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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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176 6 Appendixx B
Appendixx B
Tabless belonging to section 8.4; application of the theory of group representations to the systemm Gds(Ge,Si)4. P , : x ,, y, z P2:: 1/2-x, - y , 1/2+z P3:: 1/2+x, 1/2-y, 1/2-z P4:: - x , 1/2+y, - z P5:: 1/2-x, 1/2+y, 1/2+z P6:: x, 1/2-y, z P7:: - x , - y , - z P8:: 1/2+x, y, 1/2-z
Tablee B.l. The eight symmetry operations Pg (g = 1 to 8) of space group P nma (No. 62) obtained
fromm ref. 8.8. D ( l ) ) D(2) ) D(3) ) D(4) ) D(5) ) D(6) ) D(7) ) D(8) )
r
1 1r
2 2 l l l l - i i - i i - l l - l l l l l lr
3 3 l l - l l l l - l l l l - l l l l - l lr
4 4 l l - l l - l l l l - l l l l l l - l lr
5 5 - i i - i i - i i - l lr
6 6 l l l l - l l - l l l l l l - i i - l lr
7 7 l l - i i l l - l l - l l i i - i i l l r8 8 1 1 - 1 1 - 1 1 1 1 1 1 - 1 1 - 1 1 1 1Tablee B.2. The matrix elements D(g) (g = 1 to 8) of the eight irreducible representations T' of space groupp P nma for q = 0 obtained from ref. 8.8.
11 ( 1 , 0 , 0 ) 11 (0, 1,0) 11 (0,0, 1) 2 ( 1 , 0 , 0 ) ) 2 ( 0 ,, 1,0) 2 ( 0 , 0 ,, 1) 3 ( 1 , 0 , 0 ) ) 33 (0, 1, 0) 3 ( 0 , 0 ,, 1) 4 ( 1 , 0 , 0 ) ) 4 ( 0 ,, 1,0) 44 (0, 0, 1) P> > 11 (1, 0, 0) 11 (0, 1, 0) 11 (0, 0, 1) 2 ( 1 , 0 , 0 ) ) 22 (0, 1, 0) 2 ( 0 , 0 ,, 1) 3 ( 1 , 0 , 0 ) ) 3(0,, 1,0) 33 (0, 0, 1) 4 ( 1 , 0 , 0 ) ) 4(0,, 1,0) 4 ( 0 , 0 ,, 1) P2 2 22 (-1,0,0) 2 ( 0 , - 1 , 0 ) ) 2 ( 0 , 0 ,, 1) 11 (-1,0,0) 11 (0, - 1 , 0) 11 (0,0, 1) 44 (-1,0,0) 4 ( 0 , - 1 , 0 ) ) 4 ( 0 , 0 ,, 1) 33 (-1,0,0) 3 ( 0 , - 1 , 0 ) ) 3 ( 0 , 0 ,, 1) P3 3 3 ( 1 , 0 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 3 ( 0 , 0 , - 1 ) ) 4 ( 1 , 0 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 4 ( 0 , 0 , - 1 ) ) 11 (1,0,0) 1 ( 0 , - 1 , 0 ) ) 1 ( 0 , 0 , - 1 ) ) 2 ( 1 , 0 , 0 ) ) 2 ( 0 , - 1 , 0 ) ) 2 ( 0 , 0 , - 1 ) ) P4 4 44 (-1,0,0) 4(0,, 1,0) 4(0,0,-1) ) 33 (-1,0,0) 3(0,, 1,0) 3(0,0,-1) ) 22 (-1,0,0) 2(0,, 1,0) 2(0,0,-1) ) 11 (-1,0,0) 1(0,, 1,0) 1(0,0,-1) ) P5 5 2 ( 1 , 0 , 0 ) ) 2 ( 0 , - 1 , 0 ) ) 2 ( 0 , 0 , - 1 ) ) 11 (1,0,0) 11 (0,-1,0) 11 (0,0,-1) 4 ( 1 , 0 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 4 ( 0 , 0 , - 1 ) ) 3 ( 1 , 0 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 3 ( 0 , 0 , - 1 ) ) P6 6 11 (-1,0,0) 1(0,, 1,0) 11 (0,0,-1) 22 (-1,0,0) 2(0,, 1,0) 2 ( 0 , 0 , - 1 ) ) 33 (-1, 0, 0) 33 (0, 1, 0) 3 ( 0 , 0 , - 1 ) ) 44 (-1,0,0) 4 ( 0 ,, 1,0) 4 ( 0 , 0 , - 1 ) ) P7 7 4 ( 1 , 0 , 0 ) ) 4(0,, 1,0) 4 ( 0 , 0 , 1 ) ) 3 ( 1 , 0 , 0 ) ) 3(0,, 1,0) 33 (0, 0, 1) 2 ( 1 , 0 , 0 ) ) 2(0,, 1,0) 2 ( 0 , 0 ,, 1) 1 ( 1 , 0 , 0 ) ) 1(0,, 1,0) 1 ( 0 , 0 ,, 1) P8 8 33 (-1,0,0) 3 ( 0 , - 1 , 0 ) ) 3 ( 0 , 0 ,, 1) 44 (-1,0,0) 4 ( 0 , - 1 , 0 ) ) 4 ( 0 , 0 ,, 1) 11 (-1,0,0) 1(0,-1,0) ) 11 (0,0, 1) 22 (-1,0,0) 2 ( 0 , - 1 , 0 ) ) 22 (0,0, 1)
Tablee B.3. Symmetry operations Pg (g = 1 to 8) on the magnetic moment components at the Ac site.
