上海交通大学：《物理实验》精品课程教学资源（X光实验讲义）Compton effect：verifying the energy loss of the scattered x-ray quantum

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Fig. 1 Compton scattering of an x-ray quantum at an electron at rest. Atomic and nuclear physics X-ray physics Physics of the atomic shell LEYBOLD Physics Leaflets Compton effect: verifying the energy loss of the scattered x-ray quantum Objects of the experiment Measuring the transmission values T1 and T2 of a Cu foil for unscattered x-rays and x-rays scattered at an aluminum body. Determining the wavelength shift for scattered x-rays from the change in the transmission. Comparing the measured wavelength shift with that calculated on the basis of the Compton scattering. Principles Compton effect: In 1923, the American physicist A. H. Compton observed a wavelength shift in x-rays scattered at a scattering body, which he explained on the basis of the quantum nature of x-rays. He interpreted this effect as a collision between an x-ray quantum and an electron of the scattering material, in which the energy E = h ⋅ c l (I) h: Planck’s constant c: velocity of light l: wavelength of the x-ray quantum changes, as the kinetic energy is transferred to the electron. In the collision, energy and momentum are conserved. Before the collision, the electron can be considered as being at rest (see Fig. 1). When v is the velocity of the electron after the collision and l1 and l2 are the wavelengths of the x-ray quantum before and after collision, a relativistic formulation of the conservation of energy gives us h ⋅ c l1 + m0 ⋅ c2 = h ⋅ c l2 + m0 ⋅ c2 √1 −    v c    2 (II) m0: mass of electron The momentum of an x-ray quantum is p = h l (III) The conservation of momentum is thus h l1 = h l2 ⋅ cos q + m0 √1 −    v c    2 ⋅ v ⋅ cos w and 0 = h l2 ⋅ sin q + m0 √ 1 −    v c    2 ⋅ v ⋅ sin w (IV). q, w: collision angles (see Fig. 1). We can apply some transformations to (II) and (IV) to arrive at the following relation for the change in the wavelength: l2 − l1 = h m0 ⋅ c (1 − cosq) (V) The quantity h m0 ⋅ c = 2.43 pm (VI) is termed the Compton wavelength lC. It is identical to the wavelength of a photon having an energy equal to the resting energy of the electron. P6.3.3.7 0508-Ste 1

P6.3.3.7 LEYBOLD Physics Leaflets This experiment initially measures the counting rate Ro for x-rays scattered at an aluminum body without attenuation,and Apparatus then the counting rates R and R2 with the copper foil placed 1X-ray apparatus.············· 554811 in front of and behind the aluminum body,respectively(see 1 End-window counter Fig.3).As the counting rates are low,the counting rate R of fora,B,y and x-ray radiation·.·.··. 55901 background radiation is also taken into consideration.Using equation (VIl),the transmission values 1 Compton accessory x-ray..·.···· 554836 R1-R T1=Ro-R (I0 and T2= R2-R Verifying the wavelength shift: Ro-R (X) allow us to calculate the "mean"wavelengths1 of the un- Verification of the wavelength shift is based on an arrangement scattered x-rays and A2 of the unscattered x-rays.In accord- developed by R.W.Pohl,in which the attenuation of the ance with (VIl)to(IX),the wavelength shift unscattered x-radiation in a copper foil is compared with that of radiation scattered using an aluminum scattering body.The △λ=2-入1 ☒ transmission Tcu of the copper foil depends in the wavelength under study here is of the x-rays (see Fig.2).Therefore,a shift in the wavelength of the x-radiation due to Compton scattering becomes ap- parent as a change in the transmission or the counting rate. (In (Ro-R)-In (R2-R) △x=100pm n The evaluation is facilitated by the fact that the wavelength- dependency of the transmission of the copper foil can be formulated as (In (Ro-R)-In (R1-R) (AP (X0. Tcu=e-a100pm) (VII) with a =7.6 and n=2.75. Fig.2 Transmission of copper foil (d=0.07 mm)in the Safety notes wavelength range from 50 to 80 pm,measured at the The x-ray apparatus fulfills all regulations governing an LEYBOLD DIDACTIC laboratories x-ray apparatus and fully protected device for instructional use and is type approved for school use in Germany (NW807/97Ro). The built-in protection and screening measures reduce the 0,3 local dose rate outside of the x-ray apparatus to less than 1 uSv/h,a value which is on the order of magnitude of the natural background radiation. 入 Before putting the x-ray apparatus into operation in- spect it for damage and to make sure that the high voltage is shut off when the sliding doors are opened 0,2 (see Instruction Sheet for x-ray apparatus). Keep the x-ray apparatus secure from access by un- authorized persons. Do not allow the anode of the x-ray tube Mo to overheat. When switching on the x-ray apparatus,check to make sure that the ventilator in the tube chamber is turning. 0,1 The goniometer is positioned solely by electric stepper motors. Do not block the target arm and sensor arm of the goniometer and do not use force to move them. 0 50 60 70 80 pm 2

