THE E"₁ CENTER RE-VISITED

R. I. Mashkovtsev, D. F. Howarth, and J. A. Weil

Our previous paper [MWH2007]* on center E"₁ (deemed to be an oxygen O⁰ vacancy) has its limitations, some of which can now be addressed, in several ways. As a result of further work on the closely related defect center E"₃ also existing in alpha-quartz, various improvements in the techniques of analyzing the single-crystal EPR spectra have been attained.

(1)   One new feature is the analysis to arrive at  more accurate values of the electron-exchange energy parameter J.

We note from [MWH2007, Table IV] for E"₁ that in fact parameter J ′ ≡ J/(g_eβ_e) was close to undetermined when using only the main-line data during line-position fitting, without further effort. However, it was decided to make fitting runs while setting J ′ at selected values near the best-fit value, while keeping it constant, and thus determining the minimum RMSD. The result is shown in Figure 1a, below. The corresponding plots of RMSD vs J ′ for each of the ²⁹Si are presented in Figures 1b and 1c.

Figure 1a

Figure 1b

Figure 1c

The newly obtained matrices g₁ and g₂ are more accurate than the previous ones published [MWH2007, Table IV] and are closely similar but not identical with those.

The matrices g₁ and g₂ are far from uniaxial, but have identical isotropic components (2.00096), which also is that of the g (triplet state), and of the g matrices
of center E′′₃. Also, the average g_avg = (g₁ + g₂)/2 is exactly equal to g obtained from the triplet model [MHW2007, Table II]. The relationship between the two
matrices (if any) is not known at this point.

We note that ⁰D from the biradical analysis yields unique axis orientation ( θ , φ ) = 56.06° (1) , 56.49° (1), which is nicely close to the structural direction Si -- Si
given by 53.9°, 66.0° [MHW2007, Table I ].  The inter-Si (a, b) distance is estimated to be 0.611 nm  [MHW2007, Table VII ].

(3)    Better fitting of the ²⁹Si spectra has led to appreciable lower RMSD values, and identification of hyperfine lines in complex regions that now are realized
to arise from E"₁.  Thus more lines now contribute to the best-fit of their positions, as attained by computer program EPR-NMR.
Note that the hyperfine data describes single-occupancy by ²⁹Si. Although our previous publication did not report biradical-model matrices A(²⁹Si α or β),
these are now available, and should be compared with those derived from the triplet model [MHW2007, Table III].

Table 3. E′′₁: Biradical Model: Si_b hyperfine data. Matrices g and D, obtained from main line-position RMSD minimization,
were kept fixed, as was J ′ = -28(2) mT; only A₁ and A₂ were varied.  Matrices A₁ and A₂ were evaluated by line-position
difference minimization of 430 non-zero-weighted data points (ca. 143 for each of 3x2 sites).
Sum of weighting factors = 423.7; RMSD = 0.0227 mT.

We are pleased with the fact that each of the 3 RMSD values achieved (see Tables above) are appreciably less than the observed 1^st-derivative
linewidth (~ 0.03 mT).

The seemingly greater accuracy in [MHW2007, Table III] of matrices A (triplet state) as compared to the new ones (biradical state) may be
attributable to slight differences in the data sets used for each of the minimizations.

In the biradical model (unlike the triplet model), there are 2 hyperfine matrices (for unpaired electrons 1 and 2) for each ²⁹Si. We note that:

Si_a

For Si_a, the two matrices are literally identical, and we cannot discern whether the tabulated differences are real. The matrices are close to uniaxial,
so that the A_┴ principal directions are not that well defined.

The average unique axis directions,  62.3°  ,  34.0°   ↔   117.7°  ,  214.0°, are close to structural directions [MHW2007, Table VI].

The average principal value is 22.64 mT, i.e., ca. half that of the primary hyperfine splitting E′₁ (41.15 mT).

The A_avg data compares very well with the A_a found from the triplet-state analysis [MHW2007, Table III].

Si_b

For Si_b, the 2 matrices A are not as close to being identical as with Si_a, although their uniaxiality is closer.

Unique A_avg value (22.25 + 21.76)/2 = 22.0 mT; Triplet-state value = 21.99 mT [MHW2007, Table III].

The average unique direction:      62.2°  ,  84.3°

Triplet:    61.2°  ,  84.2°

Model:    44°     ,  86.3°  (poor resemblance of θ)

WHY:  A_┴  angles occur switched??

(4)   There is as yet no QM model for the parameters we have measured. However, the unique axis direction of D does match the quartz structure quite well, as discussed in [MHW2007].

Erratum for [MHW2007]:

Figure 6: Double ²⁹Si occupancy simulation. The magnetic field range shown is incorrect:
The low-field doublet is centered at 311.9 mT,
Of the central 6 lines, the median of the two upper-field doublets occurs at 331.8 mT.
The high-field doublet is centered at 351.7 mT.

Table I. The data therein are for 300 K, rather than 94 K.

Table IV. Symbol J shown should be J/(g_eβ_e).

Table VII. E"₃: columns 4 & 5 are wrong due to an unfortunate error in D'. These should read 2.69 and 5.37 mT, and 0.9885 nm.

* R. I. Mashkovtsev, D. F. Howarth and J. A. Weil, Biradical States of Oxygen-Vacancy Defects in α-Quartz, Phys. Rev. B 76, 214114 (2007).

12 June 2009