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The Semiempirical Formula for Atomic Masses

Prout's Rule

In the early years of the nineteenth century when atomic theory was being developed an English physician named William Prout became interested in measuring the relative masses of different types of atoms. Although he was a practising physician he found time indulge his interest in chemistry. Amedio Avodadro had asserted that equal volumes of gases under equal pressure and temperature would have equal numbers of atoms (or molecules). Therefore one could determine the relative masses of atoms by comparing the weights of equal volumes of gases. Prout did measure these relative masses of gaseous elements and made a striking discovery which he announced in 1815. Most atoms had masses that are approximately equal to an integral number of hydrogen atoms. Prout then hypothesized that all atoms were composed of hydrogen atoms. Later it was found that Prout's rule did not hold for some elements, notably chlorine.

Sir William Crookes proposed in 1871 a resolution for the discrepancies to Prout's rule; i.e., some elements are composed of atoms of different masses and the meaured atomic weights are weighted averages of atoms which obey Prout's Rule. When Francis Aston developed the mass spectrograph and confirmed Crookes' conjecture of the existence of atomic isotopes he also substantiated Prout's rule in its approximate form.

When Prout's Rule was tested more rigorously it was found that the masses of all elements fell short of the mass they would have if they were composed of an integral number of hydrogen atoms. By this time the electron had been identified and Rutherford had discovered the concentration of the mass of an atom in its nucleus. Rutherford conjectured the existence of the neutron in 1920 and its existence was confirmed by James Chadwick in 1932. This naturally led to the notion that a nucleus is composed of integral numbers of protons and neutrons.

The nuclei of atoms have a proton number corresponding to their positive charge. Nuclei have masses which are represented as relative to a standard mass, now (1/12) of the mass of the 12C isotope of carbon. The atomic masses are close to but not generally equal to integers. These integers are called the mass numbers of the nuclei. The difference between the mass number of an isotope and its proton number is called the neutron number because of the strong suspicion that the nuclei are composed of protons and neutrons.

The modern version of Prout's Rule is that the mass M of a nucleus of proton number P and neutron number N is approximately equal to sum of the masses of its constituent protons and neutrons; i.e.,

M(P,N) = mpP + mnN

where mp and mn are the masses of the proton and neutron, respectively.

But the above formula is only an approximation. The actual masses differ from this approximation in a systematic fashion.

The difference between the actual and the approximation is called the mass deficit or binding energy B(P,N); i.e.,

B(P,N) = mpP + mnN - M(P,N).

Usually the mass deficit is expressed in terms of energy units (Mev). It is significant that the binding energy for stable nuclei is always positive.

Some Strictly Empirical Considerations

At the simplest level the binding energy might be conjectured to be proportional to the numbers of protons and neutrons. The results of the regression of binding energy on the number of protons, P, and the number of neutrons, N, with no constant term based on the data set for 2932 isotopes is:

B(P,N) = 10.53476P + 6.01794N
         (0.135)           (0.095)
         [145.5]            [63.3]
R² = 0.99

The regression coefficient of 10.53476 for P indicates that for every additional proton added to the nucleus, on average the binding energy increases by 10.53476 MeV. For an additional neutron there is the lesser amount of 6.01794 MeV. The numbers in parentheses below the regression coefficents are the standard deviation of the regression coefficient estimate. The numbers in the square brackets below the standard deviations are the ratios of the coefficients to their standard deviations, the so-called t-ratios. For a coefficient to statistically significantly different from zero at the 95 percent level of confidence the t-ratio needs to be of a magnitude on the order of 2 or higher. The t-ratios for the regression coefficents indicate that they are statistically highly significantly different from zero. The coefficient of determination, R², indicates that 99 percent of the variation in binding energies of the nuclei is explained by the variations in their proton and neutron numbers.

The above regression is strictly linear. The statistical fit can be improved by including quadratic terms. Such a regression gives the following results.

B(P,N) = 8.004962P + 9.23300N + 0.000163P²
- 0.00768PN - 0.0000155N²
R² = 0.999

The standard deviations of the coefficients are not given, but they indicate that all of the coefficients are highly significant statistically except for the one for P². It is notable that 99.9 percent of the variation in binding energies of the isotope nuclei is explained by the variation in their proton and neutron numbers.

The coefficients for P and N are not much different in the above regression and this suggest that perhaps the only significant variable is the total number of nucleons, A=P+N. And perhaps the binding energy for a nucleus might simply be proportional to A, the number of nucleons or mass number. For more on the statistical analysis of binding energies in terms of proton and neutron numbers see the enigma of mass deficits.