Positionn atom 1: {x, 1/4, z); atom 2: (1/2-JC, 3/4, 1/2+z); atom 3: (1/2+x, 1/4, 1/2-z); atom 4: (-x,, 3/4, -z). 1 (1,0, 0) stands for a magnetic moment at atom 1 in the x direction.
Appendixx B
177 7
1 ( 1 , 0 , 0 ) ) 1(0,, 1,0) 11 (0,0,1) 2 ( 1 , 0 , 0 ) ) 2(0,, 1,0) 22 (0, 0, 1) 3 ( 1 , 0 , 0 ) ) 3(0,, 1,0) 3 ( 0 , 0 ,, 1) 4 ( 1 , 0 , 0 ) ) 4(0,, 1,0) 4 ( 0 , 0 ,, 1) 5 ( 1 , 0 , 0 ) ) 55 (0, I, 0) 5 ( 0 , 0 ,, 1) 6 ( 1 , 0 , 0 ) ) 6 ( 0 , 1 , 0 ) ) 6 ( 0 , 0 , 1 ) ) 7 ( 1 , 0 , 0 ) ) 7(0,, 1,0) 7 ( 0 , 0 , 1 ) ) 8 ( 1 , 0 , 0 ) ) 8(0,, 1,0) 8 ( 0 , 0 , 1 ) ) Pt t 1 ( 1 , 0 , 0 ) ) 1 ( 0 , 1 , 0 ) ) 1(0,0,, 1) 2 ( 1 , 0 , 0 ) ) 2 ( 0 , 1 , 0 ) ) 22 (0, 0, 1) 3 ( 1 , 0 , 0 ) ) 3 ( 0 , 1 , 0 ) ) 3 ( 0 , 0 , 1 ) ) 4 ( 1 , 0 , 0 ) ) 4 ( 0 , 1 , 0 ) ) 4 ( 0 , 0 , 1 ) ) 5 ( 1 , 0 , 0 ) ) 5(0,, 1,0) 5 ( 0 , 0 ,, 1) 6 ( 1 , 0 , 0 ) ) 66 (0, 1, 0) 66 (0, 0, 1) 7 ( 1 , 0 , 0 ) ) 77 (0, 1, 0) 77 (0, 0, 1) 8 ( 1 , 0 , 0 ) ) 8(0,, 1,0) 88 (0, 0, 1) P2 2 22 (-1,0,0) 2 ( 0 , - 1 , 0 ) ) 2 ( 0 , 0 ,, 1) 11 (-1,0,0) 1(0,-1,0) ) 11 (0, 0, 1) 44 (-1,0,0) 4 ( 0 , - 1 , 0 ) ) 4 ( 0 , 0 , 1 ) ) 33 (-1,0,0) 3 ( 0 , - 1 , 0 ) ) 33 (0, 0, 1) 66 (-1,0,0) 6 ( 0 , - 1 , 0 ) ) 66 (0,0, 1) 55 (-1,0,0) 55 (0,-1, 0) 5 ( 0 , 0 ,, 1) 88 (-1,0,0) 8 ( 0 , - 1 , 0 ) ) 8 ( 0 , 0 ,, 1) 77 (-1,0,0) 7 ( 0 , - 1 , 0 ) ) 7 ( 0 , 0 ,, 1) P3 3 3 ( 1 , 0 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 3 ( 0 , 0 , - 1 ) ) 4 ( 1 , 0 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 4 ( 0 , 0 , - 1 ) ) 1 ( 1 , 0 , 0 ) ) 1(0,-1,0) ) 11 (0,0,-1) 2 ( 1 , 0 , 0 ) ) 2 ( 0 , - 1 , 0 ) ) 2 ( 0 , 0 , - 1 ) ) 7 ( 1 , 0 , 0 ) ) 7 ( 0 , - 1 , 0 ) ) 77 (0, 0, -1) 8 ( 1 , 0 , 0 ) ) 8 ( 0 , - 1 , 0 ) ) 8 ( 0 , 0 , - 1 ) ) 5 ( 1 , 0 , 0 ) ) 5 ( 0 , - 1 , 0 ) ) 5 ( 0 , 0 , - 1 ) ) 6 ( 1 , 0 , 0 ) ) 6 ( 0 , - 1 , 0 ) ) 6 ( 0 , 0 , - 1 ) ) P4 4 44 (-1,0,0) 44 (0, 1, 0) 4 ( 0 , 0 , - 1 ) ) 33 (-1, 0, 0) 33 (0, 1, 0) 3 ( 0 , 0 , - 1 ) ) 22 (-1,0,0) 