Eureka! Electrons in the helium atom localised!

Michal Gryziński

I. Introduction. Researches aimed at a localisation of electrons in a matter, which have origin in works of Thomson and Rutherford and which in modern epoch have led to formulation of the free-fall atomic model concept [1-3], were not long ago crowned with some success. It has been discovered that electrons moving periodically from the one nucleus to the other form a skeleton of a solid matter [4]. These, and lot of other results obtained applying classical dynamics to interpretation of atomic experiments have shown that radial kinetics of collectively moving electrons is a characteristic feature of all atoms and molecules in the ground energy-state.

Although global shape of electron orbits in atoms and molecules is basically determined by a long distance Coulomb interaction, a short-range spin magnetic field of the electron plays at a radial motion exceptional role. The interaction of the electron with nucleus by spin magnetic field, negligibly small at a Bohr distance, in the close vicinity to the nucleus determines a whole back-scattering phase of the electron motion, see (WWW.ipj.gov.pl/~GRYZINSKI - Hydrogen atom, 19.03.1999). It is spin of the electron, which controls angular localisation of radial trajectories and as a result determines stereo-chemistry of the atom and keeps order in a lattice.

Now, trying to make a next step in our dynamical considerations on structure of the atom we will attack, taking into account spin properties of the electron, the problem of the helium atom.

II. Two, spinning electrons of the He- atom on the ground energy-state orbit. The first step to deciphering electron structure of the helium atom has been done within atomic collision physics. Experimental data on ejection of electrons by protons from helium atoms [6] delivered strong arguments that electrons in the helium atom move along radial (almost radial) trajectories [7]. Until now, however, details of this motion remained unknown. Numerical analysis of a collective motion of two spin-less electrons in the Coulomb field of nucleus has shown [8] that the problem of the helium atom cannot be solved ignoring spin properties of the electron.

But extension of theoretical formalism by non-central, velocity dependant magnetic forces complicates greatly the problem. In such a case there is a practically unlimited number of possible solutions. Symmetry requirements, however, enable to solve the problem. Quite general symmetry considerations have led to a conclusion that kinematical identity of two collectively moving electrons in the atomic shell is possible at parallel or anti-parallel oriented spin axes of both electrons.

As a result, it was possible to reduce a hopelessly difficult problem to a following relatively simple two dimensional Lagrangian:

where r is a distance between the electron and the nucleus, 2x is a distance between electrons, m is magnetic moment and r_m is a magnetic radius of the electron.

To apply the formalism defined by the given above Lgarngian to a concrete atom, one must specify the energy of the atom. In the case of the helium atom the latter is simply a sum of  the first (U_i ) and the second (U_ii) ionisation energies which respectively are:: 24.6 eV and 54.4 eV. Thus, the energy shared by one of collectively moving electrons, therefore, is:

W_He = ˝ (U_i + U_ii ).= 39.5 eV .

This information, as well the information that total angular momentum of the helium atom in the ground energy state is zero are satisfactory to start the search of  periodic solutions (closed orbits), which might represent stationary states of the helium atom. The simplest closed orbit is shown in Fig.1. This is the orbit very close to the free-fall orbit that has been succesfully used at semiquantitative interpretation of various atomic experiments. One could suppose, therefore, that the found orbit really describes electrons moving in the helium atom.

HELIUM ATOM

Three basic parameters describing orbital motion of electrons and essential from the point of view of physical properties of the helium are: the extreme distance of the electron to the nucleus r_He, the width of the orbital d_He , and the oscillation frequency w _He . Those calculated numerically respectively are:

r_He = 1.180 a₀     (0.627 ·10^-8cm),

d_He = 0.309 a₀    (0.164 ·10^-8cm)

w _He = 2.84 w ₀ , .

where a₀ is the Bohr radius and w ₀ is the Bohr frequency.

The numbers given above enable to understand why helium atoms can so easily penetrate narrow gaps and pass across solid materials like a glass. This is a needle like shape of the helium atom responsible for this unusual property of gaseous helium.

To have rights, however, to claim that we have deciphered electronic structure of the helium atom we must show that the found quasi-free-fall orbital (qff-orbital) correctly describes essential properties of the helium atom that can be observed in various experiments. And for instance, magnetic susceptibility is one of the most characteristic properties directly related to the form of the electron orbit. The another,  qualitatively different parameter directly related with motion of atomic electrons is asymptotic field of the atom, which determines a long distant interaction between atoms and may be identify by scattering in low energy collisions.

III. Experimental test of the theoretical model of the He atom. Let us start our testing procedure of the found qff-orbital with calculation diamagnetic properties of the Helium atom. As we have shown some time ago [9] contribution of the individual electron moving along a given orbit r(t) to the magnetic susceptibility of the whole atom is simply given by the integral

Calculating numerically the above integral and taking into account that in the magnetic field atom is oriented to have diamagnetic effect at maximum, one immediately gets:

a ^theor_He= 2.04× 10^-5 cm .,

while measured values are

a^exp_He = (1.94 ± 0.1)× 10^-5 cm ,

a^exp _He = (2.02 ± 0.2)× 10^-5 cm.

