Looking At Basic Atomic Physics (3)

(Continued From Part 2)

4. Radioactive Decay:

We begin with the recognition of the various types of radioactivity - in the form of their specific particles- and how these can be measured in terms of activity. Probably the last part is the most intuitive and easiest, so we begin with that: using a device called the Geiger Muller counter (shown in Fig. 1) with the general principle illustrated in Fig. 10 below:

Fig. 1: The Geiger –Muller Counter

First, note that Fig. 2 is a simplification. In actual fact the anode is a tungsten wire with extends through the cathode, which is a metal cylinder. Both are then fitted into a gas-filled envelope which is the Geiger-Muller tube. A typical low pressure gas to use for the tube is Argon (Ar) which at a pressure of about 10 cm of Hg (compared to 76 cm Hg for atmospheric) easily fills the tube via diffusion. A potential difference (p.d.) of 900- 1,000 V is then applied between the anode and cathode (A and C in Fig. 10). The applied p.d. has the effect of slightly stressing the Argon in the tube so when radiation enters its ionizing action produces more ions from the gas in the tube. In other words, the radioactive substance causes the gas atoms to lose electrons.

     The electric field produced (e.g. E = V/d) then accelerates these ions and many more ions are produced by way of collisions. In effect, an ionizing current is quickly built up and triggers the tube which is connected externally through an amplifier to a count-rate meter .

     Radioactivity basically occurs in three forms, as determined by the particles: alpha-radiation (from alpha particles or Helium nuclei), beta radiation, from beta particles or electrons, and gamma radiation, from gamma particles or very high energy photons (e.g. at very short wavelengths, hence can easily penetrate tissue).

     The simple diagram of Fig. 3, using a simple experiment, graphically shows the differences in the radiations with respect to an applied magnetic field.

 

    If one holds the thumb of one's RIGHT hand into the image (to represent the B-field direction) then the electrons (beta particles) will display a direction coincident with the curving fingers of the right hand. That is, clockwise. Now, since the alpha particles are positively charged (, e.g. He⁺⁺, as opposed to the negatively charged beta particles, e.g. e-) they will go in the opposite direction.

     The gamma rays, meanwhile, suffer no deflection in the field because they have zero charge, being photons of light. In terms of penetration power these additional differences apply:

1) Alpha ( a ) particles are absorbed by a few cm of air, or by an aluminum foil only 0.006 cm thick

2) Beta (b)  particles - while having less ionizing power than alpha particles (because of much lower mass) have 100 times more penetrating power. A sheet of aluminum at least 3mm thick is needed to absorb them.

3) Gamma rays produce little ionization since they have no electric charge but can pass through a block of iron a foot thick.

Typical Experiment using the Geiger-Muller Tube as depicted in the graph of Fig. 4.

Procedure:

1. After setting the high voltage supply to minimum high voltage, bring a beta source to within about 6” of the G-M tube.

2. Gradually increase the voltage until the counting just starts, disclosed by the clicking sound. This is the "starting potential" - so record the voltage at this point.

3.Increase the voltage to a value 100V above the starting potential - this should provide operation near the threshold (See Fig. 4)

4.Adjust the distance between the radioactive sample and the Geiger tube until the meter reads about 4,000 cpm. (Leave the sample at this distance for the balance of the plateau curve determination)

5. After returning the meter switch to the CRM position, record the count rate at the starting potential. The record the count rate corresponding to a series of voltages about 50 V apart.

6. When the continuous discharge region is reached immediately reduce the voltage.

7. Plot a curve on graph paper using cpm as the y-axis and voltage as the x-axis, The end result ought to be similar to that shown in Fig. 4. 

Another experiment:

Using a beta source, plot cpm as a function of distance between the source and counter. Note the extent to which the curve obtained obeys the inverse square law (i.e. the intensity or rapidity of the count is inversely proportional to the distance of the detector from the source).

At the end of the experiment, the student should be able to identify all the possible sources of error including systematic and measurement errors.

5. The Activity  of a radioactive source.

We define the activity of a radioactive source as:

A = dN/dt = - lN

 

 

Where  l  is the decay constant. The negative sign appended to the equation indicates that the amount N is decreasing with time t.

The units are shown in the relationship below::

A [Bq]  = dN/dt [s^-1]  = - lN[s^-1]

Where Bq is Becquerels..

 

 

The decay curve is obtained from the fundamental law of radioactive decay, based on some original number of atoms N_o decaying with an activity l over time t:

 

 

N =  N_o  exp (-lt)

 

 

Then:  ½d N/dt ½  = N_o  l exp (-lt)= R

 

 

Where  R is the decay rate, i.e. R = R_o  exp (-lt)

 

 

And:  R_o  = N_o  l is the decay rate at time t = 0.

The half-life is the time for half of the original (N_o ) atoms to disintegrate or when the point is reached such that:

 

 

N ®  N_o  /2  or  R ®  R_o  /2 

 

 

Then:

N_o  /2  =  N_o  exp (-l T _½)

 

 

Where  T _½ is specifically substituted for t.

