MgO

Journal of Magnetism and Magnetic Materials 22 (1981) 311-318 © North-Holland Publishing Company

ESR LINEWIDTH MECHANISMS IN Ni2÷/MgO J.S. THORP and M.D. HOSSAIN

Department of Applied Physics and Electronics, University of Durham, UK Received in original form 4 August 1980; in revised form 16 October 1980 The electron spin resonance linewidth of Ni2+ in single crystal MgO was examined theoretically and experimentally at 9 GHz for a Ni2+ concentration of 1400 ppm. The experimental peak-to-peak linewidth for the AMs = -+1 transition had a value of about 6 mT over the temperature range 180 to 20 K and was polar angle dependent. The calculated dipolar linewidth exceeded that observed by a factor of about five; the ratio of moments M~/4/M[/2 derived from experimental data was 1.34; the lineshape was Lorentzian and independent of polar angle. The data suggested that Ni2+ entered the lattice substitutionaUy, occupying magnesium sites, and that the linewidth for the AMs = -+1 transition was determined by exchange narrowing together with inhomogeneous broadening due to internal strain.

1. Introduction Electron spin resonance (ESR) linewidth results o f Fe 3+, Cr 3+ and Co 2+ in magnesium oxide which showed exchange interactions have recently been reported b y Thorp et al. [ 1 - 3 ] . As part o f a further study on different ions o f the iron group, a similar investigation has been made o f nickel doped magnesia (Ni2+/MgO) in an attempt to provide specimens o f k n o w n structural characteristics on which electrical conductivity and dielectric loss measurement might subsequently be made. The esr spectra o f Ni2+/MgO was first observed b y Low in 1958 [4], and the characteristic parameters o f the spin-Hamiltonian for the transition AM s = + 1, were calculated for a cubic field, he assumed that the linewidth was due to b o t h dipolar and exchange interactions. Orton et al. [5,6] first observed the double quantum line, AM s = -+2, in Ni2+/MgO in the same position as the AM s = +1 transition, the width o f the broad single quantum line varied with crystal orientation and the appearance of the double quantum line was dependent on the microwave power incident on the sample. Later on the esr linewidth behaviour o f Ni2+/MgO was observed by several authors [ 7 - 9 ] in different esr experiments; they also noticed a variation o f linewidth behaviour from sample to sample according to the sample history and reported a range of magnitude of the broadened line from about 3 to 27 mT. However, there is little detailed information available either on the question o f the sites actually occupied by dopant atoms or the nature o f the interactions between the latter. Information o f this nature has been obtained by making a comparison between observed esr linewidths and those predicted from dipolar broadening [ 1 - 3 ] . Galeeva and Kochelaev (GK) in 1977 [10] estimated the numerical value o f the;esr linewidth o f the Ni 2+ ion in MgO b y considering the spherical point defects and including the interaction ,of paramagnetic ions with symmetric point defects and the s p i n - s p i n interaction o f paramagnetic ions via the strain field. It was decided to compare the experimental linewidths for the transition AM s = + 1 o f Ni2+/MgO With these theories. The configuration o f the Ni 2+ ion is 3d s , 3 F4 and the ground term in an octahedral field is the orbital sing,let 3A 2 (t~ge~)~ thus a suitable spin-Hamiltonian is ~f = g ~ H . S [9], where S = 1. The g value for Ni2+/MgO is 2.2145 [5], the deviation from g = 2.0023 being due principally to the admixture o f the term 3T2 (t2geg) s 3 through the s p i n - o r b i t coupling [9]. Therefore the expected esr spectrum o f Ni2+/MgO is a single line at a magnetic field Ho = hv/g~, where the transitions Ms = 1 ~ 0 andMs = 0 ~ - 1 are coincident [4]. This paper presents the results o f the linewidth comparison made for the I+ 1 )~-~ 10) transitions o f Ni 2+ /MgO. 311

J.S. Thorp, M.D. Hossain / ESR linewidth mechanisms in Ni2+/MgO

312

2. Experimental techniques The doped single crystal on which measurements were made was obtained from W. and C. Spiecers Ltd. (Cheltenham), having been grown by electrofusion using pure powdered nickel oxide and pure powdered magnesia (4N) as starting materials. The nickel concentration in the specimen examined was 1400 ppm, this having been determined by optical spectrographic analysis (Johnson-Matthey Ltd.) to an accuracy of about 2%. No analysis for other trace impurities was available but the esr spectra, taken at temperatures down to 20 K, showed no detectable lines due to other magnetic impurities (e.g. Fe, Cr, Mn). The crystalline quality was good and the colours of the crystal was pale green. The esr measurements were made using a conventional 9 GHz spectrometer [ 1], which was equipped with helium cryostats having temperature control devices. Spectra were recorded by sweeping the magnetic field slowly through a known range. The magnetic field calibrations were obtained using a proton resonance magnetometer system in which the probe could be located exactly in the position normally occupied by the specimen.

