The role of the electric Bond number in the stability of pasta phases
TThe role of the electric Bond number in the stability of pasta phases
Sebastian Kubis ∗ and W(cid:32)lodzimierz W´ojcik Institute of Physics, Cracow University of Technology, Podchor¸a˙zych 1, 30-084 Krak´ow, Poland
The stability of pasta phases in cylindrical and spherical Wigner–Seitz (W–S) cells is examined.The electric Bond number is introduced as the ratio of electric and surface energies. In the case ofa charged rod in vacuum, other kinds of instabilities appear in addition to the well known Plateau–Rayleigh mode. For the case of a rod confined in a W–S cell the variety of unstable modes isreduced. It comes from the virial theorem, which bounds the value of the Bond number from aboveand reduces the role played by electric forces. A similar analysis is done for the spherical W–S cell,where it appears that the inclusion of the virial theorem stabilizes all of the modes.
I. INTRODUCTION
In a neutron star, there is a transitional region be-tween the solid crust and the liquid core, where neutron-rich nuclei could be strongly deformed. Ravenhall etal. presented the first approach to this kind of struc-ture in [1] based on the liquid drop model of nuclei. Itwas shown that the competition between the surface andCoulomb energies of nuclei immersed in a quasi-free neu-tron liquid leads to exotic shapes like infinite cylindersor flat layers called pasta phases. In order to describethe two-phase system of a proton cluster immersed ina neutron liquid, Ravenhall [1] used the Wigner–Seitzapproximation—an isolated neutral cell with a shape as-sumed to be a ball, cylinder, or slab. Comparison ofthe cell energy of different shapes will show which oneof them is preferred. When the volume occupied by theproton cluster increases, there is a sequence of phases:balls, cylinders, and slabs, followed by their inversions:the cylindrical-hole and the spherical-hole, where the pro-ton phase surrounds the neutron phase. The simplicityof this approach encouraged many authors to analyzepasta phases with regards to various neutron star prop-erties like their cooling, precession, oscillations and othertransport properties. For a review of this work, see [2].The stability of pasta phases against shape perturba-tion is still an open question. The stability analysis is,to some extent, parallel to the consideration of the elas-tic properties of the pasta and its oscillations. Elasticproperties in the liquid drop model framework were firstderived by Pethick and Potekhin in [3] and later, by otherauthors in [4, 5]. Another approach, i.e., molecular dy-namics, was presented in [6]. These works have shownthe relevance of the interplay between the Coulomb andsurface energies. However, when the elastic propertieswere analyzed, only certain classes of modes, represent-ing the relative displacements of slabs or rods in the long-wavelengthlimit, are relevant. A detailed inspection ofchanges in pasta shape is not included. In this work,we focus on pasta shape stability due to the competitionbetween the surface and Coulomb energies. ∗ [email protected] The presence of surface tension may affect the shapestability of different structures in different ways.. For aneutral system, it is commonly known that a sphericaldroplet is always stable [7] — the surface energy goes tothe global minimum under the constraint of conservedvolume. However, it appears that the same surface en-ergy may destabilize the fluid portion in a cylinder, ifits length is greater than its circumference,
L > πR .This is the well-known Plateau–Rayleigh instability [8].Because the pasta supplies the fluid with surface tensionand a charge, the question arises of whether the exoticshapes of nuclear clusters are stable. Some attempts toanalyze the stability of the pasta phases were made byPethick & Potekhin in [3] and Iida et al. [9]. In the workby Pethick and Potekhin, the elastic properties of pastaphases in the form of periodically placed slabs (lasagna)and rods (spaghetti) were considered. It was shown thatthe elastic constants are positive in both cases, whichwould guarantee structural stability; however, one mustbe cautious with this conclusion. In the case of lasagna,in order to derive the elastic constant, it was sufficientto consider one specific perturbation mode in the infinitewavelength limit. Complete stability analysis should takeinto account any type of surface perturbation with a fi-nite wavelength. Such analysis was presented recently in[10] by inspection of the second-order energy variationfor one proton slab in a unit cell