Shape stability of pasta phases: Lasagna case

Transcription

1 Eur. Phys. J. A manuscript No. will e inserted y the editor) Shape staility of pasta phases: Lasagna case Seastian Kuis a,1, W lodzimierz Wójcik 1 1 Institute of Physics, Cracow University of Technology, Podchor ażych 1, Kraków, Poland where σ and ρ are surface tension and charge difference etween the cluster and its surroundings.the constant C depends on pressure difference etween neutron and proton phases and mean value of potential C = P N P P + ρ Φ P. It means the surface curvature depends on the potential distriution and is not constant in general. In fact, the phases considered in [7] represent surfaces with H = 0 and as such they cannot e the true solution of Eq. 1). In works [7,8] it was also shown that they have larger energy than the flat slas. Nevertheless, the energy difference is not large, so it seems that such triple-periodic structures are likely to occur. Recent analysis of pasta phases, including the periodicity, ased on quantum or classical Molecular Dynamics [10, 11, 1, 13, 14] and Time-Dependent Hartree- Fock [15, 16] or Hartree-Fock with twist-averaged oundary conditions [17] have shown the existence of triplyperiodic ones in the form of gyroid, diamond, twisted spaghetti and waffles. Topological analysis of such structures was presented in [18]. The general motivation of our work is to confirm the existence of such non-trivial periodic structures also in the framework of CLDM. Therefore, we propose to examine the possiility of transition from lasagna phase to other shapes y its shape deformation. The inspection of various perturation modes could indicate into what kind of structures the lasagna is going to transform. Such consideration corresponds to the staility analysis of the lasagna phase. The answer to the staility question is not ovious. Though the contriution from surarxiv: v7 [astro-ph.he] 11 Dec 018 the date of receipt and acceptance should e inserted later Astract The staility of periodically placed slas occurring in neutron stars lasagna phase) is examined y exact geometrical methods for the first time. It appears that the slas are stale against any shape perturation modes for the whole range of volume fraction occupied y the sla. The calculations are done in the framework of the liquid drop model and otained results are universal - they do not depend on model parameters like surface tension or charge density. The results shows that the transition to other pasta shapes requires crossing the finite energy arrier. 1 Introduction The very long structures appearing in neutron star matter, called pasta phases, are a commonly accepted phenomenon. Following the seminal work [1], the pasta phases have een studied in many different manners. Many approaches are ased on the compressile liquid drop model CLDM) where the system is descried y two homogeneous phases separated y a sharp oundary with non-zero surface tension [,3,4,6]. The competition etween the Coulom and surface energy leads to different shapes which are usually descried in the Wigner-Seitz approximation, where the geometry of the phase is imposed at the eginning. The spherical and cylindrical cells are not the correct unit cells as they are not ale to fill the whole space y periodic placement. Structures oeying periodicity, in the form of gyroid and diamond-like shapes, were introduced in [7, 8]. However, they must e treated only as an approximation of a true shape, ecause they do not satisfy the necessary condition for the cell energy extremum. a skuis@pk.edu.pl It was shown in [9] that the condition relates the mean curvature Hx) of the cluster surface and electrostatic potential Φx) σhx) = C + ρ Φx), 1)

