Magnetotransport and magnetocaloric effects in intermetallic compounds - Appendix A

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Magnetotransport and magnetocaloric effects in intermetallic compounds

Duijn, H.G.M.

Publication date

2000

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Citation for published version (APA):

Duijn, H. G. M. (2000). Magnetotransport and magnetocaloric effects in intermetallic

compounds.

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Appendixx A

Tabless belonging to section 4.6; application of the theory of group representations to the

systemm (Hf,Ta)Fe2.

Pi:: x, y, z P2:: -y, x-y, z P3:: -x+y, -x, z P4:: -x, -y, Z+V2 P5:: y, -x+y, z+'/2

P

6

:: x-y, x, z+Vi

P7:: x-y, -y, -z Pg:: -x, -x+y, -z P9:: y, x, -z Pi0:: -x+y, y, -z+Vi

Pn:: x, x-y,-z+

l

/2

P12:: -y, -x, -z+Vi P13 3 Pl4 4 Pl5 5 Pl6 6 P|7 7 Pl8 8 x-y,, -y, Z+V2 -x,, -x+y, z+'/2

y,, x, z+Vi

-x+y,, y, z

x,, x-y, z

-y,, -x, z

Pi9:: x, y, -z+'/z P20:: -y, x-y, -z+'/2 P21:: -x+y, -x, -z+'/2 P22:: -x,-y,-z P23:: y, -x+y, -z P24:: x-y, x, -z

Tablee A.1. The 24 symmetry operations P

g

(g = 1 to 24) of space group P 63/mmc (No. 194) obtained

fromm ref. 4.20.

P. .

P2 2 P3 3 P4 4

p

5 5

Pe e

P

7 7 Pg g P9 9 P10 0 P11 1 P.2 2 Pl3 3 P.4 4 Pl5 5 P.6 6 P.7 7 P.8 8 P.9 9 P20 0 P21 1 P22 2 P23 3 P24 4

11 (1,0,0)

11 (0, 1, 0)

11 (-1,-1,0)

22 (-1,0,0)

22 (0, -1, 0)

2 ( 1 ,, 1,0)

11 (1,0,0)

11 (-1,-1,0)

11 (0, 1, 0)

22 (-1, 0, 0)

2 ( 1 ,, 1,0)

2(0,-1,0) )

22 (-1,0,0)

2 ( 1 ,, 1,0)

2(0,-1,0) )

1(1,0,0) )

11 (-1,-1,0)

11 (0, 1, 0)

22 (-1,0,0)

2(0,-1,0) )

2 ( 1 ,, 1,0)

1(1,0,0) )

1(0,, 1,0)

11 (-1,-1,0)

11 (0, 1, 0)

11 (-1,-1,0)

1(1,0,0) )

22 (0, -1, 0)

2 ( 1 ,, 1,0)

22 (-1,0,0)

11 (-1,-1,0)

11 (0, 1,0)

1(1,0,0) )

2 ( 1 ,, 1,0)

22 (0, -1, 0)

22 (-1,0,0)

2 ( 1 ,, 1,0)

2(0,-1,0) )

22 (-1, 0, 0)

11 (-1,-1,0)

1(0,, 1,0)

1(1,0,0) )

2(0,-1,0) )

2(1,, 1,0)

22 (-1,0,0)

11 (0, 1, 0)

11 (-1,-1,0)

1(1,0,0) )

11 (0, 0, 1)

1(0,0,, 1)

11 (0, 0, 1)

22 (0, 0, 1)

2(0,0,1) )

11 (0, 0, -1)

1(0,0,-1) )

11 (0,0,-1)

2(0,0,-1) )

22 (0, 0, -1)

1(0,0,-1) )

22 (0, 0, 1)

2(0,0,, 1)

22 (0, 0, 1)

11 (0, 0, 1)

1(0,0,, 1)

2 ( 1 , 0 , 0 ) )

2(0,, 1,0)

22 (-1,-1,0)

11 (-1, 0, 0)

11 (0, -1, 0)

1(1,1,0) )

2 ( 1 , 0 , 0 ) )

22 (-1,-1,0)

2 ( 0 , 1 , 0 ) )

11 (-1,0,0)

1(1,1,0) )

11 (0,-1,0)

11 (-1, 0, 0)

1(1,1,0) )

11 (0,-1,0)

2 ( 1 , 0 , 0 ) )

22 (-1,-1,0)

22 (0, 1, 0)

11 (-1,0,0)

1(0,-1,0) )

1(1,1,0) )

2 ( 1 , 0 , 0 ) )

2(0,, 1,0)

22 (-1,-1,0)

22 (0, 1, 0)

2(0,, 1,0)

22 (-1,-1,0)

2 ( 1 , 0 , 0 ) )

11 (0, -1, 0)

1(1,1,0) )

11 (-1,0,0)

22 (-1,-1,0)

2(0,, 1,0)

2 ( 1 , 0 , 0 ) )

1(1,1,0) )

11 (0,-1,0)

11 (-1, 0, 0)

1(1,1,0) )

1(0,-1,0) )

11 (-1,0,0)

22 (-1,-1,0)

2(0,, 1,0)

2 ( 1 , 0 , 0 ) )

1(0,-1,0) )

1(1,1,0) )

11 (-1, 0, 0)

2(0,, 1,0)

22 (-1,-1,0)

2 ( 1 , 0 , 0 ) )

2 ( 0 , 0 , 1 ) )

22 (0, 0, 1)

11 (0, 0, 1)

22 (0, 0, -1)

11 (0, 0, -1)

1(0,0,-1) )

11 (0, 0, -1)

22 (0, 0, -1)

2(0,0,-1) )

1(0,0,1) )

11 (0,0,1)

11 (0, 0, 1)

2(0,0,1) )

22 (0, 0, 1)

Tablee A 3 . Symmetry operations P

g

(g = 1 to 24) on the magnetic moment components at the 2a site.

