J. Phys. Chem. 1984,88, 5255-5260
5255
Dynamic Interchange among Three States of Phosphorus(4+) in a-Quartz. 2. Y. Uchida,? J. Isoya,t and J. A. Weil* Department of Chemistry, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 0 WO Canada (Received: January 18, 1984)
The dynamic process due to electron jumping among three states with different sp hybrid directions in the quasitetrahedral P4+center [PO4l0in a-quartz has been investigated by 10-GHz electron paramagnetic resonance, over the temperature range 40 to 400 K. The relative populations (mole fractionsh andhI) of the ground state P(1) and two degenerate thermally excited states P(I1) were determined from the measured EPR absorption line intensity ratios and from the line positions of the averaged state P(A), respectively, in the slow and fast kinetic regions. The temperature dependence of the mole fractions has been explained by considering vibrational sublevels in the potential well describing each state. The jump rate was also obtained, via EPR absorption line-width analysis based on the Bloch equations, in both the slow and the fast regions. The characteristic parameters of the dynamic process, Le., energy separation and vibrational sublevels of the ground and excited states, and barrier height between these, have been determined.
I. Introduction Ions Ge3+ and P4+both occur substitutionally for Si4+ in aquartz, each having one unpaired electron. The Ge3+centers have been found to exhibit dynamic effects in their EPR spectra, describable as being due to the unpaired electron jumping among three different states (two are energetically degenerate).’ The relationship between several known germanium-related EPR absorption lines in a-quartz is clearly explained by this process. A similar dynamic effect in the paramagnetic P4+center in aquartz has been reported in a previous paper,* hereafter referred to as I. For example, X-irradiated phosphorus-doped samples at 120 K show two kinds of P4+EPR absorption lines with unequal intensities, Le., for the ground state P(1) and for two degenerate thermally excited states P(I1). We shall denote the EPR spectra from centers in these states with respectively these same symbols. All the phosphorus EPR spectra consist of well-separated (- 100 mT) 31Phyperfine doublets. The hyperfine matrices All, of P(1) and P(I1) are found to have almost uniaxial symmetry, and in both states the unique hyperfine axes are (very close to being) along respective 0-P-0 bisector directions of the quasitetrahedral cluster, denoted by [P04]o.2 The unique hyperfine axes for the three crystallographic equivalent sites are shown in Figure 1, along with the projection of a threefold screw axis Z and of the twofold rotation axes 2, ( i = 1, 2, 3) carrying Si(P) ions. In each state, the unpaired electron is considered to be located primarily in a phosphorus (mostly 3s 3p) orbital pointed along the unique hyperfine axis observed for that state. As may be seen in Figure 1, centers in state P(1) display twofold axis symmetry, whereas centers in states P(I1) do not. Thus it follows that the P4+ions in state P(1) are located on axes d,, at least on the time-average effective here, but not so in states P(I1). At each phosphorus site i, two excited states (denoted as P(I1):site i and P(I1):site i’) are observed; these are related geometrically by operation of 2, and (neglecting hyperfine effects) are energetically degenerate in zero applied magnetic field. While their spin-Hamiltonian energies may be distinct to that EPR line position differences become evident, no population differences between the two states P(I1) have been considered in our analysis. With increasing temperature, above 130 K, the IO-GHz EPR lines first of P(I1) and then of P(1) begin to broaden and above 170 K a new spectrum labeled P(A) emerges. It is shown in I that each line position of P(A) can be described as a weighted average of corresponding line positions of P(1) and P(I1); P(A) emerges as a result of sufficiently frequent jumps between states P(1) and P(I1). At each phosphorus site i, the electron jump occurs with equal probability between the ground state P(1) site i and
+
National Institute for Research in Inorganic Materials, 1-1 Namiki, Sakura-mura, Niihari-gun, Ibaraki-ken, 305 Japan. *University of Library and Information Science, Kasuga, Yatabe-machi, Tsukuba-gun, Ibaraki-ken, 305 Japan.
