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Lineshape effects of anisotropic diffusion
Volume-6.; number 4
15 August 1970
LfNESHAPE
EFF.ECTS
OF
ANISOTROPIC
DIFFUSION
PIER LUIGI NORDIO Institute of Physical Chemistry, The University, Padova, ItaZy Received 22‘April 1970 The -effect of anisotropic rotational diffusion on the ESR lineshapes have been considered to explain the features of the solution-spectra of elongated molecules containing nitroxide groups.
I. ~-NTR~DUCTL~N In the ESR spectra of free radicals containing a nitrortide g?tnip,. the high field-component of the nitrogen hype&fine structure is always broad_er than the other two lines, having approximately.equal widths. This is due to the fact-that the width of each line is given by the relationship
sidered, the difference in width’between the low-field line (AZ = 1) and the central component (M = 0) is given by the relation:
6(l) - 6(o) = B + c = where
$+(o)
xro ++j”D(o), (I)
3
6(M)=A+BM*CM2 and, for nitrokide radicals, it turns out that in the X-band region (intensity of the magnetic field Ho = 3.300 G), IB] =lCj [l]. However, the- spectrum of 4,4’ terphenylene. nitroxide biradical dissolved in fi-azoxyanisole and registered at the nematic-isotropic transition temperature (fig. 1) shows the unusual feature that the lo+-field line is sharper than the other two components. The same ‘effect is observed in toluene solutions [Z], so that it does not depend upon the specific properties of the nematic solvent, nor can it be ascribed to the additional relaxation mechanism introdnced by the electron-electron dipole interaction in the biradical, which is negligible when the nitroxide moieties are separated by two or more benzene rings [2]. Such a behaviour can be explained. readily if the effects on the lineshape of the .anisotropic rotational diffusion of the paPamagnetic molecules are taken into account. 2; THEORY
.
According to the general the&y of li&n&s [3], if non-secular terms are neglected, and @y dipole and g-tensor interactions are: con2iic
F(m) are the irreducible (spherical) tensor c # mp’onents corresponding to the anisotropic interaction denoted by 12, and 7R is the rotational correlation time. FoIlewing Freed [4], the only effect of introducing anisotropic relaxation is to change the spherical density functions jp’, which in this case are given by the relationship:
(2) where the principal axes of the diffusion tensor are chosen as molecule-fixed axes, and each component F’(m) is relaxed with its own characteristic t&e (3) RI = R2 and R3’are the principal values of the molecular diffusion tensor, axial symmetry being assumed. F’(m) can be expressed in terms of the-principal vk.!lues Ff ):
(4) F(m) are oMain& from the values of dipole and &e&or interactions measured i& diluted mixed crystals [5]. It is fontid t&k
CHEMICAL PHYSICS LETTERS
Volume 6. number 4
I
H
Fig. 1. ESR spectrum
Fb l) = F(* ‘1 =&2) G
D
15 August 1970
D
=
of 4.4’-terphenylene
nitroxide biradical
inp-azoxyacisole
at 135OC.
0
and the principal axes of the G and D tensors coincide. In this axis system: F#’
= - f ($)1’2re@dz
;
Introduction of these expressions in (2) gives, after some manipulation:
jDG (0) = % ~(0) F(o)T + ,(*) In our case the 2 axis, taken as the axis of
the 2pz orbital centered on the nitrogen atom, makes an angle of 90’ with the long axis (2) of the molecule, i.e. #J= lr/2. Furthermore the N-O bond direction may be taken as approximately parallel to the Z’axis, and if theg-tensor principal axis parallel to the N-O direction is denoted as Xaxis, we obtain after the transformation (4):
where
5
1 7 =--4
D
G
3 TOfq7*2;
D=G
(*2) AT1
61/2 AT =7+*2
’
- 70).
(7)
Due to the near cancellation of the B and C terms as defined in eq.(l) if a single relaxation time is assumed, we obtain
6(l) - 503 = $$h"2)Ar.