Positionn atom 1: (0, 0, 0); atom 2: (0, 0, 1/2). 1 (1, 0, 0) stands for a magnetic moment at atom 1 in the

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D(l) ) D(2) ) D(3) ) D(4) ) D(5) ) D(6) ) D(7) ) D(8) ) D(9) ) D(10) ) D ( l l ) ) D(12) ) D(13) ) D(14) ) D(15) ) D(16) ) D(17) ) D(18) ) D(19) ) D(20) ) D{21) ) D(22) ) D(23) ) D(24) ) Tablee A spacee | .2.. T 'roup p -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i h ee m -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i atrix x -l l -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i elem m cc (I

(;?) )

/ww 0 \ [W** 0 I

II °

w

/

(if) )

/ww 0 \ \o\o w*/ [w** 0 | \\ 0 w /

(fi) )

(00 w ' ] \ww 0 / ( oo w\

\

w

** ° /

(fi) )

[ 00 W*| \ww 0 / // o w\ \w** o)

(fi) )

[oo w*)

\

ww

° /

// o w \

(fi) )

|| 0 w * |

\

ww

° /

// o w\ \w"" o)

(if) )

Avv 0 \ ^00 w * j (w** 0 1 \\ 0 w /

(if) )

/ww 0 \ ^00 w*/ [w** 0 | \\ 0 w / entss D(g) ( vfo.. 194) fc

(if) )

Cw-) )

(U) )

f-.

1 -

0 ,) )

/-ww o \ \ oo -W /

('»'' -t)

(fi) )

/oo w*\ \ww o ;

?)

(VoO O

// 0 -w*\ // 0 -w^

(Vo') )

(i-v) )

) )

(fi) )

(r,') )

) )

(o'°.) )

\\ o w 1

( VV 0 \

\\ 0 -w/

(if) )

ftw°-) ftw°-)

\o\o v/)

II = 1 to 24) )rr q = 0 0 ol '1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 aff th« jtaine e -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 ;; twe dd fi -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i -i i Ivee ii m m -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 r e d u c c ref. .

(if) )

/ww 0 \

\°°

w

/

\\

°

w

/

(if) )

/ww 0 \ \\ o w /

(fi) )

(00 w*l \WW 0 / // o w\ \W** 0 /

(fi) )

(o(o w*\

\

ww

° /

// 0 w\ \W"" 0 / (o(o -w"\ ^-WW 0 /

(( ° '

w

\

\-W** 0 /

\-\\-\ o)

(( 0 -w*\ \-WW 0 /

(( ° "

w

^

v

w

"" ° /

(-10) (-10) \o\o -l) ((WW 0 \ ^00 -w*^ /-W'' 0 | \\ o -w/ (-10) (-10) /-WW 0 \ ^ 00 -w*/ /-W"" 0 \ lblee represen 4.20.. W =

(if) )

(»» w°-)

(YA) )

(o'°,) )

WW 0 \

(TT

.t)

(fi) )

/ oo W'\ \WW 0 /

(.' Ï)

(V„') )

(ww r )

// 0< -w\ \-WW 0 /

(fi) )

/ oo w ' \ \ww 0 / (w-- » )

(V.') )

) ) ) )

(if) )

\ oo w /

(

W

o'' i)

(ö'-°,) )

( o . w " ) ) tationss T ' of exp(m22 / 3 ) ;

WW = exp(-m2 / 3).

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Pi Pi p? ? p? ? P4 4 p5 5 p6 6 P7 7 p? ? P9 9 P19 9 Pll l p.? ?

