LECTURE 
20 January 2003 

Language consultation: J.B., Ontario  Canada. 

HYDROGEN ATOM  ELECTRON ON A FREEFALL RADIAL ORBIT


The mistake discovered in Bohr's understanding of the structure of the atom, discussed in my previous lecture, has opened the way to describing the atom in terms of the localized electron moving according to deterministic laws of classical dynamics along a well defined orbit. The fact that


electrons in the groundenergystate atom move radially to the nucleus


immediately raised the question:


"What is the behaviour of the electron in the close vicinity of the nucleus?".


In this lecture I will present the first part of the answer to the question (the second part will be given in the next lecture).
An electron is a spinning body. Formally, as the distance of the electron to the nucleus decreasing to zero the Coulomb energy Z e^{ 2} / r tends to infinity and the problem within classical electrostatics looses its physical sense. One must remember, however, that the electron is a spinning body and has a magnetic moment m = m, which is a source of spin magnetic field H_{ s} decreasing with the third power of the distance 

(1) 

Therefore, the motion of the electron close to the nucleus, as follows from AmperFaraday's electrodynamics, is strongly influenced by the force:


(2) 

where Z e is the charge of the nucleus and v is the orbital velocity of the electron. As a result, at a distance of the order of the Compton wavelength an electron radially approaching the nucleus rapidly changes direction of motion and starts to move back to the periphery of the atom.
Since spin magnetic force is a noncentral force, mathematical formalism describing behaviour of the electron in the Coulomb field of the nucleus should be extended to spin coordinates of the electron and Euler's equations of motion describing gyroscopic effects should be solved simultaneously with Newton equation of motion. However, becouse we are not interested in details of the problem, we can neglect small changes in orientation of the spin axis of the electron and assuming that the spin axis is firmly oriented in space to carry out the analysis within so called rigid top approximation. Within the rigid top approximation, the Langrangean describing the motion of the electron in the Coulomb field of the nucleus, developed according to standard rules of classical electrodynamics has the form: 

(3) 

If the Lagrangean of the problem is known, then by a simple differentiation procedure the equations of motion can be written explicitly (those are shown later, Eqs.10a,10b,10c). Performing the integration at various initial conditions we can search for a solution which may represent motion of the electron in the atom.
Radiola  a freefall equatorial orbit. In the particular case, of a planar motion, when the nucleus at all times remains in the equatorial plane of the electron, the problem has an analytical solution. This solution, given within the HamiltonJacobi formalism, has the form: 

(4) 

where E and a_{j} are two constants of motion. The constant E represents conservation of energy


(5) 
E = 1/2 m v_{0}^{2}  Z e^{ 2} / r_{ 0} = const ,


and the constant a_{ j} conservation of angular momentum


(6) 

Calculating the derivative dS / da_{ j} we obtain the trajectory equation:
j = f (r), and calculating the derivative dS / dE we obtain the relation between the position of the electron on the orbit and the time: t = f (r) . At zero initial conditions, that is when both constants of motion, E and a_{ j} , are equal zero, the trajectory equation calculated from Eq(4) has the form: 

(7) 

where


(8) 
r_{ min} = 2 a_{ 0} a ( Z a ) ^{1/3} .


The integral (7) can be effectively calculated and we obtain an extraordinarily simple and elegant relation:


(9) 
r = r_{ min} / cos ( 3/2 j ) ,


Thus, we have obtained a very exciting result
radial asymptotes of the trajectory are, independently of the charge of the nucleus (!), inclined to each other, by exactly 120^{o}.
This means that the electron starting at some distant point from the nucleus after three reflections from the nucleus, as


D j = 1/3 ( 2 p ),


comes back to a zero velocity starting point and the trajectory is closed, see Fig.1.


Fig. 1. Electron on a freefall equatorial orbit  spin of the electron is perpendicular to the plane of the orbit.


The above shown trajectory, a looking like hyperbola, with radial asymptotes, we will call radiola.
One can suppose that this particular solution of the problem, when the electron after each of the three reflections from the nucleus comes back to the starting point and the orbit is closed, represents the hydrogen atom in the ground energy state (and in all excited states with l = 0). To see an artistic view of the hydrogen atom, please click here. The freefall atomic model solves the puzzle of spatial quantization. Here, one can of course ask the question: what rights do we have to claim that the radiola represents the orbit of the electron in the ground energy state atom? Among varius arguments, which can be presented on the issue, there is the one of special interest. The atomic model presented above solves one of the most exciting puzzles of atomic physics  it gives the answer to the question, why magnetic moment of the atom with the unpaired electron is, if placed in the magnetic field, always oriented parallel or antiparallel to magnetic field lines. To explain the phenomenon it is satisfactory to solve a trivial academic problem of classical electrodynamics and answer the question: what is the behaviour of a metallic needle placed in a magnetic field, jointed at the end to a fixed point, with oscillatory current i = i_{o} sin wt . Quite trivial considerations show that the needle performing fast azimuthal oscillations influenced by the force: 

F_{ q} ~ (i_{o}/w)^{2} sin q cos q


will slowly change orientation trying to be
oriented perpendicularly to magnetic field lines (q is the angle between the magnetic field and the axis of the needle).


