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The influence of anisotropy and nuclear couplings on the magnetic quenching of positronium complexes
Volume 182,number 5
CHEMICALPHYSICSLETTERS
9 August I99 1
The influence of anisotropy and nuclear couplings on the magnetic quenching of positronium complexes Martina Schwager and Emil Roduner Physikalisch-Chemisches
Insiitut der Universitiit
Ziirrch,
Wintrrthurerstrasc
190, CH-8057
Zurich, Swltzedand
Received 6 April I991
The magnetic field dependence of the lifetimes and energystates of positronium complexes is calculated for a three-spin-l/2 system includingan electron, a positron, and a nucleus. It is demonstrated that couplingof a magnetic nucleus and anisotropy of the system has a pronounced effect on the magneticquenchingbehaviour, in particularnear avoided crossingsof magneticenergy levels.
1. Introduction Positronium (Ps), the bound state of an electron (e) and a positron (p), is a two-spin- l/2 system. In zero magnetic field its eigenstates consist of a singlet and a set of three degenerate triplet states, split in energy by the hypertine interaction. The singlet (also called para, p-Ps, because of its antiparallel spins) and the triplet (or ortha, o-Ps) annihilate with different rates. In vacuum, the corresponding lifetimes amount to ‘I~=0.125 ns for the singlet and ?T= 140 ns for the triplet states [ 1,2]. Application of a magnetic field leaves the two triplet states characterized by the magnetic quantum numbers IV= It 1 unaffected. The singlet mixes with the remaining triplet (M= 0) state. This in principle changes the lifetimes of both states, but since the contribution for the fast decay rate given by the singlet character dominates, it means essentially that one long-lived triplet is converted to a short-lived singlet state. A lifetime plot as a function of the applied field is called a magnetic quenching curve. In a medium the magnetic quenching behaviour is slightly modified, revealing that Ps is perturbed by the interaction with its environment [ 31. For inert solvents the effect is described satisfactorily by a positron-electron hyperfine interaction which is still of spherical symmetry but reduced by a factor 9 compared with its vacuum value. This reduction amounts 0009-2614/91/$
03.50
0
1991 -
to a factor of 0.84 in benzene and as much as 0.65 in water [4]. A far more pronounced change on the quenching curve was found with solid naphthalene [ 5,6] and in particular with a solution of nitrobenzene in IZhexane [ 7,8]. This anomalous magnetic quenching effect was explained in terms of a modified o-Ps lifetime and a reversible addition of Ps to nitrobenzene, thereby forming a Ps-molecule complex [ 8,9 1.Most characteristic of the complex is a hyperfine interaction of only a few percent of its vacuum value, indicating a strong expansion of Ps by delocalization onto nitrobenzene. A calculation of the e-p distribution in the complex shows that approximately 99% of Ps is localized on the nitro group [lo]. The two particles are not equally distributed over the three atoms, however. The population on nitrogen is about 50% for the electron and 69% for the positron. We expect that the magnetic nature of nitrogen could perturb the Ps system considerably. In the analysis of the above experiments, the complex was regarded as a two-spin- I /2 system of spherical symmetry akin to Ps. A detailed discussion of chemical aspects has been given by Mogensen [ 91. Alternatively, one could view the system as the analog of a “complex” of a hydrogen atom with an unsaturated molecule, i.e. an organic free radical. In such a case, the hyperfine interaction of the H atom during addition also decreases to typically a few per-
Elsevier Science Publishers B.V. (North-Holland)
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cent of its vacuum value. This view lead us to investigate the effects of two possible complications which could occur in such complexes in general. The first one is the presence of further magnetic nuclei which are likely to be coupled magnetically to positron and electron. The multi-spin system is expected to show a different magnetic quenching behaviour which depends an a/l the couplings involved. The second one is the non-spherical nature of such complexes which leads to anisotropy of the magnetic interaction. In a non-viscous environment rapid tumbling will average this dipolar contribution to zero within the Ps lifetime, but in naphthalene or other solids this should certainly not be the case. The anisotropic Hamiltonian has additional terms which complicate the mixing of the zero-field wavefunctions and influence the lifetimes as the field is increased.