with periodic bound-ary conditions. It appeared that the second-order energyvariation is positive for all modes and all volume fractionsoccupied by the slab in the unit cell.Similarly, the same analyses can be carried out for thespaghetti phase. The second-order analysis of the candi-date structure for the minimum of energy must fulfill thenecessary condition for the extremum: vanishing of thefirst-order energy variation. Such a condition, the Euler–Lagrange equation for the total energy, is expressed bythe relation between the mean curvature, H = ( κ + κ ),of the cluster surface (where κ i are principal curvatures)and the electric potential Φ2 σH ( x ) = C + ∆ ρ Φ( x ) , (1)where ∆ ρ = ρ + − ρ − is the charge density contrastbetween phases (∆ ρ = en p , if protons are confined inclusters), σ is the surface tension, and C is a constantthat depends on the pressure difference between the a r X i v : . [ nu c l - t h ] F e b phases and the potential gauge. For more details, see[11]. In the work of [3], a periodic lattice of rods was con-sidered. In such a system, the potential, Φ( x ), becomesa complicated space-dependent function, so Eq. (1)indicates that the proton clusters cannot be cylinderswith constant curvature H = R . The charged cylinderobeys Eq. (1) only if it is placed in the cylindrical W–Scell or in vacuum. For such a structure, the first-ordervariation of the energy vanishes, and the examinationof the second-order variation can be undertaken. In thework of [9] the perturbation of a rod in a cylindricalcell was tested. The authors took only one mode intoaccount, which was the transitionally invariant modealong the axis of the rod, corresponding to the transverseflattening of the rod. In this work, we will completethe stability analysis of a single rod in vacuum and in acylindrical W–S cell by testing any kind of perturbation.Similarly, the stability of a charged ball in a sphericalW–S cell will also be considered.The structure of the work is as follows: in the sec-tion II, the choice of Green’s function is discussed, andthe generalized Green’s function is introduced as theappropriate tool for the derivation of the perturbedpotential. In the section III, the stability of a chargedrod in vacuum is presented, and the role of the Bondnumber is emphasized. In the section IV, the sameanalysis is done for a rod in a cylindrical W–S cell. Thespherical W–S cell is treated in the section V. II. THE GENERALIZED GREEN’S FUNCTION
The stability analysis consists of testing the energychange due to the cluster surface deformation (cid:15) ( x ), where x describes the position of the unperturbed proton clustersurface. Then, the second order variation of the energyis given by the surface integral over the proton clusterboundary, ∂ P δ ˜ E = 12 (cid:73) ∂ P (cid:0) − σ ( ∇ (cid:15) + B (cid:15) ) + ∆ ρ ( ∂ n Φ (cid:15) + δ Φ) (cid:1) (cid:15) dS, (2)where B is the sum of the squared principal curvatures B = κ + κ , ∂ n Φ is the normal derivative of the un-perturbed potential, and δ Φ is the potential perturbationcoming from the change in the charge distribution δρ ( x ) = ∆ ρ (cid:15) δ ∂ P ( x ) , (3)where δ ∂ P ( x ) is the surface delta function. The potentialperturbation, δ Φ, depends linearly on (cid:15) ( x ) and is thesolution of the Poisson equation ∇ δ Φ( x ) = − π δρ ( x ) . (4)The potential δ Φ depends not only on the charge pertur-bation δρ , but also on the boundary conditions assumedfor the potential, δ Φ. Because we are going to consider an isolated W–S cell, devoid of periodicity, the boundarycondition imposed for Eq.(4) becomes a matter of discus-sion. The W–S cell must be neutral, which means thatthe electric field flux over the cell boundary ∂ C is zero (cid:73) ∂ C E · n dS = 0 . (5)Therefore, it seems the most natural choice is the Neu-mann boundary condition ∂ n δ Φ | ∂ C = 0, in order to keepthe relation (5). The Dirichlet boundary condition, whichspecifies the potential value on every point of the cellboundary, is not appropriate in this case. It is suffi-cient to assume that the integral of ∂ n δ Φ | ∂ C over the cellboundary is zero. This may be achieved by the introduc-tion of the so-called generalized Green’s function whichshares some of its properties with more common Neu-mann Green’s function. The Neumann Green’s functionfulfils the following equations [12] ∇ G N ( x , x (cid:48) ) = − πδ ( x − x (cid:48) ) and ∂G N ∂n (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) x (cid:48) ∈ ∂ C = − πS c , (6)where S c is the surface area of the cell boundary, ∂ C .However, to ensure that the cell is neutral for any