Fundamentals of Rietveld Refinement

Fundamentals of Rietveld RefinementI. XRD Pattern SimulationAn Introduction to Rietveld RefinementusingPANalytical XPert HighScore Plus v3.0d

Scott A Speakman, Ph.D.MIT Center for Materials Science and Engineeringspeakman@mit.edu

OutlineCalculating a Diffraction Pattern from a Crystal StructureUsing HighScore Plus to Simulate a Diffraction Pattern Manually Refining a Model to Match Experimental Data

The position and intensity of peaks in a diffraction pattern are determined by the crystal structureThese three phases of SiO2 are chemically identical.Differences in the atomic structure are reflected in the diffraction patterns.

Quartz and cristobalite have two different crystal structuresThe Si and O atoms are arranged differently, but both have structures with long-range atomic orderThe difference in their crystal structure is reflected in their different diffraction patternsThe amorphous glass does not have long-range atomic order and therefore produces only broad scattering peaks

3The diffraction peak position is a product of the atomic bond distances in the crystalBraggs Law: = 2dhklsin

Braggs law calculates the angle where constructive interference from X-rays scattered by parallel planes of atoms will produce a diffraction peak.In most diffractometers, the X-ray wavelength l is fixed.Consequently, a family of planes produces a diffraction peak only at a specific angle 2q.The lattice parameters of the unit cell are used to calculate dhkl dhkl is a function of the size and shape of the unit celldhkl is the vector drawn from the origin of the unit cell to intersect the crystallographic plane (hkl) at a 90 angle. dhkl, the vector magnitude, is the distance between parallel planes of atoms in the family (hkl)the diffraction peak position is a product of the average atomic distances in the crystalanything that changes the average bond distances (temperature, pressure, etc.) will change dhkl and therefore the diffraction peak positions

d110The amplitude of light scattered by a crystal is determined by the arrangement of atoms in the diffracting planesThe structure factor quantifies the amplitude of light scattered by a crystal The patterns of atoms in the unit cell scatters strongly in some directions and weakly in others owing to interference of the wavelets scattered by the atomsFhkl sums the result of scattering from all of the atoms in the unit cell to form a diffraction peak from the (hkl) planes of atomsThe amplitude of scattered light is determined by:where the atoms are on the atomic planes this is expressed by the fractional coordinates xj yj zjwhat atoms are on the atomic planes the scattering factor fj quantifies the relative efficiency of scattering at any angle by the group of electrons in each atomNj is the fraction of every equivalent position that is occupied by atom j

6The scattering factor, f, quantifies the efficiency of scattering from a group of electrons in an atomX-rays are electromagnetic radiation that interact with an electron, which reradiates the X-ray as a spherical waveThe number of electrons around an atom defines how strongly it will scatter the incident X-ray beamthe initial strength of the atomic scattering factor is equal to the number of electrons around the atomThe scattering factor, f, is a product of several terms describing the interaction of the X-ray with the electrons around an atom

The atomic scattering factor f0 can be found using tables or equations determined experimentally or from quantum mechanical approximations f0 at 0 is equal to the number of electrons around the atomY and Zr are similar, but slightly different, at 0Zr and Zr4+ are slightly different at 0 Y3+ and Zr4+ are identical at 0 the variation with (sin )/ depends on size of atomsmaller atoms drop off quickerat higher angles, the difference between Y3+ and Zr4+ is more readily discernedat higher angles, the difference between different oxidation states (eg Zr and Zr4+) is less prominent

Thermal motion of the atom changes the scattering factor the efficiency of scattering by an atom is reduced because the atom and its electrons are not stationarythe atom is vibrating about its equilibrium lattice sitethe amount of vibration is quantified by the Debye-Waller temperature factor: B=82U2U2 is the mean-square amplitude of the vibrationthis is for isotropic vibration: sometimes B is broken down into six Bij anistropic terms if the amplitude of vibration is not the same in all directions. aka temperature factor, displacement factor, thermal displacement parameter

Anomalous scattering of X-rays with energies near the absorption edge of the atom changes the scattering factorscattering factor has a real and imaginary termthese terms only matter when the energy of the incident X-ray approaches the ionization energy of atomwhen the wavelength of the incident X-ray is near the absorption edge of the atomanomalous scattering describes how some photons with energy approaching the ionization energy will be scattered out of phase with the majority of photons

