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Keele University Physics/Astrophysics Laboratory 21
The Range of Beta Particles in Aluminium
- Introduction When a nucleus changes its state by beta decay the energy spectrum of the emitted beta particles extends from zero up to a maximum energy E that is characteristic of the particular isotope. If these beta particles pass through a material such as aluminium (Al) then some of them are absorbed. The number that are absorbed depends upon both the energy of the particles and upon the thickness of the material. It is possible to define a quantity R, known as the range which is equal to the distance at which all beta particles are absorbed multiplied by the density of the absorbing material. In the scientific literature Feather reports that the equation
E = 1 84. R +0 249. (1)
relates the maximum energy E of the beta particles in MeV to the range R in units of g/cm^2 of Al. The ultimate aim of this experiment is to test this equation. In this experiment the source of beta particles is a strontium(Sr)-90 source. For Sr^90 it is the daughter product of the decay which emits the most energetic beta particle according to the following decay scheme.
38 Sr^90 28 yrs^39 Y^90 +^ β-
39 Y^90 60 hrs^40 Zr^90 +^ β-^ (2.26 MeV) The value of E can, for the purposes of this experiment, therefore be taken to be exactly 2.26 MeV and if R is measured experimentally then it is possible to test Feathers equation. The experiment basically consists of using a Geiger-Muller detector to measure the number of beta particles transmitted through various thicknesses of Al. As the successive layers of absorbing material are interposed between the beta source and the detector the intensity of the transmitted flux of betas decreases rapidly at first and approaches zero almost asymptotically. Hence the endpoint is rather difficult to locate from a plot of count rate against thickness. It can be found most readily by plotting the logarithm of the count rate (corrected for background radiation) against absorber thickness. Figure 1 shows schematically the shape of such an absorption curve. An important point to note is that both beta particles and gamma rays contribute to the count rate measured in the detector, with gamma rays alone remaining after sufficient absorber has been introduced to stop all betas.
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Fig. 1 These gamma rays come from two sources (i) "bremsstrahlung" gamma rays which are produced when the beta particles are decelerated in the Al absorber and (ii) gamma rays from the daughter nucleus in the decay reaction. The absorption of gamma rays in a material varies exponentially with the thickness of the material. Therefore the tail (which is due to the gamma rays alone) of the logarithm of the intensity against thickness is linear. This linear tail can be extrapolated backwards to give the gamma ray contribution at any absorber thickness and then subtracted from the total count rate to give the count rate from betas alone. Then the range R is the thickness of absorber at which the corrected count rate of beta particles curve approaches the “vertical”.
- The use of a Geiger-Muller counter
2.1 The Physical Principles of a G-M Tube A G-M (Geiger-Muller) tube consists in essence, of a cylindrical metal cathode surrounding a coaxial wire anode with the space between containing a suitable gas at low pressure. The passage of an "ionising" particle causes an avalanche of electrons which in turn, causes an electrical pulse in the external circuit connected to the tube if the tube is biased appropriately (a few hundred volts between anode and cathode). This electrical pulse is counted by an electronic counter or "scalar". If the voltage V across a G-M tube is increased while the flux of incident particles remains constant then its rate of counting varies with V in the way shown schematically in Figure 2.
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count, then divided by 100 to get the count rate? Surely this should be related to our ten 10s counts, since if we just added up our ten 10s counts that would be one 100s count? Clearly if we did this the mean value would be identically the same, but what about the error bar? Well, without going into the statistics of nuclear physics, as long as the number of counts measured is large (~100 or greater) we can estimate the error bar in th 100s count value by taking the square root of the count value. Hence we can obtain the count rate and its error bar by dividing both the 100s count value and its error bar by 100. In section(3.b) you’ll be asked to do this and demonstrate that you get essentially the same answer as with ten 10s counts. Therefore when measuring counts as a function of Al thickness you need only measure one 100s count and use this square root “trick” to calculate your error bar.
