J. Appl. Cryst. (1969). 2 , 65-71. A Profile Refinement Method for Nuclear and Magnetic Structures H.M. Rietveld Reactor Centrum Nederland, Petten (N-H.), The Netherlands Received 29 November 1968 Abstract A structure refinement method is described which does not use integrated neutron powder intensities, single or overlapping, but employs directly the profile intensities obtained from step-scanning measurements of the powder diagram. Nuclear as well as magnetic structures can be refined, the latter only when their magnetic unit cell is equal to, or a multiple of, the nuclear cell. The least-squares refinement procedure allows, with a simple code, the introduction of linear or quadratic constraints between the parameters. Introduction The powder method has gained a new importance in neutron diffraction owing to the general lack of large specimens for single-crystal methods. Even in those cases where it proves to be possible to grow large single crystals, these may still suffer from such effects as extinction and magnetic domain structures, making a proper interpretation of the diffracted intensities unreliable. Many of these systematic effects are either nonexistent in the powder method, or become isotropic and can therefore be more easily determined. In a polycrystalline sample it is inevitable that certain information is lost as a result of the random orientation of the crystallites. A further, and in practice more serious, loss of information is a result of the overlap of independent diffraction peaks in the powder diagram. The method of using the total integrated intensities of the separate groups of overlapping peaks in the least-squares refinement of structures (Rietveld, 1966), leads to the loss of all the information contained in the often detailed profile of these composite peaks. By the use of these profile intensities instead of the integrated quantities in the refinement procedure, however, this difficulty is overcome and it allows the extraction of the maximum amount of information contained in the powder diagram (Rietveld, 1967a). Because of the importance of magnetic structures in neutron diffraction, the refinement method has been made applicable both to nuclear and to magnetic structures. The latter only comprise those structures which can be described on the nuclear unit cell or a multiple thereof. Experimental The neutron powder spectrometer in Petten operates with a wavelength of , pyrolytic graphite being used to suppress the second order contribution (Loopstra, 1966; Bergsma & van Dijk, 1967). Soller slits with angular divergences ranging from 10' to 30' can be installed between the reactor and the monochromator, and in A Profile Refinement Method for Nuclear and Magnetic Structures http://epswww.unm.edu/xrd/xrdclass/rietveld1969.ht m 1 of 12 9/24/2009 8:05 PM
Transcript
J. Appl. Cryst. (1969). 2, 65-71.
A Profile Refinement Method for Nuclear andMagnetic Structures
H.M. RietveldReactor Centrum Nederland, Petten (N-H.), The Netherlands
Received 29 November 1968
Abstract
A structure refinement method is described which does not use integrated neutron powderintensities, single or overlapping, but employs directly the profile intensities obtained fromstep-scanning measurements of the powder diagram. Nuclear as well as magnetic structures can berefined, the latter only when their magnetic unit cell is equal to, or a multiple of, the nuclear cell.The least-squares refinement procedure allows, with a simple code, the introduction of linear orquadratic constraints between the parameters.
IntroductionThe powder method has gained a new importance in neutron diffraction owing to the general lack of largespecimens for single-crystal methods. Even in those cases where it proves to be possible to grow large singlecrystals, these may still suffer from such effects as extinction and magnetic domain structures, making a properinterpretation of the diffracted intensities unreliable. Many of these systematic effects are either nonexistent inthe powder method, or become isotropic and can therefore be more easily determined.
In a polycrystalline sample it is inevitable that certain information is lost as a result of the random orientation ofthe crystallites. A further, and in practice more serious, loss of information is a result of the overlap ofindependent diffraction peaks in the powder diagram. The method of using the total integrated intensities of theseparate groups of overlapping peaks in the least-squares refinement of structures (Rietveld, 1966), leads to theloss of all the information contained in the often detailed profile of these composite peaks.
By the use of these profile intensities instead of the integrated quantities in the refinement procedure, however,this difficulty is overcome and it allows the extraction of the maximum amount of information contained in thepowder diagram (Rietveld, 1967a). Because of the importance of magnetic structures in neutron diffraction, therefinement method has been made applicable both to nuclear and to magnetic structures. The latter onlycomprise those structures which can be described on the nuclear unit cell or a multiple thereof.