Although the agreement between the theory and experiment is quite good,   the presented above numbers need some comment. Thus, on the one hand the difference between two measurements exceeds error bars, and the both values are slightly smaller than the theoretical value. Since the differences are quite small one could ignore this fact. But these small spread of results have a deep physical sense as this spread is directly related with a needle like shape of the helium atom.

It is a key point of the problem that the helium atom, in contrary to that what we have been taught at the University, is a highly aspherical object and, therefore, diamagnetic properties of the atom depend upon orientation of the latter with respect to the direction of the magnetic field. Thermal collisions between atoms destroy regular orientation of needle like atoms placed in the magnetic field. It is evident that measured values of atomic diamagnetism must depend upon  density and temperature of the gas being investigated as well as upon the strength of the magnetic field in which measurements are carried out. In this way we can understand  why theoretical value calculated for ideally oriented atoms is slightly larger then both experimental values and why differences between various measurements exceed error bars. At measurements this aspect of the problem has been simply not taken into account as in view of a widely spread opinion that noble gases have ideally spherical symmetry there was no need to check the role of the all factors which less or more influence measurements. 

Asymptotic field of the helium atom. It is well known from electrostatics that electric field of a collection of point charges is at large distances from the system basically  determined by leading electric multipoles of this system.. The two, leading terms of the expansion of the electric field of the helium atom into electric multipoles are: oscillatory dipole and a static quadrupole. Thus, the asymptotic field of the helium atom has a form:

where s_d and s_Q are unit vectors directed along dipole and quadrupole axis of the atom. Intensity of the electric field of the helium atom is determined by amplitude of dipole oscillations d and by time averaged value of the axial quadrupole Q₂.These quantities calculated numerically for the considered electron orbital respectively are::

d = 0.16 [ea ]

Q₂ = 1.72 [ea² ]

Fig.2. There is shown a passage from exact description of the helium atom to its approximate picture.

One of the basic methods of investigation external electric field of the atom are low energy small angle scattering experiments. Small angle scattering cross sections contain information on the order of leadings multipoles as well as on their values. With help of the formalism developed some time ago [ ] on a basis of classical dynamics  one can easily show, that the two multipoes describing electric field of the helium atom reproduce correctlyl experimental data..

Mysteries of He³ (spin of the nucleus and the electron shell). The fact that in very low temperatures liquid He³ behaves quite differently from liquid He⁴ is well known. Less known is the fact that atom-atom scattering at very low energies is dramatically different for He³ and for He^4. In view of the fact that atom-atom interaction is determined by properties of the electron shell of the atom one could suppose that spin of the nucleus must in some way influence motion of atomic electrons. But  at atomic distances interaction of the electron with magnetic moment of the nucleus is much insufficient to find a quantitative basis  for explanation for the effect. The situation, however,   is qualitatively different at the radial motion of atomic electrons. Increasing with the third power of a distance to the nucleus its spin magnetic field attains quite large values, sufficient to influence appreciably motion of the electron. Really, simple calculations show that magnetic field of the nucleus results in an uniform rotation of the electron shell with the speed  which is consistent with observed differences in a low energy scattering of two helium isotopes. In the case of helium 4 small angle scattering is determined by a static quadrupole, while in the case helium-3  it is with respect to a slowly moving atoms  a dynamical quadrupole ( details of this analysis we will present in a near future).

A similar situation does exist in the case of liquid phase of helium. Rotation of the electron shell of helium-3 eliminates possibility of coherent, perfectly ordered motion of helium atoms, which is possible in the case of helium-4 (to this aspect of the problem we will come back in a near future as well).

Three, qualitatively different aspects of the helium properties we have discussed above seem to be a quite satisfactory argument to suppose that the object of our investigations, He atom, looks like it is shown in Fig.1.

Although this conclusion may be easily supported by many other examples, the most impressive arguments can be formulated applying theconsidered   model of the helium atom to idescription of a condensed phase of helium.

IV. Iinteraction between helium atoms. Interaction between two helium atoms at sufficiently large distances may be in principle reduced to a dipole-dipole attractive interaction of moving in phase all four electrons of two atoms, and quadrupole-quadrupole repulsive interaction. In such a case interaction energy of the two helium atoms, as derived from a potential of a single atom, is given by a   following relation:

where

,

,

and

d² = 0.049 e² a₀²,

Q² = 3.78 e² a₀⁴

Helium molecule. The interaction energy formula presented above is quite different from commonly used in kinetic theory of gases interaction potentials. Those as a rule are spherically symmetric with much stronger dependance on distance - in the most cases 6-12 Van der Walls potential is applied. Even in considerations on a liquid helium. One can be highly surprised by the derived above potential quite precisely describes a lot of properties of liquid helium. In our case it is dipole-dipole long distance  force which attracts helium atoms. Helium atoms tend to be oriented in such a way that this force is at maximum. The equilibrium distance is determined  by this attractive force and quadrupole-quadrupole repulsive interaction.The two atoms may form, therefore, more or less stable He molecule. The equilibrium distance and potential energy at minimum depend upon relative orientation of the considered atoms. General formulas for these two parameters describing the quasi-molecular helium system obtained by differentiation of the interaction energy relation:are as follows:

L = 6.017 ( k_Q/k_d)^1/2 A,

U = 2.849  k_Q (k_Q/k_d)^3/2)  ⁰K.

