1.1 DAVISSON-GERMER EXPERIMENT

 

            The origins of Davison and Germer experiment, historically speaking, are related to a laboratory accident. In his experiments related to vacuum tubes, due to an explosion, a nickel target suffered an oxidation. They decided to clean the nickel oxide from the surface of target, heating these crystals in reductive condition. To their perplexity, the pattern of electron scattering from the newly cleaned nickel target was completely different from that before the accident.

            On examining their newly cleaned crystal carefully, they found a clue. The original target was polycrystalline - made up of a multitude of tiny crystals, oriented randomly. During the prolonged heating of the cleaning process, the nickel had recrystallized into a few large crystals. The conclusion was evident. The direction and intensity of scattered electrons is dependent on the atoms crystals arrangements. The device used for the experiments is presented in fig 1.1. In the experiment the kinetic energy of electrons can be controlled by modification of the voltage of electron gun.

            The beam of electrons is directed normally to nickel crystal and the intensity of scattering (number of electrons per unit solid angle with speeds near that of the bombarding electrons) in various directions is measured. The experimental arrangement is such that the intensity of scattering can be measured in any latitude from the equator (plane of the target) to about 20° of the pole (incident beam) and in any azimuth.

 

Figure 1.1 Experimental arrangements for the Davisson-Germer experiment.

 

            In general, if bombarding potential and azimuth are fixed and exploration is made in latitude, nothing very striking is observed. The intensity of scattering increases continuously and regularly from zero in the plane of the target to a highest value in co-latitude 20°, the limit of observations. If bombarding potential and co-latitude are fixed and exploration is made in azimuth, a variation in the intensity of scattering is always observed, but in general this variation is slight, amounting in some cases to not more than a few per cent of the average intensity. This is the nature of the scattering for electron energy in the range from 15 eV to near 40 eV. At 40 eV a slight hump appears near 60° in the co-latitude curve (fig. 1.2). This hump develops rapidly with increasing voltage into a strong spur, in the same time moving slowly upward toward the incident beam. It attains a maximum intensity in co-latitude 50° for a bombarding potential of 54 eV, then decreases in intensity, and disappears in co-latitude 45° at about 88 eV. The growth and decay of this spur are traced in fig. 2.

 

Figure 1.2: Polar plot of data for the scattered electron beam intensity as a function of scattering angle for different incident electron energies.

 

1.1.1        Wave Interference Interpretation

 

            This was the first experiment interpreted as a confirmation of the wave nature of electrons, theory expressed some years ago by Louis de Broglie.

According to the de Broglie idea, wavelength of an electron is given by:

 (1.1)

where h is Planck's constant, m the mass of electron, v the velocity. For electrons accelerated through a potential difference U, the velocity v can be obtained from the classical expression

 (1.2)

and substituted into the de Broglie relation it obtains:

 (1.3)

The Bragg condition for diffraction for small angles is

                                     (1.4)

where d = the interatomic spacing. With this formula a wavelength (l) of about 1,6 Å was established for a potential of U = 54 V. The distance between atoms in crystal are measured using Bragg maximum diffraction and the value measured was d= 1,65 Å

 

1.1.2        Proposed corpuscular interpretation

 

            We can consider that the velocities of electrons emitted by electron gun are obeyed to classical Maxwell-Boltzman distribution. Also is very important to keep in mind that only electrons which suffered an elastic interaction with crystals are counted; the electrons which loose energy during scattering are not counted.

            The comportment of electrons directed to the nickel crystal depends on the value of acceleration gained electronic gun.

            For low energy of acceleration, up to 40 eV, the electrons from the beam are scattered elastic by nickel crystals atoms due to the electronic shields which envelops the nucleus of atoms.

            At this low acceleration, we can consider an elastically scattering of electrons on a barrier of potential represented by the negative electric charge of electrons shells. The influence of nucleus is negligible. Consequently we will have a maximum of electrons returned on the same direction like electron beam, and the maximum of electrons scattered is registered at small angle of deviation. On this acceleration potential great part of electrons are scattered backwards and the deviation is small, with and angle of maximum 20º reported to the initial direction. A small number of electrons due to the Boltzman distribution of velocity can make a turn around nucleus on an elliptic trajectory.