With a crystal structure to solve Braggs Law and Fhkl, you can calculate the diffraction pattern from an ideal crystalIdeal powder diffraction patterns can be simulated if you know ...space group symmetryunit cell dimensionsatom typesrelative coordinates of atoms in unit cellatomic site occupanciesatomic thermal displacement parameters... for each and every phase in the sampleWhere to Get Crystal Structurespublications Commercial DatabasesInorganic Crystal Structure Database (ICSD)Linus Pauling File (LPF)this is included in ICDD PDF4+ Inorganic NIST Structural Database (metals, alloys, intermetallics)Cambridge Structure Database (CSD) (organic materials)Free Online DatabasesICSD- 4% available as demo at http://icsd.ill.eu/icsd/index.htmlCrystallography Open Database http://www.crystallography.net/Mincryst http://database.iem.ac.ru/mincryst/index.phpAmerican Mineralogist http://www.minsocam.org/MSA/Crystal_Database.htmlWebMineral http://www.webmineral.com/Protein Data Bank http://www.rcsb.org/pdb/home/home.doNucleic Acid Database http://ndbserver.rutgers.edu/Database of Zeolite Structures http://www.iza-structure.org/databases/A Crystal StructureZrO2Space Group Fm-3m (225)Lattice Parameter a=5.11

AtomWyckoff SitexyzBisosofZr4a0001.141O8c0.250.250.252.41site occupancy factor, sof, is equivalent to N in the Fhkl equationWyckoff Site notation is a shortcut to indicate if the type of site that the atoms occupies in the crystaleach letter in the Wyckoff notation specifies a site that sits on a different combination of symmetry elementsthe number in the Wyckoff notation indicates the number of atoms put into the unit cell if the atom is on that site4a- an atom on the a site will populate the unit cell with 4 atoms8c- an atom on the c site will populate the unit cell with 8 atomsWyckoff notation gives you a quick way to check if stoichiometry is correctly preserved when you create a crystal structureyou can deduce z, the number of molecules per unit cell, from the Wyckoff notation (z=4 for ZrO2)The Wyckoff site indicates if x, y or z are can change or if they must be fixed to preserve the symmetry of the crystalThe Site Occupancy Factor, sof, indicates what fraction of a site is occupied by a specific atomThe sof is N in the structure factor equationsof = 1, every equivalent position for xyz is occupied by that atomsof < 1, some of the sites are vacanttwo atoms occupying the same site will each have a fractional sofremember that the observed XRD pattern is an average of hundreds or thousands of unit cells that make a crystallite

For pattern calculations and Rietveld refinement in HSP, the Structures desktop is recommendedDesktops can be changed in the menu:View > Desktops

All slides in this tutorial were created with HSP in the Structures desktop

With this crystal structure information, we can build a crystal in HSPStart XPert HighScore PlusCreate a New Document File>New Analysis>Rietveld>Enter New Structure

Step 1: Enter TitleFollow the entry form from top to bottom, via the numbered steps

Step 2: Enter Space GroupEach space group has a unique number (1 thru 230)you can just enter number into the space group fieldHSP will assume the standard origin and settingYou can use the drop-down menu to select a space groupYou can open the symmetry explorer by clicking on the ... button

The symmetry explorer allows you to choose the space group and a specific origin, and to see information about the symmetry elements and special positions

Step 3: Enter Unit Celllattice parameters are automatically constrained by the space group that you chosea cubic space group: a=b=c, a=b=g=90

Step 4: Enter AtomsThe element can be an atom or ion (select oxidation state)the use of oxidation states in modifying the scattering factor for an atom is set in Customize>Program Settings, in the Rietveld tabby default it is set to Ignore Oxidation Statesname is the element by default, but it can be changed (useful if you have the same element on multiple sites)entering Wyckoff site can save you from having to type in xyzif you enter xyz, the Wyckoff site is automatically assignedWhen finished, click OK to create that crystal structure!We can now simulate the ideal X-Ray powder diffraction pattern in HSPWe have entered in all of the information necessary to simulate an X-ray diffraction patternClick on the Start Pattern Simulation button to produce the theoretical diffraction pattern

The pattern simulation parameters can be changed in Program SettingsTo change settings go to Customize> Program Settings In Simulation tab, you can change the parameters for the simulated scanDefault range for simulation is 5 to 70 2thetachange start and end angles to 25 and 90 2thetaclick OK to enter change and close windowclick Start Pattern Simulation again

The wavelength of radiation used in the simulation can be changed in Document Settings To change settings go to Customize> Document Settings In Document Wavelength tab, you can change the X-ray source usedthere is an Other option if you used a non-characteristic wavelength, such as at a synchrotronclick Start Pattern Simulation again

