LECTURE
18 December 2002
 
Language consultation: J.B., Ontario - Canada.
COLLISIONS
THE MAIN SOURCE OF INFORMATION ABOUT THE MICROSCOPIC WORLD.
                     
 
Although any physical measurement contains, in a more or less hidden way, information about properties of the microscopic world, there are three main sources of information on structure of atoms and molecules:
 
 
- mechanical and electromagnetic measurements,
- atomic and molecular spectroscopy,
- atomic collisions.
 
J.J.Thomson
and
E.Rutherford
beginning of
the atomic
collision theory.


M.Born
collision theory
on a wrong way.


M.G.
Unbelievable!
In the midle of the Twenty century the two-body collision problem not yet solved!
Among these three sources of information, atomic collisions provide us with the most direct, however, coded information on behaviour of atomic electrons. To decode information hidden in experimental data one must have a properly developed theoretical formalism. The first step towards decoding the hidden information on internal structure of the atom has been done by J.J.Thomson and E.Rutherford - the former has shown that the electron is an universal component of matter and the latter that the atom contains a point like heavy nucleus. Unfortunately, on a basis of entirely false arguments I discussed in the previous lecture, fruitful researches have been abruptly stopped by a newly born Quantum Mechanics. Considerations based on Newtonian dynamics operating with a concept of a localized electron has disappeared from atomic physics and the interpretation of atomic collision experiments has been limited to search for meaningless formal relations reproducing experimental data. In 1958 I showed that departure from Newtonian dynamics and the concept of a localized electron was premature, at least. Now, I shell present essential elements of a rigorously defined atomic collision theory developed on a basis of a thorough solution of a classical two-body problem - the theory, which enabled me to trace out electrons moving in the atom. Although, the two-body collision formalism has origin in works of J.J. Thomson and E. Rutherford, its key elements were not formulated until the break of fiftieth and sixtieth of the past century. At that time, a complete set of relations describing rigorously a two body collision problem formulated in a laboratory system of coordinates was given only in a series of papers I published in Physical Review (Phys. Rev. 1959, 1965a, 1965b, 1965c). Here, I will present a scheme of the fundamental research metod of the microscopic world.
 
 
 
Fig. 1. Shows the essence of atomic collision experiments. Macroscopically measured links between initial and final kinematical states of particles coming from the accelerator are determined by a microscopic interaction in the collision region. It is a task of the theory to figure out from macroscopic measurements behavior of particles in the microscopic region. Unfortunately, the destructive dogma of the twentieth century physics that a microscopic particle cannot have both a definite position and a definite momentum has for few decades entirely eliminated the search of causal relations between macroscopic and microscopic world.
 
Photon,
electron and proton
- fundamental carriers
of information
in the whole Universe.
Collisions - observations at the edge of the micro-world. To get information about properties and behavior of a physical object one must have a proper probing system. The change in the state of the probing system, which results from the interaction with the investigated object, is a measure of the object properties.

Existence of interaction between the investigated object and the probing system implies that any observation can be carried out with only a limited accuracy.

The most delicate, relatively compact and stable elements of matter created by nature, which in the living world are usually used for probing of material objects, are photons, which along with the electron and proton, the two smallest stable pieces of matter, stay at the bottom of the observation system of the atomic world.

Perturbations caused by probing corpuscles: photons, electrons, protons, can be neglected until there are massive objects observed. In such a case, by continuous illumination of the investigated object with a probing beam its whole history can be precisely registered. Situation becomes quite different when the investigated object has a microscopic size. Then, there is no qualitative difference between the probing particle and the investigated object and each testing act (collision between the target object and the probing particle) results in the appreciable change of kinematical state of both particles.

This means that

investigation of atomic objects can only be carried out in the way analogous to material destructive investigations

(a large number of identical objects must be destroyed to have a sufficiently good knowledge on their structure and properties). Here it is worthy to stress that at the atomic level discussions on un disturbing measurements are entirely meaningless.

Since probing corpuscles can be directed towards atomic objects with macroscopic accuracy only, and the latter is determined by accuracy with which the experimental devices producing the beam of probing particles can be made, investigations of the atomic object cannot be carried out in a systematic way. We have no possibility to check point-by-point particular fragments of the investigated object.

Statistical links between initial and final states of the probing particles arising from a chaotic illumination of a target object is the only possible form of getting information on internal structure of microscopic objects.

Cross sections. A statistical links between macro and micro-worlds. By sending a large number of probing particles (electrons, protons) on a target containing a huge number of investigated microscopic objects one cannot determine anything about the trajectory of a given individual particle inside the target - it is, in fact a black box. The observed change of direction of motion and the change of speed of the probing particle is the only information on what has happened on a microscopic scale. On macroscopic scale there are three points, which define the collision: the exit from accelerator, a target and the detector.

