Semi-empirical Nuclear Mass Formula: Simultaneous Determination of 4 Coefficients

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1 Asian Journal of Physical and Chemical Sciences 1(2): 1-10, 2016; Article no.ajopacs.1266 SCIENCEDOMAIN international Semi-empirical Nuclear Mass Formula: Simultaneous Determination of Coefficients José Luis Pinedo-Vega 1*, Carlos Ríos-Martínez 1, Mirna Patricia Talamantes-Carlos 1, Fernando Mireles-García 1, J. Ignacio Dávila-Rangel 1 and Valentín Badillo-Almaraz 1 1 Universidad Autónoma de Zacatecas, UAEN, Ciprés 10, Fracc, La Peñuela, Zacatecas, Zac. C. P , México. Authors contributions This work was carried out in collaboration between all authors. Authors JLPV and CRM designed the study. Author MPTC performed the statistical analysis, wrote the protocol and wrote the first draft of the manuscript. Authors FMG and JIDR managed the analyses of the study. Author VBA managed the literature searches. All authors read and approved the final manuscript. Article Information DOI: 10.97/AJOPACS/2016/1266 Editor(s): (1) Giannouli Myrsini, Department of Physics, University of Patras, Greece. Reviewers: (1) Airton Deppman, Universidade de São Paulo, Brazil. (2) Fatma Kandemirli, Kastamonu University, Turkey. () Shaik Babu, K L University, India. Complete Peer review History: Original Research Article Received 28 th December 2016 Accepted 6 th February 2017 Published 1 th February 2017 ABSTRACT The deduction of coefficients of the semi-empirical mass formula is presented as a function with two constants of proportionality: which relates the energy of the nuclear volume with volume and which relates volume with the mass number. Next the development of a proprietary method is presented one that permits the simultaneous calculation of of the 5 coefficients of the original semi-empirical formula. This method, which is direct and does not employ or require the use of successive approximations or iterations, is sufficiently didactic. It makes use of the experimental binding energies from 6 stable isotopes with a mass number odd-. Subsequently as validation, the coefficients are utilized for the theoretical calculation of the atomic masses of 27 stable isotopes and are compared with the experimental masses. Additionally, the calculation of the coefficients of proportionality and, the unit nuclear radius, the coefficients of nuclear surface tension, and the nuclear density are presented as well. *Corresponding author: jlpinedo@uaz.edu.mx;

2 Keywords: Nuclear mass; semi-empirical mass formula; liquid drop model; stable isotopes. 1. INTRODUCTION As is well known, experimentally, there exists a difference in the mass of the constituents of an atom,, and its atomic mass, (, ). That difference is called the Nuclear Binding Energy (, ), which is considered the energy necessary to keep the nucleons bound together, or the energy required to separate the nucleus into nucleons. It can be written as: (, ) + ( ) + (, ) (, ) (1) where,, and are the masses of the proton, neutron, and electron respectively; is the number of neutrons, is the speed of light in a vacuum, is the massenergy equivalence factor, and (, ) is the bond energy of the electrons. If the mass of the constituents is + (2) The nuclear binding energy (eq. 1) can be written as Weizsäcker []. Theoretically, it is based on the liquid drop model. According to him, the nuclear mass is basically a sum of the masses of the constituents,, on which should be applied a series of corrective terms to account for the short range forces: : volume energy, surface energy -which are similar to the intermolecular forces that are involved in a drop of liquid- Coulomb energy, asymmetry energy, and (, ) pairing energy. The structure of the formula continues to be the same since it was formulated. (, ) + [ ± (, )]/ () The set of corrective terms is equivalent to the binding energy. Said terms are a function of the atomic number and/or the mass number. (5) (6) / (7) (, ) [ (, )] () There is no single model that explains nuclear mass. The liquid drop model was, historically, the first model to describe nuclear properties. It was proposed by George Gamow [1] and developed by Niels Bohr and John Archibald Wheeler [2]. (, ) ( 2 ).5 foreven z even n 0 forodd A.5 forodd z and odd n (8) (9) The liquid drop model treats the nucleus as a drop of incomprehensible nuclear fluid of very high density. The nucleus is constituted of nucleons (protons and neutrons) that are held together by a nuclear force. Through a manner similar to the intermolecular forces in a liquid, it is assumed that the inter-nuclear forces are short range and have saturation properties. The fact that the binding energy per nucleon, is in the order of 8 MeV/nucleon for the set of isotopes is considered proof of the nuclear saturation properties. The semi-empirical mass formula is an expression that permits the approximation of various properties of the atomic nucleus in terms of the protons and neutrons. It was proposed by the German physicist Carl Friedrich von The method described in literature for the determination of the coefficients,,, and (, ) makes use of linear fits and successive approximations. For the calculation of the coefficients which corresponds to the Coulomb Energy, they drew upon the theory of alpha decay. Once was calculated, stable isobars were used to calculate the coefficient which corresponds to the asymmetry energy. The method of calculation for the rest of the coefficients is too laborious [,5]. In this paper, the deduction of the semi-empirical mass formula is reviewed, a proprietary method for the calculation of the coefficients in the formula will be developed, and the obtained coefficients and the formula will be utilized for the theoretical calculation of the atomic masses of 2

