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## C. The points H, I, J, K, and L with J$_{1}$ = − 1 . 0 and J$_{2}$ = − 1 . 5 

Typical contour plot of ( h, 0 , l ) at the points H, I, J, K, and K in the ( J$_{2}$ , J$_{3}$ ) phase diagram are shown in Fig. 13. The point H ( J$_{3}$ = − 0 . 2) is in the helical phase along the c axis. The point I ( J$_{3}$ = − 0 . 5) is in the helical 

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phase with the c axis and is on the line ( J$_{2}$ + J$_{3}$ = 2 J$_{1}$ ). The point J ( J$_{3}$ = − 0 . 667) is on the phase boundary between the helical phase with the c axis and the phase with ( h = 1 . 2 , k = 0 , l = 1 / 2). The points K ( J$_{3}$ = − 1 . 2) and L ( J$_{3}$ = − 1 . 5) are in the phase with ( h = 1 / 2 , k = 0 , l = 1 / 2). 

## D. The points M, N, O, and P with J$_{1}$ = − 1 . 0 and J$_{2}$ = − 2 

Typical contour plot of ( h, 0 , l ) at the points M, N, O, and P in the ( J$_{2}$ , J$_{3}$ ) phase diagram are shown in Fig. 14 The point M ( J$_{3}$ = − 0 . 2) is in the helical phase along the c axis. The point N ( J$_{3}$ = − 0 . 5) is on the phase boundary ( J$_{2}$ + J$_{3}$ = 2 J$_{1}$ ) between the helical phase with the c axis and the phase with ( h = 1 . 2 , k = 0 , l = 1 / 2). The points O ( J$_{3}$ = − 0 . 7) and P ( J$_{3}$ = − 0 . 8) are in the phase with ( h = 1 / 2 , k = 0 , l = 1 / 2). 

What is the three dimensional (3D) spin structure which is characterized with the wavevector of the mag- netic Bragg peaks? The vector of spin at the site R$_{i}$ of the real lattice space is given by 

where S = 3 / 2, we assume that the phase factor φ is equal to zero, and Q is defined as 

## IX. 3D SPIN STRUCTURES 

S$_{i}$ = S [cos( Q · R$_{i}$ ) e$_{x}$ + sin( Q · R$_{i}$ φ ) e$_{y}$ ] 

Q = ( ha $^{∗}$, ka$^{∗}$, lc$^{∗}$) . 

