Mathematical Aids

±

g" g]

fr

(14.52)

With ?i = 0, it follow? from equation (14.16) with equation (14.40) that Ρψ(Φ = 0) = j / ^ j ^
(14-53)

We further obtain therefrom with equation (14.38) Jf (0, 0) = if(0, /?) = j / ^ j p «U

(14-54)

With n = 0, if one considers equation (14.40), there results from equation (14.17) Pfm(0)

= (—l)m Ρψ(Φ)

(14.55)

From equation (14.20) there results Pf(n) = ( - 1 ) ' ] / ^ - ^
(14.56)

Correspondingly, one obtains from equation (14.18) Ρψ{π - Φ) = (-lf+m

Ρψ{Φ)

(14.57)

In particular, there follows therefrom P™(1L\ = 0 for

l+ m

odd

(14.58)

With equation (14.38) it follows from equation (14.57) i f (-ft) = hf (π -Φ,β

+ π) = (-If

ψ(Φβ) = ( - 1 ) ' i f (h)

(14.58a)

The recursion formula results from equation (14.23): am+lpm+l{0)

+ ΛΓ

ΡΓ-1(φ)

=

+

J^pm{0)

(14>59)

According to equation (14.38), Af (φ, β) = _ L . P f (φ) [cos mß + i sin m/?)] ]/2π

(14.60)

Mathematical Aids 359 According to equation (14.39), the Ρ^(Φ) are real. The functions $(Φ,β)=*Ρι(Φ) φ(φ,

ß)=-L J/2

(14.61) [φ{Φ, β) + ( - 1 Γ r m ( * . ß)] = - i W ) J/π

<** m/? (14.62)

^"*(Φ, 0) = ^L· [Jf (Φ, 0) - ( - l p r m ( ^ £)] = ^ ^ Γ ( Φ ) ^η mß J/2 J/π (14.63) therefore form a real basis for the spherical harmonics. The functions cos mß are symmetric and the functions sin mß are antisymmetric with respect to a mirror plane at ß = 0. Hence, the real representation of the spheri­ cal harmonics allows one to take a mirror plane at ß = 0 into account by the 'selection rule': k"m&,ß) = 0 14.3·

(14.64)

Fourier Expansion of the Ρ?η(Φ) —, —, — > interchanges the positive X-axis Δ

Δ

a I

with the positive Z-axis. For every Φ it is therefore true that

The addition theorem then yields Ρψη{Φ) =

Σ

Σ

&mni2tPf* I—) eU7t/2Pfa(0) d"*e*"*Ppf* / _ ]

s=-la=-l

\ 2/

\ 2/

inn/2

e

(14.66) ff

If, according to equation (14.16), we substitute 6sa for Pf (0), we thus obtain47 Ρ?™(φ) = e T ( M + n ) Σ Ρ™η{Φ) is thus represented as a Ρ™(φ) =

*ίπ8ΡΓ (·γ! Pfn (-J) β*φ FOURIER

2 1 af^'e"*

(14.67)

series: (14.68)

with the coefficients amns =

jM+ii+2tjMif / i l j p™ Ul\

(14.69)

360

Mathematical Aids

With the real functions Qfn(x) defined in equation (14.3) we thus obtain amns =

grns ffpj . Qns f ^

( 1 4 7 0 )

If we set

^(τ) = <3Γ

(14 71)

·

equation (14.70) can be written in the form af™ = Qf18 · Qf*

(14.72)

18

The coefficients Qf have been tabulated by BUNGE 6 7 for 0 ^ I fg 31 (see also Section 15.1.1). With the definition of the quantities Qf171 (equation 14.3) it follows from equation (14.19) that Qfmn

(_l)i+n Qmn

=

(14.73) 71

The addition theorem (equation 14.8), putting Φ = Φ' = — and taking equa­ tion (14.73) into account, leads to Σ QrQr =
(14.74)

8=-I

Under the considerations of equation (14.73), for afn~s we obtain amn-s =

(_l)m+namns

Correspondingly, for af

mn8

(14.75)

we obtain

afmns = (-l) z +* afn*

(14.76)

According to equation (14.17), it must also be true that anms =

amns =

a-m-ns

(14.77)

By consideration of equation (14.75) we can write ι Ρψη(φ) = afn0 + Σ αψη8[&8φ + (-l)m+n

e~ü0]

(14.78)

For m + n even this gives i

Ρ™(φ) = afw0 + 2 Σ a?n8 cos $Φ

(14.79)

8=1

and for m + n odd Pf»((p) = 2i Σ <*Tns sin s0

(14.80)

Mathematical Aids

361

If we set a>mnO =

azmM*

amnO

Ί

W8

= 2af

<*r° = 0

\ j

for

m + % even

| > J

for

m + w odd

}

MW

α^™ = 2Mjf

(14.81) (14.82) (14.83) (14.84)

there thus result i

Ρ™η(φ) = 2" «z'mn* cos θΦ for m + n even *=o

(14.85)

Ρ™(φ) = 2 " «i mm sin 5Φ for m + n odd

(14.86)

s=l

Similarly to equation (14.68), for the associated

LEGENDBE

_ +i Ρψ(Φ) = Σ α>Γ^*Φ

functions we set (14.87)

5=-Z

Because of equation (14.40), it is then true that amOs =

aQm8 =

i-m Ή

2

am*

(14.88)

Because of equation (14.76), it follows therefrom that af = 0 for l + s odd

(14.89)

Analogously to equations (14.81)—(14.84), we set 1

aj » = 2af* J <*im0 = 0 1 } ^ = %af* J

for

m even

(14.90) (14.91)

for

m odd

(14.92) (14.93)

Equation (14.88) is then also true for the coefficients a'™8. With these coefficients, equation (14.87) transforms into _ ι Ργ{Φ) = Σ
for

m

even

(14.94)

f

m odd

(14.95)

5= 0

_ ι P?(&) = Σ
or

5= 1

Because of equation (14.89), we are required to sum either only over even or only over odd s. The coefficients a ^ h a v e been tabulated (references 125,154).

362

Mathematical Aids

14.4.

Clebsch—Gordan Coefficients

The product of two generalized spherical harmonics can be expressed as a sum in the following manner:

Tfx^{g) · T^(g) = Thereby m = mx + m2;

w

η

Σ (hi***»* I M ( * Α * Λ 1ft»)?7%)

i=\h-h\

— ni + w2

(ΐ4.ββ> (14.97)

The CLEBSCH—GOBDAN coefficients appearing therein are defined in the following manner: (^ft m l m 2 I lm) =

" [ ^ + "+1^1)1^ + ' " ^ ^ + ^ " ^ ^ + ^ - ^

11/2

X(fi + mx)\ ft - mx)! ft + m ft - m2)! (I + m)\ (I - m)\\ m22)! )Hh

xl X

(-if

«I ft + h - I - z)\ (I + z - \ - m2)! ft + m2 - 2)!

(z + l-l2

(1498)

+ m1)\(l1-m1-z)l

In the sum z traverses positive integers greater than m2 + lx — I and Z2 — Z — m1 and smaller than lx + Z2 — i, Z2 + m2 and Zx — mv The case where one of the two indices \ or l2 is equal to 1 is particularly impor­ tant. In this case one obtains the expressions given in Table 14.311. Table 14.3 m1

CLEBSCH—GORDAN COEFFICIENTS FOB lx —

I = l2 +1

|/

(2i2 + 2) (2l2 + 1)

-|/(?2 + ^ + l)(l,--wi + l) \ ft + l)(2fe+l) -i/ft + m + lX^ + w) [/ (2i2 + 2) (2J2 + 1)

1

I = l2 (k 4- m + 1) (i2 - m ) |/ 2^2 + 1) w l/Wfc + 1) j/ttfc -,/(Z 2 -m + l)(^ + m) ]/ 2l2(k + l)

I—k—1 -, /(Z2 + m + 1) ft + ™) |/ 2ί2(2ί2 + 1) - l / * + m) (i2 — m) |/ F «2i 2 + l) , /(Z2 —ra+ 1) (h — w) ]/ 2^(2/, + 1)

The CLEBSCH—GORD AN coefficients possess the following interchange properties: (l2l1m2m1 I Im) = (-1)**+**-* (l1l2m1m2 \ Im) (ft - mm2 I Zx - mx) = ( - 1 / - + « - 1 / ^ ± J : ftZ2 ™ Λ | Im) ftZ2 — m1 — m2 IZ — m) = {—l)l*+l*~l {lyl2m1m2 | im)

(14.99) (14.100) (14.101)

Mathematical Aids

363

Also, the following relation is valid: -Hi

Σ

( ^ 2 m i m 2 | lm) ( ^ 2 m l m 2 | l'm) — ^ΙΙ'

(14.102)

Wli=—Zi

If in equation (14.96) we set nx = n2 = n = 0 and consider equation (14.41), we obtain *£(*) if*(fc) =

2" 1 ((fA"V»i Ito»))i f (*)

(14.103)

l=|lm—111

with

( ( Ζ ^ Λ | Im)) - - L · ^

( 2 Ζ ι +

2

^

2 + 1 )

(fAnyii, | ft») ( ^ 0 0 110) (14.104)

If, in addition, we set m1 — ra2 = m = 0, we obtain

5,(0) *<.(*) = ^Ψ

(*A I») ΡΑΦ)

(14.105)

with

(¥ 8 M)-|/ ( 2 ? 1 t(2?IV 1 ) ( ^ 0 0 l ; o ) 2 14.5.

