beta particle experiment

7/30/2019 beta particle experiment

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PART II LABORATORY

10 SPECTROSCOPY10.1 INTRODUCTIONThe phenomenon of-decay has been a constant topic of nuclear and particle physics. It hasled us from the discovery of the continuous electron energy spectrum and the postulation of

the electron neutrino to the success of the Fermi theory of-decay, and to the discovery ofparity violation and the modern theory ofelectroweak interaction.

In this experiment you will measure and analyse the energy spectrum of electrons ejected

from a nucleus undergoing -decay. The theory below outlines some of the physical

processes you may observe. You will also become familiar with one of the most commontypes of charged particle spectrometers: a uniform magnetic field spectrometer, and one ofthe most common charged particle detectors: the Geiger-Mllertube.

10.2 BACKGROUND

THEORY

10.2.1 THE PROCESSOF DECAYIn -decay, a nucleus decays to another nucleus of the same mass number with the emission

of an electron (the particle). If just an electron were emitted, then by conservation of energy

and momentum, all the electrons would have the same energy. This energy would bedetermined by

Eq. 10.1

(the difference in the rest mass energies of the initial and final nuclear states).

However, the energy spectrum of emitted electrons is broadened and shows that the

electrons are emitted with energies ranging from 0 to E0. It was initially thought that thisbroadening was due to experimental limitations, until even very high-precision measurementsshowed the same effect. It was therefore inferred that an additional particle is emitted, withenergy

Eq. 10.2

to satisfy conservation of energy for the decay process. A similar relationship exists for

conservation of momentum. We call this particle an electron anti-neutrino, e .

The simplest -decay is that of a single neutron to a proton. We can equivalently consider -

decay of a parent nucleus to a daughter nucleus with an increased atomic number. Theenergy spectrum of this is shown in Figure 10 .1 below

Figure 10.1 Electron energy spectrum fromnuclear-decay

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finalinitial EEE =0

electronparticle EEE = 0


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PART II LABORATORY

Taking into account the contribution from the electron anti-neutrino, the -decay of a neutron

is therefore

Eq. 10.3

and for nuclear-decay

Eq. 10.4where Mrepresents a nucleus with initial atomic number Zand mass numberA which thendecays to a nucleus with atomic numberZ+1, an electron and a neutrino. This decay can onlyoccur if the rest mass of the initial state of M is higher than the combined rest mass of the

final states ofMand e-, ie if 00 >E (we assume that the rest mass of the neutrino is zero).

In this experiment you will be studying the -decay ofPromethium into Samarium:

Eq. 10.5

The initial state of the system is the ground state of Promethium. This decays into aSamarium nucleus plus several possible states for the emitted electron and neutrino. Thesefinal states are distinguished by the way in which energy is shared between the Samarium

nucleus, the electron and the neutrino. Your measurements of the electron energy spectrumwill tell you about how the energy has been shared in these final states.

Pre-lab Question (a) How can we tell, by studying the electronenergies in -decay, that such a particle as an electron anti-neutrino is also a

product of the decay?

10.2.2 THE RELATIONSHIPBETWEEN INITIALAND FINALSTATESOF DECAYNow that you have an understanding of the physics of -decay, we can consider the

equations which govern the process. These equations will be used to analyse the energyspectrum you measure1.

Suppose that the electron is emitted with a momentum between p andp+dp. The number of

final states with electron momentum in this range (from the total range of physically possiblefinal states) is given by

Eq. 10.6

where Eis the electron kinetic energy.

Let w(p)dp be the probability per unit time2 that the decay will give an electron withmomentum betweenp andp+dp. Then it can be shown that

Eq. 10.7

where Mif is the matrix element of the dynamical interactions causing the decay. For-decay

the dynamical interaction is weak and so Mif can be considered to be constant. The aboveequation is known as Fermis Golden Rule.

Finally we incorporate an extra factor, the Coulomb correction factorF(z,p) to account forthe interaction of the electron with the electrostatic field of the nucleus. As the electron movesaway from the nucleus, it loses energy. Hence the maximum amount of energy initially

1There is a large number of equations in this section, which are included to give you an understanding of how thetheory is derived, and of the meaning of the Kurie variable. The Theory section of your lab report should notread simply as a list of equations.2 You may be wondering what the conceptual difference is between w(p) and f given that they both describe the

probability of a particular final state occurring. The quantity f is the density of final states between p and p+dp in the

region of phase space available for decay (ie the region defined by E 0 and conservation of energy and momentum),and represents a probability per unit total energy. On the other hand, w(p) is the relative probability per unit time thatthe decay to a final state with electron energy between p and p+dp occurs and so incorporates dynamical interactionscausing the decay.

