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ESR line shape of quasi-coherent triplet excitons in pairs of differently oriented molecules
Solid State Communications, Vol. 25, pp. 859-861,
Pergamon Press.
1978.
Printed in Great Britain
ESR LINE SHAPE OF QUASI-COHERENT TRIPLET EXCITONS IN PAIRS OF DIFFERENTLY ORIENTED MOLECULES P. Reineker Abteilung fti Theoretische Physik I, Universitat Ulm, Oberer Eselsberg, 7900 Ulm, Federal Republic of Germany (Received 26 September 1977 by B. Miihlschltgel) The ESR line shape of triplet excitons in pairs of differently oriented molecules is calculated. The Hamiltonian describes the coupled coherent and incoherent motion of the excitons and contains Zeeman and fmestructure terms. With decreasing coherence the four Am = l-transition lines of the pair coalesce into two lines which are first narrowed. For still lower degree of coherence the lines begin to broaden again.
ESR MEASUREMENTS at excited triplet states are a most useful method in investigating the problem whether excitons in molecular crystals move coherently or incoherently. Using this method it has been shown that at elevated temperatures triplet excitons in molecular crystals move in a quasi-incoherent manner, which may be described by a diffusion constant with an effective hopping rate being determined to a large extent also by the strength of the coherent interaction between the molecules. At temperatures of several kelvin ESR measurements of triplet excitons in pairs of differently oriented naphthalene-ha molecules, sitting substitutionally at the two inequivalent sites of the unit cell of naphthaleneds single crystals, have been performed first by Schwoerer and Wolf [4,5] in 1966 and discussed in the incoherent picture of the exciton motion. Only recently, when this pair system has been reinvestigated by spin-echo and ODMR measurements [6] and by ESR [7], it has been shown that for its description the coherent picture of the exciton motion applies and that therefore the motion should be treated theoretically as a coupled coherent and incoherent [8-lo] one. Whereas in previous calculations of the ESR line shape in molecular pairs [ 1l-l 31 only spin 3 systems have been treated for reasons of mathematical simplicity, in this paper realistic triplet systems are considered. To that end the molecular pair system is described by a Hamiltonian consisting of three parts: H = H, +H, +H,(t).
(1)
H, describes the coherent motion of the excitation
between the molecules and is given by H, = qb:bl
+ ezb;bz +J(b;b,
+ b:bz).
(2)
bf and bi are creation and annihilation operators for a localized electron hole pair. el and ez are the excitation
energies of the two molecules in the pair (which are in this calculation assumed to be equal and may therefore be nomialized to zero), and J is the coherent interaction between them. Hz contains the Zeeman energy in an external magnetic field H and terms describing the fine-structure interaction of the two spins forming the triplet state: H2
=
gpBH-S+S*M*S+&S*D*S
M = (F(’ ) + Fc2 ‘)/2 , Ab=
b;b, -bib,.
D = (F” ) - F’2’/2
(3) (4) (5)
F" )and F” ) are the fine-structure tensors at the two differently oriented molecules. In the purely coherent limit the positions of the ESR lines are given by the Am = 1 - differences of the energy eigenvalues of the time independent Schrodingerequation (Hi
+H,)JI
=
EJI.
(6)
Expanding the eigenvectors J/ into states In, s) where n = {1,2) denotes the site of the excitation and s = (1 , 0, - l} the orientation of the spin of the triplet exciton with respect to an external magnetic field, we arrive at a 6-dimensional eigenvalue problem. For its numerical solution we use the data of the naphthalene pair. The exchange interaction integral is known form optical measurements [ 141 to be J = 1.20cm-’ 9 1.28 x lo4 G. Using the values D = 1063.3G and E = 164.7 G for the fme-structure parameters [ 151 and the direction cosines [ 161 of the axes of the two differently oriented molecules with respect to the crystal axes, the fine-structure tensors FC1) and Ft2), and thus M and D may be calculated. From these calculations it is easily shown that the tensor M is diagonal in an axes-system &,, y,,, zp}, whose axis yp coincides with the b-axis of the crystal and whose axis zr, forms an angle of + 22.4’ with the crystal a-axis. For a magnetic field in the 859
860
ESR LINE SHAPE OF QUASI-COHERENT Rotation
about
x,-axis,
TRIPLET EXCITONS
No.11
e60*
s, =c2=o J=12800
G, I, =O.WOOf
H=LOOOG.
D=lt%SJG,
G E=-164.76
3LiO
Fig. 1. ESR line shape for the transitions
YP-zp-plane of strength H = 4000 G and oriented at OL= + 60” with respect to the y,-axis, the positions of the Am = 1 - ESR lines are determined by the following values: E13
E 24
= 4596.8G; Es5 = 3440.3 G upper Davydov component, = 4601.7G; EM = 3436.1G lower Davydov component.
