content_image/434391.jpg content_image/434419.jpg content_image/434414.jpg content_image/434417.jpg content_image/434418.jpg content_image/434422.jpg content_image/434415.jpg content_image/434421.jpg content_image/434423.jpg content_image/434416.jpg content_image/434420.jpg
<img src='content_image/434391.jpg'>

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 . 

## D. The contour plot in the ( h, k, l ) plane with l = 2 n + 1 

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 

<img src='content_image/434419.jpg'>

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. 

## VII. PHASE DIAGRAM IN THE ( J$_{2}$, J$_{3}$ ) PLANE 

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 

<img src='content_image/434414.jpg'>

## A. Ordered phases 

1. Helical order along the c axis 

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 

<img src='content_image/434417.jpg'>

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 

<img src='content_image/434418.jpg'>

by taking the derivative of J ( h, k = 0 , l = 0) with respect to h . The condition of the local maximum is given by 

where | J$_{1}$/J$_{3}$ | < 1. The maximum value is 

3. The ordered phase with h = 0 , k = 0 , and l = 1 

<img src='content_image/434422.jpg'>

4. The ordered phase with h = 1 / 2 , k = 0 , and l = 1 / 2 

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 

<img src='content_image/434415.jpg'>

2. Helical order along the a axis 

<img src='content_image/434421.jpg'>

<img src='content_image/434423.jpg'>

<img src='content_image/434416.jpg'>

<img src='content_image/434420.jpg'>

