upper level precession photography and the lorentz-polarization correction. part i
TRANSCRIPT
Upper Level Precession Photography and the LorentzPolarization Correction. PartIRobinson D. Burbank Citation: Review of Scientific Instruments 23, 321 (1952); doi: 10.1063/1.1746266 View online: http://dx.doi.org/10.1063/1.1746266 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/23/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Erratum: Some Comments on UpperLevel Precession Photography Rev. Sci. Instrum. 26, 888 (1955); 10.1063/1.1715337 Some Comments on Upper Level Precession Photography Rev. Sci. Instrum. 26, 612 (1955); 10.1063/1.1715260 Crystal Settings for Upper Level Photography, Precession Method Rev. Sci. Instrum. 25, 928 (1954); 10.1063/1.1771222 Upper Level Precession Photography and the LorentzPolarization Correction. Part II Rev. Sci. Instrum. 23, 328 (1952); 10.1063/1.1746267 Lorentz and Polarization Correction for the Buerger Precession Method Rev. Sci. Instrum. 22, 567 (1951); 10.1063/1.1746005
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THE REVIEW OF SCIENTIFIC
INSTRUMENTS VOLUME 23, NUMBER 7 JULY, 1952
Upper Level Precession Photography and the Lorentz-Polarization Correction. Part 1*
ROBINSON D. BURBANK
K-25 Laboratories, Carbide and Carbon Chemicals Company, Oak Ridge, Tennessee
(Received December 3, 1951)
The precession camera can be used to record the entire reciprocal lattice out to 1 rlu using one crystal orientation. For the data so recorded to be useful it is necessary to have the Lorentz polarization correction in some convenient form. The correction has been computed at about 1300 points in reciprocal space. In the present paper (Part I) the computation is presented as a series of sections showing I/L.p as a function of i; and \ at constant values of T. In the succeeding paper (Part II, by Grenville-Wells and Abrahams) the correction is presented as a function of i; and T at constant values of \, where the sections at constant \ have been obtained by graphical means from the sections at constant T.
INTRODUCTION
T HE precession camera developed by Buerger! has been recognized as a very elegant device for ob
taining unit cell constants and space group data, particularly in the monoclinic and triclinic systems. The instrument has a variety of desirable features in addition to recording an undistorted projection of the reciprocal lattice, many of which have been discussed by Buerger! and by Evans, Tilden, and Adams.2 The intensity data which one can obtain from zero level photographs have been used in a number of crystal structure determinations. However, the precession method has been considered unsatisfactory for recording the intensities of upper levels because of the blind region that occurs in the center of any such level,1·2 In spite of this, one can record the entire reciprocal lattice out to 1 rlu, and in this laboratory we have found it desirable to do so·t
UPPER LEVEL PRECESSION PHOTOGRAPHY
We have been engaged in crystal structure analyses at low temperatures following the techniques of Kaufman and Fankuchen8 ; Abrahams, Collin, Lipscomb, and
* This document is based on work performed for the AEC by Carbide and Carbon Chemicals Company, Oak Ridge, Tennessee.
1 M. J. Buerger, The Photography of the Reciprocal Lattice (ASXRED Monograph No.1, 1944).
2 Evans, Tilden, and Adams, Rev. Sci. Instr. 20, 155 (1949). t We use the symbol rlu as an abbreviation for reciprocal lat
tice unit. 3 H. S. Kaufman and I. Fankuchen, Rev. Sci. Instr. 20, 733
(1949).
Reed4; and Post, Schwartz, and Fankuchen.6 The situation has often been encountered in which a crystal has a strong tendency to grow with a particular crystal axis parallel to the axis of the capillary containing the sample. In such a case we find it reasonable to use this one crystal orientation for all x-ray photography.
The general technique for recording the entire reciprocal lattice may be illustrated by an example. Suppose that the b axis of a crystal is parallel to the sample capillary. We first record the hkO and Okllevels and all their upper levels to a height of 0.35 rlu above the zero levels, using a precession angle of JL = 30° for every level. For the uppermost levels a layer line screen is used which has an annular radius of r,=35 mm or 40 mm. These radii fall within range of the nomogram prepared by Evans, Tilden, and Adams2 for setting the precession camera. Then to record the regions that lie more than 0.35 rlu above the hkO and Okllevels we turn to levels such as h, k, hand 2h, k, h and also to as many of their upper levels of the type h, k, h±n and 2h±n, k, h as are needed. By using relatively few different classes of levels one soon has filled in all the blind spots at low angles and also obtained all the high angle reflections. In general, one must record about as many levels with Mo radiation as would be required with Cu radiation on the Weissenberg goniometer to obtain the same amount of information. However, the same crystal
4 Abrahams, Collin, Lipscomb, and Reed, Rev. Sci. Instr. 21, 396 (1950).
6 Post, Schwartz, and Fankuchen, Rev. Sci. Instr. 22, 218 (1951).
321
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322 ROBINSON D. BURBANK
TABLE 1. Values of B(~, \, T, J.L) for \=0.05 riu, J.L=30°.
Hrlu) 7=00 7=45° 7=90°
0.0781 0.500 0.583 0.666 0.10 0.575 0.571 0.591 0.20 0.658 0.581 0.508 0.30 0.666 0.583 0.500 0.50 0.648 0.578 0.518 0.70 0.610 0.572 0.556 0.90 0.557 0.572 0.609 1.0781 0.500 0.583 0.666
is used throughout, and the relative intensity scale can be held constant on all levels. In addition, the shape of the spots remains practically constant on all photographs and facilitates the visual estimation of intensities. There is a great deal of duplication of spots on different levels which makes it possible to achieve a high degree of internal correlation of the intensity measurements.
THE LORENTZ POLARIZATION CORRECTION
The intensity data are of little use unless they can be corrected easily for the Lorentz and polarization factors. We have attempted to present this complex correction in such a form that others may find it useful to utilize upper level precession photography for special problems where the Weissenberg goniometer is not always applicable.
Waser6 has recently demonstrated that in the precession method the angular velocity with which the reciprocal lattice moves through the sphere of reflection is not constant and equal to the angular velocity of precession, but that it varies with the position of the precession axis. The Lorentz factor was shown to be proportional to the product of two functions:
L(~, .\, T, fJ.) = [ 1 ] n sinT sinv sinfJ. COSfJ.
xL+tan2fJ. :in2(T+'/)) + 1+tan2fJ. :in2(T-'/))]
==x(~,.\, fJ.)Y(~,.\, T, fJ.), (1)
where ~ is the radial, .\ the axial, and T the angular cylindrical coordinates of a reciprocal lattice point P(~, .\, T). The crystal precesses with an angular velocity n at a precession angle fJ.. The angle v is half the opening angle of the diffraction cone, while T is the projection onto the zero level of the reciprocal lattice of the angle enclosed by the incident and the reflected beams. The angle'/) describes the passage of the point P(~, .\, T) through the surface of the sphere of reflection. The angles T and'/) are related to~, fJ., and v by the expression
sin2fJ.+sin2v- ~2 cosT=------
2 sinfJ. sin v
e+ sin2 fJ. - sin2v cos'/)=
6 J. Waser, Rev. Sci. Instr. 22, 563 (1951).
(2)
(3)
and v can be expressed in terms of fJ. and .\ by means of the relation
(4)
Waser7 has shown that to evaluate the Lorentzpolarization correction it is convenient to combine the polarization factor
(5)
with the function x(~,.\, fJ.) of Eq. (1). Then the Lorentz-polarization correction can be expressed as
where
and
l/L.p,,-,A(~,.\, fJ.)B(~, .\, T, fJ.),
A(~,.\, fJ.)= (l/p) (n/Xa, .\, fJ.))
B(~, .\, T, fJ.) = l/Y(~, .\, T, fJ.).
Combining Eq. (5) with the relation
we obtain
(6)
(7)
(8)
(9)
p=t[(8-4.\2+ .\4)+ (2.\2- 4)~2+~4]. (10)
From Eq. (2) we find that
[4~2 sin2fJ.- (sin2v- sin2fJ.- ~2)2J! sinT= . (11)
2 sinfJ. sinv
Thus, from Eqs. (7), (10), (1), and (11) we derive
4 CoSfJ.[ 4e sin2fJ.- (sin2v- sin2fJ.- e)2J A(t .\, fJ.)= . (12)
(8-4.\2+ .\4)+ (2.\2-4)e+~4
We are only interested in evaluating Eq. (6) for a fJ. value of 30°, and if we also hold .\ constant and let only ~ vary then Eq. (12) assumes the form
a[e- (b- e)2J! A (~, .\e, 30°) = ,(13)
c+de+~4
where a, b, c, and d are constants that can be computed from fJ.=30°, .\= .\e, and Eq. (4). From the geometry of upper level recording! we know that the minimum and maximum values that ~ can assume for any level are
~min = sinv- sinfJ., (14)
~max=sinv+sinfJ., (15)
and it can be seen by direct substitution of Eqs. (14) and (15) into Eq. (12) that these are roots of Eq. (12). For ~min~~~~max we have A(~, .\, fJ.) equal to a real positive number, corresponding to the case where Bragg reflection may occur; while for ~> ~max or ~< ~min we have A (~, .\, fJ.) equal to an imaginary number, corresponding to the case where the reciprocal lattice does not intersect the sphere of reflection. We also note that for fJ.= 30° we obtain from Eqs. (14) and (15) the condition that ~max=~min+1.
7 J. Waser, Rev. Sci. Instr. 22, 567 (1951).
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"' 0 ,.., ,.., 0 0
o o
"' 0 ,.., ,.., ci ci
UPPER LEVEL PRECESSION PHOTOGRAPHY. PART
...;
c3 ~
"' 0 "' 0 "' (\j (\j 0 "' 0 0 0 0 0 ,.., 0
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r-ex> m ci
on m ci
"' 0 ~ ~ "' N N 0 "' ci ci ci ci ci ,.., ci
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N c3 ~
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0 0
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ci ci j
323
"' '! 0
Ul m ci
<I' ex> m ci
"' m ci
Ul 0 ci
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<.N t-.:l
- - 2.0 *'" 0.35 I } I } I ) J j j I ) < \ Z I 0.35
0.30] 'T =300 ///////// I I \ \ \ \ \ ~ ~ ~::: 0.25
I 02H // / / / / / \ \ \. ""- ~ ~0.20
J/ / / / / / \ \ \ ~ ~ ~0.15 FIG.3. M 0.15....:i
0.10-; // / / / I \ \. ""- ~ ------ _______ Jt- 0.1 0
:;0 0.05 -; // / / \ \. ""- ~
-------______ ________.....+_ 0.05
0 to
0.95 0.978 0.95 Z
t )10 en 0 Z
t::1
1.8 2.0 to
00.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 c::: 0.35 I J j } ; j J I i t , ( { < I 0.35 :;0
to 0.30-; 'T .. 45 0 / 1/ / / / I \ \ \ ""- '\.. I-- 0.30 ;J>
Z
0.25j // / / / I / \ \ \ \ \ ~ ~ ~0.25 ;;:i
r '~~ // / / / / \ \ \ \ "" ~ \ ~0.20 FIG·14.
M 0.15-; // / / / / \ \ \ \. "'-- ------n-- 0.15
0.10]
0.05 /// ( I \ \ \ \~~:.~: ~-
0.95 0.985 0.95
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UPPER LEVEL PRECESSION PHOTOGRAPHY. PART I 325
vi <.i ~
ttl 0 ttl 0 U'l U'l 0 ttl 0 U'l ~ U'l N N - - 0
'" '" "l N - 0 0 0 0 0 d 0 0 0 0 ci ci ci
<Xl
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... !i .. ~
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326 ROBINSON D. BURBANK
Waser7 has demonstrated thatfor j..I.= 30°, B(~, t, T, j..I.) can be represented by the polynomial
Ba, t, T, 30°) = Kt-t COS2T cos271 - (1/6· 7) sin22T sin2271J (16)
If) 0 In 0 In 0 If)
to an accuracy of better than 0.35 percent. For constant '" '" N "! - - 0 0 ci 0 0 ci ci ci
t and j..I.=30°, Eq. (3) yields a value for 71 of the form
71= cos-I[(~2+e)/ n
'" where e is a constant.
VARIATION OF THE FUNCTION B(~, {,~, 1')
<D The function represented by Eq. (12) corresponds
'" .: to what one would obtain by extending the derivation
<t .., of Evans2 for the zero-level L.p correction to the case of Q <=:
~ upper levels. The additional complexity that is intro-<=: <t 0
duced by the non-uniform angular motion of the recip-u .., cO rocal lattice is represented by Eq. (16). Because in-<n
" elusion of Eq. (16) adds greatly to the labor involved in "E u making the L.p correction, it is worth considering what N '" rn errors would be incurred if B(~, t, T, 30°) was neglected.
'b '" In Table I we list a few values of the function for the
\I case t=0.05 rlu, T=O°, 45°, 90°, and ~ ranging from ::!.
0 "0 ~min=0.0781 rlu to ~max= 1.0781 rlu. We notice that " oj
for T= 45° the function is nearly constant so that the <n
" L.p corrections with and without function B may be > ~
considered as equivalent to within ±1 percent. At '" '" '"
1 0.. ~min, ~=O.30, and ~m= with T=O° or 90° we reach the
0 0.. ;:l
other extremity, and neglect of function B will give an '" ..2 error of approximately ±14 percent. The errors for "
<D .~ other values of ~ and T will fall between ± 1 percent
~ u and ±14 percent, and the average error for the entire ci '" t:
0 level might be estimated at about ±6 percent. The u
<t " function behaves similarly for other values of t, and the 0 0 ·n
average error for an entire reciprocal lattice should be cO N .~ ci about the same as for a single level of the lattice. How-0
oj
<3 ever, one should not overlook the fact that the occur-T-tl renee of very strong reflections in regions ~f maximum " '" error could have a more adverse effect than these '" 0 H average figures might indicate. r-:
I ..., ui COMPUTATION OF l/L.p AND ITS GRAPHICAL
" REPRESENTATION ~
The complete function represented by Eq. (6) with j..I.=30° was computed over a range of ~ values covering
0 the interval from ~min to ~max for t=O.Ol, 0.05, 0.10, 0 0> 0.15, 0.20, 0.25, and 0.35 rlu with T=Oo, 15°, 30°, 45°, " \-- 60°, 75°, and 90° for each value of t. In this way some
1300 points in reciprocal space were evaluated. The relative scale was adjusted to the scale used by Waser7
for the zero level correction which gives a maximum value of unity on the zero level. A set of 49 graphs was
If) 0 If) 2 !:? Q In prepared which gives l/L.p as a function of ~ for
'" '" N 0 constant t and T. The graphs were drawn to a scale d d d d ci d d
... S of 1 rlu = 40 cm and unit value of 1/L. p = 20 cm. From these graphs the ~ values were obtained at which l/L.p is equal to integral~multiples of 0.05. The results of this
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UPPER LEVEL PRECESSION PHOTOGRAPHY. PART I 327
procedure were combined to form a series of seven sections of 1/L.p as a function of ~ and r at constant values of T, where T=O° corresponds to the horizontal direction on a precession photograph.
The sections of 1/L.p at constant T are illustrated in Figs. 1 through 7. The horizontal edge of the figures has the magnitude of 1 rlu along the ~ coordinate. The vertical scales cover the r coordinate from 0.01 to 0.35 rlu. The circular arc that runs from the lower right corner along the right side of the figures is the trace of a sphere of unit radius with its center at the origin of the reciprocal lattice. The useful range of the precession camera is contained within this sphere and consequently the figures have been terminated at unit value of ~. The L.p correction is contoured at intervals of 0.1 and ranges from zero at the edge of the inner blind spot to values greater than 2 for large values of ~ and rand small values of T. It will be noticed that the function changes very rapidly at the edge of the blind spot. It varies even more abruptly in the region near ~max which is not included in the figures. A correction of less than 0.1 will be unreliable unless r is very small and a correction of less than 0.2 will be unreliable at large r values. Thus, if a reflection occurs near the edge of the blind region, it should be recorded on another level where the blind region has different dimensions. On the other hand, the correction will be much more reliable for points near the surface of the sphere of unit radius than is the case for the zero level. This is because ~max= ~min+ 1 for J.L= 300 and the portion of the reciprocallattice contained in a sphere of unit radius never approaches ~max very closely unless a level occurs at a very small r value.
USE OF THE CONTOUR CHARTS
The sections at constant T offer the only means of charting the complete three-dimensional behavior of the
function because of the presence of the blind region. By interpolation between a finite number of sections at constant T the correction can be found for any point in reciprocal space. With a finite number of sections at constant r an interpolation procedure will not cover all points in reciprocal space. In particular, consider two sections at ( and (', where « r". Let r min be the radius of the blind region at ( and let ~"min be the radius of the blind region at (' so that r min < ~"min. It is clear that any point which lies between ( and r" and also between rmin and ~I/min cannot be reached by interpolation between these two sections. The decision to construct sections at constant T was based on these considerations. In practice it has become apparent that the regions which cannot be reached by interpolation between a finite number of sections at constant rare quite rare. A reflection usually can be recorded on more than one reciprocal lattice level with the result that the reflection will usually occur at least once in a region where interpolation between sections of constant r will apply.
In the succeeding paper (Part II) Dr. GrenvilleWells and Dr. Abrahams present a series of sections at constant r which have been obtained by graphical means from Figs. 1 through 7. Use of these sections at constant r is recommended because of their considerably greater convenience.
It is of course assumed that one will also have Waser's chart of the zero leveF at hand to supplement the upper level charts.
ACKNOWLEDGMENT
The writer wishes to thank Dr. Jlirg Waser for permission to read his recent papers dealing with the Lorentz-polarization factor prior to their publication, and he is also indebted to Dr. Waser for helpful discussion concerning this paper.
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