2 face energy for flat sla is always positive, the Coulom energy is not and could destailize the lasagna. Another type of considerations, connected to the staility issue, were presented in the works [19, 0, 1, ] where the hydrodynamic approach was used to descrie the modes with the density perturation in pasta phases. The otained mode frequencies were always real which means that the lasagna phase is stale for this kind of collective modes. Our approach concerns a different kind of excitations - it considers only the shapechanging modes while the nucleon density is kept constant. Such analysis makes the overall staility discussion more complete. In the work [9] an overall discussion of the rigorous treatment of periodically placed proton clusters of any shape was presented. The stale cluster surface should satisfy not only the Eq. 1), which represents the necessary condition for the extremum, ut also the condition for the minimum coming from the inspection of second variation of the energy functional. The second order analysis, in a limited sense, was already carried for cylinders and alls in the [5], where a particular deformation mode was examined in the isolated Wigner- Seitz cell. Periodically placed cylinders and alls were considered in [6] and later in [3], ut we need to e aware that these structures do not represent the proper minimum determined y the Eq. 1), their shape was assumed apriori. Up to now, the only known true periodic solution of Eq. 1) is lasagna the oundaries of slas coincide with the equipotential surfaces) and thus the staility analysis of these structure may e carried out Partial staility analysis of lasagna phase was done y Pethick and Potekhin in the context of elastic properties of the phase in the work [3]. In order to determine the elasticity coefficient, they considered one particular deformation mode and moreover the expression for the energy was valid only in the limit of wavelength going to infinity. Such analysis corresponds to the staility consideration, however, eing limited to the only one type of deformation. Here, we present general, unconstrained staility analysis for any kind of deformation with finite wavelength, which, to the authors knowledge, has never een carried out. Energy variation for single sla Let us consider a proton cluster P in the shape of one sla placed in the center of unit cell with size a,, c and volume V C = ac. The sla is perpendicular to the x-axis and occupies a w fraction of the total cell volume, 0 < w < 1. The charge density contrast etween phases is ρ = ρ + ρ, so the sla is positively charged - a x 1 x ρ + Φ 0 x) a ρ - Fig. 1 The unpertured charge distriution and its potential Φ 0. with the density ρ + = 1 w) ρ and immersed in negatively charged neutron gas ρ = w ρ we assume the homogeneous electron ackground). For such charge distriution, the unpertured potential Φ 0, Fig. 1, is a function of x only and takes the form Φ 0 r) = w 1 6 a w + ) + ax + x ) a x < x 1 = π ρ w 1) 1 6 a w )w + x ) x 1 x < x w 1 6 a w + ) ax + x ) x x a, ) where x 1, = ± aw correspond to the positions of sla edges. Such configuration fulfill the necessary conditions for minimum, i.e. equations coming from vanishing variation of total energy in the first order. The sufficient condition for minimum is expressed y the positive value of total energy variation in the second order δ ε where the constraints for aryon numer and charge conservation are imposed. Such energy variation for any deformation of the proton cluster ɛ is expressed y the integral over proton cluster surface P [9] δ ε = 1 V P σ ɛ) B ɛ ) + ρ n Φ 0 ɛ + δ ɛ Φ ɛ) ) ds, C x 3) where B = κ 1 + κ = 0 is the sum of squared principal curvatures, n Φ 0 is the normal derivative of unpertured potential, δ ɛ Φ is the first order perturation of the potential caused y the surface deformation ɛ. One must rememer that here we consider only the deformation which preserves the volume of the cluster, which means ɛ ds = 0. 4) P Deformations of this kind may e expressed in terms of Fourier series on the y, z-plane. The deformations