(8)
By using the principal values of the dipole and g-tensor interactions reported by G iffith 153 it is readily famd that the product FD T0)FG(f2fis positive. The sign of AT is determined by the following consideration: for an elongated molecule, the fastest reixienhtion about the long &a implies RQ> Ry, hence it follows from (5) and (7) -*; 251
Volnme 6, nnmier ?. .-
.CHEMICAL PHYSI$XJLXTTERS ._
,thatA$~i&.negative. ~erekre~wkcan concbade tb$ the term J&t*! -FG’f,z)A+ introduced by the aqisbtrapic hiffuaionis responsibleforthe qar-~owlngofthelow-fieldline. We may Nate that if the N-C bond direction
-Gere peidendicular to the long axis of the mole&e, qe shouldhsveinterchangedthexandy indexes k(s), thus reversiagthe sign of F Wx F&a). Accordingly, the centrallinewoul f result the sharpest; This may be the case of %&Wole-N-aJride, observed in the same c%.perimental condition of our system and reported by Falls ana Iuckhurst [SJ .- Incidentally, if both dipole and g-tensor interactions had axial symmetry, the anisotropic diffusion would not change the relative width of the hyperfine lines, the only effect being a redefinition of the correlation time.
352
:
_.
15 August 1970
ACKNOWLEDGEMENTS Tlie author wishes to acknowledge the assist-. mr ;tsrcjr i.R?&&l W & z%w&zsW GizIIxYmetti, Dr:Corvaja and Dr. Brustofon. The research has been supported in part by the Italian National Research Council, Chemistry Committee. REFERENCES [IJ P. L. Nordfo. A. Scatturinand A.M. Tamburro, Ric.sCi,- 38 (19681832. [2]C. Corvaja, &. Gikometti, K. D. Kopple and Ziauddin, J. Am. Chem. Sot.. to be published. [3] J-H-Freed and G_K.Fraenkel. J. &em. Phys. 39 $968) 326. (41 J-H-Freed. J. Chem. Phys. 41 (1964) 2077. [5] O.H. Griffith, D.W. Cornell and H. M. McConnell, J. Chem. Phys. 43 (1965)2909. [6]3. R. Falle and G. R. Luckhurst. Mol. Phys. 12 {1967) 493.
15 August 1970
LfNESHAPE
EFF.ECTS
OF
ANISOTROPIC
DIFFUSION
PIER LUIGI NORDIO Institute of Physical Chemistry, The University, Padova, ItaZy Received 22‘April 1970 The -effect of anisotropic rotational diffusion on the ESR lineshapes have been considered to explain the features of the solution-spectra of elongated molecules containing nitroxide groups.
I. ~-NTR~DUCTL~N In the ESR spectra of free radicals containing a nitrortide g?tnip,. the high field-component of the nitrogen hype&fine structure is always broad_er than the other two lines, having approximately.equal widths. This is due to the fact-that the width of each line is given by the relationship
sidered, the difference in width’between the low-field line (AZ = 1) and the central component (M = 0) is given by the relation:
6(l) - 6(o) = B + c = where
$+(o)
xro ++j”D(o), (I)
3
6(M)=A+BM*CM2 and, for nitrokide radicals, it turns out that in the X-band region (intensity of the magnetic field Ho = 3.300 G), IB] =lCj [l]. However, the- spectrum of 4,4’ terphenylene. nitroxide biradical dissolved in fi-azoxyanisole and registered at the nematic-isotropic transition temperature (fig. 1) shows the unusual feature that the lo+-field line is sharper than the other two components. The same ‘effect is observed in toluene solutions [Z], so that it does not depend upon the specific properties of the nematic solvent, nor can it be ascribed to the additional relaxation mechanism introdnced by the electron-electron dipole interaction in the biradical, which is negligible when the nitroxide moieties are separated by two or more benzene rings [2]. Such a behaviour can be explained. readily if the effects on the lineshape of the .anisotropic rotational diffusion of the paPamagnetic molecules are taken into account. 2; THEORY
.