3. Experimental results Initial measurements were made at room temperature to establish the form of the spectrum. An example of this is shown in fig. 1. This spectrum at OH = 0 ° and qh-I= 0 ° shows a broad resonance line at about g = 2.215, which is the superposition of the transition I-1 ) "~ t0) and I+1 ) ~-~ 10) and a central dip with the same g value due to the transition I-1 ) ~ I+1 ) effected by the absorption of two quanta at moderate microwave power. The field value at which the transition occurred were compared with the values expected for Ni 2÷ in an octahedral site. There was close agreement and on this basis and in view of the similarity between fig. 1 and the spectrum reported in the literature [9,11 ], it was felt justifiable to attribute the spectrum to Ni 2÷ in octahedral sites. The peak-to-peak width of the dip at the centre is 1.4 mT at 293 K. A similar width has been reported in ref. [9]. It has been found that the appearance of the dip at the centre depends on the microwave power incident on the

266.?

29~.1 MAGNETIC FIELD (mT)

316.E ~'

t

272

H. =29S MAGNETIC FIELD (mT)

Fig. 1. ESR spectra of Ni2+/MgO (1400 ppm) at 293 K, 9.11 GHz, OH = 0° showing single and double quantum transitions, Fig. 2. ESR spectra of Ni2+/MgO (1400 ppm) 77 K, 9.14 GHz, OH = 0° showing single quantum transition only.

318

J.S. Thorp, M.D. Hossain / ESR linewidth mechanisms in Ni2+/MgO

313

28 DIPOLAR THEORY 2l*

20 16 -I<3

12

z

8 EXPERIMENTAL l, 0

@@@@@@0@@@@

i

I

I

1

I

i

20

/,0

60

80

100

120

POLAR ANGLE, OH [DEGREE)

Fig. 3. Variation of linewidth with polar angle OH, 9.14 GHz, 77 K, AMs = -+1, Ni2÷/MgO (1400 ppm).

sample. The appearance of the double quantum line as a function of incident power has been reported in the literature [5,7,9]. It was possible to adjust the microwave incident power such that only the AMs = -+1 transition was observed in the temperature range from 20 K to about 180 K. The appearance o f the double quantum line in the temperature range from 180 to 293 K could not be avoided. There was no change in resonance field either with temperature or polar angle. The linewidth, defined as the width between points o f maximum slopes, AHms , was obtained directly from the derivative plots. The resonance linewidth for the AMs = + 1 transition was 6.0 mT at OH = 0 ° and ¢ H = 0 ° in the temperature range 20 to 180 K and a spectrum for the AMs = +1 transition at 77 K is shown in fig. 2. Above 180 K the linewidth increases with a doublet character (as shown in fig. 1), the derivative width increasing to 10.2 mT at 293 K. Walsh [7] also observed the temperature independent linewidth in the temperature range 20 to 77 K for a powdered sample o f Ni:÷/MgO, having a concentration of about 0.1% (1000 ppm). The resonance linewidth AHrm, for the AM s = + I transition, was dependent on polar angle, and was maximum at OH = 0 ° and minimum at about 0n = 45 ° (when the magnetic field lay in the (100) plane, i.e. e H = 0°) as shown by the dotted line in figl 3. Low and Suss [12] mentioned that the occurrence o f the maximum linewidth at OH = 0 ° is a general characteristic o f the Ni 2÷ ion in alkaline earth oxide-host crystals.

