with (1, 0, 0). The detail of our phase diagram is rather different from that proposed by Yoshimori. 2 We use a nu- merical calculation approach in finding the distribution of the magnetic Bragg peaks in the fixed reciprocal lattice planes such as ( h, k, l ) with one index fixed. To this end we calculate the equi-energy contour plot of the negative sign of the Fourier transform of the spin Hamiltonian, J ( h, k, l ). The magnetic Bragg peaks are located inside the maximum equi-energy contour. The selection rule for the location of the magnetic Bragg peaks is the same as that derived by Yoshimori. 2 This numerical method has an advantage in visualizing the location of the magnetic Bragg peaks in the reciprocal lattice space. The nature of the phase transitions on the phase boundaries will be discussed. We follow the theory presented by Nagamiya. 9 We con- sider a lattice of magnetic atoms such as β -MnO$_{2}$. The unit cell can be chosen so that it contains one magnetic atom. On each magnetic atom, we assume a classical spin . Between the spin S$_{i}$ at the position R$_{i}$ and S$_{j}$ at R$_{j}$ , there is an Heisenberg-type exchange interaction. The Heisenberg spin Hamiltonian is expressed by where and The exchange interaction J ( R$_{ij}$ ) is not restricted to the nearest neighbors. We now use the Fourier transforma- tions of the exchange interaction and spin; with where N (= N$_{1}$N$_{2}$N$_{3}$ ) is the total number of spins, and S ∗ q = S$_{−}$$_{q}$ . The position vector R$_{i}$ is expressed by R$_{i}$ = n$_{1}$ a$_{1}$ + n$_{2}$ a$_{2}$ + n$_{3}$ a$_{3}$ ( n$_{1}$, n$_{2}$, n$_{3}$ are integers) , for any i . Instead of this condition, we impose a milder condition We look for the lowest minimum of Eq.(1) under the con- dition that corresponding to the first Brillouin zone. There are N$_{1}$N$_{2}$N$_{3}$ = N wavevectors in the first Brillouin zone. The spin Hamiltonian is rewritten as For convenience we assume that where This means that the wavevector q is given by The periodic boundary condition for S$_{i}$ leads to exp[ i q · ( N$_{1}$ a$_{1}$ )] = 1 , exp[ i q · ( N$_{2}$ a$_{2}$ )] = 1 , exp[ i q · ( N$_{3}$ a$_{3}$ )] = 1 . we have Noting that with where b$_{1}$ , b$_{2}$ , and b$_{3}$ are fundamental reciprocal lattice vectors and are given by where n$_{1}$ = 0 , 1 , · · · , N$_{1}$ , n$_{2}$ = 0 , 1 , · · · , N$_{2}$ , n$_{3}$ = 0 , 1 , · · · , N$_{3}$ , and a$_{1}$ , a$_{2}$ , and a$_{3}$ are the fundamental lat- tice vectors. We also define the reciprocal lattice vector G by 2 II. BACKGROUND: GENERAL THEORY FOR THE ORDERED SPIN STRUCTURE J ( − R$_{ij}$ ) = J ( R$_{ij}$ ) , R$_{ij}$ = R$_{i}$ − R$_{j}$ . J ( q ) = ∑ j ( = i ) J ( R$_{ij}$ ) exp( − i q · R$_{ij}$ ) , (2) S$_{i}$ = 1 √ N ∑ q S$_{q}$ exp( iq · R$_{i}$ ) , (3) S$_{q}$ = 1 √ N ∑ i S$_{i}$ exp( − i q · R$_{i}$ ) , NS 2 = ∑ i S 2 i = 1 N ∑ i ∑ q , q ′ S$_{q}$ · S$_{−}$$_{q}$ ′ exp[ i ( q − q $^{′}$) · R$_{i}$ ] = ∑ q S$_{q}$ · S$_{−}$$_{q}$ , (6) S 2 i = S 2 = 1 N ∑ q , q ′ S$_{q}$ · S$_{−}$$_{q}$ ′ exp[ i ( q − q $^{′}$) · R$_{i}$ ] , (5) H = − ∑ q J ( q ) S$_{q}$ · S$_{−}$$_{q}$ . (4) − N$_{1}$ 2 ≤ m$_{1}$ ≤ N$_{1}$ 2 , − N$_{2}$ 2 ≤ m$_{2}$ ≤ N$_{2}$ 2 , − N$_{3}$ 2 ≤ m$_{3}$ ≤ N$_{3}$ 2 , q$_{1}$ = m$_{1}$ N$_{1}$ , q$_{2}$ = m$_{2}$ N$_{2}$ , q$_{3}$ = m$_{3}$ N$_{2}$ . q = q$_{1}$ b$_{1}$ + q$_{2}$ b$_{2}$ + q$_{3}$ b$_{3}$ , H = − 2 ∑ i,j J ( R$_{ij}$ ) S$_{i}$ · S$_{j}$ , (1) G ( h, k, l ) · R$_{i}$ = 2 π ( n$_{1}$h + n$_{2}$k + n$_{3}$l ) = 2 π × integer . a$_{1}$ · b$_{1}$ = 2 π, a$_{2}$ · b$_{2}$ = 2 π, a$_{3}$ · b$_{3}$ = 2 π, [ a$_{1}$ , a$_{2}$ , a$_{3}$ ] = a$_{1}$ · ( a$_{2}$ × a$_{3}$ ) = a$_{2}$ · ( a$_{3}$ × a$_{1}$ ) = a$_{3}$ · ( a$_{1}$ × a$_{2}$ ) . b$_{1}$ = 2 π a$_{2}$ × a$_{3}$ [ a$_{1}$ , a$_{2}$ , a$_{3}$ ] , b$_{2}$ = 2 π a$_{3}$ × a$_{1}$ [ a$_{1}$ , a$_{2}$ , a$_{3}$ ] , b$_{3}$ = 2 π a$_{1}$ × a$_{2}$ [ a$_{1}$ , a$_{2}$ , a$_{3}$ ] , G ( h, k, l ) = h b$_{1}$ + k b$_{2}$ + l b$_{3}$ ( h, k, l are integers) ,