3 3 on each sla face are independent, so we get two series for each face located at x 1 and x. For further calculations it is convenient to introduce an expansion ased on complex amplitudes α j mk, where m and k are the mode indices and the superscript j corresponds to the face numer ɛ j y, z) = m,k= α j mk expik 0mk x), 5) where we introduce the 3-dimensional discrete wave vector πn K nmk = a, πm, πk ) 6) c and x = x, y, z). The m and k indices take integer values from to + except the case when oth of them equal zero, which is indicated y an apostrophe in the sum sign. The lack of 0, 0)-mode is consistent with the volume conservation condition, Eq. 4). The inclusion of such compression modes would require taking into account the particle density perturation. Their staility is controlled mainly y the volume compressiility coefficient K V = P ρ - quantity eing dependent on the details of nuclear interactions. The compression modes may e carried out separately as the 0, 0)-mode is orthogonal to the shape changing modes. As we are interested only in the shape staility, we postpone the compression modes for future work. Since the deformation function ɛ j y, z) must e real it means that the complex amplitudes should fulfill the following relations α j mk ) = α j m, k. 7) It is more natural to introduce the cosine ɛ mk,c cos my + kz c ) and sine modes ɛ mk,s sin my + kz c ) in the expansion Eq. 5. Then the relation etween complex and real amplitudes is ɛ j mk,c = αj mk + αj m, k 8) ɛ j mk,s = iαj mk αj m, k ). 9) In the Fig. an example of the sla deformation with real amplitudes taking the values ɛ 1 10,S = ɛ 10,C 0 and ɛ 1 10,C = ɛ 10,S = 0 is shown. Below we present and discuss the susequent contriution to the total energy variation δ ε, Eq. 3). In terms of complex amplitudes the surface energy contriution to the nd order variation takes simple form δ ε S = σ ɛ) B ɛ ) ds V C P = σ α j mk a αj m, k K 0mk. 10) j=1 mk - - ϵ 1 y ϵ - a x - a 1 x Fig. The sla deformation in the unit cell the grey rectangle). The first face is only with sine mode whereas the second face with cosine mode. It is always positive, as B = 0 for the sla and due to relations 7) we get α j mk αj m, k = αj mk > 0. That is an important fact, ecause in many cases the surface energy acts as a destailizer, like, for example, δ ε S < 0 in the case of Rayleigh-Plateau instaility. The contriution coming from the electrostatic interaction includes two terms. The first one, determined y the normal derivative of potential Φ 0, is given y δ ε norm = ρ V C = ρ a P j=1 n Φ 0 ɛ ds mk n Φ 0 x j ) α j mk αj m, k. 11) The normal derivatives at the sla faces are n Φ 0 x 1 ) = n Φ 0 x 1 ) = π1 w)wa ρ. So, finally one gets δ ε norm = π ρ w1 w) j=1 mk x α j mk αj m, k 1) which is always negative. The second term of the Coulom interaction part is associated with perturation of the potential δ ɛ Φ δ ε Φ = ρ V C P δ ɛ Φ ɛ ds, 13) where δ ɛ Φ is the first order potential perturation calculated thanks to the periodic Green function δ ɛ Φx) = ρ G P x, x )ɛx ) ds. 14) P For the three-dimensional unit cell with sizes a,, c the periodic Green function can e expressed as the sum

4 4 over discrete modes numered y three indices n, m, k G P x, x ) = 4π ac + n,m,k= expik nmk x x )) K nmk. 15) The prime sign in the summation means that n, m, k cannot vanish simultaneously, similarly as in the Eq. 5). Such Green function fulfills the Poisson equation in the unit cell with periodic oundary conditions [4, 5] we use CGS units) G P x, x ) = 4π δx x ) 1 ) 16) ac and the asence of 0,0,0) mode in the expansion 15) is a consequence of the unit cell neutrality. By joining Eqs.5,14,15) we otain the change in the energy with respect to the potential variation δ ε Φ = π ρ i,j=1 m,k α i mk α j m, k F ξ ij, χ mk ), 17) where ξ ij corresponds to the difference etween the faces location ξ ij = x i x j a 18) and χ mk is the norm of the dimensionless wave vector χ mk = πma, πka c ) πma ) ) πka χ mk = ak 0mk = +. 