2. Theory The Hamiltonian for a Ps complex is a sum of electronic and magnetic terms fi=&l$fiSPln
.
(1)
In the absence of spin-orbit coupling the solution for the wavefunction is a product V’ Ysp’“. V’ may be written in a basis of products of atomic orbitals centered at the nuclei and a hydrogen-like Is orbital for the positron. This is a very limited basis and not necessarily adequate to solve the electronic problem. However, it has the advantage that it makes the bridge to the conventional picture of a Ps complex and allows us to solve the magnetic part of the problem in the same way as is done in magnetic resonance where the solution for PC’”enters parametrically in the form of coupling constants which are proportional to the contact density at the nucleus, 1P(0) 12. In order to investigate the effect of magnetic nuclei it is sufficient to consider a system of three particles (electron, positron, and one nucleus) which are all coupled anisotropically. Its spin Hamiltonian is given by
(2)
9 August 199 I
where SC,Sp and II are the spin operators of the electron, positron and the nucleus k, and v,= vp and V~ the corresponding Larmor frequencies in the absence of g-anisotropy. &,, $, and Apk are the hyperline tensors in frequency units for the three interacting pairs. Each tensor is the sum of an isotropic Fermi contact term A,, and a dipolar spin-spin interaction D,,. For simplicity, and since it is sufficient for our purpose, we consider the case of an anisotropy of axial symmetry, as obtained for example for dipolar interaction of a nucleus with an electron in a p-type orbital [ 111. The principal values of the diagonalized tensor are called the anisotropic coupling constants. A,, corresponds to qAE/h in the nomenclature convention of Ps chemistry, where Ai?/ h is the zero field splitting of Ps in vacuum (203 GHz) and ‘1is the reduction factor of the electron contact density at the positron for Ps in solution or in a complex. The eigenvectors IV,, (dropping the superscript “spin”) are written in a basis of Zeeman product functions x, v/n= 7 c,, IX: > IXY> Ix: > = c c,,,x, . ,
(3)
For the case of isotropic coupling, the interaction of a pair has the structure S~~=AS~=AS=~=+ ^^ ). The off-diagonal terms mix basis 1‘Asj ( + _ t S-1, functions of equal M only. In the anisotropic case, the dipolar contribution leads to terms of the same type, but in addition there are terms of the structure --iDL sin 0~0s B (gf+ts+L) and - aDL sin20S+f+ (for the complete Hamiltonian we refer to ref. [ I1 ] ), where 0 is the azimuthal angle between the symmetry axis and the applied field. They lead to mixing of basis functions with 1AMI = 1 and lAMl =2, respectively. For anisotropic Ps this has the consequence that none of the states retains its long lifetime in an applied field. For evaluation of the lifetimes, the eigenvectors are transformed to a new set of basis vectors which are the tensor products of a zero field positronium eigenstate t, (representing a Ps singlet or triplet state) with a nuclear Zeeman vector xf In the new basis @,= ( I[) 11”) ), the Zeeman product functions are given by XI= c 4!,( 18 IXk>),= c 4!,@, I I
(4)
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CHEMICAL PHYSICS LETTERS
and the eigenvectors by
From the normalization condition ( v/, 1yn) = cp;, = 1 it follows that (~2, is the contribution of the jth basisvector to the eigenstate lu,. The decay rate i, of the nth eigenstate consists additively of the decay rates 2, of the contributing basis vectors @J,weighted with their coefficient ct!;, [ 31. Each basis vector decays with the rate that corresponds to its Ps part 5 when any influence of the nucleus to the singlet or triplet decay rate is neglected. When the singlet and triplet states are expressed in a Zeeman basis,
the matrix of coefficients, Q= ( qu), is given by the transposition of R= (r,,) Q=R’.