4. Theoretical linewidth 4.1. Dipolar broadening It will be assumed in this calculation that the main contribution to homogeneous line broadening is dipoledipole interaction between Ni 2÷ ions. The second moment (A6o2) o f the linewidth caused by dipolar interaction between identical atoms in the magnesium oxide lattice has been derived in the literature [1]. For Ni 2÷, s = 1 and the g-value in the MgO lattice is 2.2145 [5] and so the atomic part o f the second moment is 2.3989 X 10 -2s n (rad s-1 )2 cm 6, which leads to the fmal equation for the second moment (A602) = 2.3989 X 102°n [15.9184 - 5.175 Y4,o(0H, qh-I) -

6.218Y4,4(0H,

~bH)] •

(1)

J.S. Thorp, M.D. Hossain / ESR linewidth mechanisms in Ni2+/MgO

314

T

>-

O3 Z

z I;3 tU 03

(a) DIPOLAR THEORY,AHrns =20.3 mT (GAUSSIAN) (b) EXCHANGE THEORY,AHms =l.5rnT (LORENTZIAN) (c) EXPERIMENTAL AHms=6.0mT (LORENTZlAN)

~E

IX 0 Z

258.0

H, =295.0 MAGNETIC FIELD (mT)

T

332'0

Fig. 4. Comparison o f experimental and calculated lineshapes o f Ni2+/MgO (1400 ppm), 9.14 GHz, OH = 0°, AM s = -+1.

For q~ = 0 °, eq. (1) becomes rotally real; by substituting the experimental value of n = 0.0014 (i.e. 1400 ppm) and 0H, the theoretical curve for the variation of dipolar linewidth AHms (dipolar) with polar angle can be obtained and is shown by the full line in fig. 3. Since the shape of a dipolar line is Gaussian [13], using the Gaussian lineshape expression for a derivative line [14] as

( H-no ~ ~ ll[ H-no ,~2 }] G~(H ) = \~_z~r/--~s(d~ppolar).)exPL-~-/~½ZXHms(dipolar)) - 1._,

(2)

where Ho is the resonance magnetic field, enables the derivative line shape at OH = 0° to be calculated; this is plotted in fig. 4a.

4.2. Exchange narrowing It will be assumed in this calculation that the lineshape due to exchange narrowing is Lorentzian and that the linewidth is independent of doping concentration [1-3]. The predicted half-linewidth A of the absorption spectrum at half height due to exchange narrowing can be evaluated by adopting the Anderson-Weiss formula [ 15 ] A = (Aw 2)(dipolar)M/h,

:

(3)

where J is the exchange energy in frequency unit and (Aco2) is the second moment caused by dipolai interaction in frequency unit, which can be calculated from eq. (1). The half-linewidth ~x (in frequency units) is related to the full linewidth at half height AHI/2 (exchange) (in magnetic field units) by

A = (27rg~/h)~AH]/2 (exchange). The MgO and NiO crystals have the same lattice structure and their lattice parameters differ by less than 1% (i.e. for MgO, ao = 0.4212 nm and for NiO, ao = 0.41686 n m [ 16]). Using the relation J = 3k/2zS(S + 1), where k is Boltzmann's constant and z = 6 for NiO lattice. The exchange energy of NiO has been calculated using the value of Weiss constant 0 = 2000 K [17] and is 5.209 X 1012 Hz. One can consider this value o f J = 5.209 × 10 x2 Hz as corresponding to 50% of the available sites occupied by Ni~* in MgO (i.e. the point corresponding to NiO). Assuming that the exchange energy is linearly proportional to the doping concentration (as was found in Ci/MgO and Co/MgO [2,3] the estimated exchange energy for 1400 ppm Ni2+ in MgO is 1.458 X 101° Hz. Substituting

J.S. Thorp, M.D. Hossain / ESR linewidth mechanismsin Ni2+/MgO

315

this in eq. (3) and using the Lorentzian shape factor AHms(exch)/AHl/2(exch ) --- 0.577 [14] the peak-to-peak derivative linewidth can be evaluated as AHms(exch ) = 1.57 mT for Ni2÷/MgO (1400 ppm). This value is very comparable to the exchange narrowing linewidths in MgO single crystals doped with Fe 3÷, Cr 3+ and Co 2÷ [ 1, 2 and 3, respectively]. Using the Lorentzian lineshape expression for a derivative line [14], =16( 1 H-Ho ~/r3+[ H-Ho )2]2 Y(H) k~_z2d_/ms(exch)] [ L ~½Z3J/ms(exch) ,

L,

(4)

the predicted derivative lineshape at OH = 0 ° was calculated and plotted in fig. 4b.

