Rotation_Matricies_Derivation_Kappa_Goniometer
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The document summarizes research conducted during the spring 2006 semester to continue work from the previous summer on modifying a kappa goniometer. Goals included determining angle precision for each axis, designing and fabricating adapter plates to mount new stepper motors, presenting a poster, and deriving rotation matrices. Angle precision for each axis was measured. Adapter plate designs were completed and fabrication began. Installation and further work will continue over the summer. Derivation of rotation matrices was reviewed but not completed due to time constraints. An abstract was accepted for presentation but later withdrawn due to lack of funding.
![The Y' comDonent:
Or to find the fixed basis vectors (lab frame) in terms of the rotating vectors:
This result is commonly written in matrix notation:
cos6 - sin 6 x'
[;]= [sin 6 cos6 ][ y ]
Similarly, for clockwise rotation, the componentscan be derived and the following
matrix results:
[;]= [cosh i n61[ X I ]
- sin 6 cos6 y'
This developmentcan be extended into the derivation of the 3-d rotational matrix about
each axis. We can multiply them together to create any 3-d rotation.
We start with rotation about the z-axis. It is essentially the same as that just derived for
two axes with an identity transformation for the z-coordinate,since this component will
not change during a rotation about the z-axis:](https://image.slidesharecdn.com/4d6b6acc-ad46-4dd3-b88f-b7fa32fbe447-160907025341/85/Rotation_Matricies_Derivation_Kappa_Goniometer-9-320.jpg)
![The next rotation to consider is rotation about the x-axis. This corresponds to rotation of
the y-axis into the z-axis. A clever "trick" is to let y take the role of x and z take the role
of y (since this is analogous to the derivation of the rotation of x into y):
cosw sinw 0 y
I= c0;w
y' 0 cosw sinw x
[:j=[:s i y co;w][:]s
Finally, rotation about the y-axis plays the analogousrole of rotation about z by letting z
take the role of x, and x take the role of y:
z' sin @ 0 cos@ x
[;I= :y][+
cos@ 0 - sin @ x
[i]=[si.@ I c:4][:j=~e[i](https://image.slidesharecdn.com/4d6b6acc-ad46-4dd3-b88f-b7fa32fbe447-160907025341/85/Rotation_Matricies_Derivation_Kappa_Goniometer-10-320.jpg)
![The usual convention is to rotate about x, then y, then z. Therefore, in this case the
composite rotation matrix is:
cos@ 0 -sin@ 1 0
I[0
1 0 I[. cosw si:w][;]
sin@ 0 cos@ 0 -sinw cosw z
c o s ~ sin^ 0 COS@ sin@sinw -sin@cosw
= sin^ COSK 0 0 cosw sin w
[ 0 0 I s i n , -cos@sinw c o s ~ o s wI
cosKCOS @ cos sin@sinw+sin KCOS w sin sin w-cos sin @cosw'
-sin KCOS@ cosKCOS w-sin Ksin@sinw cosKsinw-sin K S ~@cosw
sin @ -cos@sinw cos@cosw](https://image.slidesharecdn.com/4d6b6acc-ad46-4dd3-b88f-b7fa32fbe447-160907025341/85/Rotation_Matricies_Derivation_Kappa_Goniometer-11-320.jpg)
![One of the most efficient methods to relate the relationshipof the real-spacecomposition
of a material to its diffraction pattern is through the reciprocal lattice. These must be
incorporated into the rotational matrices in order to completely return to real-space.
A nice chapter describing the bravais lattice and derivationof each reciprocallattice was
found in the appendix of Elements of X-Ray DifSraction, (3rdedition) (hardcover),B.D.
Cullity, and investigationof these was done.
Further reading was done by reviewing numerous papers written by experts on deriving
angle calculations for multi-circleinstruments. A few of these papers reviewedinclude:
Acta Cryst. (1967). 22,457-464 [ doi:10.1107/S0365110X67000970]
Angle calculationsfor 3- and 4-circle X-ray and neutron diffractometers
W. R. Busingand H. A. Levy
J.Appl. Cryst. (1993). 26,706-7 16 [ doi:10.1107/S0021889893004868]
Angle calculationsfor a six-circle surface X-ray diffractometer
M. Lohmeier and E. Vlieg
Although reading of these papers have begun, derivation of the angle calculations for a 4
or 5 circle instrument was not completeddue to time constraints.
