THE
INTERACTION OF ALPHA PARTICLES Introduction Nearly all presentday physicists are convinced of the truth of the assertion in the following quotation from Weidner and Sells’ Elementary Modern Physics^{(1)}:
However, when we study the literature of Rutherford’s era, we find that he and his associates, Geiger and Marsden, did not in fact discover the atomic nucleus. Geiger and Marsden’s paper, ”The Laws of Deflexion of a Particles through Large Angles, ”^{(2)} does present strong experimental evidence of a central repulsive force originating from atoms, but the paper does not prove that this force is electrical in nature. What their experiments did prove is that the number of particles scattered through an angle q is proportional to 1/sin^{4} (q/2) and to the inverse of the square of the kinetic energy of the particles, 1/E_{k}^{2}. Of course, the experiments did disprove the Thomson ”plum pudding ” atom model, which did not predict a strong central repulsive force; but it is one thing to disprove a theory; it is quite another to prove one. If the Rutherford model were the only alternative left, we might have to conclude that it is correctÑbut there are always other alternatives. This paper will present one such alternative: the Reciprocal System of physical theory. A new scattering equation will be derived and compared with the experimental facts as found in an uptodate version of the original experiment, that conducted by Prof. Adrian C. Melissinos and his students.^{(3)} The originator of the Reciprocal System is Dewey B. Larson; for full comprehension of this paper, the reader should first study Larson’s books.^{(4,5,6)} I. Theory A. The Repulsion Force: F 1. The Reciprocal System In the Reciprocal System, nonionized and nonmagnetized matter is subject to only two primary forces: the spacetime progression and gravitation. In the timespace region, the progression is outward and gravitation inward, whereas in the time region (inside unit space) the progression is inward and gravitation outward  a repulsion. Right at the boundary the progression is zero, but the net gravitation is not zero. Compared with the repulsive gravitational force, the attractive gravitational force is negligible. Thus at the boundary only the repulsive gravitational force is effective. Now consider what happens when an atom A, which is moving towards an atom B, reaches this boundary. According to Larson,^{(6)}
A previous paper of mine,
”Time Region Particle Dynamics, ”^{(7)} dealt with the situation
in which atom A (say an alpha particle) is assumed to continue
to move directly toward atom B (say a gold atom) in the time region. This
paper will consider the general case in which no assumption is made as
to the actual motion that takes place in the equivalent space
of the time region. All that will be considered here is the equivalent
motion that takes place at the boundary, i.e., in actual
space. Here, with the atoms separated by s_{o}, the repulsive
gravitational force F is (from Ref. 4)
The repulsion coefficient
K_{G} is expressed by
where F_{p} is the natural unit of force and the number 156.44 is the interregional ratio. The dimensionless variables t_{eff} and t’_{eff} are material constants determined by the characteristics of the interacting particles, and will be discussed further later. 2. Conventional Theory In present theory, the alpha
particles somehow avoid interacting with the cloud of electrons supposedly
surrounding each gold nucleus. The only force involved comes from the
presumed nucleus. At low energies this is a Coulombic force, given by
where z is the atomic number of the alpha particule (helium), Z is the atomic number of gold, e is the value of electric charge, e_{o} is the permittivity constant of free space, and s is the separation distance. This is an inverse square force, rather than an inverse quartic force as in the Reciprocal System. Why this Coulombic force should act between particles but not within nuclei is a question completely unanswered by current theory. B. The Impact Parameter: b
The figure shows a typical collision process. The impact parameter is the distance that the alpha particle would have passed the gold atom if there had been no force between them. 1. Reciprocal System Let m be the mass of the alpha particle and v_{o} be its initial velocity. Referring to the figure, we have
The alpha particle passes
from the timespace region, through the time region, and back into the
timespace region. For the general case, we cannot write an equation for
the actual motion in the equivalent space of the time region, but we can
write an equation for the equivalent motion in the actual space of the
timespace region. Throughout this motion, neither the angular momentum,
nor the actual spatial
separation, changes. But the angle f does change.
Therefore,
Separating dt from df in eq. (5) and substituting in eq. (4), we have
Since the kinetic energy, E_{k}, is ^{(1}/_{2}) mv_{o}^{²}, eq. (6a) can be rewritten as
Before the collision,
f = 0 and after the collision, f = pq,
so these are the limits on the integral.