Quite analogously, a "thorny" atom with radially moving electrons tends to have, as much as it is possible, radial thorns oriented perpendicularly to the magnetic field lines. The hydrogen atom, therefore, changes orientation to have three radial segments of the orbit perpendicular to magnetic field lines. Since spin axis of the electron may be oriented in two ways with respect to the plane of the orbit, the atomic beam in non uniform magnetic field becomes splited into two partially polarised atomic beames, see Fig.2.


Hydrogen atom 
the region of magnetic field H 

a beam of chaotically oriented atoms 
atoms oriented in the magnetic field 
two partially polarised atomic beams 

Fig. 2. Three radial asymptotes try to be oriented perpendicularly to magnetic field lines. Spin of the electron, which is perpendicular to them may have only two orientations with respect to magnetic field lines. The observed splitting of the beam passing through the magnetic field is, therefore, a consequence of planar geometry of the hydrogen atom and its radial kinetics!


Nature is simple but not trivial. Although, the above considered planar motion of the spinning electron in the Coulomb field of nucleus shows the essence of the internal kinetics of the hydrogen atom, one must carry out a three dimensional analysis of the problem to learn more about the properties of this fundamental element of matter. Unfortunately, in this case equations of motion must be integrated numerically. Those in a spherical system of coordinates, with zaxis oriented along spin axis of the electron, have the form:


(10a) 

(10b) 

(10c) 

where q and j define angular position of the electron, r is a distance of the electron to the nucleus (r = r / r_{ min} ) and t is current time
(t = t / t_{ 1}, t_{ 1} = l / 2 p c ). Undertaking a numerical integration it is worthy to know that integrals of motion are known. Those respectively have the form: 

(11) 

(12) 

Results of numerical integration carried out at zero initial conditions,
E = 0 and a _{j} = 0, and for the electron arriving from infinity along the asymptote inclined at the angle q _{0} to the equatorial plane, are shown in Figs.3, 4. 

Fig. 3. Results of numerical integration of the equations of motion for the electron starting with zero velocity at a very large distance from the nucleus as a function of the angle q_{ 0}  the angle between the spin axis of the electron and the radial asymptote. The angle q' .is the angle between the spin axis and the asymptote of the outgoing electron. Orbit of the electron may be closed if q' = q _{0} .


Fig. 4. The graph shows a relative azimuthal position of the initial asymptote (j = j_{ 0} = 0) and final asymptote (j' = j) of the electron trajectory as a function of the angle q_{ 0} . The orbit may be closed if j = k/n 360 ^{o} where k and n are integers.


Searching closed trajectories which may represent an orbit of the electron in the atom one must bear in mind that the spin magnetic force is a noncentral force, and that the polar angular momentum L, in contrary to azimuthal momentum, is not conserved and that in general, L_{ 0} L'. As a result, the initial asymptote and the final asymptote of the electron are situated on two different conical surfaces  the asymptote of the electron incoming with the zero polar angular momentum L_{ 0} = 0, is situated on a conical surface with the top of the cone coinciding with the position of the nucleus, while the top of the cone of the asymptote of the reflected electron is not centered on the nucleus, as in general L' 0, see Fig.5.


Fig. 5. The change of polar angular momentum L as a function of the angle q_{ 0} at initial polar angular momentum equal to zero (L = L _{ 0} = 0). The orbit may be closed if the polar angular momentum before a scattering L _{ 0 } and after scattering L' have the same value.


Inspection of all these result shows that the freefall orbit may be closedonly in one case, that is in the case of the planar orbit that was considered at the beginning.
One can supposed that this particular solution of the problem when the electron after each of the three reflections from the nucleus comes back to the starting point and the orbit is closed represents the hydrogen atom in the ground energy state (and in all excited states with l = 0). 

Spin of the electron  a source of a symmetry in Nature. A detailed analysis of a more general case of a quasifreefall motion, that is for an electron arriving from infinity with some small value of polar angular momentum, L _{ 0} 0, leads to a conclusion that the electron after two, three or four scatterings from the nucleus can come back to the starting point and the orbit may be closed providing: he initial and final asymptotes are situated on the same surface cone, however, the top of the cone may be shifted by some distance from the nucleus. This is possible if the polar angular momentum of the electron on the way to periphery of the atom changes sign. It may happen if in the vicinity of the given atom there is another atom, see drawing on the side. One can be highly surprised, but it may be the case providing


radial asymptotes are inclined to each other by the angles: 90^{o}, 109^{o} and 120^{o} , which are fundamental angles of stereochemistry.


Thus, it must be the spin of the electron which is responsible for the order and beautiful symmetries of the molecular world! To see this beatifuul world click here , there and there. In view of a deep logic hidden in the rigorous mathematics presented above which was developed for a prcisely defined physical problem one must admit that


contrary to what we have been taught at Universities the atom is a highly nonspherical starlike object with a well defined angular geometry determined by the radial segments of the orbit looking like stretched atomic arms!


Those, who want to learn a little bit more about the freefall atomic model may look at the address: www.ipj.gov.pl/~gryzinski.


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