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

Asian Journal of Physical and Chemical Sciences 1(2): 1-10, 2016; Article no.AJOPACS.31266

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

José Luis Pinedo-Vega1*, Carlos Ríos-Martínez1, Mirna Patricia Talamantes-Carlos1, Fernando Mireles-García1, J. Ignacio Dávila-Rangel1 and Valentín Badillo-Almaraz1

1Universidad Autónoma de Zacatecas, UAEN, Ciprés 10, Fracc, La Peñuela, Zacatecas, Zac. C. P. 98000, 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: 10.9734/AJOPACS/2016/31266 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. (3) Shaik Babu, K L University, India. Complete Peer review History: http://www.sciencedomain.org/review-history/17811

Received 28th December 2016 Accepted 6th February 2017 Original Research Article Published 13th February 2017

ABSTRACT

The deduction of 4 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 4 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 237 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.

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*Corresponding author: E-mail: [email protected];

Pinedo-Vega et al.; AJOPACS, 1(2): 1-10, 2016; Article no.AJOPACS.31266

Keywords: Nuclear mass; semi-empirical mass formula; liquid drop model; stable isotopes.

1. INTRODUCTION Weizsäcker [3]. Theoretically, it is based on the liquid drop model. According to him, the nuclear As is well known, experimentally, there exists a mass is basically a sum of the masses of the difference in the mass of the constituents of an constituents, , on which should be applied a series of corrective terms to account for the short atom, , and its atomic mass, (, ). That difference is called the Nuclear Binding Energy range forces: : volume energy, surface energy -which are similar to the intermolecular (, ) , which is considered the energy necessary to keep the nucleons bound together, forces that are involved in a drop of liquid- or the energy required to separate the nucleus Coulomb energy, asymmetry energy, and into nucleons. It can be written as: (, ) pairing energy.

The structure of the formula continues to be the (, ) = + ( − ) + − (, ) same since it was formulated. − (, ) (1) where and are the masses of the (, ) = + [− + + + ± (, )] / , , proton, neutron, and electron respectively; (4)

= − is the number of neutrons, is the The set of corrective terms is equivalent to the speed of light in a vacuum, is the mass- binding energy. Said terms are a function of the energy equivalence factor, and (, ) is the atomic number and/or the mass number . bond energy of the electrons.

If the mass of the constituents is = (5)

= + (2) = (6)

The nuclear binding energy (eq. 1) can be written as = (7) /

(, ) = [ − (, )] (3) ( − 2) = (8) There is no single model that explains nuclear mass. The liquid drop model was, historically, the first model to describe nuclear properties. It was −33.5 for even − z even − n (, ) = 0 for odd − A (9) proposed by George Gamow [1] and developed by Niels Bohr and John Archibald Wheeler [2]. 33.5 for odd − z and odd − n

The liquid drop model treats the nucleus as a The method described in literature for the drop of incomprehensible nuclear fluid of very determination of the coefficients , , , high density. The nucleus is constituted of and (, ) makes use of linear fits and nucleons (protons and neutrons) that are held successive approximations. For the calculation of together by a nuclear force. Through a manner the coefficients – which corresponds to the similar to the intermolecular forces in a liquid, it is Coulomb Energy, they drew upon the theory of assumed that the inter-nuclear forces are short alpha decay. Once was calculated, stable range and have saturation properties. The fact isobars were used to calculate the coefficient that the binding energy per nucleon, is in the – which corresponds to the asymmetry energy. order of 8 MeV/nucleon for the set of isotopes is The method of calculation for the rest of the considered proof of the nuclear saturation coefficients is too laborious [4,5]. properties. In this paper, the deduction of the semi-empirical The semi-empirical mass formula is an mass formula is reviewed, a proprietary method expression that permits the approximation of for the calculation of the coefficients in the various properties of the atomic nucleus in terms formula will be developed, and the obtained of the protons and neutrons. It was proposed by coefficients and the formula will be utilized for the the German physicist Carl Friedrich von theoretical calculation of the atomic masses of