Pn n

P.4 4 P.5 5

P

16 6

Pn n

P.? ? P,9 9 P?9 9 P_{?i i} P_{K K} P?? ? P24 4 1 ( 1 , 0 , 0 ) ) 1 ( 1 , 0 , 0 ) ) 2 ( 0 , 1 , 0 ) ) 33 (-1,-1,0) 44 (-1,0,0) 5 ( 0 , - 1 , 0 ) ) 6 ( 1 , 1 , 0 ) ) 4 ( 1 , 0 , 0 ) ) 66 (-1,-1,0) 5 ( 0 , 1 , 0 ) ) 11 (-1,0,0) 3 ( 1 , 1 , 0 ) ) 22 (0, -1, 0) 44 (-1,0,0) 6 ( 1 , 1 , 0 ) ) 5 ( 0 , - 1 , 0 ) ) 1 ( 1 , 0 , 0 ) ) 33 (-1,-1,0) 2(0,, 1,0) 11 (-1, 0, 0) 2 ( 0 , - 1 , 0 ) ) 3 ( 1 , 1 , 0 ) ) 4 ( 1 , 0 , 0 ) ) 55 (0, 1, 0) 66 (-1,-1,0) 1(0,, 1,0) 1(0,, 1,0) 22 (-1,-1,0) 3 ( 1 , 0 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 5 ( 1 , 1 , 0 ) ) 66 (-1,0,0) 44 (-1,-1,0) 66 (0, 1, 0) 5 ( 1 , 0 , 0 ) ) 1 ( 1 , 1 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 22 (-1,0,0) 4 ( 1 , 1 , 0 ) ) 6 ( 0 , - 1 , 0 ) ) 55 (-1,0,0) 11 (-1,-1,0) 3 ( 0 , 1 , 0 ) ) 2 ( 1 , 0 , 0 ) ) 1(0,-1,0) ) 2 ( 1 , 1 , 0 ) ) 33 (-1,0,0) 4 ( 0 , 1 , 0 ) ) 55 (-1,-1,0) 6 ( 1 , 0 , 0 ) ) 1 ( 0 , 0 , 1 ) ) 1 ( 0 , 0 , 1 ) ) 2 ( 0 , 0 , 1 ) ) 33 (0, 0, 1) 4 ( 0 , 0 ,, 1) 5 ( 0 , 0 ,, 1) 6 ( 0 , 0 ,, 1) 4 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 , - 1 ) ) 1(0,0,-1) ) 3 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 , - 1 ) ) 1(0,0,-1) ) 3 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 1(0,0,, 1) 2 ( 0 , 0 , 1 ) ) 3 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) 5 ( 0 , 0 , 1 ) ) 6 ( 0 , 0 , 1 ) ) 2 ( 1 , 0 , 0 ) ) 2 ( 1 , 0 , 0 ) ) 3(0,, 1,0) 11 (-1,-1,0) 55 (-1,0,0) 6 ( 0 , - 1 , 0 ) ) 4 ( 1 ,, 1,0) 6 ( 1 , 0 , 0 ) ) 55 (-1,-1,0) 4 ( 0 , 1 , 0 ) ) 33 (-1, 0, 0) 2 ( 1 ,, 1,0) 1(0,-1,0) ) 66 (-1,0,0) 5 ( 1 , 1 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 3 ( 1 , 0 , 0 ) ) 22 (-1,-1,0) 11 (0, 1, 0) 22 (-1,0,0) 3 ( 0 , - 1 , 0 ) ) 1 ( 1 , 1 , 0 ) ) 5 ( 1 , 0 , 0 ) ) 6 ( 0 , 1 , 0 ) ) 44 (-1,-1,0) 2 ( 0 , 1 , 0 ) ) 2 ( 0 , 1 , 0 ) ) 33 (-1,-1,0) 1 ( 1 , 0 , 0 ) ) 5 ( 0 , - 1 , 0 ) ) 6 ( 1 , 1 , 0 ) ) 44 (-1,0,0) 66 (-1,-1,0) 55 (0, 1, 0) 4 ( 1 , 0 , 0 ) ) 3 ( 1 ,, 1,0) 2 ( 0 , - 1 , 0 ) ) 11 (-1,0,0) 6 ( 1 ,, 1,0) 5 ( 0 , - 1 , 0 ) ) 44 (-1,0,0) 33 (-1,-1,0) 2 ( 0 , 1 , 0 ) ) 1 ( 1 , 0 , 0 ) ) 2 ( 0 , - 1 , 0 ) ) 3 ( 1 ,, 1,0) 11 (-1,0,0) 5 ( 0 , 1 , 0 ) ) 66 (-1,-1,0) 4 ( 1 , 0 , 0 ) ) 2 ( 0 , 0 , 1 ) ) 2 ( 0 , 0 , 1 ) ) 33 (0, 0, 1) 11 (0, 0, 1) 5 ( 0 , 0 ,, 1) 6 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) 6 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 3 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 1(0,0,-1) ) 6 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 3 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 1(0,0,-1) ) 2 ( 0 , 0 , 1 ) ) 3 ( 0 , 0 ,, 1) 11 (0, 0, 1) 55 (0, 0, 1) 6 ( 0 , 0 ,, 1) 44 (0, 0, 1) 3 ( 1 , 0 , 0 ) ) 3 ( 1 , 0 , 0 ) ) 1 ( 0 , 1 , 0 ) ) 22 (-1,-1,0) 66 (-1,0,0) 4 ( 0 , - 1 , 0 ) ) 5 ( 1 , 1 , 0 ) ) 55 (1, 0, 0) 44 (-1,-1,0) 6(0,, 1,0) 22 (-1,0,0) 1 ( 1 , 1 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 55 (-1,0,0) 4 ( 1 ,, 1,0) 6 ( 0 , - 1 , 0 ) ) 2 ( 1 , 0 , 0 ) ) 11 (-1,-1,0) 3(0,, 1,0) 33 (-1,0,0) 11 (0, -1, 0) 2 ( 1 ,, 1,0) 6 ( 1 , 0 , 0 ) ) 4 ( 0 ,, 1,0) 55 (-1,-1,0) 3 ( 0 , 1 , 0 ) ) 3 ( 0 , 1 , 0 ) ) 11 (-1,-1,0) 2 ( 1 , 0 , 0 ) ) 66 (0, -1, 0) 4 ( 1 , 1 , 0 ) ) 55 (-1,0,0) 55 (-1.-1,0) 4 ( 0 , 1 , 0 ) ) 6 ( 1 , 0 , 0 ) ) 2 ( 1 , 1 , 0 ) ) 11 (0, -1, 0) 33 (-1,0,0) 5 ( 1 ,, 1,0) 4 ( 0 , - 1 , 0 ) ) 66 (-1, 0, 0) 22 (-1,-1,0) 1(0,, 1,0) 3 ( 1 , 0 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 1 ( 1 , 1 , 0 ) ) 22 (-1,0,0) 66 (0, 1,0) 44 (-1,-1,0) 5 ( 1 , 0 , 0 ) ) 3 ( 0 , 0 , 1 ) ) 3 ( 0 , 0 , 1 ) ) 11 (0,0, 1) 2 ( 0 , 0 , 1 ) ) 6 ( 0 , 0 , 1 ) ) 4 ( 0 , 0 , 1 ) ) 5 ( 0 , 0 , 1 ) ) 5 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 1(0,0,-1) ) 3 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 1 ( 0 , 0 , - 1 ) ) 3 ( 0 , 0 , - 1 ) ) 3 ( 0 , 0 ,, 1) 11 (0,0, 1) 22 (0,0, 1) 6 ( 0 , 0 ,, 1) 44 (0,0, 1) 55 (0,0, 1) P. . P? ? P? ? P4 4 P? ? P6 6 P7 7 P8 8 P9 9 P10 0 Pn n Pi? ? P|? ? P|4 4 P|5 5 p.