Verifying the wavelength shift: Verification of the wavelength shift is based on an arrangement developed by R. W. Pohl, in which the attenuation of the unscattered x-radiation in a copper foil is compared with that of radiation scattered using an aluminum scattering body. The transmission TCu of the copper foil depends in the wavelength of the x-rays (see Fig. 2). 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 by the fact that the wavelengthdependency of the transmission of the copper foil can be formulated as TCu = e − a ⋅    l 100pm    n (VII) with a = 7.6 and n = 2.75. This experiment initially measures the counting rate R0 for x-rays scattered at an aluminum body without attenuation, and then the counting rates R1 and R2 with the copper foil placed in front of and behind the aluminum body, respectively (see Fig. 3). As the counting rates are low, the counting rate R of background radiation is also taken into consideration. Using equation (VII), the transmission values T1 = R1 − R R0 − R (VIII) and T2 = R2 − R R0 − R (IX) allow us to calculate the “mean” wavelengths l1 of the unscattered x-rays and l2 of the unscattered x-rays. In accordance with (VII) to (IX), the wavelength shift Dl = l2 − l1 (X) under study here is Dl = 100pm ⋅         ln (R0 − R) − ln (R2 − R) a    1 n      −    ln (R0 − R) − ln (R1 − R) a    1 n      (XI). Apparatus 1 X-ray apparatus . . . . . . . . . . . . . . 554 811 1 End-window counter for a, b, g and x-ray radiation . . . . . . . 559 01 1 Compton accessory x-ray . . . . . . . . 554 836 Safety notes The x-ray apparatus fulfills all regulations governing an x-ray apparatus and fully protected device for instructional use and is type approved for school use in Germany (NW 807/97 Rö). The built-in protection and screening measures reduce the local dose rate outside of the x-ray apparatus to less than 1 mSv/h, a value which is on the order of magnitude of the natural background radiation. Before putting the x-ray apparatus into operation inspect it for damage and to make sure that the high voltage is shut off when the sliding doors are opened (see Instruction Sheet for x-ray apparatus). Keep the x-ray apparatus secure from access by unauthorized persons. Do not allow the anode of the x-ray tube Mo to overheat. When switching on the x-ray apparatus, check to make sure that the ventilator in the tube chamber is turning. The goniometer is positioned solely by electric stepper motors. Do not block the target arm and sensor arm of the goniometer and do not use force to move them. Fig. 2 Transmission of copper foil (d = 0.07 mm) in the wavelength range from 50 to 80 pm, measured at the LEYBOLD DIDACTIC laboratories P6.3.3.7 LEYBOLD Physics Leaflets 2

