Solid-state physics LEYBOLD Physics Properties of crystals Leaflets P7.1.2.1 X-ray structural analysis Bragg reflection: determining the lattice constants of monocrystals Objects of the experiment Investigating and comparing Bragg reflection at an LiF and an NaCI monocrystal. Determining the lattice constant ao of NaCl and LiF. Principles Bragg's law of reflection describes the diffraction of plane set of lattice planes and is often referred to as the glancing waves at a monocrystal as the selective reflection of the waves angle. at a set of lattice planes within the crystal.Due to the periodicity In a cubic crystal with NaCl structure(cf.Fig.1),the lattice of the crystal,the lattice planes of a set have a fixed spacing planes run parallel to the surfaces of the crystal's unit cells in d.An incident wave with the wavelength A is reflected with maximum intensity when the Bragg condition the simplest case.Their spacing dcorresponds to one half the lattice constant: n.λ=2.d.sin8 n:diffraction order d号 (① λ：wavelength d:spacing of lattice planes This lets us use(I)as an equation for determining the lattice constant ao: is fulfilled (see experiment P6.3.3.1).The angle shows the direction of the incident and reflected wave with respect to the n·入=ao·sin8 (00 In other words,to determine ao we need to measure the glancing angle for a known wavelength A and diffraction Fig.1 Three-dimensional representation of the structure of NaCl order n.This method is more precise when the glancing angles d:Spacing of lattice planes in [1,0,0]-direction are also measured in higher diffraction orders. ao:lattice constant In this experiment,the molybdenum x-rays are used as radia- tion of a known wavelength.Table 1 shows its wavelengths A. Table 1:Wavelengths of the characteristic x-ray radiation of molybdenum(weighted means [1)) Line pm Ka 71.08 KB 63.09 A Geiger-Muller counter tube is used to detect the x-rays;this instrument and the crystal are both pivoted with respect to the e incident x-ray beam in 2 coupling-the counter tube is turned by twice the angle of the crystal(cf.Fig.2).The zero point 0 is characterized by the fact that the lattice planes and the
Fig. 1 Three-dimensional representation of the structure of NaCl d: Spacing of lattice planes in [1,0,0]-direction a0: lattice constant Solid-state physics Properties of crystals X-ray structural analysis LEYBOLD Physics Leaflets Bragg reflection: determining the lattice constants of monocrystals Objects of the experiment Investigating and comparing Bragg reflection at an LiF and an NaCl monocrystal. Determining the lattice constant a0 of NaCl and LiF. Principles Bragg’s law of reflection describes the diffraction of plane waves at a monocrystal as the selective reflection of the waves at a set of lattice planes within the crystal. Due to the periodicity of the crystal, the lattice planes of a set have a fixed spacing d. An incident wave with the wavelength l is reflected with maximum intensity when the Bragg condition n ⋅ l = 2 ⋅ d ⋅ sinq (I) n: diffraction order l: wavelength d: spacing of lattice planes is fulfilled (see experiment P6.3.3.1). The angle q shows the direction of the incident and reflected wave with respect to the set of lattice planes and is often referred to as the glancing angle. In a cubic crystal with NaCl structure (cf. Fig. 1), the lattice planes run parallel to the surfaces of the crystal’s unit cells in the simplest case. Their spacing d corresponds to one half the lattice constant: d = a0 2 (II) This lets us use (I) as an equation for determining the lattice constant a0: n ⋅ l = a0 ⋅ sinq (III) In other words, to determine a0 we need to measure the glancing angle q for a known wavelength l and diffraction order n. This method is more precise when the glancing angles are also measured in higher diffraction orders. In this experiment, the molybdenum x-rays are used as radiation of a known wavelength. Table 1 shows its wavelengths l. A Geiger-Müller counter tube is used to detect the x-rays; this instrument and the crystal are both pivoted with respect to the incident x-ray beam in 2q coupling – the counter tube is turned by twice the angle of the crystal (cf. Fig. 2). The zero point q = 08 is characterized by the fact that the lattice planes and the Table 1: Wavelengths of the characteristic x-ray radiation of molybdenum (weighted means [1]) Line l pm Ka 71.08 Kb 63.09 P7.1.2.1 1108-Ste 1
