Coherence length of neutron superfluids G. Lazzari a,b , F. V. De Blasio c , M. Hjorth-Jensen c , Ø. Elgarøy d and L. Engvik d a European Centre for Theoretical Studies in Nuclear Physics and Related Areas, Trento, Italy b Dipartimento di Fisica, Universit`a di Milano, Via Celoria 16, I-20133 Milano, Italy c NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark d Department of Physics, University of Oslo, N-0316 Oslo, Norway The coherence length of superfluid neutron matter is calculated from the microscopic BCS wave- function of a Cooper pair in momentum space making use of the Bonn meson-exchange potential. We find that the coherence length is proportional to the Fermi momentum-to pairing gap ratio, in good agreement with simple estimates used in the literature, and we establish the appropri- ate fitting constants using our numerical data. Our calculations can be applied to the problem of inhomogeneous superfluidity of hadronic matter in the crust of a neutron star. PACS numbers: 21.65.+f; 97.60.jd Calculations based on BCS theory with phenomeno- logical nucleon-nucleon (NN) forces indicate that neu- tron matter is superfluid in a wide region of densi- ties and temperatures. In particular, at the densities corresponding to the interior of a neutron star crust (4 × 10 11 gcm -3 <ρ< 10 14 gcm -3 ) neutrons couple in the singlet isotropic channel 1 S 0 while at larger den- sities they pair in the 3 P 2 state [1]. Although many in- vestigations have been devoted to the superfluidity and superconductivity of neutron, β-stable and nuclear mat- ter, little attention has been paid to possible effects of inhomogeneities in hadronic superfluids or superconduc- tors. In fact the interior of neutron stars, the subject of most of such calculations, is often referred to as the only existing example in nature of infinite superfluid neutron or β-stable matter, since in an atomic nucleus the wave- functions of the Cooper pairs are limited in extension by the potential well. On the other hand, superfluidity in a neutron star crust represents a case intermediate be- tween the nucleus and the idealized infinite system, since superfluid neutrons in the inner crust occupy a region where a lattice of nuclei creates strong inhomogeneities in the medium. From the point of view of astrophysi- cal observations, probably most of the visible effects of the presence of superfluids in a neutron star are due to phenomena in the crust [2]. An important length scale of the neutron superfluid is the coherence length. From a microscopic point of view the coherence length represents the squared mean dis- tance of two paired particles (a Cooper pair of neutrons) on top of the Fermi surface. The magnitude of this quan- tity affects several of the physical properties of a neutron star crust. First of all, neutrons paired in a singlet state form quantized vortices induced by the rotational state of the star. These can pin to the nuclei present in the crust, possibly leading to the observed sudden release of angu- lar momentum known as pulsar glitches. The magnitude of the pinning force depends on the size of the vortex cores, which is equal to the coherence length of the neu- tron superfluid. A second question is how properties of the neutron superfluid change due to the inhomogeneous environment of a neutron star crust, a problem related to the average thermodynamical property of neutron mat- ter [2,3]. In the inner crust, depending on density, nuclei of different shapes and sizes are present. At a density of ∼ 10 14 gcm -3 , spherical nuclei cease to be energetically favored and are replaced first by cylindrical nuclei, then slabs to end up with holes, where the roles of protons and neutrons are exchanged, see Refs. [2,4] for further details. Only at higher densities, corresponding to what is called the core of the star, do nuclei merge into the uniform medium. The fact that neutron superfluidity in neutron star matter is actually a problem of inhomoge- neous superfluidity in hadronic matter has been noticed quite recently [2,3]. According to Anderson’s theorem [5] the electron density of states in a superconductor is changed very little from a pure metal to an alloy of sim- ilar chemical properties. The physical situation we are examining here is quite different, since both the average density of states and the effective neutron-neutron ma- trix elements are changed compared to the uniform case when one considers the presence of nuclei. The typical dimension of nuclei in the inner crust of a neutron star is R N ≈ 4 - 6 fm. This number is, in an ap- propriate range of densities, comparable to the coherence length ξ as estimated from existing BCS calculations. If Anderson’s theorem holds also for neutron star matter, (limit where R N  ξ) there will be no appreciable vari- ations in the superfluid properties induced by the nuclei. On the other hand, if R N  ξ the superfluid will change its properties locally. This limit has been investigated in a recent series of papers [3] where it was found that some thermodynamical properties like e.g. the neutron specific heat may change by a very large amount. Unfortunately, the situation is complicated by the fact that R N and ξ are of the same order of magnitude. Clearly, the coherence length represents a critical pa- rameter by which one can establish the behavior of an in- homogeneous superfluid. It sets the scale for the possible spatial variation of the pairing properties of the system, 1