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237 stable isotopes, which will then be compared molecules on the surface. In this manner the to the experimental masses [6]. surface has an excess of energy in relation to the interior of the liquid. 2. A DEDUCTION OF 4 COEFFICIENTS OF THE SEMI-EMPIRICAL MASS The surface energy is defined as the FORMULA relationship between the required energy to increase the surface of a liquid, , and the increase in the surface area of the liquid, . 2.1 Volume Energy () This is stated by In a manner similar to the molecular association energy in the case of a liquid drop, the volume = (13) energy should be proportional to the nuclear volume The unit of energy per unit of surface area, in SI units is = = . This final notation realizes = (10) that the quantity of energy per unit area has units where is a proportionality constant. that are equivalent to force per unit length, by which reason this type of energy is recognized as The negative sign over in eq. (4) signifies that Surface Energy. a factor equivalent to the energy necessary to maintain the nucleons bound must be subtracted In the case of the atomic nucleus, if is matter from the mass of the constituents. and its volume, hypothetically it can be defined as a nuclear density for all or any nucleus If it is assumed that the nuclear volume is proportional to the number of nucleons or the mass number. = (14)

= (11) If nuclear volume is considered in terms of spherical volume For now substituting in eq. (10)

4 = = (10) = (15) 3

Where is an unknown proportional constant to To this volume, there is the corresponding determine that takes into account the surface area proportionality of the volume energy with respect to the volume and the number of nucleons. = 4 (16)

= (12) And by the definition of Surface Energy, eq. (13)

2.2 Surface Energy () = (13) The liquid drop model assumes that if the atomic nuclei were similar to a spherical drop, then there Substituting the eq. (16) the surface energy should exist a surface tension, in the same results in manner that occurs in the drops of a liquid. The molecules that are found on the inside are = 4 (13) attracted by all the neighboring molecules; in such a way that there is no resultant force that Arya [7] reported the coefficient of nuclear would displace them in any direction. On the surface energy as order 1010 T/mm. contrary, the molecules on the surface are attracted by the molecules on the inside of the Of the eq. (11) = , and setting it equal to liquid, generating a surface tension or surface eq. (15) then simplifying energy. In the case of drops of different sizes, the drops that are larger have a larger surface 3 = (17) area, and therefore have a larger number of 4

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Substituting in eq. (13b) results in binding energy since the excess is quantically out of reach of the other nucleons and so the fraction of nuclear volume affected is | − |/ = (18) and the total deficit is proportional to this

quantity. This is shown by Where

| − | | − 2| = = (23) = (3)(4) (19)

The eq. (18) shows that the Surface energy is a 2.5 Pairing Energy or Spin Coupling function of the mass number. Effect

2.3 Coulomb Energy The fifth factor that affects the binding energy, and therefore the mass of the nucleons, is the The deduction of the Coulomb energy is more fact that the number of the neutrons and the commonly found in literature. protons is either even or odd. The more stable

nuclei are those that have a paired number of 3 1 protons and neutrons which has been interpreted = (20) 5 4 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 Where ε = 8.85415x10 is the electrical neutrons. permissiveness in the vacuum, is the atomic number or number of protons, The effect of the spin coupling is represented by = 1.6021765653510 is the elemental the term ±(, ), which is a negative factor for electric charge, and is the nuclear radius. the case of the more stable nuclei or the ones

with the higher binding energy, so to say the Utilizing eq. (17), the Coulomb energy can ones that have even- and even- . It is expressed as 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 = (21) / or with the lowest binding energy (even- and

odd-). Where

/ 2.6 General Form of the Semi-Empirical 3 1 4 = e (22) Formula 5 4 3

Grouping the described terms from the previous The eq. (21) shows that the Coulomb energy is a sections, the semi-empirical formula of the function of both the atomic number, , and the atomic mass is mass number, .

( − 2) 2.4 Asymmetry Energy (, ) = − + + + An analysis of the isotopes shows that the more ± (, ) (24) stable isotopes have a paired proton-neutron number. The nuclei with the highest binding In literature, one can find a multitude of values energy are the ones that have = 2 . Any for the coefficients , , , and (, ) deviation from that value will lower the binding (Table 1); which signifies that there isn’t a energy. Taking that into account, a positive standardized method of calculation for the corrective term was proposed for the nucleons coefficients. As one can observe, there are that are not paired. important differences that exist between authors, with Fermi [12], Metropolis [13], Feenberg [4], Wigner [8] established that the excess in the and Evans [5] as the exceptions since they have number of nucleons | − | produces a deficit of obtained similar coefficient values.

