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Steady-state optical spin polarization. Theory of the high field anisotropic ESR intensity
c
.. ..
CHEMICAL PHYSICS LETTERS
15 Octobcs 1975
:. ‘I. .-
STE’ADY-STATE OPTiCAL SPHN POLARIZA’FIION. THEORY OF THE HIGH FIELD ANISOTROPIC ESR INTENSITY GIenn T. EVANS
17 June 1975
Reczivcd
-. Tfie cffccts of ~jsatropic for~lat~~n and depletion ISC processes on the electron let states of.spiropyrans are calculated using a density matrix theory.
2. Introduction
triplet
is dissipated
spin poIari~tion
forming
of the short lived trip-
a singlet. The dissipation
process may also be guided by the same type of seIecpaper, ~McRide and Evaas [I ] re-
In the brcceding
port steady state electron spin pol~r~ation in the triplet state of a single crystal of a spiropyran at 1.05 K. Similar observatioos on a different spiropyran have also been reported by Zubkov [2] and initially motivated this study. fn both cases, ihe ESR intensity of the spiropyrans is a highly anisotropic function of the orientation of principsl axes of ihe triplets with respect to the laboratory fixed m?s. The observed anisotropy
is not due to the anisotropy
transition moment, Rather, the tensorial character of
tion. The basic aim of this note j’sto pro&e a density matrix theory for the anisotropic signal imensity of the ‘spiropyrans. The pho!opl~ysics to be induded in the theory is well documented: (i) The notation of the triplet is assumed to occur by the agency of the spin orbit’interaction, which spin selectively connects an excited singlet state: with a molecule fixed component state of the triplet [3,5]. (ii) During the IifeGrne of the triplet, whi.ch in this case is’very short compared to the triplet electron spin lattice reiaxation time, the triplet wave-
fun~rion undergoes magnetic evolution because of the Zeeman and dipolar interactions. The Zeeman inieraction will be assumed to be dominant although fist order,corre+ns dtle to the breakdown of the’high field app~o~~atioR will be ~pIemen~ed~ fiii) The ‘.;,
_)_
. .
:, ;.
:
__
‘;
.,
for triplet
forrna-
liquid state CIDEP. 2. Theory
of the magnetic
the -intensity of the electron spin polarization is linked to the ~hotop~~s~cs of triplet product~Gn and dissipa-
&j
tion rules [5] which were responsib!e
tion, and this possibi~ty is also accoucted for. The theory presented in the next section attempts to account for these processes using procedures originally advanced by Atkins and Evans [4] in a treatment of
‘, ..
:.
:.,
2, I, Eqmztians ofrno tiun The equation of motion for the electron spin density matrix, p(Q, t), of the triplet is a Liouville equation modified to include chemical formation and depletion, dP&d
a = if’p,H],,-
+ k,,o eekf - (kq)crafpw:‘,
(1)
with H = Hz + J?~. Hi and HD are the Zeeman and dipolar interactions, respectively, which have operator forms, KC = crJ*S,
(2)
@a =g&&,
HD = se za-s,
GJ)
and high field matrix @W&V’)
=. W+WS,f,,f**.
.’
,; ”
elements
‘.’
[4] ,
(4)
Volume 36, numbcr 1
CHEMICALPHYSICSLIXI’ERS
15 October 1975
calculating the triplet intensity of the spiropyr~s owing &the short chemicai lifetime. Consequently, a relaxation tetradic has not been included in eq. (I).
(6) + E {a :,@)
f Q &(W),
and D and E being the familiar triplet spin parameters. The Euler an&s, E$,, represent the orientarion of the frame which dlagonalizes the dipolar tensor, the D frame, with respect to the laboratory fixed Zeeman frame, the 3 frame, The source term in eq. (1) represents spin selective ISC from a parent excited singlet. k can be represented by a scalar operator equivalent [4], k=kL_,ckyy+kz,_
k=&+WK*S,
0)
The intensity of an ESR tr~sition is proportional to popolatioit differences, using populations transformed to the representation which diagonalizes the total spin hamiltonian, H. To first order in the dipolar interaction, the transform, T-?HT diagonalizes H with T hav~g the simple form, T@,)
= I +- Y(fl&
Pa)
defining
The equation of motion for transformed ments of the density matrix becomes
diagonal ele-
In general, the rate tensor, K, will not be diagonal in the same frame which diagonalizes the dipolar interaction. In the principal axis frame of the K tensor, the K frame, K has matrix elements
R *&I= ($k - k,,)
-
6,&E.
