San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

A Generalization of the Theorem that the Second Difference
of the Interaction Energies of Nuclides with Respect to the
Number of Particles of a Particular Type Is Equal to the
Interaction Energy of the Last Particle of that Type with
the Next-to-Last Particle of the Same Type in the Nuclide

There is one theorem to the effect that in systems, such as nuclei, made up of two or more types of particles, such as neutrons and protons, the cross diffference in the sum of interaction energies is equal to the energy of the interaction of the last particle of one type with the last particle of another type. This is a solid and useful theorem for investigating the binding energies of nuclides and thus nuclear structure.

There is another useful theorem to the effect that in systems, such as nuclei, made up of one or more types of particles the second diffference with respect to the number of one type of particle in the sum of interaction energies is equal to the energy for the interaction of the last particle of that type with the next-to-the-last particle of the same type. What follows is a proof of that proposition subject to an additional crucial assumption.. It will apply to systems made up of particles which may be composite. For example, nuclei are ultimately composed of neutrons and protons but they may also be consideredd to be made up of neutron-neutron spin pairs, proton-proton spin pairs and neutron-proton spin pairs. These composite particles have binding energies associated with them due to their formation. Such energies will be referred to as the intrinsic interaction energies of the particles.

The situation presumed is that of a system made up of two types of particles, referred to as a-particles and n-particles. The energy of the system is the sum of the energies due to the interaction of the particles plus the intrinsic energies of the particles.

It is also assumed that there are shells for the particles. When a shell is filled any additional particles must go into higher shells where their interaction energy is lower with the particles in the lower shells.

Under the crucial assumption that the interaction energy is the same for all a-particles in the same a-particle shell, then the proposition is:

The increase in the incremental interaction energy of an a-particle as a result of an increase in the number of a-particles is equal to the interaction energy of the last a-particle with the next-to-last a-particle, provided these two are in the same a-particle shell.

The n-particles and their interaction with the a-particles do not matter. They are included to show that they do not matter and hence any number of other types of particles also would not matter.

Visual Demonstration

Consider a system such as a nucleus with a a-particles and n n-particles. The interaction energy of that system represents the net sum of the interaction energies of all a a-particles with each other, all n n-particles with each other and all mn interactions of the a a-particles with the n n-particles. Below is a schematic depiction of the interactions.

  

The black squares are to indicate that there is no interaction of a particle with itself.

What is given below is the interactions for a system of a a-particles and n n-particles overlaid with those of a system with (a-1) a-particles and n n-particles, shown in color.

  

The incremental interaction energy of an a-particle is the difference in the interaction energy of the system with a a-particles and n n-particles and that of the system with (a-1) a-particles and n n-particles. When the subtraction is carried out the interactions of the n n-particles with each other are entirely eliminated. It also eliminates the interactions of the (a-1) a-particles with each other and the (a-1) a-particles with the n n-particles. The interactions which are left after the subtraction are the squares shown in white above.

Now consider the incremental interaction energy for the nuclide with (a-1) a-particles and n n-particles. These interactions are shown in color along with those for the n n-particles in the nuclide with a a-particles and n n-particles shown in white.

 

Now consider the difference of the incremental interaction energy for a-th a-particle and n-th n-particle and the incremental interaction energy for the (a-1)-th a-particle and n-th n-particle.

 

The subtraction of the incremental interaction energy for (a-1) a-particles and n n-particles from theincremental interaction energy for a a-particles and n n-particles depends upon the magnitude of the interaction of the (a-1)-th a-particles with the different a-particles compared to the interaction of the a-th a-particle with those same a-particles. Visually this is the subtraction the values in the blue squares from the white squares on the same level. When the a-th and the (a-1)-th a-particles are in the same shell the magnitude of the interactions with any other a-particle or any n-particle are, to the first order of approximation, equal. Thus the interactions with the n n-particles are entirely eliminated. Likewise for the first (a-2) a-particles. All that is left is the interaction of the a-th a-particle with the (a-1)-th a-particle.

Algebraic Proof:

Let i and j be indices for the a-particles and k and l the indices for the n-particles. The energy of the interaction between the i-th and j-th a-particles is denoted as Iij and between the k-th and l-th n-particles as Jk l. The interaction energy between the i-th a-particle and the k-th n-particle is denoted as Kik. For now it is assumed that there are no intrinsic interactive energies for the particles. (One symbol for energy could have been used but using three makes it easier to see what is involved.) The values of Iij need be defined only for j<i to avoid double counting and noting there is no interaction of a particle with itself. The same applies for J and k and l. The same restriction is not applied to the interactions between the two types of particles given by Kik.

