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Spin polarization in the relaxed excited state of the F center in case of saturated optical pumping
Solid State Communications, Vol. 27. pp. 1009-1011. © Pergamon Press Ltd. 1978. Printed in Great Britain.
0038-1098/78/0908-1009 $02.00/0
SPIN POLARIZATION IN THE RELAXED EXCITED STATE OF THE F CENTER IN CASE OF SATURATED OPTICAL PUMPING K. Imanaka, T. Wada, T. Iida* and H. Ohkura Department of Applied Physics, Osaka City University, Osaka 558, Japan and *Department of Physics, Osaka City University, Osaka 558, Japan
(Received 26May 1978 by Y. Toyozawa) The dependence of the F center MCP on the polarization of pumping light is explained in terms of the spin polarization in the RES (relaxed excited state) which is derived by examining the spin-mixing processes; the spin-mixing parameter in the RES and that associated with the relaxation from the URES (unrelaxed excited state) to RES are evaluated. A new ESR method in the RES is proposed. IN THE PREVIOUS WORK on the vibronic theory of magnetic effects in the relaxed excited state (RES) of the F center [ 1], we have rigorously derived that the magnetic circular polarization of emission, AucP, is represented by a sum of para- and dia-magnetic components, Ap and As, respectively; the former is linearly proportional to the spin polarization in the RES, P*, and the latter is linearly proportional to the applied magnetic field, H, respectively. We have measured AMCp for KBr at 4.2 K as a function of H up to about 4 KG using a H e - N e laser of 20 roW; the result is shown in Fig. 1. It shows that the magnetic field dependence of A i c p is relevant to the polarization of steady pumping light (o., o_ and n). Almost the same result was observed at 1.9 K by Baldacchini et al. [2]. However, no explanation of this dependence of AMCP on the polarization of pumping light has been performed yet. In this letter, we show that this dependence can be explained in terms of F*, the formula of which is derived by re-examining the spin-mixing processes in the rate equations that govern the optical pumping cycle of an isolated F center. Here, unless otherwise stated, we discuss only the case of saturated optical pumping, in which the optical pumping rates by intense light power overcome the spin-lattice relaxation rates in the ground Kramers doublet. For this case, Mollenauer and Pan [3] derived that P* = 0, even if the pumping light were circularly polarized. Thus, the expected AMCp consists of only Aa; so that AMCP would be independent of polarization of pumping light. This is quite contrary to the observation. We point out that this discrepancy is partly due to their assumption that the spin-mixing process in the unrelaxed excited state (URES) is specified by a single parameter of e. In principle, the spin-mixing in the
optical transition between the Kramers doublets is represented by a single parameter, but not always for the case between the Kramers and non-Kramers doublets; notice that all of the URES's participating in the optical absorption are not the Kramers doublets. In order to precisely describe the magnetic circular dichroism in the optical absorption of the F center, Winnacker et al. [4] introduced two spin-mixing parameters of e± for the URES depending on the initial ground Kramers doublet with m8 -- + (1/2). Mauser et al. [5] evaluated the values of e± for KBr as a function of transition energy. In our argument on the rate equations, we newly introduce two more spin-mixing parameters in addition to e± ; these parameters are denoted as e* and erz. The former parameter represents the spin-mixing in the RES which is a Kramers doublet with MK = -+ (1/2) [1];the value will be evaluated later. The latter one represents the spin-mixing in the course of radiationless relaxation from the URES to the RES;it is usually assumed that, in such a process, cascade-down transitions occur successively over a great number of spin-mixed intermediate states. Although not all these intermediate states are the Kramers doublets, we tacitly assume that the spinmixing parameter at every ith state is described by a single parameter of e~(~.As for the verification of this assumption, the detailed information of the radiationless relaxation process should be required. Unfortunately, this has not been obtained yet, its study remains as a future problem. If it is assumed that er(~)'sare very small, the terms, which are higher than second order, can be neglected; this allows us to approximate the spin-mixing during cascade-down relaxation processes by a single parameter of erl "" X(0e!~. As the summation were taken over a great number of states, the resultant value of erl may become a significant amount, even if each e(~) is so small.
1009
1010
SPIN POLARIZATION IN THE F CENTER URES
KBr x10-3
z, 3 2
(-'.~-~-~.-~-~-~ A..,.. ~rt
T=4.2 K
RES
O'+pump
n•_ ~
'
~
.~
MK= ~"1/2 ,
MK=- ~"2
~ pump U+I'1
tJ
~r <3
Vol. 27, No. 10
0 -1 -2
~
I .s..
~ O'.pum p
....