Parameters for the crystal structure and the simulation can be accessed in the Refinement Control listthe parameters for the pattern simulation are shown in the Refinement Control ListThe values for global parameters and each individual crystal structure entered into the simulation are listed in a nested tree formatYou can trim the number of tabs in the Lists Pane in the View menu

The Object Inspector shows additional details for any element you select in other windowsIf you click on the phase name (eg Zirconia) in Refinement Control list, you will be able to see additional information about that phase in the Object InspectorThis includes information such as calculated density, chemical formula, and formula massYou can use these to check that you created the crystal structure correctlyYou can change details such as the phase name and color of simulated diffraction pattern

We want to keep this pattern as a reference during the next exercises Go to Scan List in the Lists PaneRight Click on the current scan that is shown in the list (which is the simulated pattern)Select Convert calc. Profile to obs. ProfileNow the original simulation will remain in the Main Graphics View and we can compare it to new simulations

We can change parameters for the crystal structure and observe how they change the diffraction patternThe unit cell lattice parameters determine where the diffraction peaks are observedthe peak intensity is determined bywhat atoms are in the crystal and their positions xyzthe site occupancy of the atomthe relative scattering strength of the atom, scattering factor, which is a product of the number of electrons around the atomthe thermal parameter of the atom

This is not just useful for these tutorial purposes, but also for anticipating what your real data might look like and for evaluating limits of detectionTo change values for the simulationIn the Refinement Control, expand the Zirconia entryExpand the Unit Cell entryChange the Unit Cell valueAfter typing, you must press Enter key to register the changeif you try recalculating the simulation without pressing enter, you may lose the changed valuerecalculate the diffraction pattern by clicking on Start Pattern Simulation button againchange the lattice parameter and recalculate the pattern several times to get a feel for the relationship

To change atom parameters

click on Atomic Coordinates in the Refinement Control ListThe atomic information appears in the Object InspectorExpand the Object Inspector window so that you can see more clearlyYou know that you are working with the atoms because the Object Inspector pane says Selected Object: Atomic CoordinatesChange the Site Occupancies (sof) for the atomsChanging the occupancy of Zr has a big effectchange the SOF of Zr to 0.5 and recalculate the patternChanging the occupancy of O has a small effectreset the SOF of Zr to 1.0 and change the SOF of O to 0.5, then recalculate the patternRemember, the atomic scattering factor, f0, is equal to the number of electrons around the atomZr atoms scatter much more strongly than O, so the XRD diffraction pattern is more sensitive to the Zr atoms

We can predict the change in peak intensities when we dope the samplereplace 20% Zr with Yright click on the Object Inspector and select add new atomTo create the new atom, type Y for the Elementtab over to Wyckoff Site and type 4a and enterthis automatically places the atom at 0,0,0make the sof 0.2make the Biso 1.14make the sof for the Zr atom 0.8This means that the 4a site is occupied by 80% Zr and 20%Yrecalculate the simulationlittle change in intensityreplace 20% Zr with Cechange the atom type of Y to Ce and recalculate the simulationhow could we quantify amount of Y doping in YSZ?

The thermal parameter, Biso, changes how scattering efficiency changes with as a function of (sin q)/lchange thermal parameter for Zrreset the SOF of Zr to 1 and the SOF of the other 4a atom (Ce or Y) to 0the diffraction pattern will be representative of ZrO2 with no dopingchange Biso and recalculate the diffraction patternchange from 1.14 to 2.3change to 5change to 0.1

These simulations do not necessarily represent the data that your diffractometer will produce There are more factors that contribute to the diffraction pattern that you collect with an instrumentSome of these factors are calculated from fundamental equationsThe simulations that we have been executing already account for multiplicity, for Ka2 radiation, and for the Lorentz polarization factorSome of these factors are approximated using empirical formulasThe simulations that we have been executing already account for the peak width due to divergence, mosaicity, resolution of the diffractometerOther Things that effect intensityIK=SMKLK|FK|2PKAKEK