3 27 stable isotopes, which will then be compared to the experimental masses [6]. 2. A DEDUCTION OF COEFFICIENTS OF THE SEMI-EMPIRICAL MASS FORMULA 2.1 Volume Energy ( ) In a manner similar to the molecular association energy in the case of a liquid drop, the volume energy should be proportional to the nuclear volume where (10) is a proportionality constant. The negative sign over in eq. () signifies that a factor equivalent to the energy necessary to maintain the nucleons bound must be subtracted from the mass of the constituents. If it is assumed that the nuclear volume is proportional to the number of nucleons or the mass number. (11) For now substituting in eq. (10) molecules on the surface. In this manner the surface has an excess of energy in relation to the interior of the liquid. The surface energy is defined as the relationship between the required energy to increase the surface of a liquid,, and the increase in the surface area of the liquid,. This is stated by (1) The unit of energy per unit of surface area, in SI units is. This final notation realizes that the quantity of energy per unit area has units that are equivalent to force per unit length, by which reason this type of energy is recognized as Surface Energy. In the case of the atomic nucleus, if is matter and its volume, hypothetically it can be defined as a nuclear density for all or any nucleus (1) If nuclear volume is considered in terms of spherical volume (10 ) (15) Where is an unknown proportional constant to determine that takes into account the proportionality of the volume energy with respect to the volume and the number of nucleons. (12) 2.2 Surface Energy ( ) The liquid drop model assumes that if the atomic nuclei were similar to a spherical drop, then there should exist a surface tension, in the same manner that occurs in the drops of a liquid. The molecules that are found on the inside are attracted by all the neighboring molecules; in such a way that there is no resultant force that would displace them in any direction. On the contrary, the molecules on the surface are attracted by the molecules on the inside of the liquid, generating a surface tension or surface energy. In the case of drops of different sizes, the drops that are larger have a larger surface area, and therefore have a larger number of To this volume, there is the corresponding surface area (16) And by the definition of Surface Energy, eq. (1) (1 ) Substituting the eq. (16) the surface energy results in (1 ) Arya [7] reported the coefficient of nuclear surface energy as order T/mm. Of the eq. (11), and setting it equal to eq. (15) then simplifying (17)

