The Single Slit Experiment: Wavelength, Diameter and Speed of Free Subatomic Massive Particle

 

The Single Slit Experiment: Wavelength, Diameter

and Speed of Free Subatomic Massive Particle

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry, 

University of Niš, pavleipremovic@yahoo.com, Niš, Serbia

aIn modern physics, it is now widely accepted that light (or in more general terms, electromagnetic radiation) and has a dual nature. Depending on the experiment it could behave as a wave (as in interference, diffraction and polarization) and it could be taken as a particle-photon as the explanation of the photoelectric effect). The particle-wave duality has been also verified experimentally for the subatomic matter, e. g. electron.

Of all the experiments in physics, it is perhaps the single slit experiment with electrons that most clearly demonstrated the particle-wave nature of matter.[1] Moreover, the single-slit experiment for the electrons is the simplest example of electron diffraction.

In this communication, we consider the relationships between the wavelength, diameter and speed of a free subatomic (massive) particle (hereinafter particle) in the single-slit thought experiment. We postulate that this particle exhibits both wave and particle properties.

Figure 1. The schematic diagram of the single slit experiment.

The single slit experiment with the particle beam is shown schematically in Fig. 1. A source of the monoenergetic particle beam is set up pointing against a wall containing a small rectangular slit of width w_s. Far behind the slit is situated a (phosphor?) screen.  The screen is excited by the particles, allowing you to see the pattern made by the diffracted particles.

Let us deal first with the beam of “non-relativistic” particles.[2] Suppose these particles are spherical and has a diameter at rest D₀. To obtain the diffraction pattern of this beam on the screen depends upon the width of the slit w_S, de Broglie’s wavelength λ of the particle and its diameter D₀.

There are four possibilities. The first is that λ and D₀ are less than w_S. In this case, the diffraction pattern on the screen would not be obtained. The beam simply travels onward in a straight line creating a spot on the screen just as it would if no slit was present. The second possibility is that λ and D₀ are greater than w_S. We would observe neither the diffraction pattern nor the spot. The third possibility is that λ is less than w_S and D₀ is greater than it. We would again observe neither the diffraction pattern nor the spot. The fourth possibility is that λ is greater than w_S and that D₀ is less than it. Following the numerous single-slit experiments with subatomic particles like electrons, we know that the best observable diffraction pattern[3] would be obtained if

λ ⪆ w_S     … (1)[4], and D₀ << w_S.

The momentum of a non-relativistic massive particle p = m₀υ where m₀ is the mass of a particle at rest and υ is its speed. Its momentum p and wavelength λ are linked by de Broglies’s relation

λ (= h/p) = h/m₀υ    … (2)

where h (= 6.63 × 10³⁴ J sec) is Planck’s constant, m₀ is the mass of a non-relativistic particle at rest and υ is its speed. Combining exp. (1) and eqn. (2) we get

λ = h/m₀υ ⪆  w_S

or

h/m₀w_S ⪆ υ   … (3).

Premović {1} derived the following expression for the minimum speed of a free non-relativistic particle

υ_min = h/m₀D₀.

A simple consideration shows that υ has to be much smaller than υ_min or

υ << h/m₀D₀    … (4).

Indeed, combining exp. (3) and exp. (4) we get

D₀ << w_S.

Examples:

(1) A neutron has about the same diameter as a proton or about 1.7 × 10⁻¹⁵ m. Their mass is about 1.675 × 10⁻²⁷ kg. Employing exp. (4) we find that the speed of free neutron (or proton) has to be << 2.3 × 10⁸ m sec^-1 (or about << 0.8c) to observe its best diffraction pattern passing through the slit width w_S >> D₀ (= 1.7 × 10⁻¹⁵ m).

(2) The mass of the electron is about 9.1 × 10^-31 kg. It is believed that its diameter[5] is much less than 10^-15 m.  A very rough estimation based on exp. (4) shows that the speed of this particle has to be relativistic (much higher than 0.1c). In this case, it requires the relativistic correction of this expression.

Special relativity predicts that the above spherical non-relativistic particle traveling at a relativistic speed would contract in the direction of motion becoming the prolate spheroid-shaped relativistic particle {1}.  Its length would be shortened in the direction of its motion, by the factor (1– υ²/c²)^1/2 but there is no contraction in the perpendicular direction. In other words, the polar axis of this spheroid would be equal to the diameter of the particle in rest, D₀.

According to Special relativity, if the rest mass of a particle is m₀ then its relativistic mass m would increase by the Lorentz factor or

m = m₀/√(1 – υ²/c²).

For this reason, exp. (4) must be relativistically corrected as follows

υ << h√(1 – υ²/c²)/m₀D₀

or 

υ/√(1 – υ²/c²) << h/m₀D₀    … (5).

After a bit of algebra and subsequent calculation, it turns out that the free electron speed has to be much less than c or non-relativistic.

It is interesting to note here that a calculation based on exp. (5) shows that for υ << υ_min = c that the free electron diameter D₀ should be >> 2.5×10^-12 m.

Reference

{1} P. I. Premović, The Minimum speed of a free massive particle. The General Science Journal, December 2021.

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[1] Of note is that all systems that pass the single slit have a dual particle-wave character. Note that the notion of “wave” in this context means de Broglie wave.

[2] In general, the “non-relativistic” particle is that whose speed υ is far less than the speed of light or υ << c (= 2.99792×10⁵ km sec^-1). Physicists usually assume that the massive particle with υ/c ≤ 0.1 (or υ ≤ 0.1c) is “non-relativistic”. They also define the massive particle with υ/c > 0.1 (or υ > 0.1c) as “relativistic”.

[3]  Of note, no device can transform back the diffracted monoenergetic electrons into the beam.

[4] Many books, papers, and other publications state that the slit width is "adequate," "comparable" to the wavelength of the particles, or "of the same order of magnitude" as that. Scientifically speaking, these are rather “vague” formulations. For that  reason, we opted for the interpretation given in exp. (1).

[5] The electron is a point-like particle with no measurable dimensions, at least within the limitations of current instrumentation capability.