The influence of the phonons on the exciton motion and thus on the ESR lines is taken into account by the stochastic part of the Hamiltonian of the Haken-Strobl model [8-lo] : ~73
=.
h 11W:h +
+
h2lW:h
h22
W:~2
+
hl2
(002
(h,(tYJ22(0= = =
2-/,w--7
%2(~N21(0 vi21
=
(tyz21
Averaging the equation operator 5
= Lp.
(11)
The Liouville-operator consists of two parts, L=LI +L2,whereL1SZ= [HI +H2,S2] isacommutator and describes the coherent part of the motion, whereas L2 is a double-commutator stemming from the fluctuating part of the Hamiltonian. With the ansatz p(t) = entp(O), (11) transforms to a 36dimensional, non-hermitean eigenvalue problem, which has been solved by computer calculations. Using the regression theorem [ 17, 181, from the solution of the density matrix equation the two-times correlation function GS,(O)S,(T)) of the spin operator S, and thus the ESR line shape may be calculated according to
(r’))
=
of motion
~21(oh2(~‘))
271 s(t -
=
t’).
(9)
for the density
(10) we arrive with p = @,>at the
= i j- dr &(O)&(T)>
cos WT.
(12)
0
(8)
ij = [H, p7 over the fluctuations,
ii
x”(w)
The nondisappearing correlation function of the 6 correlated Gaussian processes hu(t) are given by
ul,,(~h,(~‘N
following Liouville-equation
(7)
.
(hrl(0&(0=
W/G
EM and Es5 for fared value of 7r and several values of 70.
In Fig. 1 the ESR line shape for the transitions Es5 and EM, calculated in this way, has been represented for a fured value of rI and for several values of ro. A completely analogous figure may be drawn for the transitions El3 and E24. For very small values of r. we have two lines at w = 3436.1 G and at o = 3440.3 G. These lines are at the positions EM and Es5 of ESR transitions in the lower and upper Davydov components with a change in the same spin quantum numbers; they have been obtained in the purely coherent calculation above.
vol. 25, No. 11
ESR LINE SHAPE OF QUASI-COHERENT
861
TRIPLET EXCITONS
smaller with r. increasing further, the figure shows that for r. > 50 G the ESR line broadens. The broadening is connected with the loss of phase between the two molecules of the pair, i.e. with the fact that the exciton now becomes incoherent. More details of this behaviour, the influence of the strength of the non-local fluctuations 7r on the ESR line shape, and modifications on account of a Boltzmann-distribution of the occupation numbers of the excitons between the upper and lower Davydov states will be the subjects of following papers.
With increasing values of -ye the two lines broaden until at r. = 2 G, i.e. when 2~~ equals the difference in the resonance frequencies, they coalesce into a single line which narrows with r. increasing further. This behaviour is completely analogous to the motional narrowing of ESR lines; the only difference is that now the excitons do not move between different sites, but are scattered with a rate 7. between corresponding spin states of the upper and lower Davydov components. Whereas in this picture of motional narrowing one would expect the ESR line to become smaller and
REFERENCES 1.
HAARER D. &WOLF H.C.,MoZ. Ckyst. Liq. Ckysf. 10,359
2.
YARMUS L., ROSENTHAL
3.
BIZZARRO
4.
SCHWOERER M. &WOLF H.C.,Proc.
5.
SCHWOERER M. &WOLF H.C.,Mol.
6.
BOTTER B.J., NONHOF C.J., SCHMIDT J. & VAN DER WAALS J.H., Chem. Phys. Lett. 43,210
7.
HINKEL H., Thesis, University
8.
HAKEN H. & STROBL G., The Triplet State (Edited by ZAHLAN A.B.) Cambridge University Cambridge (1967).
9.
HAKEN H. & REINEKER
(1970).
J. & CHOPP M., Chem. P&s. Lett. 16,477
W., ROSENTHAL
(1972).
J., BERK N.F. & YARMUS L. P&s. Status Solidi (b) 84,27 14th Colloque Amp&e (1966). Ckysr. 3, 177 (1967).
of Stuttgart
(1976).
(1977).
P., 2. Phys. 249,253
(1972).
10.
HAKEN H. & STROBL G.,Z. Phys. 262,135
11.
REINEKER
P., Phys. Status Solidi (b) 52,439
12.
REINEKER
P., Solid State Commun. 14, 153 (1973).
13.
REINEKER
P., Z. Nuturf. 29a, 282 (1974).
14.
PORT H. (private communication).
15.
SCHWOERER M., Habilitation-Thesis,
16.
CRUICKSHANK
(1973).
17.
HAKEN H. & WEIDLICH W., 2. Phys. 205,96
18.
LAX M.,Phys. Rev. 157,213
(1972).
University
D.W.J., Acru Cryst. 10,504
(1967).
