Magnetic symmetry of the plain domain walls. in ferro- and ferrimagnets

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1 Magetic syety of the plai doai walls i feo- ad feiagets. M. Taygi ad O. V. Tycho * Kyiv Taas Shevcheo Natioal Uivesity, Radiophysics Faculty, lushov av., build.5, Kyiv, Uaie, 00 E-ail: b..taygi@gail.co E-ail: pasat@uiv.iev.ua Abstact. Magetic syety of all possible plae doai walls i feo- ad feiagets is cosideed. Magetic syety classes of o 80 degee (icludig 0 degee) doai walls ae obtaied. The doai walls degeeacy is ivestigated. The syety classificatio is applied fo eseach of all possible plae doai walls i cystals of the hexoctahedal cystallogaphic class. PACS: 6.50 Ah, Ch Keywods: doai wall type, syety tasfoatio, agetic syety class, degeeacy. Itoductio The ivestigatio of static ad dyaic popeties [,] of doai walls (Ds) i agetically odeed edia is of cosideable iteest fo the physical udestadig of ediu behavio ad it is also ipotat fo applicatios. Fo sequetial exaiatio of these popeties it is ecessay to tae ito accout the agetic syety [3,4] of the edia. Deteiatio of the D agetic syety allows *Coespodig autho. O.V. Tycho. Addess: 64 Vladiisaya st., Taas Shevcheo Kyiv Natioal Uivesity, Radiophysics Faculty Kyiv, Uaie. Tel/fax: E-ail: pasat@uiv.iev.ua, a.tycho@ail.u

2 to chaacteize qualitatively soe eleets of the D stuctue ad thei chage. The coplete syety classificatio of plae 80 degee Ds (80 0 -Ds) i agetically odeed cystals [5] ad siila classificatio of these Ds with loch lies i feoagets ad feites [6] wee caied out ealie. The plae Ds with width δ [,7] exceedig the chaacteistic size a of a uit agetic cell wee cosideed. Popeties of these Ds i feo- ad feiagets ae descibed by the desity of agetic oet M [8]. Thei syety ca be chaacteized by the agetic syety classes (MSCs) [9] of a cystal cotaiig a D [5]. The buildig of a totality of the MSCs of all possible [] plae (i.e. D with 0 >> δ, whee 0 is the cuvatue adius of the D [5]) Ds i feo- ad feiagets is the pupose of this wo.. Doai wall syety i the agetically odeed edia Let be the uit tie-odd axial vecto [9] alog the agetizatio vecto M: = M / M, whee M is the satuatio agetizatio. The ad ae uit tie-odd axial vectos alog agetizatio vectos M ad M i eighboig doais: = M / M, = M / M. The vectos ad coicide with diffeet easy agetizatio axes (EMA) of the ediu. The agle α betwee these vectos deteies the D type ( α -D): = accos( ) α. A uit pola tie-eve vecto idicates the D plae oal. It is diected fo doai with to doai with. I ode to defie the uified co-odiate syste we itoduce the vectos a ad a as well as the paaetes b = [ ] ad = [ ] [ e ~ e~, e~ ] [ a, a, ] b. The uit vectos of the co-odicate syste O ~ xy ~ ~ z ae chose as x, y z =. Hee the uit vecto ( ) (at b 0 ad = 0 b ) o [ ] with the diectio of vecto ( ) (at 0 a coicides with the diectio of the vecto a (at b = 0 o b 0 ). The uit vecto a coicides b ) o [ ] a (at b 0 ad b = 0 ) o else with a abitay diectio i the D plae ( a at b = b = 0 ). The tie-odd axial vectos ad ae deteied by equalities = ad = + espectively.