* * Pl7 7 P)? ? Pl9 9 P?P P P2l l P_{2? ?} P» » P_{?4 4} 4 ( 1 , 0 , 0 ) ) 4 ( 1 , 0 , 0 ) ) 55 (0, 1, 0) 66 (-1,-1,0) 11 (-1,0,0) 2 ( 0 , - 1 , 0 ) ) 3 ( 1 ,, 1,0) 11 (1,0,0) 33 (-1,-1,0) 2 ( 0 , 1 , 0 ) ) 44 (-1,0,0) 6 ( 1 ,, 1,0) 5 ( 0 , - 1 , 0 ) ) 11 (-1,0,0) 3 ( 1 ,, 1,0) 22 (0, -1, 0) 4 ( 1 , 0 , 0 ) ) 66 (-1,-1,0) 55 (0, 1, 0) 44 (-1,0,0) 55 (0, -1, 0) 6 ( 1 ,, 1,0) 1 ( 1 , 0 , 0 ) ) 22 (0, 1, 0) 33 (-1,-1,0) 4(0,, 1,0) 4 ( 0 , 1 , 0 ) ) 55 (-1,-1,0) 6 ( 1 , 0 , 0 ) ) 1(0,-1,0) ) 2 ( 1 , 1 , 0 ) ) 33 (-1,0,0) 11 (-1,-1,0) 33 (0, 1, 0) 2 ( 1 , 0 , 0 ) ) 4 ( 1 ,, 1,0) 6 ( 0 , - 1 , 0 ) ) 55 (-1,0,0) 1 ( 1 , 1 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 22 (-1,0,0) 44 (-1,-1,0) 6(0,, 1,0) 5 ( 1 , 0 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 5 ( 1 , 1 , 0 ) ) 66 (-1,0,0) 1(0,, 1,0) 22 (-1,-1,0) 3 ( 1 , 0 , 0 ) ) 4 ( 0 , 0 , 1 ) ) 4 ( 0 , 0 , 1 ) ) 5 ( 0 , 0 , 1 ) ) 6 ( 0 , 0 ,, 1) 1 ( 0 , 0 ,, 1) 2 ( 0 , 0 , 1 ) ) 3 ( 0 , 0 , 1 ) ) 1(0,0,-1) ) 3 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 , - 1 ) ) 1(0,0,-1) ) 3 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 , - 1 ) ) 55 (0, 0, -1) 4 ( 0 , 0 ,, 1) 55 (0, 0, 1) 66 (0, 0, 1) 1 ( 0 , 0 ,, 1) 2 ( 0 , 0 ,, 1) 3 ( 0 , 0 , 1 ) ) 5 ( 1 , 0 , 0 ) ) 5 ( 1 , 0 , 0 ) ) 6(0,, 1,0) 44 (-1,-1,0) 22 (-1,0,0) 3 ( 0 , - 1 , 0 ) ) 1 ( 1 , 1 , 0 ) ) 3 ( 1 , 0 , 0 ) ) 22 (-1,-1,0) 1(0,, 1,0) 66 (-1,0,0) 5 ( 1 ,, 1,0) 4 ( 0 , - 1 , 0 ) ) 33 (-1,0,0) 2 ( 1 ,, 1,0) 1(0,-1,0) ) 6 ( 1 , 0 , 0 ) ) 55 (-1,-1,0) 4 ( 0 , 1 , 0 ) ) 55 (-1,0,0) 6 ( 0 , - 1 , 0 ) ) 4 ( 1 ,, 1,0) 2 ( 1 , 0 , 0 ) ) 33 (0, 1, 0) 11 (-1,-1,0) 55 (0, 1, 0) 5(0,, 1,0) 66 (-1,-1,0) 44 (1, 0, 0) 2 ( 0 , - 1 , 0 ) ) 3 ( 1 ,, 1,0) 11 (-1,0,0) 33 (-1,-1,0) 22 (0, 1, 0) 1 ( 1 , 0 , 0 ) ) 6 ( 1 ,, 1,0) 5 ( 0 , - 1 , 0 ) ) 44 (-1,0,0) 3 ( 1 , 1 , 0 ) ) 2 ( 0 , - 1 , 0 ) ) 11 (-1,0,0) 66 (-1,-1,0) 5(0,, 1,0) 4 ( 1 , 0 , 0 ) ) 5 ( 0 , - 1 , 0 ) ) 6 ( 1 ,, 1,0) 44 (-1,0,0) 2(0,, 1,0) 33 (-1,-1,0) 1 ( 1 , 0 , 0 ) ) 5 ( 0 , 0 ,, 1) 5 ( 0 , 0 , 1 ) ) 6 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) 22 (0, 0, 1) 3 ( 0 , 0 ,, 1) 1 ( 0 , 0 , 1 ) ) 3 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 1(0,0,-1) ) 6 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 3 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 1 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 ,, 1) 66 (0, 0, 1) 4 ( 0 , 0 ,, 1) 2 ( 0 , 0 ,, 1) 3 ( 0 , 0 ,, 1) 1 ( 0 , 0 ,, 1) 6 ( 1 , 0 , 0 ) ) 6 ( 1 , 0 , 0 ) ) 4 ( 0 , 1 , 0 ) ) 55 (-1,-1,0) 33 (-1,0,0) 1 ( 0 , - 1 , 0 ) ) 2 ( 1 ,, 1,0) 2 ( 1 , 0 , 0 ) ) 11 (-1,-1,0) 3(0,, 1,0) 55 (-1,0,0) 4 ( 1 ,, 1,0) 6 ( 0 , - 1 , 0 ) ) 22 (-1, 0, 0) 1 ( 1 , 1 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 5 ( 1 , 0 , 0 ) ) 44 (-1,-1,0) 66 (0, 1, 0) 66 (-1,0,0) 4 ( 0 , - 1 , 0 ) ) 5 ( 1 ,, 1,0) 33 (1, 0,0) 1(0,, 1,0) 22 (-1,-1,0) 6 ( 0 , 1 , 0 ) ) 6(0,, 1,0) 44 (-1,-1,0) 5 ( 1 , 0 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 1 ( 1 , 1 , 0 ) ) 22 (-1,0,0) 22 (-1,-1,0) 11 (0, 1, 0) 3 ( 1 , 0 , 0 ) ) 5 ( 1 , 1 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 66 (-1,0,0) 2 ( 1 , 1 , 0 ) ) 11 (0, -1, 0) 33 (-1,0,0) 55 (-1,-1,0) 4 ( 0 , 1 , 0 ) ) 6 ( 1 , 0 , 0 ) ) 6 ( 0 , - 1 , 0 ) ) 4 ( 1 , 1 , 0 ) ) 55 (-1,0,0) 3 ( 0 , 1 , 0 ) ) 11 (-1,-1,0) 2 ( 1 , 0 , 0 ) ) 66 (0,0, 1) 66 (0,0, 1) 4 ( 0 , 0 ,, 1) 5 ( 0 , 0 ,, 1) 3 ( 0 , 0 ,, 1) 11 (0, 0, 1) 22 (0, 0, 1) 2 ( 0 , 0 , - 1 ) ) 1 ( 0 , 0 , - 1 ) ) 3 ( 0 , 0 , - 1 ) ) 5 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 1(0,0,-1) ) 33 (0, 0, -1) 5 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 , - 1 ) ) 6 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) 55 (0, 0, 1) 33 (0, 0, 1) 11 (0, 0, 1) 2 ( 0 , 0 ,, 1)