0022-3654/84/2088-5255$01.50/0
each of the two excited state P(1I) sites i and i’. Therefore, P(A) shows twofold symmetry. The line positions and widths of P(A) depend on the temperature. Over a wide temperature range above 170 K, the line width of P(A) depends sensitively on the spacing between the lines P(1) and P(I1) being averaged, and thus has a strong angular dependence (Figure 2). Initially, on warming from 170 K, P(A) is only detectable at some field orientations; later it is visible at all angles.*When the sample is rotated around axis dl with magnetic field B I B 1 ,the two EPR absorption hyperfine doublets, P(I1):site i and P(I1):site i’, superimpose for site 1 but not generally for sites 2 and 3. The details of the angular dependence of P(1) and P(I1) at 120 K and of P(A) at 290 K are shown in Figures 3-6 of I. The present paper will describe experimental determination of some characteristics parameters of this jump process, Le., energy separation and vibrational sublevels of the P(1) and P(I1) states and the barrier height between them. The respective spin concentrations (mole fractions) of states P(1) and P(1I) were determined in the infrequent-jump region from the EPR intensity ratio of P(I1) to P(I), and from the line position of P(A) in the frequent-jump region. The jump rate between P(1) and P(1I) was determined by performing line-width analyses based on the Bloch equations, both in the slow and the fast kinetic regions. Both the line broadening of the P(1) and P(I1) spectra and the averaging of these spectra to produce the P(A) spectrum can be considered separately for the two members of the 31Phyperfine doublet. At each temperature, the accuracy of measuring the relative population of the states and of the jump rate between them was statistically enhanced by collecting large numbers of data points, Le., rotating the crystal and utilizing both hyperfine components. 11. Experimental Section
The sample was the same as used in I, Le., a phosphorus-doped (P/Si 200 ppm) a-quartz crystal grown on a Z-plate seed (1 mm in thickness) using a weakly acidic N a H 2 P 0 4aqueous solution in a platinum tube container inside an internally heated pressure vesseL3 The crystal containing the seed had dim%nsions 10 X 5 X 3.4 mm, resp_ecLively, along Cartesian axes X (= crystal twofold axis 2,), Y,Z (= optic axis). EPR data were measured with the X-band spectrometer and cryogenic cavity system described in I and elsewhere! The sample temperature was kept constant by thermal conduction from a heat reservoir, whose temperature was controlled by either a Displex (1) J. Isoya, J. A. Weil, and R. F. C. Claridge, J . Chem. Phys., 69, 4876 (1978). (2) Y. Uchida, J. Isoya, and J. A. Weil, J . Phys. Chem., 83, 3462 (1979). (3) K. Hirota and Y . Uchida, Proceedings of the ACS/CSJ Chemical Congress, INDOR 371, Hawaii, 1979. (4) B. D.Perlson and J. A. Weil, Rev. Sci. Instrum., 46, 874 (1975); R. H. D. Nuttall and J. A. Weil, Can. J . Phys., 59, 1696 (1981).
0 1984 American Chemical Society
5256
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984
Uchida et al. measurements, and the sample temperature was obtained to f 2 K relative to T, by using previously made calibration curves. The sample alignment was carried out by observing in situ line site splittings on an oscill~scope.~ In the present work, all EPR data were measured by rotating the sample around one crystal twofold axis (E al), keeping B l d l , The magnitude B of the applied magnetic field flux density B was measured by use of a conventional proton N M R gaussmeter. The EPR microwave frequency (9.96 Iv I10.01 GHz) and the NMR frequency was measured by an electronic counter (HP5245L) outfitted with a transfer oscillator (HP5257A).
I
0: s i 0: o
(PI
a3
Figure 1. Three crystallographicallyequivalent Si(P) sites in the right a-quartz crystal structure (P3,21) shown pcojected onto the (0001) plane.
Also shown are the threefold screw axis Z (normal to the plane of the figure) and the twofold axes 2, (i = )C2, 3). The ground state P(1):site i and two energetically degenerate xcited states P(I1):site i and site i' are indicated depicting the 3s + 3 b y b r i d direction taken to be the unique axes of the hyperfine matrices , 4 3 1 ~ The positive side of the hybrid orbital was not determined experimentally and is assigned, from a simple crystal field approach, as indicated by the arrow direction (see I).