4 Substituting in eq. (1b) results in Where (18) ( ) ( ) (19) The eq. (18) shows that the Surface energy is a function of the mass number. 2. Coulomb Energy The deduction of the Coulomb energy commonly found in literature. 5 1 is more (20) Where ε x10 is the electrical permissiveness in the vacuum, is the atomic number or number of protons, is the elemental electric charge, and is the nuclear radius. Utilizing eq. (17), the Coulomb energy can expressed as Where / (21) 5 1 e / (22) The eq. (21) shows that the Coulomb energy is a function of both the atomic number,, and the mass number,. 2. Asymmetry Energy An analysis of the isotopes shows that the more stable isotopes have a paired proton-neutron number. The nuclei with the highest binding energy are the ones that have 2. Any deviation from that value will lower the binding energy. Taking that into account, a positive corrective term was proposed for the nucleons that are not paired. Wigner [8] established that the excess in the number of nucleons produces a deficit of binding energy since the excess is quantically out of reach of the other nucleons and so the fraction of nuclear volume affected is / and the total deficit is proportional to this quantity. This is shown by 2 (2) 2.5 Pairing Energy or Spin Coupling Effect The fifth factor that affects the binding energy, and therefore the mass of the nucleons, is the fact that the number of the neutrons and the protons is either even or odd. The more stable nuclei are those that have a paired number of protons and neutrons which has been interpreted as a spin coupling between nucleons of the same specie. The less stable nuclei are those that have a number odd of protons as they have of neutrons. The effect of the spin coupling is represented by the term ± (, ), which is a negative factor for the case of the more stable nuclei or the ones with the higher binding energy, so to say the ones that have even- and even-. It is equivalent to zero for the nuclei that are moderately stable or that have a mass number odd-, and it is positive for the less stable nuclei or with the lowest binding energy (even- and odd- ). 2.6 General Form of the Semi-Empirical Formula Grouping the described terms from the previous sections, the semi-empirical formula of the atomic mass is (, ) ( 2 ) ± (, ) (2) In literature, one can find a multitude of values for the coefficients,,, and (, ) (Table 1); which signifies that there isn t a standardized method of calculation for the coefficients. As one can observe, there are important differences that exist between authors, with Fermi [12], Metropolis [1], Feenberg [], and Evans [5] as the exceptions since they have obtained similar coefficient values.

5 Table 1. Comparison of the coefficients of the semi-empirical mass formula (MeV) as obtained by different authors (, ) Bethe and Bacher (196) [9] Feenberg (199) [10] Bohr and Wheeler (199) [2] Mattauch and Flugge (192) [11] Fermi (195) [12] Metropolis & Reitweisner (1950) [1] Feenberg (197) [] Fowler (197) (Cited in [5] p. 8) Friedlander & Kennedy (199) [1] /A Rosenfeld (199) [15] Canadian National Research Council (195) (Cited in [5] p. 8) Green (195) [16] Evans (1955) [5] 1.1±0.2 1± ± ±0.9 Source: [5], p. 8 If any group of coefficients is taken, the semiempirical mass formula permits values of the atomic mass that are similar to the experimental values. Nevertheless, the calculation of the difference between experimental values and calculated values presents very important differences (Fig. 1). This paper s claim was to try and elucidate the cause of these differences. For it, a new method of calculation for the coefficients was developed.. SIMULTANEOUS METHOD OF CALCULATION FOR THE COEFFICIENTS OF THE SEMI- EMPIRICAL FORMULA We start by pointing to the fact that the last five terms of the eq. (2) equal the binding energy (eq. ). That is shown by (, ) ( 2 ) ± (, ) (25) And so the binding energy per nucleon needs to be (, ) 2 ± (, ) (26) The atomic mass is of experimental character, the binding energy is obtained after subtracting the sum of the masses of the constituents of the atomic mass, and so the binding energy is also of experimental character. To calculate the first four coefficients, the binding energy of a group of stable isotopes with mass number odd- can be utilized in such a way that the coefficient (, ) 0, while at the same time avoiding proximity between the magic numbers. Given that for each one of them and are known, the eq. (26) can be written as (, ) (27) where: 1,,, and i each one of the isotopes is one of the isotopes in question. In Table 2 a list of the 6 stable isotopes that are utilized with the coefficients is presented. Utilizing the binding energies and the coefficients one can then formulate a group of simultaneous equations of the same form of eq. (8), for each isotope, whose unknowns are the coefficient,, and. And since there are only four unknowns (the four coefficients) only equations are needed. Given the data of the 6 stable isotopes, 6 groups of have been chosen to calculate the coefficients of the semi-empirical mass formula simultaneously. 5