(14 106)

·

Symmetric Generalized Spherical Harmonics

We assume that in the coordinate system KA a symmetry corresponding to the point group symmetry of the first kind GA exists, which only contains pure ro­ tations gA. Correspondingly, the symmetry GB with the elements gB exists in the second coordinate system. The presence of these symmetries requires for every function of g f(9-9A)=f(9)

(14.107)

f(9B-9)=M

(14.108)

We shall therefore also denote the symmetries GA and C?B as right-handed and left-handed symmetries. (In the case of texture representation we have called them the sample and crystal symmetries.) Equation (14.107) is not generally fulfilled by the generalized spherical harmon­ ics Tfin(g). We therefore introduce new functions 44 : TT(g)=

Σ

Är™Tfn(g)

(14.109)

n——l

and determine the coefficients Afnv so that the Τ'ΠδΟ fulfil equation (14.107). It must thus be true that ΤΓ(9 ■ 9A) =

TTto)

(14.110)

364

Mathematical Aids

With the addition theorem for generalized spherical harmonics we obtain

TTia -9A)=

Σ λγ* Σ Trig) Tfn(gA) = Trig)

n=—l

(i4.ni)

s=—l

Moreover, according to equation (14.109), ^rig)

=

Σ ArvT™ig)

(14.H2),

Since equation (14.107) is true for all g and since the functions Trig) a r e orthogonal for different s9 the coefficients of Trig) in equations (14.111) and (14.112) must agree. It must thus be true that Σ ArvTtnigA)

= Afsv

(14.113)

-
(14.114)

n=— I

It follows therefrom that Σ ^Tnv\Tln\9x)

Herein s can assume values from — I to +1, and gA traverse all elements of the group ÖA. Equation (14.114) thus represents ä system of linear homogeneous equations with the unknowns A^nv. The coefficients of this system of equations are indepen­ dent of the index m. The solutions must also be independent of m in the sense that each solution for a specific m is also a solution for every other m. We can thus omit the index m in the quantities Af1* and obtain as a definition of the righthanded symmetric functions Trig) =

Σ MvTfnig)

(i4.ii5)

where the AY must be so determined that the conditions Σ

ΛΓ[ΤΪΚ(9Α)

- L·] = 0;

9Ά€ GA;

- l ^ s ^ + l

(14.116)

n==— I

are fulfilled. The index v will thereby enumerate in some sequence the linearly independent orthogonal solutions of this system of equations. In general, we will begin the enumeration with v = 1: l ^ v ^

MA(l)

(14.117)

MA(l) naturally depends on the symmetry GA. The values MA(l) for even I and the various symmetry groups have been given in Figure 4.4. The corresponding values for the odd orders are contained in Figure 14.1. However, how one choses the linearly independent solutions from the set of all solutions is arbitrary. In most cases it is immaterial which linearly independent basis we choose for the solutions; it must only be the same within a calculation.

CD oo

SDIUOUUJDLI


luepuadapui Λμυθψί jo jequun|\|

O O U D s t N O O O l D N t N O O D C D s t CNI CNJ r*sj r>j r>j *— «— «— *— «—

25 Bunge, Texture analysis

365

366

Mathematical Aids

The point* on the right side of the function symbol Tjnv(g) will, as has been said, signify right-handed symmetry; further, if different symmetry groups are to be considered simultaneously, these will be distinguished by different numbers of points. For m = 0, equation (14.115) transforms into

Tng)= Σ Ärn*(g)= Σ ΑΓΤ^,Φ) n=—l

-. /

(u.n%)

n—~l

477-

+*

·

T?v(g) = y — - j ^ ^ l f (φ, y)

(14.119)

*f(*.y)=

(14.120)

If we set Σ ΑΓ%(Φ,γ) n=—

I

the functions &[(Φ, γ) are the symmetric spherical surface harmonics, which are invariant with respect to the rotations gA. We thus have for symmetric functions, similarly to equation (14.42),

*<» =■l/sTi*C-"-T)-V'iTi l / s T i * ( Φ · n - T ) - V'sTi*<*·rt

(14 121)

·

;roup GA was assumed to be a pure rotation rotat The symmetry group group for which every element gA further transforms a right-handed coordinate system into such a system. Every arbitrary symmetry group can now be generated from a pure ro­ tation group and an element gA>, which transforms a right-handed coordinate sys­ tem into a left-handed coordinate system. If the symmetry group GA> is not a pure rotation group, we must thus require that the spherical surface harmonics ki(0, γ) are also invariant with respect to the symmetry operation gAf. This can yield a further symmetry condition for the AY in equation (14.120) in addition to those in equation (14.116). Analogously to the right-handed symmetric functions T™v(g), we introduce the left-handed symmetric functions: +i . Tfn(9) = Σ AfTfUg) (14.122) It is true for these that nn(9B · g) = TjTia)

(14·123)

The coefficient Αψμ must fulfil the condition +

Σ Ar[Tr(gB)-dms]

wi=—I

= 0; gBeGB;

- l ^ s ^ + l

(14.124)

Mathematical Aids

367

The index μ again enumerates the linearly independent solutions: 1 ^ μ ^ MB(l)

(14.125)

For n = 0 there results Tf(g) = Tf(0, Ψ2 ~ γ ) = ] / ^ n **'* W ) (14.126) with :

+i

:

2 1 A*m»kf(0ß)

Ιο'^Φβ) =

(14.127)

where k'f(0ß) are symmetric spherical harmonics corresponding to the symmetry group GB. If we assume the group GB to be the same as 6?A, then equation (14.127) defines a set of symmetrical harmonics of symmetry GA, which however, may be dif­ ferent from the one defined in equation (14.120) because of the complex conjugate coefficients Afmtl. According to equation (14.116), these coefficients were defined as the solutions of a homogeneous system of linear equations, where the index μ defines a certain basis within the vector space of all solutions. Because of the completeness of the solution, the symmetric harmonics defined in equation (14.127) can only represent another basis within the-vector space of functions defined ac­ cording to equation (14.120). We can thus change the basis so that the functions equations (14.120) and (14.127) are the same. This requires Af** = i f

(14.128)

μ

The coefficients Αψ can thus be chosen real. This can also be concluded from equation (14.124), the left-sided symmetry condition. If we assume the symmetry group GB to be the same as GA, then equation (14.124) reads Σ

ΑΓ[ΤΓ(9Α)

-
(14.129)

n——l

Now, along with the symmetry operation gA, the inverse operation g^1 also be­ longs to the group C?A. Equation (14.129) must therefore also be fulfilled for this element. If, now, one considers equation (14.21), one obtains Σ

ΛΤ[ΤΓ(9Α)

-
(14.130)

n——l

The quantities Αψ fulfil both equation (14.116) and the corresponding equation with complex conjugate coefficients. The solutions can therefore always be chosen as real. Finally, we set Tf{9)= 25*

Σ

ΣΑψ«ΑΐνΤΓ(9)

(14.131)

368

Mathematical Aids

This can be written Tf{g) =

Σ

Är»Tr(g)

Σ ΜνΤΓ(9)

=

m——l

(14.132)

n——l

The functions Tf(g) therefore have right-handed and left-handed symmetry. The functions belonging to different values v and v are orthogonal. This re­ quires φ TTto) Tfig)

dgr =

i-

l f

T

δνν,

(14.133)

With equations (14.115) and (14.128) one obtains the orthonormal condition for the coefficients: Σ λγλγ'

= δ„>

(14.134)

n——l

If the corresponding condition is also true for the Α™μ> the two-handed symmetric functions thus fulfil the orthonormal condition φ if{g)i?»'*\g)

dgr =—^j


(14.135)

From the addition theorem (equation 14.6) it follows by multiplication with AY and summation over all n that TTv{g-gf)=

Σ

Trig) W )

(i4.i36)

«=-Z

By multiplication with A™p and summation over m there results 2TV'fiO=

Σ

mg)

mg')

(14.137)

s=-Z

If one carries out both at the same time, one thus obtains Tf(g -g')=

Σ

n8(g) W )

(14.138)

s=-l

These formulae are extensions of the addition theorem £o symmetric functions. If one sets g = e (e is the identity rotation with the EULER angles 0, 0, 0) in equation (14.115), one thus obtains ΤΤΐ9κ) = 2 T W =

Σ AfvTr(e)

(14.139)

w=—I

With equation (14.15) it follows therefrom that

τηακ) = frw = *r

( 14 · 140 )

Similarly one obtains T?n(gB) = Τμη(β) = i f

(14.141)

Mathematical Aids

369

Finally, it follows from equation (14.131) that TfdfB · 9A) = Tf{e) =

2* ÄfoÄr

(14.142)

m=— I

If both symmetries are equal, it is thus true, because of equation (14.134), that mgA)

= ff{fi) = δμν

(14.143)

From equation (14.21), it further follows that i f i g r 1 ) = Tf'Hg)

(14.144)

Finally, it follows from equation (14.4) that

n^g) =:n°(g)= H\g) = n°(g) = i

(14.145)

(The different enumeration of the indices m and n on the one hand and μ and v on the other is thereby to be considered; in other words, for I = 0 m and n can only assume the value zero; on the other hand, μ and v can only assume the value 1.) 14.5.1.