Solving for b we finally
obtain
2. Conventional Theory Refs. 1, 3, and 8 all have
derivations for b in current theory. The result is
C. Target CrossSection: s The target Cross — Section is defined as s = pb^{²}. 1. The Reciprocal System From eq. (7c), using the
above, we obtain
2. Conventional Theory With b from eq. (8),
D. Differential CrossSection: ds The differential crosssection is ds = 2 p bdb 1. Reciprocal
System
(with the minus sign dropped). The angles q
and q + dq define
two cones with the horizontal line through the gold atoms as their axis.
The differential solid angle dW between the
two cones is
Since the cosine termin
eq. (12) can be expressed as
eq. (12) becomes
The units of ds/dW are meter squared per steradian, m^{²}/sr. 2. Conventional Theory Similarly, for conventional
theory,
E. The Scattering Constant: K_{s} It is immediately seen from
eqs. (13) and (14) that for both
a constant. 1. Reciprocal System Here,
2. Conventional Theory Here,
F. The Number of Particles Scattered per Minute at the angle q: I_{s} So far we have looked at the situation involving only one alpha particle and only one gold atom. For the situation in which a beam of alpha particles strikes a gold foil, we would like to compute the number of particles scattered through a certain angle q. Let
The value of N is determined
from this equation:
where
Then the number per minute,
I_{s}, of alpha particles scattered into the detector at the angle
q is
1. Reciprocal System Here,
2. Conventional Theory Here,
Note: Both equations are based on the assumption that E is sufficiently low such that ”relativistic ” effects can be neglected and such that the gold atoms remain stationary during the interaction  this would not be the case at very high alpha particle energies. II. Experiment A. The Experimental Setup Professor Adrian C. Melissinos
carried out a modern version of the original GeigerMarsden experiment
and described his findings in Ref. 3. In this experiment
[da is differential area of
detector and L is distance of detector from foil]
Thus,
So the equation for I_{s}
is
I_{s} is measured, and then (ds/dW)is computed. For each angle q the product (ds/dW) * sin^{4}(q/2) can be found and the results plotted. From leastsquares error analysis, the best fit experimental value of K_{s} can be obtained. Melissinos states that the
above value of I_{o}, and hence also I_{s}, is subject
to at least a ± 20 percent
error in view of the approximations used and the nonuniformities in beam
density and direction. One other uncertainty is the energy of the alpha
particles. The incoming energy is 5.2 MeV, but since the particles lose
a considerable amount of energy in traversing the target, Melissinos believes
that it is more appropriate to use a mean value of E_{k} for the
calculations. He calculates the mean value to be
B. Calculation of K_{s} for the Experimental Setup 1. Reciprocal System Here,
The difficult part in calculating
K_{eff }for gold helium. The
(tentative) method used here will be different from that used in my previous
paper, Ref. 7. Gold has 3 active rotational dimensions with t = 4.5 in
all (see Ref. 4 for more details). Helium has only 1 active dimension,
with t = 3. The other two dimensions have t = 1. In the first dimension
the mean value for gold and helium is
In the other two dimensions,
the full rotational force of the gold atom is present, so instead of (4.5
* 1)^{1/2} = 2.1, we simply have t = 4.5. The mean over all three
dimensions is
Thus (tentatively),
The (tentative) value of K_{s} is then K_{s} = (1.00 * 10^{40})^{2}/[16 * (4.558816 * 10^{8})^{4} * (7.03 * 10^{13})^{2}] K_{s} = (2.93 * 10^{28} (m^{2}/sr) 2. Conventional Theory Here,
Thus,
3. Experiment The best fit experimental
value, according to Melissinos, is
Thus the theoretical and
experimental results can be summarized as follows:
Appraisal It is well known that Rutherford’s
theory of scattering fails at high energy. On the basis of Melissinos’
experiment, we must also reject this theory at low energy. But, given
the present climate of thought, Melissinos himself could not come to this
conclusion. He says
However, given the close result of the Reciprocal System, it now appears that Melissinos is too modest. Even with some uncertainty in the theoretical value of K_{G} and the experimental values of I_{o} and E_{k}, it appears that the Reciprocal System is consistent with the observed result, whereas conventional theory is not. Can an appeal to more sophisticated mathematics rescue the current theory? It cannot. From Ref. 1 (p. 223) we have this statement:
Thus, despite the thousands of books, the thousands of papers, and the thousands of lectures on the nuclear theory of the atom, the physicists are going to have to discard their cherished concept. The scattering equation of the Reciprocal System will next have to be applied to other pairs of incident and target particles and to other energy levels. References