S is an arbitrary scale factorused to adjust the relative contribution of individual phases to the overall diffraction patternM is the multiplicity of the reflectionaccounts for the fact that some observed diffraction peaks are acutally the product of multiple equivalent planes diffracting at the same position 2theta (for example, (001) (100) (010) etc in cubic)automatically calculated based on the crystal structureL is the Lorentz polarization factorP is the modification of intensity due to preferred orientationA is absorption correctionE is extinction correctionF is structure factor, which is the amplitude of scattered light due to the crystal structure (already discussed)The Lorentz polarization factor corrects for several effectsThe Lorentz factorDifferent planes remain in the diffracting condition over a different angular rangehigh angle planes remain in the diffracting position longer, increasing their apparent intensity1/sin2qcosq relationship for Bragg-Brentanovaries with geometry of the diffractometer PolarizationScattering of light by the atom polarizes the X-raysthis causes the scattered intensity to vary as 1+cos2(2q)/2Monochromator PolarizationThe monochromator uses diffraction from a crystal to remove unwanted wavelengths of radiation. This diffraction will also polarize the X-ray beam, causing intensity to vary as cos2(2q) POL is the correction factor for the monochromator the correct POL value can be found in HSP Help go to the Help Index, type in POL, and the first option describes the Lorentz polarization factor

MonochromatorPOLK-beta Filter1Cu diffracted beam graphite (point detector)0.8Cu XCelerator1.25Cr diffracted beam (point detector)0.588Setting the Polarization Correction Coefficient, POL, for your simulationin the Refinement Control list, click on Global ParametersThe Global Parameters will show up in the Object InspectorIn the General Properties expansion of the Object Inspector, you can set the Polarization Correction Coefficient (POL)

Modeling Preferred OrientationThe default calculation assumes that there are an equal number of crystallites contributing to every diffraction peakthe orientation of the crystallites is random and there are a statistically relevant number of grainspreferred crystallographic orientation produces a non-random distribution in the orientation of the crystallitesthe observed diffraction peak intensities will vary systematically

The March distribution function empirically models the preferred orientation effectThe March distribution function is a model for fitting preferred orientation of cylinder or needle shaped crystals.

k is the angle between the preferred orientation vector and the normal to the planes generating the diffracted peakr is a refinable parameter in the Rietveld methodwe will explore using the correction in our refinement on Tuesday

Modeling Preferred Orientation in HSPIn the Refinement Control list, click on the phase that has preferred orientationIn the Object Inspector, expand the Preferred Orientation settingsset the h,k, and l directions to represent the [hkl] of the preferred orientation

set the Preferred Orientation parameterthis parameter is found in the Object Inspector as Parameter (March/Dollase)this parameter is also found in the Refinement Control List as Preferred Orientation1 means the orientation of [hkl] is completely random1 means that the [hkl] direction is preferentially avoidedrecalculate the simulation to observe the change in peak intensitiesAdditional Intensity FactorsAbsorption Correctionaccounts for the transmission and absorption of the X-rays through the irradiated volume of the sample.depends on sample geometry: flat plate or cylindricaldepends on the mass absorption coefficient of the samplesignificant for cylindrical or transmission samples with a high sample absorptionsignificant for a flat plate sample with a low sample absorptionExtinction CorrectionSecondary diffraction of scattered X-rays in a large perfect crystal can decrease the observed intensity of the most intense peaksscattering by uppermost grains limits the penetration depth of a majority of X-rays, causing a smaller irradiated volumewant ideally imperfect crystals grains with enough mosaicity to limit extinctionbetter to reduce extinction by grinding the powder to a finer sizeThese two corrections are rarely used. Caution is advised. Always check to make sure that the correction values are physically meaningful. We now know how to calculate the diffraction peak intensity, but there a couple of more factors involved in simulating real diffraction dataThe intensity, Yic, of each individual data point i is calculated using the equation:

We already know how to calculate IK, the intensity of the Bragg diffraction peak k: IK=SMKLK|FK|2PKAKEKYib is the intensity of the background at point i in the patternk1 - k2 are the reflections contributing to data point i in the patternsometimes multiple Bragg diffraction peaks overlap, resulting in multiple contributions to the observed intensity at a single data pointGik is the peak profile functionthis describes how the intensity of the diffraction peak is distributed over a range of 2theta rather than at a single pointthis profile is due to instrument broadening, sample broadening, etc

44Sample attributes also contribute to the diffraction peak shape and widthmass absorption coefficient (sample transparency)sample thicknesscrystallite sizemicrostraindefect concentrationDiffraction peaks have profiles that must be empirically modeled