Transformation of the Coefficients Afv

The left-side symmetric functions T?n(g) were chosen so that symmetry condition (14.123) is true. The rotation gB is thus an element of the symmetry group of the left-side symmetry (crystal symmetry). A certain orientation of the symmetry elements relative to the crystal coordinate system is thereby assumed. One can require, for example, that the axis of highest multiplicity fall in the Z-direction of the crystal coordinate system. We now consider a different arrangement of the crystal symmetry group, for which the symmetry elements g'B show an orient­ ation rotated about g^1 relative to the crystal coordinate system. These are the rotations t/B^giffB-gr1

(H146)

They form the same symmetry group as the rotations gB. In the case of cubic symmetry the gB can, for example, be chosen so that the Z'-axis of the crystal coordinate system is a fourfold axis, while the g'B can be chosen so that the Z'axis is a threefold axis (Figure 14.2). In place of the functions T?n(g) we then naturally obtain other functions Tißn(g), which are defined similarly to equation (14.122):

^%)=

Σ MmfiTr(g)

m = —I

(14.147)

with other coefficients A'™*1. We seek the relation of the coefficients A'™** of the rotated symmetry with the coefficients Αψμ of the original symmetry. According

370

Mathematical Aids

Figure 14.2 The rotation g19 which transforms the Z'-axis of the crystal coordinate system from a fourfold axis into a threefold axis to equation (14.124), the coefficients A^ fulfil the condition +i · 9B ' 9i) = A?" Σ AptTTtor1

(14.148)

m=—l

By two applications of the addition theorem (equation 14.6) we obtain therefrom

Σ ί?μ Σ Σ τΠυϊ1)

m=—l

p=—lq=—l

T!Q(9B)

T

F(9i) = Α9μ

(14.149)

By multiplication with Tf(g^1) and summation over s there results

Σ Ar Σ Σ τηυτ1) rffee) Σ ?7%i) mgv1) m=—l

p=—lq=—l

μ

6——I

1

= Σ Μ* π(gr )

(14.150)

s=-l

With the addition theorem the summation over s on the left side is equal to Tf(e) = dqr. One thereby obtains

Σ Mmil Σ τΠοϊ1) T[r(gB) = Σ Marter1)

m=—l

p=—l

(14.151)

p=—l

Moreover, according to equation (14.124), the coefficients Af satisfy the condi­ tion

Σ irrr'foB) = M"

(14·152)

For the coefficients Α\μ, there follows directly by comparing equation (14.151) with equation (14.152) :

+1 :

Αξμ= Σ A^Tpfa1)

(14.153)

Mathematical Aids

371

By multiplication with Tf(g1) and summation over s one obtains the solution of equation (14.153) for the Α!{μ

A{» = Σ ΜμΤί(9ι)

(14.154)

Equation (14.154) is the transformation formula for an arbitrary rotation gx of the symmetry relative to the selected coordinate system. If, for example, one selects (14.155) 9l = {45, 54.7°, 0} one thus obtains an arrangement of cubic symmetry, for which a threefold axis [111] falls in the Z'-direction and a twofold [110] in the X'-direction, if in the ori­ ginal arrangement Z' was parallel to [001] and X' parallel to [100]. 14.5.2.

The Fundamental Integral

We introduce in addition to the coordinate systems KA and KB an additional system K, so that its ^''-direction has the polar coordinates Φχγ^ in system Kx and the coordinates Φ Β /?Β * n system K3 (Figure 14.3). The vectors y and h

Figure 14.3 Relation between the sample coordinate system K%9 the crystal coor­ dinate system KA and the variable coordinate system K

372

Mathematical Aids

will also be used in place of these coordinates. The rotation gx = ΙγΑ + —, Φ Α , φ2 transforms 1£Α into Κ. The angle
Σ TTfo)

' TTtei)

(14.156)

8 = - l

With definition (14.1) it follows that Tfn{g)=-

Σ e

2

~

B;

PJ«(

*"'

(14.157)

We integrate with respect to ψ2: 2π

im Β / pmO//fi η / U ~^B) \ 0ρΟη/,Λ χ J nV(AyA+2 2) / ΤΓ(9)
(14.158)

0

2π

f Trig) d
(14.159)

0-77-2

/ TT(9) #

2

= Ö T T T Φ*Φ*Ρ*) W A 7 A )

(14.160)

With the definitions of .the symmetric functions (equations 14.131, 14.120 and 14.127) it follows that 8-7Γ2

:

/ Tfig) άψ2 = — Ξ - kn*Bfo)

WA7A)

(14.161)

The index [yh] on the integral implies that the integration is over all those orient­ ations g for which the direction y of the first coordinate system KA coincides with the direction h of the second coordinate system KB and thus the crystal direction h with the sample direction y. For the completely real functions, equa­ tions (14.36) and (14.37), one obtains

X [ i f W B ) *?(ΦΑ7Α) + * ? - " W B )

Κ*Φ&Χ)\ (14.162)

Mathematical Aids

373

If one also expresses k by the real functions k' and k" (equations 14.62 and 14.63) one thus obtains

i f = - L [ t ? » + «%"»]]

(14.163)

krm = - 4 - ( - l ) w W m - M'/m]

(14.164)

|/2 It thus follows from equation (14.162), for m + n even, / T ; - % ) d ^ 2 = — — (-1)

2

(14.165) and for m -\- n odd, 8π2

/ T ; - % ) d ^ 2 = — — (-1)

*=5+l 2

(14.166) Correspondingly, there results, for m + n even, / ^ ' - ( ^ ^ ^ ( - l )

[yh]

a +

2

+

1

X K ' " W B ) ^(ΦΑ7Α) -

W B / S B ) *Γ"(ΦΑΤΑ>]

(14.167) and for m -\- n odd, 8π2

a=e±i

<4i -(- 1

[yh]

x [*i"W B ) · W A T A ) + vpitofa) *?*(<ΡΑ?Ά)] (14.168) 14.5.3.

Convolution Integrals

Let /(gr) and w(gr) be two functions of arbitrary symmetries

Kg) = Σ crh'(9)

(14-169)

Ζ,μ,ν

äfo) = 2" wf i f (9) V,0

(14-170)

374

Mathematical Aids

We calculate the integral F{g'> g") = Φί& · 9 ·
(14.171)

If we insert the series expansions (14.169) and (14.170), we thus obtain

w - g") = Σ Σ crwr*4 Tfw · g.
(14.172)

With the addition theorem it follows therefrom that W

?") =

Ofuit^Tfig9)

Σ

Tflg") φ T?{g) tifUg) <*<7 (14.173)

1,μ,ν,λ,α,β,8,σ

We express the symmetry of the second factor under the integral by the corre­ sponding symmetry coefficients: F[g\ 9") =

Σ

Ι,μ,ν,λ,α,β 8tatm,n

^ > H W ) trig")

ATAf

φ T?(g) Tf™{g) dg (14.174)

Because of the orthogonality of the generalized spherical harmonics, only the terms with λ = 1; s = m\

σ= η

(14.175)

are different from zero. One thus obtains

*V,0")=

Σ

crw^Trig^mglArÄf^-

Ι,μ,ν,α,β,ΐη,η

(14.176) ul -\- 1

If we carry out the summation over m and n, for the desired integral we thus obtain the expression F(9',9")=

Σ 1,μ,ν,«,β

CrtcFßTp'(g')T?tf')-sAri

a

"Γ

1

(14.177)

We shall consider this integral for some special cases. We first set g" = e

(14.178)

thus considering the integral hg')==tf{g'-g)™*{g)&g

(i4-i79)

With equation (14.142) it follows in this case from equation (14.177) that kg') = Σ ορτρν) Ι,μ,α

(14.180)

Mathematical Aids

375

with the coefficients 0

Γ = ΊΑΓΪ Σ Ofuf^ ZAPÄT

(14.181)

In particular, if the two symmetries (·) and ( · ) are equal, equation (14.179) thus transforms into the following frequently used convolution integral: hg')=ff(9'-9)^*(g)dg

(14.182)

With equation (14.134) it follows in this case from equation (14.181) that

( 14 · 183 )

or = 2^ηπ Σ cr«t~

We return again to the integral of equation (14.171) and consider the special case in which the function w(g) is only different from zero at the symmetrically equivalent points 9=

ft

(14-184)

We assume that the function w(g) is normalized in the customary fashion: j>ü{g)ag= 1

(14.185)

The integral of equation (14.171) then transforms into a sum which naturally, also depends, in addition to g' and g'\ on g0 (one of the equivalent points g·): Ά9', SO> 9") = Φ l to' ·ft· 9")

(14-186)

According to equation (4.19), in this case there result for the coefficients wf0^ Wf«ß

= (21 + 1) Tf(g0)

(14.187)

If we insert this expression in equation (14.177), we thus obtain 4-l/to'-ft-i")= A i=l

Σ

Of Tfig') itftio) i f

to")

(14.188)

Ι,μ,ν,α,β

In particular, therefore, -f Σ hW ■ft· 9") = Σ k*(9') ϊ?(9ο) I f & ΐ=1

to")

(14.189)

«,j8

If the two symmetries (I) and (:) are equal, because of equations (14.142) and (14.134) there results

4- Σ w ■ 9i ■ 9") = Σ rrto')^rto")

(u-uo)

376 14.6.