(7)
This yields the magnetic-field-dependent lifetimes rk of the eigenstates by evaluation of the secular determinant as a function of the magnetic field
(8)
3. Discussion of specific cases Here, the features of the three-spin-l /2 system consisting of an electron, positron and a proton as the magnetic nucleus are examined numerically with regard to the Ps lifetimes and energy eigenstates. First, the influence of adding a proton to an isotropic Ps complex is studied under both Isotropic and anisotropic conditions. For the proton hyperfine coupling constants we chose values which are typical for organic radicals. Then the influence ofthe proton was strongly increased to show some principal effects of the interacting nucleus. Since these are different in the isotropic and the anisotropic system, the two cases are discussed separately. Simulations were carried out using the parameters ~~~0.125 ns for the singlet lifetime and T,= 3.9 ns, which is about the Ps triplet lifetime in n-hexane [ 7 1.
9August 1991
This choice is quite arbitrary, but the results can be simply scaled to any other effective values of rs and rr. The isotropic electron-positron coupling constant has in all cases about 2.5% of its vacuum value, which is comparable with an observed decrease to about 1% in nitrobenzene solutions [ 121. To facilitate discussion, the lifetimes and energies are marked by their eigenfunctions in the high field limit (v,, v,>A,,,) where they are pure Zeeman functions. These are denoted by the spin quantum numbers m,, mp,mk of the three particles. They take the values + I /2, - l/2, which is abbreviated in the figures by +, -. As indicated above, there is a general behaviour of the lifetimes in high fields, where the spins are decoupled and the eigenstates of Ps or Ps complexes are pure Zeeman states. Here, the states form two groups. One group is characterized by I f t, m,) and I- -, mk), and corresponding lifetimes approach r7. The other group belongs to the states I+-,mk)and I-+,m,),wherethePspartisa linear combination of the Ps singlet (M=O) and triplet (M=O) state with equal weights. Their lifetimes become 5,=2(~~‘+z~‘)-‘s27,. The interesting behaviour therefore occurs at relatively low fields. The upper frame of fig. 1 shows the field dependence of the lifetimes for an isotropic three-spinl/2 e-p-k system with A,,=4000 MHz and Aek=Apk= 100 MHz (solid lines) and compares it with the case where the nucleus is uncoupled or absent (broken lines where they are not degenerate with the first system). The latter represents a Ps complex as normally understood in the literature. The additional spin doubles the number of states, but since Aek=Apk, this leaves the lifetimes in nearly degenerate pairs. Inspection of fig. 1 shows that introduction of the nucleus leaves the two long-lived states (i.e. the ones with maximum IM( = I I;m, 1) unchanged and independent of field, and it doubles the shortest-lived state. One pair splits significantly off the longest-lived state whereas the lifetime of another pair increases slightly from its values in the ep system. The average lifetime, which is usually the experimental observable. decreases in the region around a few hundred gauss. The lower frame of fig. 1 demonstrates the influence of a typical axial anisotropy of D, set to 10% of corresponding isotropic 447
9 August I99
CHEMICAL PHYSICS LETTERS
Volume 182, number 5
,,,,,
,
E [MHz] m,,,,
I
I+-+>
*
(it->/--i/:
3.0: I++->
O-
2.0:
Ittt>.l--->
1
:
-5ooo1.0.
