Atomic and Nuclear Physics LEYBOLD Physics X-ray physics Leaflets P6.3.3.1 Physics of the atomic shell Bragg reflection: diffraction of x-rays at a monocrystal Objects of the experiment To investigate Bragg reflection at an NaCI monocrystal using the characteristic x-ray radiation of molybdenum. To determine the wavelength for the characteristic K and Kg x-ray radiation of molybdenum. To confirm Bragg's law of reflection. To verify the wave nature of x-rays. Principles In 1913.H.W.and W.L.Bragg realized that the regular arrangement of atoms and/or ions in a crystal can be under- stood as an array of lattice elements on parallel lattice planes. When we expose such a crystal to parallel x-rays,additionally assuming that these have a wave nature,then each element in nλ=2dsin9 a lattice plane acts as a "scattering point",at which a spherical wavelet forms.According to Huygens,these spherical wave- 9 lets are superposed to create a "reflected"wavefront.In this model,the wavelength A remains unchanged with respect to the "incident"wave front,and the radiation directions which are perpendicular to the two wave fronts fulfill the condition "angle of incidence angle of reflection". Constructive interference arises in the rays reflected at the individual lattice planes when their path differences A are integral multiples of the wavelength A. △=n-λwith n=1,2,3, 0 As Fig.1 shows for two adjacent lattice planes with the spacing 10 30 d,we can say for the path differences A1 and A2 of the incident and reflected rays with the angle △1=△2=d.sin8 so that the total path difference is △=2.d.sin8. 0 (1)and(lI)give us Bragg's law of reflection: n.λ=2.d.sin8 0 The angle is known as the glancing angle In this experiment,we verify Bragg's law of reflection by investigating the diffraction of x-rays at an NaCl monocrystal in which the lattice planes are parallel to the cubic surfaces of the unit cells of the crystal.The lattice spacing d of the cubic e
Objects of the experiment To investigate Bragg reflection at an NaCl monocrystal using the characteristic x-ray radiation of molybdenum. To determine the wavelength for the characteristic Ka and Kβ x-ray radiation of molybdenum. To confirm Bragg’s law of reflection. To verify the wave nature of x-rays. Bragg reflection: diffraction of x-rays at a monocrystal 0308-Ste Atomic and Nuclear Physics X-ray physics Physics of the atomic shell P6.3.3.1 LEYBOLD Physics Leaflets Principles In 1913, H. W. and W. L. Bragg realized that the regular arrangement of atoms and/or ions in a crystal can be understood as an array of lattice elements on parallel lattice planes. When we expose such a crystal to parallel x-rays, additionally assuming that these have a wave nature, then each element in a lattice plane acts as a “scattering point”, at which a spherical wavelet forms. According to Huygens, these spherical wavelets are superposed to create a “reflected” wavefront. In this model, the wavelength l remains unchanged with respect to the “incident” wave front, and the radiation directions which are perpendicular to the two wave fronts fulfill the condition “angle of incidence = angle of reflection”. Constructive interference arises in the rays reflected at the individual lattice planes when their path differences D are integral multiples of the wavelength l. D = n ⋅ l with n = 1, 2, 3, … (I) As Fig. 1 shows for two adjacent lattice planes with the spacing d, we can say for the path differences D1 and D2 of the incident and reflected rays with the angle q: D1 = D2 = d ⋅ sin q so that the total path difference is D = 2 ⋅ d ⋅ sin q. (II) (I) and (II) give us Bragg’s law of reflection: n ⋅ l = 2 ⋅ d ⋅ sin q (III) The angle q is known as the glancing angle. In this experiment, we verify Bragg’s law of reflection by investigating the diffraction of x-rays at an NaCl monocrystal in which the lattice planes are parallel to the cubic surfaces of the unit cells of the crystal. The lattice spacing d of the cubic 1
