# a simple analytical calculation of mean-field potential in heavy nuclei[精选推荐pdf]

arXiv:nucl-th/0508030v1 16 Aug 2005A simple analytical calculation of mean-field potential in Heavy NucleiMofazzal Azam and Rajesh S. Gowda Bhabha Atomic Research Centre, Mumbai, IndiaAbstractIn the many-body theory of particles with long range interaction potential, such as the potential of Newtonian gravity and its relativistic generalization,Newton-Birkhofftheorem plays a very important role. This theorem is vi- olated in the case of short range interaction potential, such as the Yukawa potential in nuclear physics. We discuss the relevance of this in the many- body treatment of heavy nuclei. In particular, using techniques similar tothose in Newtonian gravity, we calculate analytically the mean-field potentialin heavy nuclei. The mean-field potential thus calculated depends on well known quantities such as inverse pion mass, pion-nuleon coupling constant,nuclear radius and density. The qualitative features of the mean-field poten- tial thus obtained is similiar to the well known phenomenological Wood-Saxon potential.1In this paper, using simple analytical calculations, we intend to bring out some fea- tures of motion of nucleons in a heavy nucleus [1,2]. Our calculations are based on two phenomenologically well known facts: The density of nucleons in heavy nuclei is almost con- stant and the inter-particle interactions in nuclei is given by the Yukawa potential. Theseassumptions seem to be sufficient for the qualitative understanding of the motion of nucle- ons in heavy nuclei. The central ingredient in our calculations is that the Yukawa potential violates Newton-Birkhoof theorem. In order to explain the role played by the violation ofthis theorem by the Yukawa potential in nuclear physics, we first formulate and proof the theorem in Newtonian gravity. The proof reveals why such a theorem can not be valid for Yukawa potential. In the many-body treatment of heavy nuclei with inter-particle Yukawa interactions, we consider the nucleon density to be constant. In order to justify this as- sumption, in the following section, we argue why a heavy nucleus can be treated as system of constant density. We show that the constant density has more to do with the Pauli’s exclusion principle and short range repulsive core of the nucleons rather than any ergodicityor thermodynamic equillibrium [3–7]. In the next two sections, we derive the mean field potential and indicate how the surface tension of heavy nuclei emerges from the violationof Newton-Birkhofftheorem. The mean field potential thus obtained qualitatively agrees with the phenomenological Wood-Saxon potential [8]. At the end, we make some concluding remarks.I. THE NEWTON-BIRKHOFF THEOREMNewton-Birkhofftheorem in the theory of gravitation is a very important theorem. The violation of this theorem by Yukawa potential is the central ingredient in our calculation ofthe surface tension and the mean field of heavy nuclei. Therefore, in this paper we provide a discussion of this theorem in Newtonian gravity in detail. Let us consider a spherically symmetric ball of mass M, mass density ρ and radius R. Theorem (Part-I): The potential due to the spherical ball at a distance r, where r > R is the same as the potential due to point particle of mass M placed at the centre of spherical ball. Let us consider a spherical shell of outer radius R and inner radius b, having constant mass density ρ. Theorem (Part-II): The gravitational potential due the spherical shell at any point r, where r a (> r′), and 0 ≤ r′≤ aΦN(r) =− 2πGρZa0[(r + r′) − (r − r′)]r′rdr′= −4πGρ rZa0r′2dr′= −4πa3ρG3r= −GMr(7)For r a (> r′), and 0 ≤ r′≤ aΦ1(r,a; r > a) = −2πgnρµrZa0dr′r′[e−µ(r−r′)− e−µ(r+r′)](15)After carrying out the integration, we obtain5Φ1(r,a; r > a) = −4πgnρµ2e−µr r[a cosh(µa) −1 µsinh(µa)](16)This is the potential out side the spherical ball of radius a at a distance r from its’ centre. LetA(a) = −4πgnρµ2[a cosh(µa) −1 µsinh(µa)]This allows us to write the potential outside the spherical ball asΦ1(r,a; r > a) = −A(a)e−µr rwhich has the same form as the Yukawa potential except for the fact that, now, the nuclear charge is A(a) and not g.To obtain the same potential as that of the spherical ball at distance r,r >a we need to place at centre of the ball a point nucleus of nuclear charge equal to A(a). We can estimate the value of A(a) by writing the series expansion of hyperbolic cosine and sine functions in the expression of A(a). This gives,A(a) = Ngh 1 +1 10(µa)2+1 280(µa)4+∞Xn=43 ×2n (2n + 1)!(aµ)2n−2i(17)where N =4πa3nρ 3is the number of nucleons inside the spherical ball of radius a. When radius a is large, the quantity A is much much larger than the total nucleon charge Ng inside the spherical ball. Therefore, to obtain the same potential as that of the spherical ball containing the amount of nuclear charge Ng ( at distance r, r > a ), we need to place at centre of the ball a point nucleus of nuclear charge equal to A which is larger than Ng.This clearly demostrates that the part-I of the Newton-Birkhofftheorem is violated by the Yukawa potential.However, in the limit of µ → 0, we recover result similar to the case of gravity. For r R) = −4πg2nρµ2e−µr rh R cosh(µR) −1 µsinh(µR)i (24)In the formula above the values of the the physical constants are the following: pion- nucleon coupling constant, g2/c¯ h = 0.081 , conversion constant, c¯ h = 197.328, range of two-body Yukawa potential, 1/µ = 1.4094 fm, radius of nucleus of atomic number A, R = 1.07 A1/3fm, density of nucleons in nucleus, nρ= 0.0165/fm3. We provide plots ofthe potential for three different nuclei in Fig.