-10000~ -15000-
l-++>,I--+->
0.0
0
100
200 300 H [Gauss]
400
500
Fig. I. Upper: Simulated lifetime curves for an isotropic e-p-k system withA,= MHz and&,&,,= 100 MHz (full lines), and an e-p system with/l,,=4000 MHz (broken lines, where not degenerate with the first system). The corresponding states are marked by the spin quantum numbers Im,, mp, ml,) and lnz,, mp) in high fields. Lower: Lifetime curves for an uniaxial e-p-k system with the same isotropic couplings as above, and amsotropies D, of IO%of corresponding isotropic values, with 0=45”,
values. Clearly, the lifetimes of two pairs of states decrease, and there are now no long-lived states with lifetimes independent of the field. Fig. 2 displays the energy levels and lifetimes of an isotropic system with Aek= 1500 MHz and ADk= - 2000 MHz, i.e. of the same order of magnitude as A,,=4000 MHz. The increased couplings allow us to comment on some features which are not resolved in fig. I. The first is the fact that in zero field the eight eigenstates consist of a quartet and two doublets. .4ccordingly, there are three different lifetimes, the average being less than in the absence of nuclei coupled 448
0
1000
2000 3000 H [Gauss]
4000
5000
Fig. 2. Upper: Energy levels for an isotropic e-p-k system with A,,=4000 MHz, Aek= 1500 MHz, and A,,= -2000 MHz. Lower: Corresponding lifetime curves.
to the e-p system. Secondly, the two states 1t - - > and I -- t ) undergo an avoided crossing due to A,, near 500 G. In this region, the mixing of states is a strong function of the magnetic field, and as a consequence of this the lifetimes exhibit pronounced resonances. It should be noted that there are also true crossings between the states I - - - >, I - - t ), and 1t t - ), which however have no effect on the lifetimes. A simulation of an anisotropic system is given in fig. 3. The isotropic values are the same as in fig. 2, and uniaxial anisotropies are set to 10% of their isotropic values. In the upper frame, the two states I - t t ) and I- t - ) are omitted since they are too distant and behave a lot like in fig. 2. Obviously, anisotropy has lifted the quartet degeneracy in zero
Volume 182. number 5
CHEMICAL PHYSlCS LETTERS
E[
1
o.oJ 0
1
I-+t>.l-+-> 1
1
200
I
600 400 H [Gauss]
600
4
1000
Fig. 3. Upper: Energy levels for an uniaxial e-p-k system with isotropic couplings A,,=4000 MHz, &=I500 MHz, &= - 2000 MHz, and anisotropies DI set to 10% of the corresponding isotropic constants. Lower: Lifetime curves for the system.
field to the extent that there are no four doublets in total. Accordingly, there are also four pairs of lifetimes. They show a quite dramatic behaviour as a function of field. As in fig. 2, the states 1+ - - ) and I- - + ) undergo an avoided crossing at about 500 G due to an isotropic coupling term, obeying the selection rule AM= 0. A second avoided crossing is observed between the states 1t t - ) and I- - - ) at about 600 G, obeying the selection rule Id!!1 =2 which, as seen in section 2, has to be ascribed to the anisotropic interaction. In our simulations we kept the isotropic e-p coupling constant at 400 MHz. All variations are therefore ascribed to anisotropy, and to the additional nucleus involved. In summary, the conclusions are the following. Whereas in high fields the nuclei are decoupled and the lifetimes are always the same, the
9 August 1991
influence of additional nuclei is noticeable in low fields even for relatively small couplings, in particular in the presence of anisotropy. For large couplings the average lifetime is greatly reduced in zero field and changes little as the field increases. However, resonances due to avoided level crossings may appear, with their positions depending on the size of the couplings. Similar effects are routinely observed in spectra of muon substituted organic radicals [ 131. Experimentally, it will be difficult to pick up the distribution of lifetimes of so many different states especially to fix in the fitting procedure the contributions of long-lived states, since they are field dependent in the presence of anisotropy. The present work does not invalidate the analysis of previous experimental work in this area. Rather, we wanted to point out possible pitfalls. Indeed, depending on the size of the coupling constants the consequences of further nuclei and anisotropy can be serious. It is difficult to estimate the isotropic nuclear couplings involved, since they are highly influenced by spin polarization effects. So far, there has been no experimental determination of these terms, and theoretical work in this direction would therefore be highly desirable. With the greatly improved ab initio techniques available today [141 this should not be impossible, although the inclusion of a positron represents of course a particular problem. Anisotropies are of dipolar origin and easier to estimate since they depend mostly on the magnitude of the magnetic moments and on their distance. The contribution between electron and positron will thus be by far the largest and should be similar to electron-electron anisotropies in triplet molecules of comparable size. For triplet naphthalene these amount to D= 3000 MHz and E = 4 11 MHz [ 15 ] and are thus even larger than assumed in fig. 3. It is not impossible that anisotropy effects contributed to the discrepancies noted for the r] value of Ps in the observations with naphthalene [6] since it is easy to get preferential alignment of the platelets in polycrystalline samples.