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Table 1. Comparison of the coefficients of the semi-empirical mass formula (MeV) as obtained by different authors

(, ) Bethe and Bacher (1936) [9] 13.86 13.2 0.58 19.5 Feenberg (1939) [10] 13.3 0.62 Bohr and Wheeler (1939) [2] 14 0.59 Mattauch and Flugge (1942) [11] 14.66 15.4 0.602 20.5 Fermi (1945) [12] 14.0 13.0 0.583 19.3 33.5 Metropolis & Reitweisner (1950) [13] 14.0 13.0 0.583 19.3 33.5 Feenberg (1947) [4] 14.1 13.1 0.585 18.1 33.5 Fowler (1947) (Cited in [5] p. 383) 15.3 16.7 0.69 22.6 Friedlander & Kennedy (1949) [14] 14.1 13.1 0.585 18.1 132/A Rosenfeld (1949) [15] 14.66 15.4 0.602 20.54 Canadian National Research Council 14.05 14.0 0.61 19.6 (1945) (Cited in [5] p. 383) Green (1954) [16] 15.75 17.8 0.71 23.7 Evans (1955) [5] 14.1±0.2 13±0.1 0.595±0.02 19.0±0.9 Source: [5], p. 383

If any group of coefficients is taken, the semi- atomic mass, and so the binding energy is also empirical mass formula permits values of the of experimental character. atomic mass that are similar to the experimental values. Nevertheless, the calculation of the To calculate the first four coefficients, the difference between experimental values and binding energy of a group of stable isotopes calculated values presents very important with mass number odd- can be utilized in differences (Fig. 1). such a way that the coefficient (, ) = 0, while at the same time avoiding proximity between This paper’s claim was to try and elucidate the the magic numbers. Given that for each one of cause of these differences. For it, a new method them and are known, the eq. (26) can be of calculation for the coefficients was developed. written as

3. SIMULTANEOUS METHOD OF (, ) = − − − (27) CALCULATION FOR THE COEFFICIENTS OF THE SEMI- EMPIRICAL FORMULA where: = 1 , = , = , = and i each one of the isotopes is one of We start by pointing to the fact that the last five terms of the eq. (24) equal the binding energy the isotopes in question. (eq. 3). That is shown by In Table 2 a list of the 6 stable isotopes that are ( − 2) utilized with the coefficients is presented. − (, ) = − + + + ± (, ) (25) Utilizing the binding energies and the coefficients one can then formulate a group of And so the binding energy per nucleon needs to simultaneous equations of the same form of eq. be (38), for each isotope, whose unknowns are the coefficient , , and . And since there are only four unknowns (the four coefficients) only 4 ( ) , − 2 = − − − equations are needed. ± (, ) (26) Given the data of the 6 stable isotopes, 6 groups The atomic mass is of experimental character, of 4 have been chosen to calculate the the binding energy is obtained after subtracting coefficients of the semi-empirical mass formula the sum of the masses of the constituents of the simultaneously.

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The coefficients can be obtained in various ways; that the corresponding coefficients are the most in this case matrices were utilized. If C is the representative of the set of isotopes. The fourth matrix of the terms associated with the row, corresponds to the isotopes of intermediate coefficients , B is the matrix of the binding masses from 33 ≤ ≤ 127 ( , , , energy per nucleon and A is the matrix of the ) which, strangely, presents a negative value coefficients of the semi-empirical formula a: in the case of and a very high value for .

c c c c B Which coefficient group is the most appropriate? c c c c B = ; = ; c c c c First, it would have to be verified if the c c c c B coefficients are similar to the ones reported in a books. a = (28) Comparing with Table 1, one can observe that, a with the exception of the coefficients of the group

, , , , the obtained coefficients Matrix A can be obtained by calculating the are of the same order magnitude as the ones inverse matrix of C ( ). reported in literature. The isotopes that are better distributed in the range of mass values of the = (29) stable isotopes are , , , ; and

thus it is assumed that they can be more In Table 3 the values of the obtained coefficients representative. The constants of the are presented through utilizing different corresponding coefficients are (fifth row of combinations of the stable isotopes. Table 3).

