Andrew D. Kent. 1 Introduction. p 1

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1 Compton Effect Andrew D. Kent Introduction One of the most important experiments in the early days of quantum mechanics (93) studied the interaction between light and matter; it determined the change in energy and momentum of light (x-rays photons) when they scatter off electrons. Photons carry momentum. So light (i.e. photons) can exert a force on an object, such as particle, mirror, etc. If the object s momentum changes its energy changes. By conservation of energy there must be an equal and opposite change of energy of the photon. This is known as an inelastic collision as the photon energy changes. As the energy of a photon is proportional to its frequency (E = hf) its frequency must also change during the collision. Equivalently, the wavelength of the photon must change as λ = c/f. As the photon transfers energy to the electron its frequency decreases (there is a red shift ) on scattering and thus its wavelength increases. The figure below shows the basic physics of Compton scattering: y x m 0 v p p p e Figure : Compton effect. Left: An x-ray photon incident on a particle (an electron in this experiment) is scattered through an angle θ. The initial wavelength of the x-ray is λ and the scattered x-rays wavelength is λ (λ λ ). The particle, initially at rest, acquires a velocity v in the scattering process. Right: Conservation of momentum in scattering process. The goal of this experiment is observe the Compton effect and to measure the change in wavelength of x-rays on inelastic scattering from electrons in a solid. Since in the x-ray setup we can only count the number of x-rays per unit time incident on the detector, we need to convert a change in wavelength into a change in x-ray counting rate. This will be done using filters whose attenuation depends on x-ray wavelength. By measuring the change in the count rate we will be able to infer the change in the average x-ray wavelength.

2 Theory. Compton Effect To determine the change in wavelength we need to consider the conversation of energy and momentum. We assume that an x-ray photon scatters from a particle with mass m 0, initially at rest. The conservation of energy is: p c + m 0 c = p c + E e, () where the relativistic form of the kinetic energy is used (Why is the relativistic form used? What is the maximum possible velocity of the electron in terms of the speed of light, i.e. what is v/c?). The magnitude of the photon s momentum is p = h/λ. So prior to scattering the photon has momentum p = h/λ and after the scattering event: p = h/λ. The conservation of momentum (see the right hand side of Fig. ) is: p e = p p. () As shown in Fig., θ is the angle by which the x-ray scatters and φ is the angle the particle moves in after the collision. After some algebra (see below) one finds: λ λ = h ( cos θ). (3) m 0 c The constant h/(m 0 c) =.43 pm is called the Compton wavelength λ C. This is the wavelength of a photon having an energy equal to the rest energy of the particle (an electron in this experiment). λ C is the maximum change in wavelength of the photon in the scattering process and corresponds to the photon backscattering (θ = 80 ) and transferring the maximum amount of energy and momentum to the particle.

