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Stark effect on the nuclear quadrupole resonance of 14n in an asymmetric field gradient
Journal of Molecular Structure Elsevier Publishing Company, Amsterdam. Printed in the Netherlands
STARK EFFECT ON THE NUCLEAR QUADRUPOLE 14N IN AN ASYMMETRIC FIELD GRADIENT
RESONANCE
OF
I. LINE SHAPE J. L. COLOT* Institrrrde Physique,
UniversitP de Li.+ge, 4000-LiGge Sart Tihan (Belgium)
(Received October 22nd,
1971)
ABSTRACT
Nuclear _quadrupole resonance line shapes in the presence of the Stark effect were studied in the case of asymmetric field gradients, where the exciting magnetic field forms an arbitrary angle with the electric field. This study was performed in order to interpret 14N resonance results but it can also be extended to other nuclei. INTRODUCTION
The Stark effect on the nuclear quadrupole resonance (NQR) of half-integral spin nuclei was observed in a series of polycrystalline samples by Duchesne, Read and Cornill and Dixon and Bloembergen2. The line shapes were then studied using calculations based on the linearity of the Stark effect, which had been discussed earlier by Bloembergen 3. The Stark-perturbed lines considered by these authors dealt with the special cases where the asymmetry of the field gradient is zero and where the exciting magnetic field is orthogonal or parallel to the electric field. To interpret 14N resonance data, Stark-perturbed lines were studied for a nonzero asymmetry parameter and for an arbitrary orientation of the exciting magnetic field and the electric field, since for certain substances, such as pyridine 14N resonances are characterized by high asymmetry coefficient and its derivatives,
vaIues. In addition, the approximation used for the half-integral spin halogen resonances is justified by the quadratic relation between the frequency and the low asymmetry parameter values of the sigma bonds. This approximation is not applicable to spin 1 nuclei such as 14N whose freq uencies vary linearly with the asym* The author is au Aspirant of the Fonds National de la Recherche Scientifique. J. Mol. Structure,
11 (1972)
475
metry parameter. Our primary consideration in this paper is the solution of problems concerning the Stark effect on 14N resonances. However, the following may be extended to the resonances of other nuclei in highly asymmetric field gradients. We will first consider the case of single crystals and then continue with a study of polycrystalline samples. STARK EFFECT ON SINGLE CRYSTALS
The lines given by single crystals show a frequency shift in electric fields but there is no line splitting since (i) the field gradient is assumed to be asymmetric, (ii) the quadrupole levels are not degenerate for spin 1 nuclei, and (iii) the electric field does not lift the Garners degeneracy for half-integral spin nuclei. Frequency shift
In general, the frequency of a given transition, <, is a function of the three components of the electric field E and can be developed for the first order, yielding
Aft = f#)
- f@)
= grad, fc - E
(1)
where gradJs E mo5_ The vector mog defines the direction in which the effect of the transition shift induced by the electric field is maximal; it is fixed with respect to the principa1 axis of the field gradient_ A direct result of the linearity of the Stark effect is that one can determine at the most three independent magnitudes related to the components of rn,< by observing the frequency of a transition. Another consequence is that the vectors mar of crystallites which give the same frequency shift are Iocated on a cone having the electric field as an axis. To simplify notation, we wili henceforth omit the index < characterizing the transition considered. Eqn. (1) can be transformed in order to distinguish the terms related to the orientation of the maximum Stark shift axis m, from the terms related to its module. In terms of the direction cosines of the electric field, cos Ed, and of the vector m, cosines, cos POi, eqn. (1) may be written
Af = Afo ~
COS ~oi COS Ei
i=l
Afo = IizPkfl
- FI
At this point one can relate the frequency shifts to the changes in the electric field gradient_ Frequency
shift
and the R tensor
According to the notation introduced by BIoembergen3, the changes Aqi; in the eiectric field gradient components can be expressed using the tensor Rkii Aq, 476
= Ek R,ij J. Mol.
