 1. Introduction
 2. Definition of diffractometer angles and pseudoangles
 3. Coordinate transformation and basic diffraction equation
 4. Motor angles to Miller indices and pseudoangles
 5. Reciprocal space to angles
 6. Relation to other diffractometer operation modes
 7. Relation to the sixcircle surface diffractometer
 References
 1. Introduction
 2. Definition of diffractometer angles and pseudoangles
 3. Coordinate transformation and basic diffraction equation
 4. Motor angles to Miller indices and pseudoangles
 5. Reciprocal space to angles
 6. Relation to other diffractometer operation modes
 7. Relation to the sixcircle surface diffractometer
 References
research papers
Angle calculations for a `4S+2D' sixcircle diffractometer
^{a}Argonne National Laboratory, Materials Science Division, 9700 S. Cass Ave. Argonne, IL 60439, USA
^{*}Correspondence email: hyou@anl.gov
Angle calculations are derived for the operation of a `4S+2D' sixcircle Xray or neutron diffractometer. The six
[four sampleorienting (`4S') and two detectororienting (`2D')] of the diffractometer are used to control the scattering vector and a reference vector in the laboratory frame of reference. Several modes of operation unavailable in other types of diffractometer are presented.1. Introduction
In recent years, several types of diffractometer have been developed for the study of single crystals with monochromatic Xrays or neutrons. One of the most popular instruments is the fourcircle diffractometer, which consists of one detector circle and three sampleorienting circles. Since only three ).
are needed to position a sample crystal in an orientation satisfying a desired Bragg condition, the extra degree of freedom in the crystal orienting circles is used in various modes of diffractometer operation, typically for scattering measurements of bulk singlecrystal samples. Angle calculation schemes for the fourcircle diffractometer were developed by Busing & Levy (1967A significant development in utilizing the extra degree of freedom was made by Mochrie (1988). He has shown that it is necessary to use the extra degree of freedom in the study of surfaces or interfaces to control the angle of incidence with respect to the surface. His angle calculation scheme for controlling the incident and exit angles for surface scattering has become widely used in surfacescattering studies at synchrotron sources. However, the fourcircle geometry still has some limitations for surfacescattering experiments. The accessible solid angle for Xrays is often limited in the surfacescattering chambers. Orienting the sample to a desired direction with respect to resolutions and the polarization of the incoming Xrays is not, in general, possible. To overcome the geometric limitations of the fourcircle diffractometer, a fivecircle diffractometer (Gibbs et al., 1990; Vlieg et al., 1987), in which the entire vertically scattering fourcircle diffractometer is placed on a rotating table, and a sixcircle diffractometer, in which an additional arc (Bloch, 1984) is mounted on the detector arm to provide an extra detector degree of freedom (Lohmeier & Vlieg, 1993; Abernathy, 1995), were developed. A feature of these surface diffractometers, potentially a drawback, is that the detector are coupled with the sample orientation through the table rotation. Therefore, another type of diffractometer, known as the `2S+2D' diffractometer, was specifically designed for surfacediffraction experiments (EvansLutterodt & Tang, 1995). It employs two detector both independent of the two sampleorienting circles.
Since three
are needed to determine the orientation of a singlecrystal sample, at least three sampleorienting circles are desirable to take full advantage of the two independent detector circles. In fact, the additional extra degree of freedom available with four sampleorienting circles offers the further advantage that one can avoid blind angles, particularly for a diffractometer designed to withstand heavy loading. Therefore, we have developed an angle calculation scheme, described herein, for a diffractometer with four sampleorienting and two independent detector (a `4S+2D' sixcircle diffractometer). The central idea is based on the capability, with the extra detector degree of freedom, of constraining a reference vector (the surface normal in the surfacediffraction case) to a desired orientation with respect to the laboratory frame of reference. We emphasize, however, that the reference vector is not limited to the surface normal but can be any direction of desired or broken symmetry, either structural, electronic or magnetic in nature.2. Definition of diffractometer angles and pseudoangles
The `4S+2D' sixcircle diffractometer discussed in this paper is shown schematically in Fig. 1. The notation for the angles is as consistent as possible with the notations previously used with other diffractometers. When the angles μ and ν are set to zero, the diffractometer is essentially a conventional fourcircle diffractometer for which the θ and 2θ angles are named as η and δ, respectively. The coordinate system for the laboratory frame of reference is chosen to be consistent with that used for the fourcircle diffractometer of Busing & Levy (1967) and also with the sixcircle surface diffractometer of Lohmeier & Vlieg (1993). It should be noted that onetoone correspondence with the diffractometer described by Lohmeier & Vlieg (1993) can be made with a set of simple equations, given in the last section (§7) of this paper. Therefore, all the modes described for the sixcircle surface diffractometer can be used with the simple transformation and none of those modes of operation will be specifically discussed here. Instead, a general set of equations is developed that allows a user to control the diffractometer with the constraints that most meet the users needs with respect to geometric factors, such as limited accessible solid angles of the sample chamber, the shape, resolution and polarization of the incoming beam, and the orientation of externally applied fields, while still satisfying the Bragg conditions.