Tablee A.4. Symmetry operations P

g

(g = 1 to 24) on the magnetic moment components at the 6h site.

Positionn atom I: (x, 2JC, 1/4); atom 2: (-2JC, -X, 1/4); atom 3: (*, -x, 1/4); atom 4: (-*, -2x, 3/4); atom 5:

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p3 3 P4 4 p5 5 P6 6 P7 7 P8 8 P9 9 P.o o P.. . P.2 2 Pl3 3 Pl4 4 P15 5 P.6 6 P>7 7 P.8 8 P,9 9 P20 0 P21 1 P22 2 P23 3 P24 4 11 (-1,-1,0) 22 (-1,0,0) 2 ( 0 , - 1 , 0 ) ) 2 ( 1 ,, 1,0) 3 ( 1 , 0 , 0 ) ) 33 (-1,-1,0) 3 ( 0 , 1 , 0 ) ) 44 (-1,0,0) 4 ( 1 , 1 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 22 (-1,0,0) 2 ( 1 ,, 1,0) 2 ( 0 , - 1 , 0 ) ) 11 (1,0,0) 11 (-1,-1,0) 11 (0,1,0) 44 (-1,0,0) 4 ( 0 , - 1 , 0 ) ) 4 ( 1 , 1 , 0 ) ) 3 ( 1 , 0 , 0 ) ) 3 ( 0 , 1 , 0 ) ) 33 (-1,-1,0) 11 (1,0,0) 2 ( 0 , - 1 , 0 ) ) 2 ( 1 ,, 1,0) 22 (-1,0,0) 33 (-1,-1,0) 3(0,, 1,0) 3 ( 1 , 0 , 0 ) ) 4 ( 1 ,, 1,0) 4 ( 0 , - 1 , 0 ) ) 44 (-1,0,0) 2 ( 1 ,, 1,0) 2 ( 0 , - 1 , 0 ) ) 22 (-1,0,0) 11 (-1,-1,0) 1 ( 0 , 1 , 0 ) ) 11 (1,0,0) 4 ( 0 , - 1 , 0 ) ) 4 ( 1 ,, 1,0) 44 (-1,0,0) 33 (0, 1, 0) 33 (-1,-1,0) 3 ( 1 , 0 , 0 ) ) 1(0,0,, 1) 2(0,0,, 1) 2(0,0,, 1) 2(0,0,, 1) 3(0,0,-1) ) 3(0,0,-1) ) 3(0,0,-1) ) 4(0,0,-1) ) 4(0,0,-1) ) 4(0,0,-1) ) 2(0,0,-1) ) 2(0,0,-1) ) 2(0,0,-1) ) 11 (0,0,-1) 13(0,0,-1) ) 11 (0,0,-1) 44 (0,0, 1) 4 ( 0 , 0 , 1 ) ) 4 ( 0 , 0 , 1 ) ) 3(0,0,1) ) 3(0,0,1) ) 3(0,0,1) ) 22 (-1,-1,0) 11 (-1,0,0) 1(0,-1,0) ) 1(1,1,0) ) 4 ( 1 , 0 , 0 ) ) 44 (-1,-1,0) 4 ( 0 , 1 , 0 ) ) 33 (-1,0,0) 3 ( 1 ,, 1,0) 3 ( 0 , - 1 , 0 ) ) 11 (-1,0,0) 1(1,1,0) ) 1(0,-1,0) ) 2 ( 1 , 0 , 0 ) ) 22 (-1,-1,0) 2 ( 0 , 1 , 0 ) ) 33 (-1,0,0) 3(0,-1,0) ) 3 ( 1 ,, 1,0) 4 ( 1 , 0 , 0 ) ) 4 ( 0 ,, 1,0) 44 (-1, -1,0) 2 ( 1 , 0 , 0 ) ) 1(0,-1,0) ) 1 ( 1 , 1 , 0 ) ) 11 (-1,0,0) 44 (-1,-1,0) 4(0,, 1,0) 4 ( 1 , 0 , 0 ) ) 3 ( 1 ,, 1,0) 3 ( 0 , - 1 , 0 ) ) 33 (-1,0,0) 1 ( 1 , 1 , 0 ) ) 11 (0,-1,0) 11 (-1,0,0) 22 (-1,-1,0) 2(0,, 1,0) 2 ( 1 , 0 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 3 ( 1 ,, 1,0) 33 (-1,0,0) 4 ( 0 ,, 1,0) 4 ( - l ,, -1,0) 4 ( 1 , 0 , 0 ) ) 22 (0, 0, 1) 11 (0,0, 1) 11 (0, 0, 1) 11 (0,0, 1) 4 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 3 ( 0 , 0 , - 1 ) ) 3(0,0,-1) ) 3(0,0,-1) ) 11 (0,0,-1) 11 (0,0,-1) 1(0,0,-1) ) 2 ( 0 , 0 , - 1 ) ) 2(0,0,-1) ) 2(0,0,-1) ) 3 ( 0 , 0 ,, 1) 3 ( 0 , 0 ,, 1) 3 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) P. . P2 2 P3 3 P4 4 P5 5 P6 6 P7 7 P« « P9 9 P . 0 0 P i . . Pl2 2 P13 3 Pl4 4 P i s s P l 6 6 Pl7 7 Pl8 8 P» » P20 0 P21 1 P22 2 P23 3 P24 4 3 ( 1 , 0 , 0 ) ) 3 ( 1 , 0 , 0 ) ) 3(0,, 1,0) 33 (-1,-1,0) 44 (-1,0,0) 4 ( 0 , - 1 , 0 ) ) 4 ( 1 ,, 1,0) 11 (1,0,0) 11 (-1,-1,0) 11 (0, 1,0) 22 (-1,0,0) 2 ( 1 ,, 1,0) 2 ( 0 , - 1 , 0 ) ) 44 (-1,0,0) 4 ( 1 ,, 1,0) 4 ( 0 , - 1 , 0 ) ) 3 ( 1 , 0 , 0 ) ) 33 (-1,-1,0) 33 (0, 1, 0) 22 (-1,0,0) 2 ( 0 , - 1 , 0 ) ) 2 ( 1 ,, 1,0) 1 ( 1 , 0 , 0 ) ) 1(0,, 1,0) 11 (-1,-1,0) 33 (0, 1, 0) 33 (0, 1, 0) 33 (-1,-1,0) 3 ( 1 , 0 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 4 ( 1 ,, 1,0) 44 (-1,0,0) 11 (-1,-1,0) 11 (0, 1, 0) 1 ( 1 , 0 , 0 ) ) 2 ( 1 ,, 1,0) 2 ( 0 , - 1 , 0 ) ) 22 (-1,0,0) 4 ( 1 ,, 1,0) 4 ( 0 , - 1 , 0 ) ) 44 (-1,0,0) 33 (-1,-1,0) 3(0,, 1,0) 3 ( 1 , 0 , 0 ) ) 2 ( 0 , - 1 , 0 ) ) 2 ( 1 ,, 1,0) 22 (-1,0,0) 1(0,, 1,0) 11 (-1,-1,0) 11 (1,0,0) 3(0,0,1) ) 3(0,0,1) ) 3(0,0,1) ) 3(0,0,1) ) 4(0,0,1) ) 4(0,0,1) ) 4(0,0,1) ) 11 (0,0,-1) II (0,0,-1) 11 (0,0,-1) 2(0,0,-1) ) 2(0,0,-1) ) 2(0,0,-1) ) 4(0,0,-1) ) 4(0,0,-1) ) 4(0,0,-1) ) 3(0,0,-1) ) 3(0,0,-1) ) 3(0,0,-1) ) 2(0,0,1) ) 22 (0,0,1) 2(0,0,1) ) 11 (0,0,1) 1(0,0,1) ) 1(0,0,1) ) 4 ( 1 , 0 , 0 ) ) 4 ( 1 , 0 , 0 ) ) 4 ( 0 ,, 1,0) 44 (-1,-1,0) 33 (-1,0,0) 3 ( 0 , - 1 , 0 ) ) 3 ( 1 ,, 1,0) 2 ( 1 , 0 , 0 ) ) 22 (-1,-1,0) 2(0,, 1,0) 11 (-1,0,0) 1(1,1,0) ) 11 (0,-1,0) 33 (-1,0,0) 3 ( 1 ,, 1,0) 3 ( 0 , - 1 , 0 ) ) 4 ( 1 , 0 , 0 ) ) 44 (-1,-1,0) 4(0,, 1,0) 11 (-1,0,0) 1(0,-1,0) ) 1 ( 1 , 1 , 0 ) ) 2 ( 1 , 0 , 0 ) ) 2(0,, 1,0) 22 (-1,-1,0) 4 ( 0 ,, 1,0) 4 ( 0 ,, 1,0) 44 (-1,-1,0) 4 ( 1 , 0 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 3 ( 1 ,, 1,0) 33 (-1,0,0) 22 (-1,-1,0) 2(0,, 1,0) 2 ( 1 , 0 , 0 ) ) 1 ( 1 , 1 , 0 ) ) 11 (0,-1,0) 11 (-1,0,0) 3 ( 1 , 1 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 33 (-1,0,0) 44 (-1,-1,0) 4(0,, 1,0) 4 ( 1 , 0 , 0 ) ) 11 (0,-1,0) 1 ( 1 , 1 , 0 ) ) 11 (-1,0,0) 2(0,, 1,0) 22 (-1,-1,0) 2 ( 1 , 0 , 0 ) ) 4 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) 4 ( 0 , 0 ,, 1) 3 ( 0 , 0 ,, 1) 3 ( 0 , 0 ,, 1) 3 ( 0 , 0 ,, 1) 2 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 2 ( 0 , 0 , - 1 ) ) 1(0,0,-1) ) 1(0,0,-1) ) 1(0,0,-1) ) 3 ( 0 , 0 , - 1 ) ) 3 ( 0 , 0 , - 1 ) ) 3 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 4 ( 0 , 0 , - 1 ) ) 11 (0, 0, 1) 1(0,0,, 1) 11 (0, 0, 1) 22 (0, 0, 1) 2(0,, 0, 1) 2 ( 0 , 0 ,, 1)