Structure,
11 (1972)
For the principal coordinate system of the field gradient, we have 4 =
q33
a
q11-q22
=
One then finds the changes in the principal components of the field gradient
Aq = 5 E&&s33 k=l
Am
=
;Ek(Rk~rRkm) k=l
From this, one derives the relationship between the components of the R tensor and the frequency shifts Afi when the electric field is parallel to the axes of the field gradient Afi = Afa Aj; = E,
COS ,Uoi
Ri33 + g
(Rill
1
-RiZt)
(2)
Two types of terms may be distinguished in eqn. (2) First, the R tensor defines changes in the field gradient components as a function of the electric field and depends only on electronic structures. Secondly, we have the sensitivity coefficients of the frequency to perturbations in the field gradient, that is, iTf/laqand
df/i%x.These generally depend on the spin of the resonating nucleus, on the asymmetry parameter, and on the type of transition. The relative sensitivity coefficients
aflh - Gand @Pa - q/f are expressed as a function of the asymmetry parameter for nuclei of spin 3/2 and 1 in Figs. 1A and 1B. B
Fig. 1. Sensitivity
afJ/aq- q/f
is shown
coefficient of the NQR frequency to changes in field gradient. Coefficient as a solid line; coeEcient afJaa - q/f is shown as a dotted line. A. Spin 312.
B. Spin 1. J. Mol.
Structure,
11 (1976)
477
It will be noted that for 14N, which has spin 1, af/dq - q/f and af/laa - q/f remain comparable in magnitude irrespective of the value of the asymmetry parameter. However, for nuclei of spin 3/2, the sensitivity to changes in a drops as the asymmetry parameter vanishes.
Measurements
in singIe crystals
On the basis of eqn. (2) by measuring line shifts for three orientations of the electric field relative to the field gradient axes, one can determine the vector m,. In single crystals it is possible to determine Stark shifts accurately. However, their determination involves difficulties due to the large crystai dimensions required and the fact that many substances are liquid at room temperature. The following sections will therefore be devoted to the study of the spectra of isotropic powders, since it will be easier to generalize from these experiments.
STARK EFFECT IN POLYCRYSTALLINE
SAMPLES
In order to calculate the line shapes of polycrystalline samples, it is necessary to sum the energies absorbed by the crystallites in each of the frequency bands. To this end, one must calculate the transition probability as a function of the crystallite’s position with respect to the electriG field and the exciting magnetic field. Each position corresponds to an absorption in a frequency band whose center can be calculated by means of the shift discussed above.
Calcrtlations of line shape The transition probabilities in NQR are given, for an exciting magnetic fleId H, by an expression4a5 of the following form, where the vectors xi are the field gradient axes 3
W =
c (H
-
xi)‘wi
i=1
The coe’flicients Wi depend on the nuclear spin and on the asymmetry parameter. Let us defme the coordinate system in such a way that Oz is parallel to the electric field E, and that the magnetic field vector H terminates at the point defined by (0, H cos jl, H sin /?). If the projection of the vector Xi is taken to be the origin of the angles in the polar coordinate system with axis me, its components can be written as follows, with the exception of a constant factor (sin pei, 0, cos per). Angles 8,~ and ti are de&red in Fig. 2. For an isotropic powder, the density of the orientations is proportional to the volume element dr = dS2dp = -d(cos@dqdt,G 478
(3) J. Mol. Structure, 11 (1972)
Fig. 2. Coordinate systems. The coordinates of the static electric field E and of the exciting magnetic field H are shown as a dotted line. The Oz axis of this coordinate system is parallel to the vector mo, for which the Stark effect is maximum. The angles pot define the orientation of this vector with respect to the axes of the field gradient.