On a microscopic scale a collision with a point like object is defined by:
- a distance D at which the probing particle would pass by the target particle, if they would not interact,
- an azimuth angle Q specifying orientation of the element D in a plane perpendicular to the vector of relative velocity.

A macroscopically observed effect of a collision x dependant in some way of the particle velocity D v may be symbolically written in a following way:
 
 
x (D v ) = fxFL ( initial conditions: v a , v b ; D, Q ) ,
 
 
where fxFL represents results of theoretical calculations carried out, for the given interaction law FL, at initial conditions defined, on the one hand, by initial velocities v a and v b of the colliding particles, and two geometrical variables D and Q, on the other. Since in a macroscopic experiment, microscopic geometry of the collision can not be controlled, results of macroscopic measurements may have only a statistical character resulting from the random statistics of geometrical variables D and Q. But a random <D, Q> statistics may be transformed, through the interaction law FL, into appropriate for this law experimentally observed statistics with respect to the observed fenomenon < x (D v )>. This transformation procedure is defined by the integral:
 
(1)
 
 
where d (x) is Dirac delta function. The result of integration represents a sum of these surface elements of a whole infinite D - Q plane for which argument of the d-function is zero. s x which relates random <D, Q> microscopic statistics with a measured x (D v ) statistics is called a cross section. The cross section represents the probability that at a single collision a given change D v of the kinematical state of the considered particle will be observed.

It is the primary task of a collision theory to carry out a logical analysis of experimental data and to decipher the interaction law FL hidden in the observed statistics.

If, however, the interaction law is already known than experimental results can be a source of information on kinematical state of target particles. Thus, solving the experimentally-theoretical identity:
 
 
s xexp ( v b ) = s xtheor ( v a = ?, v b ) ,
 
 
with respect to v a , for instance, we can determine the velocity of target particles. Such a combined experimentally-theoretical procedure forms the essence of atomic collision investigations. To illustrate the method let us look at historical research of J.J. Thomson and E. Rutherford with the help of the above given rigorous definition of the cross section.

E.Rutherford. Atom has a point-like nucleus. As observed by E. Marsden, the distribution of flushing points on Zn-screen produced by a stream of a-particles penetrating a thin metallic foil gave origin to one of the most important discoveries of the atomic physics. To explain accidentally observed large angle scattering events E. Rutherford has assumed that the whole positive charge of the atom is concentrated in a heavy point-like nucleus. Solving Newton's equation of motion for Coulomb interaction he found that the measured in the experiment angle J, the angle between asymptotes of the hyperbolic trajectory and the invisible impact parameter D are related in a following way:
 
(2)
J = p - 2 arctg ( D E / Q q ) ,
 
where Q is electric charge of the scattering center and q is electric charge of the scattered particle and E is its kinetic energy.

In the case of microscopic particles the established relation between J and D, which may be symbolically writen in the following way:
J = fJ (D), cannot be checked in a single collision act, as we have no possibility to control motion of the particle on the way towards the center of the atom. But it can be checked statistically, as for a given J = fJ (D) the random statistics in D results in a well defined statistics in the angle J. A transformation procedure from a random statistics in D to the observed statistics in the angle J , is determined by the following integral:
 
(3)
 
It is instructive to see the quantum way to Rutherford formula.
Performing the integration for J = fJ (D) as given by Eq.(1) one obtains the famous Rutherford formula:
 
(4)
sJRutheford = (Q q / E) 2 ( 1 + cos J ) / sin4 ( J / 2 )
 
 
The proof that Rutherford was righ is a good agreement between calculations and measurements, see Fig.2:

the whole positive electric charge of the atom is concentrated in a heavy point like nucleus.
 
 
Fig. 2. Shows experimental and theoretical results in a historically the first scattering atomic collision experiment. Existence of causal links between macroscopically observed statistics of scattered particles and being out of control in the experiment microscopic random statistics with respect to the impact parameter D enable to deduce what has happened in the microscopic region.
 
Tryvial two-body collision problem which Quantum mechanics was unable to solve.
J.J.Thomson. Atom has point-like electrons. It is a characteristic feature of scattering from the fixed centre of force that kinetic energy of the particle before and after the collision is the same and the change of direction of motion is the only result of a collision. If, however, in place of the scattering centre there is a particle of a finite mass, the scattering process is always accompanied by some transfer of energy. The target particle, which initially was at rest, after a collision moves with some velocity, carrying some amount of kinetic energy of the projectile. The amount of energy e transferred in a collision between two particles of masses m a and m b depends upon the impact parameter D in a following way:
 
(5)
 
 
where E is initial kinetic energy of the projectile and the numerical factor c has the form:
 
(6)
c = 4 m a m b / (m a + m b) 2 .
 
