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The Functional Form of the Relationship Between
the Incremental Binding Energy of Neutrons and
the Net Attractive Force They Are Subject To

In the Bohr model of hydrogen-like atoms and ions the ionization energy of an outer electron is proportional to to Z², where Z is the net positive charge; i.e., the number of protons in the nucleus less the shielding by electrons in inner shells. The implications of this model are amply born out, as witnessed by the graph of ionization energy versus the the number of protons in the nuclei.

The Bohr model works equally well for multiple electrons in the outer shell as illustrated in the the following graph.

Extention of the Model to Nuclear Structure

The Bohr model is based upon the force acting upon an electron being strictly an inverse distance squared law. The nuclear strong force does not obey such a law. Here it will be presumed that the strong force between two nucleons takes the form

F = −H*exp(−r/r0)/r²

where r is the distance between the centers of the nucleons. H and r0 are parameters of the force. This specification of the nuclear force is based upon it being carried by particles which decay over time and hence over distance. The exponential term is the proportion of the force-carrying particles which survive after traveling a distance r.

The General Case Analysis

Rather than imposing immediately the above formula let us consider the case of a general central force between a central cluster with a net attraction equal to Z nucleons. The formula for such a force can be represented as

F = −HZf(−r/r0)/r²

where f(x)≤1 for all x≥0.

As in the Bohr model it is assumed that the mass of the central cluster is infinitely greater than that of the outer neutron. Therefore the outer neutron travels in an orbit of radius r, which is the same as the distance between the center of the central cluster and outer neutron. Let ω be the angular rate of rotation of the outer neutron and m be its mass. The angular momentum of the outer neutron is then m(ωr)r and its quantization means that

mωr² = nh
and thus
ω = nh/(mr²)
and, for later use,
ω² = n²h²/(m²r4)

where n is an integer and h is Planck's constant divided by 2π.

A balance of the centrifugal force by the attractive force of the central cluster requires that

m(ωr)²/r = mω²r = HZf(r/r0)/r²
which implies that
ω² = HZf(r/r0)/(mr³)

Equating the two expressions derived above for ω² gives

HZf(r/r0)/(mr³) = n²h²/(m²r4)
which reduces to
rf(r/r0) = n²h²/(mHZ)

This is the quantization condition for r. It can be put into a dimensionless form by dividing both sides by r0 and denoting r/r0 as ρ. Thus

ρf(ρ) = n²h²/(mr0HZ)
or, grouping the constants together,
ρf(ρ) = [h²/(mr0H)](n²/Z)

Let γ denote the quantity h²/(mr0H) so

ρf(ρ) = γ(n²/Z)

For given values of n and Z a value of ρ is determined. This value would be denoted as ρn(Z). For example, if ρf(ρ) ≅ αρ (that is to say f(ρ) ≅ α) then

ρn(Z) ≅ (γ/α)(n²/Z)

For this approximation it would follow that the potential energy is then proportional to Z². But for f(ρ)=exp(−ρ) the form of ρf(ρ) is

This form implies that there is a maximum quantum number n and a maximum value for ρ. The inverse relation for ρexp(−ρ)=λ takes the form

For the Yukawa potential function the force formula implies f(ρ)=(1+ρ)exp(−ρ). For this case the relation of ρf(ρ) would be

Thus a better approximation for ρf(ρ) would be αρ − βρ². For this case then

αρ − βρ² = λ
ρ = [α − (α²−4βλ)½]/(2β)
which can be reduced to
ρ = α[1 − (1 − 4βλ/α²)½/(2β)

For small values of λ,

(1 − 4βλ/α²)½ ≅ 1 − 2βλ/α²
and therefore
ρ ≅ [2αβλ/α²]/(2α)
which reduces to
ρ ≅ (β/α²)λ

The Potential Energy

The potential energy V of the system is a function of the separation distance r. However except for special cases such as the inverse distance squared forces there are no simple explicit formula for potential energy.

The formula for the potential energy for the general force case is

V(r) = −∫rHZ[f(s/r0)/s²]ds
which reduces to
V(r) = −HZ∫r[f(s/r0)/s²]ds
or, changing the variable of
integration to q=s/r0,
V(r) = −(HZ/r0)∫r[f(q)/q²]dq

The dependence of V on Z comes from the proportionality to Z and the dependence of r on Z. For the inverse distance squared law the integral is inversely proportional to r but r is inversely proportional to Z so the integral is proportional to Z and thus potential is proportional to Z².

For the Yukawa potential the functional relationship is as shown below

The relationship appears to be a power-law relationship. This is confirmed by plotting the logarithms of the two variables, as shown below.

The slope of this line is about 2.35 instead of 2.0 as in the case of the inverse distance squared law. Thus for the Yukawa potential

V ≅ νZ7/3

where ν is a constant.

The potential energy for the force formula HZ*exp(−r/r0)/r² can be approximated through numerical integration. The graph of the potential energy versus the net attractive charge is shown below.

In logarithmic form this relationship is

The slope of this line is about 1.89 instead of 2.0 as in the case of the inverse distance squared law. Thus for the potential

V ≅ νZ17/9

(To be continued.)

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