Acknowledgement We thank O.E. Mogensen for his encouragement 449
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to look into this problem and for providing
documentation
us with
of his work.
References [I ] M. Deulsch, Progress in nuclear physics, Vol. 3, ed. O.R. Frlsch (Pergamon Press, Oxford, 1953). [2]A. Rich,Rev. Mod. Phys. 53 (1981) 127. (3lA.P. Mills. J. Chem. Phys. 62 (1975) 2646. (411. Billard, J.Ch. .4bbC and G. Duplltre, Chem. Phys. 127 (1988) 273. [ 5] A. Bisi, G. Consolati and L. Zappa, Hyperline Interaction 36 (1987) 29. [6] W. G6rniak and T. Goworek, Chem. Phys Letters I77 (1991) 23. [ 71 S. Rochanakij and D.M. Schrader, Radiation Phys. Chem. 32 (1988) 557.
450
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[R] 1. Billard, J.Ch. Abbe and CT.DuplPtre, J. Chem. Phys. 91 (1989) 1579. [9] O.E. Mogensen, Chem. Phys. Letters 163 (1989) 145. [IO] D.M. Schrader, in: Advances in chemistry series, Vol. 175. Positronium and muonium chemistry, ed. H.J. ,4che (American Chemical Society, Washmgton, 1979). [ 1I] A. Carrington and A.D. McLachlan, Introduction LO magnetic resonance (Harper, New York, 1967 ). [ 121 I.J. Billard, Influence du Champ MagncQue sur la PhysicoChlmie du Posltromum dans la Matlere Condens&, Thtse, CNRS, Strasbourg ( 1989). [ 131 E. Roduner, Lecture notes in chemistry, Vol. 49. The positive muon as a probe in free radxal chemistry. Potential and limitations ofthe pSR techniques (Springer, Berlin, 1988). [ 141 D.M. Chipman, Theorct. Chim. Acta, in press. [ 151J.E. Wertz and J.R. Bolton, Electron spm resonance (McGraw-Hill, New York, 1972).
CHEMICALPHYSICSLETTERS
9 August I99 1
The influence of anisotropy and nuclear couplings on the magnetic quenching of positronium complexes Martina Schwager and Emil Roduner Physikalisch-Chemisches
Insiitut der Universitiit
Ziirrch,
Wintrrthurerstrasc
190, CH-8057
Zurich, Swltzedand
Received 6 April I991
The magnetic field dependence of the lifetimes and energystates of positronium complexes is calculated for a three-spin-l/2 system includingan electron, a positron, and a nucleus. It is demonstrated that couplingof a magnetic nucleus and anisotropy of the system has a pronounced effect on the magneticquenchingbehaviour, in particularnear avoided crossingsof magneticenergy levels.