The values, while they seem close to each other, = 15.5933955 MeV = 0.01674019851496 ; are different. = 17.3447975 MeV = 0.01862040588 ; The first row, which corresponds to the isotopes = 0.69351989 = 0.00074452422 and with < 65, has the highest values in the = 23.6012096 MeV = 0.0253369404949 u. coefficients and (Surface energy and Coulomb energy). The second, third, and fifth Royer [17] reported: values between 15.0775 rows correspond to the isotopes in scattered and 15.93 MeV and values between 17.0068 form from Oxygen to Platinum; it could be said and 19.3863 MeV.

Table 2. Stable Isotopes with odd- that are utilized and values of the coefficients for the determination of , , and of the semi-empirical mass formula

Isotope z N A BE/A (keV) M(A,z) (u) O 8 9 17 7750.7280 16.9991 1 -0.38891 -1.46414 -0.00346 S 16 17 33 8497.6300 32.9715 1 -0.31177 -2.41855 -9.18274E-4 Mn 25 30 55 8765.0098 54.9380 1 -0.26295 -2.98811 -0.00826 Cu 29 36 65 8757.0931 64.9278 1 -0.24871 -3.21794 -0.01160 I 53 74 127 8445.4873 126.904 1 -0.19894 -4.40028 -0.02734 Pt 78 117 195 7926.5655 194.965 1 -0.17245 -5.38034 -0.04000

Table 3. Coefficients for the semi-empirical mass formula (MeV) obtained through formulated simultaneous equations

,, , 16.6688061 19.459436 0.8601476 26.214231 ,,, 14.3169813 14.866376 0.4513931 34.583932 , , , 15.5028903 17.180071 0.6724054 24.897948 , ,, 13.3979686 16.076480 -0.0751606 76.251848 ,, , 15.5933955 17.344797 0.6935199 23.601209

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In Fig. 1, presented is a comparison between from Evans are similar to those of Fermi, experimental atomic masses and theoretical Metropolis, and Feenberg. Graphically, the masses for 237 stable isotopes. The theoretical differences between the experimental atomic masses have been calculated utilizing the mass and the theoretical masses are coefficients of Evans [5], Arya [7], and the imperceptible, nevertheless if the differences previous coefficients, which were obtained between the experimental values and the utilizing the isotopes the , , , calculated values are compared, important (UAEN). As is verified by Table 1, the coefficients differences are observed (Fig. 2).

200 Evans Arya UAEN Experimental 150

A (u) A 100

0 0 20 40 60 80 z

Fig. 1. Experimental masses and calculated masses through the semi-empirical formula, of 237 stable isotopes

0.015 UAEN Arya 0.010 Evans

0.005

0.000 (u) calc -0.005 -M exp

M -0.010

-0.015

-0.020

-0.025

-0.030 0 20 40 60 80 z

Fig. 2. Comparison of the differences between the experimental masses and the calculated masses of the 237 stable isotopes through the coefficients of Arya and Evans and the coefficient obtained with the isotopes ,, and (UAEN)

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Table 4. Comparison between the average absolute values through the differences between experimental mass and the calculated mass

Coefficients | − | UAEN 0.00228 u = 2.124MeV 0.00259u = 2.412 MeV Arya 0.00212 u = 1.975 MeV 0.00293 = 2.729 MeV Evans 0.00623 u = 5.803 MeV 0.0033 u = 3.070 MeV

In Table 4, presented are the average absolute 4. RADIUS, COEFFICIENT OF SURFACE values of the differences between the ENERGY, AND NUCLEAR DENSITY experimental masses and the calculated masses, and the standard deviations for 237 stable Once the coefficients of the Semi- isotopes. Empirical Formula are evaluated, it is possible to

calculate the proportional constant which The absolute average value of the obtained gives an account of the relationship differences with the determined coefficients in between nuclear volume and the mass number. this paper (UAEN)—so to say the ones Given that calculated with the isotopes , , , – is slightly higher than that of the calculations / 3 1 4 with the coefficients of Arya (there is a difference = e ( 22) of 0.1490 MeV). What this means is that the 5 4 3 calculated masses with Arya’s coefficients are Simplifying slightly better focused than the ones that use our method. 4 3 1 = e ( 30) Nevertheless the standard deviations are slightly 3 5 4 better for the case of our method (UAEN) (there is a difference of 0.3167 MeV with respect to Substituting = (0.69351989 ) = Arya’s method). So to say that with the 1.11114131510 , = 8.8541510 dispersion, it is less with the denominated method UAEN. and = 1.6021765653510 , the constant of proportionality results in