111. Results and Discussion Boltzmann Distribution. Since states P(1) and P(I1) are considered to be in thermal equilibrium, the mole fraction& of state P(1) andfII of state P(I1) will be described by the Boltzmann distribution; thusfiI/2& = exp(-A/kT). The factor occurs since states P(I1):site i and P(I1):site i'are taken to be energetically degenerate;&, describes the sum of the populations in these two states. Heref, and&, can be determined by EPR measurements of P(I), P(II), and P(A), and hence the apparent energy separation A between states P(1) and P(I1) can be obtained as a function of temperature. In the infrequent-jump region ( T 5 170 K for 10-GHz EPR), in which the P(1) and P(I1) spectra are separately observable, the intensity ratio of the P(1I) and P(1) absorption lines givesfII/& directly. The line shape for P(1) and for P(I1) was found close to being Lorentzian, and hence each absorption line intensity was assumed to be proportional to the product of the first-derivative peak-to-peak amplitude and the square of the peak-to-peak line width. In practice, data from the 31P hyperfine doublets at orientation BllZ were used. To obtain the correct intensity ratio, the effects of microwave power saturation and of magnetic field modulation broadening were minimized in the experiments. Below 80 K, the P(I1) signal was too weak for thef,,/f, ratio to be measured quantitatively. In the frequent-jump region ( T > 170 K), in which P(I1) or both P(1) and P(I1) have already disappeared and P(A) has appeared for at least some field orientations, the P(A) line positions yield the mole fractions by using the appropriate calculated line positions of P(1) and P(II), via the relation2 BA,i
50P
= hB1,i+
~ ? h I ( B I I , i + BII,it)
--* 0.2mT
Figure 2. The high-field EPR line of the 31P(A)doublet (site 1, v 19.99 GHz, T = 278 K) as a function of crystal rotation angle.'6 Here B l i j , , and 9 = 0 denotes blli. The line positions have been superimposed to place the emphasis on the angular dependence of line width. The angular dependence of the P(A) line positions ( T 290 K) was illustrated in I.
-
helium gas refrigerator with heater, liquid nitrogen bath with heater, dry ice-alcohol bath with heater, or a heated oil bath. In these heat reservoirs, except with the oil bath, the heaters were attached to a heat-conducting metal block located between the cold bath and the sample. The temperature T, of the heat reservoir was monitored at the metal block just above the cavity during EPR
i = 1, 2, 3
(1)
relating the line positions, valid for each 31Phyperfine component and at each crystal orientation. Here the relative unpaired ele'ctron average residence times in P(1) and P(I1) are expressed respectively as the mole fractionsf, andf,,. The line positions are functions of polar angle I9 and aziyuthal angle C givin_gthe direction of the applied magnetic field B relative to X , Y, Z . The line positions of P(1) and P(I1) needed in eq 1 can be calculated from the spin-Hamiltonian matrices previously measured2 at 120 K together with appropriate thermal correction for (none was needed for g ) ; the corrections were assumed to be the same for P(1) and P(I1). When P(A) is observable at many orientations of B l d , (Le., for T > 270 K), the mole fractions and the small thermal correction terms 6A,,, and 6b for A3Ipcan be simultaneously determined by least-squares fitting, using the line positions of the P(A) hyperfine doublets (sites 1 and 2, 3) measured at a set of rotation angles, say, every 30' (-24 data points). Details of this treatment were given in I for data taken at 290 K. In the temperature region 170 < T < 270 K, P(A) is observable only at those angles at which the line separation between P(1) and P(I1) is sufficiently small; as shown in Figures 3 and 4 of I, this situation 75, 135' and 40, 165' for the site 1 occurs near angles I9 low-field and high-field components of the 31Phyperfine doublet, respectively. At those angles Omin at which P(A) site 1 exhibits minimum line width (note Figure 2), the line separation between the P(1) and P(I1) spectra was assumed to be zero. These orientations were utilized to estimate the above-mentioned thermal correction factors, for the latter temperature range. At angles near Omin, line position data used in conjunction with eq 1 yielded values of the mole fractions.