P7.1.2.1 LEYBOLD Physics Leaflets Apparatus 1X-ray apparatus.············· 554811 1 End-window counter fora&,B,y and x-ray radiation···.·.· 55901 1 LiF monocrystal for Bragg reflection... 55477 additionally required: 1 PC with Windows 9x/NT Fig.2 Schematic diagram of diffraction of x-rays at a mono- crystal and 2 coupling between counter-tube angle and scattering angle(glancing angle) 1 collimator,2 monocrystal,3 counter tube axis of the counter tube are parallel to the incident x-ray beam. As the lattice planes are seldom precisely parallel to the sur- face of the crystal,the zero point of each crystal must be calibrated individually. Setup Setup in Bragg configuration: Fig.3 shows some important details of the experiment setup. To set up the experiment,proceed as follows (see also the Instruction Sheet for the x-ray apparatus): Safety notes The x-ray apparatus fulfills all regulations governing an Mount the collimator in the collimator mount (a)(note the guide groove). x-ray apparatus and fully protected device for instructional use and is type approved for school use in Germany(NW Attach the goniometer to guide rods(d)so that the distance 807/97Ro). s1 between the slit diaphragm of the collimator and the target arm is approx.5 cm.Connect ribbon cable (c)for The built-in protection and screening measures reduce the controlling the goniometer. local dose rate outside of the x-ray apparatus to less than Remove the protective cap of the end-window counter, 1 uSv/h,a value which is on the order of magnitude of the place the end-window counter in sensor seat (e)and natural background radiation. connect the counter tube cable to the socket marked Before putting the x-ray apparatus into operation in- GM TUBE. spect 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 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 Fig.3 Experiment setup in Bragg configuration 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. d 2
axis of the counter tube are parallel to the incident x-ray beam. As the lattice planes are seldom precisely parallel to the surface of the crystal, the zero point of each crystal must be calibrated individually. Setup Setup in Bragg configuration: Fig. 3 shows some important details of the experiment setup. To set up the experiment, proceed as follows (see also the Instruction Sheet for the x-ray apparatus): – Mount the collimator in the collimator mount (a) (note the guide groove). – Attach the goniometer to guide rods (d) so that the distance s1 between the slit diaphragm of the collimator and the target arm is approx. 5 cm. Connect ribbon cable (c) for controlling the goniometer. – Remove the protective cap of the end-window counter, place the end-window counter in sensor seat (e) and connect the counter tube cable to the socket marked GM TUBE. Apparatus 1 X-ray apparatus . . . . . . . . . . . . . . 554 811 1 End-window counter for a, b, g and x-ray radiation . . . . . . . 559 01 1 LiF monocrystal for Bragg reflection . . . 554 77 additionally required: 1 PC with Windows 9x/NT 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 Schematic diagram 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. 3 Experiment setup in Bragg configuration P7.1.2.1 LEYBOLD Physics Leaflets 2
LEYBOLD Physics Leaflets P7.1.2.1 By moving the sensor holder(b),set the distance s2 be- In coupled scanning mode,move the target back by 10.2 tween the target arm and the slit diaphragm of the sensor (even if this takes you into the negative range!). seat to approx.6 cm. Save the positions of the target and the sensor as the "zero Mount the target holder(f)with target stage. position of the measuring system"by pressing TARGET, Manually align the target and sensor arm horizontally using COUPLED and B LIMITS simultaneously. the ADJUST knob and save these positions as the"zero position of the measuring system"by pressing TARGET, Recording the diffraction spectrum: COUPLED and B Limits at the same time(see Instruction Sheet for x-ray apparatus). Start the software "X-ray Apparatus",check to make sure that the apparatus is connected correctly,and clear any existing measurement data using the button or the F4 Preparing the PC-based measurement: key Connect the RS-232 output and the serial interface on your Set the measuring time per angular step At =10s and the PC (usually COM1 or COM2)using the 9-pin V.24 cable angular step width△β=0.1°. (supplied with x-ray apparatus). Press the COUPLED key to activate 2 coupling of