3 Deriving the Compton formula (Eqn. 3). First take the square of both sides of Eqn.. This makes this vector equation an equation for scalars (i.e. just numbers) which are far easier to work with: ( p) e = ( p p ) (4) p e = p + p p p cos θ. (5) x-rays scattered at an aluminum body without attenuation, and Here and throughout theapparatus lab notes we use the usual convention then the counting that rates letters R and R with the copper a vector foil placed X-ray apparatus in front of and behind the aluminum body, respectively (see sign, e.g. p, represent the magnitude of the vector, e.g. p Fig. = 3). p As the. counting Thenrates take are low, the counting square rate R of End-window counter background radiation is also taken into consideration. Using (Eqn. ): for,, and x-ray radiation (p c p c + m 0 c ) = Ee = (m 0 c equation ) (VII), the + (p e c) transmission values Compton accessory x-ray (6) and Now eliminate p e in this equation (Eqn. 6) by substituting p e from Eqn. 5 into Eqn. 6 and divide both sides of the resulting equation by c : T = R R (IX) R 0 R Several terms cancel to give: allow us to calculate the mean wavelengths of the unscattered x-rays and of the unscattered x-rays. In accord- developed by R. W. Pohl, (p p + m 0 c) in which the = (m 0 c) attenuation + p of the + p ance pwith p(vii) cos to (IX), θ the wavelength shift (7) m 0 c m 0c = cos θ. = 00pm ln (R 0 R) ln (R R) n (8) p a Finally, substitute p = h/λ and p = h/λ and multiply by h/(m 0 c) to give Eqn. n 3! In this experiment x-rays with a = 7.6 are and scattered n =.75. off an aluminum crystal. As aluminum is a metal its valence electrons (the outermost electrons of Al atoms) behave as nearly free electrons. We thus expect x-rays to scattering inelastically off the valence electrons in the Al. 3 Experiment P LEYBOLD Physics Leaflets Verifying the wavelength shift: Verification of the wavelength shift is based on an arrangement unscattered x-radiation in a copper foil is compared with that of radiation scattered using an aluminum scattering body. The transmission T Cu of the copper foil depends in the wavelength of the x-rays (see Fig. ). Therefore, a shift in the wavelength of the x-radiation due to Compton scattering becomes apparent as a change in the transmission or the counting rate. The evaluation is facilitated p by the fact that the wavelengthdependency of the transmission of the copper foil can be formulated as T Cu = e a 00pm Safety notes The x-ray apparatus fulfills all regulations governing an In this experiment the attenuation of unscattered x-ray x-ray apparatus and fully protected device for instructional use and is type approved for school use in Germany radiation passing through a copper foil is compared to (NW 807/97 Rö). that of radiation scattered The built-in fromprotection aluminum. and screening measures The reduce transmission T Cu of the copper foil Sv/h, depends a value which is on the the order wavelength of magnitude of the the local dose rate outside of the x-ray apparatus to less than natural background radiation. of the x-rays (see Fig. ). Therefore, Before putting the ax-ray shift apparatus in into the operation wavelength of the x-radiation due to Compton scattering in- inspect it for damage and make sure that the high voltage is shut off when the sliding doors are opened (see Instruction Sheet for x-ray apparatus). duces a change in the transmission and thus x-ray counting rate. In other words, the average x-ray transmission Keep the x-ray apparatus secure from access by unauthorized persons. Do not allow the anode of the x-ray tube Mo to overheat. through the Cu foil is usedwhen toswitching determine on the x-ray the apparatus, average check to make x- sure that the ventilator the tube chamber is turning. ray wavelength both before and after scattering from the The goniometer is positioned solely by electric stepper aluminum. The average motors. change in x-ray wavelength as Do not block the target arm and sensor arm of a function of scattering angle goniometer is compared and do not use force to to theoretical move them. expectation for Compton scattering (Eqn. 3). The transmission of the copper foil depends on wavelength as follows: T Cu = exp [ ( ) n ] λ a λ 0 with a = 7.6 and n =.75 and λ 0 = 0. nm. n 3 (VII) (9) This experiment initially measures the counting rate R 0 for T = R R R 0 R (VIII) = under study here is Fig. ln (R 0 R) ln (R R) a Transmission of copper foil (d = 0.07 mm) in the wavelength range from 50 to 80 pm, measured at the LEYBOLD DIDACTIC laboratories (X) (XI). Figure : Transmission of a copper foil (d = 0.07 mm) in the wavelength range from 50 to 80 pm. Figure from Leybold physics leaflet P

4 Four different counting rates are measured to determine the wavelength shift:. R 0 : The counting rate when the x-rays are scattered from the Al without being attenuated.. R : The counting rate when the x-rays are attenuated by a Cu filter before scattering from the Al. 3. R : The counting rate when the x-rays are attenuated by a Cu filter after scattering from the Al. 4. R B : The background x-ray counting rate; the count rate the detector measures when the x-ray tube is off. With these measurements one can then compute the average λ from the transmission of the x-rays when they are attenuated before scattering from the Al: T = R R B R 0 R B. (0) One can also compute the average λ from the transmission of the x-rays when they are attenuated after scattering from the Al: T = R R B R 0 R B. () (Why is it necessary to include the background x-ray counting rates in the above expressions?) One then can rearrange Eqn. 9 to determine the average wavelength before and after scattering in terms of the measured x-ray transmission rate: ( λ = λ 0 ln(t ) ) /n. () a 3. Setup The main components of the x-ray experiment are: X-ray tube (left side of instrument) X-ray optics: slits to collimate and filter the x-rays Goniometer: computer controlled sample positioner X-ray detector Samples: single crystals of NaCl, LiF and Al Figure. 3 has a schematic of the x-ray setup. A source of x-rays is collimated with slit. The x-rays strike the sample and then pass through another collimating slit before begin detected an x-ray counter 3. A more detailed view of the setup is shown in Fig. 4. Caution: X-rays are also not good for the human body, as they can ionize matter and lead to cell damage. In this experiment the x-rays are contained in the apparatus using lead lined glass and other materials that absorb the x-rays. Of course, care must be taken to ensure that the glass window is closed during the measurements, and that you are not in close proximity to the unit when it is running. The unit should also be off when you are not taking data. 4

