FIG. 4: (Color online) (a)(b) Distribution of magnetic Bragg points in the ( h, k = 0 , l ) reciprocal lattice plane. J$_{1}$ = − 1, J$_{2}$ = − 1 . 68469. J$_{3}$ = − 0 . 2. p$_{3}$ = J$_{3}$/J$_{1}$ = 0 . 2. (c)(d) Distribution of magnetic Bragg points in the ( h, k = 0 , l ) re- ciprocal lattice plane. J$_{1}$ = − 1, J$_{2}$ = − 1 . 68469. J$_{3}$ = − 0 . 3. p$_{3}$ = J$_{3}$/J$_{1}$ = 0 . 3. (e)(f) Distribution of magnetic Bragg points in the ( h, k = 0 , l ) reciprocal plane around h = 1 and l = 2. J$_{1}$ = − 1, J$_{2}$ = − 1 . 68469. J$_{3}$ = − 0 . 4. p$_{3}$ = J$_{3}$/J$_{1}$ = 0 . 4. in the reciprocal lattice plane of ( h, k = h, l ), where ϵ = 0 . 29771. Using the contour plot of the Mathematica, we find the location of the magnetic Braggs in the ( h, k, l = 2 n ) reciprocal lattice, where The location of the magnetic Bragg peaks is given by Using the contour plot of the J ( h, k = h, l ), we find the location of the magnetic Braggs in the ( h, h, l ) reciprocal lattice, where where ϵ = 0 . 29771 and is independent of p$_{3}$ ( p$_{3}$ < 0 . 5935). Note that the essential results are independent of the choice of p$_{3}$ at least for 0 . 2 ≤ p$_{3}$ ≤ 0 . 4. takes a maximum, where J$_{1}$ = − 1, J$_{2}$ = p$_{2}$J$_{1}$ = − 1 . 68469. Figures 4(a)-(f) show the contour plot of J ( h, 0 , l ) in the reciprocal lattice plane of ( h, 0 , l ), where J$_{3}$ = p$_{3}$J$_{1}$ ( p$_{3}$ < 0 . 5935) and J$_{3}$ is changed as a param- eter ( J$_{3}$ = − 0 . 2 , − 0 . 3 , − 0 . 4). As is predicted above, we find that the magnetic Bragg peaks appear at Using the numerical calculation of the contour plot us- ing the Mathematica, we find the location of the magnetic Bragg peaks in the ( h, 0 , l ) reciprocal lattice, where VI. NUMERICAL CALCULATION A. The contour plot of J ( h, 0 , l ) in the ( h, 0 , l ) reciprocal lattice plane 7 J ( h, 0 , l ) = 8 J$_{1}$ cos( πh ) cos( πl ) + 2 J$_{2}$ cos(2 πl ) + 2 J$_{3}$ [cos(2 πh ) + 1] , (0 , 0 , 1 ± ϵ ) , (0 , 0 , 3 ± ϵ ) , (0 , 0 , 5 ± ϵ ) , (0 , 0 , 7 ± ϵ ) , . . . , (1 , 0 , 0 ± ϵ ) , (1 , 0 , 2 ± ϵ ) , (1 , 0 , 4 ± ϵ ) , (1 , 0 , 6 ± ϵ ) , . . . , (2 , 0 , 1 ± ϵ ) , (2 , 0 , 3 ± ϵ ) , (2 , 0 , 5 ± ϵ ) , (2 , 0 , 7 ± ϵ ) , . . . , J ( h, h, l ) = 8 J$_{1}$ cos $^{2}$( πh ) cos( πl )+2 J$_{2}$ cos(2 πl )+4 J$_{3}$ cos(2 πh ) . B. The contour plot of J ( h, k = h, l ) in the ( h, k = h, l ) reciprocal lattice plane C. The contour plot in the ( h, k, l ) plane with l = 2 n (0 , 0 , 1 ± ϵ ) , (0 , 0 , 3 ± ϵ ) , (0 , 0 , 5 ± ϵ ) , (0 , 0 , 7 ± ϵ ) , . . . , (1 / 2 , 1 / 2 , 1 / 2) , (1 / 2 , 1 / 2 , 3 / 2) , (1 / 2 , 1 / 2 , 5 / 2) , (1 / 2 , 1 / 2 , 7 / 2) , . . . , (1 , 1 , 1 ± ϵ ) , (1 , 1 , 3 ± ϵ ) , (1 , 1 , 5 ± ϵ ) , (1 , 1 , 7 ± ϵ ) , . . . , (3 / 2 , 3 / 2 , 1 / 2)(3 / 2 , 3 / 2 , 3 / 2) , (3 / 2 , 3 / 2 , 5 / 2) , (3 / 2 , 3 / 2 , 7 / 2) , . . . , J ( h, k, l = 2 n ) = 8 J$_{1}$ cos( πh ) cos( πk )