chargedistribution, we have to use the generalized Green’s func-tion which fulfils another pair of equations [13] ∇ G gen ( x , x (cid:48) ) = − π (cid:18) δ ( x − x (cid:48) ) − V c (cid:19) , (7) ∂G gen ∂n (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) x (cid:48) ∈ ∂ C = 0 , (8)where V c is the cell volume.Now the perturbed potential δ Φ is derived by δ Φ( x ) = (cid:90) C G gen ( x , x (cid:48) ) δρ ( x (cid:48) ) d x (cid:48) . (9)Green’s function for periodic system fulfills the same con-ditions as Eqs. (7,8). For more details, see [14]. Here-after, the Green’s functions used in this work for a W–S cell will be understood to be the generalized Green’sfunction. III. THE SHAPE PERTURBATION OF A RODIN VACUUM
Before we proceed with the stability analysis for theW–S cell, we present the charged rod in vacuum. Al-though the case does not correspond exactly to the struc-ture of the pasta phases, because of the lack of a neutralcell, it does show an interesting interplay between electricand surface forces. Moreover, a comparison of the casein vacuum with the case of a W–S cell emphasizes therole played by the boundary conditions. The formalismpresented in the previous section can be applied to thiscase, with the sole difference of using the vacuum Green’s flattening hourglass gyroidFIG. 1. The considered perturbation shapes. function which ensures that δ Φ → r → ∞ , whichmeans that G vac ( x , x (cid:48) ) | r (cid:48) →∞ → x . The charge contrast is thecharge density of the rod ρ = ∆ ρ and the unperturbedpotential isΦ( r ) = − πρR (cid:18) r R − (cid:19) + ln( R ) 0 < r ≤ R ln r R < r < ∞ . (11)It is sufficient to consider the deformations that are peri-odic along the z -axis, with mode wavelength L . Then theGreen’s function is also periodic in the z -coordinate. Inthe cylindrical coordinates, r, z, φ , the Green’s functionis represented by the following sum G cyl ( x , x (cid:48) ) = (12)2 L ∞ (cid:88) n =0 ∞ (cid:88) m =0 γ nm g nm ( r, r (cid:48) ) cos k n ( z − z (cid:48) ) cos m ( φ − φ (cid:48) ) , where k n = 2 π/L . The coefficient, γ nm , has the proper-ties γ = 1 γ n = γ m = 2 and γ mn = 4 for m, n > g nm ( r, r (cid:48) )functions and their form depends on the boundary con-ditions. In vacuum, they are g vac nm ( r, r (cid:48) ) = − ln r > n = m = 012 m (cid:18) r < r > (cid:19) m n = 0 , m > I m ( k n r < ) K m ( k n r > ) n > , m > . (13)The presence of surface tension means that the cylinder ofneutral fluid is always unstable. The cylindrical portionof the fluid fragmentizes into pieces by because of localnarrowing in the shape of an hourglass. The wavelengthof the most unstable mode is L = 2 πR . Here we showhow the situation changes when the cylindrical portionof the fluid in vacuum is charged. It appears that therelevant modes of deformation are: (cid:15) ( z, φ ) = α cos(2 φ ) flattening α cos(2 πz/L ) hourglass α cos(2 πz/L + φ ) gyroid , (14) where α is perturbation amplitude. Their shapes areshown in Fig. 1. From the derivation of the subsequentterms in Eqs.(2), we determine the expression for thesecond variation of the total energy. It is a quadraticfunction of the deformation amplitude α . In order to testthe sign of the energy variation, it is convenient to express δ ˜ E in terms of a dimensionless function of dimensionlessvariables: f ( x, ζ ) = δ ˜ E π α R ρ , (15)where x = L/ πR is dimensionless wavelength and ζ = πR ρ σ (16)is a parameter which measures the ratio of electrostaticforces to surface forces. Such a quantity can be directlyrelated to the well known quantity which appears in thephysics of electrofluids, the so-called electric Bond num-ber [15]. It is defined byBo e = | E | R πσ , (17)where R is the characteristic size of a structure, and E is the typical electric field If we include the size of theproton cluster and the electric field on its surface then theelectric Bond number is exactly equal to the ζ parameterBo e = ζ . (18)The stability function f takes the following form for thethree deformations considered: f vac ( x, ζ ) = x (cid:18) ζ − (cid:19) flattening x ζ (cid:18) x − (cid:19) − x + 2 xI ( x ) K ( x ) hourglass12 xζ − x + 2 xI ( x ) K ( x ) gyroid . (19)When f ( x, ζ ) < x, ζ ). For somemodes, these regions can overlap; then the most unstablemode is the one with the most negative value for f ( x, ζ ).The regions for the most unstable modes are depictedindicated in the Fig. 2 by their names or colors. As ex-pected, for large Bond numbers ( ζ (cid:29) Here we use Gaussian units, in SI units the electric Bond numberis expressed as Bo e = ε | (cid:126)E | R/σ . = L π R B o e = π R ρ σ flatteninggyroidhourglassstable FIG. 2. The stability map (color version online). The pointon the x-axis represents the Plateau–Rayleigh instability. out to diminish the electrostatic energy, which can occurfor any mode wavelength. This kind of instability corre-sponds to the transition from the spaghetti phase to thelasagna phase. For very small Bond numbers, the maininstability is represented by the hourglass mode, wherethe surface energy drives the deformation. In the caseof an uncharged cluster, i.e., ζ →