6 The coefficients can be obtained in various ways; in this case matrices were utilized. If C is the matrix of the terms associated with the coefficients, B is the matrix of the binding energy per nucleon and A is the matrix of the coefficients of the semi-empirical formula a : c c c c c c c c c c c c c c c c a a a ; B B B ; (28) Matrix A can be obtained by calculating the inverse matrix of C ( ). (29) In Table the values of the obtained coefficients are presented through utilizing different combinations of the stable isotopes. The values, while they seem close to each other, are different. The first row, which corresponds to the isotopes with < 65, has the highest values in the coefficients and (Surface energy and Coulomb energy). The second, third, and fifth rows correspond to the isotopes in scattered form from Oxygen to Platinum; it could be said that the corresponding coefficients are the most representative of the set of isotopes. The fourth row, corresponds to the isotopes of intermediate masses from 127 (,,, ) which, strangely, presents a negative value in the case of and a very high value for. Which coefficient group is the most appropriate? First, it would have to be verified if the coefficients are similar to the ones reported in books. Comparing with Table 1, one can observe that, with the exception of the coefficients of the group,,,, the obtained coefficients are of the same order magnitude as the ones reported in literature. The isotopes that are better distributed in the range of mass values of the stable isotopes are,,, ; and thus it is assumed that they can be more representative. The constants of the corresponding coefficients are (fifth row of Table ) MeV ; MeV ; and MeV u. Royer [17] reported: values between and 15.9 MeV and values between and MeV. Table 2. Stable Isotopes with odd- that are utilized and values of the coefficients determination of,, and of the semi-empirical mass formula for the Isotope z N A BE/A (kev) M(A,z) (u) O S E- Mn Cu I Pt Table. Coefficients for the semi-empirical mass formula (MeV) obtained through formulated simultaneous equations,,, ,,, ,,, ,,, ,,,

7 In Fig. 1, presented is a comparison between experimental atomic masses and theoretical masses for 27 stable isotopes. The theoretical masses have been calculated utilizing the coefficients of Evans [5], Arya [7], and the previous coefficients, which were obtained utilizing the isotopes the,,, (UAEN). As is verified by Table 1, the coefficients from Evans are similar to those of Fermi, Metropolis, and Feenberg. Graphically, the differences between the experimental atomic mass and the theoretical masses are imperceptible, nevertheless if the differences between the experimental values and the calculated values are compared, important differences are observed (Fig. 2) Evans Arya UAEN Experimental A (u) z Fig. 1. Experimental masses and calculated masses through the semi-empirical formula, of 27 stable isotopes UAEN Arya Evans M exp -M calc (u) z Fig. 2. Comparison of the differences between the experimental masses and the calculated masses of the 27 stable isotopes through the coefficients of Arya and Evans and the coefficient obtained with the isotopes,, and (UAEN) 7

8 Table. Comparison between the average absolute values through the differences between experimental mass and the calculated mass Coefficients UAEN u 2.12MeV u 2.12 MeV Arya u MeV MeV Evans u 5.80 MeV 0.00 u.070 MeV In Table, presented are the average absolute values of the differences between the experimental masses and the calculated masses, and the standard deviations for 27 stable isotopes. The absolute average value of the obtained differences with the determined coefficients in this paper (UAEN) so to say the ones calculated with the isotopes,,, is slightly higher than that of the calculations with the coefficients of Arya (there is a difference of MeV). What this means is that the calculated masses with Arya s coefficients are slightly better focused than the ones that use our method. Nevertheless the standard deviations are slightly better for the case of our method (UAEN) (there is a difference of MeV with respect to Arya s method). So to say that with the dispersion, it is less with the denominated method UAEN. The largest absolute average differences are the ones obtained with the coefficients of Evans, which are similar to those of Fermi, Metropolis, and Feenberg. In conclusion, the coefficients obtained in this paper permit the obtainment of a good approximation of the atomic masses with respect to the experimental values. The most recent values of the binding energies and the nuclear masses have been utilized, as was software that is much more precise than what could have been used in the 50 s. Even so, it can t be concluded that the model is finished. Although the approximation between the calculated values and the experimental ones is surprising, the dispersion that is associated to the said differences gives an account that the theory of the semi-empirical formula has not taken into account the totality of the phenomena or the forces that are at play in the constitution of the nucleus.. RADIUS, COEFFICIENT OF SURFACE ENERGY, AND NUCLEAR DENSITY Once the coefficients of the Semi- Empirical Formula are evaluated, it is possible to calculate the proportional constant which gives an account of the relationship between nuclear volume and the mass number. Given that 5 Simplifying 1 5 e 1 / ( 22) e ( 0) Substituting ( ) , and , the constant of proportionality results in ( 1) After the eq. (17) the nuclear radius can be calculated for any isotope of mass The unit nuclear radius results in / / (2) ( ) Royer [17] reported the charge radius between 1.22 and 1.2 fm. Of eq. (18) the coefficient of nuclear Surface Energy can be simplified ( ) / ( ) / () 8

