8 the maximum of by taking the derivative of J ( h = 0 , k = 0 , l ) with respect to l . The condition of the local maximum is given by where | J$_{1}$/J$_{2}$ | < 1. Then the maximum of J ( h = 0 , k = 0 , l ) is given by which does not depend on the index n . Figures 5(a) show the location of magnetic Bragg peaks in the in- plane ( h, k, l = 2 n ) reciprocal lattice plane. We find that the magnetic Bragg peaks appear for h + k = even. This selection rule is true for l = 2 n with any integer n . for the helical order with ( h = 0, k = 0, and l ) where l = 0 . 70228, The helical spin order where the spin vector in the same bc plane are parallel and they screw along the a axis. We find the maximum of Using the contour plot of the Mathematica, we find the location of the magnetic Braggs in the ( h, k, l = 2 n + 1) reciprocal lattice, where by taking the derivative of J ( h, k = 0 , l = 0) with respect to h . The condition of the local maximum is given by which does not depend on the index n . Figure 5(b) show the location of the magnetic Bragg peaks in the in-plane ( h, k, l = 2 n + 1) reciprocal lattice plane, where n is the integer. The magnetic Bragg peaks appear for h + k = odd for l = 2 n + 1. where | J$_{1}$/J$_{3}$ | < 1. The maximum value is The phase diagram for J$_{1}$ = − 1 can be determined us- ing the programs of the ContourPlot and FindMaximum. The phase diagram consists of the four phases; (i) the he- lical order along the c axis, (ii) the helical order along the a axis, (iii) the ordered phase with h = 1 / 2, k = 0, and l = 1 / 2, and (iv) the ordered phase with h = 0, k = 0, l = 1 is shown in the phase diagram Fig. 6. The spin vectors in the same ab plane are parallel, i.e., h = 0 and k = 0. They screw along the c axis. We find 1. Helical order along the c axis This phase corresponds to the MnF$_{2}$-type antiferro- magnetic structure (named by Yoshimori $^{2}$). J ( h = 1 / 2 , k = 0 , l = 1 / 2) is evaluated as 4. The ordered phase with h = 1 / 2 , k = 0 , and l = 1 / 2 3. The ordered phase with h = 0 , k = 0 , and l = 1 2. Helical order along the a axis A. Ordered phases VII. PHASE DIAGRAM IN THE ( J$_{2}$, J$_{3}$ ) PLANE D. The contour plot in the ( h, k, l ) plane with l = 2 n + 1 FIG. 5: (Color online) (a) Magnetic Bragg points in the ( h, k, l = 2 n ) reciprocal lattice plane. (b) Magnetic Bragg points in the ( h, k, l = 2 n + 1) reciprocal lattice plane. J$_{1}$ = − 1, J$_{2}$ = − 1 . 68469. J$_{3}$ = − 0 . 3. p$_{3}$ = 0 . 3. + 2 J$_{3}$ [cos(2 πh ) + cos(2 πk )] + 2 J$_{2}$, J ( h = 0 , k = 0 , l ) = 4 J$_{3}$ + 8 J$_{1}$ cos( lπ ) + 2 J$_{2}$ cos(2 lπ ) , cos( πl ) = − J$_{1}$ J$_{2}$ , J$_{max}$ ( h = 0 , k = 0 , l ) = 4 J$_{3}$ − 4 J 2 1 J$_{2}$ − 2 J$_{2}$, (15) J ( h, k = 0 , l = 1) = 2 J$_{2}$ +2 J$_{3}$ +8 J$_{1}$ cos( πh )+2 J$_{3}$ cos(2 πh ) , J ( h, k, l = 2 n + 1) = − 8 J$_{1}$ cos( πh ) cos( πk )) + 2 J$_{3}$ [cos(2 πh ) + cos(2 πk )] + 2 J$_{2}$, cos( πh ) = − J$_{1}$ J$_{3}$ , J$_{max}$ ( h, k = 0 , l = 1) = 2 J$_{2}$ − 4 J 2 1 J$_{3}$ . (16) J ( h = 0 , k = 0 , l = 1) is given by J ( h = 0 , k = 0 , l = 1) = − 8 J$_{1}$ + 2 J$_{2}$ + 4 J$_{3}$. (17) J ( h = 1 / 2 , k = 0 , l = 1 / 2) = − 2 J$_{2}$. (18)