P6.3.3.1 LEYBOLD Physics Leaflets Apparatus 1X-ray apparatus.·············· 554811 1 End-window counter for,B,y and x-ray radiation.·...··· 55901 additionally required: 1 PC with Windows 9x or Windows NT face-centered NaCl crystal is half the lattice constant ao.We can thus say [1] Fig.1 Diagram of the reflection of x-rays at the lattice planes of a 2.d=a0=564.02pm monocrystal. △i,△z:path differences, The measurements are conducted using the built-in goniome- glancing angle. ter of the x-ray apparatus(554 811).The x-rays are detected d:spacing of lattice planes using a GM counter tube (end-window counter)which is swiveled in tandem with the NaCI crystal in a 2_coupling with respect to the incident light;this means that the counter tube always advances by an angle which is twice that of the crystal Table 1:Energy E,frequency v and wavelength A of the char- (cf.Fig.2). acteristic x-ray radiation of molybdenum (weighted mean The x-ray radiation consists of the bremsstrahlung continuum values [1]) and several sharply defined lines which correspond to the characteristic x-ray radiation of the Mo anode and which 总 EHz pm originate in the Ka and Kg transitions of the molybdenum atoms.This characteristic radiation is particularly suitable for Ka 17.443 4.2264 71.080 investigating Bragg's law.Its properties are known from the KB 19.651 4.8287 63.095 literature [2]and summarized in table 1.Table 2 shows the corresponding glancing angles at which the diffraction maxima keV =103 eV,EHz =1018 Hz,pm 10-12 m of the characteristic radiation are to be expected for scattering at an NaCl monocrystal (d =282.01 pm)up to the third diffraction order. Table 2:Glancing angle of the characteristic x-ray radiation of molybdenum for diffraction at an NaCl monocrystal up to the third order Safety notes 8(K (KB) The x-ray apparatus fulfills all German regulations govern- 1 7.24° 6.42 ing an x-ray apparatus and fully protected device for instructional use and is type approved for school use in 2 14.60° 12.93° Germany (NW 807/97 Ro). 3 22.21° 19.61° The built-in protection and screening measures reduce the 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. Fig.2 Diagram showing the principle of diffraction of x-rays at a Before putting the x-ray apparatus into operation,in- monocrystal and 2 coupling between counter-tube angle spect it for damage and check to make sure that the and scattering angle(glancing angle) high voltage shuts off when the sliding doors are 1 collimator,2 monocrystal,3 counter tube opened(see Instruction Sheet of 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. 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. 2
face-centered NaCl crystal is half the lattice constant a0. We can thus say [1] 2 ⋅ d = a0 = 564.02 pm The measurements are conducted using the built-in goniometer of the x-ray apparatus (554 811). The x-rays are detected using a GM counter tube (end-window counter) which is swiveled in tandem with the NaCl crystal in a 2_ coupling with respect to the incident light; this means that the counter tube always advances by an angle which is twice that of the crystal (cf. Fig. 2). The x-ray radiation consists of the bremsstrahlung continuum and several sharply defined lines which correspond to the characteristic x-ray radiation of the Mo anode and which originate in the Ka and Kb transitions of the molybdenum atoms. This characteristic radiation is particularly suitable for investigating Bragg’s law. Its properties are known from the literature [2] and summarized in table 1. Table 2 shows the corresponding glancing angles at which the diffraction maxima of the characteristic radiation are to be expected for scattering at an NaCl monocrystal (d = 282.01 pm) up to the third diffraction order. E keV n EHz l pm Ka 17.443 4.2264 71.080 Kb 19.651 4.8287 63.095 keV = 103 eV, EHz = 1018 Hz, pm = 10–12 m Table 1: Energy E, frequency n and wavelength l of the characteristic x-ray radiation of molybdenum (weighted mean values [1]) n q(Ka) q(Kb) 1 7.248 6.428 2 14.608 12.938 3 22.218 19.618 Table 2: Glancing angle q of the characteristic x-ray radiation of molybdenum for diffraction at an NaCl monocrystal up to the third order Apparatus 1 X-ray apparatus . . . . . . . . . . . . . . . 554 811 1 End-window counter for a, β, g and x-ray radiation . . . . . . . . 559 01 additionally required: 1 PC with Windows 9x or Windows NT Safety notes The x-ray apparatus fulfills all German 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 check to make sure that the high voltage shuts off when the sliding doors are opened (see Instruction Sheet of 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 Diagram showing the principle of diffraction of x-rays at a monocrystal and 2q coupling between counter-tube angle and scattering angle (glancing angle) 1 collimator, 2 monocrystal, 3 counter tube Fig. 1 Diagram of the reflection of x-rays at the lattice planes of a monocrystal. D1, D2: path differences, q: glancing angle, d: spacing of lattice planes P6.3.3.1 LEYBOLD Physics Leaflets 2
