The Derivation of Estimates of the Parameters of the Nuclear Force Formula from the Best Estimate of the Separation Distance of the Centers of the Nucleons in a Deuteron

San José State University

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Thayer Watkins
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The Derivation of Estimates of the
Parameters of the Nuclear Force Formula
from the Best Estimate of the
Separation Distance of the Centers
of the Nucleons in a Deuteron

The separation distance of the proton and neutron in a deuteron provides a way of estimating the parameters in a formula for the nuclear strong force. It is the separation distance of the centers which is crucial. This quantity can be obtained by subtracting the sum of the radii of the proton and neutron from the diameter of the deuteron. There are various concepts of the diameters and radii of particles and nuclides. One is the root-mean-square (rms) charge radius. There are others, such as the mass radius and the magnetization radius, but so long as the same concept is used for the deuteron, proton and neutron the separation distance of the centers of the proton and neutron should be the same.

A group of physicists under the editorship of Savely G. Karshenboin published in 2008 a book devoted to the compilation of the best estimates of physical properties of particles, simple atoms and simple molecules (Precision Physics of Simple Atoms and Molecules, Springer-Verlag).

The best estimate of the rms-charge diameter of a deuteron from page 70 of the above mentioned work is 4.260 fermi with a margin of error of ±0.02 fermi. The recommended estimate of the rms-charge radius of the proton, given on page 49 of the above work, is 0.895 fermi. Precision Physics of Simple Atoms and Molecules does not give an estimate for the radius of the neutron. Another source gives the rms-radius of the neutron as 1.11 fermi.

Thus the separation distance of the centers of the nucleons is

s = 4.260−0.895−1.113=2.252 fermi.

Derivation of Estimates of the
Parameters of the Nuclear Strong Force

The nuclear force is assumed to be given by the formula

F = −He−s/s0/s²

where H and s0 are parameters to be estimated. A value of s0 can be derived from the Yukawa Relation and the mass of the pi mesons. Its value is 1.522 fermi. The reduced mass of the proton and neutron is denoted as μ and its value is 0.13288984×10−28 kg.

The separation distance of the nucleons is an element of the transcendental equation

s*e-s/s0 = σ
which can be put into
the more convenient
form for solution of
(s/s0)*e-s/s0 = σ/s0
or, more succinctly,
with z=s/s0
z*e-z = (σ/s0)

where σ is a natural unit of length given by

σ = h²/(μH)

For a known value of the separation distance s the value of z can be determined and from that σ is determined. From σ, H can be determined.

For a separation distance s=2.252 fermi, z is then equal to 2.252/1.522=1.4796. This means that

(σ/s0) = 1.4796e-1.4796=0.33694
and thus
σ = (0.336944)(1.522 fermi)= 0.5128 fermi.

Since σ = h²/(μH)

H = h²/(μσ)
= (1.11212132×10-68/8.368746×10−28)/σ
= 1.3288984×10-41
= 2.591310×10-26 kg*m³/sec²

Thus the nuclear force between two neutrons, or a protron and neutron, is given by

F = −He−s/s0/s²
 
where
 
H = 2.591310×10-26 kg*m³/sec²
and
s0 = 1.522 fermi


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