1. Introduction Positronium (Ps), the bound state of an electron (e) and a positron (p), is a two-spin- l/2 system. In zero magnetic field its eigenstates consist of a singlet and a set of three degenerate triplet states, split in energy by the hypertine interaction. The singlet (also called para, p-Ps, because of its antiparallel spins) and the triplet (or ortha, o-Ps) annihilate with different rates. In vacuum, the corresponding lifetimes amount to ‘I~=0.125 ns for the singlet and ?T= 140 ns for the triplet states [ 1,2]. Application of a magnetic field leaves the two triplet states characterized by the magnetic quantum numbers IV= It 1 unaffected. The singlet mixes with the remaining triplet (M= 0) state. This in principle changes the lifetimes of both states, but since the contribution for the fast decay rate given by the singlet character dominates, it means essentially that one long-lived triplet is converted to a short-lived singlet state. A lifetime plot as a function of the applied field is called a magnetic quenching curve. In a medium the magnetic quenching behaviour is slightly modified, revealing that Ps is perturbed by the interaction with its environment [ 31. For inert solvents the effect is described satisfactorily by a positron-electron hyperfine interaction which is still of spherical symmetry but reduced by a factor 9 compared with its vacuum value. This reduction amounts 0009-2614/91/$
03.50
0
1991 -
to a factor of 0.84 in benzene and as much as 0.65 in water [4]. A far more pronounced change on the quenching curve was found with solid naphthalene [ 5,6] and in particular with a solution of nitrobenzene in IZhexane [ 7,8]. This anomalous magnetic quenching effect was explained in terms of a modified o-Ps lifetime and a reversible addition of Ps to nitrobenzene, thereby forming a Ps-molecule complex [ 8,9 1.Most characteristic of the complex is a hyperfine interaction of only a few percent of its vacuum value, indicating a strong expansion of Ps by delocalization onto nitrobenzene. A calculation of the e-p distribution in the complex shows that approximately 99% of Ps is localized on the nitro group [lo]. The two particles are not equally distributed over the three atoms, however. The population on nitrogen is about 50% for the electron and 69% for the positron. We expect that the magnetic nature of nitrogen could perturb the Ps system considerably. In the analysis of the above experiments, the complex was regarded as a two-spin- I /2 system of spherical symmetry akin to Ps. A detailed discussion of chemical aspects has been given by Mogensen [ 91. Alternatively, one could view the system as the analog of a “complex” of a hydrogen atom with an unsaturated molecule, i.e. an organic free radical. In such a case, the hyperfine interaction of the H atom during addition also decreases to typically a few per-
Elsevier Science Publishers B.V. (North-Holland)
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CHEMICAL PHYSICS LETTERS
cent of its vacuum value. This view lead us to investigate the effects of two possible complications which could occur in such complexes in general. The first one is the presence of further magnetic nuclei which are likely to be coupled magnetically to positron and electron. The multi-spin system is expected to show a different magnetic quenching behaviour which depends an a/l the couplings involved. The second one is the non-spherical nature of such complexes which leads to anisotropy of the magnetic interaction. In a non-viscous environment rapid tumbling will average this dipolar contribution to zero within the Ps lifetime, but in naphthalene or other solids this should certainly not be the case. The anisotropic Hamiltonian has additional terms which complicate the mixing of the zero-field wavefunctions and influence the lifetimes as the field is increased.
2. Theory The Hamiltonian for a Ps complex is a sum of electronic and magnetic terms fi=&l$fiSPln
.