5 d = 8.0 pm: lattice plane spacing of NaCl PHYS-UA 74 Intermediate Experimental Physics II Compton Effect Figure 3: Diagram showing the basic components of the x-ray experiment.. Collimator,. Sample and 3. Detector slit and x-ray counter. In this experiment the incident and scattered angle are not necessarily he angular width and the angular spacing Fig. 3 Diagram showing the diffraction of x-rays at a monocrystal nsity maxima. collimator, monocrystal, 3 counter tube equal. LEYBOLD Physics Leaflets P tus fulfills all regulations governing an nd fully protected device for instructional proved for school use in Germany (NW tion and screening measures reduce the tside of the x-ray apparatus to less than hich is on the order of magnitude of the d radiation. the x-ray apparatus into operation inmage and to make sure that the high t off when the sliding doors are opened n Sheet for x-ray apparatus). y apparatus secure from access by unrsons. Fig. 3 Leybold leaflet P node of the x-ray tube Mo to overheat. g on the x-ray apparatus, check to make entilator in the tube chamber is turning. The wavelength interval of two lines thus corresponds to the angular spacing = a) Without copper filter: n d cos (III), which increases with the diffraction order. It is important to distinguish between the angular spacing and the angular width of an intensity maximum. This latter should be smaller than the angular spacing so that the two lines can be observed separately (see Fig. ). The angular width is determined by the opening slit of the counter tube (see Fig. 3), its distance from the crystal and the divergence of the incident x-ray beam, and remains constant even for higher diffraction orders. Thus, the K doublet can be resolved in the diffraction order n = 5. Figure 4: Experimental setup for Compton scattering. Left: without Copper filter, Center: with Cu filter b) Copper filter front of aluminum scattering body: (a) in front of Al scattering body (b), Right: with Cu filter behind the Al scattering (b). Figure from 3. Setup Experiment setup for verifying the wavelength shift due to Compton scattering Left: without copper filter Center: with copper filter (a) in front of aluminum scattering body (b) Right: with copper filter (a) behind aluminum scattering body (b) Set the measuring time per angular step to t = 60 s. Start the measurement with the SCAN key and display the mean counting rate R after the measuring time elapses by pressing REPLAY. Record the result as counting rate R 0. Place the copper filter on the collimator. Increase the measuring time per angular step to t = 600 s. Start the measurement with the SCAN key and display the mean counting rate R after the measuring time elapses by pressing REPLAY. Record the result as counting rate R. s positioned solely by electric stepper c) Copper filter behind aluminum body: Setup Mount the copper filter on the sensor seat. Set up the experiment as in Fig. 3. Carry out the Start the measurement with the SCAN key and display the the target arm and Setsensor up the arm experiment following of thesteps: as shown in Fig. 4. mean counting rate R after the measuring time elapses by d do not use force to move them. Remove the collimator Setup pressing REPLAY. Record the result as counting rate R. Carry out the followingand steps: mount the zirconium filter supplied with the x-ray apparatus on the radiation inlet aperture side of the collimator. d) Background effect: Remove Mount the the collimator Setup together in Bragg and with the mount configuration: zirconium filter. the zirconium Set the emission filter current supplied I = 0. with the x-ray apparatus oncounter the (see radiation the Fig. Instruction 4 shows inlet Sheet some of the aperture x-ray important apparatus). sidetails of mean the of counting the collimator. experiment rate R after the setup. measuring time elapses by If necessary, mount the goniometer and end window Start measurement with the SCAN key and display the Set a distance between To set the up collimator the experiment, and target of approx. proceed pressing as follows REPLAY. Record (see the also result the as counting rate R. 5 cm and a distance between the target and the detector Mount the inlet diaphragm collimator Instruction of approx. (in 4 cm. Sheet the correct for x-ray orientation!) apparatus): together with the zirconium filter. Mount the aluminum scattering body from the Compton accessory x-ray as the target. Press the TARGET key and manually set the target angle to Measuring example If necessary, mount the goniometer and end window counter (see the Instruction Sheet 0 using the ADJUST knob. U = 30 kv, I =.00 ma of the x-ray Press the apparatus SENSOR key and manually on the set the labsensor website). angle to 45 using the ADJUST knob. Target angle = 0, sensor angle = 45 a) R 0 = 3.88 s t = 60 s Set a distance between the collimator andb) target R = to s about t = 6009 s cm and a distance between Carrying out experiment c) R = s t = 600 s the target and the detector also to approximately 6 cm. Set the tube high voltage U = 30 kv and the emission d) R = 0.8 s t = 600 s current I =.00 ma. Set the angular step width = 0.0. Mount the aluminum scattering body from the Compton accessory x-ray as the target. 5 3 Evaluation and results From (VIII) and (IX), we can calculate the respective transmission values as T = 0.9 and T = 0.07