ExperimentalThe neutron powder spectrometer in Petten operates with a wavelength of , pyrolytic graphite beingused to suppress the second order contribution (Loopstra, 1966; Bergsma & van Dijk, 1967). Soller slits withangular divergences ranging from 10' to 30' can be installed between the reactor and the monochromator, and in
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front of the counter. The monochromator normally consists of a Cu crystal with its (111) plane in thereflecting position. The sample is contained in a thin-walled vanadium tube, approximately 6 cm long and with adiameter of 1, 1.5, or 2 cm. The maximum scattering angle is and the step width usually ranges from2.16' to 8.64' depending on the divergences of the Soller slits. The counter scans through the diffraction peaksand at each step, at position , measures a number of counts for a preset monitor count. The background of
the recorded diagram is evaluated graphically at different positions and used to obtain, by linear interpolation,background corrections for each intensity . To the corrected intensity, , is assigned astatistical weight , where . Because is obtained by graphical means, its
variance is not known and is arbitrarily set equal to zero. With the variance of from counting statistics equalto , becomes .
Peak shapeThe measured profile of a single powder diffraction peak is dependent on the neutron spectral distribution, themonochromator mosaic distribution, the transmission functions of the Soller slits, and the sample shape andcrystallinity. While each of these contributions can have a form not necessarily Gaussian, it still is an empiricalfact that their convolution produces almost exactly a Gaussian peak shape (Fig.1). Assuming this Gaussian peakshape for each Bragg peak, one can now write its contribution to the measured profile at position as
where
Because the absorption of neutrons in the sample is generally negligibly small, no corresponding correctionfactor has been included in the above expression.
Putting equation (1) can be simplified to
At very low scattering angles, however, the peaks begin to show a pronounced asymmetry. This is mainlybecause of the use of finite slit heights together with finite sample heights. This vertical divergence effect (Klug& Alexander, 1959) causes the maximum of the peak to shift to lower angles, but does not affect the integratedpeak area. Introduction of a semi-empirical correction factor in equation (2) gives a good approximation to theasymmetric peak profile (Fig.2):
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where P is the asymmetry parameter and s =+1,0,-1 depending on the difference being positive,
zero, or negative respectively. As can be seen from Fig.2, this correction has two main effects: the peak isshifted to lower angles and the Gaussian peak shape is made slightly asymmetric.
The peak widthThe formula given by Caglioti, Paoletti & Ricci (1958) to express the angular dependence of the halfwidths ofthe diffraction peaks can be simplified to
where U, V, and W are the halfwidth parameters. This simple formula also takes account of the peak broadeningresulting from the particle-size effect and describes very adequately the experimentally observed variation ofhalfwidth with scattering angle (Fig.3).
Initial and approximate values for these parameters are found by graphically measuring the halfwidth ofselected single peaks in the diagram and finding a least-squares fit to these observed quantities through equation(4).
Preferred orientation correctionPlate-like crystallites have a tendency, at least in part of the sample, to align their normals along the axis of thecylindrical sample holder. When this effect is not too pronounced, a condition which is generally fulfilled inneutron diffraction where large samples are common, the intensity corrected for preferred orientation can bewritten as
where is the acute angle between the scattering vector and the normal to the crystallites. G is the preferredorientation parameter and is a measure for the halfwidth of the assumed Gaussian distribution of the normalsabout the preferred orientation direction.
Method of calculationEquation (3) can also be written as
where
is a measure of the contribution of the Bragg peak at position to the diffraction profile at position
.
Because the tails of a Gaussian peak decrease rather rapidly, no large error is introduced by assuming the peakto extend no further than one and a half times its halfwidth on both sides of its central position. In the case of
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overlap, more than one Bragg peak contributes to the profile intensity, , i.e.
where the summation is over all reflections which can theoretically contribute to on the basis of their position and their halfwidth . For larger scattering angles and for crystals with a low symmetry, this summation
can easily be over more than ten terms. On the other hand, there may be regions of the diagram where no peakscan possibly contribute and these regions are therefore left out of the calculations as containing no relevantinformation.
