Electron Interference with Divergent Electron Beam

Introduction

            The angular distribution of electrons is controlled by two physical entities.

First there is the wave character of the probability density of the electron’s position.  This wave character results in a definitive distribution of the electrons after many electrons have been accumulated at the display screen. 

The second is the particle character of the electrons.  This says that each electron can add to the accumulation at any angular location.  However, the overall accumulation after many electrons will be the same as predicted by the wave character.

So here is the quandary: After accumulating many electrons at various angles, must I redistribute the electrons to at least approximately match the wave character prediction?  That is not a physically acceptable answer.  What we must do is to use the computer’s random number generator to select from an angularly weighted probability distribution.  That weighted probability distribution and the way it is random selected will be discussed below.

Figures

In order to handle all the variation it is necessary to do a rather complicated integral over the input wave function width taking into account the tilt of the wave fronts. Let the tilt angle be qi and the observation angle be qd Also let the length of the input column be L and its left center be (xi0,yi0). See Figure 1 and Figure 1a.

Figure 1: Simple diagram of the experiment showing the tilt angle of the electron wave function.

 

Figure 1: Diagram of the experiment.  The items involved are reasonably well labeled.  It must be understood that there is a very large distance between the slitted screen and the Display screen.  That is depicted by the well-labeled spatial seperator.

The Interference Peak Angles and Number

It is well known that the angles at which diffraction peaks occur is:

                                                                                                   (1)

where d is the slit spacing, l is the wavelength, and qi is the angle of incidence on the left side of the grating. Of course the absolute value argument of the arc sine in the above expression has to be less than 1.0.  Therefore the limits of the m values are

           

                                             

Effect of Number of Slits, N

Equation 1 makes the assumption that the phase differences between the light emanating from adjacent slits is 2p.  When qd is different from qm the phase difference is 2p+f where

                                                                                                             (2)

  So the sum that normally results in N becomes:

                                                                                                                     (3)

The sum in equation 2 is the geometric series and it results in:

                                                                                             (4)

The intensity at the angle that corresponds to f is the magnitude squared of equation 4 and becomes:

                                                                                                            (5)

Equation 5 indicates that the intensity will drop to half of its peak value when

                                                        

Effect of Finite Slit Width

                                                                                               (6)

where Ssw(qd)=1 when qd=0.  Of course, we cannot have partial electrons at any of the qm peaks.  We therefore must compute what I call a weighted probability function.  This is done as discussed in the Appendix.

 

 

Appendix:

Use of Uniform Random Number Generator to Access Weighted Probability Distribution

Introduction

            The impetus for this document is an electron diffraction animation.  I wanted to show the buildup of the wave-directed electrons in the far field of a slitted screen upon which electrons are incident one-by-one.  After many electrons have passed, the envelope of the histogram of number of electrons Vs far-field angle will be a product of two sinc functions.  This is the weighted probability distribution of the title. Since there is no way to divide individual electrons among the entire display area, I needed to apply this weighted probability distribution to each electron as it arrives.

Method

            The method that I came up with is to compress the distribution of the index numbers at which one might plot the probability density.  Each of the discrete blocks in the new index number distribution will have width proportional to the value of the probability that that block represents. This way, when we choose uniform random variables on the entire numerical domain of the blocks, we will be choosing fewer values where the original probability was low.

Example with a Gaussian Distribution

            Let the original distribution be written as:

                                                                                       (1)

where s is the width at 1/e of the distribution and dP/dx is the probability that x will be in the interval dx.

 

We will remap the numbers describing the distribution such that the range of numbers corresponding to P(x) is proportional to the value of P(x).  i.e.

                                                            

with NT>>1.

 

                                                           

where xn varies in units of (xmax-xmin)/M where M>>1.

Having computed ki, and saved it to an array, then we use the computer’s uniformly distributed random number generator to select a number, grandom, from the range 0 to kmax.  Then we use the following algorithm to compute the i value to which g corresponds:

int i=0;

                     do

                     {

                           i++;

                     }

                     while(k[i]<grandom);

                     return i;  

The final result for the Gaussian appears as shown in Figure 1:

Figure 1: Both the original Gaussian probability distribution and the weighted profile (really a frequency distribution) generated by the compression technique described above.