0, the first term in thestability function for the hourglass mode, Eq. (19), domi-nates completely. Then we may recover the already men-tioned Plateau–Rayleigh instability, which is indicated by a point in Fig. 2. However, the most intriguing modeis the gyroidal mode, which can be unstable for boththe surface energy-dominated region (small Bo e ) and thecharge-dominated region (large Bo e ). When the surfaceenergy dominates, the gyroidal mode is unstable only ifit is sufficiently long. When the electric charge energydominates, the gyroidal mode is unstable for short wave-length. The region of the ( x, ζ )-space where the chargedcylindrical cluster is stable is represented by the whitecolor in Fig. ?? . The narrow region for stable modes,located between the hourglass and gyroid instability re-gions, declines with increasing mode wavelength, x → ∞ .In summary, an unexpected and important outcome isthat for any value of the electric Bond number, there isalways at least one type of unstable mode. The chargedrod in vacuum is always unstable , but this work showsthat the type of unstable mode may be controlled by itscharge.This consideration concerning the single rod may betreated as an approximation to the spaghetti pasta phasewhen the distance between rods is much larger than theirperpendicular size. The case of a single rod in an isolatedWigner–Seitz cell is presented in the next section. IV. THE SHAPE PERTURBATION OF A RODIN A W–S CELL
Here we consider the cylindrical W–S cell which is elec-trically neutral as whole and has volume V c = π R c L .The positively charged rod with ρ + = (1 − w )∆ ρ occupies w fraction of the cell volume V c and is surrounded by thenegatively charged medium ρ − = − w ∆ ρ . The volumefraction, w , depends on the cluster radius w = R /R c .For the generalized Green’s function in cylindrical coordi-nates the expansion given by Eq. (12) is still valid. Theonly change concerns the radial g -functions which nowhave the form: g gen nm ( r, r (cid:48) ) = r + r (cid:48) R c − ln (cid:18) r > R c (cid:19) n = m = 012 m (cid:18) r < R c (cid:19) m (cid:18)(cid:18) R c r > (cid:19) m + (cid:18) r > R c (cid:19) m (cid:19) n = 0 , m > I m (cid:0) πn r < L (cid:1) (cid:32) K m (cid:0) πn r > L (cid:1) − K (cid:48) m (cid:0) πnR c L (cid:1) I m (cid:0) πn r > L (cid:1) I (cid:48) m (cid:0) πnR c L (cid:1) (cid:33) n > , m > , (20)where the W–S cell radius, R c = R/ √ w appears explic-itly.As in the case of the single rod in vacuum, the secondvariation of the total energy Eq.(2) of the cell may be calculated. The stability function f , defined by Eq.(15),is now a function of three variables: the mode wavelength x , the Bond number ζ and the volume fraction w occu-pied by the charged rod. The value of the Bond number = L /( π R ) w = stable hourglass gy r o i d = L /( π R ) w = stable hourglassflattening gy r o i d = L /( π R ) B o e = π R ( w - ) Δ ρ / σ w = stable hourglassflattening gy r o i d FIG. 3. The stability map for a charged rod in a cylindrical W–S cell. The colors (online) represent the regions of the ( x, ζ )space where the most unstable mode occurs. The value of the volume fraction w is shown in the panels. is still defined by Eq. (17) but in the W–S cell, the elec-tric field distribution is different than in vacuum, and the Bond number is now expressed byBo e = ζ = πR ( w − ∆ ρ σ . (21)For the three types of perturbation mode the stabilityfunctions are given by Eqs. (22) f ws ( x, ζ, w ) = x w + 2 w − ζ ( w − ) flattenig2 xI ( x ) I ( x √ w ) (cid:104) I ( x ) K ( x √ w ) + K ( x ) I ( x √ w ) (cid:105) + x ( w − − ζx ( w − ( x −
1) hourglass2 xI ( x ) (cid:20) I ( x )( K ( x √ w )+ K ( x √ w )) I ( x √ w )+ I ( x √ w ) + K ( x ) (cid:21) + x ( w −
1) + 12 ζx ( w − gyroid . (22)In Fig. 3, the regions of instability for the three basicmodes are shown in the contour map. For the W–S cell,the stability functions are functions of the three variables x, ζ , and w . In order to compare the results with thosein vacuum, we plot the contours of f ( x, ζ, w ) < x, ζ space with w fixed. We have chosen w = 0 . , .