2 ( 0 , 1 , 0 ) ) 2 ( 0 , 0 , - 1 ) ) 11 (-1,0,0) 1(0,, 1,0) 1 ( 0 , 0 , - 1 ) ) 88 (-1,0,0) 88 (0, 1, 0) 8 ( 0 , 0 , - 1 ) ) 77 (-1,0,0) 77 (0, 1, 0) 7 ( 0 , 0 , - 1 ) ) 66 (-1,0,0) 6(0,, 1,0) 6 ( 0 , 0 , - 1 ) ) 55 (-1,0,0) 55 (0, 1, 0) 5 ( 0 , 0 , - 1 ) ) Ps s 5 ( 1 , 0 , 0 ) ) 5 ( 0 , - 1 , 0 ) ) 55 (0, 0, -1) 6 ( 1 , 0 , 0 ) ) 6 ( 0 , - 1 , 0 ) ) 6 ( 0 , 0 , - 1 ) ) 7 ( 1 , 0 , 0 ) ) 7 ( 0 , - 1 , 0 ) ) 7 ( 0 , 0 , - 1 ) ) 8 ( 1 , 0 , 0 ) ) 8 ( 0 , - 1 , 0 ) ) 88 (0, 0, -1) 1 ( 1 , 0 , 0 ) ) 1(0,-1,0) ) 11 (0, 0, -1) 2 ( 1 , 0 , 0 ) ) 2 ( 0 , - 1 , 0 ) ) 2 ( 0 , 0 , - 1 ) ) 3 ( 1 , 0 , 0 ) ) 3 ( 0 , - 1 , 0 ) ) 3 ( 0 , 0 , - 1 ) ) 4 ( 1 , 0 , 0 ) ) 4 ( 0 , - 1 , 0 ) ) 4 ( 0 , 0 , - 1 ) ) P6 6 66 (-1,0,0) 6(0,, 1,0) 6 ( 0 , 0 , - 1 ) ) 55 (-1,0,0) 5(0,, 1,0) 5 ( 0 , 0 , - 1 ) ) 88 (-1,0,0) 8 ( 0 , 1 , 0 ) ) 8 ( 0 , 0 , - 1 ) ) 77 (-1, 0, 0) 7(0,, 1,0) 7 ( 0 , 0 , - 1 ) ) 22 (-1,0,0) 2(0,, 1,0) 2 ( 0 , 0 , - 1 ) ) 11 (-1,0,0) 1(0,, 1,0) 1(0,0,-1) ) 44 (-1,0,0) 4(0,, 1,0) 4 ( 0 , 0 , - 1 ) ) 33 (-1,0,0) 33 (0, 1, 0) 3 ( 0 , 0 , - 1 ) ) P7 7 7 ( 1 , 0 , 0 ) ) 7(0,, 1,0) 7 ( 0 , 0 ,, 1) 8 ( 1 , 0 , 0 ) ) 8(0,, 1,0) 88 (0, 0, 1) 5 ( 1 , 0 , 0 ) ) 55 (0, 1, 0) 5 ( 0 , 0 , 1 ) ) 6 ( 1 , 0 , 0 ) ) 66 (0, 1, 0) 6 ( 0 , 0 , 1 ) ) 3 ( 1 , 0 , 0 ) ) 3 ( 0 , 1 , 0 ) ) 33 (0, 0, 1) 4 ( 1 , 0 , 0 ) ) 4 ( 0 , 1 , 0 ) ) 4 ( 0 , 0 , 1 ) ) 1 ( 1 , 0 , 0 ) ) 1 ( 0 , 1 , 0 ) ) 11 (0,0,1) 2 ( 1 , 0 , 0 ) ) 2(0,, 1,0) 2 ( 0 , 0 , 1 ) ) P8 8 88 (-1,0,0) 8 ( 0 , - 1 , 0 ) ) 8 ( 0 , 0 , 1 ) ) 77 (-1,0,0) 7 ( 0 , - 1 , 0 ) ) 7 ( 0 , 0 ,, 1) 66 (-1,0,0) 6 ( 0 , - 1 , 0 ) ) 6 ( 0 , 0 , 1 ) ) 55 (-1,0,0) 5 ( 0 , - 1 , 0 ) ) 55 (0, 0, 1) 44 (-1,0,0) 4 ( 0 , - 1 , 0 ) ) 4 ( 0 , 0 ,, 1) 33 (-1,0,0) 3 ( 0 , - 1 , 0 ) ) 3 ( 0 , 0 ,, 1) 22 (-1,0,0) 2 ( 0 , - 1 , 0 ) ) 22 (0, 0, 1) 11 (-1,0,0) 1(0,-1,0) ) 1 ( 0 , 0 , 1 ) )Tablee B.4. Symmetry operations P
g(g = 1 to 8) on the magnetic moment components at the $d site.