LEYBOLD Physics Leaflets P6.3.3.1 General remarks Adjust the sensor seat(b)until the distance s2 between the In principle,you can conduct measurements in both manual target arm and the slit diaphragm of the sensor seat is scan and autoscan modes of the x-ray apparatus (see the approx.6 cm. Instruction Sheet of the x-ray apparatus).You can record the Attach the target holder with target stage (f). measured values manually by reading the values from the Loosen knurled screw (g).lay the NaCI crystal flat on the display field and writing them in a table,using a chart recorder target stage,carefully raise the stage as far at it will go and or via a PC. then tighten the knurled screw with care(press against the The fastest and most convenient measurement is in autoscan screw lightly to prevent it from stripping). mode with simultaneous registration of measured values and Adjust the zero position of the goniometer measuring sys- subsequent evaluation on a Windows 9x/NT PC.This type of tem as necessary(see Instruction Sheet of x-ray appara- measurement is described in your Instruction Sheet. tus). The data is transmitted to the PC via the RS-232 serial inter- Notes: face on the x-ray apparatus.The software"X-ray Apparatus" NaCI crystals are hygroscopic and fragile.Store the crystals in supplied with the device,enables you to record,display and a dry place.Avoid mechanical stresses on the crystal;handle evaluate the data stream supplied by the x-ray apparatus.The the crystal by the short faces only. program contains detailed online help which you can access by pressing F1.Please refer to the Instruction Sheet of the If the counting rate is too low,you can reduce the distance s x-ray apparatus for details on installing the software. between the target and the sensor somewhat.However,this distance must not be too small,as otherwise the angular The Instruction Sheet also describes recording data under resolution of the goniometer is no longer great enough to Windows 3.1. separate the characteristic K and Kg lines. Setup Preparing a PC-based measurement: Setting up the Bragg configuration: Connect the RS-232 output to the serial interface on the PC (usually COM1 or COM2)using the 9-pin V.24 cable Fig.3 shows some important details of the experiment setup. (included with the x-ray apparatus). Specifically,you need to carry out the following steps (see also If you have not already done so,install the software"X-ray Instruction Sheet of x-ray apparatus): Apparatus"under Windows 9x/NT (see Instruction Sheet Mount the collimator in the collimator mount(a)(note the of x-ray apparatus)and select the desired language. guide groove). Attach the goniometer to the guide rods(d)in such a way that the distance s1 between the slit diaphragm of the collimator and the target arm is approx.5 cm.Connect the ribbon cable (c)for controlling the goniometer. Remove the cap of the end-window counter,insert the end-window counter in the sensor seat (e)and connect the counter tube lead to the socket marked GM-Tube. Carrying out the experiment Start the program"X-ray Apparatus",check to make sure that the x-ray apparatus is properly connected and delete any existing measurement data by clicking the button or pressing F4. Set the x-ray high voltage U=35.0 kV,emission current / 1.00 mA,measuring time per angular step At =10s and angular step width△B=o.1°. Fig.3 Experiment setup in Bragg configuration Press the COUPLED key on the device to enable 25 cou- pling of the target and sensor;set the lower limit value of the target angle to 2 and the upper limit to 25. Press the SCAN key to start the measurement and data transmission to the PC. When the measurement is finished,save the measurement series to a file under a suitable name using the button or F2. Measuring example Fig.4 shows the measured diffraction spectrum 3
