. . . . . . . .. . . .. . . .. . .. . . .. . . .. . . ... . . . .. . . .. ... . ... . .. . . .. . .. . . . .. . .. . . .. . . . .. . .. . .. . . .. ... . . .. . .. . . .. . ... . . .. . . ... . . .. . . .. . . .. . ... . . ... . . . . .. . .. . . .. . . . . ... . ... ... . .... . . . . . . .. . . .. .. . .. . .. . . . ... . .. . . ... . . .. . .... . . .. . . .. . ... . .. . .... . ... . . . .. . .. . .... . . .. . . . .. . . . .. . . . . . . . . . . . . .. . . .. . .. . ... T 2 ture dependence of the precession frequency in the Gd- doped sample is almost identical to that in the pure sam- ple, demonstrating that the order parameter is following the intrinsic magnetism in the EuO host and is relatively insensitive to low levels of doping. In fact, at the 0.6% level of Gd-doping, 93% of the Eu ions have their full complement (twelve) of nearest-neighbour Eu ions, with most of the remaining Eu ions having only one of those nearest neighbours replaced by Gd. For both samples, the relaxation rate of the oscillatory component rises as T approaches T$_{C}$ from below and the low-temperature re- laxation rate of the Gd-doped sample is larger than that of the pure sample The muon spin precesses around a local magnetic field, B$_{µ}$ (with a frequency ν = ( γ$_{µ}$/ 2 π ) | B$_{µ}$ | , where γ$_{µ}$/ 2 π = 135 . 5 MHz T − $^{1}$). This local field ( B$_{µ}$ ≈ 0 . 22 T at T = 0) is a sum of various terms, including the Lorentz field B$_{L}$ , the hyperfine field B$_{hf}$ , the demagnetizing field B$_{demag}$ and the dipolar field B $_{dip}$. The latter quantity is a func- tion of the muon-site r µ and can be written as a sum over the magnetic ions in the crystal; the magnetic moment of the i th ion is m $_{i}$. In Eq. (1), D αβ i ( r $_{µ}$) is the dipolar tensor given by D αβ i ( r $_{µ}$) = µ 0 4 πR 3 i ( 3 R α i R β i R 2 i − δ αβ ) , where R i ≡ ( R x $_{i}$, R y z $_{i}$) = r µ − r $_{i}$. The behaviour of this tensor is dominated by the arrangement of the nearest- neighbour magnetic ions and leads to a non-zero local magnetic field for almost all possible muon sites [20]. On electrostatic grounds, the likely muon site in EuO is at the 1 4 1 4 1 4 position [see Fig. 2(a)], equidistant from four Eu cations and four oxygen anions. Because of the mag- netic anisotropy [21], the easy axis for the Eu moments is along the 〈 111 〉 set of directions and for this moment alignment the 1 4 1 4 1 4 position is a point at which the dipo- lar magnetic field actually vanishes. The value of B$_{dip}$ has been calculated for the case in which the muon is displaced from the 1 4 1 4 1 4 site and its position is allowed to vary along the [111] direction, see Fig. 2(b). The dipolar field vanishes at both 1 4 1 4 1 4 (muon site) and 1 2 1 2 1 2 (oxy- gen site) and increases sharply as the site moves away from these special positions of high symmetry. The two curves show the cases in which the muon displacement is the same or a different choice of 〈 111 〉 direction. In the former case, the dipolar field at the muon site is parallel to the moment direction; in the latter case (for which there are three possibilities), it lies along one of the crys- tallographic axes and its amplitude is reduced by a factor of √$_{3. Thus if the muon site is displaced from the}$ 1 4 1 4 1 4 position towards a particular oxygen anion, then there would be a contribution to the dipole field resulting in two precession frequencies with a ratio of √$_{3, and with}$ amplitudes in a 3:1 ratio. Since only a single frequency is observed, we conclude that the muon site is indeed at the FIG. 1: (Color online.) (a) Raw µ SR data for EuO. Above T C (at 70 K), P z ( t ) relaxes. An oscillating signal develops below T C (data shown for 1.5 K). (b) The same for Eu 0 . 994 Gd 0 . 006 O. (c) Plot showing FFT of the muon data for EuO as a function of temperature. The single precession frequency which follows the order parameter is clearly visible. (d) The extracted pre- cession frequency for EuO as a function of temperature. (e) The same experiment repeated for Eu 0 . 994 Gd 0 . 006 O. The line through the data in (d) and (e) is identical for the two samples and is a best fit of the data to the phenomelogical function ν ( T ) = ν (0)(1 − ( T /T C ) $^{α}$)$^{β}$, producing α ≈ 1 . 5, β ≈ 0 . 4). A more reliable extraction of the critical exponent β , focussing only on the critical regime, is given later in the paper. B α $_{dip}$( r $_{µ}$) = ∑ i D αβ i ( r $_{µ}$) m β i , (1)