 
Calculating the integral:
 
(7)
 
 
which transforms a random statistics with respect to the impact parameter D into the observed in the experiment statistics with respect to the transferred energy e, one gets:
 
(8)
 
 
With help of the found cross section s e one can calculate one of the most important cross sections of atomic physics. Calculating the integral
 
(9)
 
 
one obtains the famous Thomson's ionisation cross section describing ejection of electrons by fast charged particles from atoms:
 
(10)
Q i = Z 2 s 0 / c E U i (1 - U i / c E ) ,
 
 
where U i is the ionisation of the atom, Z represents the electric charge of the projectile and the numerical coefficient, if U i and E are in eV, has the value:
 
(11)
s 0 = p e 4 = 6.56 . 10-14 e V 2 cm 2 .
 
 
The formula given above did play a very important role in the early days of atomic physics. Bombarding atoms by electrons and observing production of free electric charges in the target one could directly measure binding energies of atomic electrons. It is satisfactory to measure the threshold energy for production of positively charged fragments of the considered atom. Measured in this way ionisation potentials are shown in Fig.3.
 
A key information on electronic structure of the atom obtained with atomic collision technique.
 
Fig. 3. Ionization potentials of atoms as measured with help of atomic collision technique and atomic spectroscopy. A far reaching correlation between measured ionisation potentials and the position of the element in Mendeleev table resulted in a claim difficult to question: that inside the atom must exists a perfect order.
 





We are comming back to rigorous rules of Newtonian dynamics.

































M.G.
classical two-body
collision problem solved!
It is interestic to note that, with help of atomic collision technique J.J.Thomson arrived at the conclusion, not trivial at that time,that number of electrons in the atom is roughly equal to one half of the mass number the atom.

A two-body collision - the bedrock of all collision physics. Since the time of Rutherford it became clear that electrons in the atom cannot stay at rest and atomic collision formalism must take into account kinetics of the moving electrons. Theoretical analysis of a collision with a system containing some number of electrons being in a continuous motion is not an easy task. The problem was undertaken already in the early twenties by Bohr, but essential step towards formulation of a more accurate atomic collision theory was done only a few years later, in 1927. At that time Thomas and Williams applying a statistical Hartree-Fock model of the atom successfully described losses of energy of charged particles moving in a gaseous media. Unfortunately, with formulation of quantum mechanics, which has questioned the notion of a localized electron, fruitful research based on classical dynamic has been entirely stopped and misty formalism introduced by Born has entirely dominated the physics community.

Almost half century ago, in 1957, I decided, ignoring restrictions of quantum philosophy and going against mainstream research, to come back to the banned concept of a localized electron. I was highly surprised, when I discovered that the fundamental problem of all collision physics, a collision between two elementary particles interacting by a central force, still remained unsolved, while Scientific Journals were full of sophisticated quantum collision calculations on atomic, nuclear and relativistic collisions. It was a great satisfaction to derive some fundamental relations of collision physics and among them: an equivalent of Rutherford formula and an equivalent of Thomson energy exchange cross section s e. others extended on particles of arbitrary masses and arbitrary velocities. And for instance, the cross section, for which quantum theory has been unable to formulate any alternative, has the form:
 
(12)
 
 
where m a b is the reduced mass of two particles arriving from infinity with a relative velocity V
 
Two-body collision problem, sollution of which lies beyond possibilities of Quantum theory.
V  =  v a - v b ,

and moving with a total linear momentum P

P  =  p a + p b .

The unit function H(x ) defined in a following way
 
 
H ( x ) =
 1      if      x > 0

 0      if      x < 0
 
 
where the argument x given by
 
(13)
 
 
determines limits of possible transfers of energy between colliding particles. The two formulas, (12) and (13) given above form the essence of the modern classical atomic collision formalism. A rigorous solution of a two-body collision problem enabled me to discover one of the biggest mistakes done in the early days of atomic physics, which dramatically influenced further development of all physics. Now few words about the way towards a historical discovery.