4.3. G.aleeva and Kochelaev (GK) estimation The esr linewidths of paramagnetic ions due to the strain field have been estimated on the basis o f isotropic elasticity theory by Galeeva and Kochelaev (GK) in 1977 [I0] ; the total linewidth has been given, on the assumption of a Lorentzian lineshape as ATGK = A1 + A2 + A3,

(5)

where At is the linewidth due to interaction o f paramagnetic ions with lattice defects, A2 is the linewidth due to interaction of paramagnetic ions, and A3 is the tinewidth due to spin-spin interaction due to the strain field (these are concentration dependent). The expression for A1 has been given as AI = Nkf20(3 - 2 . ) G 1 1 1 4 ~ G ( 1 - v)h,

(6)

where N is the concentration of paramagnetic ions per cm3;k/G = 2(1 + v)/3(1 - v), * where v is the Poisson ratio; GI 1 is the s p i n - p h o n o n coupling constant; ~ o is a tensor with the dimensions of volume and has been given by ~-'0

4 3 3 ~/r(RNi2+ -- RMg2+),

where RNi2+ and RMg2+ are the radii of the ions Ni 2+ and Mg2+, respectively. It has been reported by Shannon and Prewitt [18] that the effective ionic radius o f an ion depends on the coordination o f both cation and anion and the electronic spin state. According to their calculations, based on an effective ionic radius R (v102-) = 0.14 nm, the value for RNi2+ is 0.07 nm and for RM~2+ is 0.072 nm. On substituting these values in the expression for ~2o, eq. (6) becomes negative indicating the linewidth AI due to interaction of nickel ions with the lattice defects would be zero. The linewidth A2, due to spin-spin interaction has been given for Ni2+/MgO [10] as A2 = 4~2Ng 2/32/9x/~h = 1.64 X 10 -I 3N Hz. The expression for the contribution to the esr linewidth of the spin-spin interaction due to the strain field is given by GK [10] as rrN (3[(a + b ) s/2 - (a - b) s/~ ] - 30(v - 3)[(a + b ) 3/2 - (a - b) 3/2 ] A3 = 30 s/2 X 8G(1 - v)h

+302v[(a+b)l/2 _ ( a _ b ) , / 2 ] } G,~4 6

'

•

(7)

where a = 9 + 3v, b = x/81 + 9v 2 - 6v, G = po 2 . For Ni2+/MgO, G44 = 3600 m -1 , p = 3.6 X 103 kg/m 3 and u = 6 × 103 m/s. As a result A3 = 11.34 × 10 -16N Hz = 0.0113 X 10 -13N Hz which shows that A 2 is greater than A s . Therefore the estimated total GK linewidth of Ni2+/MgO is ATGK = (1.64 + 0.01134)NX 10 -13 Hz = 1.65134NX 10 -13 Hz. • Corrected formulae following private communication with authors [10].

(8)

J.S. Thorp, M.D. Hossain / ESR linewidth mechanisms in Ni2+/MgO

316

Table 1 Comparison between predicted linewidth of N1.2+/MgO on the basis of different theories at OH = 0° and q~H = 0° together with experimental value Concentration (ppm)

~qms(dipolar) (mT)

AHms(exch) (mT)

AHms(GK) (mT)

AHms(expt) AMs = 1, 77 K (mT)

1400

20.20

1.57

0.231

6.0

The linewidth estimated in eq. (8) is an absorption linewidth at half-height, which can be converted to the derivative linewidth AHms(GK). Therefore the derivative linewidth for a concentration of 1400 ppm Ni 2÷ ions in MgO according to the modified GK estimation becomes AHms(GK) = 0.231 mT. This is less than the values predicted from other calculations and also much less than the observed linewidth. The comparison of the predicted linewidths with the experimental linewidth is given in table 1 and their shapes in fig. 4.