Mathematical Aids Symmetric Spherical Surface Harmonics

The symmetric spherical surface harmonics k(&>ß)=

Σ Ärm&*ß)

(14.191)

defined by equations (14.118)—(14.121) satisfy the orthonormal condition φ k\{0, β) ψ'(Φ, β) sin Φ άΦ άβ =
(14.192)

The orthonormal condition (equation 14.134) is valid for the coefficients Afv. It follows from the addition theorem (equation 14.136) with m — 0 that Η(Φ'>β') =

Σ

s=-i

ΜΦ>β) ΪΓ(9)

(14.193)

Φ', β' denote the spherical angular coordinates of the direction Φ, ß in the coordi­ nate system rotated through gr 1 . One obtains the same result from the addition theorem of the usual spherical surface harmonics (equation 14.45) by multipli­ cation by Afv and summation over all n. From equation (14.191) it follows with equation (14.54) that ίξ(Φ = 0, β) = j / ^ i Äf

(14.194)

Correspondingly, there results Μ(Φ = π,β) = ( _ l ) ' y ü ± i i *

(14.195)

If we introduce a new coordinate system Θ, ψ (see Figure 5.2), such that its pole Θ = 0 has the coordinates rt = [Φν β^ in the original system Φ, β, we can thus write the function Α£(Φ, β) in this system: Μ{Φ>β)=

Σ n=-l

Αϊ"%(θ,ν)

(14.196)

If we form the integral φ'ϊονι(φ,β)άψ=

Σ

Α'ιηνφ^(θ,ψ)άψ

we thus obtain from equation (14.38) φ %{Φ, β) άψ = ]/2^ Α?νΡ?(Θ)

(14.197)

(14.198)

ν

If we express ΑΊ° according to equation (14.194), we thus obtain φ if (Φ, β) dip = 2π j / ä ^ Μ θ = 0, ψ) Ρ?(Θ)

(14.199)

Mathematical Aids 377 Finally it follows that / k\{0, β) άψ = 2π l / ä ^ H(Ί) F?m If we express the spherical harmonics ^(Φ,β) to equation (14.60), we thus obtain

(14.200) in equation (14.191) according

l l — _ r _ X ΑψΡξ(Φ) + Σ Α[ηνΡγ{Φ) cos nß + i Σ Α'ι'ηνΡγ{Φ) sin ηβ

L

n=l

n=l

(14.201) with real coefficients ΑΊην = [Afv + (-l)n Afnv]

(14.202)

Αϊην = [Afv - ( - 1 ) * ^ Γ η Ί

(14.203)

If the symmetry group contains a mirror plane in the plane β = 0, then the sin terms in equation (11.201) must vanish; thus we have from equation (14.203) Afnv = {-lfAfnv

(14.204)

Thus in this case A'{nv = 0

(14.205)

and equation (14.201) transforms into if(Φ, β) = A? -JLJ?() + - i γ2π

£ i r W )

^

(14.206)

J/2TZ η=ι

We now set Äf · - ^ L r = J3f ]/2π

(14.207)

i f . - 4 - ^ ^ Γ γ2π Equation (14.206) then transforms into i

cos

__

&ί(Φ> I») = Σ Β?νΡ?(Φ) cos n—0

(14.208)

raß

(14.209)

From equation (14.58a) it follows that harmonics of odd order must vanish for all those directions h which are symmetrically equivalent with their opposite direc­ tion kh+i(h) = 0

for

* e q u i v · ~h

(14.209a)

This equivalence holds for all directions h when a centre of inversion is present in the symmetry group. In fact, this is the definition of the centre of inversion.

378

Mathematical Aids

Hence, harmonics of odd order must be zero everywhere, thus leading to the selection rule equation (14.226). The equivalence assumed in equation (14.209a) holds also for all directions perpendicular to a twofold axis (Figure 14.4). Hence, all planes perpendicular to twofold axes must be zero planes to odd harmonics of all orders. In Figure 14.5 these zero planes are shown for the various symmetry groups in stereographic projection.

Figure 14.4 A direction h perpendicular to a twofold axis is symmetrically equivalent with its opposite direction —h 14.7.

The Symmetric Functions of the Various Symmetry Groups

The symmetry properties of the symmetric generalized spherical harmonics are, as we have said, completely determined by those of the symmetric spherical surface harmonics of the non-centrosymmetric groups. The latter ones can be rela­ tively easily surveyed204. We first consider individual symmetry elements in par­ ticular positions. (1) An wth-order rotation axis in the direction Φ = 0. If the function (14.210) Jf(
(14.211)

monoclinic

2

tetragonal

4, 4

hexagonal

6, 6

orthorhombic 222 cubic

23

tetragonal

422

hexagonal 622

trigonal 32

cubic 432

Figure 14.5 The planes perpendicular to twofold axes in the various non-centrosymmetric symmetry groups are zero planes of the odd harmonies of the corresponding symmetry

as

~3

to

380

Mathematical Aids

where μ is a positive or negative integer. Equation (14.211) represents a 'selection rule' for the indices m. The linearly independent functions of this symmetry are contained among the customary functions. They are distinguished from the latter only in that the 'selection rule' for the indices must be followed. If we represent the linearly independent functions in the previously used form *f(
Σ

Αψμ*Τ(Φ,β)

[(14.212)

m=— I

the coefficients in this case thus assume the form AT = < W ,

(14.213)

If the rotation axis is of infinite order, k™(0, ß) thus transforms into itself for every rotation ß, so that we must have m = 0

(14.214)

The rotationally symmetric functions thus have the form $(Φ,β) = ψ==-Ρι(Φ)

(14.215)

(2) A mirror plane perpendicular to the direction Φ = 0. In this symmetry the function must be unchanged if Φ is replaced by π — Φ. According to equation (14.57), this is the case for I + m even. The selection rule is thus I + m = 2V

(14.216)

(3) A mirror plane in the plane β — 0. If in the expression (equation 14.191) for the symmetric functions we group the terms with +m and —ra,we thus obtain Μ{φ,β)=-]^ΛΥΡ1{Φ) ]/2π l 1 · + - = Σ ^ Τ ( φ ) [AfW1? γ2π m=i

+ (-If* Armfle-imß]

(14.217)

The symmetry requires that this expression tranforms into itself if we replace +β by — β. It must thus be true that Afmf> = (-l)mÄfmf

(14.218)

We can so choose the linearly independent functions that in each case only one coefficient is different from zero 4 + ""-=77Τ·4«.; i f 0 = dm0

·<>,-"·» = ( - ! ) " - ^ . y

(14.219) (14.220)

Mathematical Aids 381 We thus have the linearly independent functions kf(0, ß) = - « L . Ί>ϊ(Φ) [eM + e - ^ ] = - L Ργ(φ) cos μβ )/4π ]/π

(14.221)

#(Φ,0)=

(14.222)

* Ρ,(Φ) )/2π

(Note that here another enumeration of the linearly independent functions has been used.) Thus (in contrast to the definition of equation 14.125) 0^ μ^ I

(14.223)

(4) A 2^-fold rotation reflection axis in Φ = 0. The point Φ,β will be transform­ ed into π — Φ, π/η + β by this symmetry operation. If we insert this in equa­ tion (14.38), we thus obtain with equation (14.57) ψ(π

-Φ, — + β\ = (_i)H-m+m/n ^(Φ,β)

(14.224)

We thus obtain the selection rule m = m'n\

I + m\n + 1) = 2Γ

(14.225)

For a twofold rotation reflection axis, i.e. a centre of symmetry, n'=1. There thus results from equation (14.225) 1 = 2Γ

(14.226)

(5) A twofold rotation axis in the direction Φ = 90°, β = 0°. This transforms the point Φ, β into the point π — Φ, —/?. One sees that the condition i f » " = (-1)* 4 + " *

(14.227)

must then be fulfilled. This yields the linearly independent functions ■ίγ(φ, β) = -=,Ρ^(φ) [e^ + (-lf+v e-^] |/4π 1 ^°(Φ,/?) = 7 7 = - Ρ ?Λ( Φ ) ;

j/ΐΓ

? = 2Γ

(14.228) (14.229)

We thus see that, in addition to 'pure' selection rules (equations 14.213, 14.216 and 14.225), we also obtain conditions such that two coefficients with equal but opposite m are different from zero (14.218) and (14.227). The linearly in­ dependent functions in this case thus have the form of either equation (14.62) or equation (14.63). In the case of a real basis for the spherical harmonics, conditions such as, e.g., equation (14.227) thus lead to 'pure' selection rules. We shall therefore group them with the selection rules in the general sense. 26 Bunge, Texture analysis

382

Mathematical Aids

14.7.1.

