3 FIG. 3: Precession frequency extracted from µ SR data as a function of temperature close to T C for (a) EuO and (b) Eu 0 . 994 Gd 0 . 006 O. FIG. 2: (Color online.) (a) Muon site in EuO. The crystal structure for one-eighth of the unit cell is shown (the side of the cube is a/ 2). (b) Calculated dipole field along the 〈 111 〉 axes in EuO. The muon is located at ξξξ and the Eu moments are all aligned along [111] (solid black line) or one of [ ¯ 111], [1 ¯ 11] or [11 ¯ 1] (dashed red line). values of T$_{C}$ and the critical exponent β are similar in each case, though the value of β is quite sensitive to the precise value taken for T$_{c}$ . Due to the difficulty in sta- bilising the temperature better than ≈ 10 mK, we do not believe that the difference between the two values of β is significant. They are both close to 0.36–0.37 obtained using neutron scattering [26] and 0.38 obtained from a second order ϵ expansion for the Heisenberg ferromagnet with dipolar interactions [27]. Above T$_{C}$ , we observe simple exponential relaxation [Fig. 1(a,b)] with a relaxation rate λ . For zero-field re- laxation of muons initially polarized parallel to z , λ can be written in terms of field-field correlation functions us- ing λ = γ 2 µ 2 ∫ ∞ $_{−∞}$d t ( 〈 B$_{x}$ (0) B$_{x}$ ( t ) 〉 + 〈 B$_{y}$ (0) B$_{y}$ ( t ) 〉 ) [18]. When each Eu spin component fluctuates, it produces a field fluctuation via the resulting modulation of the dipo- lar and hyperfine couplings. Our measurements of the zero-field relaxation rate for the EuO sample are plotted in Fig. 4 (a less complete set of data for the Gd-doped sample is also shown). There is a small rise in λ as the temperature is lowered towards T$_{C}$ but apart from this λ remains just below ≈ 2 MHz for both samples across the entire range studied. 1 4 1 4 1 4 position in which the dipolar field is zero, so that the observed local field ( B$_{µ}$ = 0 . 22 T at T = 0) is due to a sum of the Lorentz field ( B$_{L}$ = µ$_{0}$M/ 3 = 0 . 80 T at T = 0), B$_{demag}$ and B$_{hf}$ . Since the sample is polycrys- talline and multidomain, we neglect B$_{demag}$ and deduce that the hyperfine field B$_{hf}$ < 0 (antiparallel to the mag- netization), as found for EuS [22], and takes the value B$_{hf}$ = − B$_{L}$ ± B$_{µ}$ , and so either − 0 . 58 or − 1 . 02 T. For both samples, the amplitude of the oscillatory compo- nent is reduced from the full value at low temperature [see Fig. 1(a,b)], but recovers on warming towards T$_{C}$ , so a fraction of muons may implant in some additional state which depolarizes the muon very rapidly. We note that a recent experiment [23] on SmS has shown evidence for the formation of a bound magnetic polaron consisting of an electron around the implanted muon, in which the electron localization is stabilised by exchange energy. This occurs in the paramagnetic state in which a ferromagnetic droplet is localized in the para- magnetic host. A similar effect has been noted in EuS [24] although the larger magnetization ensures it occurs at temperatures ≫ T$_{C}$ [25] and the same will be true in EuO in which the magnetization is even larger. There- fore such a muon-related polaron is not relevant for EuO in the studied temperature regime. The temperature dependence of the precession fre- quency for both EuO and Eu$_{0}$$_{.}$$_{994}$Gd$_{0}$$_{.}$$_{006}$O was followed near T$_{C}$ and the results are plotted in Fig. 3. The fitted These results can be compared with calculations on a localized Heisenberg ferromagnet which have been per- formed with EuO in mind [15, 16] (Fig. 4). The theory of Lovesey and Engdahl [15] includes only the dipolar coupling and has been evaluated for temperatures above 1 . 3 T$_{C}$ , assuming a muon site of 1 4 1 4 1 $_{4}$. Though underes- timating the observed experimental values, this theory does remarkably well in providing a good estimate of the rough size of λ , the discrepancy perhaps being due to neglecting the hyperfine contribution. It is known that critical fluctuations enhance the role of the hyperfine cou- pling over the dipole coupling [14] because B$_{dip}$ = 0 at the muon site in the ordered state and the peak in the susceptibility is at k = 0. Nevertheless, when the dipo- lar calculation is extended into the critical regime (just above, and very close to, T$_{C}$ ) it predicts a divergence in λ which is not observed. An earlier mode-coupling ap- proach [16] also predicts a very sharp increase in λ on