( h , k , 5), ( h , k , 7),... (ii) The distribution of the magnetic Bragg peaks is the same in the reciprocal lattice planes , ( h , k , 0), ( h , k , 2), ( h , k , 4), ( h , k , 8),... When k = 0 and h is a fixed integer, J ( h = fixed integer, k = 0, l ) is rewritten as which indicates that the location of the magnetic Bragg peaks along the ( h = fixed integer, k = 0, l ) direction depends only on J$_{1}$ and J$_{2}$ , and is independent of J$_{3}$ . The following selection rules can be derived. (i) The location of the magnetic Bragg peaks along the l direction is the same for (1, 0, l ), (3, 0, l ), (5, 0, l ), (7, 0, l ), · · · . (ii) The location of the magnetic Bragg peaks along the l direction is the same for (0, 0, l ), (2, 0, l ), (4, 0, l ), (8, 0, l ), · · · . For h = k = half-integer (= 1/2, 3/2, 5/2, · · · ), we have leading to the appearance of the magnetic Bragg peaks appear at for h = k = 1 / 2 , 3 / 2 , 5 / 2 , 7 / 2 , · · · . For any l , J (0 , 0 , l ) can be expressed by with θ = πl . This indicates that the location of the magnetic Bragg peaks along the (0 , 0 , l ) direction, is in- dependent of J$_{3}$ . The derivative of J (0 , 0 , l ) with respect to θ is obtained as Then we get the two solutions. (i) Ferromagnetic or antiferromagnetic configurations respectively. Since | J$_{1}$/J$_{3}$ | > 1 in MnO$_{2}$, we have only sin( πh ) = 0. Since J$_{1}$ < 0, we have the following selection rule. The magnetic Bragg appears only at along the h -direction. and where ϵ is defined as If the conditions J$_{2}$ < 0 and | J$_{1}$/J$_{2}$ | < 1 are satisfied, the magnetic Bragg peaks appear at the wavevectors denoted by Then the helical spin order appears when The difference is calculated as under the condition of | J$_{1}$/J$_{2}$ | < 1, where (ii) Helical spin configuration where B. Magnetic Bragg peaks along the l direction with h = fixed integer and k = 0 C. Magnetic Bragg peaks along the l direction with h = k = half inetgers D. Magnetic Bragg peaks along the (0 , 0 , l ) direction E. Magnetic Bragg peaks along the ( h, 0 , 0) direction J ( θ = 0) = 8 J$_{1}$ + 2 J$_{2}$ + 4 J$_{3}$ J ( θ = π ) = − 8 J$_{1}$ + 2 J$_{2}$ + 4 J$_{3}$. cos θ = − J$_{1}$/J$_{2}$, θ = θ$_{0}$ = arccos( − J$_{1}$/J$_{2}$ ) , J ( θ = θ$_{0}$ ) = − 4 J 2 1 J$_{2}$ − 2 J$_{2}$ + 4 J$_{3}$. J ( h, 0 , l ) = 8( − 1) $^{h}$J$_{1}$ cos( πl ) + 2 J$_{2}$ cos(2 πl ) + 2 J$_{3}$, J ( θ = θ$_{0}$ ) − J ( θ = π ) = − 4 J$_{2}$ ( J$_{1}$ J$_{2}$ − 1) $^{2}$, J ( θ = θ$_{0}$ ) − J ( θ = 0) = − 4 J$_{2}$ ( J$_{1}$ J$_{2}$ + 1) $^{2}$. J ( θ = θ$_{0}$ ) > J ( θ = π ) , J ( θ = θ$_{0}$ ) > J ( θ = 0) . ( h, k, l ) = (0 , 0 , 1 ± ϵ ) , (0 , 0 , 3 ± ϵ ) , (0 , 0 , 5 ± ϵ ) , (0 , 0 , 7 ± ϵ ) , · · · , cos( πϵ ) = J$_{1}$ J$_{2}$ . J ( h, h, l ) = 2 J$_{2}$ cos(2 πl ) − 4 J$_{3}$, l = 1 / 2 , 3 / 2 , 5 / 2 , · · · J ( h = 0 , k = 0 , l ) = J ( θ ) = 8 J$_{1}$ cos( θ )+2 J$_{2}$ cos(2 θ )+4 J$_{3}$. dJ ( h, k = 0 , l = 0) dh = 8 π [ J$_{1}$ − J$_{3}$ cos( πh )] sin( πh ) , J ( h, k = 0 , l = 0) and its derivative are given by J ( h, k = 0 , l = 0) = 8 J$_{1}$ cos( πh )+2 J$_{3}$ cos(2 πh )+2 J$_{2}$ +2 J$_{3}$, dJ ( θ ) dθ = − 4[ J$_{1}$ + J$_{2}$ cos( θ )] sin( θ ) = 0 . sin θ = 0 , θ = 0 , π. (1 , 0 , 0) , (3 , 0 , 0) , (5 , 0 , 0) , (7 , 0 , 0) , · · · 5