-
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5257
Interchange among Phosphorus States in a-Quartz
r
2000
r
t
E
E1
I
&
EO
0 1 -
,
100
O O
~
"
T(K)
'
'
'
400
300
200
1
'
P(I)
/
500
(ARBITARY UNIT)
4
Figure 3. The temperature dependence of the apparent energy separation A between states P(1) and P(I1). The full curve shows the calculated
values obtained by considering vibrational sublevels in each P(1) and P(I1) potential well (see text).
-
TABLE I: Temperature Dependence of the P(1) P(I1) Process Parameters: Mole Fraction Ratio f1,/2f1,Apparent Energy Separation A between States P(1) and P(II), Average Lifetime 7,and the Effective Transverse Relaxation Time T2 T, K frI/2fr Afhc, cm-l lo%, s 108T2,s 82 104 121 142 148 162 178 193 206 228 253 279" 290 314 356 388 405b
0.006 0.020 0.033 0.056 0.062 0.097
297 283 287 285 286 263
0.158 0.187 0.254 0.314 0.376 0.406 0.453 0.527 0.594 0.631
248 240 217 204 190 182 173 159 140 130
51.0 15.0 2.9 0.37 0.20 0.10 0.033 0.014 0.01 1 0.0063 0.0043 0.0022 0.0019
59 58 54 57 53 56 52 49 54 49 32 24 23 20 17 15 14
'At 279 K, the thermal correction terms for Anip were found* to be 6Ais, = 0.345 mT, Ab = -0.136 mT. bAt 405 K, 6A,,, = 0.788 mT, 66 = -0.814 mT. The values obtained for the ratios fII/2fI and the apparent energy separation A are included in Table I. Figure 3 shows the temperature dependence of A, which is seen to decrease monotonically with increasing temperature. Clearly the temperature dependence of the mole ratio fII/2fI cannot be explained by a Boltzmann distribution with constant A. We ascribe this effect to differences in the potential energy curves of the states P(1) and P(I1). For simplicity, we shall assume that each potential curve allows only single-frequency harmonic vibrational modes, whose energy is expressed as ( m 1/2)hvIand ( n + 1/2)hvIIfor the states P(1) and P(II), respectively. The ratiof11/2fr was calculated by assuming validity of the Boltzmann distribution among these vibrational sublevels, giving the relation
+
no
fII/2fI =
2 exP(-[(m + '/Z)h.ll/W)
m=O
Figure 4. The configuration coordinate diagram of states P(1) and P(I1); Eo/hc = 370 cm-I, E I / h c = 1750 cm-', E I l / h c= 1380 cm-I, u I / c = 170 cm-I, and uII/c = 70 cm-'.
energy curves of the states P(1) and P(I1) are depicted by the configuration coordinate diagram in Figure 4. For simplicity, each potential well is assumed to be parabolic. The vertical axis shows the energy relative to the minimum of the P(1) potential well. The horizontal axis is a distortion coordinate Q; for instance, any change of an angle 0-P-0 may be such a coordinate. In each of the three states P(1):site i, P(I1):site i, and P(I1):site i', the phosphorus hybrid orbital containing the unpaired electron orients along each of three particular 0-P-0 bisector directions. Jumping of the electron from one such orbital to another results in nuclear displacements so as to attain the atomic geometric configuration corresponding to minimum total energy. Comparison of the above frequency parameters with known lattice vibration data of a-quartz is probably none too meaningful, since substitution of phosphorus for silicon changes the force constant of each chemical bond, and since our treatment assumes only single harmonic vibrational modes. Electron Jump Kinetics. As described in section I, the electron jumps at each phosphorus site i occur between the ground state P(1):site i and two energetically degenerate excited states P(I1):site i and P(I1):site .'i Actually, in principle, there are two kinds of kinetic processes; one is jumping between P(1):site i and P(I1):site i (or P(I1):site i'), and the other is direct jumping between P(I1):site i and P(I1):site i'. For brevity, the first of these processes will hereafter be referred to as P(1) s P(II), and the latter as P(I1,i) 3 P(I1,i'). Herein, process P(1) s P(I1) is treated with the site 1 data, which can be analyzed as a case of two interconverting chemical species. Since P(1) and P(I1) are each describable by an appropriate spin Hamiltonian without invoking any interaction term linking states P(1) and P(II), as described in I, the method based on the Bloch equations is useful in both the slow and fast kinetic regions. In this for an assembly of independent unpaired electrons each resident in thejth state (j = I, 11), the complex transverse magnetization M,(t) is described by the solution of the Bloch equations. The averaged total transverse magnetization when a dynamic process interconverting two chemical species occurs is given by eq 18 in ref 5. The EPR absorption line shape is proportional to the imaginary component
