The Minimum Speed of a Free Massive Particle

 

The Minimum Speed of a Free Massive Particle

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

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

Abstract. The equations for the minimum speed and kinetic energy of a free massive particle are derived within the non-relativistic and special relativistic frameworks. These equations are based on de Broglie's relation between momentum and wavelength of this particle.

Keywords: Particle, de Broglie’s wavelength, speed, kinetic energy, relativistic.

The particle-wave duality of the matter is one of the biggest mysteries in science because a massive particle and a wave are opposite each other in every way. Indeed, this particle is a discrete entity enclosed to a relatively very small volume of space, while a wave propagates over a large region of space.

Introduction. In modern physics, it is now widely accepted that light (or in more general terms, electromagnetic radiation) has a dual nature. The wave-like nature of light explains most of its properties: reflection, refraction, diffraction, interference and Doppler effect. On the other hand, the photoelectric effect and Compton effect can be only explained based particle-photon nature of light. The wavelength of this photon λ can be expressed as follows

λ = h/p    … (1)

where h (= 6.62×10^-34 J sec^-1) is Planck’s constant and p is the momentum of photon.

Elementary physics shows that the momentum of a free massive particle p is

p = mυ

where m is the relativistic mass of this particle and υ is its speed relative to an observer at rest. De-Broglie postulated that a free massive particle shows the dualistic wave-particle character just like light and the eqn. (1) can be also applied to this particle

                                                                             λ = h/mυ    … (2).

aIn the first part of this paper, we will deal with a free[1] “non-relativistic” massive particle (hereinafter “non-relativistic” particle) and in the second part we will deal, in general with a free massive particle 

(hereinafter a free particle).

In general, the “non-relativistic” particles are those 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 particles with υ/c ≤ 0.1 (or υ ≤ 0.1c) are “non-relativistic”. De Broglie’s particle-wave duality has been also verified experimentally for the “non-relativistic” (e. g. electron or neutron diffraction) and relativistic (e. g. electron diffraction) massive particles.

Derivation and Discussion. Suppose now that the “non-relativistic” particle is spherical and has a diameter and a mass at rest: D₀ and m₀. Obviously, de Broglie’s wave needs a medium through which to spread and that is of course - matter. From analogy with light, we hypothesize that the maximum de Broglie’s wavelength λ_max of a “non-relativistic particle” must be equal to its diameter.

Mathematically speaking

λ_max = D₀.

Applying the formula (2), we get

λ_max = h/m₀υ_min

where υ_min is the minimum speed of a “non-relativistic” particle. Substituting λ_max of this equation with

D₀ and after some rearrangement we get

υ_min = h/m₀D₀ = β    … (3).

According to this equation, in order for a massive “non-relativistic” particle to have a minimum velocity β equals zero its diameter must be infinitely large, which is, of course, impossible.

The minimum kinetic energy of a “non-relativistic” particle is

KE_min = 1/2(m₀β²)    … (4).

In sum, eqn. (3) appears also to set a lower limit for its speed, momentum and kinetic energy.

Examples:

(1) Alpha (α)-particle consists of two protons and two neutrons. The mass of the α-particle is about 6.65×10^-27 kg and its diameter is about 3.6 × 10^-15 m.  Employing eqn. (3) we calculate that the minimum speed of this particle is about 2.8 × 10⁷ m sec^-1 (or about 0.093c). In other words, this is roughly the lowest possible speed of the free α-particle.

The kinetic energy of the free α-particles emitted by the uranium isotopes (²²⁶U - ²³⁸U) ranges from about 4.2 MeV (= 6.7×10^-13 J) up to about 7.6 MeV (= 12.1×10^-13 J). Using the expression for the “non-relativistic” kinetic energy KE = 1/2(mυ_α²) we estimated that the speed of their α-particles υ_α range is about 1.4 × 10⁷ m sec^-1 up to about 1.9 ×10⁷ m sec^-1. These values agree in order of magnitude with the above rough estimate of the lowest possible speed of the free α-particle.

Alpha particles are relatively big and heavy and are not able to penetrate very far through a medium. As a result of scattering collisions with various nuclei of the medium through which the free α-particle passes, its kinetic energy is reduced and usually becomes a helium atom capturing two electrons from its surroundings.

(2) Macroscopic non-relativistic objects would have a minimum speed extremely low. For the golf ball with a mass of about 0.05 kg and a diameter of 0.05 m we estimate [using eqn. (3)] its minimum speed would be about 2.7×10^-31 m. The average speed of a golf ball is about 50 m sec^-1.

In the next part of this paper, we will consider a relativistic particle and the consequences of that consideration on the case of a “non-relativistic” particle.

Strictly speaking, all free massive particles are relativistic. Even if their speed is much less than the speed of light (or υ << c), they are still relativistic. So, there is only a relativistic free massive particle or better to say a free massive particle, in general. This is why the term non-relativistic is put under quotation marks.

Special theory of relativity sets the light speed c as an upper limit to the speed of a massive particle. According to this theory, the mass of a relativistic particle

m = m₀/√(1 – υ²/c²)   … (5).

However, Special relativity states 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, Fig 1.  Its diameter at rest D₀ would be shortened in the direction of its motion, by the factor (1– υ²/c²)^1/2.

 

In equation form,

L = D₀√(1 – υ²/c²)    … (6)

where L is the length of a relativistic particle[1] (hereinafter length) along its direction of motion Fig.

1.

Multiplying eqns. (5) and (6) we have

 

mL = m₀D₀.

Eqn. (6) shows that in contrast to the previous non-relativistic particle whose diameter D₀ is assumed to be constant; the length of a relativistic particle L depends on the speed of this particle υ. Obviously, its wavelength λ cannot be larger than the length L. In equation form,

a

Fig. 1.  The shape and dimensions of the free spherical particle at relativistic speed.

λ ≤ L = D₀√(1 – υ²/c²).

Combining eqn. (2) and the left end of this equation we have

υ ≥ h/mL.

Substituting into this equation m₀D₀ instead mL we obtain

υ ≥ h/m₀D₀

having a minimum value

υ_min  = h/m₀D₀ = β.

So, its minimum kinetic energy

KE_min [= (1/2mυ_min²)] = 1/2(m₀β²)

These equations are identical to eqn. (3) and eqn. (4) derived for υ_min and KE_min of a “non-relativistic” particle. In other words, our approach to the minimum speed and kinetic energy of  “non-relativistic” massive particle in the first part of this work sounds reasonable.

Conclusion. The equations for the minimum speed and kinetic energy of a free massive particle are derived. These equations are based on the non-relativistic and special relativistic formulations using de Broglie^’s relation between linear momentum and wavelength of this particle.

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[1] It is a free particle in the sense that it is experiencing no net force. 

[2] In fact, its equatorial axis.