where we use Under this milder condition, the minimum of Eq.(1) is obtained simply by taking only that q for which J ( q ) has the maximum. Denoting this q by Q ( q = - Q being equally allowed), we have the minimum value of Eq.(4) as We also obtain The condition (6) can be written as for any R$_{i}$ . This indicates that Here we assume that where R$_{Q}$ and I$_{Q}$ are real vectors. Then we have Then the vector R$_{Q}$ is perpendicular to the vector I$_{Q}$ , and the magnitude R$_{Q}$ is the same as the magnitude I$_{Q}$ . From Eq.(8), we get The minimum energy E$_{min}$ is obtained as The spin vector is expressed by Figure 1 shows the structure (rutile) of β -MnO$_{2}$ where a = b = 4 . 396 ˚ A , c = 2 . 871 ˚ A , and u = 0 . 302. 8 The ex- change interactions J$_{1}$ , J$_{2}$ , and J$_{3}$ are defined by Fig. 1. The definition of J$_{1}$ , J$_{2}$ , and J$_{3}$ are the same as that used by Yoshimori. The N´eel temperature T$_{N}$ is equal to 92 K. Each Mn 4+ ion and surrounding six O 2 − ions form a cation-occupied deformed octahedron (see Sec. X). Here we calculate J ( q ) for MnO$_{2}$, where the wavenum- ber q , the lattice vectors a ( n ) ( n = 1, 2,. . . , 4) and c are given by The spin structure thus derived is the most fundamen- tal spin structure and is realized as a result of the mini- mum energy state in the classical spin system. The fer- romagnetic state ( Q = 0) and antiferromagnetic state ( Q = the zone boundary of the first Brillouin zone) are the special case of the spin structures. In general, Q is not related to the crystal structure, but is related to the details of the exchange interactions. where φ is the angle between ˆ R$_{Q}$ and x axis. where ˆ R$_{Q}$ and ˆ I$_{Q}$ are the unit vectors which are perpen- dicular to each other. For convenience, ˆ R$_{Q}$ and ˆ I$_{Q}$ are in the x - y plane. The z axis is perpendicular to the x - y plane. Then using the unit vectors e$_{x}$ and e$_{y}$ , we get the final result ∑ i exp[ i ( q − q $^{′}$) · R$_{i}$ ] = Nδ$_{q}$$_{,}$$_{q}$ ′ . − J ( Q )( S$_{Q}$ · S$_{−}$$_{Q}$ + S$_{−}$$_{Q}$ · S$_{Q}$ ) . S$_{i}$ = 1 √ N [ S$_{Q}$ exp( i Q · R$_{i}$ ) + S$_{−}$$_{Q}$ exp( − i Q · R$_{i}$ )] . NS 2 = 2 S$_{Q}$ · S$_{−}$$_{Q}$ + S$_{Q}$ · S$_{Q}$ exp[2 i Q · R$_{i}$ ] + S$_{−}$$_{Q}$ · S$_{−}$$_{Q}$ exp[ − 2 i Q · R$_{i}$ ] , S$_{Q}$ · S$_{Q}$ = 0 . (7) S$_{Q}$ = R$_{Q}$ + iI$_{Q}$, S$_{−}$$_{Q}$ = R$_{Q}$ − iI$_{Q}$, S$_{Q}$ · S$_{−}$$_{Q}$ = ( R$_{Q}$ + i I$_{Q}$ ) · ( R$_{Q}$ − i I$_{Q}$ ) = R$_{Q}$ · R$_{Q}$ + I$_{Q}$ · I$_{Q}$ S$_{Q}$ · S$_{Q}$ = ( R$_{Q}$ + i I$_{Q}$ ) · ( R$_{Q}$ + i I$_{Q}$ ) = R$_{Q}$ · R$_{Q}$ − I$_{Q}$ · I$_{Q}$ + 2 iR$_{Q}$ · I$_{Q}$ = 0 . (8) R$_{Q}$ = I$_{Q}$ R$_{Q}$ · I$_{Q}$ = 0 . R 2 $_{Q}$= I 2 $_{Q}$= 1 4 NS $^{2}$. E$_{min}$ = − NS $^{2}$J ( Q ) . (9) q = ( q$_{x}$, q$_{y}$, q$_{z}$ ) , a ( n ) = a cos[ π 2 ( n − 1)] , a sin[ π 2 ( n − 1)] , 0 , c = (0 , 0 , c ) . S$_{i}$ = 1 √ N [( R$_{Q}$ + i I$_{Q}$ ) exp( i Q · R$_{i}$ ) + ( R$_{Q}$ − i I$_{Q}$ ) exp( − i Q · R$_{i}$ )] = 2 √ N [( R$_{Q}$ cos( Q · R$_{i}$ ) − I$_{Q}$ sin( Q · R$_{i}$ )] = S [( ˆ R$_{Q}$ cos( Q · R$_{i}$ ) − ˆ I$_{Q}$ sin( Q · R$_{i}$ )] , S$_{i}$ = S [cos( Q · R$_{i}$ + φ ) e$_{x}$ + sin( Q · R$_{i}$ + φ ) e$_{y}$ ] , (10) FIG. 1: (Color online) The position of Mn 4+ ions (denoted by solid circles) and O 2 − (denoted by open circles) in β -MnO$_{2}$ (rutile-type structure, space group P4$_{2}$/mmm). a = b = 4 . 396 ˚ A . c = 2 . 871 ˚ A . J$_{1}$ , J$_{2}$ , and J$_{3}$ are the exchange in- teractions between Mn 2+ spins. One of the O atom is located at ( ua, ua ), where u = 0 . 302. 3 III. HELICAL SPIN ORDER FOR β -MnO$_{2}$ A. Calculation of J ( q ) for β -MnO$_{2}$