19) c The vector χ mk descries the manner in which the face is virating - the indices m and k determine the numer of wavelengths eing placed in the face sizes and c. The function F ξ, χ) comes from the summation over the n index. As the n-numered modes do not depend on x, the summation over n can e done separately and defines a function F F ξ, χ) df = n= e iπnξ πn) + χ. 0) The aove series can e evaluated [6] to a closed form F ξ, χ) = cosh χ 1 ξ )) χ sinh χ ). 1) The function is positive for any ξ and χ. In some special cases the function simplifies to F 0, χ) = coth ) χ, ) χ F 1, χ) = 1 χ sinh χ ). 3) Taking together all energy terms we get the total energy variation expressed in terms of mode amplitudes α mk δ ε = i,j=1 m,k { σ a 3 χ mkδ ij π ρ w1 w)δ ij + 4) + π ρ F ξ ij, χ mk ) } α i mkα j m, k. As the δ ε is the quadratic form of the amplitudes α mk the competition etween the terms in curly rackets decides aout the staility of the face surface. There are three characteristic terms: one from the surface energy eing positive and two from Coulom interactions: the first is negative and the second is always positive. It seems that the staility consideration depends on the values of surface tension σ or charge contrast ρ ut it appears that these parameters may e removed from our analysis. One of the conditions for the minimum of the energy for unit cell is the virial theorem [9]. The theorem takes the form of relation etween the surface and Coulom energy of the cell. For the cell with high symmetry, it has the simple form E S = E Coul, 5) from which we may get the relation etween σ and ρ σ = 1 6 πa3 ρ w 1) w. 6) Finally the total energy variation may e written as the quadratic form of α i km δ ε = π ρ i,j=1 m,k with its coefficients A ij mk given y A ij mk αi mkα j m, k 7) A ij mk = { 1 1 w 1 w) χ mk w1 w)) δ ij + F ξ ij, χ mk )}. 8) The otained general expression, Eq. 7) for the second order variation of the total energy allows us to determine whether the sla is stale with respect to any deformations preserving its volume. It is worth noting that A ij mk coefficients, which decide aout staility, do not depend on the strength of interactions eing determined y surface tension σ and charge contrast ρ. In this way, we otained an interesting result that the staility of pasta depends only on the geometry of phase and mode under consideration and not on the details of strong or electromagnetic interactions. Before general discussion of the staility of a single sla we show how the aove results work in the staility analysis for particular class modes.

5 5 where w is the volume fraction and χ determines the wavelength of the mode in comparison to the cell size χ = πa/. The inspection of eigenvalues and eigenvectors of ˆM allows for a complete staility analysis for the deformation we have chosen. The matrix ˆM given y Eq. 34) possesses the two-fold degenerated two eigenvalues. Further, we call the ˆM-eigenvalues λ l, l = as the staility function. For our concrete form of ˆM the eigenvalues and their eigenvectors are λ 1,3 = w) w χ 1 w1 w) + F 0, χ) F w, χ) Fig. 3 The staility functions λ i w, χ) for two types of deformations: snaky left) and hourglass-shaped right) mode. 3 An example of staility analysis Let us consider the simplest surface perturation consisting of comination of sine and cosine modes going along the y-axis for each face, which corresponds to m = ±1 and k = 0. Then, for the i-th face, the only non-vanishing complex amplitudes α i mk are: α i 10 = 1 α i 10 = 1 ɛ i C i ɛ i ) S, 9) ɛ i C + i ɛ i ) S, 30) where we introduced the real amplitudes ɛ i S and ɛi C corresponding to the functions sin πy πy ) and cos ). The deformation ɛ i for the i-th face is then a function of y only ɛ i y) = ɛ i C cos πy ) + ɛi S sin πy ). 