Compton Scattering of Gamma Rays Sherman Ip Department of Physics and Astronomy, University College London [email protected] Date submitted 12th November 2012 Using the gamma rays emitted from Caesium-137, gamma rays were scattered by colliding the gamma rays or photons with electrons in the scattering rod. As a result some of the photon energy was transferred to electrons and this transfer of energy is known as the Compton Effect. The experiment aims were to collect evidence of the Compton Effect and hence an experimental value of the electron mass was obtained. PHAS 3440 - 1 - Sherman Ip I. INTRODUCTION Einstein postulated that light is 'quantized' as photons which can behave like particles or a wave.4 A photon has energy which corresponds to the Plank’s energy, Eq. (1), and momentum which corresponds to the de Broglie’s relationship, Eq. (2).4 (1) (energy of photon E, reduced Planck’s constant , angular frequency ω) (2) (momentum p, wave vector k) A. H. Compton demonstrated that photons can behave like particles. Compton conducted an experiment which involved scattering x-ray photons by colliding of photons with electrons as shown in Fig. (1).4 BEFORE AFTER Fig. (1) The photon before collision, γ1, collides with an electron at rest, e. After the collision the photon transfers some of its energy to the electron. As a result the electron, e, and photon, γ2, and recoils at an angle and respectively, known as Compton scattering.3, 4 From the conservation of energy and momentum and using Eq. (1) and Eq. (2), Eq. (3) was obtained, which showed how the wavelength of the photon changed before and after the collision, due to the Compton Effect.3, 4 (3) (wavelength of photon before collision , wavelength of photon after collision , mass of electron m, speed of light c, photon scattering angle ) The main aims of this experiment were to provide experimental evidence of the Compton Effect by showing that the model of the Compton Effect, Eq. (3), fitted with the experimental data. By providing such evidence, an experimental value of the electron mass m was obtained. II. METHOD The energies of photons were measured using a scintillation counter which consisted of a 50 mm diameter NaI scintillation crystal. This was shielded in a lead collimator of 20 mm aperture and coupled to a photomultiplier tube, supplied at 0.7 kV. The scintillation counter was connected to a Cassy Lab Multichannel Analyser and to a computer which ran a Cassy Lab PHAS 3440 - 2 - Sherman Ip program. Cassy Lab recorded the energies of each photon, detected in the scintillation counter, with a resolution of 1024 possible values of energy, ie 10 bits. Caesium-137 (Cs-137) emits gamma rays of about 662 keV. 3 A Cs-137 source, with activity of about , was placed in a lead block with a diameter hole. This hopefully has produced a collimated beam of gamma rays, which was aimed directly at a cylindrical brass scattering rod with properties: mass , radius and height . The lead block was mounted on a rotating arm, of length surrounded by an external aluminium shield as shown in Fig. (2). Radius = Scintillation Counter Brass scattering rod Cs-137 source External aluminium shield Fig. (2) The setup of the equipment used to detect the scattered photon energy from the Cs-137 source, which may be rotated as shown by the red arrows. Lead blocks were placed such that it prevented gamma rays from the Cs-137 source traveling directly to the scintillation counter without being scattered by the brass scattering rod as shown in Fig. (3). The function varied the most around so most of the measurements were taken around . The energy spectrum detected in the scintillation counter was recorded for 10 minutes at each of these angles: . Cassy Lab program was used to attempt to fit a Normal curve on an energy peak in each energy spectrum at each angle, which produced the estimated parameters of the Normal curve, the mean and variance. This was done 3 times at each angle to obtain 3 energy spectrums for each angle, hence 3 values of the mean and variance of the energy peak. The mean and error of the scattered photon energy detected at each angle were worked out using weighted means of the 3 values of the mean parameter of the Normal curve with the reciprocal of the variance as its weights. PHAS 3440 - 3 - Sherman Ip Fig. (3) Lead blocks were placed between the Cs-137 and the scintillation counter to prevent gamma rays from the Cs-137 travelling directly into the scintillation counter without being scattered. An equation of the energy of photons before and after scattering was needed to model the Compton Effect. Eq. (1) was substituted into Eq. (3) to obtain an equation for the change in the energy of the photon due to the Compton Scattering as shown in Eq. (4). (4) (energy of photon before collision , energy of [ ] photon after collision ) For Cs-137, and which implied a model for the Compton Effect for Cs-137 was given in Eq. (5). 