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eepn ++

e

A

Z

A

Z eMM ++

+1

eeSmPm ++147

62

147

61

( ) dpEEpch

f

2

0

2

36

216

=

fifMh

dppw 2

24

)( =


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S PECTROSCOPY

available to the electron must be greater than E0. This is a dynamical correction, ie it is2

ifM which is modified and it can be shown that

Eq. 10.8

where

Eq. 10.9

where =h/2and is the final velocity of the electron. F(Z,p) is called the Fermi function.Hence

Eq. 10.10

Let us now define a new variable, the Kurie variableK(Z,p), to be

Eq. 10.11

then

Eq. 10.12

and a graph ofK(Z,p) against Ewill be a straight line plot of negative slope intersecting theelectron kinetic energy axis at E=E0. Such a graph is called a Kurie plot. The Kurie plot forthe spectrum in Figure 10 .1 is shown below.

In practise you will measure the number of counts at each electron energy and momentum,

N(p), in a fixed time interval t. Since N(p) w(p) we can define an experimentalKurie

variable to be

Eq. 10.13

By plotting the Kurie variable against the electron energy, we can obtain a value for E0 (the

intercept on the Eaxis). The accepted value ofE0 is (224.5 0.4) keV for147

61Pm .

Pre-lab Question (b) What is the value of the Kurie variablewhen the electron energy is equal to E0?

Pre-lab Question (c) How can you explain this in terms of the

probability of an electron with energy E0 being emitted?

Figure 10.2 Kurie plot of the

spectrum shown in Figure 10 .1.

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( )

21

2,

=e

pZF

0

2

4

Ze=

( ) ( )dppZFEEpMch

dppw if ,64

)(2

0

22

37

4

=

( )( )

( )pZFp

pwpZK

,

,2

=

( ) ( )EEMch

pZK if = 02

32

7

8,

( )),(

)(,

2pZFp

pNpZK =


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42222 cmcpET +=

PART II LABORATORY

Pre-lab Question (d) The expression forw(p)dp shown above isderived on the assumption that the neutrino has zero rest mass (m=0). Without this

assumption, the expression becomes

Eq. 10.14

What is the maximum kinetic energy that the electron may have now?

10.3 APPARATUS

10.3.1 THE MAGNETIC SPECTROMETERA particle of mass m, charge q, and velocity v , moving in a uniform magnetic field B , willexperience a force F , where

Eq. 10.15

If B is always perpendicular to v , then the particle will move in a circle of radius rwhere

Eq. 10.16

and will have a momentum

Eq. 10.17

Also recall that

Eq. 10.18

and

Eq. 10.19

Constant magnetic field spectrometers use this principle in order to momentum analyse a

source of charged radiation. The magnetic spectrometer you will be using is shown below.

Figure 10.3 Spectrometer chamber with source and detector. Note that r=(3.800.01) cm.

Pre-lab Question (e) For the direction of electron motion shownin u will be using is shown below., what is the direction of the magnetic field?

For the first part of the experiment, you will use a duplicate spectrometer chamber as shownbelow.

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( ) ( ) dp

EE

cmpZFEEpM

ch

dppw if

2

0

2

0

22

37

4

1,64

)(

=

BvqF =

Bqvr

mv=

2

qBrp =

kineticrestmassT EEE +=


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S PECTROSCOPY

Figure 10.4 Spectrometer chamber used for calibration.

This chamber has identical dimensions to the spectrometer, without the source and detector.A port in the top of the chamber allows a Hall Probe to be inserted, for measuring themagnetic field.

10.3.2 THE GEIGER-MLLER TUBEThe detector used in this experiment is a Geiger-Mller tube. A schematic diagram of itsoperation is given in Figure 10 .5.

A Geiger-Mller tube consists of an easily ionisable gas contained within a thin-walledchamber. A high voltage is applied between the central anode and the walls of the chamber,creating a very high electric field within the chamber. After entering the detector, an electronionises the gas molecules. Each collision generates another electron which is acceleratedtoward the anode, together with the original electron. The high electric field provides extraenergy to the electrons as they travel toward the anode. Due to the very small mean free pathin the gas, the electrons undergo many ionising collisions before reaching the anode. Thus,the original electron initiates an electron cascade which results in a detectably large chargereaching the anode. This charge is collected within the detector electronics.

While the electrons move rapidly toward the anode, the positive ions created in the cascadetake considerably longer to travel to the cathode, due to their much greater mass. This meansthat if another electron enters the detector before a significant number of positive ions havebeen discharged at the cathode, another cascade will not occur, due to the positive ionsrecombining with the electrons within the chamber. Therefore, the tube will not detect anothereven until the previous event has been cleared from the chamber. The time it takes to clearan event from the chamber is called the dead time. Usually this time is considered to beconstant for a particular detector (and experimental arrangement) and is denoted by the

symbol .