target If necessary,install the software"X-ray Apparatus"under and sensor and set the lower limit of the target angle to 4 Windows 9x/NT(see Instruction Sheet for x-ray apparatus) and the upper limit to 34. and select the desired language. Start measurement and data transfer to the PC by pressing the SCAN key. When you have finished measuring,save the measure- ment series under an appropriate name by pressing the button里or the F2key. a)Bragg reflection at an NaCl monocrystal: Press the ZERO key to return the target and sensor to the Carrying out the experiment current zero position. Notes: Remove the LiF crystal and carefully mount the NaCI crystal in its place. NaCl and LiF crystals are hygroscopic and extremely fragile. Store the crystals in a dry place;avoid mechanical stresses on Determining the zero position of the measuring system the crystals;handle the crystals by the short faces only. In coupled scanning mode,set the target to about 7.2 If the counting rate is too low,you can reduce the distance sa using the ADJUST knob. between the target and the sensor somewhat.However,the Switch on the tube high voltage with HV on/off. distance should not be too small,as otherwise the angular Leave the target position unchanged and,in sensor scan- resolution of the goniometer is no longer sufficient to separate ning mode,manually find the counting rate maximum for the characteristic K and Ke lines. the first reflection maximum of the Ka line. Leave the sensor unchanged in the maximum counting rate position and manually find the maximum of the count- a)Bragg reflection at an LiF monocrystal: ing rate in target mode. Switch between sensor and target modes and check Loosen knurled screw(g),place the LiF crystal flat on the whether you have found the counting rate maximum. target stage,carefully raise the target stage with crystal all In coupled scanning mode,move the target back by 7.2 the way to the stop and gently tighten the knurled screw (even if this takes you into the negative range!). (prevent skewing of the crystal by applying a slight pres- sure). Save the positions of the target and the sensor as the"zero Set the tube high voltage U=35.0 kV and the emission position of the measuring system"by pressing TARGET, COUPLED and B LIMITS simultaneously. current I 1.00 mA. Recording the diffraction spectrum: Determining the zero position of the measuring system In coupled scanning mode,set the target to about 10.2 Start the software "X-ray Apparatus"or clear any existing using the ADJUST knob. measurement data using the button or theF4 key. Switch on the tube high voltage with HV on/off. Press the COUPLED key to activate 20 coupling of target Leave the target position unchanged and,in sensor scan- and sensor and set the lower limit of the target angle to 4 ning mode,manually find the counting rate maximum for and the upper limit to 24. the first reflection maximum of the Ka line. Start measurement and data transfer to the PC by pressing Leave the sensor unchanged in the maximum counting- the SCAN key. rate position and manually find the maximum of the count- ing rate in target mode. When you have finished measuring,save the measure- Switch between sensor and target modes and check ment series under an appropriate name by pressing the whether you have found the counting rate maximum. button or the F2 key. 3
– By moving the sensor holder (b), set the distance s2 between the target arm and the slit diaphragm of the sensor seat to approx. 6 cm. – Mount the target holder (f) with target stage. – Manually align the target and sensor arm horizontally using the ADJUST knob and save these positions as the “zero position of the measuring system” by pressing TARGET, COUPLED and b Limits at the same time (see Instruction Sheet for x-ray apparatus). Preparing the PC-based measurement: – Connect the RS−232 output and the serial interface on your PC (usually COM1 or COM2) using the 9-pin V.24 cable (supplied with x-ray apparatus). – If necessary, install the software “X-ray Apparatus” under Windows 9x/NT (see Instruction Sheet for x-ray apparatus) and select the desired language. Carrying out the experiment Notes: NaCl and LiF crystals are hygroscopic and extremely fragile. Store the crystals in a dry place; avoid mechanical stresses on the crystals; handle the crystals 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, the distance should not be too small, as otherwise the angular resolution of