alpha-particle, mean life

\(\alpha\)-particle:

It is the nucleus of Helium atom its charge is +2e and mass is 4amu=4unit. The moving \(\alpha\) -particle is deflected by electric and magnetic field. The moving \(\alpha\) -particle ionises the medium through which it is travelling it's denoted by \(2He^4\).

Range of \(\alpha\) -particle:

The maximum distance covered by \(\alpha\) -particle in a medium before coming to test by R. The range of \(\alpha\) -particle depends upon

1. Initial kinetic energy of \(\alpha\) -particle
2. Ionisation potential of the gas of medium.
3. The density (or pressure) of medium.

Experimentally it is found that the range of \(\alpha\) -particle at particular pressure & temperature is found to be directly proportional to cube of speed of \(\alpha\) -particle.

i.e \(R\propto V^3\dotsm(1)\)

We have kinetic energy of \(\alpha\) -particle=1m\(V^2\)

or, \(E=\frac12 mv^2\)

\(\Rightarrow\;\; v\propto E^{\frac12}\)

From (1) & (2)

\(R\propto [E^\frac12]^3\)

\(R\propto E^\frac32\)

\(R=aE^\frac32\dotsm(3)\)

Where a is proportionality constant.

This is known as Geiger law.

Geiger-Nuttal Law:

Experimentally it is found that long life lived radioactive substances emits \(\alpha\) -particle of least energy. Geiger-Nuttal plot the graph of long (\(lambda|)) along Y -axis and long(R) along X -axis for different radioactive series and found that the graph is straight line for each each series.

Mathematically,

\(log\lambda=A_BlogR\dotsm(4)\)

Fig

Taking log on both sides of equation (3)

\(logR-loga +\frac32logE\)

\(\Rightarrow log(R)=\alpha log(E)\dotsm(5)\)

From (4) & (5)

\(log (\lambda)\propto log(E)\)

\(\Rightarrow log\frac{0.693}{T_\frac12}\propto log(E)\dotsm(6)\)

From (6) we can say that long life radioactive substance emits least energetic \(\alpha\) -particle.

Importance of Geiger-Nuttal law:

1. By measuring the value of slope of each graph & it's Y-intercept decay constant oat different radio-active substance can be determined.
2. If the graph of long(\(\lambda\)) and log(R) is straight line, then the law of radioactive disintegration i.e verified.

\(\alpha\) -Disintegration Energy:

The total kinetic energy of emitted active substances is called as \(\alpha\) -Disintegration Energy.

If parent nucleus of mass M at rest disintegrates into and \(\alpha\) -particle of mass m and daughter nucleus of mass M then from the conservation of linear momentum.

mv+MV=0

u=0

M X 0= 0

i.e mv=MV ----(1) taking magnitude only.

v= speed of \(\alpha\) -particle

V= Speed of daughter nucleus

\(\alpha -disintegration = \frac12 mv^2+ \frac12 MV^2\dotsm(2)\)

Substituting v=\(frac{mv}{M}\) in equation (2)

We get,

\(\alpha\) -disintegration energy=\(\frac12 mv^2+\frac12 M\cdot\frac{m^2v^2}{M^2}\)

\(=\frac12 mv^2[1+\frac mM]\dotsm(3)\)

=(intial K.E of particle) [1+\(\frac{mass\;of\alpha -particle}{mass\; of\; daughter \;nucleus}]\)

Mean life time or average life time:

It is the raatio of life time fo all the radioactive atoms in the sample to the total number of atoms in the sample

\(\bar T\)=sum of life of all atoms in the sample/tootal number of atoms

Let, dn, be the number of atoms decay in between t and t+dt (dt\(\rightarrow 0\)). The life time of dN atoms= dN.t---(5)

we have,

$$N=N_\circ e^{-\lambda t}$$

$$\frac{dN}{dt}= -\lambda N_\circ e^{\lambda t}$$

$$\Rightarrow |dN|= \lambda N_\circ e^{\lambda t}dt\dotsm(6)$$

Therefore life time of all atoms in the sample

\(=\int_0^\infty\lambda N_\circ e^{-\lambda t}\cdot t\cdot dt\)

\(=\lambda N_\circ\int_0^\infty e^{-\lambda t}\cdot dt\)

The average life time (\(\bar T\))=\(\frac{\lambda N_\circ\int_0^\infty e^{-\lambda 6\cdot t}\cdot dt}{N_\circ}\)

\(\bar T=\lambda\int_0^\infty e^{-\lambda t}\cdot t\cdot dt\)

=\(\lambda \biggl[\frac{e^{-\lambda t}\cdot t}{-\lambda}\biggr]_0^\infty -\int_0^\infty\frac{e^{-\lambda t}}{-\lambda}\biggl(\frac{dt}{dt}\biggr)\cdot dt\)

=\(\int_0^\infty\frac{e^{-\lambda t}}{-\lambda}\cdot dt\)

=\(\biggl[\frac{e^{-\lambda t}}{-\lambda}\biggr]_0^\infty\)

=\(-\biggl[\frac{e^{-\infty}}{\lambda}-\frac{e^0}{\lambda}\biggr]=\frac{1}{\lambda}\)

\(\therefore\bar T=\frac{1}{\lambda}\dotsm(7)\)

Hence, average life time of a radioactive sample is the reciprocal of decay constant.

Relation between (\(\bar T\)) and (\(T_\frac12\)):

we have,

\(T_\frac12=\frac{0.693}{\lambda}\)

and \(\bar T=\frac{1}{\lambda}\Rightarrow\lambda=\frac{1}{\bar T}\)

or, \(T_\frac12=\frac{0.693}{\frac{1}{\bar T}}\)

\(T_\frac12=\bar T\times 0.692\dotsm(8)\)

Hence, half life time of a radio active substance is 69.3% of its mean life time.

Reference:

Reviews of Modern Physics. Lancaster, P.A.: Published for the American Physical Society by the American Institute of Physics, 1952. Print.

Wehr, M. Russell, and James A. Richards. Physics of the Atom. Reading, MA: Addison-Wesley Pub., 1984. Print.

Young, Hugh D., and Roger A. Freedman. University Physics. Boston, MA: Pearson Custom, 2008. Print.

Adhikari, P.B. A Textbook of Physics. 2070 ed. Vol. II. Kathmandu: Sukunda Publication, 2070. Print.