9 And given that MeV (5) Arya [7] reported the coefficient of nuclear surface energy of the order T/mm. Given that the previous quantity is equal to Of eq. (12) one can simplify, the constant of proportionality between the energy of the volume and the volume (12 ) Given that MeV (6) This establishes the relationship between the energy of the volume and the nuclear volume. Of the eqs. (1 and 15) the nuclear density is (7) If the unit nuclear radius is associated with to a unit of atomic mass , the nuclear density results in (8) Arya reported a nuclear density of the order 10 5 T/mm. 5. CONCLUSIONS In this paper the theoretical fundamentals of the semi-empirical mass formula are reviewed. Five sets of coefficients were determined from the semi-empirical formula, which, with the exception of one, are similar to some of the 1 groups of coefficients found in literature. The coefficients that were most representative were selected and were tried in the calculation of the masses of the 27 stable isotopes. To verify the reproducibility ability, the absolute differences between the experimental values and theoretical or calculated ones were determined, among the associated standard deviation. The results were compared to the calculations made utilizing the coefficients and Arya s semiempirical formula and that of Evans. It was observed that the set of coefficients that corresponded to the isotopes,, and produced the lowest standard deviation between the experimental values and the calculated values, in comparison to the standard deviations that were obtained with the coefficients of Arya and Evans. Additionally, the deduction of the formula gave evidence to the independent character of three of the five coefficients of the formula,, and, according to what is shown by the eqs. (12, 18, and 6). The three contain the factor of proportion between the volume and the mass number,. Presented is the form of calculation for, and once evaluated, the coefficient of surface energy and the constant of proportionality of the volume energy is determined. These coefficients are not known in literature, one form of verifying their coherence is the evaluation of the unit radius,, and the nuclear density,, and to compare them with the corresponding values that are reported in literature. One of the benefits of this paper was to make explicit a form of calculation of the coefficients, which is also not published in literature. This paper explains the theoretic fundamentals of the semi-empirical mass formula. Nevertheless, it cannot be concluded that the model is finished. The found coefficients although similar, form a part of many attempts to justify the semi-empirical formula. The order of dispersion of the difference indicates that there must be factors that are not taken into account in the base theories of the semi-empirical mass formula. For now, the semi-empirical mass formula is only an approximation. COMPETING INTERESTS Authors have declared that no competing interests exist. 9

10 REFERENCES 1. Gamow G. Mass defect curve and nuclear. Proc. R. Soc. 190;126: Bohr N, Wheeler JA. The mechanism of nuclear fission. Phys. Rev.; Von Weizsäcker CF. Theorie de kernmassen. Z. Phys. 195;96: Feenberg E. Semi-empirical theory of the nuclear energy surface. Rev. Mod. Phys. 197;19(): Evans RD. The atmoc nucleus. McGraw- Hill Book Company Inc.; IAEA. Live chart of nuclides; Arya AP. Introduction to nuclear physics. Allyn and Bacon; Wigner EP. On the consequences of the symmetry of the nuclear hamiltonian on the spectroscopy of nuclei. Phys. Rev. 197;51(2): Bethe HA, Bacher RF. Nuclear physics A. Stationary states of nuclei. Rev. Mod. Phys. 196;8: Feenberg E. On the shape and stability of heavy nuclei. Phys. Rev. 199;55: Mattauch J, Fluegge S. Nuclear physical tables and an introduction to nuclear physics. Inter. Pub. Inc.; Fermi E, Notes compiled by Orear F, Rosenfeld AH, Schluter RA. Nuclear Physics. University of Chicago Press.; Metropolis N, Reitwiesner G. NP-1980; Friedlander G, Kennedy JW. Introduction to radiochemistry. John Wiley and Sons. Inc.; Rosenfeld L. Nuclear forces. Intersciences Publisher, Inc.; Green AES. Coulomb radius constant from nuclear masses. Phys. Rev. 195;95, Royer G. On the coefficients of the liquid drop model mass formulae and nuclear radii. Nucl. Phys. A. 2008;807( ): Pinedo-Vega et al.; This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Peer-review history: The peer review history for this paper can be accessed here: 10