M.G. Atomic electrons in a radial free-fall motion. t is evident that results of collision of the charged particle with the atom must, more or less, depend on behavior of atomic electrons. Let us consider, for instance, ionization of atoms by electrons. Threshold energy relation for ejection of the atomic electron, bound with some energy U i , according to Eq.(13), is given by:
 
(14)
E thr = U i ( E e - U i ) / ( E e sin 2 q - U i ) ,
 
 
where q is the angle between velocity vectors of the ionising electron and the atomic electron just before a collision. In the case of gaseous targets, atoms are randomly oriented in space and the angle q may have any value between 0 and p. In the case when q = p / 2, minimal energy is needed to eject the atomic electron, and is just equal to the ionisation potential U i..

The situation is, however, quite different if the mass of the ionising particle is much greater then the mass of the electron. Then, the threshold energy E thr = m e v 2 thr / 2, as follows from Eq.(13), depends on speed of the atomic electron v e , and is minimal for head-on collisions, that is for q = 0, and depends on speed of the electron in a following way:
 
(15)
2 m e v thr (v thr + v e ) = U i .
 
 
If the atomic electron would stay on a circular orbit, then speed of the electron would be a constant quantity, and according to the above relation a well-defined threshold for the ionisation process should exist. But the experiment does not show any evidence of the threshold! There is only monotonous decrease of the number of produced ions with a decrease of the projectile energy, see Fig.4.
 
 
Fig. 4. K-shell ionisation cross section of Ag by protons at very low energies as measured and calculated for atomic electrons of various speed. A lack of the threshold was strong evidence that electrons in the atom come into close vicinity of the nucleus.
 
 
The lack of the threshold could be interpreted only in the following way:

electrons in the atom move radially towards the nucleus.

Only in such a case may electron velocities reach high values. In this way in 1965 the author have arrived to a free-fall atomic model concept. Although the lack of the threshold for ionisation of atoms by heavy charged particles was quite a strong argument that electrons in the atom move along radial orbits it was necessary to find some other examples supporting this important conclusion. The possibility of giving the exact proof on radial motion in the atom appeared when Helbig and Everhardt carried out the famous experiment with electron capture in head-on collisions of protons with hydrogen atoms. In this case the process appeared to be sensitive to the form of electron orbit. Now, results of calculations performed on the basis of rigorously formulated mathematical algorithm and compared with the experiment did not left any room for doubts that classical dynamics works at the atomic level and atomic electron moves radially to the nucleus.
 
 
Fig. 5. A comparison of measurements and exact numerical three-body calculations for electron capture in head-on collisions of protons with hydrogen atoms. Measurements do not leave any room for doubts that electron in the hydrogen atom moves a long radial trajectory towards the nucleus.
 
 
To show that radial kinetics is a characteristic feature of all atoms, energy spectra of electrons ejected by protons from helium atoms were calculated and compared with the experiment. Calculations were carried out for two extreme forms of collective motion of two electrons: circular motion and radial motion. confrontation of the theory with the experiment does not leave any doubts: electrons in helium atom move radially.
 
 
Fig. 6. Results of theoretical calculations for ejection of electrons from helium atoms by protons for two different atomic models.
 
 
If only motion of electrons in the atom is known any atomic collision problem, at least numerically, can be immediately solved, see figer 7.
 
 
Fig. 7. Results of exact three-body numerical calculations for p ( e ) + H collisions.
 
 
Quantum collision theory - sophisticated mathematics pretending to describe a physical reality. The corpuscular-wave duality puzzle, I discussed in the first lecture, has led physicists to irresponsible speculations. The most destructive step has been done by Max Born, who in quite an artificial way applied Schrödinger formalism valid for analysis of radiation-less states of the bound electron upon a qualitatively quite different kind of physical phenomena. To adjust the wave formalism to describe free states of the electron, Born quite arbitrarily and violating fundamental laws of physics - including energy and momentum conservation laws, connected in a probabilistic way the wave function y with a position of the electron in space. With time, Born original probabilistic interpretation of the wave function, evolved into an abstract quasi-probabilistic interpretation, as the original could not withstand physical reality. As a result, the electron has been entirely deprived of a physical existence, and its ghost, present at the same time in an entire infinite volume, may be brought to life by the act of observation and instantaneously materialize in some point of space. It is difficult to understand, how the whole philosophy introduced originally by Born could be taken seriously. Nevertheless, it has been.

To show false moments in Born formalism let us trace, step by step, the calculation procedure of some fundamental and at the same time the simplest problem of atomic collision physics. It is the scattering of elementary charged particle from the fixed center of force. To obtain Rutherford scattering formula on the grounds of classical dynamics one needs two trivial steps: one must solve Newton equation of motion for the Coulomb potential to have a relation between the impact parameter and the scattering angle, see Eq.(2), and one must calculate the elementary integral, see Eq.(3), to transform random statistics with respect to the impact parameter into the observed intensity of scattered particles.