Conclusion:
In all, satisfaction was found in the amount of work accomplishedduring this semester
even though more work remains. Some of this work will probably be performedover the
summer months until the finished project is realized.](https://image.slidesharecdn.com/4d6b6acc-ad46-4dd3-b88f-b7fa32fbe447-160907025341/85/Rotation_Matricies_Derivation_Kappa_Goniometer-12-320.jpg)
Rotation_Matricies_Derivation_Kappa_Goniometer
- 1. Introduction: The reason for conducting research this semester was to follow up on research done during the summer 2004 REU program. A list of goals was laid down at the start of the semester, of which, most were met. These goals included: 1) Determination of angle precision for each of the four axes on the kappa goniometer. 2) Design of adapter plates with allowance for backlash setting using pre-determined gear ratios and stepper motors to mount to a kappa goniometer. 3) Review of these designs and prints with Gerry Becker, a UWM machinist, and have him begin production and fabrication of these parts. 4) Install adapter plates, attach gears and stepper motors, and begin instrument alignment 5) Author and present a poster to be given at the SRMS-5. 6) Derive rotational matricesfor existing four circle instrument. Do the same for a five circle instrument using the future addition of a detector arm as the fifth axis of rotation. Investigation: Angle Precision Using the gear trains and stepper motors selected during the REU summer 2004 program, angle precisionfor each of the kappa goniometer axes was determined. .go lo .5rev 1) @ : -0 - deg-= 0.00125- (adding 20 & 40 tooth gears) + goniometer step l ~ v step head-axis 1:360 1:Zgearing instrument ratio .go lo .5rev 2) 5 : - 0 - deg-=0.00125- (adding 20 & 40 tooth gears) kappa step 366 & stepblock-axis 1 : % 0 1:2~e&ing instrument ratio 1.8" 1"0-0 lrev 3) SZK : - = 0.0005* (adding 1:10gear reducer) + s t e ~ 360" lorev step 0- block-axis 1:360 l:i0 instrument near-reducer ratio 1.8" 1"0- 0 lrev 4) 28 : -+ =0.00l* (adding 1:10 gear reducer) det ecror step g8 @-3 step i:iso Go instrument gear-reducer raIio
- 2. AdapterPlates The stepper motor mounting hole patterns that will be replacing the original motors do not match the pre-drilledand tapped holes on the kappa goniometer. Therefore, adapter plates were required to accommodate these hole mismatches. The slotted holes, where the stepper motors mount to on both the kappa and phi axes were designed to facilitate backlash settings and adjustment. These drawings required 48 hours of design time and were completed just before spring break started. After careful measurementsand design, the following prints were developed using Autocadsoftware.
- 6. The following entries represent what was accomplished during the remaining 8 weeks of the semester: Review of Prints with Machinist During a meeting with Professor Lyman, Gerry Becker, and myself, it was determined that the adapter plates for both the kappa and phi axes were sufficient and fabrication would begin immediately accordingto the first two prints above. Further inquiry into the two theta and omega axes demonstrated that fabricationof the adapter plates for these two axes may not be the most efficient method. Therefore, it was decided that modification of the pre-existing shaft and motor mount block to accommodatethe hole pattern of the new stepper motor would be acceptable. Shortly thereafter, the material was ordered and machining of the components began. Installation of Parts As of the end of the spring semester, the parts for the goniometer were still in the process of being fabricated. Therefore, these parts will be installed probably during the summer months. Abstract for SRMS-5 The Synchrotron Radiation in Materials Scienceconference sent a request for abstracts of potential presentationsand posters. It was decided that there was enough research completed and there would be enough material to present a poster at the conference. The following is a copy of the abstract submitted to the SRMS-5 acceptancecommittee:
- 7. Appendageof a Two-Degree-of-Freedom Detector with a Conventional Kappa Goniometer R.J. Moriai. P.F.Lyman btrocluctlou A flexible new diffixctaeta a~rongculcutis &scribed. The gco~lletryco~ubiiesthe opclmcs wd accessibilityto the sample of the kapp coufiguration with the ability to move the point detector out of the traditiml fixed scatteringpbne. This extra degree of frrcdom greatly facilitates sample positiouiug for investigationof s~ufacoand nlterfnces. hkthocls and Materials The dctcctor will be nttachcd to a two-degree-of-6ecdonlcircle. which itself will coincide with the verticd axis of an Eurnf Nonius kappa goniomctcr as d o w n in Fig. 1. Thir will create a (3r2d) difiactometer. siuce tbe sample %--illhave thee aud the detector will have two degrees of fsccdon~respectively. Iu nddition to the ring being hee to rotate about kappa's vertical nxis. the detector a m is free to rotate nbo11ta hhorizontal axis a5 sbvu in figure 2. Fu~hmuol.c.besides the iutroduction of this fifth circle. the original sclvomotorsare beiug rylaced with stepper n~oto~sa d appropriategear rrductiola. These compuents allow for mm'c accurateand precise sarllyleploceme~~tmd detection. Preloaded gear wains will rrdwe the baclilash prescut in the origiual &sign. Fig. 1. The 5-circle kappa r-voy d~@acronteter. n ~ i sview represents home posacinort where aN angles are set to Iero and theprimary x-rqv beantpr~oagatestmrnrds the ringcenter. Results I1uplen1entntionof a nvo-dcprcc-of-freedom dctccto~.coinciding with kappa's priu~uyaxis will facilitate detection of out-of. surface-plane rcatt&g vectorr, thereby nwking @&ug incident and or exit angles possible holding the san~ple nmml in the Ilorkoutal plme. This will fi~rtlmreduce the intensity rcqui~rdof the p r h a y x-my sotuce wd fire up all mt~ictiomto the reciprocal Iatticcspace. Disrutsiou Augle calcnlationsof the mtarioualmatrices for each axis along with pmcdurc for in5hmcut aligwcut is c~urcutlyt&lg investigated nud will be documeutcd to allow n ~ t i u c comprlterizedncccss to a~%itrarypoints in reciprocalspace. Fig. 2. The dcfiacrorneter as slronwr irr sconerittg our-ofplane derectiorr mode. This abstract was accepted and was scheduled to be included into the SRMS-5 conference, however funding to help pay for the entrance fee could not be secured. It was unfortunatelydecided that a request be made to withdraw the abstract from the conference and present it once funding could be available.