The intensity equation Ik predicts that diffraction peak intensity occurs at a single point 2theta but in reality the diffraction peak intensity is spread out over a range of 2thetaThe total area of the diffraction peak profile is the predicted intensity Ik Diffractometers contribute a characteristic broadening and shape to the diffraction peak based on their optics and geometryeach separate combination of divergence slit apertures, Soller slits, beta filter or monochromators, and detectors will have their own characteristic instrument profile4445Profile ParametersRietveld refinement assumes that peak profiles vary systematically as a function of 2thetaPeak profile may also vary according to crystallographic direction [hkl]

Peak WidthPeak Shape45The Cagliotti equation describes how peak width varies with 2thetaHk is the Cagliotti function where U, V and W are refinable parametersHSP treats the Cagliotti equation as a convolution between instrument and sample broadening functionsThe change in U and W between an instrument profile standard (eg NIST 640c) and your sample can indicate the amount of nanocrystallite size and microstrain broadening

47

The systematic variation of peak width with 2theta47For a determination of the resolution, we again used the NIST Standard Reference Material SRM1976 (Aluminium Oxide; corundum) and measured this overnight from 20 to 120 degrees 2Theta. We determined the Full Widths at Half Maximum (FWHMs) of the reflections using XPert Graphics and Identify. We plotted all reflections with an intensity exceeding 0.5% of the largest reflection as a function of 2Theta and fitted the data points with the Cagliotti function, a well-known description of the instrumental aberrations of a powder diffractometer. It is clear from the graph that the resolution in the diffractogram is excellent: a FWHM at the lowest reflections below 0.04 degrees 2Theta is easily obtained. We tried to obtain the same results with the traditional detection geometry: a receiving slit and a proportional counter but did not carry out the actual experiment because it would cost us more than one month of continuously scanning. It is clear from this result that the resolution of the XCelerator equals the traditional systems, or even improves it because it becomes possible to obtain higher-quality data in the same time as the traditional system. 48Diffraction Peaks from laboratory diffractometers are a mixture of Gaussian and Lorentzian profilesPosition [2Theta]45.5046Counts01000200030004000Gaussian profile shapePosition [2Theta]4244464850Counts010002000Lorentzian profile shape48Several different functions can be used to describe the mixing of Gaussian and Lorentzian contributionsThe profile function is set in the Object Inspector for Global Parameters (found in Refinement Control list)the same profile function is used for all phases in a single patternPseudo Voigt and Pearson VII are the most commonly usedPseudo VoigtPseudo Voigt with FJC Asymmetry correctionPearson VIIVoigt

The Pseudo Voigt equation uses up to 3 parameters to describe Gaussian and Lorentzian componentsThe parameters g1, g2, and g3 are the refinable shape parameters, listed as Peak Shape 1, Peak Shape 2, and Peak Shape 3 in the Refinement Control list Usually Peak Shape 1 is the only one usedHk is the Cagliotti peak width functionC1 is 4 ln 2

Xkj is

51Background can be empirically fit with an equation, manually fit, or evaluated with blankrefinement of a polynomial functionrefinement using a type I or II Chebyshev polynomial function By amorphous sinc functionlinear interpolation between base pointsindependent data collection run without the sample in place

51Systematic error in diffraction peak positions can also be calculatedThe systematic error functions are found in Global ParametersThey apply to all phases in the samplezero shiftspecimen displacement

53The Parameters of a Pattern SimulationGlobal parameters affect the entire sample rather than individual phasesBackground functionZero ShiftSpecimen DisplacementPOL (lorentz polarization factor)Sample Geometryselecting the Profile FunctionOther parameters are set individually for each phase scale factorpreferred orientation (March-Dollase function)unit cellatomic parameters are for each atom in the phaseposition, SOF, thermal parameterPeak Profile parametersCagliotti peak width function parameters U,V, and WProfile shape parameterspeak asymmetry profileanisotropic broadening function53What is Physically Meaningful?Wel (1975)54

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YY(3+)ZrZr(4+)O(2-)(sin q)/lfo

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Sheet2

Sheet3

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YY(3+)ZrZr(4+)O(2-)(sin q)/lfo