In the following, SK-Sshall be written ashy, has hi& geld mat&v elemen:s
w’hich
(M~&&)IM’)
defming I$@) analogous to DQ(R) except that we replace D and E by R and 1? respectively, with 2K=k,,
‘kvy
- 2kzs,
21=kw
-k,,.
The Euler a@es, 52~) relate the reIative orientation of the K and B frames. The rate operator, k,, describes the first Grder lass of triplets and may also-be anisotropic [5]. If the final ISC from a component state of the triplet to an excited singlet is spin orbit allowed, the operator form of k, will be isomorphic to k, differing only in the K and I values. In the following treatment, kq will bE: given a general vlgutar dependence omitting for the moment an ~xp~cit’mat~ represe~tatioR. The effects of spin relaxation ze ~~~~a~t in
CkqW,)+ IkpK)~ W%)l
L.&wr
(10)
using g = T-l pT. Eq, (1 O} has the long time solution go_
r+ +w3-w -t &&w H&,)1 = __-_____Y {&#@ -+ k+‘-K), HD(%)l f,
$I
I)
Using eq. (I I)1 the polarized contribution-to the ESR intensity can be calculated from the appropriate differences in level populationsRather than to treat the general case in which formation and depletion of the triplet are both anisotropic, we shall instead consider the simpler Iirniting cases in which the selection rules are operative in only one of the two steps. In the case of symmetry guided formation of the 9 tripIet and isotropic destruction, the intensity, It,, ad the frequency, vn, of the triplet transitions are r, = sgn (Fz)($)“Mo
(Q& (12)
Vo!umc 36, npmbcr 1
=‘-1. for the M = -1 +M = 0 txuxition. TheK frame and the D fratie are related thrdugh the identity for Wigner functjons,
where aDlc express and K frames. If the destruction
the relative orientations
of the D
is proportional
to
where J4 and K:(2) are the quenching analogues aild &(a), respectively.
to k
3. Discussion In the high field approximation, the spin polarization contributions to the two triplet transitions are equal in intensity and opposite in phase for either spin selective populitiofi or dep’opulntion .of the triplet. The tensor character of the spin polarization arising. from either process differs significantly and to that we now turn. In the case of isotropic quenching: the leading term in eq. (12) is a second rank tensor. The correction terms, bracketed in eq. (I’_), diiplace both lines equally and will in general consist of zeroth, second and fourth rank tensors. In liquid state CIDEP, only the
scalar term survives rptational averaging. Since the fourth rank term is a highly oscillatory function of the Euler angles, it is doubtFul_if its contribution can be -distinguished from the experimental scatter in the observed intensities. The resultant intensity is suitably approximated by 2 scdar and a second rank term. For
1975
the especially simple case of an axially symmetric molecule with coaxial K and D tensors and isotropic depletion, eq. (12) simplifies (16) r,,(8)=K{~E+‘[sgn(,*)+qE]~~(B)&EP4(e)}, with E = D,/wOI If D/o0 = $ tien the carrection terms to the signed second rank contribution are of the order of 10% and I,,(0)
of the triplet is the sole anisotropic step, then the ESR intensity, neglecting higher order
dipolar contributions,
15 October
CHEMICAL PHYSICS LETTERS
greater
than
the magic
has its nodes
at an angle
3O
angle.
If the destruction step is anisotropic, then I,! is asymptoticdy second rank for small anisotropies in the ISC rate, i.e.., K%q and Iq/kq < 1. For the general case in which KS or 19 are comparabie to kq, then 1, is not well described by a small xt of Wigner functions. Finally, in the work of McBride and Evans, the spin polarization is large and describable by a second rank tensor with minor deviations. As 2 consequence, the theory presented here indicates that the formationai IX process is the source of spin polarization and apparently, in the case of the spiropyran, chemical quenching of the triplet takes place without any selection rules.
Acknowledgement This work .was supported by Professor hlarsl12ll Fixman under a National Institute of Health Grant, GM-13556.
References [I] J.&l. hlcEridc and G.T. Evans, Chem. Phys. Letters 36 (1975) 41. [2] A.V. Zubkov, Dokl. Fhys. Chcm. 216 (1974) 558. [3] MS. de Groot, I.A.hl. Hesselmann and J.H. yan der Waals, Mol. Phys. 12 (1967) 259. [4] P.W. Atkins gnd G.T. Evans, Mol. Phys. 27 (1974) 1633; 1Chem. Ptiys. Letters 35 (1974) 108. [5] M.A. El-Saycd, Account: Chem. Res. 4 (i971) 23.