Taking into account eqn. (l), the frequency shift Afcan be written as follows using the notation defined above Af = AfO cos 0
The transition probability frequency can be written
W(e) =
for a cone of angle 8 corresponding
J2’J2’$ [H(a, q=O
#=O
i=l
to a given absorption
X&i)]” * “idqdti
CP, $I) m
By integrating eqn (4) and normalizing it in terms of the maximum one can deduce the spectral absorption density A (Af/Afo)
=p
3
8 5
3
+=I =I
co2 p
(1 +cos2 /foi)+(l
-3
shift AfO.
cos2 /lOi)
j=l
+2 sin 2j?
sin’ iuoi+(2-3 [
Af
sin2 IrOi) ( Ml0
HI 2
- Wi
(5)
Analysis of the spectral density5 A shows that the Stark broadening is a parabola bounded by two vertical axes which are symmetrically located with respect to the line at zero electric field. Eqn. (5) becomes simplified for certain nuclear spin values and certain resonant site symmetries which we shall discuss for the cases of half-integral spin and spin 1.
Half-integral spin nuclei If q = 0, for the case of spin 3/2, the sensitivity coefficient to changes in a is zero, as shown in Fig. 1A. In addition, the symmetries for which the field gradient possesses a zero asymmetry parameter cause certain components of the tensor Rkii J. Mol. Sfructure, 11 (1972)
479
vanish. In particular, only RJ3 s is involved in the frequency shift if q = 0. Consequently, by making the exciting magnetic field and the eIectric field orthogona1 or parallel (/? = 0, /I = a~), one obtains the same Stark-spectra of halfintegral spin nuclei for a zero asymmetry field gradient as were found earlier2. If the asymmetry parameter is no longer negligible, or if the fields make an arbitrary angle, al1 the terms of eqn. (5) must be taken into account. to
Spin I nuclei
For spin 1 nuclei, the frequency shifts for an electric field parallei to the axes of the field gradient (eqn. (2)) become, taking into account the equations for the sensitivity coefficients,
In addition, the transition probabilities6 are such that each line only involves one of the terms wl, IV,, w3 of sum eqn. (4), as shown in Table 1. TABLE
1
TRANSITION
Transition to-+
(Ot-) t++-_)
+I
PR0BABILl-i-Y
OF
SPIN
1 NUCLEI
Wl
w2
w3
1 0 0
0 1 0
0 0 1
According to eqn. (5), the broadenings of lines (0 -+ + ), (0 -P - ) and (+ --, -) are different if the terms (Ri1, - Ri22) are not zero: one can thus observe the changes in the terms as a function of the asymmetry parameter in the case of spin 1 nuclei. The lines corresponding to orthogona1 exciting magnetic fields and electric fiekls are shown in Fig. 3. Also shown (as dotted lines) are the absorption lines and their derivatives and (as a solid Iine) the convolution products of these lines by a Gaussian whose width is equal to the natural line width. The derivatives correspond to recordings made by the frequency modulation technique. Fig. 4 shows‘the lines obtaihed when the electric field is parallel to the exciting magnetic field_ 480
J. Mol. Structure, 11 (1972)
0”
30’
60”
90’
Fig. 3. Stark effect on the NQR of spin 1 nuclei, for an exciting magnetic field orthogonal to the electric field. The curves are parametrized in terms of the angles ,uO1which define the orientatidn of the vector m,-, with respect to the axes of the field gradient. (a) Absorption curves, (b) derivatives_
Fig. 4. Stark effect on the NQR of spin nuclei, for an exciting magnetic field parallel to the electric field. The curves are parametrized in terms of the angles par which define the orientation of the vector mo with respect to the axes of the field gradient. (a) Absorption curves, (b) derivatives.