E*n~/. I (l-U'ln~ ',
U_ft.
i m,o%
~o
H(kG)
'~=~"2
Fig. I. AMCP is plotted as a function of magnetic field up to about 4 kG for KBr at 4.2 K under fixed polarization of pumping light at o+, o_ and ft. Now, with reference to Fig. 2, the rate equations of the isolated F center are written as follows, dn±/dt = - - u ± n ± + [ ( 1 - e*)n*± + e*n*]/r,
%:.v2 ms=- I/2
n.
Fig. 2. Energy levels and transition rates including spinmixing rates for the optical pumping cycle of the F center under saturated optical pumping condition in the presence of magnetic field. Kramers doublets of the ground state and the RES are shown at the lower left and the upper right hand side, respectively. The spinmixing rate from the URES to the RES is represented by erz.
and argument. With these relations, one can derive the following relations,
dn*/dt = (1 -- err)A± + erv4~ - - n * / r , with A± = (1 --e±)u±n± + e~u~n~,
P*(o+pump) = - - P * ( o _ p u m p ) d= 0 (1)
where n± and n±* are the populations in the Kramers doublet with m e = + (1/2) in the ground state and in that with M r = -+ (1/2) in the RES, respectively, u± are optical pumping rates from the m s = + (1/2) levels; r is the radiative life time of the RES. From the stationary state solutions of equation (1), the spin polarization in the RES, P*, which is defined as (n*_ -- n*)/(n*_ + n*), is derived as follows, P* = (1 -- 2ert)(e_ -- e+)/[(1 -- 2err)(1 -- e_ -- e+) x (1 -- 2e*) -- 1 ].
(2)
P* is not zero except the case when e_ = e+ = (1/2)e and/or err = (1/2): the former corresponds to the case in [3]. It is found that P* is determined solely by the spin-mixing parameters in the cycle and is independent of pumping power. Notice that P* is irrelevant to temperature and magnetic field in the case of saturated pumping. Now, let us explain the observed dependence of AMCP on the polarization (of pumping light) by checking the dependence of P* in equation (2) on the polarization. Although e* and err are postulated to be independent of the polarization, e± are dependent on the polarization in the manner of e+(o± pump) = e_(o~ pump) and e+0r pump) = e_(Tr pump), respectively; these were derived in [5] from the symmetry
and P*(~ pump) = 0.
(3)
Thus, by using the above-mentioned relations [1 ] of AMCP = Aa + Ap and Ap ccp*, it is found that, AMCP (7r pump) = Ad
(4)
and
AMCP (O± pump) = &a -+ -~p-
(5)
Equation (4) shows that when the pumping light is 7r-polarized, AMc P must be linearly proportional to the magnetic field. This zs consistent with the experimental results in Fig. 1. A slope of AMCp(~ pump) is found to be (16 -+ 2) x 10 -s in [G] -l, which agrees sufficiently with (17 + 2) x 10 -a in [G] -l obtained in [2]. From equation (5), it is found, consistently with the experiment, that the difference between IAMcp(O± pump)] and [AMCP(TTpump)[ is given by Ap. The intercepts of extrapolation of AMcp(o± pump) at H = 0 are (1.4 + 0.1) × 10-3; the value is almost equal to (1.1 + 0.1) x 10 -a which is the value of Ap measured by modulated pumping method in [2]. From Ap, the value of P* is estimated [1] as 4.17 x 10 -3. Experiment also shows the value of AMcv(rt pump) at the magnetic field where AMCe (o_ pump) = 0 is the same as the value of Ap obtained above; the fact endorses the consistency of equations (4) and (5).