3 The MSC (hee is a MSC ube) of a α -D is the agetic syety goup icludig all syety tasfoatios (hee ad heeiafte all taslatios ae cosideed as uit opeatios) that do ot chage the spatial distibutio of agetic oets i the cystal with D [5]. The aboveetioed goup is a subgoup of the agetic (Shubiov s) syety goup of the cystal paaagetic phase [0]. These tasfoatios do ot chage D bouday coditios ad ca be classified by two types [5]. The fist type tasfoatios ( ) g do ot chage the diectios of the vectos 3, ad : ( ) g =, ( ) g =, g ( ) =. The secod type tasfoatios ( g ) chage these diectios: ( ) g, = ( ) g =, g ( ) =. I cofoity with the teiology of [6] the MSC of D bouday coditios is the totality of all tasfoatios of the agetic syety goup of the cystal paaagetic phase that satisfy the etioed six coditios. It is the MSC of the axiu possible syety of a α -D i the give cystal fo a paticula utual oietatio of the vectos, ad. The othe possible MSCs of a α -D with fixed diectios of the vectos, ad esult by eueatio of the subgoups of :, whee P is the MSC P of the cystal paaagetic phase. The utual oietatio of the vectos, ad is deteied by the set of paaetes a ( ), a = ( ), a ( ) = C =, b ad C b, whee tie-eve axial vecto is deteied by equality = [ ] C C. The possible MSCs α -Ds with α 80 0 ae peseted i table. ( 4) of Ds wee foud ealie [5]. All possible MSCs of Fo a cetai α -D the diffeet MSCs ae diffeet goups of agetic poit syety tasfoatios. Thei epesetatios [,5] ae witte i the co-odiate syste O ~ xy ~ ~ z. All epeseted MSCs ae ot iteelated by a otatio ove a abitay agle aoud. Also the aboveetioed MSCs ae ot educed with each othe by uit vectos tasfoatio a a. The possible tasfoatios ( ) g o ( ) g (colu Syety eleets of table ) of α -Ds with α 80 0 ae otatios aoud two-fold syety axes, o, o else, that ae

4 colliea with the uit vectos o a o else a, espectively, eflectios i plaes, o o else 4, that ae oal to the above etioed vectos, espectively, otatios aoud thee-, fou-, sixfold syety axes 3, 4, 6 that ae colliea with the vecto, otatios aoud thee-, fou-, sixfold ivesio axes 3, 4, 6 that ae colliea with the vecto, ivesio i the syety cete ad idetity (syety eleet ). Hee a accet at syety eleets eas a siultaeous use of the tie evesal opeatio R [9]. Fo MSCs with 4 39 ad 5 64 oly geeative syety eleets [] ae epeseted i table. Thee is a coespodece betwee MSCs of 80 -Ds (i.e. at = []), 0 -Ds (i.e. at = [3]) ad α -Ds with o-colliea oietatio of vectos ad [] (heeiafte the last Ds will be aed as α -Ds). The above etioed deteiatios of citeios fo tasfoatios ( ) g ad ( ) g ca be epeseted i aothe idetical fo: ( ) g =, ( ) g =, ( ) g = ad ( ) g, = ( ) g =, g ( ) =. These citeios estict a eseble of MSCs syety tasfoatios fo a abitay α -D. e have = 0 ad = 0 fo 0 - ad 80 -Ds, espectively. A pai fo the above etioed citeios does ot estict the MSCs syety tasfoatios of 0 - o 80 -Ds. Theefoe the agetic syety of α -Ds does ot exceed the agetic syety of 0 - ad 80 -Ds geeically. The MSCs of 80 -Ds ae the MSCs of α - Ds if thei tasfoatios do ot bea the syety of the vecto of the α -D (i.e. these MSCs ust be subgoup of the goup /, whee the ifiite-fold syety axis is colliea with the vecto ). Thee is a aalogy betwee MSCs of 80 - ad 0 -Ds: thei tasfoatios ( ) g ae the sae sice they belog to a subgoup of axial tie-odd vecto syety goup (MSC / ), whee the ifiite-fold syety axis is colliea with MSCs cosist of the tasfoatios o fo 80 - o 0 -Ds, espectively. Theefoe if ( ) g oly the these MSCs ae coo fo 80 - ad 0 -Ds. They ae aed with sig - i colu D cete of table. A covesio of MSC of 80 -D ito MSC