As eqn. (5) demonstrates, the sum weighted by coefficients wl, w2 and w3 of the three lines shown in Figs. 3 and 4 makes it possible to deduce the line shapes for all spin values. Let. us turn again to spin 1 nuclei_ If we consider, for example, the resonance of sites of symmetry C,,, certain components of the tensor Rkii are zero’. One obtains Afl In
= Af2 = 0
other words, for angles po3 : PO3
(0
PO3
(+
-+
+-I --,
=
3n:
-I=
0
Figs. 3 and 4 show that the convexity of lines (0 + +) and (0 + -) are then opposed to the convexity of Iine (-t -+ -). The broadenings of the two lines (0 -+ A-) and (0 -+ -) are clearly different is not zero, but for symmetry C,, they retain provided that the term (Ri,, -I&) the same shape_ J. Mol. Structure, 11 (1972)
481
In the general case where the angle between the exciting magnetic field and the electric field is arbitrary, it is clear from eqn. (5) that the lines can be deduced by linear combination of the distributions shown in Figs. 3 and 4. CONCLUSIONS
For an asymmetric field gradient, the line shapes in the presence of the Stark effect on NQR are parabolic, irrespective of the spin of the resonating nucleus and the orientation of the exciting magnetic field with respect to the electric field. The shapes of these lines may differ considerably from those studied previously for half-integral
spin nuclei and zero asymmetry,
as shown in Figs. 3 and 4. For
spin 1 nuclei, one can observe the changes in the asymmetry parameter by measuring the difference between the broadenings of lines (0 + t) and (0 + -), as we have demonstrated for sites of symmetry C,,.
ACKNOWLEDGEMENTS
It is a pleasure to thank Professor
Jules Duchesne
subject proposed for study and for the kind hospitality
for the interest of the
offered in his laboratories.
Thanks are also due to Drs. J. Depireux, A. Gerard and M. Read for the we help provided during the work and for numerous fruitful discussions.
REFERENCES
J. DUCHESNE, M. READ AND P. CORNIL, J. Phys. Chem. Solids, 24 (1963) 1333. R. W. DIXON AND N. BLOEMBERGEN,J. Chem. Phys., 41 (1964) 1739. N. BLOEMBERGEN,Science, 133 (1961) 1363. T. P. DAS AND E. L. HAHN, Nuclear QaadrupoIe Resonance Spectroscopy, Supplement 1 of F. SEITZ AND D. TURNBULL (Editors), Solid Stare Physics, Academic Press, New York, 1958. 5 M. TOYAMA, J. Phys. Sot. Jap., 14 (1959) 1727. 6 L. GUIBE, Ann. Phys. (Leipzig), 7 (1962) 177. 7 J. F. NYE, PropriPtPs Physiques des Cristaux, Dunod, Paris, 1961, p- 125.
1 2 3 4
482
J. ~Mof. Swucrare, I1 (1972)
STARK EFFECT ON THE NUCLEAR QUADRUPOLE 14N IN AN ASYMMETRIC FIELD GRADIENT
RESONANCE
OF
I. LINE SHAPE J. L. COLOT* Institrrrde Physique,
UniversitP de Li.+ge, 4000-LiGge Sart Tihan (Belgium)
(Received October 22nd,
1971)
ABSTRACT
Nuclear _quadrupole resonance line shapes in the presence of the Stark effect were studied in the case of asymmetric field gradients, where the exciting magnetic field forms an arbitrary angle with the electric field. This study was performed in order to interpret 14N resonance results but it can also be extended to other nuclei. INTRODUCTION
The Stark effect on the nuclear quadrupole resonance (NQR) of half-integral spin nuclei was observed in a series of polycrystalline samples by Duchesne, Read and Cornill and Dixon and Bloembergen2. The line shapes were then studied using calculations based on the linearity of the Stark effect, which had been discussed earlier by Bloembergen 3. The Stark-perturbed lines considered by these authors dealt with the special cases where the asymmetry of the field gradient is zero and where the exciting magnetic field is orthogonal or parallel to the electric field. To interpret 14N resonance data, Stark-perturbed lines were studied for a nonzero asymmetry parameter and for an arbitrary orientation of the exciting magnetic field and the electric field, since for certain substances, such as pyridine 14N resonances are characterized by high asymmetry coefficient and its derivatives,
vaIues. In addition, the approximation used for the half-integral spin halogen resonances is justified by the quadratic relation between the frequency and the low asymmetry parameter values of the sigma bonds. This approximation is not applicable to spin 1 nuclei such as 14N whose freq uencies vary linearly with the asym* The author is au Aspirant of the Fonds National de la Recherche Scientifique. J. Mol. Structure,
11 (1972)
475