Now let us look at the calculation procedure, as given in one of the most famous academic text-books on Quantum mechanics - "Quantum mechanics" by L.Landau and E.Lifshitz.

The first step - the step, which stays at the beginning any quantum calculation procedure on scattering. says that the y function must have the following form:
 
(16)
 
 
where the first term represents the incident plane wave and the second term a scattered spherical wave. The first term imitates initial conditions of the collision problem, as in the experiment there is not an infinite plane but a narrow beam of particles moving to the target. The second term anticipates the solution of the problem, as a spherical flux of scattered particles is already the result of interaction with a target. In this way Quantum mechanics omits the fundamental question of the collision problem: what has happened in the black box that particles change direction of motion going through the box? In classical dynamics it is the interaction law, which gives the effect of scattering - in the Young experiment it is the electric field at the edges of slits, and in Rutherford theory it is the Coulomb field of nucleus.How to explain the change of direction of motion of particles before the scattering potential has been defined? The potential could result, for instance, in a parallel shift of the beam and then one would have any spherical wave.

The second step - the first stage of manipulation procedure adjusted to the problem being solved. In the case of Young experiment it is said that two spherical waves, in fact in the experiment we observe extremely narrow beam of photons (electrons), interfere with each other, and in consequence we observe at the given directions modulations of the intensity on the screen. But according to Quantum mechanics photons move in the vacuum and Schrödinger equation cannot describe fluid oscillations. Moreover, photons (electrons) arrive to the screen, one by one, how do they interfere if they do not interact? The fact wave like pictures on a screen have a common measure with the periodic solution of the Schroedinger equation is not a proof that those arise from the interference. A whole verbal explanation is needed to make the impression that the problem is solved. In the case of Rutherford scattering in L.Landau and E.Lifshitz book we read: we will search the solution of the Schroedinger equation for the angular part f(Q) of the outgoing spherical wave in parabolic system of coordinates. Everybody, however, knows that proper system of coordinates simplifies solution of the problem since some characteristic features of the investigated problem are included, it contains, in fact, some fraction of the searched solution. And for instance, solving a Kepler problem in an elliptic system of coordinates one would automatically have the final solution of the problem. A parabolic system of coordinates required in the solution procedure of the wave equation introduces some information on physics of the problem - a parabola represents a particular case of the scattering problem.

The third step, when some formal sophisticated elements of manipulation are needed: A parabola, correctly represents Coulomb scattering in the close vicinity of the scattering center and the Schrödinger equation is already prepared to produce a correct solution. It is necessary only to modify slighttly the arms of the parabola. It is said that y function should in infinity be consistent with the incoming plane wave and the outgoing spherical wave function in infinity should have only radial component. With the help of quite sophisticated mathematics one finally gets the wave function satisfying the needed requirement.

The fourth step, terms we do not like are being removed on the basis of verbal arguments. However, the exact solution of so prepared differential equation is still not the needed solution. The exact solution is expanded into series and two terms of the expansion are taken into account - the rest is simply dropped out, on a basis of verbal arguments that these terms in infinity give negligibly small correction. The authors admit that the incoming plane wave, as a result of whole these transformations, contains terms decreasing too slowly, but they are not troubled much by this fact since the outgoing spherical wave is correct.

Really, the final result is correct, they have got Ruherford formula.
If one wants to see details of exact, as the authors write, quantum calculations and compare them with the above presented classical calculations which, as they say, are only accidentally in agreement with exact quantum result, please click here. To realize the absurdity of the Born concept one must know that scattering from a fixed center of force is the simplest problem of the whole collision physics and Quantum collision theory, with its artificial formalism is unable to go beyond this trivial case. The entire class of two body collisions, for particles with arbitrary masses and arbitrary velocities, for quantum theory do not exists. Neither quantum equivalent of Rutherford formula for the center with finite mass:
 
(17)
 
 
nor quantum eqivalent of Rutherford formula for two moving charged particles:
 
(18)
 
 
where
 
(19)
 
 
A quantum equivalent of energy transfer cross section even in its primitive form, as given by J.J.Thomson, does not exist. And what about the transfer energy cross section in its general form as given by Eq.(5)? On a basis of the wave formalism it is impossible to calculate collision cross section for transfer of energy in a two-body collision. It is impossible since the physical essence of the transfer of energy is based on the energy conservation law and for this law there is no place in wave formalism. It is hard to believe but Quantum mechanics is unable to describe collisions between two crossed proton (electron) beams produced in a school device!!!
 
 
If the theory is unable to describe two-body collision between two protons (electrons) in keV range how it can be used to high energy collisions or for collisions between systems of charged particles?
 
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