Vol. 27, No. 10
SPIN POLARIZATION IN THE F CENTER
Now, let us evaluate the spin-mixing parameters to explain the estimated P* which is seemingly a very small amount. The spin-mixing parameter in the RES, e*, is defined as follows,
e* = w( + ~ - ) / [ w ( + ~ - ) =
w(--~+)llW(--~+)
+ w(+ --, +)] +
w(-~-)],
(6)
where W(+ ~ 4-) and W(-- --> 4-) are the radiative transition rates from M K 4- (1/2) levels in the RES to the Kramers doublet of m s = 4- (1/2) in the unrelaxed ground state, respectively (see Fig. 2). By using equation (5) in [1] which was derived for the zeroth moment, we can calculate W(+ ~ 4-) as follows, ----
"0
n
M
x (1/2 ÷, + 1/2))12 ,
(7)
where 77= (o+, o_ and n). W(-- ~ +) are obtained simply by replacing ~ l in equation (7) as ~1 (1/2 ÷, -- 1/2). By adopting the parameter values of vibronic and spinorbit interactions determined in [1 ], the values of e* for KF, KCI, and KBr are computed from equations (6) and (7) as 5.5 x 10 -4, 5.5 x 10 -3 and 5.1 x 10 -2, respectively; they are a very small amount. From the comparison of the thus estimated value ofF* for KBr with equation (2), the value of erl is estimated as 0.43 by
1011
using values of thus calculated e* and those of e_+at the H e - N e line of 6328 ~, which are adopted from [5]. Based on this scheme, we propose a new method for the electron spin resonance (ESR) in the RES; this will be observed by monitoring the change of AM ca(O+_pump) caused by the ESR transition. When the ESR transition is induced in the RES, the rate equations are reformed by simply adding terms of IV~ ( n ~ - n_+*) to the second equations in equation (1), where W~ is the ESR transition probability in the RES. From the stationary solutions of the rate equations, the spin polarization under the ESR transition, P ~ , is represented by replacing the denominator of equation (2) as [(1 -- 2er, )(1 -- e_ -- e+)(1 -- 2e*) -- 1.-- 2W~r]. Therefore, when the ESR occurs, it is predicted that IP~I is decreased; so that IAMCp(O_+pump)l is reduced down to IAMcv(Tr pump)l. Here, it is ascertained that the change of P* induced by the ESR of ground state is very small in comparison with that of '.he RES [6]. This implies that the ESR signal of the RES will be avoided from the mixing of the ESR signal of the ground state. This is an advantageous point in this proposed method. The work is scheduled in our laboratory. Details of this Letter will be published.
Acknowledgements - We are obliged to Dr. Y. Mori for his efforts on experimental set-up. The work is partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education in Japan.
REFERENCES
1.
IMANAKA K.,IIDA T.& OHKURA H.,J. Phys. Soc. Japan 44,1632 (1978).
2.
BALDACCHINI G., GRASSANO U.M. & TANGA A., Phys. Rev. B16, 5570 (1977).
3.
MOLLENAUER L.F. & PAN S., Phys. Rev. B6, 772 (1972).
4.
WINNACKER A., MAUSER K.E. & NIESERT B., Z. Phys. B26, 97 (1977).
5.
MAUSER K.E., NIESERT B. & WINNACKER A., Z. Phys. B26, 107 (1977).
6.
HAHN K., MAUSER K.E., REYHER H.J. & WINNACKER A., Phys. Lett. 63A, 151 (1977).
0038-1098/78/0908-1009 $02.00/0
SPIN POLARIZATION IN THE RELAXED EXCITED STATE OF THE F CENTER IN CASE OF SATURATED OPTICAL PUMPING K. Imanaka, T. Wada, T. Iida* and H. Ohkura Department of Applied Physics, Osaka City University, Osaka 558, Japan and *Department of Physics, Osaka City University, Osaka 558, Japan
(Received 26May 1978 by Y. Toyozawa) The dependence of the F center MCP on the polarization of pumping light is explained in terms of the spin polarization in the RES (relaxed excited state) which is derived by examining the spin-mixing processes; the spin-mixing parameter in the RES and that associated with the relaxation from the URES (unrelaxed excited state) to RES are evaluated. A new ESR method in the RES is proposed. IN THE PREVIOUS WORK on the vibronic theory of magnetic effects in the relaxed excited state (RES) of the F center [ 1], we have rigorously derived that the magnetic circular polarization of emission, AucP, is represented by a sum of para- and dia-magnetic components, Ap and As, respectively; the former is linearly proportional to the spin polarization in the RES, P*, and the latter is linearly proportional to the applied magnetic field, H, respectively. We have measured AMCp for KBr at 4.2 K as a function of H up to about 4 KG using a H e - N e laser of 20 roW; the result is shown in Fig. 1. It shows that the magnetic field dependence of A i c p is relevant to the polarization of steady pumping light (o., o_ and n). Almost the same result was observed at 1.9 K by Baldacchini et al. [2]. However, no explanation of this dependence of AMCP on the polarization of pumping light has been performed yet. In this letter, we show that this dependence can be explained in terms of F*, the formula of which is derived by re-examining the spin-mixing processes in the rate equations that govern the optical pumping cycle of an isolated F center. Here, unless otherwise stated, we discuss only the case of saturated optical pumping, in which the optical pumping rates by intense light power overcome the spin-lattice relaxation rates in the ground Kramers doublet. For this case, Mollenauer and Pan [3] derived that P* = 0, even if the pumping light were circularly polarized. Thus, the expected AMCp consists of only Aa; so that AMCP would be independent of polarization of pumping light. This is quite contrary to the observation. We point out that this discrepancy is partly due to their assumption that the spin-mixing process in the unrelaxed excited state (URES) is specified by a single parameter of e. In principle, the spin-mixing in the