(1977).
of Stuttgart,
(1957). (1967).
1972.
Press,
Pergamon Press.
1978.
Printed in Great Britain
ESR LINE SHAPE OF QUASI-COHERENT TRIPLET EXCITONS IN PAIRS OF DIFFERENTLY ORIENTED MOLECULES P. Reineker Abteilung fti Theoretische Physik I, Universitat Ulm, Oberer Eselsberg, 7900 Ulm, Federal Republic of Germany (Received 26 September 1977 by B. Miihlschltgel) The ESR line shape of triplet excitons in pairs of differently oriented molecules is calculated. The Hamiltonian describes the coupled coherent and incoherent motion of the excitons and contains Zeeman and fmestructure terms. With decreasing coherence the four Am = l-transition lines of the pair coalesce into two lines which are first narrowed. For still lower degree of coherence the lines begin to broaden again.
ESR MEASUREMENTS at excited triplet states are a most useful method in investigating the problem whether excitons in molecular crystals move coherently or incoherently. Using this method it has been shown that at elevated temperatures triplet excitons in molecular crystals move in a quasi-incoherent manner, which may be described by a diffusion constant with an effective hopping rate being determined to a large extent also by the strength of the coherent interaction between the molecules. At temperatures of several kelvin ESR measurements of triplet excitons in pairs of differently oriented naphthalene-ha molecules, sitting substitutionally at the two inequivalent sites of the unit cell of naphthaleneds single crystals, have been performed first by Schwoerer and Wolf [4,5] in 1966 and discussed in the incoherent picture of the exciton motion. Only recently, when this pair system has been reinvestigated by spin-echo and ODMR measurements [6] and by ESR [7], it has been shown that for its description the coherent picture of the exciton motion applies and that therefore the motion should be treated theoretically as a coupled coherent and incoherent [8-lo] one. Whereas in previous calculations of the ESR line shape in molecular pairs [ 1l-l 31 only spin 3 systems have been treated for reasons of mathematical simplicity, in this paper realistic triplet systems are considered. To that end the molecular pair system is described by a Hamiltonian consisting of three parts: H = H, +H, +H,(t).
(1)
H, describes the coherent motion of the excitation
between the molecules and is given by H, = qb:bl
+ ezb;bz +J(b;b,
+ b:bz).
(2)
bf and bi are creation and annihilation operators for a localized electron hole pair. el and ez are the excitation
energies of the two molecules in the pair (which are in this calculation assumed to be equal and may therefore be nomialized to zero), and J is the coherent interaction between them. Hz contains the Zeeman energy in an external magnetic field H and terms describing the fine-structure interaction of the two spins forming the triplet state: H2
=
gpBH-S+S*M*S+&S*D*S
M = (F(’ ) + Fc2 ‘)/2 , Ab=
b;b, -bib,.
D = (F” ) - F’2’/2
(3) (4) (5)
F" )and F” ) are the fine-structure tensors at the two differently oriented molecules. In the purely coherent limit the positions of the ESR lines are given by the Am = 1 - differences of the energy eigenvalues of the time independent Schrodingerequation (Hi
+H,)JI
=
EJI.
(6)
Expanding the eigenvectors J/ into states In, s) where n = {1,2) denotes the site of the excitation and s = (1 , 0, - l} the orientation of the spin of the triplet exciton with respect to an external magnetic field, we arrive at a 6-dimensional eigenvalue problem. For its numerical solution we use the data of the naphthalene pair. The exchange interaction integral is known form optical measurements [ 141 to be J = 1.20cm-’ 9 1.28 x lo4 G. Using the values D = 1063.3G and E = 164.7 G for the fme-structure parameters [ 151 and the direction cosines [ 161 of the axes of the two differently oriented molecules with respect to the crystal axes, the fine-structure tensors FC1) and Ft2), and thus M and D may be calculated. From these calculations it is easily shown that the tensor M is diagonal in an axes-system &,, y,,, zp}, whose axis yp coincides with the b-axis of the crystal and whose axis zr, forms an angle of + 22.4’ with the crystal a-axis. For a magnetic field in the 859
860
ESR LINE SHAPE OF QUASI-COHERENT Rotation
about
x,-axis,
TRIPLET EXCITONS
No.11
e60*
s, =c2=o J=12800
G, I, =O.WOOf
H=LOOOG.
D=lt%SJG,
G E=-164.76
3LiO
Fig. 1. ESR line shape for the transitions
YP-zp-plane of strength H = 4000 G and oriented at OL= + 60” with respect to the y,-axis, the positions of the Am = 1 - ESR lines are determined by the following values: E13
E 24
= 4596.8G; Es5 = 3440.3 G upper Davydov component, = 4601.7G; EM = 3436.1G lower Davydov component.