metry parameter. Our primary consideration in this paper is the solution of problems concerning the Stark effect on 14N resonances. However, the following may be extended to the resonances of other nuclei in highly asymmetric field gradients. We will first consider the case of single crystals and then continue with a study of polycrystalline samples. STARK EFFECT ON SINGLE CRYSTALS
The lines given by single crystals show a frequency shift in electric fields but there is no line splitting since (i) the field gradient is assumed to be asymmetric, (ii) the quadrupole levels are not degenerate for spin 1 nuclei, and (iii) the electric field does not lift the Garners degeneracy for half-integral spin nuclei. Frequency shift
In general, the frequency of a given transition, <, is a function of the three components of the electric field E and can be developed for the first order, yielding
Aft = f#)
- f@)
= grad, fc - E
(1)
where gradJs E mo5_ The vector mog defines the direction in which the effect of the transition shift induced by the electric field is maximal; it is fixed with respect to the principa1 axis of the field gradient_ A direct result of the linearity of the Stark effect is that one can determine at the most three independent magnitudes related to the components of rn,< by observing the frequency of a transition. Another consequence is that the vectors mar of crystallites which give the same frequency shift are Iocated on a cone having the electric field as an axis. To simplify notation, we wili henceforth omit the index < characterizing the transition considered. Eqn. (1) can be transformed in order to distinguish the terms related to the orientation of the maximum Stark shift axis m, from the terms related to its module. In terms of the direction cosines of the electric field, cos Ed, and of the vector m, cosines, cos POi, eqn. (1) may be written
Af = Afo ~
COS ~oi COS Ei
i=l
Afo = IizPkfl
- FI
At this point one can relate the frequency shifts to the changes in the electric field gradient_ Frequency
shift
and the R tensor
According to the notation introduced by BIoembergen3, the changes Aqi; in the eiectric field gradient components can be expressed using the tensor Rkii Aq, 476
= Ek R,ij J. Mol.
Structure,
11 (1972)
For the principal coordinate system of the field gradient, we have 4 =
q33
a
q11-q22
=
One then finds the changes in the principal components of the field gradient
Aq = 5 E&&s33 k=l
Am
=
;Ek(Rk~rRkm) k=l
From this, one derives the relationship between the components of the R tensor and the frequency shifts Afi when the electric field is parallel to the axes of the field gradient Afi = Afa Aj; = E,
COS ,Uoi
Ri33 + g
(Rill
1
-RiZt)
(2)
Two types of terms may be distinguished in eqn. (2) First, the R tensor defines changes in the field gradient components as a function of the electric field and depends only on electronic structures. Secondly, we have the sensitivity coefficients of the frequency to perturbations in the field gradient, that is, iTf/laqand
df/i%x.These generally depend on the spin of the resonating nucleus, on the asymmetry parameter, and on the type of transition. The relative sensitivity coefficients
aflh - Gand @Pa - q/f are expressed as a function of the asymmetry parameter for nuclei of spin 3/2 and 1 in Figs. 1A and 1B. B
Fig. 1. Sensitivity
afJ/aq- q/f
is shown
coefficient of the NQR frequency to changes in field gradient. Coefficient as a solid line; coeEcient afJaa - q/f is shown as a dotted line. A. Spin 312.
B. Spin 1. J. Mol.
Structure,
11 (1976)
477
It will be noted that for 14N, which has spin 1, af/dq - q/f and af/laa - q/f remain comparable in magnitude irrespective of the value of the asymmetry parameter. However, for nuclei of spin 3/2, the sensitivity to changes in a drops as the asymmetry parameter vanishes.
Measurements
in singIe crystals
On the basis of eqn. (2) by measuring line shifts for three orientations of the electric field relative to the field gradient axes, one can determine the vector m,. In single crystals it is possible to determine Stark shifts accurately. However, their determination involves difficulties due to the large crystai dimensions required and the fact that many substances are liquid at room temperature. The following sections will therefore be devoted to the study of the spectra of isotropic powders, since it will be easier to generalize from these experiments.