optical transition between the Kramers doublets is represented by a single parameter, but not always for the case between the Kramers and non-Kramers doublets; notice that all of the URES's participating in the optical absorption are not the Kramers doublets. In order to precisely describe the magnetic circular dichroism in the optical absorption of the F center, Winnacker et al. [4] introduced two spin-mixing parameters of e± for the URES depending on the initial ground Kramers doublet with m8 -- + (1/2). Mauser et al. [5] evaluated the values of e± for KBr as a function of transition energy. In our argument on the rate equations, we newly introduce two more spin-mixing parameters in addition to e± ; these parameters are denoted as e* and erz. The former parameter represents the spin-mixing in the RES which is a Kramers doublet with MK = -+ (1/2) [1];the value will be evaluated later. The latter one represents the spin-mixing in the course of radiationless relaxation from the URES to the RES;it is usually assumed that, in such a process, cascade-down transitions occur successively over a great number of spin-mixed intermediate states. Although not all these intermediate states are the Kramers doublets, we tacitly assume that the spinmixing parameter at every ith state is described by a single parameter of e~(~.As for the verification of this assumption, the detailed information of the radiationless relaxation process should be required. Unfortunately, this has not been obtained yet, its study remains as a future problem. If it is assumed that er(~)'sare very small, the terms, which are higher than second order, can be neglected; this allows us to approximate the spin-mixing during cascade-down relaxation processes by a single parameter of erl "" X(0e!~. As the summation were taken over a great number of states, the resultant value of erl may become a significant amount, even if each e(~) is so small.
1009
1010
SPIN POLARIZATION IN THE F CENTER URES
KBr x10-3
z, 3 2
(-'.~-~-~.-~-~-~ A..,.. ~rt
T=4.2 K
RES
O'+pump
n•_ ~
'
~
.~
MK= ~"1/2 ,
MK=- ~"2
~ pump U+I'1
tJ
~r <3
Vol. 27, No. 10
0 -1 -2
~
I .s..
~ O'.pum p
....
E*n~/. I (l-U'ln~ ',
U_ft.
i m,o%
~o
H(kG)
'~=~"2
Fig. I. AMCP is plotted as a function of magnetic field up to about 4 kG for KBr at 4.2 K under fixed polarization of pumping light at o+, o_ and ft. Now, with reference to Fig. 2, the rate equations of the isolated F center are written as follows, dn±/dt = - - u ± n ± + [ ( 1 - e*)n*± + e*n*]/r,
%:.v2 ms=- I/2
n.
Fig. 2. Energy levels and transition rates including spinmixing rates for the optical pumping cycle of the F center under saturated optical pumping condition in the presence of magnetic field. Kramers doublets of the ground state and the RES are shown at the lower left and the upper right hand side, respectively. The spinmixing rate from the URES to the RES is represented by erz.
and argument. With these relations, one can derive the following relations,
dn*/dt = (1 -- err)A± + erv4~ - - n * / r , with A± = (1 --e±)u±n± + e~u~n~,
P*(o+pump) = - - P * ( o _ p u m p ) d= 0 (1)
where n± and n±* are the populations in the Kramers doublet with m e = + (1/2) in the ground state and in that with M r = -+ (1/2) in the RES, respectively, u± are optical pumping rates from the m s = + (1/2) levels; r is the radiative life time of the RES. From the stationary state solutions of equation (1), the spin polarization in the RES, P*, which is defined as (n*_ -- n*)/(n*_ + n*), is derived as follows, P* = (1 -- 2ert)(e_ -- e+)/[(1 -- 2err)(1 -- e_ -- e+) x (1 -- 2e*) -- 1 ].
(2)
P* is not zero except the case when e_ = e+ = (1/2)e and/or err = (1/2): the former corresponds to the case in [3]. It is found that P* is determined solely by the spin-mixing parameters in the cycle and is independent of pumping power. Notice that P* is irrelevant to temperature and magnetic field in the case of saturated pumping. Now, let us explain the observed dependence of AMCP on the polarization (of pumping light) by checking the dependence of P* in equation (2) on the polarization. Although e* and err are postulated to be independent of the polarization, e± are dependent on the polarization in the manner of e+(o± pump) = e_(o~ pump) and e+0r pump) = e_(Tr pump), respectively; these were derived in [5] from the symmetry
and P*(~ pump) = 0.