3 [ ] (5) Eq. (4) was rearranged to form a linear regression as shown in Eq. (6), with intercept Eq. (7) and gradient Eq. (8). (6) (dependent variable y, independent variable x, ( ) ⏟ intercept a, gradient b) ⏟ ⏟ ⏟ (7) (8) PHAS 3440 - 4 - Sherman Ip Therefore the expected gradient and intercept was worked out as shown in Eq. (9) and Eq. (10) to 5 significant figures. (9) (population or expected intercept , population or accepted electron mass ) ( to 3 significant figures)3 (10) (population or expected gradient ) Eq. (8) was subtracted from Eq. (7) which obtained an equation Eq. (11) which yielded a value for the experimental electron mass, which used both the gradient and intercept with a given error Eq. (12). Even though the value of was given of a value of , the standard deviation must be estimated to consider all possible sources of error. 3 The mean and standard deviation of were estimated using the mean and error respectively on measured at 0 degree. (11) ( ⁄ ) (12) (error on the variable x ) √ ( ⁄ ) III. RESULTS Figs. (4) showed a sample of the energy spectrums of the scattered photon at different angles. For each energy spectrum, a Normal curve was fitted on the highlighted red peaks to obtain the estimated mean and variance of the energy peak. The weighted mean and error of the energies of the scattered photons were plotted with the corresponding scatter angle in Fig. (5). Fig. (4a) Energy spectrum of the scattered photons at 0 degree. By using Cassy Lab Normal curve fit, the mean of the curve was found which was used to calibrate the scintillation counter. Because a sample of 3 values of at 0 degrees were obtained using the Cassy Lab Normal curve fit, the degree of freedom for the student t distribution is .1 PHAS 3440 - 5 - Sherman Ip Fig. (4b) Energy spectrum of the scattered photons at 30 Fig. (4c) Energy spectrum of the scattered photons at 40 degrees degrees Fig. (4d) Energy spectrum of the scattered photons at 50 Fig. (4e) Energy spectrum of the scattered photons at 60 degrees degrees Fig. (4f) Energy spectrum of the scattered photons at 70 Fig. (4g) Energy spectrum of the scattered photons at 80 degrees degrees Fig. (4h) Energy spectrum of the scattered photons at 90 Fig. (4i) Energy spectrum of the scattered photons at 120 degrees degrees PHAS 3440 - 6 - Sherman Ip In Figs. (4), there was always a low energy peak at around ~100keV highlighted with a blue dotted line. It was assumed this was just background radiation or back scattering and was ignored. However this seemed unjustified because no evidence of background radiation was collected. An energy spectrum of the scintillation counter collecting photons from the background should have been recorded and compared if this low energy peak corresponded to background radiation. Also noted was a high energy peak highlighted in a green dotted line in Fig. (4b) centred about ~670 keV. This peak seemed to be in the same position, but shorter, as the highlighted red peak in Fig. (4a), which suggested the highlighted green peak in Fig. (4b) corresponded to the unscattered photon energy of Cs-137. Because the highlighted green peak was shorter but distinguishable from the highlighted red peak in Fig. (4b), this suggested the lead blocks have done well to prevent the photons from travelling directly from the Cs-137 to the scintillation counter without being scattered. 700.0 600.0 500.0 400.0 300.0 200.0 Energy Energy of scatteredphoton(keV) 100.0 -20.0 0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 Angle (degrees) Fig. (5) Experimental values of the energy of the scattered photon were plotted, with the corresponding error bars, at varying angle. The model Eq. (5) was plotted as shown the black curve and the model Eq. (13) was plotted as shown as the blue curve. The blue curve is essentially the black curve with an offset error added of 10.2 keV. IV. DATA ANALYSIS From Fig. (5), the test statistics was obtained and tested as shown in Fig. (8). The experimental test statistic was found not to be significantly large therefore there was evidence at the 5% significant level that the model Eq. (5) fitted with the experimental data.1 As a result, it was feasible to process the experimental data to produce a linear regression. PHAS 3440 - 7 - Sherman Ip The processed data were plotted as shown in Fig. (6) with the expected linear regression.