'Lower' Symmetry Groups (Non-cubic)

Each of the symmetry elements mentioned above by itself indeed represents a symmetry group — i.e. a simply cyclic group. Additional symmetry groups may be generated by combination of the symmetry elements. Combination of the symme­ try elements leads to combination of the corresponding selection rules. All sym­ metry groups thus generated can be satisfied with the normal spherical surface harmonics by satisfying appropriate selection rules for the indices. We shall call these 'lower' symmetry groups. If only these groups existed, we would not need to introduce special symmetric functions. We would simply have to specify the 'selection rules' belonging to each group. The following belong to the lower sym­ metry groups: (1) The groups (i}l having an w-fold rotation axis, which we place in the direc­ tion Φ = 0. They require the selection rule (equation 14.211) m = n- μ

(14.230)

In particular, cylindrical symmetry g ^ , which in this sense is thus a lower sym­ metry, belongs hereto. (2) The group ®w results from (£w, if one further adds a twofold axis in the direction Φ = 90°, β = 0°. The conditions of equations (14.211) and (14.227) are thus to be simultaneously fulfilled. m = n · μ; Afm» = ( - 1 ) 1 Afmii

(14.231)

(3) The groups S 2 ^ have 2n-iold rotation—-reflection axes. They require ful­ filment of condition (14.225). m = m'-n;

I + m'(n + 1) = 2Z'

(14.232)

(4) The groups SnÄ contain, in addition to n-iold rotation axes in the direction Φ = 0, a mirror plane perpendicular thereto. Equations (14.211) and (14.216) must be fulfilled. τη = η-μ;

l + m = 2V

(14.233)

(5) The groups ©wA result from φ^ by addition of a mirror plane perpendicular to Φ = 0. They thus require conditions (14.211), (14.216) and (14.227). ηι = η·μ:

l + m=2V;

Afmtl = (-1)1 Af™"

(14.234)

DV

- (6) The groups 5)wd result from @2w addition of a mirror plane in β = 0 and a twofold rotation axis in Φ = 90°, β = 0. Accordingly, they require conditions (14.218) and (14.225), which leads to m = n - m;

I + m\n + 1) = 2Γ; Arm" = ( - l ) m Afmrt

(14.235)

All lower symmetry groups are thereby exhausted. One thus obtains Table 14.4 for the coefficients Αψμ for some frequently used lower symmetry groups. In the

SYMMETRY COEFFICIENTS Αψμ

Table 14.4

Orthorhombic symmetry £)27i

Tetragonal symmetry %±h \

m

0

±2

±4

±6...

1

1

0

0

0

1

2

°w

0

0

2

0

3

3 ,4

M(l)

0 0

0

1

w 0

FOR SOME LOW SYMMETRY GROUPS

1

w

4

M(l)

Hexagonal symmetry Φ6Λ

0

±4

±8

±12...

1

0

0

0

1

0

0

2

0

3

0

0

4

0

0

·£ 0 0

0 0

1

w 0

1

w

M(l)

Cylindrical symmetry © ^ 0

±1

±2

±3...

1

1

0

0

0

0

2

0

0

0

0

£·

3

0

0

0

0

0

±6

±12

±18...

1

0

0

0

0

°w

•w

384

Mathematical Aids

particularly important case of cylindrical symmetry only one linearly independent function thus results for each order I: Ίο}(φ,β)=Λ^ψι(φ)

(14.236)

γζπ

This is naturally — except for the different designation of the index μ of the sym­ metric functions, compared with the index m of the customary functions — identi­ cal with equation (14.215). 14.7.2.

'Higher' Symmetry Groups (Cubic)

Groups of higher symmetry cannot be satisfied by selection rules of the indices alone. The linearly independent functions of these symmetries can only be re­ presented by linear combinations with coefficients different from zero23'191. The symmetry groups of the tetrahedron, the octahedron and the icosahedron belong to these groups. The icosahedral groups can not occur as crystal symmetries. Instead of higher and lower symmetries, one can thus also divide the symmetry groups into cubic and non-cubic. The diagonal threefold rotation axis, which can not be satisfied by a simple selection rule for the indices, is characteristic of the cubic symmetry groups. If one arranges the cubic symmetries so that the threefold axis falls in the direction Φ = 0, other twofold and fourfold axes will lie diagonally, so that one cannot fulfil all symmetry elements of this group by selection rules in any arrangement. We must therefore write the linearly independent functions of cubic symmetry in the form74 +z

. 1 . Σ Αψ*--=ζ17ηβΡΐι{Φ) (14.237) m——i ]/2π and choose the Afμ so that the symmetry in question is fulfilled. We express the Ρ™(Φ) according to equation (14.87) by a FOURIER series and obtain Ι$(φ9β)=

1

tf(Φ, β) = - = γ2π

+Z

Σ

+Z

.

Σ Αγμαγ*^&φ

m=—is=—i

(14.238)

The threefold axis now requires ίγ(φ = oc, β = 0) = *f (Φ = 90°, β = oc)

(14.239)

By correspondingly renaming the indices, this yields the condition +1

Σ

-fZ

+1

+1

Σ Äfvaf*^801^ Σ Σ ΜμafV^/V**

m=— ls=— I

(14.240)

n=—ls——l

This must be true for all oc, so that coefficients of eWÄ must agree: +i . . +z Σ Α?μα?* = Af Σ »Γ θ ί η π / 2

(14.241)

Mathematical Aids 385 If we set bj=

Σ

afn&nn^

(14.242)

n=— I

we thus obtain the following equation for determination of the coefficients Αψμ: Σ

i f > r - f] = 0

(14.243)

m=— I

Different selection rules are now to be added to this basic condition for the differ­ ent cubic groups. Another straight foreward calculation of the coefficients Αψμ has been proposed by ESLING and BUNGE 1 2 8 . 14.7.2.1. Tetrahedral Groups If one adds a twofold axis in the direction Φ = 0, one obtains the tetrahedral group Ϊ . This is the pure rotation group of the tetrahedron. The twofold axis re­ quires selection rule (1) for m with n = 2. If one further considers equation (14.89), for even I one thus obtains

Σ irt«r - *mM\ =

I = 21' m o == 2m'

m=— I

(14.244)

s = 2s'

and for odd I

Σ Αψμαγ* m=—l

[m = 2m' I = 2V + 1 s = 2s' + 1

(14.245)

The group %d results from %, if one now adds a mirror plane in β = 45°. In addi­ tion to equations (14.244) and (14.245), it must also be true that Afm^

= imAf^ = ( - l ) ^ / 2 Αψμ

(14.246)

The group %h results from % by addition of a centre of inversion, i.e. a twofold rotation — reflection axis. This yields selection rule 4, with n = 1. We must thus have I = 2V. 14.7.2.2. Octahedral Groups One obtains the octahedral group D, if one adds to the threefold axis a fourfold axis in Φ = 0. Because of selection rule (1), m must therefore be a multiple of 4. Now the octahedral group can also be generated by a fourfold axis in Φ = 0 and an additional fourfold axis in Φ = 90°, β = 90°. This fourfold axis requires that the function must be 'fourfold' along β = 0. The coefficients on the left side

386 Mathematical Aids of equation (14.240) must vanish for non-fourfold s. We thus obtain the condition ΣΑψ"αψ*

= 0; m = 4m'

(14.247)

Thereby either 1 = 2Γ; s = 4s' + 2

(14.248)

l=2V+l;

(14.249)

or s = 2s' + l

Finally we turn to the highest cubic symmetry £)A, so that we must add a centre of symmetry to the group £). We thus obtain the additional condition l = 2V

(14.250)

We have thus exhausted all cubic symmetry groups. There remain now only the two icosahedral groups ty and 3)A, which, however, cannot occur as crystal sym­ metries and are therefore not of interest here. 14.7.3.

