Half life

La Average life is the average life of a core or a free subatomic particle before disintegrating. It is represented with the Greek letter Δ Δ {displaystyle tau } (Tau). The disintegration of particles is a probabilistic process (specifically follows the Poisson law) so this does not mean that a certain core will take exactly that time to disintegrate. Average life should not be confused with the period of semi-disintegration, semi-period, half-life or half-life: they are related but different concepts. In particular, the semi-disintegration period applies only to radioactive substances and not to free particles as it is said.

It has been verified that the isotopes of radioactive elements present different degrees of instability over time because each isotope decays or transforms into others following a particular radioactive series. To refer to the speed with which nuclear disintegrations occur, we use the concept of half-life.

Calculation of τ {displaystyle tau }

Notation: In what follows, atoms means atoms of a given radioactive isotope.

• Δ Δ {displaystyle tau } is the average time of the sample.
• N(t){displaystyle N(t)} is the number of atoms in the sample at the time t.
• N0{displaystyle N_{0}} is the initial number (when t = 0) of atoms in the sample.
• λ λ {displaystyle lambda } is the constant disintegration.

During a time interval dt, the number of atoms that disappear from the sample dN is equal to the population change of the sample (note the negative sign that means negative increase or decrease):

− − dN=N(t)⋅ ⋅ λ λ ⋅ ⋅ dt{displaystyle -dN=N(t)cdot ;lambda cdot ;dt,}

The solution of this differential equation gives us the exponential variation of the population of radioactive atoms with time:

N(t)=N0e− − λ λ t{displaystyle N(t)=N_{0}e^{-lambda t},}

The average time Δ Δ {displaystyle tau }, that is, the average duration of a radioactive atom in the sample results from the following evaluation:

Δ Δ =1N0∫ ∫ 0∞ ∞ tdN=− − ∫ ∫ 0∞ ∞ λ λ te− − λ λ tdt=1λ λ {displaystyle tau ={frac {1}{N_{0}}}}}{int _{0}{infty }t,dN=-int _{0}{infty }lambda ,t,e^{-lambda t},dt={frac {1}{lambda }}}}}}}}}}}}}}}}}}}}}{

Relationship between half-life and half-life

Average time or average life (Δ Δ {displaystyle tau }) is equal to the reverse of the constant disintegration (λ λ {displaystyle lambda }).

So it also turns out that Δ Δ {displaystyle tau } is equal to the time necessary for the number of atoms to be reduced in a factor e; and it relates to the period of semi-deintegration, half-life, hemivida or semi-life, according to the following formula:

t1/2=Δ Δ ⋅ ⋅ ln 2{displaystyle t_{1/2}=tau cdot ln 2}

Half-period

Half-period is confusing. For example, in the description of particle accelerators it is said:

• The magnetic field adjusts so that the time needed to travel the semicircular path within the electrode is equal to the semiperiod of the oscillations. Consequently, when the ions return to the middle region, the electric field will have reversed their sense and the ions will then receive a second increase in speed as they move into the other 'D' [1].

• Loaded particle accelerator. Cyclothron - The magnetic field adjusts so that the time needed to... Consequently, when ions return to the middle region, the field...[2].

• The cyclotron... at the speeds of the ions, the time needed for the tour... We can calculate the semiperiod, considering that the time you... nestorc/elecmagnet/ciclotron/cycle.html (breakable link available on the Internet Archive; see history, first version and last)..

• The magnetic field adjusts so that the time needed to travel the semicircular path within the electrode is equal to the semiperiod of the Cyclothron oscillations.