(1)
In the absence of spin-orbit coupling the solution for the wavefunction is a product V’ Ysp’“. V’ may be written in a basis of products of atomic orbitals centered at the nuclei and a hydrogen-like Is orbital for the positron. This is a very limited basis and not necessarily adequate to solve the electronic problem. However, it has the advantage that it makes the bridge to the conventional picture of a Ps complex and allows us to solve the magnetic part of the problem in the same way as is done in magnetic resonance where the solution for PC’”enters parametrically in the form of coupling constants which are proportional to the contact density at the nucleus, 1P(0) 12. In order to investigate the effect of magnetic nuclei it is sufficient to consider a system of three particles (electron, positron, and one nucleus) which are all coupled anisotropically. Its spin Hamiltonian is given by
(2)
9 August 199 I
where SC,Sp and II are the spin operators of the electron, positron and the nucleus k, and v,= vp and V~ the corresponding Larmor frequencies in the absence of g-anisotropy. &,, $, and Apk are the hyperline tensors in frequency units for the three interacting pairs. Each tensor is the sum of an isotropic Fermi contact term A,, and a dipolar spin-spin interaction D,,. For simplicity, and since it is sufficient for our purpose, we consider the case of an anisotropy of axial symmetry, as obtained for example for dipolar interaction of a nucleus with an electron in a p-type orbital [ 111. The principal values of the diagonalized tensor are called the anisotropic coupling constants. A,, corresponds to qAE/h in the nomenclature convention of Ps chemistry, where Ai?/ h is the zero field splitting of Ps in vacuum (203 GHz) and ‘1is the reduction factor of the electron contact density at the positron for Ps in solution or in a complex. The eigenvectors IV,, (dropping the superscript “spin”) are written in a basis of Zeeman product functions x, v/n= 7 c,, IX: > IXY> Ix: > = c c,,,x, . ,
(3)
For the case of isotropic coupling, the interaction of a pair has the structure S~~=AS~=AS=~=+ ^^ ). The off-diagonal terms mix basis 1‘Asj ( + _ t S-1, functions of equal M only. In the anisotropic case, the dipolar contribution leads to terms of the same type, but in addition there are terms of the structure --iDL sin 0~0s B (gf+ts+L) and - aDL sin20S+f+ (for the complete Hamiltonian we refer to ref. [ I1 ] ), where 0 is the azimuthal angle between the symmetry axis and the applied field. They lead to mixing of basis functions with 1AMI = 1 and lAMl =2, respectively. For anisotropic Ps this has the consequence that none of the states retains its long lifetime in an applied field. For evaluation of the lifetimes, the eigenvectors are transformed to a new set of basis vectors which are the tensor products of a zero field positronium eigenstate t, (representing a Ps singlet or triplet state) with a nuclear Zeeman vector xf In the new basis @,= ( I[) 11”) ), the Zeeman product functions are given by XI= c 4!,( 18 IXk>),= c 4!,@, I I
(4)
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CHEMICAL PHYSICS LETTERS
and the eigenvectors by
From the normalization condition ( v/, 1yn) = cp;, = 1 it follows that (~2, is the contribution of the jth basisvector to the eigenstate lu,. The decay rate i, of the nth eigenstate consists additively of the decay rates 2, of the contributing basis vectors @J,weighted with their coefficient ct!;, [ 31. Each basis vector decays with the rate that corresponds to its Ps part 5 when any influence of the nucleus to the singlet or triplet decay rate is neglected. When the singlet and triplet states are expressed in a Zeeman basis,
the matrix of coefficients, Q= ( qu), is given by the transposition of R= (r,,) Q=R’.
(7)
This yields the magnetic-field-dependent lifetimes rk of the eigenstates by evaluation of the secular determinant as a function of the magnetic field
(8)
3. Discussion of specific cases Here, the features of the three-spin-l /2 system consisting of an electron, positron and a proton as the magnetic nucleus are examined numerically with regard to the Ps lifetimes and energy eigenstates. First, the influence of adding a proton to an isotropic Ps complex is studied under both Isotropic and anisotropic conditions. For the proton hyperfine coupling constants we chose values which are typical for organic radicals. Then the influence ofthe proton was strongly increased to show some principal effects of the interacting nucleus. Since these are different in the isotropic and the anisotropic system, the two cases are discussed separately. Simulations were carried out using the parameters ~~~0.125 ns for the singlet lifetime and T,= 3.9 ns, which is about the Ps triplet lifetime in n-hexane [ 7 1.