STARK EFFECT IN POLYCRYSTALLINE
SAMPLES
In order to calculate the line shapes of polycrystalline samples, it is necessary to sum the energies absorbed by the crystallites in each of the frequency bands. To this end, one must calculate the transition probability as a function of the crystallite’s position with respect to the electriG field and the exciting magnetic field. Each position corresponds to an absorption in a frequency band whose center can be calculated by means of the shift discussed above.
Calcrtlations of line shape The transition probabilities in NQR are given, for an exciting magnetic fleId H, by an expression4a5 of the following form, where the vectors xi are the field gradient axes 3
W =
c (H
-
xi)‘wi
i=1
The coe’flicients Wi depend on the nuclear spin and on the asymmetry parameter. Let us defme the coordinate system in such a way that Oz is parallel to the electric field E, and that the magnetic field vector H terminates at the point defined by (0, H cos jl, H sin /?). If the projection of the vector Xi is taken to be the origin of the angles in the polar coordinate system with axis me, its components can be written as follows, with the exception of a constant factor (sin pei, 0, cos per). Angles 8,~ and ti are de&red in Fig. 2. For an isotropic powder, the density of the orientations is proportional to the volume element dr = dS2dp = -d(cos@dqdt,G 478
(3) J. Mol. Structure, 11 (1972)
Fig. 2. Coordinate systems. The coordinates of the static electric field E and of the exciting magnetic field H are shown as a dotted line. The Oz axis of this coordinate system is parallel to the vector mo, for which the Stark effect is maximum. The angles pot define the orientation of this vector with respect to the axes of the field gradient.
Taking into account eqn. (l), the frequency shift Afcan be written as follows using the notation defined above Af = AfO cos 0
The transition probability frequency can be written
W(e) =
for a cone of angle 8 corresponding
J2’J2’$ [H(a, q=O
#=O
i=l
to a given absorption
X&i)]” * “idqdti
CP, $I) m
By integrating eqn (4) and normalizing it in terms of the maximum one can deduce the spectral absorption density A (Af/Afo)
=p
3
8 5
3
+=I =I
co2 p
(1 +cos2 /foi)+(l
-3
shift AfO.
cos2 /lOi)
j=l
+2 sin 2j?
sin’ iuoi+(2-3 [
Af
sin2 IrOi) ( Ml0
HI 2
- Wi
(5)
Analysis of the spectral density5 A shows that the Stark broadening is a parabola bounded by two vertical axes which are symmetrically located with respect to the line at zero electric field. Eqn. (5) becomes simplified for certain nuclear spin values and certain resonant site symmetries which we shall discuss for the cases of half-integral spin and spin 1.
Half-integral spin nuclei If q = 0, for the case of spin 3/2, the sensitivity coefficient to changes in a is zero, as shown in Fig. 1A. In addition, the symmetries for which the field gradient possesses a zero asymmetry parameter cause certain components of the tensor Rkii J. Mol. Sfructure, 11 (1972)
479
vanish. In particular, only RJ3 s is involved in the frequency shift if q = 0. Consequently, by making the exciting magnetic field and the eIectric field orthogona1 or parallel (/? = 0, /I = a~), one obtains the same Stark-spectra of halfintegral spin nuclei for a zero asymmetry field gradient as were found earlier2. If the asymmetry parameter is no longer negligible, or if the fields make an arbitrary angle, al1 the terms of eqn. (5) must be taken into account. to
Spin I nuclei
For spin 1 nuclei, the frequency shifts for an electric field parallei to the axes of the field gradient (eqn. (2)) become, taking into account the equations for the sensitivity coefficients,
In addition, the transition probabilities6 are such that each line only involves one of the terms wl, IV,, w3 of sum eqn. (4), as shown in Table 1. TABLE
1
TRANSITION
Transition to-+
(Ot-) t++-_)
+I
PR0BABILl-i-Y
OF
SPIN
1 NUCLEI
Wl
w2
w3
1 0 0
0 1 0
0 0 1
According to eqn. (5), the broadenings of lines (0 -+ + ), (0 -P - ) and (+ --, -) are different if the terms (Ri1, - Ri22) are not zero: one can thus observe the changes in the terms as a function of the asymmetry parameter in the case of spin 1 nuclei. The lines corresponding to orthogona1 exciting magnetic fields and electric fiekls are shown in Fig. 3. Also shown (as dotted lines) are the absorption lines and their derivatives and (as a solid Iine) the convolution products of these lines by a Gaussian whose width is equal to the natural line width. The derivatives correspond to recordings made by the frequency modulation technique. Fig. 4 shows‘the lines obtaihed when the electric field is parallel to the exciting magnetic field_ 480
J. Mol. Structure, 11 (1972)
0”
30’
60”
90’
Fig. 3. Stark effect on the NQR of spin 1 nuclei, for an exciting magnetic field orthogonal to the electric field. The curves are parametrized in terms of the angles ,uO1which define the orientatidn of the vector m,-, with respect to the axes of the field gradient. (a) Absorption curves, (b) derivatives_
Fig. 4. Stark effect on the NQR of spin nuclei, for an exciting magnetic field parallel to the electric field. The curves are parametrized in terms of the angles par which define the orientation of the vector mo with respect to the axes of the field gradient. (a) Absorption curves, (b) derivatives.