5 of 0 -D is siply a chage of the citeio ( ) g = by the citeio g ( ) 5 =. The tasfoatios of coespodig MSCs of these α -Ds ae diffeet by the substitutio g ( ) ( ) g oly. Theefoe, if a pai of MSCs of 80 -Ds ad a pai of MSCs of 0 -Ds is coected by the above-etioed substitutio, the these MSCs ae coo fo 80 - ad 0 -Ds. As a esult the lists of MSCs of 0 -, 80 - ad α -Ds ae itesected i geeal. Total ube of MSCs of a α -D with abitay α value (icludig α =80 0 ) i feo- ad feiagets is equal to 64. eeal eueatio of MSCs of 80 -Ds cotais 4 MSCs: 4 [5]. This eueatio holds also fo MSCs of α -D with α 80 (MSC ubes ae bold type i colu MSC ube of table ). Thee ae 0 MSCs of α -Ds: 7 3 ad 6 8. The geeal list of MSCs of 0 -Ds icludes all 4 MSCs of table : =, 6 3, R, =, 4, 6, 30, 3, 37, 39 ad 3. Doai wall stuctue The α -Ds with δ >> a i feo- ad feiagets ae descibed by the acoscopic desity of agetic oet M( z ~ ) [5]. The tasfoatios ( ) g ad ( ) ( ) g ( g ; ( ) g ) ipose estictios o the id of coodiate depedece of ( z~ ) copoets ( ( ~ z ) ( ~ z ) ( ~ z ) ( ~ ~ + ~ + ~ z ) = ) i the D volue ad allow to fid this depedece [5]. Fo the deteiatio of the id of coodiate depedece of ( z~ ) copoet of 0 - ad α -Ds fo each MSC (colu Coodiate depedeces of ( z~ ) x y z copoets i table ) the ext ules ae used: a) if a axial tie-odd vecto alog uit vectos e ( x ~, y ~ o z ~ ) is ot a ivaiat of the tasfoatio ( ) g the thee is o copoet ( ~ z ) (figue (-) i colu Coodiate depedeces of ( z~ ) copoets of table ); b) if the axial tie-odd vecto alog e is iveted by the tasfoatio ( ) g the the copoet ( ~ z ) is a odd (A) fuctio of coodiate z ~ ; c) if the axial tie-odd vecto alog e is a ivaiat of the tasfoatio ( ) g the ( ~ z ) is a eve (S) fuctio of coodiate ~ z ; d) if the axial tie-odd vecto alog the tasfoatio ( ) g the tasfoatio ( ) g does ot estict the id of fuctio ( ~ z ) e is a ivaiat of (A,S).

6 If the MCS of a α -D icludes tasfoatios that taspose adjacet agetic doais the this D has a cete of syety [5]. These MSCs eclose the syety tasfoatios g aed by coodiate ~ z = 0 i colu D cete of table. ( ) 6. They ae As i the case of Ds [5], the Ds ca be pulsatig (i.e. D with colliea diectios of vectos M ad M cost i its volue [5]) Ds. The MSCs with =, 6, 9-45, descibe syety of pulsatig Ds oly. I cotast with 80 - ad 0 -Ds thee ae o pulsatig Ds aog the α -Ds, sice α -Ds equie the pesece of two ozeo ( z~ ) copoets. The α -Ds ae otay (i.e. D with M =cost i its volue) o sei-otay [5] Ds oly. Aog otay o seiotay Ds thee ae Ds with oly loch (i.e Ds with M =cost) [,4] ( =7, 8 o 46) ad oly Neel (i.e. Ds with otatio i the plae cotaiig ) [,5] ( =9,, 7 o 47) laws of otatio i thei volue. Cystal agetic odeig is accopaied by phase tasitio ad chage of cystal agetic syety [3]. I a agetically odeed cystal q -ultiply degeeate α -Ds with fixed α ca be obtaied [6], whee q = od( )/od( ). Fuctios od ( ) ad ( ) P P od give the ode [] of the agetic poit goup of the cystal paaagetic phase [9,0] ad of a α -D i this cystal, espectively. These α -Ds have the sae eegy but diffeet stuctues (agetizatio distibutio, plae oietatio, etc.). The iiu value of tie evesal opeatio R. q is i accodace with the ivaiace of eegy fo At epesetatio of the P as the totality of (with fixed value ad diffeet syety eleets oietatios) the lost tasfoatios (ebes of adjacet classes) l g [6,] iteelate the above etioed q -ultiply degeeate α -Ds (i.e. ito aothe). l g opeatio covets a oe of such α -Ds The degeeacy q of a α -D ca be witte i the fo q = qq ( q q ), whee ( )/od( ) q = od is the ube of equal-eegy α -Ds with fixed bouday coditios,