5. Discussion

Two salient features emerge from initial comparison between the experimental result for the as grown crystal and those predicted on the basis of dipolar broadening. In the first place the predicted dipolar linewidth is about 5 times larger than the observed linewidth at OH = 45°; secondly the angular dependence of observed linewidth gives a maximum at 0n = 0 °, whereas the predicted dipolar linewidth shows a maximum at OH = 45 ° • The difference in the magnitude of the linewidths is rather tess than those encountered for other ions doped in MgO in which similar comparisons have been made and where the linewidths were determined by exchange narrowing (table 2). To determine the nature of the linewidth mechanisms, the observed derivative spectra were integrated numerically giving the integrated lineshapes and the lineshape factor AHms(obs)/AH1/2(obs) was 0.52.5. The moment ratio M~4/4/M~/2(M2 and )1//4 are the second and fourth moments, respectively) was 1.34. Both the lineshape factor and moment ratio were found to be almost independent of polar angle. The values of the lineshape factor and the moment ratio are similar to the figures quoted for Fe3÷/MgO, Cr3÷/MgO and Co2+/MgO as given in table 2. This shows that the lineshape is Lorentzian. Taken together with the result for the position of the angular depen-

Table 2 Comparison of predicted dipolar linewidth with observed linewidth and lineshape of different ions doped in MgO Doped ions

Predicted linewidth (dipolar)

Fe 3÷

~-I/4. t~Ar1:2 lvl4 liV12

A/-/ms/AHI/2 less than

Lineshape

Refs.

greater than

100 times larger than observed

1.42

0.577

Lorentzian

[1]

160 times larger than observed

1.33

0.577

Lorentzian

[2]

Co 2+

50 times larger than observed

1.35

0.577

Lorentzian

[31

Ni2+

5 times larger than observed

1.34

0.577

Lorentzian

present work

Cr a+

J.S. Thorp, M.D. Hossain / ESR linewidth mechanisms in Ni2+/MgO

317

12'0 10.0 8.0 6.0

0

0

0

0

"1"

0

0

0 0

0

I

I

I

20

40

0

0

0

~.0 itl Z

.~ 2.0

H/Ill00)

t

,

i

60

80

H//11101

•

90

t

HIll010)

POLAR ANGLE, eH (DEGREE) Fig. 5. The angular dependence of the linewidths for Ni2+/MgO ( ] 4 0 0 ppm), 9.14 GHz, 77 E, OH = 0 °. Open circles, peak-to-peak

linewidth, z.X//ms;crosses, absorption-half-amplitude linewidth, zXnl/2; solid lines, dependence predicted by eq. (9).

dence maximum, the data proves that dipolar mechanisms cannot be dominant. The observed Lorentzian lineshape strongly indicates that exchange narrowing is important. This suggests that the lineshape is Lorentzian and that exchange narrowing is important. Although the GK lineshape is the same as the observed Lorentzian lineshape, the magnitude of the strain broadened linewidth, due to point defects considered by GK, is about 25 times less than the magnitude of the observed linewidth. This proves that the GK estimation of linewidth, is insufficient to account for the width observed for this concentration if Ni 2÷ in MgO. On this assumption it is suggested that the principal operative mechanism is exchange narrowing. It is observed that the linewidth is about four times larger than that estimated from exchange narrowing, and we have examined the angular dependence of the linewidth in order to determine whether it is due to the presence of inhomogeneous strain in the crystal. The equation for linewidth due to internal strain (or stress) given by Feher et al. [19,20] of any transition for any angle (0, 4) of the magnetic field with respect to the cube axes of the crystal can be written as z 2 t a l / 2 = [a(1 - 3PO + bF] ~/2,

with

F = (sin20H COS20H + sin2$1-1 COS2¢H sin*0H),

(9)

where z2t/-/1/2 is the esr linewidth at half intensity; a and b are constants determined by the elastic constants and strain (or stress) distribution in the crystal lattice. A good fit to this equation has been found with experimental data as shown by the solid line in fig. 5; where a = 172 mT 2 , b = 124 mT 2 a n d F = sin20H -- sin40H (in the experimental situation ~H = 0 ° ) • Therefore the linewidth, AH 1/2, when H / / [100] (i.e. OH = 0 °, $H = 0°) is: ~ k n l / 2 [100] =

a 1/2

= 11.5 mT,

(lOa)

When H / / [111] (i.e. OH = 54.5 °, ~bH= 45°): ZM-/1/2 [111] = {b/3} 1/2 = 6.4 mT,

(10b)

and w h e n H / / [110] (i.e. OH = 45 °, $H = 0°): ~r/i/2 [110] = {(a +b)/4) t/2 = 8 mW.

(10c)

Therefore the values of &/-/1/2 with H along the directions [ 111 ], [ 110] and [ 100] are in the ratios 1 : 1.25 : 1.8.