v(x) =
(n=O C expi-[(n + )/z)hv11 + EOl/kTJ)/(
P(I1)
Q-
(2)
The values of integers moand no were chosen to give satisfactory convergence of the respective sums. The full curve in Figure 3 shows the calculated A vs. T using the best-fit parameters v I / c = 1 7 0 cm-I, v I I / c= 7 0 cm-I, and Eo/hc = 370 crn-l. Here Eo denotes the energy difference between the potential well minima of states P(1) and P(I1). The low-temperature limit of A/hc is seen to be 320 cm-'. The parameters characterizing the potential
+
x2 - ax b x4 cx2 dx e
+
+
+
(3)
of this magnetization. Variable x is the angular frequency w wo = gefie(B- B o ) / h measured from the midpoint wo = gefieBo/h between the site 1 resonance line of P(1) and P(I1) in the absence of jumping. Here g, is the free electron Zeeman splitting factor and fie is the Bohr magneton. Parameters a, b, c, d , and e as functions offr -frI, 7-1 = 71-l q I - l , 6w, and T2 are given in Appendix A. Here 71 and 711 are respectively the mean lifetimes
+
(5) H. S. Gutowsky and A. Saika, J . Chern. Phys., 21, 1688 (1953).
5258 The Journal of Physical Chemistry, Vol. 88, No. 22, 1984
08
I
Uchida et al. Ob -
07 -
//
(C) 206 K
A
I-
05
-
0 4 -
03
9-
02
m 4
0'
a
A
-
E
( b ) 193K
06
-
0
E
01
v
n I n
0
m
a
31
0
( 0 ) 162
K
66 (rnT)--+ Figure 5. The first-derivativepeak-to-peak line width AB, vs. the line separation 6B between P(1) and P(II), at 162, 193, and 206 K. The squares and triangles show the site 1 data and site 2, 3 data, respectively. The lines from site 2, 3 of P(I), site 2, 3' of P(II),-site 2', 3 of P(I1) are pairwise magnetically equivalent in the rotation B I B 1 . The 6B values for sites 2, 3 were taken as the separation between line P(1) and the midpoint of lines P(I1) site 2, 3' and 2', 3. The dashed line indicates the line separation &Bo,Le., c = 0 in eq 3 (see text). For 6B < 6Bo,the line width plotted is that of P(A); for 6B > 6Bo,the line width is that of P(1). The curved lines were calculated with eq 3 and with the T value bestfitted to the site 1 data at each temperature. Although, at 162 K, P(I1) was observable at some orientations, no data from these lines were included. Above 170 K, the peaks assignable to P(I1) were not observed.
of states P(1) and P(I1). Parameters 6w = g&B/h is the separation between P(1) and P(I1) in the absence of jumping, in units of angular frequency, anisotropic in our case. Parameter T2 is the inverse line width 2/Awlj2 where Awl12 is the full-width at half-height of the basic absorption curve, Le., in the absence of jumping. Function u for c > 0 describes the presence of a single absorption line P(A), and two absorption lines P(1) and P(I1) for c < 0. Since at the lower temperatures, below 130 K, the line widths of P(1) and P(I1) are observed to be the same within experimental error, it has been assumed that TZp(I) and T2p(11) are identical (=T2) also at higher temperatures. Here the transverse relaxation time T2 can be estimated, using the Lorentzian lineshape approximation, from the line width measured at crystal settings for which 6B = 0, at which the P(1) G P(I1) process does not contribute to the observed line width. For a Lorentzian, = 31/2Aw,p, where the latter is the first-derivative peak-to-peak width. In eq 3, line shape u is given as a function of the EPR transition energy expressed in terms of angular frequency w = ~ P Y . In practice, our EPR spectra were taken in the field-sweep mode. In using the EPR line position data measured at constant frequency conditions, linearity between w and the magnetic field B was assumed by using w = g,P,B/ h , since the maximum separation between lines of P(1) and P(I1) is relatively small (ca. 8 mT) and since g g,. In obtaining data for the jump rate analysis, it proved unnecessary to carry out complete line-shape analyses, Le., com-
-
Figure 6. The peak-to-peak line width AE,, of P(A) vs. the separation 6B between the lines of P(1) and P(II), at 228, 253, and 279 K. The squares and triangles show the site 1 and site 2, 3 data, respectively.
