TIME REGION PARTICLE DYNAMICS Mr. Larson has worked. out the static relations between particles in the time region; specifically, he has calculated the equilibrium interatomic distances for all the elements and many compounds (see pages 27-49 of The Structure of the Physlcal Universe). This paper will explore he dynamic relations between particles in the time region. Consider a particle (say an alpha particle) moving directly towards a stationary atom (say a gold atom fixed in thin foil). Initially the particle has a velocity v_{o}. Once it enters the tlme region, that is, when its distance is less than one natural unit of space, two forces are encountered: the progression and gravitation. In the time region, the progression acts to bring particles closer together, whereas gravitation acts to repel particles — the reverse of gravitation in the time-space region. The progression is stronger until the equilibrium distance is reached, then the gravitational force becomes stronger. I believe that the equation of motion is
where
Dividing by m gives
The right hand side reduces to
Thus
Separating variables and integrating, we have
or
There are two cases of interest with this equation. Case 1: Suppose we want to know the initial velocity required to bring the particles to a certain distance apart from each other. Equation (3) is solved for v_{o}, letting v_{f }be zero.
Case 2: Suppose we want to know the final separation between two particles, given v_{o}. Let
Equation (3) becomes
With v_{f} = 0, and putting the terms irevolving x_{sep} on one side a_{f} the equation, we have
Define the following coefficients:
With these coefficients, the result is a quartic equation:
This equation can then be solved by the usual means. Now, going back to equation (3) we can solve for v as a function of x:
Separating variables and integrating we have
The integral can be evaluated numerically by Romberg’s method. Example Consider an alpha particle moving directly towards a gold atom in a foil, at an initial velocity of 2.06x10^{7} meters/sec. What is the distance of closest approach? How long does it take to get there? What happens afterward.?
Now,
For gold, t_{A} =4.5; for helium, t_{B} = 3. But helium has only one active dimension so the force is multiplied by 1/3. Thus
(This assumes that since helium is inert, the electric displacements of gold have no bearing on the motion). The coefficients are next calculated:
The quartic equation is
The only physical solution is
Note that his is cons:lderably greater than that predicted by use of classical atomic theory and Coulomb’s law: 2.581 x 10^{14} m.
The average velocity of the particle to the point of closest approach is
The initial velocity having been dissipated, the particle goes back to the equilibrium point. Of course, at room temperature, helium is a gas, and so the particle would not remain in the time region! Situations in which the particle is not moving directly towards the atom will be treated in a future paper. |