6 The Al should be pressed firmly against a ridge in the sample holder to be at proper height. Turn on the x-ray apparatus! Press the TARGET key and manually set the target angle to 0 by using the ADJUST knob. Press the SENSOR key and manually set the sensor angle to 0 by using the ADJUST knob. Gradually and carefully increase the sensor angle (steps of 5 ) to 40 watching to be sure the GM tube does not touch the inner left side of the instrument which would damage the sensor. 3.3 Carrying out the experiment Set the tube high voltage V T = 30 kv and the emission current I =.00 ma. Set the angular step width θ = 0.0; you will not be varying the scattering angles in the next series of steps, only counting x-rays at a fixed scattering angle. a) Without the copper filter Set the measuring time per angular step to t = 60 s. R 0. b) With the copper filter in front of the Al scatterer Place the copper filter on the collimator. Increase the measuring time per angular step to t = 300 s. (You can also use t = 600 s to get better statistics.) R. c) With copper filter behind the Al scatterer Mount the copper filter on the sensor seat. R. d) Background counts Set the emission current I = 0. R B. 6

7 3.4 Results and further studies From the data you have taken compute T and T. Use the results to find the average wavelength shift on scattering from the Al body. Compare you results with the theoretical expectations. Do you they agree? If not, what are possible reasons for the discrepancy? In addition, vary the scattering angles (at least two additional angles) and repeat the above experiment. The aim of this study is to determine the angular dependence of the average wavelength shift and compare it to that expected for Compton scattering, Eqn. 3. Determine the wavelength shift for these additional angles and plot the wavelength shift versus the scattering angle θ. Plot the theoretically expected wavelength shift as a function of angle on the same graphic. Do the results agree? If not, what are possible reasons for the discrepancy? There are several other experiments that you can try. Here are some suggestions, but please feel free to propose and conduct your own experiment! Conduct at least one more experiment that uses the x-ray apparatus. Determine the average wavelength shift when a Bragg condition is satisfied for x-ray scattering from Al. Note that to find the Bragg condition you need to conduct a angle scan and also set the angle of incident to be equal to the angle of reflection, using the COUPLED mode of the setup. How does the average wavelength shift compare to those of your previous studies? Is the wavelength shift smaller or larger and why? Determine the average wavelength of the x-rays as a function of the tube voltage V T. Are the results what you expect? Why or why not? Repeat the experiment without the copper filter 0 times and determine the standard deviation of the counting rate. Does your resulting standard deviation agree with that expectations for a Poisson distribution, σ R = R/t c 4 Finishing Up Turn off the x-ray apparatus and remove the crystals and place it back in plastic bin with the other components of the experiment. 5 Final Note Include an analysis of the uncertainties in your determination of the wavelength shift in your lab reports. Also, answer all the questions in this lab writeup in your laboratory reports! 7

Atomic and nuclear physics

Atomic and nuclear physics X-ray physics Attenuation of x-rays LEYBOLD Physics Leaflets P6.3.2.2 Investigating the wavelength dependency of the coefficient of attenuation Objects of the experiment To measure