We now lay out a simple scattering geometry with angles that relate to the laboratory frame of reference. These angles are the most convenient angles to follow the scattering angles and sample orientations. However, these angles often do not correspond to the diffractometer circles, which are generally based on mechanical considerations. These angles will be used in defining the orientations of vectors such as the scattering vector (Q), a reference vector (n), or the outgoing vector (k^{f}). Since these angles do not have a onetoone relation with any diffractometer circles, we will call them pseudoangles. Some of the pseudoangles are defined and shown in Fig. 2. The most fundamental pseudoangle among all the pseudoangles is θ. In the fourcircle case, θ is defined to be half the detector angle, but in the sixcircle case it is defined as a pseudoangle by the scattering equation
As shown in Fig. 2, the angle θ sets the radius of the sphere that intersects the We fix the origin of the laboratory frame of reference at the center of this sphere. We denote the intersecting circle as Ck, which has its plane normal to the incoming beam. It is then evident that the angle of Q with respect to the xz plane is θ. We can specify the orientation of Q by defining an additional azimuthal angle ϑ (we call it qaz for convenience). Similarly, we define two pseudoangles, α and φ (similarly, naz) for the reference vector n ( is shown in Fig. 2). When the surface normal is defined to be the reference vector, then α is the incident angle as defined by Mochrie (1988). In addition, two other pseudoangles, τ and ψ, which are the longitude and azimuth of measured with respect to the scattering vector and (a plane formed by the incident beam and the scattering vector), will be used in our calculations. The angle ψ has been previously defined by Mochrie (1988) and originally by Busing & Levy (1967). However τ is a new pseudoangle, but it is a simple invariance defined by the relation, . Therefore, the only significant new pseudoangles defined here are the two azimuthal angles φ and ϑ.
3. Coordinate transformation and basic diffraction equation
The format and the angle definitions used in the next two sections were adapted from those of Abernathy (1995), Lohmeier & Vlieg (1993), and Busing & Levy (1967). Some of the algebra is identical or similar to that in the earlier papers but the essential equations are reproduced in order to prepare the readers for the later sections.
Let h be the column vector describing a momentum transfer in the with a righthanded coordinate system so that
where b_{i} are the reciprocallattice vectors and h_{i} are generally referred to as h, k and l. Then we can construct a matrix, B, using the Cartesian components of b in the laboratory frame so that equation (2) can be rewritten as
Note that matrix B is not necessarily orthonormal as the crystal symmetry is not necessarily cubic. Now let us define the vector h_{ϕ} which represents the orientation of the reciprocal vector h_{c} when the sample is mounted on the diffractometer with every diffractometer circle set to zero. In this case, h_{ϕ} is simply related to h by a matrix U, known as the orientation matrix, which corrects the misalignment between the Cartesian axes of the reciprocal crystal space and those of the laboratory frame of reference. The orientation matrix is orthonormal because it simply rotates or reorients the Cartesian coordinate system. Therefore, we can describe h_{c} as h_{j} after a rotation by the jth sampleorienting circle. Then we can write the relation of h to h_{M} as a simple equation,
where , X, H and M are the matrices representing the rotation of the corresponding circles. Note that all but B are orthonormal matrices. The matrices corresponding to the rotation of the circles are explicitly written, using the senses of rotation shown in Fig. 1, as
Now let us define the detector position in the frame of reference attached to the detector circle δ. When the momentum transfer is zero, the two detector angles are zero and the final outgoing vector is equal to the incoming vector, and is given as
where the wave number . Following a similar scheme used for the sixcircle surface diffractometer (Lohmeier & Vlieg, 1993; Abernathy, 1995), this vector is obtained in the laboratory frame of reference for nonzero detector angles by the following two transformations:
where the explicit forms of and are
Note that H, and are defined by lefthanded angles and M, X, and by righthanded angles. Also note that the x axis is defined along the vertical μ and ν axes and the y axis is defined along the incoming beam direction. Now then
In order to satisfy the diffraction condition, h_{M} should be equal to the desired diffraction vector determined by the detector circles in the laboratory frame of reference. Using the subscript L to emphasize that the vector is the final orientation in the laboratory frame of reference, the diffraction equation becomes
where
described entirely by the sampleorienting circles, and
described only by the detector circles. Here , i.e. the lattice vector in the frame of reference attached to the ϕ table of the diffractometer. The total rotational operation made by the sampleorienting circles for the vector transformation, , is
This matrix is used in the later sections for the angle caculations.
