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RADIOACTIVE DECAY
Kholkhodjayev B.A.
Dots.
Dilmurodov J.J.
Student.
Tashkent State Technical University,
Asatullayeva D.B.
Student
Uzbekistan State Word Languages University
Tel.: +998 90 082 81 73
https://doi.org/10.5281/zenodo.15344042
Annotation.
The term 'radioactive decay' is used in the Uzbek language, although its
core terminology originates from international scientific nomenclature. This
article explains the fundamental concepts and calculations related to radioactive
decay.
Keywords.
Radioactive decay, alpha decay, beta decay, gamma decay, half-life, electron
capture, exponential functions, differential equation, radiocarbon method, C-14
isotope, nuclear physics, radiation, ionizing radiation, mass reduction.
Introduction.
To analyze the origin of this term, attention must be given to its
components:
Radioactive:
This
word
derives
from
the
Latin
'radius,'
meaning
'ray.'
The phenomenon of radioactivity was discovered at the end of the 19th century
and the term became widely used in scientific literature of that time.
Decay:
This is an Uzbek word meaning 'fragmentation' or 'disintegration.'
It is used to describe the process of atomic nuclei of radioactive substances
breaking apart.
Radioactive decay is the process by which unstable atomic nuclei break
down into smaller nuclei or other particles, emitting ionizing radiation. This
process can occur naturally or be induced artificially.
Main Part. Types of Radioactive Decay
1.
Alpha Decay
An alpha particle (²⁴He, composed of 2 protons and 2 neutrons) is emitted
from the nucleus.
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As a result, the atomic number decreases by 2 units, and the mass number
decreases by 4 units.
Example:
2.
Beta Decay
A neutron transforms into a proton, emitting a beta particle (electron).
The atomic number increases by 1 unit, but the mass number remains
unchanged.
Example:
3.
Gamma Decay
The nucleus releases excess energy in the form of gamma radiation.
The
mass
number
and
atomic
number
remain
unchanged.
Gamma decay often occurs after alpha or beta decay.
4. Positron Emission
A proton transforms into a neutron, emitting a positron (e⁺).
The atomic number decreases by 1 unit, while the mass number remains
unchanged.
4.
Electron Capture
The nucleus captures an electron from the surrounding electron cloud,
converting a proton into a neutron.
The atomic number decreases by 1 unit, and the mass number remains
unchanged.
Applications
In Medicine: Radiotherapy, X-ray diagnostics.
In
Energy:
Power
generation
at
nuclear
power
plants.
In
Archaeology:
Dating
using
the
radiocarbon
(¹⁴C)
method.
In Military: Nuclear weapons.
Radioactive Decay Equation
Let m(t) represent the mass of a radioactive substance at time t. It is known
from physics that the rate of radioactive decay is proportional to the remaining
mass of the substance, that is:
m' = -km (1)
where k = const > 0 is the proportionality coefficient.
Thus, m = m(t) satisfies the separable differential equation (1). Solving it yields
the general solution:
m = ce^(-kt)
If the initial mass at time t = 0 is m₀ > 0, then c = m₀, and the mass changes
according to:
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m = m₀e^(-kt) (2)
Over time, the mass decreases exponentially towards zero.
Half-Life
The half-life T is the time required for half of the initial radioactive
substance to decay.
Setting m(T) = m₀/2 gives:
m₀/2 = m₀e^(-kT)
which leads to:
T = (ln2)/k or k = (ln2)/T.
This formula allows determining k if T is known (which is relatively easy to
measure).
Application to Radiocarbon Dating
It is known that living organisms absorb C-14 isotopes from the
atmosphere along with stable C-12 carbon. During the life of an organism, the
ratio of C-14 to C-12 remains constant at some value m₀. Upon death, the intake
of C-14 stops, and its amount begins to decrease.
The half-life of C-14 is approximately 5570 years. Therefore:
k = ln2/5570 ≈ 1.24 × 10⁻⁴ year⁻¹.
Thus, the mass of C-14 at time t after death is:
m(t) = m₀e^(-t/8000).
If m(t) is determined (for instance, by measuring the emitted particles),
then the time elapsed since the death of the organism can be calculated by:
t = 8000 × ln(m₀/m(t)).
This formula allows determining the age of ancient organic materials.
References:
1. Shokhamidov Sh.Sh. “Elements of Applied Mathematics,” Tashkent,
Uzbekistan, 1997.
2. Fikhtengolts G.M. Fundamentals of Mathematical Analysis.
3. Azlarov T., Mansurov H. Fundamentals of Mathematical Analysis, 2005.
4. Berman: Problem Book in Mathematical Analysis, 1989