2.3 Sources of systematic errors (a) The Background Count Rate This arises from sources other than the Sr^90 source, such as radioactive nuclei in nearby materials and from cosmic rays. This count rate can be measured with the Sr^90 source removed. Ensure as far as possible that the counting rates in your experiment proper (with the Sr^90 source present) are much greater than the background rate. (b) Scattering from the holder As shown schematically in figure 3 it is possible for the beta particles to be scattered by the walls of the container in such a way that they corrupt the count rate. Position the G-M tube at different distances from the Al sheets to investigate the scattering of betas into the G-M tube.
G-M Tube
Source
Wall
Al plate
Fig. 3
(c) Dead time losses in the G-M tube
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After each and every count the G-M tube is insensitive for a short time, the so-called “dead time” dt. Consequently, if the mean count rate is N then the counter is insensitive for a portion N dt of each second and the observed mean count rate is less than the true count rate. Hence the true mean count rate is given by
N (^) true = (^1) - NN dt (5)
If the condition N dt << 1 holds true then N (^) true ≈ N ( 1 + dt ). Always check as to whether the
term N dt is significant or not in your measurement. For your G.M. tube the dead time dt, is 100 microseconds.
- Experimental Procedure The beta sources used in this experiment have half lives of many years (e.g. for Sr^90 the half life is ~ 25 yrs) and so may be considered to have a constant activity during the course of the experiment. If the source is used sensibly the level of this activity is low enough to present no personal danger, even without shielding. However following the principle that unnecessary exposure to radiation should be avoided even at low levels the sources are supplied inside shields. Try to ensure that the covering cap is only removed after the source has been placed inside the castle which holds the G.M. tube. Similarly ensure that the cap is replaced before removal from the castle and secured in place when the source is left on the open bench. Never leave unshielded sources lying about on the bench giving unnecessary radiation exposures to yourselves and others. Always return the source to the laboratory technician for safe keeping at the end of each laboratory session.
(a) Measuring the Voltage Dependence of the G-M Tube Place the source centrally on the base of the castle and remove the cap. Investigate the counting characteristics of your G.M. tube by measuring the count rate as a function of the supply voltage V. To avoid unnecessary problems with the dead time losses, described in section (2.3c) use a thin sheet of Al as an absorber if necessary in order to keep the maximum count rate below 1000 counts/second. Plot a graph of count rate against voltage similar to that sketched in figure 2. Set the supply voltage for all subsequent measurements at a value in the middle of the plateau region where the count rate is least sensitive to variations in the supply voltage.
DO NOT LET THE SUPPLY EXCEED 550 V OTHERWISE THE TUBE MAY BE DAMAGED. DO NOT TOUCH THE WINDOW OF THE TUBE - IT IS FRAGILE.
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down. In cell G1 enter the formula = E1 / ( 1. - E1 * 0.0001 ), which corrects the count rate for the dead time effect using equation(9), and fill down. You also need to correct the error bar in the count rate for the dead time effect, so in cell H1 enter the formula = F1 / ( 1. - E1 * 0.0001 ) and fill down. Thus in columns G and H you now have your count rate and error bar values. Finally you need to take the natural logarithm of these values. In cell I1 enter the formula = LN(G1) and in J1 the formula = H1/G1 and then fill down both columns. Save your speadsheet to diskette and print off a copy to paste into your notebook. Columns I and J are respectively the y-values and y-errors you need to enter into the Linefit program, along with the relevant thicknesses of Al as the x-values. Use Linefit to plot the data (but don’t click the fit button) and print off a graph of the data.
From your graph make a sensible estimate of the value of the thickness R (in mg cm-2 ) at which all beta particles are absorbed. Assign an error to this (by estimating the range of possible values of thickness over which your graph changes from a curve to a straight line). From equation(1) calculate a value for E and compare it with the tabulated value for Sr^90 of 2.26 MeV.