31) It is convenient to introduce the vector ɛ uilt of deformation amplitudes ɛ = ɛ 1 C, ɛ C, ɛ 1 S, ɛ S). 3) Then the total energy variation for such deformation may e written down in the matrix form δ ε = π ρ ɛ ˆM ɛ T, 33) where the dimensionless matrix ˆM is A B 0 0 ˆM = B A A B 34) 0 0 B A and its elements are: A = w) w χ 1 w1 w) + F 0, χ), 35) B = F w, χ), ɛ 1 = 1, 1, 0, 0), ɛ 3 = 0, 0, 1, 1) λ,4 = w) w χ 1 w1 w) + F 0, χ) + F w, χ) ɛ = 1, 1, 0, 0), ɛ 4 = 0, 0, 1, 1). 36) The staility functions λ i do not depend on the details of interactions σ, ρ ut only on the geometry of our system which is descried y volume fraction w and mode wavelength χ. In Fig. 3 the staility functions are plotted in the w, χ parameter space for their eigenvectors. These vectors represent two classes of modes. We may call them as snaky and hourglass-shaped modes. The snaky modes occur when the deformations on oth faces are in phase ɛ 1, ɛ 3 ) and hourglass modes occur when the deformations on faces are out of phase ɛ, ɛ 4 ). As we may notice, for all values of volume fraction and mode wavelengths the staility functions are positive, which means, that for oth classes of modes the sla is stale. However the staility is not too strong. Careful inspection of λ 1 and λ shows that those functions go to 0 when χ and w approach to some specific values λ 1 χ 0 0 for any w 37) and λ χ 0 for w 0 or 1. 38) It means that snaky modes ecome unstale in the limit of very long waves regardless of sla thickness, whereas the hourglass modes ecome unstale for very thin sla and very short waves. To sum up, we may say the sla ecomes asymptotically unstale for very long mode or for very short mode when the sla ecomes very thin in comparison to unit cell width. One should note that, the case when w 0 or w 1 must e treated with caution ecause in reality the cluster surface has finite thickness and for very thin sla the validity of the liquid drop model could e questioned. Nevertheless, the first case of asymptotic instaility, Eq.37), is

6 6 worthy of careful inspection as it is connected to the macroscopic deformation and allows for determination of elastic properties of lasagna phase, which is shown in the next section. 4 Elastic properties Nuclear pastas share their elastic properties with liquid crystals. If the wavelength of the snaky mode is very large in comparison to the cell size, it corresponds to the so-called splay deformation of the liquid crystal. It allows to determine elastic constant K 1 for the lasagna phase what was shown y Pethick and Potekhin in [3]. By definition, the constant K 1, relates the deformation energy with the transverse derivative of the deformation field when the mode wavelength ecomes very large. In our notation the relation takes the form δ ε = K 1 yɛ), where the rackets.. mean the average over the sla surface. Taking the definition of K 1 in the limit of the very long mode, we get K 1 = 4a 4 δ ε lim χ 0 χ 4 ɛ, 39) here ɛ denote the mode amplitude only). The deformation energy for the snaky mode is δ ε snaky 1 = π ρ 1 1 w) w χ 1 w)w ) + F 0, χ) F w, χ)) ɛ. 40) Applying the limit in Eq.39) we get K 1 = πa4 ρ w 1) w 1 + w w ), 41) which exactly corresponds to the result of [3] if the ρ is replaced y the unpertured Coulom energy of the cell ε C,0 = 1 6 πa ρ 1 w) w. In comparison to [3] our approach represents an improvement. Pethick and Potekhin used a triple sum for the deformation energy, we mean Eq.8) in [3], which gives correct result only in the limit of the very long mode χ 0. In fact, that series represents an asymptotic expansion in the powers of mode wavelengths and the leading term gives correct result. However the expansion is divergent for finite χ. The detailed discussion of this divergence is shown in the Appendix. The approach, presented here, allows to avoid any divergences and is not limited to the case χ 0. 