the goniometer is no longer sufficient to separate the characteristic Ka and Kb lines. a) Bragg reflection at an LiF monocrystal: – Loosen knurled screw (g), place the LiF crystal flat on the target stage, carefully raise the target stage with crystal all the way to the stop and gently tighten the knurled screw (prevent skewing of the crystal by applying a slight pressure). – Set the tube high voltage U = 35.0 kV and the emission current I = 1.00 mA. Determining the zero position of the measuring system – In coupled scanning mode, set the target to about 10.28 using the ADJUST knob. – Switch on the tube high voltage with HV on/off. – Leave the target position unchanged and, in sensor scanning mode, manually find the counting rate maximum for the first reflection maximum of the Ka line. – Leave the sensor unchanged in the maximum countingrate position and manually find the maximum of the counting rate in target mode. – Switch between sensor and target modes and check whether you have found the counting rate maximum. – In coupled scanning mode, move the target back by 10.28 (even if this takes you into the negative range!). – Save the positions of the target and the sensor as the “zero position of the measuring system” by pressing TARGET, COUPLED and b LIMITS simultaneously. Recording the diffraction spectrum: – Start the software “X-ray Apparatus”, check to make sure that the apparatus is connected correctly, and clear any existing measurement data using the button or the F4 key. – Set the measuring time per angular step Dt = 10 s and the angular step width Db = 0.18. – Press the COUPLED key to activate 2q coupling of target and sensor and set the lower limit of the target angle to 48 and the upper limit to 348. – Start measurement and data transfer to the PC by pressing the SCAN key. – When you have finished measuring, save the measurement series under an appropriate name by pressing the button or the F2 key. a) Bragg reflection at an NaCl monocrystal: – Press the ZERO key to return the target and sensor to the current zero position. – Remove the LiF crystal and carefully mount the NaCl crystal in its place. Determining the zero position of the measuring system – In coupled scanning mode, set the target to about 7.28 using the ADJUST knob. – Switch on the tube high voltage with HV on/off. – Leave the target position unchanged and, in sensor scanning mode, manually find the counting rate maximum for the first reflection maximum of the Ka line. – Leave the sensor unchanged in the maximum countingrate position and manually find the maximum of the counting rate in target mode. – Switch between sensor and target modes and check whether you have found the counting rate maximum. – In coupled scanning mode, move the target back by 7.28 (even if this takes you into the negative range!). – Save the positions of the target and the sensor as the “zero position of the measuring system” by pressing TARGET, COUPLED and b LIMITS simultaneously. Recording the diffraction spectrum: – Start the software “X-ray Apparatus” or clear any existing measurement data using the button or the F4 key. – Press the COUPLED key to activate 2q coupling of target and sensor and set the lower limit of the target angle to 48 and the upper limit to 248. – Start measurement and data transfer to the PC by pressing the SCAN key. – When you have finished measuring, save the measurement series under an appropriate name by pressing the button or the F2 key. LEYBOLD Physics Leaflets P7.1.2.1 3
P7.1.2.1 LEYBOLD Physics Leaflets Measuring example a)Bragg reflection at an LiF monocrystal: X-Ray Apparatus 回☒ D色回 Bragg Planck Transmission Moseley log R wl-in Fig.4 Diffraction spectrum of x-rays in Bragg reflection to the third diffrac- n=3 tion order at an LiF monocrystal with logarithmic display of counting 20 30 rate R. 8/ Parameters of x-ray tube: by Leybold Didactic GmbH.1998 U=35 kV,I=1 mA b)Bragg reflection at an NaCl monocrystal: X-Ray Apparatus 问☒ 凸西路色都回 Bragg Planck Transmission Moseley og R Fig.5 Diffraction spectrum of x-rays in Bragg reflection to the third diffrac- tion order at an NaCl monocrystal with logarithmic display of counting 20 0 rate R. 8/ Parameters of x-ray tube: by Leybold Didactic GmbH,1998 U=35 kV,/=1 mA Evaluation In each diagram,click the right mouse button to access the For each glancing angle calculate the values sin and evaluation functions of the software "X-ray Apparatus"and and plot these value pairs in a diagram (see Fig.6). select the command "Calculate Peak Center"to evaluate In each case,the results lie along a straight line through the the diffraction spectra. origin:in accordance with (Ill),its slope corresponds to the Using the left mouse button,mark the "full width"of each lattice constant ao. peak and write down the center values in a table as the glancing angle (see tables 2 and 3)
