content_image/407106.jpg content_image/407113.jpg content_image/407111.jpg content_image/407112.jpg content_image/407117.jpg content_image/407118.jpg content_image/407109.jpg content_image/407115.jpg content_image/407116.jpg content_image/407114.jpg content_image/407103.jpg content_image/407104.jpg content_image/407105.jpg content_image/407107.jpg content_image/407108.jpg content_image/407119.jpg content_image/407110.jpg
( 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),... 

## B. Magnetic Bragg peaks along the l direction with h = fixed integer and k = 0 

When k = 0 and h is a fixed integer, J ( h = fixed integer, k = 0, l ) is rewritten as 

<img src='content_image/407106.jpg'>

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 ), · · · . 

## C. Magnetic Bragg peaks along the l direction with h = k = half inetgers 

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 , · · · . 

## D. Magnetic Bragg peaks along the (0 , 0 , l ) direction 

For any l , J (0 , 0 , l ) can be expressed by 

<img src='content_image/407113.jpg'>

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 

<img src='content_image/407111.jpg'>

<img src='content_image/407112.jpg'>

<img src='content_image/407117.jpg'>

<img src='content_image/407118.jpg'>

where 

(ii) Helical spin configuration 

under the condition of | J$_{1}$/J$_{2}$ | < 1, where 

The difference is calculated as 

Then the helical spin order appears when 

If the conditions J$_{2}$ < 0 and | J$_{1}$/J$_{2}$ | < 1 are satisfied, the magnetic Bragg peaks appear at the wavevectors denoted by 

<img src='content_image/407109.jpg'>

where ϵ is defined as 

## E. Magnetic Bragg peaks along the ( h, 0 , 0) direction 

<img src='content_image/407115.jpg'>

<img src='content_image/407116.jpg'>

and 

<img src='content_image/407114.jpg'>

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. 

<img src='content_image/407103.jpg'>

<img src='content_image/407104.jpg'>

<img src='content_image/407105.jpg'>

<img src='content_image/407107.jpg'>

<img src='content_image/407108.jpg'>

<img src='content_image/407119.jpg'>

<img src='content_image/407110.jpg'>