The general nature of the plot of the ratio of the binding energy B to the mass number A=P+N is shown in the graph below. Beyond A=40 the curve is very close to a straight line with a negative slope. The deviation between the actual values and this straight line is inversely proportional to A or some power of A. The binding energy ratio might be approximated by a function of the form:

B(P,N)/A = c0 - c1A - c2(1/Aε)

When ε is taken to be 1/3 for reasons given later and B, the binding energy, is measured in Mev a regression of the 2930 atomic masses yields the following:

B/A = 1.343 - 0.014A - 166.7A-1/3
         (0.057) (0.000162) (0.173)
R2 = 0.76378

Again, the number in parentheses under a coefficient is its standard deviation. The coefficient of determination, R2, is good but there is still about 24 percent of the variation in B/A that is not explained by the variables in the equation.

If the proton number P and the neutron number N are used separately in the equation instead of being combined into the mass number A the regression result is:

B/A = 13.650 - 0.020P - 0.010N - 16.93A-1/3
     (0.00060)    (0.00107)    (0.00065)     (0.176)
R2 = 0.76681

The coefficients of P and N are different and the difference is probably statistically significant at the 95 percent level of confidence but a precise determination cannot be carried out without an estimate of the covariance of the estimates of the two coefficients.

Statistical Results Using the Binding Energy B Rather the Ratio B/A

If the equation for B/A in terms of A is multiplied by A the result is:

B(P,N) = c0A - c1A2 - c2A1-ε.

If the masses of the protons and neutrons are added to this equation the result is:

M(P,N) = mpP + mnN + c0A - c1A2 - c2A1-ε.

Consider what the formula would imply for a non-atom, P=0 and N=0. If ε=1 the formula would imply that M(0,0) = -c2. On the other hand, if ε < 1 then M(0,0)=0.

If B(P,N) is regressed on P, N, P2, N2, PN and A1-ε, for some exogenously given ε the equation

B(P,N) = c0A - c1A2 - c2A1-ε.
would imply that
B(P,N) = c0P + c0N - c1P2 - 2c1PN - c1N2 - c2A1-ε.

That is to say, if the proper variable is A, the number of nucleons, rather than P and N separately the statistical results should show:

If these predictions are borne out then the equation can be re-estimated using A and A2 instead of P, N etc. The cube root of A, A1/3, is proportional to the radius of the nucleus and A2/3 is then proportional to the surface area of the nucleus and thus the number of nucleons exposed at that surface.

The results of the indicated regression are

B = 16.115P + 12.649N
- 0.0729P2 - 0.0135PN + 0.000001N2 - 18.1106A2/3

R2 = 0.9995

The regression coefficients are all highly significant statistically. The predictions do not appear to be borne out by the regression although a precise determination would require more complete statistics that are given by the regression program used (Excel). The differing signs for the coefficients for P² and N² weigh against a dependence on (P+N) or (P-N).

It is worthwhile to look at the effect of incremental changes in P and N as ∂B/∂P and ∂B/∂N. These are:

∂B/∂P = 16.115 - 0.1458P - 0.0135N - 12.7A-1/3
∂B/∂N = 12.649 - 0.000003N - 0.0135P - 12.7A-1/3

For P=1 and N=1 these evaluate to

∂B/∂P = 5.87 MeV
∂B/∂N = 10.30 MeV

The regression of B on A, A2 and A2/3 is:

B = 11.81354A - 0.010655A2 - 12.7A2/3

R2 = 0.999164

The incremental effect of an increase in A is given by

∂B/∂A = 11.81354 - 0.02131A - 7.1799A-1/3
which for A=2 is equal to 6.09 MeV.

The Semi-Empirical Formula

The semi-empirical approach was first formulated by Carl-Friedrich von Weizsäcker in 1935 based upon the liquid drop model of the nucleus. Hans Bethe and R. Bacher simplified von Weizsacker formulation in 1936. E. Wigner extended the formula in 1937. This approach explains the difference between the mass of a nucleus and the mass of its constituent protons and neutrons, the binding energy, as being the result of energies associated the interaction of the nucleons. These adjustment terms are as follows:

The functional form of the terms is based upon theory and the magnitudes of the coefficients aV, aS, aC, aa and ap are found statistically. They are chosen to give the best fit of the formula to the data.

The semi-empirical formula is thus:

B(P,N) = aVA - aSA2/3 - aCP2/A1/3 - aa(N-P)2/A + apδA-3/4

The regression estimates of the coefficients are:

B(P,N) = 15.282A - 16.060A2/3 - 0.6876P2/A1/3
- 22.409(N-P)2/A - 16.738δA-3/4

R2 = 0.99993


A German physicist, Haro von Buttlar, made some rough approximation of these terms from theoretical principles and concluded that:

The volume term coefficient and the surface term coefficient are not independent. M.A. Preston derives the relationship that should prevail if the nuclear force is zero beyond a distance of rn. This relationship is:

aS = - (9/16)(rn/r0)aV

where r0 is the coefficient in the expression for nuclear radius in terms of the cube root of the mass number; i.e., rA = r0A1/3. The value of r0 is about 1.2 fm.