4. Motor angles to and pseudoangles
In this section, the calculation of h, k and l from arbitrary detector and sampleorienting angles is given. The same equation can be used to set the orientation matrix when h, k and l are known for particular reflections. From equations (10) and (12), the value of h_{ϕ} satisfying the diffraction condition is found to be given by
where
Then, from the definition
we can calculate the components h_{i} (h, k and l) for a given orienation, or when h, k and l are known for two or more reflections, we can determine the orientation matrix.
Now we shall show how to find the pseudoangles from motor angles. The most fundamental pseudoangles are θ and ϑ, since the scattering vector Q can be written either in terms of pseudoangles or the motor angles. Therefore, Q alone produces the relation
or
and from this relation we obtain
where is zero when ν is zero. The pseudoangles α and φ can be found from a similar equation for the reference vector,
From this equation we obtain
Using equations (18) and (20), the angle τ can be determined from the identity as
Since α is the incident angle when n is chosen to be the surface normal in surfacescattering experiments, we can conveniently define another pseudoangle, the exit angle β, from
From the definitions of α and β, we can see that
Therefore,
We can now imagine a new primed frame of reference where the vector Q lies along the axis and is along the axis in order to define the remaining pseudoangle ψ. In this frame the axis is normal to the We define the zero of the azimuth angle ψ to be when the reference vector n lies in the on the side closer to the axis. Under this condition, the vector points to the intersection of circles CQ and CS as shown in Fig. 2. By writing vectors in the Q frame of reference as
the incident and exit angles are determined from the azimuth by
and
or conversely, the azimuth ψ can be obtained as
or
as derived from the above equations. This completes the calculation of
and all the relevant pseudoangles from only motor angles and a given reference vector.5. to angles
Since the diffractometer has six , although this is not a unique way of describing the various modes of operation. In Table 1, the first two columns of angles are associated with the detector circles and pseudoangles and the last three identical columns relate to the sampleorienting circles.
and determination of a sample orientation in the laboratory frame of reference requires only three angular the three remaining angles need to be given in advance. Because of the extra there are many modes of diffractometer operation, far more than in the case of a fourcircle diffractometer. These modes can conveniently be described by using Table 1

Each mode of operation requires that three columns be chosen out of the five columns of angles shown, and that one angle in each chosen column be given. Of course, one may not choose a particular sampleorienting angle more than once. Some of these modes will be discussed explicitly below.
5.1. One of the detector angles is given
The first three angles in the `detector' column of Table 1, δ, ν and ϑ, are for the detector circles. Given any one of these angles, only one detector degree of freedom remains. Since the angle θ is known from equation (1), both detector angles can be determined by inverting equation (19). Either ν or δ given in advance leads to a quasihorizontal (δ is frozen) or vertical (ν is frozen) scattering geometry. If the value of the frozen angle is zero, the configuration becomes a true horizontal or vertical scattering mode. As one can see, the pseudo angle ϑ sets the azimuthal angle of the In other words, the is horizontal for ϑ = 0 and vertical for . When one of the detector angles is given, one needs to provide another pseudoangle and one sampleorienting circle (see §5.3), or two sampleorienting circles (see §5.4) to complete the determination of angular positions for all six circles.
5.2. One or two of the pseudoangles for the reference vector are given
As shown in Fig. 2, the reference vector n can point anywhere on the circle CQ, which requires that the pseudo angle ψ be constrained to determine the orientation of the sample crystal uniquely. Alternatively, the pseudoangles α or φ can be constrained from equation (26) and equation (23). However, one needs to be cautious when the pseudoangles α or φ are given instead of ψ because their available angular range is limited due to their rotation axes not neccessarilly being aligned with Q. These cases belong to the modes where one pseudoangle and two sampleorienting circles are given, and are discussed in §5.4.
Alternatively, both α and φ can be given in advance, thereby predefining the orientation of the reference vector in the laboratory frame. In this case, Q_{L} can be found at the intersections of the circles Ck and Cn. The point of the intersection can be uniquely determined since the pseudoangle ϑ can be found using the identity equation (23):
Once ϑ is known, the detector angles are determined from equation (19). Alternatively α and ϑ can simply be given instead. In either case, with the two pseudoangles, one sample orienting circle needs to be given in advance to determine the sample orientation. This case is discussed in §5.3.