5 General discussion of staility In the Section 3 the staility analysis for the simplest sla deformation was carried out. The full staility analysis would require the inclusion of modes for all multiplicities m and k. Writing down the matrix ˆM for all modes it appears to take the lock-diagonal form... A B 0 0 ˆM = B A A B, 4) 0 0 B A... where in the mk-th position we get the same matrix as in Eq. 34) with A and B elements taking the same form as in Eq. 36) ut with the replacement χ χ km. Such a form of ˆM means that the modes with different multiplicity m, k and the cosine-like and sine-like modes do not couple. So, taking fixed m, k one may repeat the discussion from the Section 3 and finally conclude that the single sla is stale for any mode keeping in mind the asymptotic cases, descried y the Eqs. 37,38). 6 Multi-sla modes In previous sections the cell with only one proton layer was considered. However, one may consider many slas which are placed in a cell with periodic oundary conditions. Such system allows to test a larger class of perturations and makes our analysis of staility more complete. Let us suppose we have N slas in one cell. The expressions derived earlier for energy variation 10,11,17) take the same form, except the summation over the surfaces which now takes the range i, j = 1..N, as we have N surfaces for N slas. Moreover, the value of the normal derivative of the electric potential scales with the numer of slas N according to the rule n Φ 0 x i ) = π 1 w)wa ρ. 43) N As was discussed in section 5 it is enough to test the only one type of mode per surface. Then the staility matrix ˆM for one given mode has dimension N N. As an example we show results for the case of two slas, N = and cosine-mode with m = 1, k = 0. Then we have four distinct eigenvalues of staility matrix ˆM : λ 1 = D + F 1, χ) F w, χ) F w+1, χ) λ = D F 1, χ) + F w, χ) F w+1, χ) λ 3 = D F 1, χ) F w, χ) + F w+1, χ) λ 4 = D + F 1, χ) + F w, χ) + F w+1, χ) 44)

7 7 Fig. 4 The slas deformations and their staility functions λ i w, χ) in the case of two slas per cell. where D is D = 1 19 w 1) w χ + 1 w 1)w + F 0, χ). 45) 4 All of these eigenvalues are positive. Their dependence on volume fraction w and mode wavelength χ and the corresponding sla deformations are shown in the Fig.4. As one may see, the eigenmodes of the deformation of multi-sla system are always the comination of snaky and hourglass modes. 7 Conclusions In this work, y use of analytical methods, we have shown that proton clusters having the form of slas placed periodically in space are stale for all values of volume fraction occupied y the cluster. It is a compelling result. All works ased on CLDM see for example [1,4,6,7]), show that different shapes of pasta are preferred for different values of w. Here, we have shown that the lasagna phase is stale in the whole range of w. One must rememer that our analysis means that the lasagna phase represents merely a local minimum. The transition to another geometry is not totally locked, ut requires finite size deformation in order to exceed the energy arrier. It may e interpreted as the fact that, at least for some range of volume fraction the lasagna phase is metastale. That could e quite interesting for the dynamics of pasta appearance during the neutron star formation. Our analysis is ased on small deformations so it cannot state at which range of w the lasagna represents gloal minimum of the cell energy. First, the gloal minimum statement requires the knowledge of exact solutions of Eq. 1) for other kind of shape than flat sla. So far, we have not known such solutions in the CLDM approach. The seeking of them marks out the direction of further research of pasta y differential geometry methods. We are also conscious that the approach, ased on the CLDM, has its limitations and the inclusion of such effects like finite thickness