J.S. Thorp, M.D. Hossain / ESR linewidth mechanisms in Ni2+/MgO

318

These ratios agreed well with the corresponding ratios 1 : 1.1 : 1.5 reported by Orton et al. for the esr linewidths of Ni2÷/MgO [6]. Several mechanisms of inhomogeneous line broadening due to strain have been discussed by Stoneham [21 ], such as strain broadening by point defects (which is not likely to make an important contribution) and strain broadening by line defects which may be the possible reason here. Lewis et al. [8] deduce a relation between the width o f the spin resonance line and the width of the strain distribution (irrespective of the shape) as hAOOoox -:GlxeE,

hAOOlll =G446T,

(11)

where Aoo denotes the linewidth at half intensity in frequency units corresponding to the cube axes; G l l and G44 are the coupling coefficients for Ni2+/MgO, G11 = 5700 m -1, G44 = 3600 m - l [22] ; eE and eT are the full width of the strain distribution at half intensity corresponding to H / / [ 100] and H / / [ 111 ], respectively. According to Stoneham's notation [23] eE = ~ e o o l ,

eT = ~ e , l l .

(12)

Using eqs. (I I) and (12) together with the linewidths obtained from eqs. (10a) and (10b), the ratio o f e o o l / e l 11 becomes 3.01, which is close to the theoretical value o f 2.3 given by Stoneham [23] for strain broadening due to dislocations. This value for Ni2+/MgO is very comparable with the values calculated by Stoneham [2 I] for Mn 2+, Fe 3+, Fe 2+ and Co 2+ (which lie in the range 1.0 to 4.0) and suggests that the line broadening is due to the presence o f strain which arises from in_homogeneous dislocations.

Acknowledgements We are particularly grateful to Professor B.I. Kochelaev and to Dr. NNI. Galeeva (Kazan State University, U.S.S.R.) for their communications. M.D. Hossain also wishes to thank the University of Rajshahi, Bangladesh, for the award o f a research scholarship.

References [1] J.S. Thorp, R.A. Vasquez, C. Adock and W. Hutton, J. Mater. Sci. 11 (1976) 89. [2] J.S. Thorp, M.D. Hossain and L.J.C. Bluck, J. Mater. Sci. 14 (1979) 2853. [3] J.S. Thorp, M.D. Hossain, L.J.C. Bluck and T.G. Bushell, J. Mater. Sci. 15 (1980) 903. [4] W. Low, Phys. Rev. 109 (1958) 247. [5] J.W. Orton, P. Auzins and J.E. Wertz, Phys. Rev. Lett. 41 (1960) 128. [6] J.W. Orton, P. Auzins, J.H.E. Griffiths and J.E. Wertz, Proc, Phys. Soc. London 78 (1961) 584. [7] M.M. Walsh, Phys. Rev. 122 (1961) 762. [8] M.F. Lewis and H.M. Stoneham, Phys. Rev. 164 (1967) 271. [9] S.R.P. Smith, F. Dravnicks and J.E. Wertz, Phys. Rev. 178 (1969) 471. [10] N.M. Galeeva and B.I. Kochelaev, Soviet Phys. Solid State 19 (1977) 787. [ 11 ] J.E. Wertz and J.R. Bolton, Electron Spin Resonance (McGraw-Hill, New York, 1972) p. 292. [12] W. Low and J.T. Suss, Solid State Commun. 2 (1964) 1. [13] P. Swarp, Can. J. Phys. 37 (1959) 848. [14] C.P. Poole, Electron Spin Resonance (John Wiley, New York, 1967) p. 775. [15] P.W. Anderson and P.R. Weiss, Rev. Mod. Phys. 25 (1953) 269. [16] R.W.C. Wyckoff, Crystal Structure V-1 (Interscience, New York, 1965) Chs. V and VI. [17] C.R.C. Handbook for Physics and Chemistry, 57th ed. (1977-78) p. E-120. [18] R.D. Shannon and C.T. Prewitt, Acta Cryst. B25 (1969) 925. [19] E. Feher and M. Weger, Bull. Am. Phys. Soc. 7 (1962) 613. [20] E. Feher, Phys. Rev. 136A (1964) 145. [21] A.M. Stoneham, Rev. Mod. Phys. 41 (1969) 82. [22] G. Watkins and E. Feher, Bull. Am. Phys. Soc. 7 (1962) 29. [23] A.M. Stoneham, Proc. Phys. Soc. London 89 (1966) 909.