The interactive energy of the system is given by

E(a, n) = Σ i=1aΣj=1i-1Iij + Σ k=1nΣl=1j-1Jkl + Σ i=1aΣj=1nKik

The incremental interactive energy with respect to the number of a-particles is given by

ΔaE(a, n) = E(a, n) − E(a-1, n) = Σj=1a-1Iaj + Σj=1nKak

The second difference is the increment with respect to the number of a-particles of the increment in interactive energy with respect to the number of a-particles. The expression for the first difference for (a-1) and n is

ΔaE(a-1, n) = E(a-1, n) − E(a-2, n) = Σj=1a-2I(a-1) j + Σj=1nK(a-1) k

The second difference is then given by

Δ2E(a, n) = ΔaaE(a, n))
Δ2E(a, n) = Ia,(a-1) + Σj=1a-2[Iaj−I(a-1) j] + Σj=1n[Kak−K(a-1) k]

Under the assumption that

Iaj = I(a-1) j
and
Kak = K(a-1) k

the above reduces to

Δ2E(a, n) = Ia,(a-1)

That is to say, the second difference in interaction energy with respect to the number of a-particles is equal to the interaction of the last a-particle with the next-to-last a-particle.

The Effect of Intrinsic Energy

Now suppose each a-particle has intrinsic energy μ and each n-particle intrinsic enery ν. The total interactive energy H is then the interactive energy E given above and the total intrinsic energy:

H = E + mμ + nν

The increment in H with respect to the number of a-particles is then

ΔaH(a, n) = ΔaE(a, n) + μ

The subtraction of ΔaH(a-1, n) eliminates the intrinsic energy μ of an a-particle. Therefore

Δ2H(a, n) = Δ2E(a, n) = Ia,(a-1)

Thus the existence of intrinsic energy of the particles has no effect on the value of the second difference and thus on the second difference being equal to the interaction between the last a-particle and the next-to-last a-particle.

On the Matter of the Equality of the
Interaction of Particles in the Same Shell

There is no question but the theorem is true given the crucial assumption. The question is whether the crucial assumption is valid for the empirical world. Unfortunately there is no way of getting the value of the interaction of two particles of the same type except through the theorem that is presented in this webpage and that theorem presumes the equality of interaction terms in the same particle shell. However it is still possible to check for consistency.

One of the previously displayed diagram will help conceptualize the quantities involved.

The theorem says that the values in the red squares should be the second differences. The crucial assumption of the theorem is that the elements in the last two columns on the right are equal. The theorem works just as well if the column sums on the right are equal.

The relative values in the red and green squares for each row are relevant for the values in the last two columns. The transition from one red square to the one in the next higher row can be considered a movement to the green square above it and then a movement to the red square on the left. The change in energy between the green square and the red square to its left is about one half of the change from one red square to the another. The change in the binding energy from one red square to another is the third difference in the binding energy computed within a shell. The sum of the differences is just the difference in the value of the second difference at the beginning and end of the shell.

The Second Differences of the
Interaction Energies for the Alpha Nuclides

In nuclei when ever possible neutrons and protons form spin pairs. But the formation of spin pairs is exclusive in the sense that one neutron can form a spin pair with only one other neutron and with one proton. The same applies for protons. Thus a nucleus is composed of chains involving modules of the form -n-p-p-n-, or equivalently -p-n-n-p-. The binding energies of the nuclides which consist entirely of alpha modules can be compiled. These can be called the alpha nuclides. From these the incremental binding energies of alpha modules can be computed.

Here are the incremental binding energies and second differences of binding energies with respect to the number of alpha modules in nuclides that could be composed entirely of alpha modules.

The first (incremental) and second differences in the Binding Energy
with respect to the number alpha modules in nuclides which could
be composed entirely of alpha modules
Number of
Alpha Modules
Incremetal
Binding Energy
Second
Difference
in Binding Energy
1 28.295674
2 28.203836 -0.091838
3 35.662218 7.458382
4 35.457608 -0.20461
5 33.025523 -2.432085
6 37.612031 4.586508
7 38.28 0.667969
8 35.24377 -3.03623
9 34.93504 -0.30873
10 35.3363 0.40126
11 33.4227 -1.9136
12 35.9873 2.5646
13 36.235 0.2477
14 36.291 0.056
15 31.004 -5.287
16 30.958 -0.046
17 30.45 -0.508
18 30.7 0.25
19 31 0.3
20 31.7 0.7
21 31.1 -0.6
22 30.5 -0.6
23 30.7 0.2

The first difference is nearly constant over the shell that extends for 15 alpha modules to 23. It just happens that the change in the second difference over this range is zero. In this case the crucial assumption of the theorem is vindicated.

The second difference can also be identified with the slope of the relation between incremental interaction energy and the number of the particles of the same type in the nuclide. Here is an example of three of those types of relationships.

The relationships are notably linear, notably parallel and notably evenly spaced. This lends credence to the proposition that the interaction between the m-th particle and the j-th particle is the same as the interaction between the (a-1)-th particle and the j-th particle for all j between j=1 and j=a -2. There is an exhaustive presentation of such relationships at neutrons and protons

Conclusion

The second difference of the interaction energies of nuclides with respect to the number of particlesof a particular type Is equal to the interaction energy of the last particle of that type with the next-to-last particle of the same type in the nuclide.


HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins