APPENDIX II

SELECTED PAGES FROM "CRYSTAL ORIENTATION MANUAL" BYE.A. WOOD

5. Miller Indices, "Forms," Multiplicity

Mil/tr indices, hkl, arc the reciprocals of the intercepts of a face on the crystal axes. Consider the figure in §3. The intercepts of the plane arc 2, 4, 2, naming the axes in the usual order. Since \\'C arc only interested in orientation of the plane. the interccpu I, 2, I, vvhich arc smaller, \Viii describe it just as ,,·ell. Its Miller indices will be I/I, 1/2, 1/1, or I, 1/2, I. Multiplying by 2, to elimi­nate fractions, '"c have (2 12) as the indices of the plane (called "t,\'o, one. t\,•o"). If the plane had been parallel to c, it would be said to meet ' at infinity. Intercepts, 1, 2, co give us MilJef indices (2 i 0), customarily given in paicnthescs. Ncg~tivc Miller indices (bar above number) refer to intercepts at the negative citds of the crystallographic axes.

The [210] direction (square- brackets) is the direction ofa line from the origin to a point whose coordinates arc 2, I, 0. Note that it is only in the case of

<1101 planes cutting equal-. orthogonal axes (cubic crystals ,....-r----,'?l"-"-b or MO plan~ 'of tetragonal crystals), or planes per­

-'

•

,,.. pcndicular to a single axis and para11cl to t\\'O others, that the direction [h,k,11] is normal to the plane (h1k11i)· The figure illustrates this for a (110) plane

'-.. and [I IO] direction in an orthorhombic crystal [110]

(y = 90°,''1' ~ 90°} .

The direction [uvw] of.the line of intersection of any two planes (hkl) and (h1k1/1) is u = k/1 - lk., u = lh1 - h/1,ond w = hk, - kh., conveniently obtained by subtracting the light-diagonal products froJTI the. dark- h k l h k l diagonal products in the accompanying diagram. T.hc result x·x x [uow] is the designation of the %4nt to . which these two .., planes belong. A zone comprises aJl those planes that arc h, k, t, h, k, t. parallel to a given direction: A plane that is common to t\\•O zones may be found from the similar relations}tip h - vw1 - wv1, k ~ wu1 - uw1, I ·= uv1 -

vu1• If (hkl) is parallel to (uvw], uh + ok + lw - 0. The use of these relation· ;hips is illustrated in §18.

A form, {hkl}, curly brackets, is made up of all those faces required by the S)'mmctry, once a given face is n'cntioncd. In order to kno'v ho"' many faces there arc in ;i form (''the 1nultiplicity0

), it is of cour$C nCCC$Sary to kno\v the symmetry of the crystal. E.g., { IOO}.,. = ( 100), (0.10), (00 I), (ioO), (Oio), (OOi).

{100),.~ = (100). (010), (lOO), (Olo).

The planes of a form arc called "erystallographically cqui\'alent" planes. Tltt)' haue tlze saint ph;·sical propertiu.

(1 l 1} bears a similor relation to [ 111 ], referring in class m3m to directions [I 11 J, [ill], [Ill], [Iii] , [iiq, [ iii], (In], [ii"ii. All these di1-ections, being related by symmetry, have the same physical properties.

(8]

.. 8. X-Ray Reflection from Atomic Planes in a

Crystal-Missing Reflections in the Cubic System

c" Cl)

When x rays ( I) and (2) reflect from atomic planes which arc a distance ti apar1, 1hey will be in phase (and therefore nor

1 exti nguished) only ifihe distance traveled by -------~.'-----'--- ray (2) is an integral number ofwa\:elcngths

nJ. = Ztf sin 8 greater than 1he distance traveled by ray (I), so that when it is again traveling with ray (1) it is "in step," i.e., in phase. In the figure, AB = BC, sin 8 - AB/ti, therefore AB - ti sin 8. If n is an integer and i. is the wavelength of the x rays, 2tfsin 8 (AB T BC) must equal nl for "reflection" to take place. If a similar plane of atoms is added, just halfway between 1hese two, the exira path length to it will be just half of ABC, 1he reflected ray will therefore be just out of phase with ray (1) and will cancel ii. For this reason body·cemcred cubic crystals (i.e., crystnls with exactly the same niom or gro11ping of atoms a t 1hc center of the cell as at the corners) have no {100) reflection; in fact, it turns out that 1hcy have only 1hosc reflections for which the sum h - k - I (and therefore h' + k' + ft) is an even number. Note that only 1/8 of each corner atom lies in the unit cell (sec figure, below). There arc thus only two atoms in 1he unit cell of vanadium.

The various cubic lattices for which each reflection is "allowed" arc listed under "lattice" in §23. All reflections arc allowed for a primitive lattice, which lrns a unit cell in which no point is exactly like those a t the corners of the cell. An example of a structure for each 1ype of cubic lanice is sketched below .

Primitiu ( P) (c.~ .. C.Ct; all t'llowcd)

./ f', \ ,, ~ ... ~ ... ,

./

Body-unltrttf ( /) (c:.g. , vanadium;

only h + k + l even)

Faa-etnltrttf (F) Diamond (D) (e.g., copper; (e.g., siliccn;

Jr, k, and I mu.st be facc-<cntcrcd :..11 odd or :ill even) cubic 'atticc)

lattice: The 1erm lattice is used by crystallographers to refer to one of the 14 a,,,.ais space laniees (sec, for example, C. S. Barren, Structure of Mtla/J, 1952). These arc all of 1he three-dimensional rcpca1 pauerm of crystals-every crysi~I structure is made up of units reputed according to one of these pallerns. lne lattice points mark the periodicity of the repetition. They do not neces­sarily mark the positions of atoms.

( 12]

17. How to Take Back-Reflection Laue Photographs Back-rcOcction Laue photographs arc preferred to forward Laue photographs for two reasons: the sample orientation u more easily determined from them and they can be taken with a sample of any thicl<ness and almost any shape .

Clwitt of tubt: The x rays from each tube have a range of wavelengths and vary in intensity with wavelength ~. as shown in the figure. The wavelengths at which the P and x peaks occur are characteristic of the particular target metal used and arc therefore called characttTistic radiation. I The continuous background, by analogy to white light , s with its continuous ra.nge of wavelengths, u called whitt z

w radiation. Laue photographs require white radiation (§18). !

At a given voltage and tube current, the tungsten tube gives the most white radiation, and is therefore best for the Laue photographs. Other tubes are poorer but can be used. With a tungsten tube operated at 40 kv, 25 ma, a specimen to film distance of3 cm, and a collimator open- ~-

•

ing of about 0.02 in., typical exposure time for a back-rcAcction Laue photo­graph would be 25 minutes.

Sptcimtn·to-film distanu: This determines the scale of the photograph. A trans­parent Greninger net for angle measurement (§19) i~ a\·ailable from ::-J.P. Nies, 969 Skyline Drive, Laguna Beach, California. This net u for a specimen-to-film distance of 3 cm. The Bond modification of the Greninger net is printed in §19. This net is for a specimen-to-film distance of5 cm. !fuse is to be made of either of these, the specimen·to·film distance should be appropriately controlled.

A shorter distance results in shorter exposure time and more spots on the film. A longer distance permits more accurate angular measurement (§19) and facilitates study of the texture of the spots whic.h gives some information about crystal perfection (Appendix II).

Undtsirablt botkground: When this results from scattered radiation or Auorescent radiation (if the specimen is just below the tube-target material in the periodic table) it can sometimes be reduced by the insertion of one or two sheets of aluminum foil on top of the film. Kitchen grade is satisfactory but wrinkles must be avoided.

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~ 1fll l.M ·' ''- ..

(} 16.l (b) '"

OritntatiQn of tl1t samplt rtlatict to tltt film: This orientation must be accurately known. The Laue photograph gh·es the orientation of the Cl")"tal lattice of the sample relath·e to the film (a). The orieniation of the physical sample relative to the film (b) must be xnown in order to know the orieniation of the crystal lattice ·relative to the sample (c), which is what is sought. It may be preferable to begin by mounting the cryst:tl on t.he barrel holder (§15). if it is planned to complete the orientation on the goniomeicr (§16} after the preliminary Laue photograph. In that case a V-block will be needed whose orientation relative to the film is accurately known. The film wiU give the orientation of the crystal as it was on the x-ray machine (mark the upper right corner of the film so as not to invert or reverse it). It is necessary to ensure in some way that this orientation is not lost when the crystal is removed from the machine.

... flfi:r·---- ., .. : . -An x-ra;· tuht, a simplt hacJ.-rtfat1ion uut camtra, and a spuimtn on a Bond ham/ holdtr suitably mou111td for a uut photograph

The x ra)'S pass through a hole in the S x 7 in, film which iJ prottttcd by bla.ck paper, chrous;h the collimator which restricts the bra.m, and fall on the spe<i· men, from \Y"hich chq• are diffracted ("rcRectcd") back onto the lilm. Back· reflection Laue cameras arc available rrom commercial supplic.n of x-ray

diffrJ..ction equipment.

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8. How to Interpret Back-Reflection Laue Photographs

Su Appendix /I for illustrations.

X-ray rencction occurs when n). - 2d sin 0 (§7, §8, and §9) where n = an integer, ). - wavelength or x rays in Angstrom units, d - spacing between atomic planes (also in A), and 8 =angle of reflection. The d ~fixed by the crystal. When the crystal is not moving, 0 is fixed. Each plane therefore chooses the right ). from the white radiation for reflection. A plane that happens to reflect radiation of a wavelength that is strong in the incident beam (sec beam spectrum in §17) will give a strong spot even though it is not an "important" plane. If it is due to the characteristic radiation, such a spot will be extremely strong. If a group of symmetrically cqui,·alent planes all reflect the characteristic radiation, then the symmetry axis relating them is parallel to the incident beam. In general, each spot on a Laue photograph comes from a different )., These rtOcctions can be thought of as though the atomic planes were bright facets reflecting the x rays like visible light back to the film. A zone consists of all planes parallel to a single line, the zone axis. Note how the renections from a zone will fall on a curve which will be a straight line if the zone axis is parallel to the / ......-:: ..>--.....:o:-::--film. These curves intersect in reflections from /@ ~ --

. -~--~ ' important planes. Angles between these planes may be used to determine which planes they ........__ '. arc. ZOHt AXIS' ~

Angle m<a.surement: On a Laue photograph this is accomplished with the Cm1ingtr net (§22). This is calibrated to read the angles of normals or planes giving the reflections. Spacing between lines is 2°.

Tiu symmetry of the pattern of reflections from the crystal is of course consistent with the symmetry of the crystal. If the x-ray beam looks down a 4-fold axis of the cry•tal, the Laue photograph will have a 4-fotd axis of symmetry. Since x-ray reflection is the same from either side of a set of atomic planes, the x rays " add a center of S)"mmctry" to the symmetry of the crystal. The symmetry of a crystal with a center of symmetry added is called the !Aue SJmm<try. In the point-group chart (§4) the Laue symmetry of any crystal is shown by the fir<r

[30)

diagram below its point group in the chart that has a center of symmetry (letter c in upper right corn<r).

To bring on important symrrutry axis to till "'"" of till film: When you think you have found such a direction, away from center, it may hclp to attach a short stick to thc specimen with w:tx, about where you believe the axis tu lie. Then tum the specimcn until the axis aims along the beam.

Angle measurements between spots on Laue photographs, together with a knowledge of intcrplanar angles (§2 1 for cubic crystals, §22 for other systems) form a powerful tool for determining crystal orientation. (Sec §20.)

For a fuller discussion of the use of Laue photographs for determining crystal orientation, sec C. S. Barrett, Structurt of Mttals (1952), page 185. Appendix II shows examples of some back-reffection Laue photographs.

•

•

•

01':1 •

•

Bock-rt}ftttion Uiut photograph of a single tungsttn crystal

Beam approximaidy along (001 ]. Specimen-lo-film ditlancc 5 c:m. Spou labeled wilh the ~filler indices o( the planes r~sponsi~c for the re· fleeted beams produc1n1 the spots. The {103) and It 14) >pots were iden­tified from Crcninscr-nct angle meas .. urcmcnts {fl9) and comparison with the cubic angle cable (§21). The (1251 .spou were idcntifi«I (as m.1.ny othcn could be) by ion.i.I rcla1iomhips as rollow>. Spol> (plannj ( 103) and {11 4) belong to a ionc [•«lD] which can be found, as dcs.cribcd in §S, 10 be [li IJ . Similarly, •poi. ( I i4) and (O i3) define zone [di]. The •pot (plane) common lo both thC'i¢ :r:ones can then be found, u dnc:ribcd in §5, to be (2iS). (Noic that a ll si(N may be reverted and still rc(cr to the same pl~cs ot atoms.) Size of original photograph : 4i x 61 in.

[31]

19. The Greninger Net1

The location of each spot on a back-rencction L:>ur photogr:>ph iridic:>tes the orientation of a set of atomic planes relative to that of the direct beam. If Mis the distance along the film from the point where the direct beam passed through it (usually, though not always, precisely at the center of the hole) to the point where the renectcd beam made the spot, and the specimen-to-film distancc:is R, then M{R - tan ( 180° - 28) and the angle between the atomic plane normal and the direct beam is 90 - 8 (sec Fig. I). Clearly, from this a net could be

' .o·-· ,.. ....... ........ , ..

... W(A.SUREO 09Sf Af'fCf.

FROM Cl:HnJt O' Fll.M

• 90+...1

>P---r-~~-'--,.~~~~i

f-- ·--- - -R· - · · -- • 5Pf.CO••(H TO 111..W

CKSTA.HQ

Fig. I

X RAYS

made for any given R which could be placed over the Laue photograph, making possible angular orientation readings without repetitious calculation. The data for constructing such a net were given by A. 8. Greninger in L it­schrifl j/Jr Kristallographie, Volume 91 ( 1935}, page 424, and a copy of his net is given, scaled I : I, in §19. This ntl is for a sptcimm-to-film tlistan&t of 3 cm. It will

of course give wrong readings for any other specimen-to-film distance. l .l\lthough a.n c-fforl has bttn m~dc to reproduce both the Ctcningcr n.ct and the CrtningC"r­

Bond nc1 without distortion, they may be slightly d istorted due to paper shrinkage. for areatcr accuracy. ncu may be drawn dirccdy rrom the equ•tions given in Grcningcr's paper. Chee.le points for the two ncu arc : Crmiit11r nc:l dittancc: from cc:nltt down to meridian 32• - 6.15 <:m, distance from «:ntcr to meridian to•, parallel 28• - ~.07 cm; G11nU.1n-BoNI net distance from the center co the oulctmost point on either che ~-ertical or the horiiontal line through the: CC1'1tCT - 6.'4 cm. A line: at 45• to thac: linc:s should go through grid comcn tO the outermost corner. N. P. Nies, 1495 Coolidge A\'enuc, PMadena, California, sells a Crcningcr net (or a !l cm spccimcn·to-6lm distance that ii not mcaa.urably distorted.

[32)

21. Angles between Planes .

Crystals of the 1n Cubic System (R. M. Bozorth, Phy1. Rto. 26:390, 1925)

\,\,4,1, I A t•,.t,1,1 All v.all.lO o(d~c angln bttwccn (A1.t,11) a.nd (A~.11) 100 /\ 100 o• "°' 110 <)' 90•

111 ,, ........ • 210 264 34' 63,'26" 90•

211 ~·16' 6)·~·

221 •8'11' 70'32'. 110 !8' 26' 11· 1·· 90' 311 2.s•1•· n-27' 320 3.3•41 • )6'19' 90• 321 36' 42' S7't2' 74'30'

' 110 /\ 110 . O' 60' 90' 111 3.)'16' 90• 210 111'26' )o-46' 7!'134' 211 30' s.t•t•• 73'13' 90• 221 19'28' u · 76"22 .. 90• 310 26'34' 47')2' 63"26' 77'3' 311 31 ' 29' &4'46' 90' 320 11•19· .S3'S8' 66·~· 78' 41' 321 1916' 40'S4' )S"28' 67•48• in·

111 A 111 O' 70'32' 210 39'W 7)'2' 211 19'28' 61 .. 52' 90 Tll W <8" s.t'+l' 78')4' 310 43')' &a·w 311 29'30' S8'31' 19'.).8' 320 16'.+8' ao••1·· 321 22' 12' )1') 3' 72' 1' 90•

210 A 210 O' 36')2"' sn· 66'2)' 78"28' 90• 211 24'6' 43•5· 56'47' 79'29' 90• 221 26'34' •1'49' S3'24' 63'26' n•39• 90• 310 8'8' 31•.s1· 4)' 64' )4' n'34' a1 ·s2· 311 19' 17' 47'36' 66'8' e2·1s· 320 ,.,. 29•.45• 41'.S.S' &o-W 68'9' 7)'38' 82')3' 321 11• 1· 3S'l3' S3' 18' 61"26' 70' 1S' 83' 8' 90'

211 A 211 O' '3'33' 48'11' 60' 70'32' 80'24' 221 17' 43' 35'lb' .. ,.,. 6S'S4' 74'12' 82'12' 310 2)"21' 49'48' 58')4' 7)'2' 82' 3)' 311 10'1' 42'24' 60"30' l.S'4.S' 90' 320 25'4' 37'34' .ss•31 • 63'S' 83'30~

321 IO'S4' 29'12' 40' 12' 49'6' 56'$6' 10')<' 77'24' 83'44' 90'

221 A 221 O' 27'16' :SS'S7' 63'37' 83' 37' 90• 310 32°31' i2"27' S8'12' 6$•4' 93•57• 311 25• 1•· 4S•l 7' )9')0' n ·2r 84•t4· 320 22'2i' •2•1a· •9'40' 68'18' 79'2 I' 84°42' 321 11 '29' 21• 1· 16°'42' ~,.·1· 63'33' 7+'30'

79• .... · a .. ·~3·

310 A 310 O' 2s•s1· 36'32' )3'8' n'33· 84°16' 311 17•33· 40°17' W6' 67°3$' 79'1' 90' 320 WU' 37'S2' S2'8' 74'4$' a+•sa· 321 21'"37' 12• 19· 40•29• 47'28' S3°+4' S9'32'

6S' 1s• 1g· 8)'9' 90'

( 41 J

ANGLES 8£TWE£!'l P L AN£$ IN C RYSTALS OF THE CUBIC SYSTEM

ALONC SELECTED ZONES

(LAu.~indu spots u.•11, in gtnrral, be thi strongrr ones) Zone W. (100) Zon< .. ;, (IOI)

l'tom (001) lo (011) "°"' ( l ii) lo (Ill)

(001) /\ (018) (017) (016) (OIS) (029) (014) (072) (013) (02S) (037) (o+9) (012) (0S9) (o+7) (03S) (023) (0S7) (034) (o+S) (OS6) (078) (Oil)

1• 01.s· 8 08 9 27.S II 18.S 12 31.S 14 02 IS S6.S 18 26 21 48 23 12 23 S1.S 26 34 29 03.s 29 +4.S 30 ~ 3' +2 3S 32.S 36 S2 38 39.S 39 48.S 41 11 4S 00

T.,.....i (010) Zoo< a.xis(! 10] Crom (001 ) 10 ( ti I ) (001) /\ (119) 8 S6

(118) 10 01.S (117) II 2S.S (116) 13 16 (llS) IS 47.S (229) 17 26.S (114) 19 28.S (227) 22 00 (113) 2S l+.S (338) 27 S6.S (22S) 29 30 (337) 31 13 (112) 3S 16 (+47) 38 S6.S (33S) 40 19 (223) 43 19 (334) 46 41 (SS6) 49 41 (Ill) S4 ff

Tow• «! [ llO)

(Ill ) /\ (SiS) S' +6' (4HJ 7 19.s (323) 10 01.S (S~S) 12 16.S (747) 13 IS.S (212) IS 47.S (SfS) 19 28.S (313) 22 00 (727) 23 so.s

' . (414) 2S 14.S (SIS) 27 13 (616) 28 32.S (717) 29 30 (818) 30 12 ( IOI) 3S 16

(IOI) /\ (818) S 03 (717) s 46 (616) 6 43.S (SIS) 8 03 (414) 10 01.S (727) 11 2S.S (313) 13 16 (S2S) IS 47.S (212) 19 28.S (747) 22 00 (S3S) 22 S9.S (323) 2S 14.S (434) 27 .16.s (S4S) 29 30 (lit) 3S 16

Zone ax;, ( lit ) (<om ( IOI ) to (0 11) ( IOI) /\ (314) 13 S4

(213) 19 06.S (32S) 23 24.S (112) 30 00 (23S) 36 ss.s ( 123) 40 S3.s (011) 60 00

Note that the sums of indices of two planes (h1 + Ir,, k, + k,. /1 + I,) give the indices of a plane lying between them in the same zone, e.g., (213) lies between (IOI) and (112), but (123) lies between (011) and (ll2).

[•2)

ANCL£S 8E T\'t' £E.:-t SOM E. HJCHER.-lNDE.X PLAN.£$

DI THE CUBIC SYSTEM : h,k1/ 1 AJr.X1/2

014 UIS UIG 017 018 02S 027 029 03• 03S 037 00 OH osi; 057 II< llS 116 117 118 223

(001) (011) (II I)

li.O"l' II 18.5 9 21.S 8 08 7 07.S

21 48 IS )l;.5 12 31.5 36 S2 30 58 23 12 :18 39.S 29 « .5 39 •8.5 35 32.5 19 18.S IS 47.S 13 16 II 2S.S 10 01.S '3 19

'°~ ss· 33 41.5 3S 32.S 36 St 37 S2.5 23 12 29 03.5 32 28.5 8 08

14 02 21 48

6 20.S IS 15.S 5 11.S 9 27.5

33 33.5 3S 16 )G 35 37 37 38 26 30 58

45' 33.5' 47 12.5 •8 22 •9 13 49 52.S 41 22 « 27.5 46 27.5

36 °' 37 " 40 42 35 45.5 38 01.5 35 35.5 36 21 35 15.5 38 SG.5

" 28 43 18.5 " •t.5 11 25

(00 1) (011) (111)

This 1ablc will be of mos1 help if i1 is used together wi1h the stereographic projccrions (§25). For example, no1c 1hat 1hc information (014) A (001) -14° 02', implies, by symmetry, the following addi1ional information : (104) A (001) - 14°02', (T04) A (001) - 14° 02' (plus 1he (Mi} equivalents with (OOT)] and fur1hcr 1hat (OH ) A (010) - 90° - 14° 02', (104) A (100) = 90° - 14° 02', c1c., and further that (041) A (010) = 14° 02', (401) A (100) = 14° 02', etc., and further that (041) A (001) - 90° - 14° 02', (401) A (001) - 90° -14° 02', etc.

Refrrencc: Ang/ts btlwun Plants in Cubic Crystals, R. J. Pcavl<r and J. N. 1.rnu•ky, publishrd by Amer. ln•t. Mer. Eng., 1962, as Inst. Met. Div. Series No. 8.

(43]

(+Ill

25. The Stereographic Projection

tNTROOUCTJON

The stereographic projection provides a useful way of showing the thrcc­dimcnsional relationships among planes and directions in a crystal on our 1w<>-dimcnsional piece of paper.

The first figure on the opposite page shows a cubic crystal centered in a sphere. From the oentcr, a normal to each face is erected and extmded until it mecu the sphere in a point. These points and the normals themselves arc called "poles" of the faces. This assemblage of points is a sphtrical prajtttian of the faces of the crystal. It would look the same even if growth conditions had caused the crystal to be elongated or flattened in one direction . It thus extracts from the crystal the important information concerning its faces, namely, their angular relations to each other.

To transfer this information in useful form to our two-dimensional paper we need one more step. In the second diagram a line connects the (OOi ) point of the spherical projection to each point on the opposite half of the sph<rc. Where these lines cut the plane on which the opposite half of the sphere is based, they produce a sci of poinu which arc the sttrtagraphie projtctian of the planes of the upper hair of the crystal. This is shown in perspective as the stippled plane of the second diagram and, undistorted, as the finished stereographic projection in the third diagram. Comparing this with the first diagram, note that it is as though you were looking down on the top face of the crystal whose pole comes straight up at you in the center of the projection. The facu parallel to your line of sight have poles on the outer edge of the project.ion and the sloping faces have poles part way in, whose angular measurement will be discussed later. Figure 2 is a stereographic projection including many more planes of a cubic crystal.

Now consider the circles that outline the sphere in the first diagram. They arc drawn solid on the near side, dashed on the far side. Each defines a symmetry plane of the crystal. These have been transferred 10 the plane of the stereographic projection by connecting every point along the line with the "bottom point" of the sphere as before. Examine the resulting lines in the finished stereographic projection. Notice that the straight lines arc the projec­tion of the "vertical planes," those planes that meet in the pole of the (001) face. The "horizontal" plane of symmetry, normal to your line of sight, forms the circular boundary of the projection . The "45° planes" form arcs of great circles through the {110) planes.

These arc the planes shown in the stereographic projection of point group o. - m3m on the §4 chart, which also shows the poles of a general form, (hkl) (§5), such as {123). The planes in the upper half of the crystal arc projected as dots. Those in the lower half arc projected upward (by inverting the process just described) as circles. Herc the two coincide because the crystal has a horizontal plane of symmetry.

(49)

Frg 2 Sttttograp!trc pro;tttron of plants cubr.c !JJltm (00/)

[55]

"'

,,,

.,, ,,, "' •

,,. .,

'"' "'

"' '"

•fl

"'

(bl

· 1111 .. , ••• ... . ,.

~I

"' "' ,,, ..

" "'

.,.

"'

"'

., 1•

""

"' ""' ... ••

'f' '?' .~ '='

• #I ••• .,.,

. .,

,., ·~·

.,

,., '"

" .,

m

'" ' ...

,,,

Fig. Stmogrophic projtttion.s of some of lht plan11 in a cubic cry1lal u1twtd along '·[Ill) and (b) (110)

[59)

LAU£ PATT£RNS OF FACt•C£STtR£D AND

80DY•C£STERED CUBIC CRYSTALS

In race-centered cu bic crysials, rcRcc· tio11"will only occur when h, le, and I arc all odd or all even (with icro considered as an even number). Sec §23. Thus, the first-order reAcc1ion from the (103) plane, for example, cannot occur. Laue spots that occur in 1hc position for the ( 103) planes will be due to second-order rcAcc-1ion (with higher even-order rcAections superimposed, resulting from different wavelengths; see §10 and the first para· graph or§J8). Such a spot is bbelcd (206) in the diagram below, to call attention to thU fact, but in the photographs on the following pages the order or the rcAcction will not be indicated. In body-centered cubic crystals, reflection will only occur when 1he sum or the Miller indices is even.

The center regions or the Laue patterns or a body-ccnlcrcd cubic crystal and a race-centered cubic crystal arc shown here. Note 1hat because of the restrictions mentioned above, the zones made up or {hOI) reAcctioru, such as (1 03), extend farthest toward the center or the pattern in the body-centered case, while the zones made up of{M/} rcAcction.s, such as ( 114), extend farthest toward the center of the pattern in the face-centered case. Com­pare Fig. I with Fig. 2 on the following page ..

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F'ig. 2. BoiJ·<tnfntd tu hit tTJlfa/ (lt11t,fS/t1t) Beam tippro:w.imatdy par.Utl to [DOI).

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Fr,z. J. FM~-<Olk'rr4 t11.6i< 07st•I (UJl1lit1) Beam •PPfO.u Matety ~nlkl 10 llll). C.O.. ~re fig . ...

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Fig. 5. B•<(J•ttntertd cubic '1)Jtol ( tungstl11) Beam approJdmatcly pa.raUcl to [110]. The (IOOJ dirc«ion io •5•from (110) tow.,.d (Ii OJ .

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Fig. 6. Bod.)-lt11tmd cu6it<'}ltol (tungsten) Beam approximatcty pa.ralkl to (112). Thi• p.a11etn don not have ;, 1wo-(old iymmctry uis normal IO c.hc film alt.houah at fine gla.n<it it ~pptars to have. The [011 J dirrttion is 30• from (112) toward (i IOJ. The (001) dirl!Ction .. 35'16' fn>m (112) towud (ii2).

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