LEYBOLD Physics Leaflets P6.3.3.7 GM-TUBE GM-TUBE GM-TUBE Fig.3 Experiment setup for verifying the wavelength shift due to b)Copper filter in front of aluminum scattering body: Compton scattering Place the copper filter on the collimator. Left:without copper filter Center:with copper filter(a)in front of aluminum scatter- 、 Increase the measuring time per angular step to At=600 s ing body(b) Start the measurement with the SCAN key and display the Right:with copper filter(a)behind aluminum scattering mean counting rate R after the measuring time elapses by body (b) pressing REPLAY.Record the result as counting rate R1. c)Copper filter behind aluminum body: Setup Mount the copper filter on the sensor seat. Set up the experiment as shown in Fig.3.Carry out the Start the measurement with the SCAN key and display the following steps: mean counting rate R after the measuring time elapses by pressing REPLAY.Record the result as counting rate R2. Remove the collimator and mount the zirconium filter sup- plied with the x-ray apparatus on the radiation inlet aperture side of the collimator. d)Background effect: Mount the collimator together with the zirconium filter. Set the emission current /=0. If necessary.mount the goniometer and end window Start the measurement with the SCAN key and display the counter (see the Instruction Sheet of the x-ray apparatus). mean counting rate R after the measuring time elapses by Set a distance between the collimator and target of approx. pressing REPLAY.Record the result as counting rate R. 5 cm and a distance between the target and the detector inlet diaphragm of approx.4 cm. 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 20using the ADJUST knob. U=30kV,/=1.00mA Press the SENSOR key and manually set the sensor angle to 145 using the ADJUST knob. Target angle-2o°，sensor angle=145 a)R=3.88s-1△t=60s b) R1=0.669s-1△t=600s Carrying out the experiment c R2=0.496s-1△t=600s Set the tube high voltage U=30 kV and the emission d R=0.218s-1△t=600s current /1.00 mA. -Set the angular step width△B=O.o°. a)Without copper filter: Evaluation and results Set the measuring time per angular step to At =60 s. Start the measurement with the SCAN key and display the From (VIll)and(IX),we can calculate the respective transmis- mean counting rate R after the measuring time elapses by sion values as pressing REPLAY.Record the result as counting rate Ro T1=0.119andT2=0.071 3

Setup Set up the experiment as shown in Fig. 3. Carry out the following steps: – Remove the collimator and mount the zirconium filter supplied with the x-ray apparatus on the radiation inlet aperture side of the collimator. – Mount the collimator together with the zirconium filter. – If necessary, mount the goniometer and end window counter (see the Instruction Sheet of the x-ray apparatus). – Set a distance between the collimator and target of approx. 5 cm and a distance between the target and the detector inlet diaphragm of approx. 4 cm. – 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 208 using the ADJUST knob. – Press the SENSOR key and manually set the sensor angle to 1458 using the ADJUST knob. Carrying out the experiment – Set the tube high voltage U = 30 kV and the emission current I = 1.00 mA. – Set the angular step width Db = 0.08. a) Without copper filter: – Set the measuring time per angular step to Dt = 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 R0. b) Copper filter in front of aluminum scattering body: – Place the copper filter on the collimator. – Increase the measuring time per angular step to Dt = 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 R1. c) Copper filter behind aluminum body: – Mount the copper filter on the sensor seat. – 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 R2. d) Background effect: – Set the emission current I = 0. – 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. Measuring example U = 30 kV, I = 1.00 mA Target angle = 208, sensor angle = 1458 a) R0 = 3.88 s–1 Dt = 60 s b) R1 = 0.669 s–1 Dt = 600 s c) R2 = 0.496 s–1 Dt = 600 s d) R = 0.218 s–1 Dt = 600 s Evaluation and results From (VIII) and (IX), we can calculate the respective transmission values as T1 = 0.119 and T2 = 0.071 Fig. 3 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) LEYBOLD Physics Leaflets P6.3.3.7 3