Tablee A.5. Symmetry operations P

g

(g = 1 to 24) on the magnetic moment components at the 4/site.

Positionn atom 1: (1/3,2/3, z); atom 2: (2/3, 1/3, z+1/2); atom 3: (2/3, 1/3,-z); atom 4:

(1/3,, 2/3, -z+l/2).

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r r

r' '

r

2 2

r

3 3

r

4 4

r

5 5

r

6 6

r

7 7

r

8 8

r

9 9

pp 10

p l l l p l 2 2

Basiss function

ƒ// = {1(0,0,0); 2(0,0,0)}

tftf = i{l(0,0,B);2(0,0,B)}

/i

33

= {1(0,0,0); 2(0,0,0)}

/,

44

= i{l(0,0,B);2(0.0,-B)}

yj

55

= 11 i /(l. W)A, - ('V3A, 0); 2[(W-1)A, t'V3A, oj 1

/

255

= I { i^(i_w)A, iV3A, 0); 2((W-l)A,-iV3A,0) }

AA

55

=fl+fi==fl+fi= |{l(A,0,0);2(-A,0,0)}

/bb ='(f\ ~ fl) = \ { 1 (V3A, 2V3A, 0); 2(-V3A, -2V3A, o) }

/,

66

= - j 1 ((1-W)A, -/V3A, 0]; 2[(l-W)A, -/V3A, o]] }

/

266

= I ( 1 ((1-W)A, /V3A, O) ; 2 [(1-W)A, j'V3A, o) }

/a

6

=/l

6

+/2

6

={{l(A,0,0);2(A,0,0)} }

fbfb = Hff - fl) = | { I (V3A, 2V3A, 0); 2(V3A, 2V3A, o) }

/!