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PART II LABORATORY

Figure 10.5 Operation of a Geiger-Mller tube

Pre-lab Question (f) Explain, in your own words, the physical

cause of dead time in the Geiger-Muller tube.Pre-lab Question (g) Show that if the tube has a dead time ,and Ncounts are observed in a time t, the true number of events Nis given by

Eq. 10.20

The dead time for the detector you will be using has been measured, at a count rate of

approximately 1000 counts/second, to be (200.5) s. To a good approximation, the dead

time may be assumed constant over the spectrum you will measure.

10.3.3 OTHER APPARATUSNIM modules amplifier, SCA, counter, timer DC power supply stabilisedDigital multimeter for measuring currentElectromagnet 3 pole faces, 3kG maximum field (see Magnets and Magnetic

Fields notes)147Pm source point source mounted on the end of a bolt

Hall probe and meter for measuring magnetic field strength (see Magnets and MagneticFields notes)

Magnet degausser generates an AC field across the magnet poles which slowlyreduces to zero

Vacuum pump rotary type (see Energy Loss notes)Vacuum gauge thermistor typeCRO for observing amplified detector pulses

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N

NN

t

=1


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Initialising the Hall Probe

Place the tip of the probe in the zero field case and switch the meter on. After a few seconds the number1973 should appear on the display. After this number appears, press the ENTER/RESET button. Three dashes (- - -) should appear on the display. Now press the PB button. Press the orange UP/DOWN keys next to RANGE until KGAUSS is

selected.

The Hall Probe is now ready for use. Insert it into the duplicate chamberwithout switching the meter off.

If the meter is switched off, it must be initialised again. If it is switchedoff in the middle of taking a set of data, you must re-take all the datawith the new initialisation.

It is therefore imperative that you do not switch off the meter until

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10.4 EXPERIMENT: CALIBRATINGTHE MAGNET

10.4.1 PROCEDURECarefully mount theduplicate spectrometerchamber between thepoles of theelectromagnet.Initialise the Hall Probe(see box).Remove any remnant fieldby applying thedegausser.If the magnitude of thefield is greater than 10G, repeat the degaussing procedure.

If the field remains high after several degaussing cycles, consult your demonstrator beforeproceeding with the rest of the experiment.

Question (a) Explain how the degausser removes the remnant field.

Now connect the magnet to the stabilised DC supply, again using the multimeter to measurethe current. You should connect the supply such that the current is flowing into the red plugon the magnet and out of the black plug. Before switching on the supply check that

the current knob is turned to a maximum

the voltage knob is turned to a minimum

the multimeter is on the 0-2A DC scale.

Question (b) How will you calculate uncertainties in the current andmagnetic field measurements?

Measure the magnetic field as a function of current from 0 to 500 mA in 20 mA steps.Precautions:

never reduce the current: backtracking at any stage will render the calibration

useless

never change scales on the hall probe or multimeter

allow a few seconds for the field to stabilise before taking a measurement.

Question (c) Why would backtracking render the calibration useless?You will notice a small black button on the side of the hall probe. When the magnetic field isentering the hall probe from this side, the gaussmeter will give a positive reading.

Question (d) What is the direction of the magnetic field for this direction of

current flow?

10.4.2 ANALYSISAfter you have completed your reading, slowly reduce the current to zero and repeat theprocedure to remove the remnant field. Then use SigmaPlot to fit a straight line to your dataand produce a calibration curve.

Check the slope and intercept with your demonstrator.

Question (e) What is the maximum kinetic energy you wish to measure?

Question (f) What magnetic field does this correspond to?

Question (g) What current would produce that magnetic field?

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PART II LABORATORY

10.5 EXPERIMENT: MEASURINGTHE SPECTRUMYou are now able to determine the magnetic field inside the spectrometer chamber bymeasuring the current in the electromagnet. By measuring the number of counts hitting the

detector for various magnet currents, a -decay spectrum can be measured and analysed.

10.5.1 PROCEDUREEnsure that you have disconnected the power supply and degaussed the magnet.Carefully remove the Hall Probe, switch it off and replace it in its case.Remove the duplicate chamber from between the magnet poles and replace it with thespectrometer chamber containing the source and detector.

All the data for this part of the experiment must be taken in one go . It cannot be takenhalf on the first day and half on the second. It is recommended that you take themeasurements on the first day and do the analysis on the second day, thus leaving time to re-take the data should any problems arise.

Question (h) Why must all the data be taken in one day?