Subgroups

One turns from a symmetry group to one of its subgroups, when one omits certain symmetry elements, so that the symmetric spherical harmonics of the subgroup have fewer symmetry conditions to fulfil, and the number of linearly independent functions is thus increased. New functions occur, which indeed satisfy the lower symmetry of the subgroup — not, however, the higher symmetry. It is then fre­ quently appropriate to choose the functions occurring so that they are orthogonal to those of the higher symmetry. In the case of lower symmetry groups this is easy to attain. By omission of a selection rule there occur either functions of the form of equation (14.38) or of the form of equations (14.62) and (14.63), which are orthogonal to each other at the outset. If both the group and its subgroups belong to cubic symmetry, they can thus be distinguished only by symmetry elements, which lead to selection rules. The orthogonality of the functions of the subgroups to those of the higher symmetry is thus given in the same way as in the case of functions of the lower symmetry groups. It is a somewhat different matter if we are concerned with a higher symmetry group, while the subgroup is of lower symmetry. The linearly independent func­ tions of the higher symmetry can then only be represented as linear combinations of functions of lower symmetry, which can then naturally not be orthogonal to all functions of lower symmetry, since otherwise all coefficients of the linear com­ bination must vanish. We therefore introduce a new orthonormal basis for the functions of lower symmetry: Jcf(0,ß)=

Σ m=—l

ΑΓ"φ(Φ9β)

(14.251)

Mathematical Aids 387 The k™(0, ß) are functions of lower symmetry, represented by customary spher­ ical surface harmonics and selection rules, and the &f (Φ, ß) functions of the same symmetry but in general representation. The coefficients Αψμ must fulfil the orthonormalization condition Σ

Α?μΑ?μ'

=
(14.252)

m=—l

Equation (14.251) represents an orthogonal transformation in the function space of the subgroup. There are naturally very many such transformations. We now choose one, for which the first M functions agree with those of the higher sym­ metry. I t must then be true that Αψμ = Ä?μ for 1 ^ μ ^ M

(14.253)

where M denotes the number of the linearly independent functions of the higher symmetry. The coefficients A™μ for M < μ ^ M' must then also be determined so that the orthonormalization condition of equation (14.252) is fulfilled. These functions, which then have only the symmetry of the subgroup, are, however, orthogonal to those of the higher symmetry. Thus, for example, the functions of the highest cubic symmetry £)Λ are most simply expressed by tetragonal functions with m = 4m'. It is thus frequently convenient to introduce such a basis among the tetragonal spherical harmonics which is orthogonal to the cubic functions. 14,7.4.

Explicit Representation of Symmetric Generalized Spherical Harmonics

We consider right- and left-handed symmetries which are at least orthorhombic (or higher). This requires twofold rotation axes in the directions Φ = 0 and φ = 90°, γ — 0° in both coordinate systems. According to the first selection rule, equation (14.211), m and n must thus be even, and equation (14.227) requires AfmiA = ( - l ) z Äf™*,

Äfnv = (-l)lAfnv

(14.254)

The symmetric generalized spherical harmonics then depend only on the quantities Smn =

ymn

+

rp-m-n +

^_^l

rpm-n

+

( _ 1 ) Z /yy-fmi]

(14.255)

According to equation (14.34), it follows that Sf1n = — Pfln(0) [cos mcp2 cos ηφχ — sin m
388 Mathematical Aids If we express the Ρψη(Φ) according to equation (14.85) by its EOUEIEB coefficients a'mm a n ( j cQnsiciCT equations (14.76) and (14.77), we thus obtain t gmn __

£

a'mns C Q g 8φ

«=0,2,... I

—

Σ

a m

c o g m

cos s

T

^

CQg

^

n

sm m

9^2 s m ηΨι

&

(14.257)

5=1,3,...

If both symmetries are lower symmetries, the symmetric generalized spherical harmonics are thus given by Tf = smenSr

(14.258)

em and εη are normalization factors, which are defined in the following manner: fl ε

for

m= 0

- = \γ2 ior

(14 259)

m^0

·

The relations given in Table 14.4, \m\ = Ζ{μ-1);

\n\ = Z(v - 1)

(14.260)

exist between m and μ and between n and v, where Z is the multiplicity of the ro­ tation axis in the direction Φ = 0. If the left-hand symmetry is 'high' and the right-hand symmetry 'low', one thus obtains Tf = εη γ2π' Σ Bf^Sf™*

(14.261)

The Bfm,ß are related to the A\m,lx of cubic symmetry according to equations (14.207) and (14.208). The Bfm^ are given in Table 15.2.1 With equation (14.257) and the index n instead of v according to equation (14.260) the cubic orthorhombic functions may be written with

ΤΓ(ΨιφΨ2) = ΡΓ(φΨ2) ,

cos η

Ψι + 9ιη(φΨ2)

1/4,

ΡΪη(Φφ2) = εη ν%π Σ

Β

I

:

ι™,μ Σ

m=0

$=0,2

,

qin(0(p2) = -εη γ2π

1/4

Σ

Β

m=0

:

ι™,μ

sin η

Ψι

(14.262)


Σ

a

'i*m'n's c o s S& s i n

4m

^2

(14.263) (14.264)

«=1,3,...

Hence, the functions Τ?η(φ1Φφ2) are essentially determined by the functions <ΡΪη(Φφ2) = Τΐη(0Φφ2)

(14.265)

qf"(0
(14.266)

Φφ2)

Table 14.5

CUBIC-ORTHORHOMBIC GENERALIZED SPHERICAL HARMONICS FOR I <^ 6

• cos Φ

• cos 2Φ

• cos 3Φ

• cos 4Φ

• cos 5Φ

• cos &Φ

T? =

+0.10741 +0.17900

+0.23868 -0.23868

+0.41768 +0.05966

T? =

-0.16010 -0.26686

-0.21349 +0.21349

+0.37359 +0.05337

• cos 22

T? =

+0.21184 +0.35301

+0.07061 +0.01008

• cos Αφχ • cos 4φχ cos 4^2 • sin 4φ1 sin 4^2

+0.21349

-0.21349 -0.28242 +0.28242

-0.56484 T? =

+0.03452 -0.19013

-0.05003 n*=+0.21013 +0.05482 n* =-0.23021

-0.08069 +0.07252 -0.07252

+0.08701 +0.37702

+0.15952 -0.15952

-0.08505 +0.08505

-0.03003 -0.13007

+0.16511 -0.16511

• cos 2g?j

+0.06028 -0.06028

• cos 4φ1 • cos 4φτ cos 4φ2 • sin 4g?x sin 4g?2

+0.00742 -0.00742

• cos βφ± ' cos 6φ± cos 4φ2 • sin βφχ sin 4ψ2

-0.36025

-0.08006

^6

=

-0.07422 +0.31167

+0.43846 +0.11132 -0.11132

-0.35622

+0.44029 -0.14250 -0.61751

+0.02740 -0.02740 +0.17539

U

• cos 4
+0.32155 -0.04452 -0.19296

+0.29683

+0.05936

• cos 4^2 • cos 2
390 Mathematical Aids From equation (14.257) one deduces the symmetry relation H ΐΐη{ψν Φ, 45 + a) = ( - 1 ) 2 Tf (90° - φν Φ, 45 - *)

(14.267)

from which it follows that it is sufficient to represent the functions in the angular range or

0^^^90°;

0^Φ^90ο;

0 <: ψ2 S 45°

(14.268)

0^^^45°;

0^Φ^90°;

0 ^ φ2 ^ 90°

(14.269)

Finally, the cubic-cubic functions have the form 1/4

^ Γ = 2π Σ

Z/4

.

Σ Bfm>»Bfn>vStm>±n

(14.270)

»ι=0η=0

Hence, they are linear combinations of the cubic-orthorhombic ones Tf =

ζ/4 ]/2π y - Σ Bfn>vT^n

£\n

(14.271)

n=0

Hence, with n being a multiple of 4, it follows from equation (14.267) that Tf( 45 + a) = 27(90° -

Vl,

Φ, 45° - oc)

(14.272)

They can thus be restricted to the same angular ranges given in equations (14.268) and (14.269). Furthermore the cubic-cubic functions are symmetric with respect to φχ and φ2, leading to the symmetry relation Τΐ(ΨιΦφ2)

= Τη<Ρ2Φ<Ρι)

(14.273)

The functions with both sides of löwer symmetry are thus given essentially by expression (14.257), whose coefficients are known, e.g., according to Table 15.1.2. For the cubic-orthorhombic functions — i.e. the functions necessary for the re­ presentation of sheet textures of cubic metals — one obtains, according to equation (14.261), the explicit representation given in Tables 14.5 and 15.5.1 and Figure 16.3.1. Since the orthorhombic symmetry contains the cylindrical symmetry as a special case, the cylindrically symmetric functions T]} and TQ1, which are inde­ pendent of
Representation of the Cubic Spherical Harmonics by Products of Powers of Cubic Polynominals

In order to obtain the cubic spherical harmonics, we started with the functions Ρψ{Φ) cos mß, which are spherical harmonics but do not possess the cubic sym­ metry. We then formed linear combinations of these functions and selected the

H

CD

Θ ^

CO

6«

CM

©

O

O O

jj

i

"S

*QQ £ £ ^H r *

+ TiH

£ 8 8 .S

«f

i > rH

CM (M CO IH ©'

OS CM CM CO iH

+

t> CO !>CM N ©*

O

O P

8 »I i

8 8 .ä

O iO © ©*

o o o O Λ © ©'

+ I+

CO iO © ©*

©

©'

CM CO © CO CO CO © CO CO ©

©

1

TH © TH

+++

©

+ 1+

CO CO CO iO iO GO CM CM © ©# © © ' ©* © '

©

CO

CO ©

CM CM CO CM i-l τΗ ι θ TH © © CM © ©* © *

II

+1 +

CO CO

1

© ©*

© CM

co

+++

CO © ιθ © © © ©©' ©* CO

CM CM 00 TH ©

cxi t^ CO CO H ©

+1©

CO © CM 00 O ©"

