R LEP Compton scattering of X-rays 5.4.07

Related topics 2. The measurement of problem 1. is to be repeated, this time X-rays, Bragg equation, absorption, transmission, Compton for a limited wavelength interval with maximum angular res- effect, Compton wavelength, rest energy, conservation of olution. momentum and energy conservation principle, relativistic elec- 3. By using a scatterer, the intensity of the X-rays scattered trons, Einstein’s mass/energy relation. under 90° is determined; then, the intensity attenuation is determined for an aluminium absorber in two different posi- tions. Principle and task 4. The Compton wavelength is determined from the different X-rays strike a scatterer and are then scattered according to transmissions yielded from problem 3. and compared with Compton. By means of a counter tube, the portion scattered the theoretical value. under 90° is recorded. By positioning an absorber in front of as well as above a scattering body, the Compton wavelength can be determined from a pre-recorded and measured transmission Set-up and procedure curve due to the varying intensity attenuation of the X-rays. The experiment is set up as shown in Fig. 1. The aperature of d = 2 mm is introduced into the outlet X-raxs. By pressing the “zero key”, the counter tube and crystal holder device are Equipment brought into starting position. The crystal holders are mount- X-ray unit. w. recorder output 09056.97 1 ed with the crystal surface set horizontally. The counter tube, Compton attachment 09052.01 1 with horizontal slit aperture, is mounted in such a way that the Counter tube, type A, BNC 09025.11 1 mid-notch of the counter tube output of the X-ray machine is Geiger-Müller-Counter 13606.99 1 connected to the corresponding input of the digital counter. Screened cable, BNC, l 750 mm 07542.11 1 The counter tube voltage is app. 500 V. First the zero pulse rate N0[Imp/sec] is determined at a coun- ter tube anode voltage of Ua = 0 V. Problems Beginning at a glancing angle of ␽ = 10° and at maximal 1. The transmission cuve of an aluminium absorber as a func- anode voltage, the X-ray pulse rate reflected by the crystal N1 tion of the wavelength is determined by means of Bragg (␽) is determined in steps of 1° up to ␽ Ϸ 18°, using the digi- scattering and plotted graphically. tal counter.

Fig. 1: Experimental set-up with Compton scattering device.

PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany 25407 1 R LEP Compton scattering of X-rays 5.4.07

This is done at a synchronized rotation of the crystal and Fig. 3: Scattering geometry of the Compton effect. counter tube in an angular relationship of 1:2. Due to the nec- essary exactitude, however, a number of pulses ≥8000 should be measured. If the measured number of pulses is ⌱, the rela- tive measuring error is given by the ratio: ⌬⌱ ͙⌱ 1 = = ⌱ ⌱ ͙⌱ The aluminium absorber is then inserted between the X-ray outlet and the crystal. Using this arrangement, the measure- ␽ ments are repeated to determine the pulse rate N2 ( ). At high pulse rates N, not all incoming photons are recorded due to the dead time ␶ = 100 ␮s of the counter tube. Determination of the true pulse rates NЈ follows with the help Theory and evaluation of the relation: A schematic representation of the scattering geometry of the Compton effect is shown in Fig. 3. The incident photon with an N NЈ = (1) energy loss under scattering angle ␣ is veered away from its 1 – ␶N original direction, while the formerly idle, free electron is emit- By means of the Bragg equation ted from the collision point with an energy gain under angle ␤ with respect to the direction of incidence of the photon. 2d sin ␽ = ␭ lattice constant (2) (d = 2.014 · 10-12 m), The energy conservation principle yields the wavelength length ␭ is calculated as a function of the 2 2 glancing angle ␽. hƒ1 + m0c = hƒ2 + mc (3) From the ratio of the corrected pulse rates, the transmission values are calculated as a function of the wavelength and plot- h = Planck’s quantum of action ted graphically (Fig. 4). ƒ1/ƒ2 = photon frequency before/ Subsequently, in order to determine with adequate precision after collision the wavelength change due to scattering from the transmis- m/m = electron mass/electron rest mass sion curve, the measurement is repeated in the interval of 0 10.0° ≤ ␽ ≤11.2° in steps of ⌬␽ = 0.2° (Fig. 5). c = velocity of light Before the Compton attachment – complete with counter and screening tube – is installed, remove the crystal and the counter From the momentum conservation principle follows: tube slit diaphragm, turn the counter tube holder to its 90° final Ǟ Ǟ ␯Ǟ (4) position, and insert the aperture of d = 5 mm. Then determine P1 = P2 + m Ǟ Ǟ the following pulse rates at maximum anode voltage (Fig. 2): P1/P2 = photon impulse before/after collision N : with plexiglass scatterer but without A1-absorber Ǟ 3 ␯ = electron velocity N4: with scatterer and absorber in position 1 N : with scatterer and absorber in position 2 5 By using the cosine formula, Fig. 3 yields: If necessary, dead time and background radiation must also m2␯2 = P 2 + P 2 – 2 p p cos ␣ (5) be taken into account. 1 2 1 2 h Note: The counter tube should never be exposed to primary If photon impulse p =␭ is inserted in (5), the result is: radiation for any longer period of time. 2 2 2 2␯2 h h – 2 h cos ␣ (6) m = ␭2 + ␭2 ␭ ␭ 1 2 1 2 Squaring (3) yields the necessary formula: 1 c2 (m – m )2 = ͓(hƒ )2 ϩ (hƒ )2 – 2 h2ƒ ƒ ͔ (7) 0 c2 1 2 1 2 2 2 2 h h – 2h = ␭2 + ␭2 ␭ ␭ 1 2 1 2 Subtracting (7) and (6) produces:

2 2␯2 2 2 2h ␣ m – c (m–m0) = ␭ ␭ (1–cos ) (8) 1 2

4h2 ␣ = sin2 ␭ ␭ 2 Fig. 2: Schematic representation of the 90° Compton scatter- 1 2 ing attachment Taking into account the relativistic electron mass S = Scatterer m0 A = A1-absorber in positions 1 and 2 m = ␯2 (9) ͙1– D = Detector. c2

2 25407 PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany R LEP Compton scattering of X-rays 5.4.07

yields from (8) Fig. 4: Transmission of the A1-absorber as a function of the 2 2 2 2 2 2 wavelength. c (m –m0 )–c (m–m0) = 2 c m0 (m–m0) (10) 2 4h 2␣ = ␭ ␭ sin 1 2 2 Eliminating electron mass m by means of (3) yields:

1 1 2h 2 ␣ m0 c ( ␭ – ␭ ) = ␭ ␭ sin (11) 1 2 1 2 2 By rearraging (11), the desired wavelength change of the pho- ton as a function of scattering angle ␣ is finally attained. ␣ ⌬␭ ␭ ␭ 2h 2 = 2 – 1 = sin (12) m0c 2 The wavelength change and energy transfer attain maximum values for central collision (180° back-scattering). The change in wavelength for 90° scattering is called the Compton wavelength. The following holds true for this: h ␭ = = 2.426 · 10-12 m (13) c m0c h = 6.626 * 10-34 Js -31 m0 = 9.109 * 10 kg c = 2.998 * 108 ms-1

␭ A photon with Compton wavelength c contains the energy.

h · c 2 Ec = hƒc = ␭ = m0 c (14) c

i.e., Ec is the electron rest energy. Fig. 4 shows the trans- This value corresponds acceptably well with the theoretical mission T as a function of the wavelength ␭. There after, T value of the Compton wavelength. decreases almost linearly with greater wavelenghts (smaller photon energy). The later ascent is caused by the 2nd order of Bragg scattering. By allowing the X-rays to collide with a scatterer, it is possible to determine the pulse rate N3 scattered in a 90° direction and then the pulse rates N4 (absorber in front of the scatterer) and N5 (absorber behind the scatterer). It can be seen that

N4 N5 T1 = >T2 = N3 N3 The means that the wavelength of the scattered radiation is greater than the wavelength of the primary radiation. For the scattering rates, the following values have been found: ⌱ ⌱ = 8329 ⌱mp/39.5 s ; ⌬ 3/⌱ ϳ ± 1.1% 3 3 ⌱ Ј N3 = 210.9 mp/s ; N 3 = 215 Imp/s ⌱ ⌱ = 8125 ⌱mp/236.2 s ; ⌬ 4/⌱ ϳ ± 1.1% 4 4 ⌱ Ј N4 = 34.4 mp/s ; N 4 = 34.2 Imp/s ⌱ ⌱ = 7275 ⌱mp/261.5 s ; ⌬ 5/⌱ ϳ ± 1.2% 5 5 ⌱ Ј N5 = 27.8 mp/s ; N 5 = 27.6 Imp/s

(NЈ value = dead time and background radiation corrected) Ј ⌱ ⌬ N 4 34.2 mp /s T1 ϳ T1 = Ј = ⌱ = 0.159; ± 2.2% N 3 215 mp /s T1 Ј ⌱ ⌬ N 5 27.6 mp /s T2 ϳ T2 = Ј = ⌱ = 0.128; ± 2.3% N 3 215 mp /s T2 The T-values recorded in Fig. 5 yield within their error limits the following: Fig. 5: Transmission of the A1-absorber as a function of the ⌬␭ ␭ -12 = c = (2.35 ± 0.25) * 10 m wavelength with higher angular resolution.

PHYWE series of publications • Laboratory Experiments • Physics • PHYWE SYSTEME GMBH • 37070 Göttingen, Germany 25407 3