7 q = od( )/od( ) is the ube of possible bouday coditios. Hee ( ) P 7 od is the ode of the poit goup of the axiu agetic syety of the α -D i the give cystal. The α -Ds of MSC 6 (MSC ) have the axiu degeeacy q. Fo 80 - ad α -Ds it is equal to 6 (cystallogaphic class ), 48 (cystallogaphic class 6/) ad 96 (cystallogaphic class 3) i cystals of lowe, ediu ad highe syety sigoies (i cofoity with teiology of []), espectively. The 0 -Ds ae foed i spatially ihoogeeous edia [3]. Coditios of occuece ad existece of such Ds dead to tae ito accout ediu peculiaities. 4. Magetic syety classes of doai walls i hexoctahedal cystals As a exaple let's coside MSCs of all possible Ds i agetically odeed cystals of hexoctahedal class (cystallogaphic poit syety goup 3 i the paaagetic phase [3]). This class is assued to exhibit the lagest vaiety of possible Ds. Futheoe it ecopasses widely ivestigated ad used agetic edia (all cubic syety etals, specifically io ad icel [6], agetic oxides, specifically feites with stuctues of spiel [4] ad gaet [6], peovsite, agetite ad othes). The agetic aisotopy (MA) eegy e K is the ivaiat of the iitial paaagetic phase of cystal. Fo the 3 cystal this eegy is give by e K ( α, α, α3 ) = K s + K p + 3 K s + K 4 sp +..., whee K, K, K 3 ad K 4 ae fist, secod, thid ad fouth MA costats, s = α α + α α3 + α α3, p = α α α3, α, α ad α 3 ae the diectio cosies of [6]. The absolute iiu of this eegy coespods to EMAs. Sigs of MA costats ad elatio betwee thei values deteie EMAs diectios. I the faewo of the ( K, K, K 3 ) appoxiatio the EMAs diectios ca coicide with both high-syetic ad low-syetic cystallogaphic diectios [7]. I the faewo of the twocostat ( K, K ) appoxiatio the EMA diectios ca coicide oly with high-syetic <> o <0> o else <00> lie cystallogaphic diectios at K - K 3 o 0 K - K o else K 0 espectively [,8]. At that 7 0 -, ad Ds o , , ad Ds o else ad