(1). The qualitative feature of this potential is similiar to the phenomenological Wood-Saxon potential [8]. The qualitative features is similiar even near the boundary of the nucleus. This is a bit of a surprising result because the nucleon density near the boudary of the nucleus decreases drastically where as in our derivation we have assumed constant density through out the nucleus. This suggests thatthe mean field potential is not very sensitive to the distribution of nucleons near the bound- ary. From eq.(23), it is now easy to infer that in the nuclear matter limit, i.e., when R is very large,U(r) = −4πg2nρµ2. Therefore, the quantity, fσ introduced in section-II is equal to 4πg2/µ2.IV. THE ORIGIN OF SURFACE TENSION IN HEAVY NUCLEIIn this section we sketch some arguments to show that the surface tension in heavy nucleihas its’ orgin in the violation of Newton-Birkhofftheorem. Actual calculation of the surface7tension will require the true density profile of the nucleons in the nucleus. However, in the frame work of the present paper the density is assumed to be constant, and therefore, we will not attempt to calculate the surface tension. We rather point out that , even under the assumption of constant density, one can observe the origin of surface tension. The truedensity profile influences only the numerical value of the surface tension.To make the role played by the violation Newton-Birkhofftheorem more transparent,we briefly recall some of the derivations. The total potential at any point, r, in the region between a sphere of radius a and a spherical shell of inner radius b and outer radius R ( a R~F = −g~∇Φout(r) = −4πg2nρµ2(1 + µr)e−µr r2h R cosh(µR) −1 µsinh(µR)i ~ nr(26)where ~ nris a unit vector directed along the radius towards the centre of the nucleus.The depth of the mean field potential varies very slowly in the central region (r ≈ 0) of the nucleus ( see Fig.(1) ). However, near the outer surface, there is a sharp decline in the depth of the total potential. Therefore, the force will have very small in magnitude (practically zero) in the central region but a broad peak near the bounday of the nucleus. This is what we observe in Fig.(2), where we have plotted the total force,~F as a function of r. The direction of this force at any point r near the boundary is towards the centre along the radius connecting the point. Every nucleon in the outer shell is, therefore, radially pulled towards the centre of the nucleus. Since the force is practically zero in the central part ofthe of the nucleus, there is a net pressure difference between the outer shell and the inner core. This is the origin of surface tension in nuclei. We believe that the surface tension in most normal liquids have similar origin . Our believe is based on the fact that potentials of the type −1/rnwhich describe the properties liquid state very well, can be obtained by the exchange of continuum of Yukawa modes [9,10] with some spectral weight ρ(µ). In fact, the most commonly used Leonard-Jones potential in liquids can be obtained with spectral weight ρ(µ) = µ4.−1 r6= −1 4!Z∞0dµ µ4e−µr r(27)8V. CONCLUDING REMARKSIn this paper, we have tried to emphasise the importance of the violation of Newton-Birkhofftheorem by Yukawa interaction for the emergence of mean field shell model potential as well as the surface tension in heavy nuclei. The only input from the nuclear physics thatwe have used is the constant density profile of nucleons in heavy nuclei. Some very basic and general assumptions, such as Pauli’s exclusion principle, short range character of the attractive Yukawa potential and the existence of a repulsive hard core in nucleons, ensures theemergence of constant density profile in heavy nuclei. We have derived the mean field shell model potential analytically. This potential qualitatively agrees wtih the phenomenologicalWood-Saxon potential. Our formula explicitly shows how the mean field potential depends on pion-nucleon coupling constant, range of the two-body Yukawa potential, nuclear size andthe density. Solution of the Schroedinger equation with the mean field potential so derived leads to the emergence of shell model structure. We have also pointed out how the violationof the Newton-Birkhofftheorem leads to the emergence of surface tension in heavy nuclei. Our derivations, although semi-quantitative in nature, strongly suggest that the qualitative features which are attributed either to shell model or liquid drop model can be derived from the Yukawa interaction of nucleons having repulsive hard core, and some basic principles of quantum mechanics.VI. ACKNOWELEDGEMENTSThe present work evolved from a conversation with Prof. M.Sami at the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India. We would like to thank him for discussion, suggestions as well as hospitality at IUCAA. M.A. would also like to thank Dr. S.G.Degweker of Bhabha Atomic Research Centre, Mumbai, India, for critical comments and suggestions. Thanks are also due to Dr. S.Ganesan of Bhabha Atomic Research Centre, Mumbai, India, whose encouragement played an important role in completion of this work.9REFERENCES[1] Bernard L. Cohen, ”Concepts of Nuclear Physics”, McGraw Hill, 1971 [2] A. deShalit and H. 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D 72 (2005) 02402410-2024681012141618-70-60-50-40-30-20-10010Figure 1: Plot of potential energy of a single nucleon in the mean-field, inside and outside the nucleus, as a function of radial distance from the center of the nucleus.Radius in fermi forCu Sn Pb 4.28 5.28 6.34Radius = 1.07 A1/3=Cu Sn PbPotential Energy U (MeV)Radial distance (r fm)024681012-20-15-10-50Figure 2: Plot of mean-field force F on a single nucleon inside and outside the nucleus as a function of radial distance r from the centre of the nucleus. The minima in the plot occur at the radius R=1.07A1/3 for the respective nuclei.Cu Sn PbF = - dU/dr (MeV/fm)Radial distance (r fm)