Connect the DC power supply to the magnet.Question (i) In which direction are the particles flowing (you may wish toconsider your answers to Pre-lab Question (e) and Question (d) do theyproceed from the source to the detector?

Ensure that the vacuum pump is connected to the chamber and switch on the pump andgauge. After a few minutes the pressure in the chamber should have fallen below 10 -2 Torr.

Connect the electronics as shown in Figure 10 .6.

Figure 10.6 Schematic diagram of apparatus.

Check that the electronic settings are correct (see below), then switch on the NIM crate andthe Geiger supply.

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S PECTROSCOPY

At very high count rates, the size of the Geiger output pulses is reduced because of theincreased current being drawn from the supply. You should monitor the output of the amplifieron the CRO to see when this occurs.It is important that these reduced pulses are still counted, so the following settings arerecommended:

Amplifier coarse gain = 16fine gain = 6input = positiveoutput = unipolar

SCA ULD = 10VLLD = 0.20-0.30Vnormal operation

Counter discriminator = 3.0VTimer 100s

Start measuring the number of counts for 100s intervals in the current range 0 to 400mA.

For 0 to 100mA, measure every 20mA.

For 100 to 300 mA, measure every 10mA.

For 300 to 400 mA, measure every 20mA.Once again, never back track, always increase the current and allow a few seconds for thefield to stabilise before making a measurement.

As you take the data, observe the output pulses.

Question (j) Draw, on scaled axes, the shape of the output pulses. Atwhat current do they appear to be smallest in size?

Check, for a current of approximately 100mA, that the output pulses do not exceed the upperlevel on the SCA.

Question (k) Explain the shape of the pulses from the Geiger tube, interms of how the Geiger tube and electronics function.

Question (l) Are the pulses consistent with the dead time quoted?

After you have taken your measurements, increase the current to 500mA and take a furthermeasurement. This will be the background measurement, which must be subtracted fromeach of your readings.

Question (m) For electrons of a given energy, what happens to the radiusof curvature of the electron beam as the current is increased?

Question (n) Why do we choose a current of 500mA for the backgroundreading?

Question (o) What are the sources of this background?

10.5.2 ANALYSISYou should now have a table of measured counts at various different currents, as well as theiruncertainties (ask your demonstrator for assistance if you are unsure how to calculate these).

The analysis section requires a series of calculations, and great care must be taken to ensurethe transforms are done correctly. The steps you have taken should be made very clear comment the transforms by putting a semi-colon at the start of a comment line.For example:

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;Magnetic field B in Tesla (1 Tesla = 10^4 Gauss)B=col(4)*0.1

;Now calculate the momentum, p, in kg m per second

p=B*q*r


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PART II LABORATORY

If the results do not follow the expected form, it is then very easy to check the transformcalculations for errors.

Choice of units is, as always, very important. All calculations require SI units (metres, Tesla,eV, seconds and so on).

a) Kinetic EnergyUsing SigmaPlot, correct the number of counts for background and dead time, and calculatethe electron kinetic energies (look back at Question (e), Question (f) and Question (g) forideas). Plot the corrected counts against the kinetic energy (in keV).

Question (p) Explain the shape of the spectrum:a) below 50keVb) above the predicted maximum energyc) around the broadened peak in the center.

In this explanation, you may wish to consider effects of the thin mica windows abovethe source and detector, the effect of residual air in the chamber, and the efficiency ofthe Geiger tube as a function of energy.

b) Kurie VariableAgain using SigmaPlot, calculate the Kurie variable (including uncertainty analysis).Fit a straight line to the region of interest and obtain a value for E0.

Question (q) How does your measured value compare with the expectedvalue? Suggest reasons for any discrepancy.

Further considerations:

Question (r) What region of the spectrum would be most useful instudying neutrino mass?

Question (s) What problems would have to be overcome to perform sucha measurement?

10.6 USEFUL DATAaccepted value ofE0 for

147

61Pm (224.5 0.4) keV

PlancksConstant, h 6.6256x10-34 Js

Speed of Light, c 2.997925x108 ms-1

Permittivity of free space, 0 8.85416x10-12 Fm-1

Electron charge, e 1.60210x10-19 C

Electron-Volt, eV 1.60210x10-19 J

Electron rest mass, ER 511 keV

10.7 REFERENCESIntroduction to Nuclear Physics, H. Enge, Addison-Wesley, 1981 (chapter 11)Fundamentals of Nuclear Physics, N.A. Jelley, Cambridge University Press, 1990 (chapter 3)Quantum Mechanics, E. Merzbacher, John Wiley & Sons, 1970Table of Isotopes, C.M. Lederer, V.S. Shirley (Eds), John Wiley & Sons, 1978S.T. Hsue et al, Nuclear Physics, 80, 1966

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