Fig.1. Comparison of a measured diffraction peak, ....., with a calculated Gaussian peak profile, ---.
Fig.2. Comparison of an asymmetric diffraction peak with a symmetric and an asymmetry-correctedcalculated profile: ..... measured intensities, - - - symmetric Gauss curve,--- asymmetric curve.
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Fig.3. Variation of peak width with Bragg angle; ..... measured halfwidths,--- calculated curve.
The structure factorBy writing
the preferred orientation correction [see equation (5)] and the overall temperature factor are kept outside theexpressions for the nuclear and magnetic structure factors. Q is here the overall isotropic temperatureparameter.
{ The expression for can be written as where
and
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The occupation number for fully occupied lattice sites is equal to m/M where m is the multiplicity of theposition, special or general, and M is the multiplicity of the general position in the particular space group. Thevalue of m ranges in general from 1 to M.
The magnetic coherent scattering cross section can be expressed as
where is the unit vector in the direction of the scattering vector and , the magnetic
structure factor (Halpern & Johnson, 1939).
can be resolved into its components:
where
The above formulae for the magnetic cross section are applicable to all magnetic structures with a unit celldefined by , , and where u, v, and w are integers and , , and are the original nuclear unit-cellvectors.
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The expression (6) is valid for all reflections, equivalent or independent, and can therefore equally well be usedwith powder or single-crystal methods. In powder diffraction, the number of calculations can be significantlyreduced by computing only one average cross section for each set of equivalent reflections (Shirane, 1959; vanLaar, 1968). For a magnetic structure with a uniaxial configurational spin symmetry, this average cross sectionis
where is the angle between the unique axis and the scattering vector . The unique axis is assumed to be the
[001] axis.
For magnetic structures with cubic configurational spin symmetry (Shirane, 1959) this expression becomes
Least-squares parametersThe least-squares parameters can be divided into two groups. The first group, the profile parameters, define thepositions, the halfwidths, and the possible asymmetry of the diffraction peaks in addition to a property of thepowder sample i.e. preferred orientation. These parameters are
The second group, the structure parameters, define the contents of the asymmetric unit cell:
In order to describe the contents of the complete unit cell, it suffices to give, in addition to the contents of theasymmetric unit, the set of symmetry operations to generate the remaining positions and magnetic vectors in thecell. The symmetry operations on the nuclear positions consist of a rotation and a translation, i.e.
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where and are respectively a rotation matrix and a translation vector describing the ith
equivalent position.
The magnetic vector can undergo a rotation only, i.e.
where is a rotation matrix describing the rotation of the magnetic vector of the ith atom on goingfrom the asymmetric unit to the ith equivalent position. The subscript i on this matrix indicates that not allmagnetic vectors have to undergo the same rotation when being transformed to the ith equivalent position.Each magnetic atom in the asymmetric unit may therefore have its own set of rotation matrices.
Least-squares refinementThe principle of the profile refinement method is best demonstrated by the form of the function M which has tobe minimized with respect to the parameters. While for the normal refinement procedure on separatedintegrated intensities this function is
and on the integrated intensities of groups of overlapping reflections,
this function becomes, in the case of profile refinement,
where
A computer program, based on the above described method, carries out the least-squares refinement in theusual manner. Because the problem is not linear in the parameters, approximate values for all parameters arerequired for the first refinement cycle. These are refined in subsequent refinement cycles until a certainconvergence criterion has been reached.
The program provides the possibility of keeping any parameter constant during refinement or of introducingconstraints between any number of them (Rollett, 1965; Rietveld, 1967b). The input procedure allows theconstraint functions to be either linear or quadratic in all parameters. The latter type of function, for instance,enables the length of a magnetic vector to be kept constant while refinement is carried out on its direction. Thisintroduction of constraint functions, however, increases the size of the, in this case bordered, normal matrix and
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may lead to possible instabilities in the subsequent inversion when the matrix becomes too large. For verysimple linear constraints, therefore, the program allows a substitution method which in effect decreases the sizeof the matrix.