The influence of the phonons on the exciton motion and thus on the ESR lines is taken into account by the stochastic part of the Hamiltonian of the Haken-Strobl model [8-lo] : ~73
=.
h 11W:h +
+
h2lW:h
h22
W:~2
+
hl2
(002
(h,(tYJ22(0= = =
2-/,w--7
%2(~N21(0 vi21
=
(tyz21
Averaging the equation operator 5
= Lp.
(11)
The Liouville-operator consists of two parts, L=LI +L2,whereL1SZ= [HI +H2,S2] isacommutator and describes the coherent part of the motion, whereas L2 is a double-commutator stemming from the fluctuating part of the Hamiltonian. With the ansatz p(t) = entp(O), (11) transforms to a 36dimensional, non-hermitean eigenvalue problem, which has been solved by computer calculations. Using the regression theorem [ 17, 181, from the solution of the density matrix equation the two-times correlation function GS,(O)S,(T)) of the spin operator S, and thus the ESR line shape may be calculated according to
(r’))
=
of motion
~21(oh2(~‘))
271 s(t -
=
t’).
(9)
for the density
(10) we arrive with p = @,>at the
= i j- dr &(O)&(T)>
cos WT.
(12)
0
(8)
ij = [H, p7 over the fluctuations,
ii
x”(w)
The nondisappearing correlation function of the 6 correlated Gaussian processes hu(t) are given by
ul,,(~h,(~‘N
following Liouville-equation
(7)
.
(hrl(0&(0=
W/G
EM and Es5 for fared value of 7r and several values of 70.
In Fig. 1 the ESR line shape for the transitions Es5 and EM, calculated in this way, has been represented for a fured value of rI and for several values of ro. A completely analogous figure may be drawn for the transitions El3 and E24. For very small values of r. we have two lines at w = 3436.1 G and at o = 3440.3 G. These lines are at the positions EM and Es5 of ESR transitions in the lower and upper Davydov components with a change in the same spin quantum numbers; they have been obtained in the purely coherent calculation above.
vol. 25, No. 11
ESR LINE SHAPE OF QUASI-COHERENT
861
TRIPLET EXCITONS
smaller with r. increasing further, the figure shows that for r. > 50 G the ESR line broadens. The broadening is connected with the loss of phase between the two molecules of the pair, i.e. with the fact that the exciton now becomes incoherent. More details of this behaviour, the influence of the strength of the non-local fluctuations 7r on the ESR line shape, and modifications on account of a Boltzmann-distribution of the occupation numbers of the excitons between the upper and lower Davydov states will be the subjects of following papers.
With increasing values of -ye the two lines broaden until at r. = 2 G, i.e. when 2~~ equals the difference in the resonance frequencies, they coalesce into a single line which narrows with r. increasing further. This behaviour is completely analogous to the motional narrowing of ESR lines; the only difference is that now the excitons do not move between different sites, but are scattered with a rate 7. between corresponding spin states of the upper and lower Davydov components. Whereas in this picture of motional narrowing one would expect the ESR line to become smaller and
REFERENCES 1.
HAARER D. &WOLF H.C.,MoZ. Ckyst. Liq. Ckysf. 10,359
2.
YARMUS L., ROSENTHAL
3.
BIZZARRO
4.
SCHWOERER M. &WOLF H.C.,Proc.
5.
SCHWOERER M. &WOLF H.C.,Mol.
6.
BOTTER B.J., NONHOF C.J., SCHMIDT J. & VAN DER WAALS J.H., Chem. Phys. Lett. 43,210
7.
HINKEL H., Thesis, University
8.
HAKEN H. & STROBL G., The Triplet State (Edited by ZAHLAN A.B.) Cambridge University Cambridge (1967).
9.
HAKEN H. & REINEKER
(1970).
J. & CHOPP M., Chem. P&s. Lett. 16,477
W., ROSENTHAL
(1972).
J., BERK N.F. & YARMUS L. P&s. Status Solidi (b) 84,27 14th Colloque Amp&e (1966). Ckysr. 3, 177 (1967).
of Stuttgart
(1976).
(1977).
P., 2. Phys. 249,253
(1972).
10.
HAKEN H. & STROBL G.,Z. Phys. 262,135
11.
REINEKER
P., Phys. Status Solidi (b) 52,439
12.
REINEKER
P., Solid State Commun. 14, 153 (1973).
13.
REINEKER
P., Z. Nuturf. 29a, 282 (1974).
14.
PORT H. (private communication).
15.
SCHWOERER M., Habilitation-Thesis,
16.
CRUICKSHANK
(1973).
17.
HAKEN H. & WEIDLICH W., 2. Phys. 205,96
18.
LAX M.,Phys. Rev. 157,213
(1972).
University
D.W.J., Acru Cryst. 10,504
(1967).
(1977).
of Stuttgart,
(1957). (1967).
1972.
Press,