3, and0 . w . We also ana-lyzed larger values of w , and when the rod fills the wholecell, w →
1, all unstable regions move towards very largeBond numbers (note the logarithmic scale for ζ ). Theonly exception is the hourglass mode, which shrinks to apoint x = 1 in the region of very small ζ , corresponding to the Plateau–Rayleigh instability.The three variables x, ζ , and w are not independent.The so-called virial theorem says that minimization withrespect to the cell size makes the relation between thesurface and Coulomb energies E s = 2 E Coul . This leadsto the relation between the charge contrast, ∆ ρ , and thesurface tension σσ = 12 π ∆ ρ R ( w − ln w − . (23)Thus, the Bond number in the case of the W–S cell isuniquely determined by the volume fraction w Bo vire = ζ ( w ) = 2( w − w − ln w − . (24) w spherical W - S cylindrical W - S FIG. 4. The relation between the Bond number and volumefraction for different W–S cell shapes.
The above equation shows that the virial theorem con-strains the Bond number. It now ranges between 0 and 4,as it is shown in Fig. 4. The fact that the Bond numberin the W–S cell never goes to a higher value means thatthe electric forces are always smaller or at most compa-rable to the surface forces. Large values of ζ are relevantfor the instability of the flattening and gyroid modes (seeFig. 2). These modes are driven by the spread of chargeand may be unstable if the electric forces alone are suffi-ciently large. Thus, when the Bond number is boundedfrom above, such modes become stable inside the W–Scell. The region where f vir ( w, x ) < w in Fig. 5) and occurs onlyfor the hourglass mode, which is shown in Fig. 5. x = L /( π R ) w = V p / V w s hourglassstable FIG. 5. The stability map for the cylindrical W–S cell withthe virial theorem included.
V. THE SHAPE PERTURBATION OF A BALLIN A W–S CELL
The analysis of shape stability for a spherical W–S cellis simpler than for the cylindrical cell. The perturbation modes are now expressed by the Legendre polynomials (cid:15) ( θ ) = αP l (cos θ ) (25)and there is no additional scaling factor, unlike in thecase of the cylinder, where the mode wavelength had tobe introduced. In the spherical coordinate system r, θ, φ ,the unperturbed potential isΦ( r ) =2 π ∆ ρ (cid:26) r ( w − − R ( w / −
1) 0 ≤ r ≤ R r w + R r − R w / R < r ≤ R c , (26)where the cell radius is R c = R/w / . For the sphericalcell, the Bond number is ζ = 4 π ∆ ρ R ( w − σ (27)and the Green’s function takes the form G sph ( x , x (cid:48) ) = ∞ (cid:88) l =0 g l ( r, r (cid:48) ) P l (cos γ ) , (28)where P l are the Legendre polynomials, γ is the anglebetween the points x and x (cid:48) , and is given by cos γ =cos θ cos θ (cid:48) + sin θ sin θ (cid:48) cos( φ − φ (cid:48) ) [12]. As was previ-ously discussed, the radial functions g l ( r, r (cid:48) ) depend onthe boundary conditions imposed on the cell surface andfor the generalized Green’s function these are g gen l ( r, r (cid:48) ) = r > + r + r (cid:48) R c − R c , l = 0 r l< r l +1 > (cid:34) l + 1 l (cid:18) r > R c (cid:19) l +1 + 1 (cid:35) , l > , (29)where R c is now the spherical cell radius. Thestability function defined by the expression f =25 δ ˜ E/ (4 π ∆ ρ R α ) is now a function of only the vol-ume fraction, w , and the Bond number, ζ . Below wepresent the explicit form for the three lowest deforma-tion modes: f ( w, ζ ) = (cid:0) w / + 10 w − (cid:1) + ζ ( w − l = 2 (cid:0) w / + 7 w − (cid:1) + ζ ( w − l = 3 (cid:0) w + 12 w − (cid:1) + ζ ( w − l = 4 . (30)In Fig. 6, the instability regions for the different modesare shown. As may have been expected for w →