5.3. When one sampleorienting angle and two other angles are given
Two angles from the first two columns of Table 1 will uniquely set the detctor position for a given scattering vector and orientation of the reference vector. There is only one sample orientation satisfying the orientations of both the scattering and reference vectors (as long as they are not collinear). So three sample must be used to orient the sample. Therefore the position of one sampleorienting circle must be given in advance.
The vectors and are fixed in the laboratory frame as defined in equations (18) and (20). An orthonormal matrix (N_{L}) based on these two vectors can be constructed by the three vectors , and [known as Gram–Schmidt orthonormalization; also used by Busing & Levy (1967)] in the laboratory frame of reference as
where indicates the normalized x component of vector a, i.e. a_{x}/ a. This matrix is related to another orthonormal matrix (N_{ϕ}) formed by the corresponding two vectors h_{ϕ} and n_{ϕ} in the ϕ frame through the matrix Z as
when N_{ϕ} is constructed from h_{ϕ} and n_{ϕ} (components are known) in an identical manner. Since from equation (32), the sampleorienting circles are determined by comparing the matrix components in the following relations:
For μ fixed (μ = μ_{0} or ), is known, and its components are to be compared to
Only the matrix components explicitly shown above are needed to determine ϕ, χ and η [in the same manner as Busing & Levy (1967)]:
Likewise, for ϕ fixed, is known and compared to
with μ, χ and η similarly determined as
For η fixed ( = or ) or χ fixed, is known and needs to be compared to
From the comparison of V_{13}, χ and η are simply determined as
and
The other two angles, μ and ϕ, are similarly determined by examining the (V_{33}, V_{23}) and (V_{11}, V_{12}) pairs:
and
5.4. When two sampleorienting angles and one detector or one pseudoangle are given
For the case in which two sampleorienting angles and one of the angles in the first two columns of Table 1 are given, it is desirable that the frame attached to the scattering and reference vectors is specified by successive rotations of the laboratory frame about the x axis by ψ, then about the z axis by −θ, and finally about the y axis by . Then the orthonormal matrix is , and we rewrite equation (32) as
where
and
with .
For the case in which χ, ϕ and ψ are given, it is convenient to rewrite equation (43) as
The variables on the righthand side are all known and we call the righthand side V. Now let us express the lefthand side in terms of the unknown angles η, ϑ and μ as
Its components should then be equal to the components of . By comparing, we obtain
When ψ, μ and η are given, the following form is most useful:
We again define the righthand side as V. Then the lefthand side becomes
These components should match those of V. By comparing V_{32} to the corresponding component of this matrix, we have
Since we know the values of η and μ,
An examination of the matrix components V_{31} and V_{33} yields
and from this equation we obtain ξ using the value of χ obtained above,
We can similarly solve for ϕ from the matrix components V_{12} and V_{22} also using the value of χ.
When ϑ, η and μ are given, the equation becomes
where all the components of the lefthand side are known and we again call it V. Then the righthand side is
By comparing the V_{21} component we can see that
Therefore we obtain
As performed above, an examination of the displayed components yields
and
For given ϕ, χ and , the equation becomes
From the lefthand side we obtain
By defining the righthand side as V, we can similarly solve for η, μ and ψ as before. By comparing the V_{31} component, we find
From the ratio of V_{11} and V_{21}, we can obtain the valueof η,
Similarly, we obtain ψ from the ratio of V_{32} and V_{33},
Other conditions such as fixing the values of ϕ and μ, χ and μ, ϕ and η, or χ and μ along with a pseudoangle can be similarly obtained.
5.5. When three sampleorienting angles are given
Now let us consider the case when three sampleorienting angles are given. Although we see no advantage in using such a mode under normal scattering conditions, one can imagine, nevertheless, a situation where three sampleorienting circles have only limited angular ranges, for example, because of geometric constraint due to a sample chamber. In these modes, the reference vector is not necessary to obtain the solution. There are two ways to solve these cases: solving without the reference vector or solving with a conveniently chosen reference vector as discussed in §6. Since the latter case can be treated in the same way as the previous cases in which two sampleorienting circles are given, we will outline the former case, i.e. without using the reference vector.
Here, we use the vector equation
From equations (10) and (18) we obtain
We then rearrange the y component of this equation to the following form:
where denotes the unknown angle (one of ϕ, χ, η or μ). The unknown angle can then be found as
Using this angle, we can additionally find
with which one can subsequently set the angles for the detector circles according to equation (19).