Chart239.9683239.9683239.9683239.921017219539.917025317439.881116156239.780865587539.764956423339.622059948139.552990751639.517409074139.198610936139.245471093139.182728546738.622540291138.868698856338.771648472737.909027262238.434592066938.29647629337.075556171337.955744217537.770215986236.140743407637.44459990637.205719698735.123221917336.912734130536.614948643434.040690549136.370293336.008402835132.909202282835.825630979435.394752913931.742725671635.285145184734.780679960830.552974903934.753300594934.170904919729.349471774434.232801281933.568371063428.13978148133.724869080732.974532467826.929854936433.229579733432.389698835925.724412276932.746212448331.813391101224.527312804532.273576725431.244671117423.341872189131.810291084230.682420131822.171104820831.354999631230.125552510321.017884796230.906522589129.573161639319.885031219430.463944895129.024603003718.775331421230.026652301328.479524694417.69151944429.594327115227.937858135216.636227408529.16691622127.399782115.61192524228.744582920626.865671771914.620860778128.327652074326.336042353613.665008380927.916555582425.811494162512.746030666927.511782853125.292663673211.865254986427.113838842924.78018287111.023664232626.723210654524.274647687310.221900237126.340342562623.77659519469.460277357825.9656186623.2864885948.738803698525.599351977722.8047087288.057207542925.241778843822.33155079447.414966905224.893057309121.86722502566.811340486324.553268612721.41186026356.245398711324.222420837220.965509555.716053873923.900454075220.52815703675.222088709223.587246582720.09972567424.762182954123.282621520819.68008527724.334937639622.986353986419.26906065843.938897000222.698178107118.86643961013.572567988322.417794032218.47198056553.234437456522.14487469418.08541982142.922987117821.879072245117.70647823672.636706432121.620024100217.33486734942.374103581621.367358531716.97029487442.133714708521.120699784416.61246956561.914111587520.879672685816.2611054381.713907903720.643906742515.91592535881.531764291720.413039721115.57666402831.366392286220.186720722415.24307037671.216557316819.96461276214.91490941431.081080868119.746394880914.59196357370.958841913419.531763809814.27403358820.848777715319.320435219213.96093895420.749884079919.112144587213.65251802420.661215133918.906647719613.34862777820.581882693818.703720959213.04914332270.511055280218.503161120612.75395715910.447956830818.304785185112.46297826890.391865155718.10842979312.17613105280.342110178117.913950564311.89335416320.298071994417.721221281811.61459926240.259178792717.530132964511.33982973770.224904656417.340592859211.06901939990.194767285917.152523375110.80215118790.168325663716.965860982910.53921589920.14517768816.780555097710.28021096190.124957797516.596566964210.02513926220.107334609216.41386855539.77400803660.092008586116.23244150019.52682783720.078709753516.05227604789.28361157430.067195476315.87337007689.04437364030.057248311415.69572815458.80912911560.048673944315.51936065248.57789305650.041299219315.34428291878.35067986380.034970268715.17051450988.12750272970.02955074614.99807848167.90837315860.024920166414.82700073977.69330055940.020972355614.65730944687.48229190440.017614006814.48903448657.27535144890.014763344714.32220697927.07248050910.012348894514.15685884976.8736772910.010308352713.99302244116.67893676730.008587555213.83073017376.48825059630.007139539113.67001424496.30160707910.005923692413.51090636636.11899115060.004904987413.35343753665.94038439990.004053290913.19763784495.76576511650.003342747713.04353630385.59510835890.002751229712.89116070735.42838604240.002259847112.74053751245.26556704260.001852514912.59169174225.10661731240.001515571312.44464690614.95150001030.001237441412.29942493774.80017563680.001008342412.15604614664.6526021780.00082002712.01452918384.50873525410.00066555911.87489101724.36852827140.000539119211.73714691844.23193257640.000435837411.60131045754.09889761110.000351647311.46739350533.96937106720.000283161711.33540624293.84329904090.000227566311.2053571763.72062618430.000182527811.07725315543.6012958550.000146116510.95109940073.4852502630.000116740110.82689952873.37243061320.000093087910.70465558423.26277724440.000074083210.58436807383.15622976410.000058843910.46603600283.05272717830.000046648710.34965691272.95220801670.000036909310.23522692182.8546104530.000029146810.12274076622.75987241970.000022972410.01219184292.66793171780.00001807119.90357225282.57872612160.00001418819.79687284552.49219347750.00001111819.69208326432.40827179790.00000869569.58919199142.32689934940.00000678799.48818639412.24801473630.00000528869.38905277082.17155697820.00000411269.29177639672.09746558280.0000031929.19634157072.02568061370.00000247279.1027316611.95614275290.00000191199.01092915091.88879335880.00000147548.92091568521.82357451950.00000113648.83267211441.76042910070.00000087378.74617854091.69930079060.00000067048.66141436231.64013413890.00000051348.57835831641.58287459320.00000039258.4969885241.52746852980.00000029958.41728253171.47386328210.0000002288.33921735431.42200716430.00000017338.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B=0B=1B=10(sin q)/lf

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