General remarks In principle, you can conduct measurements in both manual scan and autoscan modes of the x-ray apparatus (see the Instruction Sheet of the x-ray apparatus). You can record the measured values manually by reading the values from the display field and writing them in a table, using a chart recorder or via a PC. The fastest and most convenient measurement is in autoscan mode with simultaneous registration of measured values and subsequent evaluation on a Windows 9x/NT PC. This type of measurement is described in your Instruction Sheet. The data is transmitted to the PC via the RS−232 serial interface on the x-ray apparatus. The software “X-ray Apparatus”, supplied with the device, enables you to record, display and evaluate the data stream supplied by the x-ray apparatus. The program contains detailed online help which you can access by pressing F1. Please refer to the Instruction Sheet of the x-ray apparatus for details on installing the software. The Instruction Sheet also describes recording data under Windows 3.1. Setup Setting up the Bragg configuration: Fig. 3 shows some important details of the experiment setup. Specifically, you need to carry out the following steps (see also Instruction Sheet of x-ray apparatus): – Mount the collimator in the collimator mount (a) (note the guide groove). – Attach the goniometer to the guide rods (d) in such a way that the distance s1 between the slit diaphragm of the collimator and the target arm is approx. 5 cm. Connect the ribbon cable (c) for controlling the goniometer. – Remove the cap of the end-window counter, insert the end-window counter in the sensor seat (e) and connect the counter tube lead to the socket marked GM-Tube. – Adjust the sensor seat (b) until the distance s2 between the target arm and the slit diaphragm of the sensor seat is approx. 6 cm. – Attach the target holder with target stage (f). – Loosen knurled screw (g), lay the NaCl crystal flat on the target stage, carefully raise the stage as far at it will go and then tighten the knurled screw with care (press against the screw lightly to prevent it from stripping). – Adjust the zero position of the goniometer measuring system as necessary (see Instruction Sheet of x-ray apparatus). Notes: NaCl crystals are hygroscopic and fragile. Store the crystals in a dry place. Avoid mechanical stresses on the crystal; handle the crystal by the short faces only. If the counting rate is too low, you can reduce the distance s2 between the target and the sensor somewhat. However, this distance must not be too small, as otherwise the angular resolution of the goniometer is no longer great enough to separate the characteristic Ka and Kb lines. Preparing a PC-based measurement: – Connect the RS−232 output to the serial interface on the PC (usually COM1 or COM2) using the 9-pin V.24 cable (included with the x-ray apparatus). – If you have not already done so, install the software “X-ray Apparatus” under Windows 9x/NT (see Instruction Sheet of x-ray apparatus) and select the desired language. Carrying out the experiment – Start the program “X-ray Apparatus”, check to make sure that the x-ray apparatus is properly connected and delete any existing measurement data by clicking the button or pressing F4. – Set the x-ray high voltage U = 35.0 kV, emission current I = 1.00 mA, measuring time per angular step Dt = 10 s and angular step width Db = 0.18. – Press the COUPLED key on the device to enable 2q coupling of the target and sensor; set the lower limit value of the target angle to 28 and the upper limit to 258. – Press the SCAN key to start the measurement and data transmission to the PC. – When the measurement is finished, save the measurement series to a file under a suitable name using the button or F2. Measuring example Fig. 4 shows the measured diffraction spectrum. Fig. 3 Experiment setup in Bragg configuration LEYBOLD Physics Leaflets P6.3.3.1 3
P6.3.3.1 LEYBOLD Physics Leaflets Rontgengerat-p6331 回☒ b凸鸟都回鱼 Bragg Planck Transmission Moseley R 1:K-alpha 2000 1:K-beta 1000- 2:K-alpha 2 K-beta 3.K-alpha 3:K-beta 20 ©by Leybod Didactic GmbH.1998么/ 图Rontgengerat-p6331 回☒ ·鱼陷凸回) Bragg Planck Transmission Moseley gR35 1:K-alpha K-beta 3、 2:K-alpha 2:K-beta 25 3 K-alpha 0 8/ ©by Leybod Didaclic GmbH,1998么 Fig.4 Diffraction spectrum of x-ray radiation for Bragg reflection to the third order at an NaCl monocrystal Top:linear representation of counting rate R Bottom:logarithmic representation of counting rate R Parameters of x-ray tube:U=35 kV and /1 mA
Fig. 4 Diffraction spectrum of x-ray radiation for Bragg reflection to the third order at an NaCl monocrystal Top: linear representation of counting rate R Bottom: logarithmic representation of counting rate R Parameters of x-ray tube: U = 35 kV and I = 1 mA P6.3.3.1 LEYBOLD Physics Leaflets 4