of the cluster surface or the temperature fluctuations could change the final conclusion concerning lasagna phase staility. Appendix The Coulom energy in the work [3], Eq.8), was finally expressed y the power series S = 1) j a j ξ) kr c /π) j, j=1 A.1) here we keep the notation used y Pethick and Potekhin: r c is the cell size, k is the mode wavenumer and ξ is the dimensionless amplitude of perturation. Coefficients a j ξ) are determined y the doule sum a j ξ) = 8 π 3 w m=1 sin mπw) m 4+j n j Jnmξ). n=1 A.) The series given y S is divergent. It is enough to take only one term from the doule sum, Eq.A.) for m = 1 and n = j/ and then every coefficient a j ξ) is ounded from elow y the expression 8 π 3 w j ) j sin πw)j j ξ) < a j ξ). A.3) The j -th Bessel function for small amplitude ξ < 1), again, may e ounded y its Taylor expansion ξ/) j 1 + j ξ/) ) j + 1)! < J j ξ). A.4)

8 8 So, finally, all the coefficients of the power series S are underestimated y the new ones ã j ã j ξ) = ) j eξj j + ξ 4) sin πw) 4 4πjj + ) < a j ξ), where we removed factorial y use of the Stirling formula. The power series with coefficients ã j is however divergent ecause for large j the ã j ehaves like j j and the convergence radius is zero r conv = lim j ã j ã j+1 = 0. A.5) This divergence comes from the introduction of the sum over j in the expression for Coulom energy. The Coulom energy, was initially expressed y the convergent doule sum over m, n indices, Eq.7) of [3], then the third summation over j was introduced in the following way screening was neglected k T F = 0) 1 mπ/r c ) + nk) = rc ) 1) j nkrc mπ mπ j=0 ) j A.6) and then the order of summation sequence was changed n,m j j n,m. The change in the summation order is allowed only if all su-series are convergent, however, the expansion given y Eq.A.6) is convergent only if n < π kr c m. A.7) That means that the indices are not independent: the summation sequence is not interchangeale. The authors of [3] got a correct result ecause they took the doule limit: ξ 0 small amplitude of deformation) and π kr c very long mode). Summarizing, we want to emphasize that such expression for the Coulom energy, Eq.8) in [3], is valid only in the limit of the very long mode, kr c K. Nakazato, K. Iida, K. Oyamatsu, Phys. Rev. C 83, ). 9. S. Kuis, W. Wójcik, Phys. Rev. C 94, ). 10. G. Watanae, T. Maruyama, K. Sato, K. Yasuoka, T. Eisuzaki, Phys. Rev. Lett. 94, ). 11. A. S. Schneider, C. J. Horowitz, J. Hughto, D. K. Berry, Phys. Rev. C 88, ). 1. A. S. Schneider, D. K. Berry, C. M. Briggs, M. E. Caplan, C. J. Horowitz, Phys. Rev. C 90, ). 13. D. K. Berry, M. E. Caplan, C. J. Horowitz, G. Huer, A. S. Schneider, Phys. Rev. C 94, ). 14. P. N. Alcain, P. A. Giménez Molinelli, C. O. Dorso, Phys. Rev. C 90, ). 15. B. Schuetrumpf, M. A. Klatt, K. Iida, J. Maruhn, K. Mecke, P. G. Reinhard, Phys. Rev. C 87, ). 16. B. Schuetrumpf et al., Phys. Rev. C 91, ). 17. B. Schuetrumpf, W. Nazarewicz, Phys. Rev. C 9, ). 18. R. A. Kycia, S. Kuis, W. Wójcik, Phys. Rev. C 96, ). 19. L. Di Gallo, M. Oertel, M. Uran, Phys. Rev. C 84, ). 0. M. Uran, M. Oertel, Int. J. Mod. Phys. E 4, ). 1. D. N. Koyakov, C. J. Pethick, Zh. Eksp. Teor. Fiz. 154, ).. D. Durel, M. Uran, Phys. Rev. C 97, ). 3. C. J. Pethick, A. Y. Potekhin, Phys. Lett. B 47, ). 4. S. L. Marshall, J. Phys.: Cond. Matt., 1, ). 5. S. Tyagi, Phys. Rev. E 70, ). 6. I.S. Gradshteyn, I.M. Ryzhik Tale of Integrals, Series, and Products New York, Academic Press, 1994). References 1. D. G. Ravenhall, C. J. Pethick, J. R. Wilson, Phys. Rev. Lett. 50, ).. G. Baym, H. A. Bethe, C. Pethick, Nucl. Phys. A 175, ). 3. M. Hashimoto, H. Seki, M. Yamada, Prog. Theor. Phys. 71, ). 4. K. Oyamatsu, Nucl. Phys. A 561, ). 5. K. Iida, G. Watanae, K. Sato, Prog. Theor. Phys. 106, ). 6. R. D. Williams, S. E. Koonin, Nucl. Phys. A 435, ). 7. K. Nakazato, K. Oyamatsu, S. Yamada, Phys. Rev. Lett. 103, ).