            If we grow the acceleration of electrons beam, majority of them can penetrate the electronic shield of atoms and we can consider that electrons from the gun present a ,,cometary comportment”. This means that electrons as individual particle pass through electronic shields of nucleus, rotate around nucleus, and return on a curve specific to the initial acceleration of electrons. Part of the electrons from initial beam are scattered also by interaction with electrons shells of atoms, but this electrons are not counted. The changes in the initial direction of electrons beam movements depend in a first approximation, on the angle of incidence and also on the modality of arrangements of atoms in the crystal.

            Consequently, with energy of electrons beam, greater then energy described up for potential barrier (40 eV), and having a normal incidence of electrons beam, and specific cut of nickel crystal, scattered electrons follow, generally, an cometary trajectory. They pass trough electronic shield, they make an orbital rotation around the nucleus and they return on a specific dependent energy curve.

            When we increase again the energy of acceleration, from 40 and 54 eV, trajectories of electrons from fascicle beam are changed from elliptic to parabolic, respectively hyperbolic form. At 54 eV for nickel crystal, and specified arrangements of incidence beam and specific cut of face of crystal, we have a maximum of scattering corresponding to an azimuthally angle of 55º. Taken into consideration this maximum of scattering and using classical physics low it is possible to make measurements of crystal distance between atoms.

            If we increase again the acceleration (between 54 and 70 eV) the trajectory of accelerated electrons is represented by large hyperbola. But with increasing of energy and extension of hyperbola form, the scattered trajectories intersect the electronic shells of neighborhood atoms. So with the increasing acceleration of electron beam the trajectory is perturbed by the neighboring electrons shells, and the maximum of scattering at a specific angle decrease till disappear completely. Also with increasing energy it is possible to have an interaction of electrons with the second layer of atoms in crystal.

The different form of trajectory of electron at scattering on crystal is suggested also by the changes in the position of maximum angle of scattering. So, at 40-44 eV the hump develops at d=60º angle of scattering (fig 1.3), corresponding to an extended ellipse or parabola, but if energy is increased, after scattering the trajectory is broadened and the angle for maximum scattering became d=50º. If the energy increase again the angle of maximum scattering decrease at d=45º, corresponding to a large hyperbola.

            Keeping the same configuration of incidence angle and the crystal, and increasing the energy over the 75-80 eV the scattering of electrons is a little bit more complicated. The electrons movement is more perturbed from the neighboring atoms and it is possible of having scattering due to the second layer of atoms from crystals.

 

1.1.3              Determination of impact parameter

 

We are interested to measure the ,,impact parameter” b for an electron scattering around nucleus considered in the M position (fig 1.3).

 

 

Figure 1.3 Scattering of electrons by nucleus

 

In case of electric forces between two charges Q and q

 (1.5)

and E>0 we want to find the deflection parameter δ. The mass m is considered ,,coming” from infinity with velocity v∞ and an  impact parameter b (“distance”) and passing the center of force (mass M) caused by the attractive force. For the deflection angle δ, it holds that

 (1.6)

The quantities φ and Θ are further related by

Insertion in   yields

 (1.7)

The radius in polar coordinates is given by

 (1.8)

For r=∞, it follows from  that

 (1.9)

            In then holds that

(1.10)

With the constants E (energy) and h=L/m (constant of angular momentum):

(1.11)

 (1.12)

 

 ( 1.13) (see chapter atomic structure)

 

Insertion of and into  (1.14) then yields:

 (1.15)

 

            Using the value obtained for deflection angle we can estimate the b parameter (the value of g is approximately 1):

                         (1.16)

For nickel atoms in Davisson Germer experiment:

 

b=9*109*28*(1.6*10-19)2/(2*54*1.6*10-19*0.97*1.7)=2.26*10-10m

 

            The obtained value is a little bit different from the value obtained by wave diffraction, but this is not so important; more important is the fact that Davisson Germer experiment can be explained in a corpuscular theory.

            The scattering of electrons is high dependent on the angle due to relative position of atoms reported at fascicle beam. Therefore it is possible at the same energy to rotate the crystal and the angle of scattering is modified.