Breaking stress of Coulomb crystals in the neutron star crust
aa r X i v : . [ a s t r o - ph . H E ] N ov Breaking stress of Coulomb crystals in the neutronstar crust
A A Kozhberov
Ioffe Institute, Politekhnicheskaya 26, St. Petersburg, 194021, RussiaE-mail: [email protected]
Abstract.
It is generally accepted that the Coulomb crystal model can be used to describematter in the neutron star crust. In [1] we study the properties of deformed Coulomb crystals andhow their stability depends on the polarization of the electron background; the breaking stressin the crust σ max at zero temperature was calculated based on the analysis of the electrostaticenergy and the phonon spectrum of the Coulomb crystal. In this paper, I briefly discuss theinfluence of zero-point and thermal contributions on σ max .
1. Introduction
Neutron stars are born in the final stages of stellar evolution. The structure and composition ofmatter in neutron star has been widely discussed, but remain not completely clear, especiallyfor the core (e.g., [2, 3]). The structure of the neutron star crust ( ρ . . × g cm − ) seemsto be more clear (e.g., [2, 3, 4, 5, 6]). At ρ > g cm − the crust consists of an electron gasand fully ionized atomic nuclei. The outer layer of the crust is composed of iron nuclei ( Fe).With increasing density ( ρ & × g cm − ) nuclei become progressively more neutron-rich.At ρ ≈ (4 − × g cm − free neutrons appear in matter but they have a rather weak effecton the dynamic behavior of electrons and ions.The interaction between i th and j th point-like ion with charge Ze and mass M in the neutronstar crust can be described by the screened Coulomb potential U ( r ij ) = Z e exp( − κr ij ) r ij , (1)where r ij is the distance between two ions; κ ≡ p πe ∂n e /∂µ e is the electron screeningparameter; n e , µ e , and e are the electron number density, chemical potential, and charge,respectively. It is more convenient to use the dimensionless screening parameter κa , where a ≡ (4 πn/ − / is the ion sphere radius, n = n e /Z is the number density of ions. For degenerateelectrons κa ≈ . Z / (1 + x ) / x / , (2)where x ≈ . ρZ/A ) / is the electron relativity parameter, A is the mass number of ions, ρ is the density (in g cm − units).If κa .
1, the ions form a crystal, which is usually called a Coulomb or Yukawa, atΓ ≡ Z e / ( aT ) ∼
200 [7]. The distance between two ions in a Coulomb crystal is r ij = R i − R j + u i − u j | , where R i is the equilibrium position of the i th ion in the crystal, and u i is its displacement. At Γ ≫ κa = 4 .
76 was calculated in [8, 9]. For Fe ions and degenerate electrons this value gives ρ ≈ − , for which the Coulomb crystal model is inapplicable. In the most part of theneutron star crust κa .
1. Note that I am using a bcc lattice because this lattice possesses thelowest electrostatic energy at κ = 0 (e.g., [5]).As it shown in [1] the deformed lattice becomes unstable at κa less than 4.76 and itsignificantly depends on the direction of deformation. Here I consider only one deformationof the bcc lattice, which translates the vector R i as a l ( n , n , n ) → a l (cid:18) n + ǫ n , n + ǫ n , n − ǫ / (cid:19) , (3)where a l is a lattice constant, ǫ being a deformation parameter, and n , n , and n are arbitraryintegers. In addition to [1] this tensile-shear deformation was considered in [10, 11], wheredeformed crystals in neutron stars were studied via molecular dynamic (MD) simulations.Our investigations of the phonon spectrum show that the bcc lattice remains stable at ǫ ≤ ǫ max . The dependence of the maximal deformation parameter ǫ max on κa was discussedin [1]. For a crystal with a uniform electron background, it is equal 0.11090. Here I will considera lattice with κa = 4 /
7, for it ǫ max = 0 .