Positionn atom h(x,y,z); atom 2: (1/2-JC, -y, 1/2+z); atom 3: (1/2+jc, 1/2-y, 1/2-z); atom 4:
(-x,(-x, \ll+y,-z); atom 5: (1/2-JC, 1/2+y, 1/2+z); atom 6: (JC, \l2-y, z); atom 7: (-x, -y, -z); atom 8:
178 8
Appendixx B
r r
r
1 1r
2 2r
3 3r
4 4r
5 5r
6 6r
7 7 r8 8Basiss function
/J11 = {1 (0, B, 0); 2 (0, -B, 0); 3 (0, -B, 0); 4 (0, B, 0 ) } /LL = {l (A, 0, C); 2 (-A, 0, C); 3 (-A, 0. C); 4 (A, 0, C)}ff{{ = {l (A, 0, C); 2 (A, 0, -C); 3 (A, 0, -C); 4 (A, 0, C)} f*f* = {l (0, B, 0); 2 (0, B, 0); 3 (0, B, 0); 4 (0, B. 0 ) } fyfy = {l (A. 0, C); 2 (-A, 0, C); 3 (A, 0, -C): 4 (-A, 0, - C ) } ff{{ = {l (0, B, 0); 2 (0, -B, 0); 3 (0, B, 0); 4 (0, -B, 0 ) } flfl = {l {0, B, 0); 2 (0, B, 0); 3 (0, -B, 0); 4 (0, -B, 0 ) } ffxx = {l (A, 0, C); 2 (A, 0, -C); 3 (-A, 0. C); 4 (-A, 0, - C ) }
Magneticc structure
bb axis AF
aa axis AF; c axis F
aa axis F; c axis AF
bb axis F
aa axis AF; c axis AF
bb axis AF
bb axis AF
aa axis AF; c axis AF
Tablee B.5. Basis functions /
K' belonging to irreducible representation T' for the Ac site. The resulting
magneticc structures are given in the third column.
r
1 1r
1 1r
2 2r
3 3r
4 4r
5 5r
6 6r
7 7r
8 8Basiss function
ƒ// = {l(D,E,F); 2(-D,-E,F); 3(D,-E,-F); 4{-D,E.-F): 5(D,-E,-F); 6(-D,E.-F); 7(D,E,F); 8(-D,-E,F)} /jj = {l(D,E,F); 2(-D,-E.F): 3(-D.E,F); 4(D,-E,F); 5(-D,E,F); 6(D.-E.F); 7(D,E,F); 8(-D,-E.F)}
/ j33 = {l(D,E,F); 2(D,E.-F); 3(D,-E,-F); 4(D,-E,F); 5(D,-E.-F); 6(D,-E.F); 7(D,E,F); 8(D,E.-F)}
/ ,44 = {l(D,E,F); 2(D,E,-F); 3(-D.E,F); 4(-D,E,-F); 5(-D,E,F);6(-D,E,-F); 7(D,E,F); 8(D,E,-F)} /jj = {l(D,E,F);2(-D,-E,F);3(D,-E,-F);4(-D,E.-F);5(-D,E,F);6(D,-E,F);7(-D,-E.-F);8(D,E,-F)}
fyfy = {l(D,E,F); 2( -D, -E,F); 3( -D,E,F); 4(D, -E,F); 5(D. -E, -F); 6( -D,E, -FJ; 7( -D. -E, -F); 8(D,E,-F)}
/ j77 = {l(D,E,F); 2(D,E,-F); 3(D,-E,-F); 4(D,-E,F); 5(-D,E.F); 6(-D,E.-F): 7(-D.-E.-F); 8( -D.-E.F)} /jj = {l(D.E.F): 2(D.E.-F); 3(-D.E,F); 4(-D,E,-F); 5(D.-E,-F); 6(D,~E,F); 7(-D,-E,-F): 8( -D,-E,F)}
Magn.. struct.
a,a, b, c axis AF
a,a, b axis AF;
cc axis F
aa axis F;
b,b, c axis AF
a,a, c axis AF;
bb axis F
a,a, b, c axis AF
a,a, b, c axis AF
a,, b, c axis AF
a,a, b, c axis AF
Tablee B.6. Basis functions /
K' belonging to irreducible representation T
1