77

= {1(0,0,0); 2(0,0,0)}

/,

88

= {1(0,0,0); 2(0,0,0)}

/,

99

= {1(0,0,0); 2(0,0,0)}

/t

100

= {1(0,0,0); 2(0,0,0)}

ƒ,"" = {1(0,0,0); 2(0,0,0)}

/

2nn

= {1(0,0,0); 2(0,0,0)}

/t

122

= { 1(0,0,0); 2(0,0,0)}

flfl

22

= { 1(0,0,0); 2(0,0,0)}

Magneticc structure

— —

Ferroo // c axis

— —

Antiferroo // c axis

Antiferroo in

basall plane

Ferroo in

basall plane

— —

Tablee A.6. Basis functions f'

K

belonging to irreducible representation T' for the 2a site. By taking

suitablee linear combinations of the basis functions of the irreducible representations T

5

and T

6

new

basiss functions f

la

, and f\ are obtained, that are real and remain orthogonal. The resulting magnetic

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r r

r

1 1

r

2 2

r

3 3

r

4 4

r

5 5

r

6 6

r

7 7

r

8 8

r

9 9

PP 10 p l l l r1 2 2

ƒ// = { 1(0, 0,0); 2(0, 0,0); 3(0, 0, 0); 4(0, 0,0); 5(0, 0, 0); 6(0, 0,0) }

tftf = i {1(0, 0, B); 2(0, 0, B); 3(0, 0, B); 4(0, 0, B); 5(0, 0, B); 6(0,0, B)}

fifi = i { 1(A, 0, 0); 2(0, A, 0); 3(-A, -A, 0); 4(A, 0, 0); 5(0, A, 0); 6(-A, -A, 0)}

f\f\ = { % 0, 0); 2(0, 0, 0); 3(0,0, 0); 4(0,0,0); 5(0, 0,0); 6(0, 0, 0)}

/,

55

= | { 1(0, 0, B); 2(0,0, WB); 3(0, 0, WB); 4(0, 0, B); 5(0, 0, WB); 6(0, 0, WB)}

/255 = i J 1(0, 0, B); 2(0, 0, WB); 3(0, 0, W*BI 4(0, 0, B); 5(0, 0, WB); 6J0, 0, W*BJ}

/a

55

= A

5 +

fl=\{ 1(°'°.

2B

); 2(0,0,-B); 3(0,0 ,-B); 4(0,0,2B); 5(0,0,-B); 6(0,0,-B) }

fbfb ='"(f\ ~ fi) = i f { 1(0A0);

2

(°AB); 3(0,0,-B); 4(0,0,0); 5(0,0,B); 6(0,0,-B) }

/J

66

=11 l(A,0,0);2(0,wA,0);3(-wA,-wA,0);4(A,0,0);5(0,wA,0);6(-wA,-wA,0) J

/

266

= 1 1 l(A,0,0);2(0,wA,0);3(-wA,-wA,OJ;4(A,0,0);5(0,wA,0);6(-wA,-wA,o)]

/a

66

= f\ + /2

6

= \ { 1(2A,0,0); 2(0,-A,0); 3(A,A,0); 4{2A,0,0); 5(0,-A,0); 6(A,A,0) }

fbfb = ''(A

6

~ f2 ) = ^ {1(0,0,0); 2(0, A, 0); 3(A, A, 0); 4(0, 0, 0); 5(0, A, 0); 6(A, A, 0)}

ff 1 = { 1(0, 0, 0); 2(0, 0, 0); 3(0, 0, 0); 4(0, 0, 0); 5(0, 0, 0); 6(0, 0, 0) }

ƒ/** = I { 1(A, 0, 0); 2(0, A, 0); 3(-A, -A, 0); 4(-A, 0,0); 5(0, -A, 0); 6(A, A, 0) }

/

L99

= I { 1(0, 0, B); 2(0, 0, B); 3(0,0, B); 4(0,0, -B); 5(0,0, -B); 6(0, 0, -B) }

/,

100

= { 1(0, 0, 0); 2(0, 0, 0); 3(0, 0, 0); 4(0, 0,0); 5(0, 0, 0); 6(0, 0, 0) }

ƒ/

ll

_{= 111(A,0,0); 2(0,wA,0); 3(-wA,-wA,0); 4(-A,0,0); 5JO,-wA,o); 6(wA,wA,0)}}

fjfj

ii

= 1 j l(A,0,0);2(0,wA,0);3(-wA,-wA,0\4(-A,0,0);5(0,-wA,0);6(wA,wA,o)}

fafa

l

= f\

l

+ / 2

l

= \ { 1(

2A

'°'°); 2(0,-A,0); 3(A,A,0); 4(-2A,0,0); 5(0,A,0); 6(-A,-A,0) }

/

buu

= ,(ƒ/' - /

2u

) = ^{l(0,0,0);2(0,A,0);3(-A,-A,0);4(0,0,0);5(0,-A,0);6(A,A,0)}

ff I

2

= I { 1(0,0, B); 2(0,0, wB); 3(0,0, wB); 4(0,0, -B); 5(0,0, -WBV 6(0,0, -wB) I

fjfj

22

_{= I J 1(0,0, B); 2(0,0, wB); 3(0,0, WBV 4(0,0, -B); 5(0, 0, -wB); 6(0,0, -WB) }}

/a

122

= A

12

+ / 2

2

= { {1(0,0,2B); 2(0,0,-B); 3(0,0,-B); 4(0,0,-2B); 5(0,0,B); 6(0,0,B) }

fbfb

22

='(A

12

" /2

2

) = ^j- { 1(0,0,0); 2(0,0,B);3(0,0,-B);4(0,0,0); 5(0,0,-B);6(0,0,B) }

— —

Ferroo // c

Triangular r

antiferroo in

basall plane

— —

'Antiferro' '

lie lie

Non-collinear r

ferroo in basal

plane e

— —

Triangular r

antiferroo in

basall plane

Antiferroo // c

— —

'Antiferro'' in

basall plane

'Antiferro' '

He He

Tablee A.7. Basis functions f\. belonging to irreducible representation T

1

for the 6h site. By taking

suitablee linear combinations of the basis functions of the irreducible representations T

5

, T

6

, T

11

and

r

1 22

new basis functions f

la, and fb

are obtained, that are real and remain orthogonal. The resulting

magneticc structures are given in the third column.