 

 

After dividing N_o into both sides and taking natural logarithms we get:

 

 

l T_½  =  ln 2 = 0.693

 

 

Or:

        T _½  =  ln 2/ l  =   0.693/ l

 

 

Using this basis any sample or fossil with even a minuscule amount of radioactive material can be dated. All we need know is that over the period defined as T½  half of the number of the remaining atoms decay and the activity is in Becquerels (Bq).  Thus, if  T_½ = 15,000 yrs.  for  l = 200 Bq then if l  = 50 Bq now the sample is 45,000 years old.

A graphical depiction of generic radionuclide decay is shown below in Fig. 13.  Here the vertical axis shows a relative scale for the amount or mass  of some, unnamed decaying isotope  which commences decay at some initial specified value, e.g. 1 gram then decreases to half that original amount in one half life:

Fig. 5: Radioactive Decay for a radionuclide with time T in millions of years (mY)

    It is easy to see from the graph shown that the half-life of this nuclide is about 1 million years, so: T½ =  10⁶ yrs.

     The problem of dating radioactive fossils or other specimens  (e.g. the cloth of Turin) is really a problem of finding a radioactive isotope that is most appropriate, or one that enables the maximum accuracy for the time scale desired.  Hence, for ancient fossils one would look for isotopes that have half-lives in the millions rather than thousands of years. Failing that one would wish to have available some kind of correction method, say to correct for extraneous effects such as the atmosphere might ipose on samples.

 

 

     In many ordinary fossil dating applications, potassium-argon methods are employed, based on the relative compositions of Potassium -40 to Argon-40 gas. Typically when rocks or other items are tested the sample is split between the Potassium-40 content on the one hand and the Argon on the other. The instrument of choice to assess the ratio: K40/Ar 40 is the mass spectrometer.

 

 

     Exotic isotopes of carbon can also be used is the measurement technique is sufficiently refined. In a recent use of the isotope d ¹³C, for instance, evidence has been found for the existence of life on Earth at least 3,850 million years ago. For this purpose, quartz (zircon, zirconium) crystals have been found to be of use since they may harbor small amounts of thorium and uranium at the level of parts per billion.

 

 

     For dating samples in the millions of years, particularly for igneous rocks and samples embedded within, isotopes of lead and strontium may be used – being the ‘daughters’ from millions of years of radioactive decay.

 

 

    Now, we examine another application of the G-M tube is to find the half-life of a particular source or the activity (decay rate) - say at a place.


Practical:


A Geiger-Muller tube measures the background count at a given place to be 20/min. Over a period of time, the readings shown in the two row table below are then obtained after an unknown source is placed at the location.

Time hrs	0	6	8	10.5	20
Count/min	120	70	60	50	30

Obtain the corrected counts, determine the half -Life (T½) of the unknown source, and find the decay constant.

Solution:

First, we obtain the corrected counts by subtracting the background count of 20/m from each of the values above, to obtain:

Correct/cpm

100

50

40

30

10

We can easily see from inspection, that since the activity drops by half (from 100 to 50 cpm) in 6 hours, that 6 hrs is the half life.

Hence: T½  = 6 h = 21600 s

The decay constant l is found from:

l = ln 2/ (T½) = 0.693/ (21600s) = 3.2 x 10^-5 /s

     This can also be verified by sketching a plot of the half life ((T½) ) on the vertical axis against the time in hours - to obtain the radioactive decay curve. 



Other Problems:

1) Sketch the radioactive decay curve for the practical problem above and confirm that the half life is 6 hrs.

2) A point source of gamma radiation has (T½) = 30 mins. The initial count rate recorded by a G-M tube is 360/s. Find the count rate that would be recorded after 4 half lives. Sketch the decay curve and determine the activity, A.


3)  Find the half life of the beta particle emitting nuclide: 

 ³²P ₁₅,

If the activity A = 5.6 x 10^-7 /s.

 

 

4) A radionuclide sample of N = 10¹⁵ atoms undergoes decay at the constant average rate of dN/dt =  6.00 x  10¹¹  /s.  From this information, find:

a) The Activity A

b) The decay constant l

c) The half life of the sample in minutes.

 

 

5) (a) Find the wavelength of the first line of the ten-times ionized sodium (Na) atom.   

 

 

(b) Obtain the shortest x-ray wavelength which can be obtained using an accelerating voltage of 10 ⁴ V.

 

 

6) The activity of a radio-nuclide is given as:

A =  A_o  exp (-lt)

Where  A_o is the decay rate at time t = 0, and A refers to the decay rate at some time t thereafter. If a particular radio-nuclide has  A_o  = 1. 1 x 10 ¹⁰  decays/sec and a half life T_1/2 = 28.0 years, find:

a) the decay constant, l ,

b) The activity  A after 1 hour,  after 2 hours.

c) The activity  A  after 49 years,