.. ..
CHEMICAL PHYSICS LETTERS
15 Octobcs 1975
:. ‘I. .-
STE’ADY-STATE OPTiCAL SPHN POLARIZA’FIION. THEORY OF THE HIGH FIELD ANISOTROPIC ESR INTENSITY GIenn T. EVANS
17 June 1975
Reczivcd
-. Tfie cffccts of ~jsatropic for~lat~~n and depletion ISC processes on the electron let states of.spiropyrans are calculated using a density matrix theory.
2. Introduction
triplet
is dissipated
spin poIari~tion
forming
of the short lived trip-
a singlet. The dissipation
process may also be guided by the same type of seIecpaper, ~McRide and Evaas [I ] re-
In the brcceding
port steady state electron spin pol~r~ation in the triplet state of a single crystal of a spiropyran at 1.05 K. Similar observatioos on a different spiropyran have also been reported by Zubkov [2] and initially motivated this study. fn both cases, ihe ESR intensity of the spiropyrans is a highly anisotropic function of the orientation of principsl axes of ihe triplets with respect to the laboratory fixed m?s. The observed anisotropy
is not due to the anisotropy
transition moment, Rather, the tensorial character of
tion. The basic aim of this note j’sto pro&e a density matrix theory for the anisotropic signal imensity of the ‘spiropyrans. The pho!opl~ysics to be induded in the theory is well documented: (i) The notation of the triplet is assumed to occur by the agency of the spin orbit’interaction, which spin selectively connects an excited singlet state: with a molecule fixed component state of the triplet [3,5]. (ii) During the IifeGrne of the triplet, whi.ch in this case is’very short compared to the triplet electron spin lattice reiaxation time, the triplet wave-
fun~rion undergoes magnetic evolution because of the Zeeman and dipolar interactions. The Zeeman inieraction will be assumed to be dominant although fist order,corre+ns dtle to the breakdown of the’high field app~o~~atioR will be ~pIemen~ed~ fiii) The ‘.;,
_)_
. .
:, ;.
:
__
‘;
.,
for triplet
forrna-
liquid state CIDEP. 2. Theory
of the magnetic
the -intensity of the electron spin polarization is linked to the ~hotop~~s~cs of triplet product~Gn and dissipa-
&j
tion rules [5] which were responsib!e
tion, and this possibi~ty is also accoucted for. The theory presented in the next section attempts to account for these processes using procedures originally advanced by Atkins and Evans [4] in a treatment of
‘, ..
:.
:.,
2, I, Eqmztians ofrno tiun The equation of motion for the electron spin density matrix, p(Q, t), of the triplet is a Liouville equation modified to include chemical formation and depletion, dP&d
a = if’p,H],,-
+ k,,o eekf - (kq)crafpw:‘,
(1)
with H = Hz + J?~. Hi and HD are the Zeeman and dipolar interactions, respectively, which have operator forms, KC = crJ*S,
(2)
@a =g&&,
HD = se za-s,
GJ)
and high field matrix @W&V’)
=. W+WS,f,,f**.
.’
,; ”
elements
‘.’
[4] ,
(4)
Volume 36, numbcr 1
CHEMICALPHYSICSLIXI’ERS
15 October 1975
calculating the triplet intensity of the spiropyr~s owing &the short chemicai lifetime. Consequently, a relaxation tetradic has not been included in eq. (I).
(6) + E {a :,@)
f Q &(W),
and D and E being the familiar triplet spin parameters. The Euler an&s, E$,, represent the orientarion of the frame which dlagonalizes the dipolar tensor, the D frame, with respect to the laboratory fixed Zeeman frame, the 3 frame, The source term in eq. (1) represents spin selective ISC from a parent excited singlet. k can be represented by a scalar operator equivalent [4], k=kL_,ckyy+kz,_
k=&+WK*S,
0)
The intensity of an ESR tr~sition is proportional to popolatioit differences, using populations transformed to the representation which diagonalizes the total spin hamiltonian, H. To first order in the dipolar interaction, the transform, T-?HT diagonalizes H with T hav~g the simple form, T@,)
= I +- Y(fl&
Pa)
defining
The equation of motion for transformed ments of the density matrix becomes
diagonal ele-
In general, the rate tensor, K, will not be diagonal in the same frame which diagonalizes the dipolar interaction. In the principal axis frame of the K tensor, the K frame, K has matrix elements
R *&I= ($k - k,,)
-
6,&E.