8 80 0 -Ds ae ealized i a 3 cystal, espectively []. The MSCs ad degeeacy q of a α -D bouday coditios with α > 90 ad α 90 ae peseted i tables ad 3 espectively. The ealie obtaied MSCs of eely 80 -Ds (bold type ubes i table ) iclude eleets [5]: = - (,,, ) (, ); =4 - (,,, ); =5 - (,,, ) ; =4 - (,, ), ; =5 - (, ). =3 - (,,, ) (, ); =9 - ( 3, ); =34 - ( 4,, ) 8 ; Oly geeative syety eleets ae peseted fo =9 ad 34. Othe MSCs of tables ad 3 ae peseted i table. I these tables the D plae oietatio is assiged by diffeet Mille idexes h,,l>. A siultaeous chage o egative ad/o cyclic peutatio of all idexes does t chage MSCs. Thee ae o coo MSCs of axiu syetical 80 - ad α -Ds i the 3 cystal. It is coected with the pesece of the tasfoatio ( α -D vecto is chaged by this tasfoatio) i the MSCs of such 80 -D. 5. Coclusios The full agetic syety classificatio of all possible doai walls i feo- ad feiaget cystals icludes 64 agetic syety classes: 4 classes of Ds, 0 classes of α -Ds with 0 0 < α <80 0 ad 4 classes of Ds. Lists of agetic syety classes of all above etioed types of Ds ae itesected i geeal case Ds ca be pulsatig, otay o sei-otay Ds. The α -Ds with 0 0 < α <80 0 ae otay o sei-otay Ds oly. Aog otay o sei-otay Ds thee ae Ds with loch o Neel laws of agetizatio otatio i thei volue. Pulsatig, otay o sei-otay Ds ca have a cete of syety i thei volue. All possible ad α -Ds with 0 0 < α <80 0 have eve degeeacy (its value is betwee ad 96 i geeal case). Magetic syety classes of axiu syetical 80 -Ds do ot eet with such classes of α -Ds with 0 0 < α <80 0 i a 3 cystal.

9 Refeeces 9 [] A. Hubet, Theoie de Doaewade i eodete Mediele (Theoy of Doai alls i Odeed Media), Spige, eli, Heidelbeg, New Yo, 974 A Hubet ad R. Shafe, Magetic Doais. The Aalysis of Magetic Micostuctues, Spige, eli, 998 [] V. oov ad V. Volov, Physics of the Solid State 50 (008)98 [3] L. Shuvalov, Sov. Phys. Cystallog. 4(959)399 [4] L. Shuvalov, Mode Cystallogaphy IV : Physical Popeties of Cystals, Spige, eli, 988 [5] V. ayahta, V. Lvov ad D. Yablosy, JETP 87(984)863 [6] V. ayahta, E. Koteo ad D. Yablosy, JETP 9(986)9 [7]. Lilley, Phil.Mag. 4(950)79 [8] A. Adeev ad V. Macheo, JETP 70(976)5 [9] L. Ladau, E. Lifshitz ad L. Pitaevsii, Couse of Theoetical Physics, vol.8. Electodyaics of Cotiuous Media, Pegao Pess, Lodo, 984 [0] V. A. Kopci, Xubiovsie uppy: Spavoqi po sietii i fiziqesi svostva. istalliqesih stutu [Shubiov s goups: Hadboo o the syety ad physical popeties of cystallie stuctues, i Russia], Izdatel stvo Mosovsogo Uivesiteta, Moscow, 966 A.V. Shubiov ad N.V. elov, Coloed syety, Pegao Pess, Lodo, 964. Tavge ad V. Zaitzev, JETP 3(956)430 []. Vashtei, Mode Cystallogaphy : Syety of Cystals, Methods of Stuctual Cystallogaphy, Spige, eli, 994 [] E. ige, oup Theoy ad its Applicatio to the Quatu Mechaics of Atoic Specta, Acadeic Pess, New Yo, 959 [3] L. Heydea, H. Niedoba, H. upta ad I. Puchalsa, J. Mag. Mag. Mate 96(99)5. R. Vahitov, A Yuaguzi, J. Mag. Mag. Mate. 5-6(000)5 [4] L.Ladau ad E.Lifshitz, Sov.Phys. 8(935)53

10 [5] L. Neel, Copt.ed. 49(955)533 0 [6] A. Paoletti, Physics of Magetic aets, Esevie, Asteda, 978 [7] U. Atzoy ad M. Daiel, Phys. Rev. 3(976)4006 [8] K.P. elov, A.K. Zvezdi, R.Z. Leviti, A.S. Maosya,.V. Mill, A.A. Muhi ad A.P.Peov, JETP 4(975)590