2. The breaking stress
When a neutron star evolves, its crust undergoes various deformations (e.g.,[12, 4, 13, 14, 15]).For magnetars, these deformations are primarily associated with the magnetic field (e.g.,[16,18, 19]); for pulsars they are thought to be connected with glitches (e.g.,[20]). Investigationsof deformed lattices are important for understanding different processes in the neutron starsinterior.In most of the previous studies, deformed Coulomb crystals were used to calculate elasticproperties and effective shear modulus µ (e.g.,[21, 22, 23, 24, 17]). Less attention was paidto the stability of deformed crystals in the neutron star crust and the breaking (maximum)stress. Analytically it has been done in [13, 25] for a bcc lattice with a uniform electronbackground. Deformations of lattices with κa > κa values. The first analytical investigation ofthe stability of a deformed bcc Coulomb crystal with κa > ǫ can be calculated as σ ( ǫ ) = ∂E int /V∂ǫ , (4)while the breaking stress is σ max ≡ σ ( ǫ max ) , (5)where E int is the internal energy of the crystal.At zero temperature in the harmonic approximation, the internal energy is a sum ofelectrostatic U M and zero-point vibration E energies. It can be written as U M + E ≡ N Z e a ζ + 1 . N h ω i = N ω p (Γ p ζ + 1 . u ) , (6)here ζ is the Madelung constant [27], h ω i is the phonon frequency averaged over the firstBrillion zone, u ≡ h ω/ω p i is the first momentum of the phonon spectrum, ω p ≡ p πnZ e /M is the ion plasma frequency, and Γ p ≡ Z e / ( aω p ). Then σ max = nω p (cid:18) Γ p ∂ζ∂ǫ + 1 . ∂u ∂ǫ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ǫ max . (7)The values of the first momentum and the Madelung constant at ǫ = 0 are − . . ǫ = 0 . − . . u changecoincides with the scale of the ζ change, while for the solid neutron star crust Γ p & κa = 4 / ∂ζ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ max ≈ . , ∂u ∂ǫ (cid:12)(cid:12)(cid:12)(cid:12) ǫ max ≈ − . . (8)Consequently, the contribution of zero-point vibration energy to σ max is a few percent or lessand its influence on the breaking stress can be mostly neglected. For T = 0 it is appropriate towrite σ max = 0 . n Z e a . (9)At high temperatures ( T ≫ T p , where T p ≡ ~ ω p ) the total internal energy in the harmonicapproximation is U M + E + F ≈ N T [Γ ζ + 3 u ln − T /T p )] , (10)where F ≡ N T h ln(1 − exp( − ~ ω/T )) i is the thermal contribution, u ln ≡ h ln( ω/ω p ) i . Then σ max = nT (cid:18) Γ ∂ζ∂ǫ + 3 ∂u ln ∂ǫ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ǫ max . (11)At κa = 4 / σ max = n Z e a (cid:18) . − . (cid:19) . (12)In this form the zero temperature and thermal contributions at T ≫ T p satisfactorily convergeswith the result of molecular dynamic simulations. The MD studies presented in [10, 11]determine ǫ max and σ max by direct simulations of the crystal evolution with increasing ǫ . Theyuse the same deformation as I do here and in [1]. In [10] it was reported (it was obtained fromapproximation MD simulations) that at κa = 4 / σ MDmax = n Z e a (cid:18) . − . − (cid:19) . (13)Therefore, σ MDmax = 0 . nZ e /a at T = 0 and the difference with my result is less than1%. The slight difference in the results at T ≫ T p can be explained by the insufficient accuracyof our calculations (in contrast to u , the u ln moment requires more precise studies) and MDsimulations (see, Figures 1-3 in [11]). In any case, the agreement between results supports thestatement that the complex-valued phonon modes control the crystal stability.Note that with the accuracy in use, the difference between the thermal contribution at κa = 4 / κa = 0 is negligible, while the zero temperature contribution at κa = 0 is σ max = 0 . nZ e /a . Thus, this gives a breaking stress σ max , = n Z e a (cid:18) . − . (cid:19) . (14)ccording to molecular dynamic simulations [10] at κa = 4 / σ max can be neglected at Γ & . × (changes become less than 1 %, see Eq. (13)) and at such highΓ the zero temperature approach can be used. Whereas at Γ . . × the thermal correctionto the breaking stress should be investigated together with the anharmonic corrections.It is also instructive to mansion about the influence of a magnetic field ( B ). At T ≫ T p thebreaking stress depends on two parameters: ζ and u ln , but both parameters are independent onthe magnetic field therefore at high temperatures σ max is independent on B . At T = 0 and athigh magnetic field, the first momentum is determined only by the properties of the magneticfield and does not depend on the deformation, hence its contribution to σ max is absent and thebreaking stress remains the same as at B = 0. AcknowledgmentsReferences [1] Kozhberov A A and Yakovlev D G 2020
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