Measuring example a) Bragg reflection at an LiF monocrystal: b) Bragg reflection at an NaCl monocrystal: Evaluation – In each diagram, click the right mouse button to access the evaluation functions of the software “X-ray Apparatus” and select the command “Calculate Peak Center” to evaluate the diffraction spectra. – Using the left mouse button, mark the “full width” of each peak and write down the center values in a table as the glancing angle (see tables 2 and 3). – For each glancing angle q, calculate the values sin q and and plot these value pairs in a diagram (see Fig. 6). In each case, the results lie along a straight line through the origin; in accordance with (III), its slope corresponds to the lattice constant a0. Fig. 4 Diffraction spectrum of x-rays in Bragg reflection to the third diffraction order at an LiF monocrystal with logarithmic display of counting rate R. Parameters of x-ray tube: U = 35 kV, I = 1 mA Fig. 5 Diffraction spectrum of x-rays in Bragg reflection to the third diffraction order at an NaCl monocrystal with logarithmic display of counting rate R. Parameters of x-ray tube: U = 35 kV, I = 1 mA P7.1.2.1 LEYBOLD Physics Leaflets 4
LEYBOLD Physics Leaflets P7.1.2.1 Table 3:Glancing angle of the NaCl crystal sin Line n .x pm 200 6.41° 0.112 KB 1 63.06 pm 7.23 0.126 Ka 1 71.08 12.91 0.223 K8 2 126.12 100 14.57 0.252 Ka 2 142.16 19.55 0.335 Ke 3 189.18 22.15 0.377 Ka 3 213.24 0 0 0,1 0.2 0.3 0.40.5 sin Results Fig.6 Value pairs as a function of sin LiF:squares,slope of line =404.5 pm a)LiF crystal: NaCl:circles,slope of line =565.2 pm Measurement result: ao=404.5pm Literature value [2]: a0=402.7pm lon radii [3]: 68pm(i).133pm(F) Sum of ion radii: 201pm b)NaCl crystal: Measurement result: a0=565.2pm Table 2:Glancing angle of the LiF crystal Literature value: a0=564.02pm lon radii [3]: 98 pm (Na),181 pm(Cl-) sin Line n.x Sum of ion radii: 279pm pm Conclusion:the LiF lattice shows a significantly smaller lattice 8.95 0.156 4 constant than the NaCl lattice,as the radii of the ions involved 63.06 are smaller. 10.10 0.175 Ka 71.08 18.17 0.312 2 126.12 Literature 20.54 0.351 2 142.16 [1]C.M.Lederer and V.S.Shirley,Table of Isotopes,7th Edition,1978,John Wiley Sons,Inc.,New York,USA. 27.91 0.468 KB 189.18 [2]Handbook of Chemistry and Physics,52nd Edition(1971- 72),The Chemical Rubber Company,Cleveland,Ohio,USA. 31.82 0.527 Ka 3 213.24 [3]Charles Kittel,Introduction to Solid State Physics,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 o by Leybold Didactic GmbH Printed in the Federal Republic of Germany lechnical alterations reserved
Results a) LiF crystal: Measurement result: a0 = 404.5 pm Literature value [2]: a0 = 402.7 pm Ion radii [3]: 68 pm (Li+), 133 pm (F–) Sum of ion radii: 201 pm b) NaCl crystal: Measurement result: a0 = 565.2 pm Literature value: a0 = 564.02 pm Ion radii [3]: 98 pm (Na+), 181 pm (Cl–) Sum of ion radii: 279 pm Conclusion: the LiF lattice shows a significantly smaller lattice constant than the NaCl lattice, as the radii of the ions involved are smaller. Literature [1] C. M. Lederer and V. S. Shirley, Table of Isotopes, 7th Edition, 1978, John Wiley & Sons, Inc., New York, USA. [2] Handbook of Chemistry and Physics, 52nd Edition (1971− 72), The Chemical Rubber Company, Cleveland, Ohio, USA. [3] Charles Kittel, Introduction to Solid State Physics, John Wiley & Sons, Inc. New York, USA Table 2: Glancing angle q of the LiF crystal q sin q Line n n ⋅ l pm 8.958 0.156 Kb 1 63.06 10.108 0.175 Ka 1 71.08 18.178 0.312 Kb 2 126.12 20.548 0.351 Ka 2 142.16 27.918 0.468 Kb 3 189.18 31.828 0.527 Ka 3 213.24 Table 3: Glancing angle q of the NaCl crystal q sin q Line n n ⋅ l pm 6.418 0.112 Kb 1 63.06 7.238 0.126 Ka 1 71.08 12.918 0.223 Kb 2 126.12 14.578 0.252 Ka 2 142.16 19.558 0.335 Kb 3 189.18 22.158 0.377 Ka 3 213.24 0 0,1 0,2 0,3 0,4 0,5 sin ϑ 0 100 200 nλ pm Fig. 6 Value pairs as a function of sin q LiF: squares, slope of line = 404.5 pm NaCl: circles, slope of line = 565.2 pm 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 P7.1.2.1