The Dimensionality of the Geiger-Nuttall Law: a simple but interesting note

The Dimensionality of the Geiger-Nuttall Law: a simple but interesting note

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

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

Abstract. The natural logarithm form of the empirical Geiger-Nuttall law is not dimensionally correct. A new type of this form and four new non-exponential forms of this law are suggested.

Keywords: Geiger-Nuttall law, quantum mechanics, dimensionality, alpha, emitter.

Derivations and Discussion. One of the milestones of quantum mechanics was formulating the Geiger-Nuttall law {1, 2}. This law states that there is a simple empirical relation between the α-decay energy Q_α and the half-life of heavy alpha (α)-emitter t_1/2

logt_1/2 = A + B/√Q_α

where A and B are constants. Let us convert this equation into a natural logarithm (LN) form

lnt_1/2 = a + b/√Q_α

where a = 2.303A and b = 2.303B.[1] The kinetic energy, E_α, of the emitted α-particle is slightly less (about 0.4 % for heavy α-emitters, such as the uranium α-emitters) than Q_α. Therefore, the last equation can be written as follows

lnt_1/2 = a + b/√E_α    … (1)

After the conversion of this equation into the exponential form we have

t_1/2 = e^{a + b/√Eα}    … (2).

The dimension[1] of the left side of this expression is T but its right side is dimensionless. Hence, eqn. (2) is dimensionally incorrect. The simplest way to make this equation dimensionally correct is to multiply its right side with the constant θ equal or close to 1 and expressed in time unit of t_1/2 (e.g. seconds)

t_1/2 = e^{a + b/√Eα} × θ.

Recently, Premović {3} derived a new mathematical form for the Geiger-Nuttall law. This is expressed by the following equation

t_1/2 = u × w × 1/√E_α

where u and w are the constants. According to Premović [1], the dimension of u is L and the dimension of (w × 1/√E_α) is [L⁻¹ × T] or the dimension of w is M Therefore, the left side of this expression is dimensionally consistent with its right side.

Referenes

[1] H. Geiger and J.M. Nuttall, The ranges of the α particles from various radioactive substances and a relation between range and period of transformation. Phil. Mag., 22, 613-621 (1911).

[2] H. Geiger and J.M. Nuttall, The ranges of α particles from uranium, Phil. Mag., 23, 439-445 (1912).[3] P. I. Premović, The lifetimes of ²³⁸U nuclei, the escaping attempts of their alpha particles and the Geiger-Nuttall law. The General Science Journal, December 2021.

---

[1] lnt_1/2 = 2.303logt_1/2.

[2] Just to remind the reader that M, L and T are the symbols for basic dimensions:  mass, length and time, respectively.

[3] Since  E_α = 1/2mv² (where m is the mass of the α-particle and v is its speed) and the dimension of v is LT^-1 then the dimension of E_α is ML²T^-2