LEYBOLD Physics Leaflets P6.3.3.1 Evaluation Table 5:Mean value and literature value [2]for the charac- Access the evaluation functions of the software "X-ray teristic wavelengthλ Apparatus"by clicking the right-hand mouse button and select the command"Calculate Peak Center" A(Ka) X(Kg) Using the left mouse button,mark the "entire width"of the pm pm peaks;if desired,insert the calculated peak center B and Mean value 71.07 63.08 the peak width o in the diagram with Alt+T and note the center as the glancing angle in the measurement table (see Literature value 71.08 63.09 tables 3 and 4). Save your measurements and evaluations to a suitably named file with the buttonor by pressing F2. Using the glancing angle and the lattice plane spacing d =282.01 pm,calculate the wavelength A using Bragg's law of reflection(IV)(see tables 3 and 4). Results Find the mean values for the individual diffraction orders of The close agreement of the experimentally determined the measured wavelengths (see table 5). wavelengths for the characteristic lines with the literature values in table 5 verify the validity of Bragg's law.This simul- Table 3:Measured glancing angles of the Mo K line and the taneously confirms the wave nature of x-rays,as this property calculated wavelengths A for the first through third diffraction was assumed in the process of deducing this law. orders (K.) A(Ka) pm 1 7.24° 71.08 Additional information 2 14.60° 71.09 The characteristic K and Ke lines actually consist of multiple 3 22.20° 71.04 adjacent discrete lines,which can be observed separately at higher diffraction orders(see Physics Leaflet P 6.3.3.4).Table 1 shows the weighted mean values of the respective individual lines from this substructure. Table 4:Measured glancing angles of the Mo Ke line and the calculated wavelengths A for the first through third diffraction orders n a(Kp) X(Kg) pm Literature 6.42° [1]Handbook of Chemistry and Physics,52nd Edition 63.07 (1971-72),The Chemical Rubber Company,Cleveland, 3 12.94° 63.15 Ohio,USA. 19.58° 63.01 [2 C.M.Lederer and V.S.Shirley,Table of Isotopes,7th Edition,1978,John Wiley Sons,Inc.,New York,USA. 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 PHiniednthefeieeRsReaaeame☒
Evaluation – Access the evaluation functions of the software “X-ray Apparatus” by clicking the right-hand mouse button and select the command “Calculate Peak Center”. – Using the left mouse button, mark the “entire width” of the peaks; if desired, insert the calculated peak center b and the peak width s in the diagram with Alt+T and note the center as the glancing angle in the measurement table (see tables 3 and 4). – Save your measurements and evaluations to a suitably named file with the button or by pressing F2. – Using the glancing angle q and the lattice plane spacing d = 282.01 pm, calculate the wavelength l using Bragg’s law of reflection (IV) (see tables 3 and 4). – Find the mean values for the individual diffraction orders of the measured wavelengths (see table 5). Results The close agreement of the experimentally determined wavelengths for the characteristic lines with the literature values in table 5 verify the validity of Bragg’s law. This simultaneously confirms the wave nature of x-rays, as this property was assumed in the process of deducing this law. Additional information The characteristic Ka and Kb lines actually consist of multiple, adjacent discrete lines, which can be observed separately at higher diffraction orders (see Physics Leaflet P 6.3.3.4). Table 1 shows the weighted mean values of the respective individual lines from this substructure. Literature [1] Handbook of Chemistry and Physics, 52nd Edition (1971−72), The Chemical Rubber Company, Cleveland, Ohio, USA. [2] C. M. Lederer and V. S. Shirley, Table of Isotopes, 7th Edition, 1978, John Wiley & Sons, Inc., New York, USA. n q(Ka) l(Ka) pm 1 7.248 71.08 2 14.608 71.09 3 22.208 71.04 Table 3: Measured glancing angles of the Mo Ka line and the calculated wavelengths l for the first through third diffraction orders n q(Kb) l(Kb) pm 1 6.428 63.07 2 12.948 63.15 3 19.588 63.01 Table 4: Measured glancing angles of the Mo Kb line and the calculated wavelengths l for the first through third diffraction orders l(Ka) pm l(Kb) pm Mean value 71.07 63.08 Literature value 71.08 63.09 Table 5: Mean value and literature value [2] for the characteristic wavelength l 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 LEYBOLD Physics Leaflets P6.3.3.1