6. Relation to other diffractometer operation modes
The modes of operation discussed above include most of the known diffractometer settings and modes of operation. Since it may not be apparent how the modes described here correspond to some of the previously known modes of operation, we will discuss the relation of the previously popular modes of diffractometer operation to the modes discussed above.
The most popular mode for the fourcircle diffractometer is the socalled ω = 0 mode or bisecting mode. In this mode, the scattering vector is parallel to the plane of the χ circle. For either horizontal or vertical scattering condition, one can reproduce the bisecting mode of the fourcircle case simply by setting ϑ = 0 or ϑ = 90° and by setting μ = ν/2 or η = δ/2, respectively. Unlike the fourcircle case, there is an extra degree of freedom available for orienting the sample, such as choosing a desirable ψ, α or ϑ. Therefore, we define ω as the angle of the scattering vector (Q) with respect to the plane of the χ circle. Since the axis of the χ circle is on when η = μ = 0, we obtain the relation and from this we obtain
When the axis of the χ circle is within the as expected for a fourcircle diffractometer, we obtain an additional equation,
from the condition . We can now obtain a set of equations for the angles η and μ from these two equations:
and
Another popular mode of operation of the fourcircle diffractometer is the zone mode. In this mode, a zone (a plane including the origin of reciprocal space) is specified by two reciprocallattice vectors and the specified zone is moved to the p_{ϕ} and r_{ϕ} and define the reference vector . The zone mode can be achieved by setting and for a given value of ϑ. This condition constrains n_{L} to the xz plane and normal to the There are several ways to satisfy this condition but we will describe a way that is most similar to the zone mode of the fourcircle diffractometer. First, we find ϕ and χ that satisfy the conditions ϑ = 0 and , as if the diffractometer is a horizontal fourcircle diffractometer. From this condition we obtain the known solution:
In this condition, one can easily navigate within the plane of without changing the orientation of instrumental resolution, the polarization of incoming Xrays or neutrons, or the direction of the external field applied to the sample with respect to the zone. Let us call the two input reciprocal vectorsand
We can use ϑ, χ_{z} and ϕ_{z} to determine the remaining angles, μ and η. For either the horizontal or vertical scattering condition, one can reproduce the zone modes of the fourcircle case by setting ϑ = 0 and η = 0, or ϑ = 90 and μ = 0°. The accessible reciprocal vectors in the zone modes should, of course, be limited to since reciprocal vectors that are not in the zone can also be reached in this manner due to the extra degree of freedom.
The above two modes are related just as in the fourcircle case. Once χ and ϕ are frozen, the vectors in the zone are reached by changing ω only, just as in the fourcircle case. The change of ω is ensured by setting ϑ and finding the corresponding η and μ. The η and μ then satisfy the relation given in equation (73).
There are several modes of operation for which the definition of the reference vector n is not needed. Examples are the modes where ν, δ or ϑ and two of the sampleorienting circles are given. Nevertheless, the pseudoangles for a specified reference vector can be calculated as long as the scattering vector is not parallel to the reference vector. In order to avoid accidentally setting the reference vector parallel to the scattering vector using an arbitray reference vector, we again use the Gram–Schmidt orthonormalization procedure (see Busing & Levy, 1967) to find a suitable reference vector. The following matrix can be formed solely by the scattering vector either in the laboratory frame or in the ϕ frame:
We can set the second (or third) column of this matrix to be the reference vector n.
7. Relation to the sixcircle surface diffractometer
In this section, we derive the relation between the motors of the sixcircle surface diffractometer and our `4S+2D' type of sixcircle diffractometer. A part of the algebra has essentially been reported by Vlieg (1998) but is reproduced here with appropriate notations for clarity. Note that the two diffractometers differ only by the detector circles. That is, the circles γ and δ′ for the surface diffractometer sit on the μ table, thereby coupling the γ and δ′ rotations to the μ rotation. The final wavevector in the surface diffractometer is given by
From this identity we obtain the relations
and
to convert from the `4S+2D'type sixcircle diffractometer to the sixcircle surface diffractometer, or
and
to convert from the sixcircle surface diffractometer to the `4S+2D'type sixcircle diffractometer.
Acknowledgements
The author thanks Dr Y. Chu for useful discussions and Dr G. Swislow for testing the formulae presented in computer code and for a careful reading of the manuscript. It is acknowledged that Ch. 2 of a thesis by Dr D. L. Abernathy was helpful in the initial derivation of the calculation schemes. The calculation schemes described in this paper are implemented as a part of the control program SPEC (Certified Scientific Software, 1998). This work was supported by DOE under contract W31–109ENG38.
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