11 Table. Magetic syety classes of the plae α -Ds with α 80. MSC ub. Mutual oietatios of the vectos, ad Syety eleets Coodiate depedeces of ( z~ ) copoets D ( ~ ~ z ) ( ~ ~ z ) ( ~ ~ z ) y x z cete Iteatioal MSC sybol b = a = 0,,, 6 a = a = 0 a = C, (-) (A,S) (-) - (-) (A,S) (-) - 7 a = a = 0,,, (A) (S) (-) ~ z = 0 8 a = a = 0, (A,S) (A,S) (-) - 9 a C = a = b = 0,,, (A) (-) (S) ~ z = 0 0 a = 0, (A) (S) (S) ~ z = 0 a C = a = b = 0, (A) (A) (S) ~ z = 0 a C = 0, (-) (A,S) (A,S) - 3 a = 0, (A) (S) (A) ~ z = 0 6 Abitay (A,S) (A,S) (A,S) - 7 a C = a = b = 0,,, (-) (S) (A) ~ z = 0 8 a C = a = b = 0, (S) (S) (A) ~ z = 0 9 b = b = 0, (-) (-) (A,S) - b = b = 0,,, (-) (-) (A,S) - 4 b = 0 b = 6 b = 0 b = 3, 30 b = 0 b = 3 b = 0 b = 4, 37 b = 0 b = 39 b = 0 b = 6, (,,, ) (, ) 43 b = a = 0 44 b = a = 0,,, 3 (-) (-) (A,S) - 3 (-) (-) (A,S) (-) (-) (A,S) - 4 (-) (-) (A,S) (-) (-) (A,S) - 6 (-) (-) (A,S) - 6 (-) (S) (-) ~ z = 0 (-) (S) (-) ~ z = 0 45 a = b = a = 0,,, (-) (S) (-) ~ z = 0 / 46 a = b = a = 0,,, (S) (S) (-) ~ z = 0 / 47 a = b = 0,,, (-) (S) (S) ~ z = 0 / 48 a b = 0, (S) (S) (S) ~ z = 0 =

12 Table. Magetic syety classes of the plae α -Ds with α 80 (cotiue). MSC ub. Mutual oietatios of the vectos, ad Syety eleets Coodiate depedeces of ( z~ ) copoets D ( ~ ~ z ) ( ~ ~ z ) ( ~ ~ z ) y x z cete Iteatioal MSC sybol 49 a = b = b = 0,,, (-) (-) (S) ~ z = 0 / 50 a = b = b = 0,,, (-) (-) (S) ~ z = 0 5 b = b = 0 (,,, ) (, ) 5 b = b = 0 53 b = b = 0, 54 b = b = 0 6, 55 b = b = 0 3, 56 b = b = 0, 57 b = b = 0, 58 b = b = 0,, 59 b = b = 0 60 b = b = 0 4, 6 b = b = 0, 6 b = b = 0, 63 b = b = 0,, 64 b = b = 0 (-) (-) (S) ~ z = 0 6 (-) (-) (S) ~ z = (-) (-) (S) ~ z = 0 3 (-) (-) (S) ~ z = 0 6 (-) (-) (S) ~ z = (-) (-) (S) ~ z = 0 4/ 4 (-) (-) (S) ~ z = (-) (-) (S) ~ z = 0 4 / 4 (-) (-) (S) ~ z = 0 4 (-) (-) (S) ~ z = (-) (-) (S) ~ z = 0 6/ 6 (-) (-) (S) ~ z = (-) (-) (S) ~ z = 0 6 / 3 (-) (-) (S) ~ z = 0 3