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r

1 1

r

l l

r

2 2

r

3 3

r

4 4

r

5 5

r

6 6

r

7 7

r

8 8

r

9 9

pp 10 p . 1 1 r. 2 2

Basiss function

flfl = { l ( 0 , 0 , 0 ) ; 2 ( 0 , 0 , 0 ) ; 3 { 0 , 0 , 0 ) ; 4 ( 0 , 0 , 0 ) } fifi = 6 { l ( 0 , 0 , B ) ; 2 ( 0 , 0 , B ) ; 3 ( 0 , 0 , B ) ; 4 ( 0 , 0 , B ) } flfl3,3, = { l ( 0 , 0 , 0 ) ; 2 { 0 , 0 , 0 ) ; 3 { 0 , 0 , 0 ) ; 4 ( 0 , 0 , 0 ) } f*f* = 6 { 1 (0, 0, B); 2 (0, 0, - B ) ; 3 (0, 0, B);4 (0, 0, -B) }

/ i55_{= {l((i-w)A,-iV3A,0): 2((W-i)A,/V3A,0) 3((i-W)A,-i\/3A,0) 4((W-i)A,/'V3A,0)}}

ƒ // = {l((i-w*)A,iV3A,0);2((W*-l)A,-iV3A,0)3{(l-W*)A1»V3A,0)4((W*-l)A,-/V3A,0)} fafa = fl + fl = 3 { 1 (A, 0, 0 ) ; 2 ( - A , 0, 0);3 (A, 0, 0 ) ; 4 ( - A , 0, 0) }

fbfb = «fl ~ fl) = ^3 { 1 (A, 2A, 0); 2 (-A, -2A, 0); 3 3 (A, 2A, 0); 4 4 {-A, -2A, 0) }

/ i66_{= { l((i-W)A,-iV3A,0) 2((l-W)A}

t-iV3A,0)s 3((l-W)A,-iV3A,0); 4((l-W)A,-iV3A,0)} ffff = { l((i-w*)A,/V3A,0) 2((1-W*)A,/V3A,0); 3((l-W*)A,iV3A,0); 4((l-W*)A,iV3A,0)} fa=flfa=fl6+6+flfl = 3 { 1 (A, 0, 0);2(A, 0,0);3 (A, 0, 0);4(A, 0, 0) } fbfb = «fl6 - /26) = V3 { 1 (A, 2A, 0); 2 (A, 2A, 0) -, 3 (A, 2A, 0); 4 (A, 2A, 0) } flfl = 6 { l ( 0 , 0 , B ) ; 2 ( 0 , 0 , B ) ; 3 { 0 , 0 , - B );4 ( 0 , 0 , - B ) }

fyfy = { l ( 0 , 0 , 0 ) ; 2 ( 0 , 0 , 0 ) ; 3 ( 0 , 0 , 0 ) ; 4 { 0 , 0 , 0 ) } fyfy = 6 { l ( 0 , 0 , B ) ; 2 ( 0 , 0 , - B ) ; 3 ( 0 , 0 , - B ) ; 4 ( 0 , 0 , B ) }

/i'°° = { l ( 0 , 0 , 0 ) ; 2 ( 0 , 0 , 0 ) ; 3 ( 0 , 0 , 0 ) ; 4 ( 0 , 0 , 0 ) }

/i1 11 = {l((i-W)A,-iV3A,0)t 2((W-1)A,/V3A,0); 3((W-1)A,/VJA,0) 4((1-W)A,-/V3A,0)} /2nn = {l((i-w")A,/V3A,0) 2((W*-!)A,-i73A,0) 3((W*-l)A,-/V3A,0) 4((l-w*)A,/V3A,0)} fafa11 = fl1 + fl[ - 3 { 1 (A, 0, 0 ) ; 2 ( - A , 0, 0);3 (-A, 0, 0);4(A, 0, 0) }

fbfb'' =«(fi l ~ fll) = V3 { 1 {A, 2A, 0), 2 (-A, -2A, 0); 3 (-A, -2A, 0); 4 (A, 2A, 0) }

flfl11 = { l((i-W)A,-iV3A,0) 2(ü-W)A,-iV3A,0);3((W-l)A,/V3A,0) 4(<w-l)A,iV3A,0)}

/2122 = { l((l-W*)A,;V3A,0); 2((l-W*)A,iV3A,0); 3((W*-l)A,-iV3A,0); 4((W*-1)A,-/V3A,0)} fafa22 = f\2 + fl2 = 3 { 1 (A, 0, 0); 2 (A, 0, 0); 3 (-A, 0, 0); 4 4 (-A, 0, 0) }

fbfb22 ='(fl2 - fl2) = VÏ { 1 (A, 2A, 0); 2 (A, 2A, 0)-, 3 3 (-A, -2A, 0); 4 (-A, -2A, 0 ) }

Magn.. struct.

— —

Ferroo // c

— —

Antiferroo // c

Antiferroo in

basall plane

Ferroo in

basall plane

Antiferroo // c

— —

Antiferroo // c

— —

Antiferroo in

basall plane

Antiferroo in

basall plane

Tablee A.8. Basis functions ƒ\. belonging to irreducible representation T' for the 4 / site. By taking suitablee linear combinations of the basis functions of the irreducible representations T5_{, T}6_{, r " and}

TT '2 new basis functions fa, and f\ are obtained, that are real and remain orthogonal. The resulting