parisons of observed (derived from eq 3) values of AB,, sufficed. At some orientations, at intermediate temperatures, spectra showing averaging not near the slow motion or fast motion limits were observable; these broad asymmetric EPR signals (generally having poor signal/noise ratios) were not used. Rather, at such a temperature, other crystal orientations yielded spectra suitable for the simpler analysis. We analyzed the line-width data under the assumption that the two 31Phyperfine components can be treated separately. The quantity 6B is the line separation of P(1) and P(I1) corresponding hyperfine components. The values of 6B were generally not the same for the two hyperfine components. No difference in the line width between the two hyperfine components of each state was observed in the slow kinetic region. In the fast kinetic region, the line widths of the two hyperfine peaks of P(A) were found tp be different: for example (Figure 1, ref 2), at 290 K and at BI(2, AB, = 0.12 mT for the low-field peak (6B = 3.90 mT) and AB, = 0.05 mT for the high-field peak (6B = 2.09 mT). Presumably these widths become equal at still higher temperatures. Figures 5-7 show the observed widths AB, plotted vs. the line separation 6B between P(1) and P(II), for various temperatures T > 160 K. At each temperature, a number of (AB,,, 6B) points is collectable by rotating the crystal and by utilizing both hyperfine lines. As shown in Figures 5-7, various stages of motional narrowing were observed at each temperature. The spacings 6B were obtained by calculating the P(1) and P(I1) line positions at each applied magnetic field direction using the 120 K spin-Hamiltonian matrices with the appropriate thermal correction for d31p. The line separation for which c = 0 (see Appendix A) at each temperature is denoted by 6Bo and is shown in Figure 5 by a dashed line. For 6B < 6Bo (Le., c > 0) it is the line width of P(A) which may be measured, and for 6B > 6Bo ( c < 0) the line width of P(1) and of P(I1) may be measured. Least-squares fittings to measured AB,, vs. 6B data were carried out, varying T in eq 3. This was done for either P(1) or P(A), using values of fI - fII and T2 determined as discussed, and using the site 1 data. The ABpp vs. 6B functions calculated with the best-fit T values are shown as curved lines in Figures 5-7. The best-fit T values are presented in Table I, along with T2. We note that a single T value for each temperature can describe all data points derived from different orientations and including both hyperfine components. In the fast
The Journal of Physical Chemistry, Vol. 88. No. 22, 1984 5259
Interchange among Phosphorus States in a-Quartz
IO“ -
04
( a ) 290 K 10‘0
-
io9
-
IOB
-
io7
-
106
-
02
-t I-
€
Y
a I a
G
m
a
I
( sec- I 1
01
0 01 O
0
1
2
W
3
4
8B
’
5
I
6
I
7
io5 -
I
8
(mT)-
0
Figure 7. The peak-to-peak line width ABpp of P(A) vs. the separation 6B between the lines of P(1) and P(I1) at 290, 314, 356, 388, and 405 K. The squares and triangles show the site 1 and site 2, 3 data, re-
2
4 IOOO/T
6
0
(K-l)
spectively.