0, thequadrupole mode recovers the well-known Bohr–Wheeler We use the same letter, r , for the radial coordinate in the cylin-drical and spherical systems. The correct meaning is determinedby the context in which it is used. instability for a nucleus in the liquid drop model [16]. TheBohr–Wheeler condition corresponds to a Bond number ζ = 103 . If the virial theorem is applied, the relation between thesurface tension and the charge contrast is σ = 415 π ∆ ρ R ( w − w / + 2) (31)and again similar to the cylindrical W–S cell, the Bondnumber becomes bounded. For the spherical W–S cell,the value of the Bond number takes the formBo vire = ζ ( w ) = 5( w − w − w / + 2) . (32)The relation is shown in Fig. 4 and the values of the Bondnumber are confined to the range < Bo vire <
5. Therestriction of the Bond number values makes the stabilityfunction f vir ( w ) positive for any modes and any valuesof the volume fraction, w . The spherical charged ball ina spherical W–S cell is always stable. VI. CONCLUSIONS
The stability of pasta phases in the cylindrical andspherical Wigner–Seitz cell approximation was analyzed.The stability of a given mode was determined by in-specting the second-order energy variation with respectto the proton cluster shape perturbation. As an illustra-tion, we have also examined the charged rod with finitesurface tension placed in vacuum. This case could betreated as a limiting case when the rod placed in a cylin-drical W–S cell becomes very thin. In the cylindricalcase, different perturbation modes are unstable in dif-ferent regions of parameter space. The relevant param-eters are the following: the mode wavelength and the B o e = π R ( w - ) Δ ρ / σ Bohr - Wheeler qu a d r upo l e o c t upo l e s i x t upo l e stable FIG. 6. The stability map for a charged ball in a sphericalW–S cell. The colors (online) represent the regions of the( w, ζ ) space with the most unstable mode. electric Bond number, which is the measure of the mag-nitude of the electric forces. Due to the virial theorem,the Bond number is no longer a free parameter and isstrictly related to the ratio of the proton cluster volumeto the cell volume, w . Because of that, the Bond numbernever grows to large values, electric forces are reduced,and most of the modes cannot grow. The only unstablemode is the Plateau—Rayleigh mode ,which is unstablefor very small values of the volume fraction, w . Thesetwo cases, vacuum and in the W–S cell, lead to oppositeconclusions about cluster stability, which have their rootsin the boundary conditions.Similar considerations were also conducted for thespherical W–S cell. Again, the virial theorem constrainsthe value of the electric Bond number and completelystabilizes the charge ball inside the W–S cell. The onlyinstability concerns the quadrupole perturbation, in thelimit of w → ACKNOWLEDGMENTS
One of the authors (SK) would like to acknowledgeStefan Typel who encouraged us to research the pastastability issue. [1] D. G. Ravenhall, C. J. Pethick, and J. R. Wilson, Phys.Rev. Lett. , 2066 (1983).[2] A. Schmitt and P. Shternin, Astrophys. Space Sci. Libr. , 455-574 (2018).[3] C. J. Pethick and A. Y. Potekhin, Phys. Lett. B , 7(1998). [4] D. Durel and M. Urban, Phys. Rev. C , 065805 (2018)[erratum: Phys. Rev. C , 029901 (2020)] .[5] C. J. Pethick, Z. Zhang and D. N. Kobyakov, Phys. Rev.C , 055802 (2020).[6] M. E. Caplan, A. S. Schneider and C. J. Horowitz, Phys.Rev. Lett. , 132701 (2018). [7] J. L. Barbosa and M. Carmo, Math. Zeit. , 339(1984).[8] Strutt, J. W. Lord Rayleigh, Proc. London Math. Soc. , 4 (1878) .[9] K. Iida, G. Watanabe, and K. Sato, Prog. Theor. Phys. , 551 (2001); [erratum: Prog. Theor. Phys. , 847(2003)].[10] S. Kubis and W. W´ojcik, Eur. Phys. J. A , 215 (2018). [11] S. Kubis and W. W´ojcik, Phys. Rev. C , 065805 (2016).[12] J. D. Jackson, Classical electrodynamics
John Wiley &Sons (1998).[13] D. G. Duffy,
Green’s functions with applications , 2nd edi-tion, Taylor & Francis Group (2015).[14] S. L. Marshall, J. Phys.: Cond. Matt., , 4575 (2000) .[15] K. Brakke and J. Berthier, The Physics of Microdroplets ,Chapter 3, John Wiley & Sons (2012).[16] N. Bohr and J. A. Wheeler, Phys. Rev.56