These figures already show that we are on energy-length scale at the values characteristic for liquid helium (lattice constant for body centred cubic cell is equal  to 4.018 A, while melting temperature of helium is 5.2 ⁰K).

V. Helium clusters. Although helium atom can join together in quite different ways forming different molecular systems there are some configurations when atoms are particularly tightly bound to each other. The mostly tight configuration form two atoms paralelly oriented and with quadrupole axes perpendicular to each other. This configuration, as well as some other, ares shown in Fig.3 . This tightly bound helium molecule is chemically very active, as its dipole field is two times greater then the field of a single atom. This molecule gives origin to a three atomic cluster, which   is chemically inert, as its effective dipole moment is equal zero. The another chemically inert molecular system is a quadratic cluster,. These clusters can join togeteher and form more sophisicated spatial configurations.

Trigonal and quadratic clusters can join together giving origin to a body centered cubic cell and a face centred cubic cell, see Figs. 4 and 5. The most impressive result of the present considerations undoubtidily is the fact that

dimensions of bcc and fcc cells calculated from parameters of the electron orbital of the helium atom appeared to be almost identical with that measured in the experiment.

Dimensions of the cell are basically determined by dimensions of the clusters which form a skeleton of liquid and a solid body. The length of a trigonal cluster calculated from potential energy formula is equal to 4.018 A, while calculated from a measured density of a liquid helium  is equal to 4.036A. Analogical situation does exist in the case of bcc lattice. The experimental value of the length of the cell is ~ 4.51 A , while the lentgh of the side of quadratic cluster calculated from potential energy formuls is ~ 4. 42  A. Here it is worthy to note that trigonal clusters are bound by electrostatic quadrupole interaction, while quadratic clusters are bound to each other by dipole-dipole resonance force.

Fig.4. A face centered cubic cell build up with basic building block , which the trigonal cluster is.

One can be highly surprised that trivial in fact calculations based on interaction of two multipoles, with absolute values rigorously defined by parameters of the electron orbit  of the atom, may give so far going agreement with the experiment.

Here it is interesting to note that the two bound clusters may be identified with famous rotons invented by Landau in the early days of the theory of liquid helium.. Really, they have needed masses: ( 7 - 8 ), and the excitation energy of the roton is few kelvins – it is energy needed to split two clusters bound to each other in a corner.

Although on a basis of the derived potential energy function it is possible to explain a lot of problems connected with a liquid state of helium, there is a problem worthy a short comment. It is a problem of a surface tension. The high value of surface energy is related with the fact that at the sutface atoms are not so strongly limited by the neighbours and can take energetically more suitable orientation then inside the liquid. The two first atomic layers at the surface can ajust their orientation to satisfy requirement of energy minimum. The atoms of two surface layers are joined in pairs bound by resonance force with dipole vector perpendicular to the surface, and qudrupole vectors perpendicular to each other. The surface energy comes out directly from the potential energy formula. The obtained in this way value of surface energy is just the value measurred in the experiment (one must know that quite recently carried out experiments has shown that at the surface helium atoms oscillate perpenducularly to the surface).

There arises, however, the question: why the surface energy of liquid helium 3 is essentially smaller then surface energy  of helium 4. The answer to the question is of course hidden  in nuclear magnetic  moment of helium 3. As we already do know in a free helium-3 atom electron orbital rotates around the dipole axis of the atom. As a result the condition of energy minimum, which takes place if quadrupole axes of the atoms of the first and second layer are oriented perpendicularly to each other cannot be satisfied.  Relative orientation of quadrupole axes of the two layers is determined by equilibrium of two moments trying to reorient the electron shell of the he 3 atom. Those are: magnetic moment arising in a spin orbit interaction of the electron with the magnetic moment of the nucleus :and the moment produced by electrostatic interaction of qudrupole moments of the considered atoms.:Equaling these two moments one gets equilibrium position of atoms in two considered surface ayers. Having the value of this angular shift one can find a respective  decrease of surface energy. A rough estimate of the phenomenon has appeared to be quite promissing..

Closing this announcement on electronic structure of helium atom one can express the opinion that the way to localisation of electrons in the rest of atoms has been opened. In the near future we should have periodic table of elements with precisely localised electrons. One can hope that localisation of electrons in molecules and a solid body will  be not too difficult as well. There are reasons to think, for instance, that Cooper electron pair in superconducting state moves in a similar way as move two electrons of the helium atom.

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5. Rudd M. E. , Sautter C. A. , Bailey C.L. Phys.Rev. 115,20 (1966),

6. Gryziński M. and Okopi nska A. Proc. of VIII ICPEAC, Belgrade 1973.

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