- 8. Derivation of the Rotation Matricies To begin deriving the rotational matricies for each axis, analysis of coordinate transformationwas done first as a review: CoordinateTransformationfor Counter-Clockwise Rotation Recall: Using the dot product to find componentsin terms of rotated unit vectors. Let the x-y coordinates be the lab frame: The x' component: A A A A i* if = cost? & j* if = cos
- 9. The Y' comDonent: Or to find the fixed basis vectors (lab frame) in terms of the rotating vectors: This result is commonly written in matrix notation: cos6 - sin 6 x' [;]= [sin 6 cos6 ][ y ] Similarly, for clockwise rotation, the componentscan be derived and the following matrix results: [;]= [cosh i n61[ X I ] - sin 6 cos6 y' This developmentcan be extended into the derivation of the 3-d rotational matrix about each axis. We can multiply them together to create any 3-d rotation. We start with rotation about the z-axis. It is essentially the same as that just derived for two axes with an identity transformation for the z-coordinate,since this component will not change during a rotation about the z-axis:
- 10. The next rotation to consider is rotation about the x-axis. This corresponds to rotation of the y-axis into the z-axis. A clever "trick" is to let y take the role of x and z take the role of y (since this is analogous to the derivation of the rotation of x into y): cosw sinw 0 y I= c0;w y' 0 cosw sinw x [:j=[:s i y co;w][:]s Finally, rotation about the y-axis plays the analogousrole of rotation about z by letting z take the role of x, and x take the role of y: z' sin @ 0 cos@ x [;I= :y][+ cos@ 0 - sin @ x [i]=[si.@ I c:4][:j=~e[i
- 11. The usual convention is to rotate about x, then y, then z. Therefore, in this case the composite rotation matrix is: cos@ 0 -sin@ 1 0 I[0 1 0 I[. cosw si:w][;] sin@ 0 cos@ 0 -sinw cosw z c o s ~ sin^ 0 COS@ sin@sinw -sin@cosw = sin^ COSK 0 0 cosw sin w [ 0 0 I s i n , -cos@sinw c o s ~ o s wI cosKCOS @ cos sin@sinw+sin KCOS w sin sin w-cos sin @cosw' -sin KCOS@ cosKCOS w-sin Ksin@sinw cosKsinw-sin K S ~@cosw sin @ -cos@sinw cos@cosw
- 12. One of the most efficient methods to relate the relationshipof the real-spacecomposition of a material to its diffraction pattern is through the reciprocal lattice. These must be incorporated into the rotational matrices in order to completely return to real-space. A nice chapter describing the bravais lattice and derivationof each reciprocallattice was found in the appendix of Elements of X-Ray DifSraction, (3rdedition) (hardcover),B.D. Cullity, and investigationof these was done. Further reading was done by reviewing numerous papers written by experts on deriving angle calculations for multi-circleinstruments. A few of these papers reviewedinclude: Acta Cryst. (1967). 22,457-464 [ doi:10.1107/S0365110X67000970] Angle calculationsfor 3- and 4-circle X-ray and neutron diffractometers W. R. Busingand H. A. Levy J.Appl. Cryst. (1993). 26,706-7 16 [ doi:10.1107/S0021889893004868] Angle calculationsfor a six-circle surface X-ray diffractometer M. Lohmeier and E. Vlieg Although reading of these papers have begun, derivation of the angle calculations for a 4 or 5 circle instrument was not completeddue to time constraints. Conclusion: In all, satisfaction was found in the amount of work accomplishedduring this semester even though more work remains. Some of this work will probably be performedover the summer months until the finished project is realized.
- 13. Robert Morien 990415960 Undergraduate Research Participation Physics 391 Class # 17227 Sect. 010 Spring 2006