++ +

5i

rH<© ::

391

392 Mathematical A ids coefficients so that cubic symmetry is fulfilled. Conversely, one can now proceed from polynomials, which indeed possess cubic symmetry but are not spherical harmonics, and then represent the cubic spherical harmonics as linear combinations of these functions. Let xv x2, #3 be the rectangular coordinates of a point on the unit sphere. It is then true that φ2 = x\ + x\ + 4 = 1

(14.274)

We further set 9?4 = x\x\ + x\x\ + x\x\ φ6 = x\x\x\

(14.275) (14.276)

The functions
*f = Σ ΚφΙφΙ

(14.277)

«,ß

Here the indices oc and β assume all positive integers subject to the condition 4* + 6/? ^ I

·

(14.278)

For I rg 10 there result explicitly K = »4[fy4 - 1] *β = ηβ[23ΐ9?β - 2 % + 2]

(14.279) (14.280)

k8 = η8[δ2ψ6 - 6δφΙ + 18^4 - 1]

(14.281)

*ιο = η10[3δδ3 φ^φ^ - 979<ρ6 - 187<ρ| + 55ςρ4 - 2]

(14.282)

Since for the direction [100] it is true that
(14.283)

one obtains for the normalization factors n± to n10 η± = -]fe4(100) = +0.646360

(14.284)

n6 = + - ί &6(100) = +0.359601

(14.285)

Δ

n8 = —fc8(100)= +0.835193 = +0.531858 1 10 - - -ό-*ιο(100)

,

(14.286) (14.287)

Similar expressions also exist for the odd harmonics (see reference 193). 14.7.6.

Space Groups in the Euler Space

In Section 2.1.1 it was shown that the EULER space (i.e. the representation of the angles in Cartesian coordinates) is three-dimensionally periodic with period 2π in all three directions. Furthermore, there is an m glide plane perpendiEULER

Mathematical Aids

393

cular to Φ at Φ = π with glide components π in both directions φχ and φ2. The elementary cell is thereby subdivided into two equivalent asymmetric units, and the space group is Pn. Every arbitrary orientation distribution function must thus be three-dimensionally periodic and have the space group Pn. If now there exists in the sample coordinate system the symmetry corresponding to group GA and in the crystal coordinate system the symmetry corresponding to group GB, a series of symmetrically equivalent orientations ρΛ will be generated for each orientation g. In many cases (except for cubic symmetry) the transformations of the orientation space which transform the symmetrically equivalent points g^ into one another are linear transformations, which can be represented by symmetry elements in EULER space. There thus result additional symmetry elements in the EITLER space whereby the elementary cell or at least the asymmetric unit is reduced. The sym­ metry can moreover correspond to another space group. Every combination of two points symmetry groups {GA, GB} of the sample and crystal symmetries thus gives rise to a space group 6?E in EULER space: {£A>#B}->#E

(14.288) 15 16

For its derivation, according to BAKER, ' one must first of all consider the sym­ metries induced in EULER space by individual symmetry elements of GA and GB (i.e. simple cyclic groups). The symmetry elements induced in the EULER space by different rotation axes of the sample and crystal symmetries are compiled in T a b l e 14.7

SYMMETRIES I N E U L E R SPACE, GENERATED B Y ROTATIONS O F THE CRYSTAL AND SAMPLE SYMMETRIES. ACCORDING TO B A K E R 1 5 » 1 6

Symmetry element

Equivalent orientation

Symmetry element in O D F

E U L E R angle identity

φ1-\-

Glide plane J_ Φ axis a t φ == 0, π, 2π. Translation components: Δ ^ = π, Δφ2 = π

£7-fold rotation axis parallel crystal z'-axis


Diad axis parallel crystal #'-axis F-fold rotation axis parallel specimen z-axis Diad axis parallel specimen #-axis


π, — Φ, w2-\- π

2π

Φ + π, —φ2

2π

Ύ


Vl>

,Φ,φ2

Φ + π, <ρ2

U n i t cell of ODF reduced 2π t o — in φ22 direction U Glide plane J_ w2 axis a t φ2 = 0. Translation ΔΦ = π Unit cell of ODF reduced 2π to — in φλ direction Glide plane J_ axis a t wx = 0. Translation com­ ponent ΔΦ = π

394

Mathematical A ids

Table 14.7f while the symmetry elements generated by mirror planes are given in Table 14.8. Mirror planes must always occur simultaneously in the crystal and sample systems, since they transform a right-handed coordinate system into a left-handed coordinate system. This must thus always happen simultaneously for both coordinate systems (sample and crystal systems). If one combines the symmetry elements from Tables 14.7 and 14.8 for symmetry groups GA and 6rB,

Table 14.8

SYMMETRIES ΓΝ E U L E R SPACE, GENERATED B Y MIRROR PLANES O F THE CRYSTAL AND SAMPLE SYMMETRIES. ACCORDING TO B A K E R 1 5 » 1 6

Mirror plane J_ crystal axis

r

X' X

— Φ> ~Ψ2 Diad parallel Φ axis at 0, · , 0

Ζ'

— φ ν Φ9π — φ2 Diad parallel Φ π axis at 0, · , — u

Mirror plane J_ specimen axis

Y

π — φΐ9 Φ, — φ2 Diad parallel Φ π axis at — , · , 0 φν π — Φ, π — <ρ2 Diad parallel φ± 71 71

axis at · , — , — 2 2

71

7t

axis at — , — ,

Inversion centre at 0, 0, 0

—φΐ9 π — Φ, φ2 Diad parallel φ2 _ π axis at 0, — , ·

ψΐ9 π — Φ, — φ2 Diad parallel φχ π axis at · , — , 0

<Ρν ~Φ> Ψζ Mirror plane J_ Φ axis at Φ = 0

-

- Φ > —<Ρ2

Δ

Z

π — φ19 π — Φ, <ρ2 Diad parallel φ2 ·

one thus obtains Table 14.9 (due to BAKER) of the space groups and asymmetric units in EITLER space. A graphical representation of these groups 166 is given in Figure 14.6. A complete group-theoretical derivation was given by POSPIECH, 2X 2 2X GNATEK and FICHTNER 2 3 4 . Figure 14.7 shows the space group P— — - y which is induced in

space by trigonal crystal symmetry 3m and monoclinic 2 sample symmetry — for the second definition of the EITLER angles (ιρΘΦ). This m symmetry case was used on natural quartz samples. Figure 14.8 shows the sym­ metry for cubic crystal and orthorhombic sample symmetries (sheet symmetry). This is the most frequently treated symmetry to date. For the cubic symmetry the threefold axis will not be considered, so that for this symmetry actually only the tetragonal subsymmetry will be considered. EITLER

Mathematical Aids Table 14,9

395

SYMMETRIES IN EULEB SPACE, GENERATED BY VARIOUS SYMMETRIES OF CRYSTAL AND SAMPLE SYMMETRY. ACCORDING TO BAKER 1 5 ' 1 6

Specimen symmetry Crystal Ϊ symmetry

2/m

mmm

Ϊ

Ρ Φ η 2±

ΡΦψ22χ

7%, 71

71, 71, 71

Pn 7i,

2ΤΪ,

2/ra

2ΤΪ

Ρ2ιηΦ 71, 71, 2τΐ

mmm

P 2 l φ1 Φ 71, 71, 7t

3m

Ρ2ιηΦ 71, 71,

4/mm

§lmmm

2π/3

2, 2 2, P-i Φ η Φ

Ρ

71, 71, 71

7t, 7t/2, 7t

pfL-ii. Φ φ Φ

PJLJLJL

71, 71/2, 7t

πβ, π/2, π

±

2, 2 2, Ρ— Φ η Φ

2 2 2 Φ φ2 Φ

π, π/2, 2πβ

π/2, π\2, 2πβ

ph.ll.

71, 71, 71/2

Φ φχ Φ π, π/2, π/2

π/2, π\2, π/2

2Χ 2 2 Ρ— Φ ψχ Φ

2 2 2 Φ m Φ

ρ 2 ι Ψι Φ

7t, 7t\2, 7t/ö

Ρ2χ

Ψι

Φ

71, 71 71/2

2Χ 2 2 Φ ψχ Φ π, π/2, π\2

14.7.7.

Φ m Φ

Ρ 2 ι ψχ Φ

71, 71, Ttjo

m 3m

2 2 2Χ Φ φ2 Φ

Cubic

2 2 2 Φ m Φ

π/2, π/2, πβ 2 2 2 Φ m Φ π/2, πβ, π/2

Symmetry

If one considers t h e full cubic symmetry, one must include t h e threefold axis in t h e consideration. Since it is inclined t o t h e selected axes of t h e crystal coordinate system X' = [100], Ύ' = [010], Z ' = [001], its application gives rise to compli-

396 ^φ=0,π

<ί>2

2π ,

2ΤΪ

Φ,

7|φ*ο.π

Φ2 2ττ -

—

1

J

TC 2

Ρη

1 1 Γ τι τι

2 TC

π

π2

2

Ρφη2,

1

^ | Φ=ο,τι

Ψ2

υ

^ Ι φ=0,τι

t

f

L 2TC

2TC

*l — 3 I

t

OO,TC !

t

♦

■ i

—τ-

f φ,—

piliil

Ρ2,ψ,φ

Ρ2,ηφ

2

φ2

—* ^ι

2τυ, U

TC 2

2

Ρ φ Ψ 722Μ

"~1 φ=0,π

φ2

~ Ί φ=0,π

Φ2

φ ηφ

Ψ2 II

7

4-

··♦. ^

*Ί *

I

■it 2

'

t <» f - r "2

i

ρill.