(3)
Thus, by using the above-mentioned relations [1 ] of AMCP = Aa + Ap and Ap ccp*, it is found that, AMCP (7r pump) = Ad
(4)
and
AMCP (O± pump) = &a -+ -~p-
(5)
Equation (4) shows that when the pumping light is 7r-polarized, AMc P must be linearly proportional to the magnetic field. This zs consistent with the experimental results in Fig. 1. A slope of AMCp(~ pump) is found to be (16 -+ 2) x 10 -s in [G] -l, which agrees sufficiently with (17 + 2) x 10 -a in [G] -l obtained in [2]. From equation (5), it is found, consistently with the experiment, that the difference between IAMcp(O± pump)] and [AMCP(TTpump)[ is given by Ap. The intercepts of extrapolation of AMcp(o± pump) at H = 0 are (1.4 + 0.1) × 10-3; the value is almost equal to (1.1 + 0.1) x 10 -a which is the value of Ap measured by modulated pumping method in [2]. From Ap, the value of P* is estimated [1] as 4.17 x 10 -3. Experiment also shows the value of AMcv(rt pump) at the magnetic field where AMCe (o_ pump) = 0 is the same as the value of Ap obtained above; the fact endorses the consistency of equations (4) and (5).
Vol. 27, No. 10
SPIN POLARIZATION IN THE F CENTER
Now, let us evaluate the spin-mixing parameters to explain the estimated P* which is seemingly a very small amount. The spin-mixing parameter in the RES, e*, is defined as follows,
e* = w( + ~ - ) / [ w ( + ~ - ) =
w(--~+)llW(--~+)
+ w(+ --, +)] +
w(-~-)],
(6)
where W(+ ~ 4-) and W(-- --> 4-) are the radiative transition rates from M K 4- (1/2) levels in the RES to the Kramers doublet of m s = 4- (1/2) in the unrelaxed ground state, respectively (see Fig. 2). By using equation (5) in [1] which was derived for the zeroth moment, we can calculate W(+ ~ 4-) as follows, ----
"0
n
M
x (1/2 ÷, + 1/2))12 ,
(7)
where 77= (o+, o_ and n). W(-- ~ +) are obtained simply by replacing ~ l in equation (7) as ~1 (1/2 ÷, -- 1/2). By adopting the parameter values of vibronic and spinorbit interactions determined in [1 ], the values of e* for KF, KCI, and KBr are computed from equations (6) and (7) as 5.5 x 10 -4, 5.5 x 10 -3 and 5.1 x 10 -2, respectively; they are a very small amount. From the comparison of the thus estimated value ofF* for KBr with equation (2), the value of erl is estimated as 0.43 by
1011
using values of thus calculated e* and those of e_+at the H e - N e line of 6328 ~, which are adopted from [5]. Based on this scheme, we propose a new method for the electron spin resonance (ESR) in the RES; this will be observed by monitoring the change of AM ca(O+_pump) caused by the ESR transition. When the ESR transition is induced in the RES, the rate equations are reformed by simply adding terms of IV~ ( n ~ - n_+*) to the second equations in equation (1), where W~ is the ESR transition probability in the RES. From the stationary solutions of the rate equations, the spin polarization under the ESR transition, P ~ , is represented by replacing the denominator of equation (2) as [(1 -- 2er, )(1 -- e_ -- e+)(1 -- 2e*) -- 1.-- 2W~r]. Therefore, when the ESR occurs, it is predicted that IP~I is decreased; so that IAMCp(O_+pump)l is reduced down to IAMcv(Tr pump)l. Here, it is ascertained that the change of P* induced by the ESR of ground state is very small in comparison with that of '.he RES [6]. This implies that the ESR signal of the RES will be avoided from the mixing of the ESR signal of the ground state. This is an advantageous point in this proposed method. The work is scheduled in our laboratory. Details of this Letter will be published.
Acknowledgements - We are obliged to Dr. Y. Mori for his efforts on experimental set-up. The work is partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education in Japan.
REFERENCES
1.
IMANAKA K.,IIDA T.& OHKURA H.,J. Phys. Soc. Japan 44,1632 (1978).
2.
BALDACCHINI G., GRASSANO U.M. & TANGA A., Phys. Rev. B16, 5570 (1977).
3.
MOLLENAUER L.F. & PAN S., Phys. Rev. B6, 772 (1972).
4.
WINNACKER A., MAUSER K.E. & NIESERT B., Z. Phys. B26, 97 (1977).
5.
MAUSER K.E., NIESERT B. & WINNACKER A., Z. Phys. B26, 107 (1977).
6.
HAHN K., MAUSER K.E., REYHER H.J. & WINNACKER A., Phys. Lett. 63A, 151 (1977).