The largest absolute average differences are the ones obtained with the coefficients of Evans, = 8.09890873610 ( 31) which are similar to those of Fermi, Metropolis, and Feenberg. After the eq. (17) the nuclear radius can be calculated for any isotope of mass In conclusion, the coefficients obtained in this / paper permit the obtainment of a good 3 = (32) approximation of the atomic masses with respect 4 to the experimental values. The most recent values of the binding energies and the nuclear The unit nuclear radius results in masses have been utilized, as was software that is much more precise than what could have been 3 / = = 1.24579322710 ( 33) used in the 50’s. Even so, it can’t be concluded 4 that the model is finished. Although the approximation between the calculated values Royer [17] reported the charge radius between and the experimental ones is surprising, the 1.22 and 1.23 fm. dispersion that is associated to the said differences gives an account that the theory of Of eq. (18) the coefficient of nuclear Surface the semi-empirical formula has not taken into Energy can be simplified account the totality of the phenomena or the ( ) forces that are at play in the constitution of the = / / 34 (3) (4) nucleus.

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And given that = 17.3447975 MeV The coefficients that were most representative were selected and were tried in the calculation of = 1.42487843810 = 1.42487843810 (35) the masses of the 237 stable isotopes. To verify the reproducibility ability, the absolute differences between the experimental values and Arya [7] reported the coefficient of nuclear 10 theoretical or calculated ones were determined, surface energy of the order 10 T/mm. Given among the associated standard deviation. The that 1 = 9.8 the previous quantity is equal to results were compared to the calculations made 1.4510 . utilizing the coefficients and Arya’s semi- empirical formula and that of Evans. It was observed that the set of coefficients that Of eq. (12) one can simplify , the constant of proportionality between the energy of the volume corresponded to the isotopes ,, and and the volume produced the lowest standard deviation between the experimental values and the calculated values, in comparison to the standard = (12) deviations that were obtained with the coefficients of Arya and Evans. Given that = 15.5933955 MeV Additionally, the deduction of the formula gave evidence to the independent character of three of = 3.08478261210 (36) the five coefficients of the formula, , and , according to what is shown by the eqs. (12, 18, This establishes the relationship between the and 36). The three contain the factor of energy of the volume and the nuclear volume. proportion between the volume and the mass number, . Presented is the form of calculation Of the eqs. (14 and 15) the nuclear density is for , and once evaluated, the coefficient of surface energy and the constant of proportionality of the volume energy is = (37) 4 determined. These coefficients are not known in 3 literature, one form of verifying their coherence is

the evaluation of the unit radius, , and the If the unit nuclear radius is associated with nuclear density, , and to compare them with the = 1.24579322710 to a unit of atomic corresponding values that are reported in mass = 1 = 1.6605387310 , the literature. nuclear density results in One of the benefits of this paper was to make = 2.0503240410 explicit a form of calculation of the coefficients, which is also not published in literature. This = 2.0503240410 = paper explains the theoretic fundamentals of the semi-empirical mass formula. = 2.0503240410 (38) Nevertheless, it cannot be concluded that the model is finished. The found coefficients Arya reported a nuclear density of the order 105 3 although similar, form a part of many attempts to T/mm . justify the semi-empirical formula. The order of dispersion of the difference indicates that there 5. CONCLUSIONS must be factors that are not taken into account in the base theories of the semi-empirical mass In this paper the theoretical fundamentals of the formula. For now, the semi-empirical mass semi-empirical mass formula are reviewed. formula is only an approximation.

Five sets of coefficients were determined from COMPETING INTERESTS the semi-empirical formula, which, with the exception of one, are similar to some of the 14 Authors have declared that no competing groups of coefficients found in literature. interests exist.

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