Figure 8. The temperature dependence of the lifetimes T~ and q1for process P(1) 9 P(1I). The best-fit lines were calculated with eq 6 (see
kinetic region, the anisotropy of the P(A) line width decreases as the temperature increases, Le., as the complete averaging situation is approached. The remaining anisotropy of the P(A) line width at 405 K shows that the averaging is not complete at this temperature (ABpp = 0.07 and 0.06 mT respectively for the low- and high-field hyperfine components, at BllZ). Between -80 and -170 K, both P(1) and P(I1) signals were observable, at least at some orientation(s). Below 120 K, the line width of P(1) is equal to that of P(I1) and the jump rate was not measurable. At 142 and 148 K, T was estimated from the relation that the line width ABpp of P(I1) is approximately equal to 2(3-’/’)(T2-’ + f , ~ - ’ ) h / g ~ P ~ . In eq 3, we have treated jump processes P(I1):site 1 + P(1):site 1 s P(I1):site 1’ as a case of two interconnecting chemical species, i.e., P(1):site 1 G P(I1):site 1 P(I1):site 1’). The rate equations for the forward and reverse P(1) G P(I1) jumps are6
text).
-
+
d . / d t = -kIh
df,I/dt =
(4)
Rate constants kI and kIIare related to each other via the thermal equilibrium condition kdI = kIJII between P(1) and P(I1). Presence of process P(I1,i) + P(I1,i’) will not affect the thermal equilibrium attained by P(1) G P(I1). W i t h f , +fII = 1, the average liftime of each species is6
Here, the definition of T is the same as used in eq 3. Rate constants kI and kIIare each expressed, via the Arrhenius relation for a thermally activated process, as follows:
kI = CI e x p ( - ~ d k T )
k1, = CI, exp(-E,,/kT)
(6)
where E, and E,, are the effective energy differences between states P(1) and P(II), respectively, and the energy barrier between them, as depicted in Figure 4. The temperature dependence of T~ and T,,, as shown in Figure 8, obeys the Arrhenius relation to within (6) H. S . Gutowsky and C. H. Holm, J . Chem. Phys., 25, 1228 (1956).
experimental accuracy. Note that is smaller than T I C 1 over most of the observed temperature range, consistent with the low-temperature limit T~ = 0 3 . The cross-over visible in Figure 8 is consistent with the high-temperature limit T~~ = 2 q . The best-fit temperature-independent parameters, obtained with eq 6, are E I / h c = 1750 cm-’, EII/hc = 1380 cm-l, CI = 2 X l O I 3 s-I, and CII= 4 X 10l2s-l. We see that the difference (E, - EII)/hc = 370 cm-’ is larger by 50 cm-’ than the low-temperature limit of A/hc, presumably because of inaccuracies in our mole fraction and line-width analyses. We note that frequency factors CI and CIIand effective vibration frequencies uI and vII are of comparable magnitude. So far, we have treated the P(1) s P(I1) process using only site 1 data, taken in plane B l i i l , thus removing the need to consider the direct process P(I1,i) P(II,i’). Since the P(I1):site i and P_(II):sitei’ (i = 2 or 3) lines do not in general superimpose when B l i i , , the process P(I1,i) s P(I1,i’) could contribute broadening of the P(A):site 2, 3 lines, in addition to the line-width changes caused by P(1) s P(I1). In Figures 5-7, the line widths of P(1):site 2, 3 and P(A):site 2, 3 are plotted (open triangles) vs. the line separation between P(1) and-the averaged line positions of P(1I):site i and P(I1):site i’, all for B l i i l . These widths agree well with those derived from the jump rate 7-l of P(1) s P(I1) determined from site 1 data only. However, this agreement does not imply that P(I1,i) + P(I1,i’) does not oc_cur: any extra line broadening is likely to be very small since, for B l i i , , the separation between the P(I1):site i and P(I1):site i’ lines is very small in comparison with the P(1)-P(I1) line separation, as shown in Figures 5 and 6 of 1. The deviation of the line width of P(A):site 2, 3 from the calculated curve at 279, 290, and 3 14 K seems to be larger than at the other temperatures. However, it is unclear how P(I1,i) + P(I1,i’) could be responsible for this result. As discussed, three states of the p4+ center have been observed. These are thought to correspond to occupation of phosphorus 3s + 3p orbitals pointed along three of the six 0-P-0 bisectors of the PO4clusters (Figure 7 of I). The two ( u p ? such directions of P(I1):site i and P(I1):site i’ are related via crystal twofold rotation axis ii,, as are two (w,w’)of the bisectors not serving as directions for observed occupied orbitals. Thus there are ener-
5260 The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 400
69
I 5
,
-
TIK)
100
80
60
I
1
I
40
’\
J
62
0
5
j
I
ID
IOOO/T
-
15
ITINK)
20
25
Figure 9. The temperature dependence of the intrinsic transverse relaxation time T2derived from the line width at zero separation (6B = 0) between the lines of P(1) and P(II), using the Lorentzian line-shape approximation.