13 Table.. Nube (degeeacy q ) of MSC of bouday coditios of abitay oieted plae α -D ( α > 90 ) i the cubic 3 D plae 80 -D [00], [ 00] cystals at selected doai agetizatio diectios. α -D bouday coditios 80 -D [0], [ 0] 80 -D [], [ ] 0 -D [0], [ 0 ] 09 -D [], [ ] (00) 34 (6) 4 (4) 4 (4) 6 (96) 9 (4) (00) () 4 (4) 4 (4) 3 (48) 3 (48) (00) () () 4 (4) 6 (96) 3 (48) () 4 (4) 4 (4) 9 (8) 6 (96) (48) ( ) 4 (4) 4 (4) 4 (4) 3 (48) (48) ( ) 4 (4) 4 (4) 4 (4) 6 (96) 0 (48) ( ) 4 (4) 4 (4) 4 (4) 3 (48) 0 (48) (0) 4 (4) 3 () 4 (4) 6 (96) 6 (96) (0) 4 (4) 5 (48) 4 (4) (48) 6 (96) (0) () 5 (48) 4 (4) 6 (96) 7 (4) ( 0) 4 (4) () 5 (4) 6 (96) 6 (96) ( 0) 4 (4) 5 (48) 5 (4) 3 (48) 6 (96) ( 0 ) () 5 (48) 5 (4) 6 (96) 7 (4) (hhl) 5 (48) 4 (4) 4 (4) 6 (96) 6 (96) (hh) 5 (48) 5 (48) 4 (4) 6 (96) 6 (96) (h) 4 (4) 5 (48) 4 (4) 6 (96) (48) ( h hl) 5 (48) 4 (4) 5 (48) 6 (96) 6 (96) ( h h) 5 (48) 5 (48) 5 (48) 3 (48) 6 (96) ( h ) 4 (4) 5 (48) 5 (48) 6 (96) 0 (48) (h0), ( h 0) 4 (4) 4 (4) 5 (48) 6 (96) 6 (96) (h0l), ( h 0 l) 4 (4) 5 (48) 5 (48) 6 (96) 6 (96) (0l), ( 0 l) 4 (4) 5 (48) 5 (48) 6 (96) 3 (48) (hl), ( h l), ( h l), ( h l ) 5 (48) 5 (48) 5 (48) 6 (96) 6 (96) 3

14 Table.3. Nube (degeeacy q ) of MSC of bouday coditios of abitay oieted plae α -D ( α 90 ) i the cubic 3 D plae cystals at selected doai agetizatio diectios. 90 -D [00], [ 0 0] α -D bouday coditios 90 -D [0], [ 0] 7 -D [], [ ] 60 -D [0], [0] (00) (48) 9 (4) 7 (4) 6 (96) (00) (48) 7 (4) 0 (48) 0 (48) (00) 7 (4) 7 (4) 0 (48) 6 (96) () 3 (48) 6 (96) (48) 0 (48) ( ) 0 (48) 6 (96) (48) 6 (96) ( ) 0 (48) 6 (96) 3 (48) 0 (48) ( ) 3 (48) 6 (96) 3 (48) 6 (96) (0) 7 (4) (48) 6 (96) 6 (96) (0) 6 (96) 0 (48) 6 (96) 0 (48) (0) 6 (96) 3 (48) 9 (4) 6 (96) ( 0) 9 (4) (48) 6 (96) 6 (96) ( 0) 6 (96) 0 (48) 6 (96) 8 (48) ( 0 ) 6 (96) 3 (48) 7 (4) 6 (96) (hhl) 3 (48) 6 (96) 6 (96) 6 (96) (hh) 6 (96) 6 (96) 6 (96) 0 (48) (h) 6 (96) 6 (96) (48) 6 (96) ( h hl) 0 (48) 6 (96) 6 (96) 6 (96) ( h h) 6 (96) 6 (96) 6 (96) 6 (96) ( h ) 6 (96) 6 (96) 3 (48) 6 (96) (h0), ( h 0) (48) (48) 6 (96) 6 (96) (h0l), ( h0 l) 6 (96) 0 (48) 6 (96) 6 (96) (0l), ( 0 l) 6 (96) 3 (48) 0 (48) 6 (96) (hl), ( h l), ( h l), ( h l ) 6 (96) 6 (96) 6 (96) 6 (96) 4