content_image/444098.jpg content_image/444099.jpg content_image/444102.jpg content_image/444107.jpg content_image/444096.jpg content_image/444097.jpg content_image/444101.jpg content_image/444104.jpg content_image/444105.jpg content_image/444100.jpg content_image/444103.jpg content_image/444109.jpg content_image/444108.jpg content_image/444106.jpg
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 

<img src='content_image/444098.jpg'>

The condition (6) can be written as 

<img src='content_image/444099.jpg'>

for any R$_{i}$ . This indicates that 

Here we assume that 

where R$_{Q}$ and I$_{Q}$ are real vectors. 

<img src='content_image/444102.jpg'>

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 

<img src='content_image/444107.jpg'>

<img src='content_image/444096.jpg'>

<img src='content_image/444097.jpg'>

<img src='content_image/444101.jpg'>

<img src='content_image/444104.jpg'>

<img src='content_image/444105.jpg'>

<img src='content_image/444100.jpg'>

<img src='content_image/444103.jpg'>

<img src='content_image/444109.jpg'>

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 

<img src='content_image/444108.jpg'>

where φ is the angle between ˆ R$_{Q}$ and x axis. 

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. 

## III. HELICAL SPIN ORDER FOR β -MnO$_{2}$ 

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 

<img src='content_image/444106.jpg'>

## A. Calculation of J ( q ) for β -MnO$_{2}$ 

