. 2.5 The minimum magnetic
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