P6.3.3.7 LEYBOLD Physics Leaflets Using (VIl),we can then calculate the wavelengths of the We can use the abbreviations unscattered and scattered x-radiation as λ1=62.6 pm and入2=67.5pm _1_.(100pm)” Ao3.0 pms. and the wavelength shift as △λ=(4.9±1.3)pm. 1.(100pm)" Ao222.6 pms (See below for calculation of the statistical error). Comparison with theory of Compton scattering: 1.100pm) A1=R1-Rn:=24 pm.s and The value to be expected according to(V)at a scattering angle 8=145°is△amda=4.42pm. 1,100pm)」 A234 pm.s to reformulate the partial derivations appearing here as: 22=-A01+Ao2=-0.4pm·s, aRo A=A1=24pm·5, aR Additional information aAX=-A22=-34 pm.s and aR2 Statistical error: =-A11+A22-Ao1+Ao2=+10pm:s When measuring a counting rate R at a time interval At, aR N=R.△t As expected,the errors of the two counting rates R and R2 make up the largest part of this statistical error,while the effect events are counted each time.The statistical error of the of the counting rate Ro can be largely ignored. counting rate R here is thus From the numerical values of the measuring example,we -YN-VR obtain 0R At At CA 1.3 pm For the counting rates given in the measuring example,it This value can be halved by increasing the measuring times for follows that R and R2 by a factor of four. R=0.3s-1,R,=0.03s-1,R=0.03s1 and oR=0.02s1 Corrections: These errors carry over into the determination of the wavelength shift AX.For this statistical error x,we can say In addition to inelastic Compton scattering,it would also be appropriate to take the elastic scattering of x-rays into con- j器j钱j sideration,in which by definition no wavelength shift occurs. The measured transmission T2 is reduced less with respect to 7 than the transmission 72'for purely inelastic scattered 器j器j x-rays which we are actually trying to detect.The evaluation thus estimates the wavelength shift AX due to Compton scat- tering too low. LEYBOLD DIDACTIC GMBH.Leyboldstrasse 1.D-50354 Hurth.Phone (02233)604-0.Telefax(02233)604-222.Telex 17 223 332 LHPCGN D by Leybold Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved

Using (VII), we can then calculate the wavelengths of the unscattered and scattered x-radiation as l1 = 62.6 pm and l2 = 67.5 pm and the wavelength shift as Dl = (4.9 ± 1.3) pm. (See below for calculation of the statistical error sDl). Comparison with theory of Compton scattering: The value to be expected according to (V) at a scattering angle q = 1458 is Damda = 4.42 pm. Additional information Statistical error: When measuring a counting rate R at a time interval Dt, N = R ⋅ Dt events are counted each time. The statistical error of the counting rate R here is thus sR = √N Dt = √R Dt . For the counting rates given in the measuring example, it follows that sR0 = 0.3 s −1, sR1 = 0.03 s−1, sR2 = 0.03 s−1 and sR = 0.02 s−1 These errors carry over into the determination of the wavelength shift Dl. For this statistical error sDl, we can say   sDl  2 =    Dl R1 ⋅ sR1    2 +    Dl R2 ⋅ sR2    2 +    Dl R0 ⋅ sR0    2 +    Dl R ⋅ sR    2 We can use the abbreviations A01 = 1 R0 − R ⋅ (100 pm) n n ⋅ a ⋅ l1 n−1 = 3.0 pm ⋅ s, A02 = 1 R0 − R ⋅ (100 pm) n n ⋅ a ⋅ l2 n−1 = 2.6 pm ⋅ s A11 = 1 R1 − R ⋅ (100 pm) n n ⋅ a ⋅ l1 n−1 = 24 pm ⋅ s and A22 = 1 R2 − R ⋅ (100 pm) n n ⋅ a ⋅ l2 n−1 = 34 pm ⋅ s to reformulate the partial derivations appearing here as: Dl R0 = − A01 + A02 = − 0.4 pm ⋅ s, Dl R1 = A11 = 24 pm ⋅ s, Dl R2 = − A22 = − 34 pm ⋅ s and Dl R = − A11 + A22 − A01 + A02 = + 10 pm ⋅ s As expected, the errors of the two counting rates R1 and R2 make up the largest part of this statistical error, while the effect of the counting rate R0 can be largely ignored. From the numerical values of the measuring example, we obtain sDl = 1.3 pm This value can be halved by increasing the measuring times for R1 and R2 by a factor of four. Corrections: In addition to inelastic Compton scattering, it would also be appropriate to take the elastic scattering of x-rays into consideration, in which by definition no wavelength shift occurs. The measured transmission T2 is reduced less with respect to T1 than the transmission T2‘ for purely inelastic scattered x-rays which we are actually trying to detect. The evaluation thus estimates the wavelength shift Dl due to Compton scattering too low. LEYBOLD DIDACTIC GMBH ⋅ Leyboldstrasse 1 ⋅ D-50354 Hürth ⋅ Phone (02233) 604-0 ⋅ Telefax (02233) 604-222 ⋅ Telex 17 223 332 LHPCGN D © by Leybold Didactic GmbH Printed in the Federal Republic of Germany Technical alterations reserved P6.3.3.7 LEYBOLD Physics Leaflets

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