In the following, SK-Sshall be written ashy, has hi& geld mat&v elemen:s
w’hich
(M~&&)IM’)
defming I$@) analogous to DQ(R) except that we replace D and E by R and 1? respectively, with 2K=k,,
‘kvy
- 2kzs,
21=kw
-k,,.
The Euler a@es, 52~) relate the reIative orientation of the K and B frames. The rate operator, k,, describes the first Grder lass of triplets and may also-be anisotropic [5]. If the final ISC from a component state of the triplet to an excited singlet is spin orbit allowed, the operator form of k, will be isomorphic to k, differing only in the K and I values. In the following treatment, kq will bE: given a general vlgutar dependence omitting for the moment an ~xp~cit’mat~ represe~tatioR. The effects of spin relaxation ze ~~~~a~t in
CkqW,)+ IkpK)~ W%)l
L.&wr
(10)
using g = T-l pT. Eq, (1 O} has the long time solution go_
r+ +w3-w -t &&w H&,)1 = __-_____Y {&#@ -+ k+‘-K), HD(%)l f,
$I
I)
Using eq. (I I)1 the polarized contribution-to the ESR intensity can be calculated from the appropriate differences in level populationsRather than to treat the general case in which formation and depletion of the triplet are both anisotropic, we shall instead consider the simpler Iirniting cases in which the selection rules are operative in only one of the two steps. In the case of symmetry guided formation of the 9 tripIet and isotropic destruction, the intensity, It,, ad the frequency, vn, of the triplet transitions are r, = sgn (Fz)($)“Mo
(Q& (12)
Vo!umc 36, npmbcr 1
=‘-1. for the M = -1 +M = 0 txuxition. TheK frame and the D fratie are related thrdugh the identity for Wigner functjons,
where aDlc express and K frames. If the destruction
the relative orientations
of the D
is proportional
to
where J4 and K:(2) are the quenching analogues aild &(a), respectively.
to k
3. Discussion In the high field approximation, the spin polarization contributions to the two triplet transitions are equal in intensity and opposite in phase for either spin selective populitiofi or dep’opulntion .of the triplet. The tensor character of the spin polarization arising. from either process differs significantly and to that we now turn. In the case of isotropic quenching: the leading term in eq. (12) is a second rank tensor. The correction terms, bracketed in eq. (I’_), diiplace both lines equally and will in general consist of zeroth, second and fourth rank tensors. In liquid state CIDEP, only the
scalar term survives rptational averaging. Since the fourth rank term is a highly oscillatory function of the Euler angles, it is doubtFul_if its contribution can be -distinguished from the experimental scatter in the observed intensities. The resultant intensity is suitably approximated by 2 scdar and a second rank term. For
1975
the especially simple case of an axially symmetric molecule with coaxial K and D tensors and isotropic depletion, eq. (12) simplifies (16) r,,(8)=K{~E+‘[sgn(,*)+qE]~~(B)&EP4(e)}, with E = D,/wOI If D/o0 = $ tien the carrection terms to the signed second rank contribution are of the order of 10% and I,,(0)
of the triplet is the sole anisotropic step, then the ESR intensity, neglecting higher order
dipolar contributions,
15 October
CHEMICAL PHYSICS LETTERS
greater
than
the magic
has its nodes
at an angle
3O
angle.
If the destruction step is anisotropic, then I,! is asymptoticdy second rank for small anisotropies in the ISC rate, i.e.., K%q and Iq/kq < 1. For the general case in which KS or 19 are comparabie to kq, then 1, is not well described by a small xt of Wigner functions. Finally, in the work of McBride and Evans, the spin polarization is large and describable by a second rank tensor with minor deviations. As 2 consequence, the theory presented here indicates that the formationai IX process is the source of spin polarization and apparently, in the case of the spiropyran, chemical quenching of the triplet takes place without any selection rules.
Acknowledgement This work .was supported by Professor hlarsl12ll Fixman under a National Institute of Health Grant, GM-13556.
References [I] J.&l. hlcEridc and G.T. Evans, Chem. Phys. Letters 36 (1975) 41. [2] A.V. Zubkov, Dokl. Fhys. Chcm. 216 (1974) 558. [3] MS. de Groot, I.A.hl. Hesselmann and J.H. yan der Waals, Mol. Phys. 12 (1967) 259. [4] P.W. Atkins gnd G.T. Evans, Mol. Phys. 27 (1974) 1633; 1Chem. Ptiys. Letters 35 (1974) 108. [5] M.A. El-Saycd, Account: Chem. Res. 4 (i971) 23.