6 FIG. 3: (Color online) Plot of | Θ | /T$_{N}$ as a function of p$_{3}$ , where p$_{2}$ = 1 . 68469. FIG. 2: (Color online) Plot of ϵ as a function of J$_{2}$ . J$_{1}$ = − 1 B. Ratio p$_{3}$ = J$_{3}$/J$_{1}$ V. DETERMINATION OF J$_{2}$ AND J$_{3}$ FROM EXPERIMENTAL DATA ON β -MnO$_{2}$ A. The ratio J$_{2}$/J$_{1}$ The ratio J$_{3}$/J$_{1}$ cannot be determined uniquely from the experimental results on the magnetic neutron scat- tering or x-ray magnetic scattering. The ratio | Θ | /T$_{N}$ can be calculated as We now calculate the value of ϵ , at which J ( h = 1 , k = 0 , l = 2 + ϵ ) takes a maximum, as a function of J$_{2}$ , where J$_{1}$ = − 1. The value of ϵ is uniquely determined as a function J$_{2}$ as shown in Fig. 2. When J$_{2}$ = − 1 . 6039, we find ϵ = 2 / 7 (Yoshimori $^{2}$). Experimentally, the magnetic Bragg peak is observed at ( h = 1 , k = 0 , l = 2 . 29771) for β -MnO$_{2}$. Here we note that l = 2 . 29771 = 2 + ϵ , where ϵ = 2 / 7 + 0 . 0120. The value of l is slightly deviated from the value of 2 + 2/7, which is predicted by Yoshimori. 2 J ( h = 1 , k = 0 , l ) and its derivative are given by and respectively. J ( h = 1 , k = 0 , l ) has a maximum when or where Since the magnetic Bragg peak appears at l = 2 . 29771, we have indicating that J$_{2}$ < 0 since J$_{1}$ < 0. where p$_{2}$ = J$_{2}$/J$_{1}$ , p$_{3}$ = J$_{3}$/J$_{1}$ , Θ is the Curie-Weiss temperature (the value is negative for β -MnO$_{2}$) and T$_{N}$ is the N´eel temperature. The value of p$_{3}$ can be determined from the ratio | Θ | /T$_{N}$ since p$_{2}$ is already determined. We make a plot of | Θ | /T$_{N}$ as a function of p$_{3}$ in Fig. 3, where p$_{2}$ = 1 . 68469. As far as we know, there have been several reports on the experimental values of Θ and T$_{N}$ for β -MnO$_{2}$; Θ = − 316 K, T$_{N}$ = 84 K [Bizette and B. Tsai (1949)], 10 Θ = − 1050 K, T$_{N}$ = 92 K [Ohama and Hamaguchi (1971)], 3 Θ = − 783 K, T$_{N}$ = 92 K [Sato et al.(2000)], 8 respectively. Using the values of T$_{N}$ , Θ and the value of p$_{2}$ (= 1.68469) determined above, the value of p$_{3}$ can be calculated as p$_{3}$ = 0.537, 0.986, 1.091 using the data of Bizette and Tsai, 10 Sato et al., 8 and Ohama and Hamaguchi, 3 respectively. We note that the boundary of the helical phase is de- scribed by (the derivation will be shown later) The helical order can exist only when p$_{3}$ < 1 /p$_{2}$ . When p$_{2}$ = 1 . 68469 for MnO$_{2}$, p$_{3}$ should be lower than 0.5935. In other words, the ratio | Θ | /T$_{N}$ should be lower than 4.07853. Since T$_{N}$ = 92 K, this means that | Θ | should be smaller than 375.2 K. The value of | Θ | by Bizette and Tsai seems to be reasonable, while the values of | Θ | obtained by Sato et al. 8 and Ohama and Hamaguchi 3 are much higher than 375.2 K. Note that Yoshimori 2 predicts that the boundary is described by p$_{3}$p$_{2}$ = 1 / 2. This expression is not correct according to our calculation. | Θ | T$_{N}$ = J (0) J ( θ$_{0}$ ) = 4 + p$_{2}$ + 2 p$_{3}$ 2 p$_{2}$ + p$_{2}$ − 2 p$_{3}$ , (13) p$_{3}$ = 1 /p$_{2}$ or J$_{2}$J$_{3}$ = J 2 $_{1}$. (14) J ( h = 1 , k = 0 , l ) = − 8 J$_{1}$ cos( πl ) + 2 J$_{2}$ cos(2 πl ) + 4 J$_{3}$, J ( h = 1 , k = 0 , l ) dl = 4 π [2 J$_{1}$ sin( πl ) − J$_{2}$ sin(2 πl )] , cos( πl ) = J$_{1}$ J$_{2}$ . l = 0 ± ϵ, 2 ± ϵ, 4 ± ϵ, 6 ± ϵ, · · · , cos( πϵ ) = J$_{1}$ J$_{2}$ . J$_{2}$ J$_{1}$ = 1 . 68469 ,