As eqn. (5) demonstrates, the sum weighted by coefficients wl, w2 and w3 of the three lines shown in Figs. 3 and 4 makes it possible to deduce the line shapes for all spin values. Let. us turn again to spin 1 nuclei_ If we consider, for example, the resonance of sites of symmetry C,,, certain components of the tensor Rkii are zero’. One obtains Afl In
= Af2 = 0
other words, for angles po3 : PO3
(0
PO3
(+
-+
+-I --,
=
3n:
-I=
0
Figs. 3 and 4 show that the convexity of lines (0 + +) and (0 + -) are then opposed to the convexity of Iine (-t -+ -). The broadenings of the two lines (0 -+ A-) and (0 -+ -) are clearly different is not zero, but for symmetry C,, they retain provided that the term (Ri,, -I&) the same shape_ J. Mol. Structure, 11 (1972)
481
In the general case where the angle between the exciting magnetic field and the electric field is arbitrary, it is clear from eqn. (5) that the lines can be deduced by linear combination of the distributions shown in Figs. 3 and 4. CONCLUSIONS
For an asymmetric field gradient, the line shapes in the presence of the Stark effect on NQR are parabolic, irrespective of the spin of the resonating nucleus and the orientation of the exciting magnetic field with respect to the electric field. The shapes of these lines may differ considerably from those studied previously for half-integral
spin nuclei and zero asymmetry,
as shown in Figs. 3 and 4. For
spin 1 nuclei, one can observe the changes in the asymmetry parameter by measuring the difference between the broadenings of lines (0 + t) and (0 + -), as we have demonstrated for sites of symmetry C,,.
ACKNOWLEDGEMENTS
It is a pleasure to thank Professor
Jules Duchesne
subject proposed for study and for the kind hospitality
for the interest of the
offered in his laboratories.
Thanks are also due to Drs. J. Depireux, A. Gerard and M. Read for the we help provided during the work and for numerous fruitful discussions.
REFERENCES
J. DUCHESNE, M. READ AND P. CORNIL, J. Phys. Chem. Solids, 24 (1963) 1333. R. W. DIXON AND N. BLOEMBERGEN,J. Chem. Phys., 41 (1964) 1739. N. BLOEMBERGEN,Science, 133 (1961) 1363. T. P. DAS AND E. L. HAHN, Nuclear QaadrupoIe Resonance Spectroscopy, Supplement 1 of F. SEITZ AND D. TURNBULL (Editors), Solid Stare Physics, Academic Press, New York, 1958. 5 M. TOYAMA, J. Phys. Sot. Jap., 14 (1959) 1727. 6 L. GUIBE, Ann. Phys. (Leipzig), 7 (1962) 177. 7 J. F. NYE, PropriPtPs Physiques des Cristaux, Dunod, Paris, 1961, p- 125.
1 2 3 4
482
J. ~Mof. Swucrare, I1 (1972)