getically four types of orbitals along the six 0-P-0 bisectors in the PO4 cluster, two of which are doubly degenerate. Hence it is expected that there are three as yet undetected excited states, P(II1) and P(IV), in which the unpaired electron is located in orbitals along the third and the fourth type of 0-P-0 bisector directions, respectively. SincehI/2fI is relatively large (0.63) at 405 K,the mole fraction of the P(II1) state might be expected to be sufficiently large to be observable. However, no extra EPR line assignable to P(II1) was found at 405 K. One possible reason for this may be that the P(II1) lines have already begun broadening due to jumps to states P(1) or P(I1); another is that P(II1) may exist at much higher energy than P(1) and P(I1) and hence is not yet populated. In Figure 9, the effective transverse relaxation time T2data in Table I, determined at 6B = 0, are plotted as a function of temperature. As described above, this T2 is not affected by P(1) g P(I1). We see that an unexplained line-width increase occurs above 200 K. If one takes this extra broadening to be caused by process P(1,II) + P(III), then one may expect that TT1 -fIII/# exp(-EIII/kT). The activation energy estimated from Figure 9 is 630 cm-’. This value seems considerably too small to be EIII,since E I / h c = 1750 cm-’ in the case of P(1) F! P(I1). N o detailed measurements of the spin-lattice relaxation time T I have yet been made. Thus, at present, we do not have an ex-
-
planation for the extra line broadening and the nondetection of state P(II1).
i
t il
66
61
200
Uchida et al.
IV. Summary The following aspects of the dynamic P(1) G P(I1) interchange process were elucidated by analyzing the EPR lines from P(I), P(II), and P(A) over the temperature range 40-400 K: (1) Information about the energy difference A between the ground state P(1) and the excited state@)P(I1) was obtained from the mole fraction analysis. In particular, the temperature dependence of A determined by our EPR measurements is found to contain information about the potential well of each state. (2) The barrier height for the process P(1) P(I1) was estimated from the temperature dependence of the jump rate, obtained by line-width analyses based on the modified Bloch equations. Other aspects of the dynamical properties of the P+center are still to be clarified. Measurements at other magnetic field directions, at which the line separation between P(I1):site i and P(I1):site i’is larger, may be useful to investigate the P(I1,i) + P(I1,i’) process. To further examine the extra line broadening observable at 6B = 0, it will be necessary to measure P(A) at still higher temperatures, to see how this broadening changes with temperature. A detailed vibronic state calculation of the p+center will be necessary to examine the experimentally determined parameters and to investigate the distortion coordinate in Figure 4.
Acknowledgment. The research reported here was supported by the Natural Sciences and Engineering Research Council of Canada. The authors are greatly indebted to Dr. K. Hirota (National Institute for Research in Inorganic Materials, Japan) for his help in growing the phosphorus-doped quartz crystals. Appendix A
a=
b = y4[(1
cfr -fI1)6w
+ 7-lT2) - 7-’T2(f1- f I I ) ’ l ( d ~ ) ~ + ( 7 - 1 + T2-1)2
+ 27-lT2-l + 2T2-2]- X(~W)’ d = + 2T2-’)7-’6W e = ‘/16(6~)4+ 72[(7-1T2-1+ T2-2)+ y2(fI -frI)27-2](6~)2 + + T2-1)2T2-2 c =
[T-~
-fII)(7-*
(7-1
Registry No. P, 7723-14-0; a-quartz, 14808-60-7.