φΨι c

"Ίφ'Ο,τι

Ψ2 — Τ .

» οθ,π

t

? i i —f-Φΐ— o ♦

Φ

" Ί φ=0,π A-JjL ▼ 2

* *Μ

·"♦·· ό ■■·♦■■■ TU

ψ2 2ΤΕ A A Τ " Τθ,τιΤ

Τΐ ^

—

n

TU

τι

φΨ2 Φ

Figure 14.6 Possible space groups in the E U L E R space in the representation according to the International Tablesim as given by B A K E R 1 6

Mathematical Aids

Figure 14.7 Symmetry P symmetry15'16

397

/or trigonal crystal and monoclinic sample

cated non-linear relations between the EULER angles of equivalent points. If we consider the threefold axis in Figure 14.9, we recognize that it forms the straight line A—A'" in the orientation space {Figure 14.10) — i.e. orientations for which the Z = ND direction lies in the [111] direction are represented by points on this line (see reference 232). This line will therefore be transformed into itself by the threefold rotation axis. One further sees that, in general, orientations with the same coordinates Φφ2 must have equal coordinates Φ'φ^ after application of the threefold axis — i.e. a line parallel to ψλ transforms into a similar line, but in another fundamental region defined by the points A—F, so that the axis A—A'" is a pseudo-screw axis (screw axis with distortion of the space). In parti­ cular, the boundary lines of the region AB up to AG for ψ1 = 0 transform into the lines A'B' up to A'G', A " B " up to A"G" and finally A ' " B ' " up to A'"G'", which all have the same projections, namely A ' " B " up to A'"G"'. The cylindrical sur­ faces defined by the lines AB and A ' " B ' " up to AG and A'"G'" thus define regions in the orientation space, which in each case are transformed completely into themselves. Since three equivalent points must lie in each of the four cubic regions of the orientation space of Figure 14.10, one can· thus select the boundaries of 27 Bunge, Texture analysis

398

2

2

2

2

Figure 14.8 Symmetry P i

,

sample symmetry

*-[m'L

for cubic (tetragonal) crystal and orthorhombic Φ m

Φ

r~"

ζ·=[οοι:

Figure 14.9 The threefold axis of cubic crystal symmetry

Mathematical Aids

399

Figure 14.10 The orientation space and the symmetrically equivalent regions for cubic orthorhombic symmetry. After POSPIECH 232

asymmetric regions, as is represented by shading in Figure 14.11. This region, however, contains a singular plane Φ = 0, on which each orientation is represented by a line. One can therefore also choose either of the two other elementary regions of Figure 14.11 as asymmetric units. The course of the boundary lines in the Φ—φ2 plane is shown in Figure 14.12. 27*

ore cos

Figure 14.11 The elementary region of cubic-orthorhombic symmetry

"T

1

1

1

1

1

r

10° 20° 30°

φ40° φ =arctan [1/cos φ ] 2 50°

0°

10°

20° '

30°

40°

50°

60°

70°

80°

90°

—►ΨΣ. Figure 14.12 The course of the boundary lines of the elementary regions for cubicorthorhombic symmetry in the φτ = 0 section

Mathematical Aids 401 14.8.

CLEBSCH—GORDAN

Coefficients for Symmetric Functions

The product of two symmetric generalized spherical harmonics can be expressed as a sum of generalized spherical harmonics of the same symmetry. We therefore set ..

T^Tff*

\h+h\

=

Σ

M(l) N(l)

Σ

ζ=μ 2 _ζ χ |μ=ι

Σ # 1 ^ 2 I ¥} {hhviv* I lv) TJT

ν=ι

(14.289)

We express the symmetric functions by their symmetry coefficients: Jm^

l '

mljma,n1,n2

L\

1%

1%

ίχ

1%

= l'z zi{h^i^\^}{hkviv2\iv}ArArTr

(14.290)

l=\l2— h\ μ,ν ™>,η

If now we transform the left side with the help of equation (14.96) and equate the coefficients of the functions Tf™ on both sides, we obtain μ,ν

=

Σ

( Ά ™ Λ I M (*Α»ι«8 I &») Af^Äf^A^Ä^'

(14.291)

This equation is fulfilled if we set | Ιμ) Α?μ =

ΣΨΜιμζ

2 1 (hh™i™z I M Af^Af**

2" ft^xvg 1 lv) i f = 2" (ii^»i"a I &») i f ' 1 4 ? ' *

(14.292) (14.293)

where m == mx + m2;

n- = wx + w2

(14.294)

If we multiply the first of these equations by Af1**' and sum over all m, because of equation (14.134) we obtain {IMiPz

IW =

+lx Σ

+h . . . Σ (hh™i™2 I &») ^ ^ ^ Γ ·

(14.295)

#1!=—Z x w 2 = — Za

The quantities {1^2μχμ2 I W (which are analogous to the CLEBSCH—GOBDAN coef­ ficients) for the symmetric functions are expressed by CLEBSCH—GOEDAN coeffi­ cients themselves and the symmetry coefficients Αψμ. In the case of lower sym­ metry the latter have the form of the selection rule: Αψμ = δΜμ

(14.296)

so that it is naturally true, as it must be, that {l1l2m1m2 I lm) = (l1l2m1m2 | Im)

(14.297)

402 Mathematical Aids It therefore follows from the interchange relation for the coefficients (equation 14.99) that

CLEBSCH—GOBDAN

{1211μ2μ1 | Ιμ} = (-ljM-*·-* {Ιχ12μλμ2 \ Ιμ]

(14.298)

One further finds that {ß2W2|W=

= Σ

Σ

m=—l

Σ

m2=—lz

(^mm2 \Ιλπι + m2) Af Μ ^ £ + * · *

Σ (-if*+m>

m——lmi=—lt

X l / 2 * + | ( ^ 2 - (w + m2) m2 | Z - m) A^Xf^Af+m^

(14.299)

If we assume it is true that AfmfX = Afmfi

(14.300)

when we replace m2 by — m2, we thus obtain

{ B ^ A l W = ( - i ) i - l / ^ T 2" Σ (-1)" X (^ 2 - (m - m2) - m2 \l - m) Äf^Äf^Äf^^

(14.301)

With equations (14.97) and (14.101), equation (14.301) transforms into

^mlW^-iW-ä^T 2" 2" (-if

{11

X ^2mxm2

| Zm) 4 f * ^ f ^if 1 *

(14.302)

It follows finally for even m2 that

A m I W = (-&-' ]/§ψγ &W* IW

( 14 · 303 )

For Zx ^ 6, l2 ^ 6 one obtains the values given in Table 14.10 for the coefficients of the symmetry £)h. If in equation (14.289) we change to one-sided symmetric functions, we obtain with equation (14.297) .

:

T^TJF*

\h+h\ M(l)

= Σ

Σ {hhwH I ¥) (V2*i*21 &0 Tr

(14·304)

If now we set ^ = w2 = w = 0 and change to the symmetric spherical harmonics, we obtain .

.

|z f +y M{i)

KK = Σ

Σ {{ΙΜιμ* | Ιμ}} *f

(14·305)

Mathematical Aids

403

with t h e coefficients

{{hhw* IW} = γ^ ]/(2?1t2?+2i2) +

1) {ΙΑμιμ21 ΐμ}

(llk 110)

°°

(14.306) For \ ^ 6, l2 ^ 6 one obtains the values given in Table 14.11 for t h e cubic sym­ m e t r y DÄ. Table 14.10

Ij /2 1 = 0

THE COEFFICIENTS {/X lzll\l

Z= 4

I= 6

7= 8

μ} FOR CUBIC SYMMETRY FOR Z1)2 ^ 6

2= 9

Z = 10

? = 12 u= l

μ = 2

4 4 +0.33333 +0.47795 -0.63563 +0.50637 4 6 -0.52888 -0.44330 -0.29693 -0.55282 +0.36059 6 6 +0.27735 -0.36885 +0.04740 +0.61853 0.00000 +0.44482 0.00000 -0.45200 Table 14.11

\

l2 Z = 0

THE COEFFICIENTS {{L l211

?= 4

Z= 6

1= 8

11 μ}} FOR CUBIC SYMMETRY FOR l12 <* 6

J= 9

/ = 10

Z = 12 μ = 1

4 4 +0.28218 +0.16272 4 6 +0.20117 6 6 +0.28218 -0.14517

+0.20117 -0.14517 -0.01618

μ

= 2

+0.19239 +0.09464 0.00000 +0.14204 +0.19745 0.00000 -0.14433 0.00000 -0.18628