The above estimates of aS and rV indicate that (9/16)(rn/r0) should be 1.051 and thus rn should be 2.24 fm. This implies that the value of A which would make the radius of the nucleus equal to the range of the nuclear force is about 6.5. At a value of A=6 or less the nuclear force of one nucleon would affect all of the other nucleons in the nucleus, but beyond a value of A=6 some nucleons would be unaffected by some of the other nucleons in the nucleus. Thus the nuclear force would reach saturation at A=6.

Modifications of the Semi-Empirical Mass Formula

The volume term in the semi-empirical formula represents the correction in mass for the energy due to interactions among the nucleons in a nucleus. For A nucleons the number of interactions would be A(A-1)/2. These presumes that a nucleon interacts with all other nucleons excepts itself. (It seems that quantum mechanically a nucleon may interact with itself so the number of possible interactions may in fact be A2/2 instead of A(A-1)/2.) But not all of the possible interactions occur. There is said to be a saturation of the nuclear forces. This saturation could result from a particle only interacting with its nearest neighbors or its nearest and next nearest neighbors. This limited range of interaction could be from a shielding of the force or it could come from a restricted range for the nuclear force.

On the basis of the above considerations, the saturation-nonsaturation of the nuclear force and the non-self-interaction, the volume and surface terms would be of the form:

a'VAAI and a'SAAI
AI = A-1 if (A-1)< Asat
AI = Asat if (A-1)≥ Asat

where Asat is the saturation level of interactions for the nuclear force.

Graphically the interaction term for the volume term in the formula is as shown below:

The regression estimates of the coefficients for this model when the saturation level is 10 nucleons are:

B(P,N) = 1.521AAI - 1.585A2/3AI - 0.6831P2/A1/3
- 22.240(N-P)2/A - 15.592δA-3/4

R2 = 0.999915

With a saturation level of 20 nucleons the regression estimates are:

B(P,N) = 0.73884AAI - 0.72336A2/3AI - 0.6546P2/A1/3
- 21.208(N-P)2/A - 15.852δA-3/4

R2 = 0.999786

With a saturation level of 6 nucleons the regression estimates are:

B(P,N) = 2.5447AAI - 2.6698A2/3AI - 0.68665P2/A1/3
> - 22.3727(N-P)2/A - 15.0547δA-3/4

R2 = 0.999932

The statistical performance of the models are virtually the same, but the coefficient of determination, R2, is higher the lower the saturation level. The traditional formula in not taking into account a lack of saturation for low levels of A, in effect, makes the saturation level one nucleon.

The concerns about non-self-interaction of nucleons in the volume and surface terms also apply to the protons for the Coulomb term. The functional form for the Coulomb term was derived from considering the energy required to bring P units of charge to a sphere of radius r in infinitesimal increments. This energy is proportional to P2/r. But adding (P-1) units of charge to an existing unit of charge would give a term proportional to P(P-1). The traditional formula for the Coulomb term presumes the charge would be concentrated on the surface of a sphere whereas experimental investigations of the distribution of charge indicates a distribution throughout the body of the nucleus.

The use of P2 rather than P(P-1) is justified on the basis of the self-energy of the proton. But the use of P2 rather than P(P-1) on the basis of the self-energy of the proton is not valid in explaining binding energy because any self-energy of the proton is accounted for in the masses of the protons which are subtracted in the computation of binding energy. This argument also applies to the use A(A-1) versus A2 for the volume and surface terms; i.e., any self-interaction energy is included in the masses of the nucleons and is thus allowed for in the computation of the binding energies.

The regression estimates when P(P-1) is used instead of P2 are as follows:

B(P,N) = 1.524AAI - 1.625A2/3AI - 0.6854P(P-1)/A1/3
- 22.000000558(N-P)2/A - 15.736δA-3/4

R2 = 0.999915

This regression is based upon a saturation level for the nuclear force of 10 nucleons. The statistical estimates are only slightly different from the one based up a Coulomb term involving P2 rather than P(P-1).

The statistical results are slightly better for a saturation level of 6; i.e.,

B(P,N) = 2.549AAI - 2.736A2/3AI - 0.6888P(P-1)/A1/3
- 22.111111885(N-P)2/A - 15.199δA-3/4

R2 = 0.999931

The preponderance of the error in the estimates of the bindingg energy is for the light nuclei and the very large errors are for special cases such as 4He. This reflects some sort of internal shell structure. These phenomena are shown in the plot of the B/A ratios versus A for the lighter nuclei. The differences between the actual binding energy and that calculated from the regression equations seems to be related to an internal shell structure of the nuclei.

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