According to the general the&y of li&n&s [3], if non-secular terms are neglected, and @y dipole and g-tensor interactions are: con2iic
F(m) are the irreducible (spherical) tensor c # mp’onents corresponding to the anisotropic interaction denoted by 12, and 7R is the rotational correlation time. FoIlewing Freed [4], the only effect of introducing anisotropic relaxation is to change the spherical density functions jp’, which in this case are given by the relationship:
(2) where the principal axes of the diffusion tensor are chosen as molecule-fixed axes, and each component F’(m) is relaxed with its own characteristic t&e (3) RI = R2 and R3’are the principal values of the molecular diffusion tensor, axial symmetry being assumed. F’(m) can be expressed in terms of the-principal vk.!lues Ff ):
(4) F(m) are oMain& from the values of dipole and &e&or interactions measured i& diluted mixed crystals [5]. It is fontid t&k
CHEMICAL PHYSICS LETTERS
Volume 6. number 4
I
H
Fig. 1. ESR spectrum
Fb l) = F(* ‘1 =&2) G
D
15 August 1970
D
=
of 4.4’-terphenylene
nitroxide biradical
inp-azoxyacisole
at 135OC.
0
and the principal axes of the G and D tensors coincide. In this axis system: F#’
= - f ($)1’2re@dz
;
Introduction of these expressions in (2) gives, after some manipulation:
jDG (0) = % ~(0) F(o)T + ,(*) In our case the 2 axis, taken as the axis of
the 2pz orbital centered on the nitrogen atom, makes an angle of 90’ with the long axis (2) of the molecule, i.e. #J= lr/2. Furthermore the N-O bond direction may be taken as approximately parallel to the Z’axis, and if theg-tensor principal axis parallel to the N-O direction is denoted as Xaxis, we obtain after the transformation (4):
where
5
1 7 =--4
D
G
3 TOfq7*2;
D=G
(*2) AT1
61/2 AT =7+*2
’
- 70).
(7)
Due to the near cancellation of the B and C terms as defined in eq.(l) if a single relaxation time is assumed, we obtain
6(l) - 503 = $$h"2)Ar.
(8)
By using the principal values of the dipole and g-tensor interactions reported by G iffith 153 it is readily famd that the product FD T0)FG(f2fis positive. The sign of AT is determined by the following consideration: for an elongated molecule, the fastest reixienhtion about the long &a implies RQ> Ry, hence it follows from (5) and (7) -*; 251
Volnme 6, nnmier ?. .-
.CHEMICAL PHYSI$XJLXTTERS ._
,thatA$~i&.negative. ~erekre~wkcan concbade tb$ the term J&t*! -FG’f,z)A+ introduced by the aqisbtrapic hiffuaionis responsibleforthe qar-~owlngofthelow-fieldline. We may Nate that if the N-C bond direction
-Gere peidendicular to the long axis of the mole&e, qe shouldhsveinterchangedthexandy indexes k(s), thus reversiagthe sign of F Wx F&a). Accordingly, the centrallinewoul f result the sharpest; This may be the case of %&Wole-N-aJride, observed in the same c%.perimental condition of our system and reported by Falls ana Iuckhurst [SJ .- Incidentally, if both dipole and g-tensor interactions had axial symmetry, the anisotropic diffusion would not change the relative width of the hyperfine lines, the only effect being a redefinition of the correlation time.
352
:
_.
15 August 1970
ACKNOWLEDGEMENTS Tlie author wishes to acknowledge the assist-. mr ;tsrcjr i.R?&&l W & z%w&zsW GizIIxYmetti, Dr:Corvaja and Dr. Brustofon. The research has been supported in part by the Italian National Research Council, Chemistry Committee. REFERENCES [IJ P. L. Nordfo. A. Scatturinand A.M. Tamburro, Ric.sCi,- 38 (19681832. [2]C. Corvaja, &. Gikometti, K. D. Kopple and Ziauddin, J. Am. Chem. Sot.. to be published. [3] J-H-Freed and G_K.Fraenkel. J. &em. Phys. 39 $968) 326. (41 J-H-Freed. J. Chem. Phys. 41 (1964) 2077. [5] O.H. Griffith, D.W. Cornell and H. M. McConnell, J. Chem. Phys. 43 (1965)2909. [6]3. R. Falle and G. R. Luckhurst. Mol. Phys. 12 {1967) 493.