experiment 6 1 the compton effect physics 2150 experiment no. 6
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The Compton Effect Physics 2150 Experiment No. 6
University of Colorado Introduction In some situations, electromagnetic waves can act like particles, carrying energy and momentum, which they may impart to other particles with which they interact. According to Plancks hypothesis, an electromagnetic wave of frequency ! carries an energy given by:
! = !! (1) where ! is a positive integer and is Plancks constant. The minimum amount of energy which can be carried by an electromagnetic wave is thus that carried by one photon. Further, according to the de Broglie relation, if the wavelength of the radiation is !, then the photon has a momentum given by
! = !!= !!
! for an electromagnetic wave.
Consider a photon which collides with an electron at rest. In Fig. 1, let ! be the angle
at which the photon comes off, and let ! be the angle at which the electron comes off. Since
Figure 1: Collision of a Photon with an Electron Initially at Rest the photon emerging from the collision has a momentum different from that of the incident photon, it will have a different wavelength, which we shall denote by !. Applying the relativistic law of conservation of energy to this collision
!!!+ !!!! =
where the mass of the electron after that collision is
! = !!!! !
In this equation, ! is the electrons velocity in the laboratory reference frame. Applying the laws of conservation of momentum in the plane of the collision gives !
!! cos! +!" cos!, (5)
and 0 = !
!! sin! !" sin!. (6)
It follows from these three equations (see any text in modern physics for the derivation) that the change in wavelength of the scattered photon is given by: !! ! = !
!!! 1 cos! . (7)
This result can be expressed directly in terms of the energies ! and ! of the scattered and unscattered gamma rays, respectively, by using ! = !/!: !
!!!! 1 cos! . (8)
This change of gamma-ray wavelength (or energy) upon scattering from an electron is called the Compton Effect, and the object of this experiment is to measure the change of the energy of the scattering and to compare the results with Eq. (7) or (8). The apparatus used in this experiment consists of one strong 137Cs source used in parts 1, 2, and 3. Both its scattered and unscattered energies will be determined. Three weak calibrating sources (137Cs, 22Na, and 60Co) will be used solely for calibrating a pulse-height analyzer. In addition to these sources, there is a sodium iodide detector, a scaler-time-counter, associated power supplies, and a computer that contains a pulse-height analyzer. The geometry of the scatterers is based on a theorem from plane geometry that states that, in effect, if a triangle is inscribed inside a circle with a fixed chord of the circle as one side, then the angle opposite the fixed chord is a constant (see Fig. 3). Thus, if a source is placed at one end of the chord, and a photon emitted from this point is scattered
Figure 2: Geometry of the Source, Detector, and the 90 Scattering Plates
Figure 3 If a triangle is inscribed in a circle with a fixed chord of the circle as one side, then the opposite angle of the triangle is a constant. by electrons placed on the circumference of the circle, and if the photon is subsequently detected by a counter placed at the other end of the chord, then the scattering angle is fixed by the geometry. If a circular plate is shaped to fit the circumference of the circle and the photons are allowed to scatter off the entire plate, then any photon which is detected after one scattering must have been scattered through the same angle. The particular setup used in this experiment has been constructed in such a way that the two different scattering angles may be used: 69 and 90. These angles were arbitrarily chosen and there is no reason in principle why one could not construct
scatterers for any chosen angle. The scatterers used are thin aluminum plates because aluminum has relatively high density of free electrons. Referring to Eq. (7) or (8), the constants , !! , and ! are known fundamental constants. Therefore, in order to verify the correctness of these predictions, we must measure the energies of both the unscattered (!) and the scattered (!!) photons for one or more particular angles. Since the mass absorption coefficient of lead is a known function of energy, the energy measurement in this experiment is performed by measuring the mass absorption coefficient of the gamma rays in lead. For a given absorbing material, the absorption probability is proportional to the number of atoms of absorber per unit of cross sectional area. Therefore, it is convenient to measure absorber thickness in cm. A series of cylindrical lead absorbers is provided. These are constructed so that they may be placed in successively thicker layers around the detector. A beam of gamma rays of initial intensity !! is attenuated upon passing through a thickness ! (in g/cm2) of absorber according to the equation: ! = !! !"# (!") (9) where ! is the mass absorption coefficient in cm2/g. The intensities ! and !! are in terms of number of gamma rays per square centimeter per second. For a particular geometrical configuration of source and scatterer, the mass absorption coefficient is determined by measuring the counting rate (intensity) as a function of the thickness of the absorbers. This functional relationship may be then plotted either on semi-log paper or by using a weighted linear-regression computer program such as Wlinfit or its equivalent. Wlinfit gives both the slope of the resulting straight line and its uncertainty. If semi-log paper is used to make the plot, the slope and its uncertainty are determined graphically. The energy may then be determined from the curve provided (Fig. 6). The sodium iodide (NaI) scintillation detector used in this experiment is an important and widely used device for detecting gamma rays. As shown in Fig. 4, it consists of a crystal of NaI to which a small amount of thallium has been added in order to make it an effective scintillator. The cylindrical NaI (2 x 1 inch diameter) is hermetically encased in an aluminum cup with a glass window on one end. The glass window of the detector is mounted on the face of a 10-stage photo-multiplier tube. Gamma rays entering the NaI detector will interact by either the photoelectric or Compton process. Gamma rays with energy greater than 1.02 MeV can also produce an electron-position pair. The photoelectric process results in the gamma ray imparting essentially all of its energy to one of the bound electrons in the crystal. The Compton process is assumed to occur on a free electron, and as we have just seen, results in only a fraction of the gamma-ray energy being carried off by an electron. Electrons from either process will quickly lose their energy in the crystal by causing ionizing events with the atoms of the crystal with the
net effect that photons in the visible region will be produced in the de-exitation of the atoms. The number of visible photons produced is directly proportional to the energy deposited by a gamma ray in the crystal. The photons are reflected by a coating of MgO surrounding the crystal except for the end covered by the glass window and many of them enter the photomultiplier.
Figure 4 Gamma-ray detection and counting equipment, configured for parts one and two. Event (a) is a Compton scattering event and event (b) is a photoelectric process. The photons entering the photomultiplier will strike a photocathode surface present on the inside face of the phototube. Low energy electrons will be ejected from the photocathode and some of them will be incident on the first dynode of the tube. The dynode provides amplification of the number of electrons by the secondary emission process (more electrons are emitted than are incident on a given dynode). The number of electrons is increased by nice successive dynode stages and finally the electrons are collected on the anode where a reasonably large voltage pulse will result. The amplitude of the voltage pulse will again be directly proportional to the energy deposited in the NaI.
After going through a preamplifier at the tube base, the voltage pulse is carried via a cable to the scaler on a table outside of the room where the Compton scattering experiment is located. Procedure: Part 1
1. Energy of the Unscattered Gamma Ray The radioactive source used in this part of the experiment is 137Cs. It has a half life of about 30 years and emits a gamma ray with an energy of 0.662 MeV. Before attempting to measure the energy of the Compton scattered gamma rays, an absorption measurement will be made of the direct or unscattered gamma ray in order to become familiar with the techniques.
a. The AC and DC power should be turned on for the two power supplies in the
room with the scattering experiment. The high voltage for the photomultiplier should be +850 volts. No adjustment of either power supply should be necessary. A small lead source holder and collimator is available and this should be placed 6 to 8 inches from the NaI detector with the collimation hole aimed directly at the detector. No cyli