In order to be able to make some quantitative judgment of the agreement between observed and calculatedintegrated intensities instead of profile intensities, a fair approximation to the observed integrated intensities canbe made by separating the peaks according to the calculated values of the integrated intensities, i.e.
where
From the values one can now obtain values for ), , and and define the
following R values:
and
ResultsOver the past year many structures, nuclear as well as magnetic, have been refined with the method. Anexample of a nuclear structure is (Loopstra & Rietveld, 1969). Its powder diagram (Fig.4) showshardly any overlap and the structure may have been as successfully refined with integrated intensities. Thereason for inclusion of this diagram, however, is the fact that it demonstrates the nearly perfect fit obtainable onthe assumptions of a Gaussian peak shape [equation (1)] and a quadratic relation between halfwidth andscattering angle [equation (4)].
The powder diagram (Fig.5) of exhibits severe overlap. At large angles, more than ten reflections maycontribute to a profile intensity. While the information content of this part of the diagram is low, the agreementbetween observed and calculated profiles shows that even this amount has been used to the fullest extent. Someof the other structures are shown in Table 1 with their R values. The large values in this list arecaused by the fact that the nuclear scattering generally overrides the magnetic contributions, making the above-described method of separating peaks [equation(7)] very unreliable for these magnetic intensities. This,however, does not indicate that the magnetic moments found by this least-squares procedure suffer from the
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same inaccuracy, their relative statistical errors being in the range of . It only demonstrates the potential
of the method for extracting the maximum available information from a powder diagram as opposed to anypeak-separation method. It may also be remarked here that an increase in the resolution of the diagram, whileof no use in peak separation methods if it does not produce separable peaks, is always profitable in the profilemethod because it results in a higher information content owing to increased detail in the profile.
Fig.4. Neutron powder diffraction diagram of measured at ;--- calculated profile, . . . . measured profile.
Fig.5. Neutron powder diffraction diagram of measured at ;--- calculated profile, . . . . measured profile.
Table 1. R Values of various nuclear and magnetic structures.
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Computer programThe least-squares refinement program has been written in Algol 60 to be run on an Electrologica X-8 computerhaving 48 K words (26 bits) core storage and magnetic tape units. Because of the necessity of comparinggraphical data in the form of calculated and observed diffraction diagrams, instead of the usual integratedintensities, extensive use is being made of an automatic incremental plotter.
A detailed description of the program can be obtained from the author (Rietveld, 1969).
ConclusionIn all instances the profile refinement procedure has proved to be superior to any other method involving eitherpeak separation or the use of the total integrated intensity of groups of overlapping peaks. It is felt that in thisway one of the inherent drawbacks of the powder method, i.e. the loss of information as a result of overlap, hasbeen effectively overcome and that the method in many instances can now compete with single-crystalmethods, especially when these are subject to systematic errors.
The method can in principle also be extended to X-ray powder diagrams, if a satisfactory function can be foundto describe the peak profiles. However, the method will remain best suited for neutron powder- techniquesbecause of the nearly exactly Gaussian shape of the diffraction peaks and the possibility of describing thevariation of halfwidth with Bragg angle in terms of a simple quadratic function.
The author wishes to thank Drs B.O. Loopstra and B. van Laar for their suggestions and helpful criticism.
References
1BERGSMA, J. & DIJK, C. van (1967). Nucl. Instrum. Meth. 51, 121.
2CAGLIOTI, G., PAOLETTI, A. & RICCI, F. P. (1958). Nucl. Instrum. 3, 223.
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3ENGELSMAN, F. M. R., LAAR, B. van, WIEGERS. G. A. & JELLINEK, F. (1969). To be published.
4HALPERN, 0. & JOHNSON, M. H. (1939). Phys. Rev. 55, 898.
5KLUG, H. P. & ALEXANDER, L. E. (1959). X-ray Diffraction Procedures, 2nd ed. p. 251. New York:John Wiley.
6LAAR, B. van (1968). Reactor Centrum Nederland Research Report RCN-92
7LAAR, B. van, RIETVELD, H. M. & YDO, D. J. W. (1969). To be published.