9August 1991
This choice is quite arbitrary, but the results can be simply scaled to any other effective values of rs and rr. The isotropic electron-positron coupling constant has in all cases about 2.5% of its vacuum value, which is comparable with an observed decrease to about 1% in nitrobenzene solutions [ 121. To facilitate discussion, the lifetimes and energies are marked by their eigenfunctions in the high field limit (v,, v,>A,,,) where they are pure Zeeman functions. These are denoted by the spin quantum numbers m,, mp,mk of the three particles. They take the values + I /2, - l/2, which is abbreviated in the figures by +, -. As indicated above, there is a general behaviour of the lifetimes in high fields, where the spins are decoupled and the eigenstates of Ps or Ps complexes are pure Zeeman states. Here, the states form two groups. One group is characterized by I f t, m,) and I- -, mk), and corresponding lifetimes approach r7. The other group belongs to the states I+-,mk)and I-+,m,),wherethePspartisa linear combination of the Ps singlet (M=O) and triplet (M=O) state with equal weights. Their lifetimes become 5,=2(~~‘+z~‘)-‘s27,. The interesting behaviour therefore occurs at relatively low fields. The upper frame of fig. 1 shows the field dependence of the lifetimes for an isotropic three-spinl/2 e-p-k system with A,,=4000 MHz and Aek=Apk= 100 MHz (solid lines) and compares it with the case where the nucleus is uncoupled or absent (broken lines where they are not degenerate with the first system). The latter represents a Ps complex as normally understood in the literature. The additional spin doubles the number of states, but since Aek=Apk, this leaves the lifetimes in nearly degenerate pairs. Inspection of fig. 1 shows that introduction of the nucleus leaves the two long-lived states (i.e. the ones with maximum IM( = I I;m, 1) unchanged and independent of field, and it doubles the shortest-lived state. One pair splits significantly off the longest-lived state whereas the lifetime of another pair increases slightly from its values in the ep system. The average lifetime, which is usually the experimental observable. decreases in the region around a few hundred gauss. The lower frame of fig. 1 demonstrates the influence of a typical axial anisotropy of D, set to 10% of corresponding isotropic 447
9 August I99
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Volume 182, number 5
,,,,,
,
E [MHz] m,,,,
I
I+-+>
*
(it->/--i/:
3.0: I++->
O-
2.0:
Ittt>.l--->
1
:
-5ooo1.0.
-10000~ -15000-
l-++>,I--+->
0.0
0
100
200 300 H [Gauss]
400
500
Fig. I. Upper: Simulated lifetime curves for an isotropic e-p-k system withA,= MHz and&,&,,= 100 MHz (full lines), and an e-p system with/l,,=4000 MHz (broken lines, where not degenerate with the first system). The corresponding states are marked by the spin quantum numbers Im,, mp, ml,) and lnz,, mp) in high fields. Lower: Lifetime curves for an uniaxial e-p-k system with the same isotropic couplings as above, and amsotropies D, of IO%of corresponding isotropic values, with 0=45”,
values. Clearly, the lifetimes of two pairs of states decrease, and there are now no long-lived states with lifetimes independent of the field. Fig. 2 displays the energy levels and lifetimes of an isotropic system with Aek= 1500 MHz and ADk= - 2000 MHz, i.e. of the same order of magnitude as A,,=4000 MHz. The increased couplings allow us to comment on some features which are not resolved in fig. I. The first is the fact that in zero field the eight eigenstates consist of a quartet and two doublets. .4ccordingly, there are three different lifetimes, the average being less than in the absence of nuclei coupled 448
0
1000
2000 3000 H [Gauss]
4000
5000
Fig. 2. Upper: Energy levels for an isotropic e-p-k system with A,,=4000 MHz, Aek= 1500 MHz, and A,,= -2000 MHz. Lower: Corresponding lifetime curves.
to the e-p system. Secondly, the two states 1t - - > and I -- t ) undergo an avoided crossing due to A,, near 500 G. In this region, the mixing of states is a strong function of the magnetic field, and as a consequence of this the lifetimes exhibit pronounced resonances. It should be noted that there are also true crossings between the states I - - - >, I - - t ), and 1t t - ), which however have no effect on the lifetimes. A simulation of an anisotropic system is given in fig. 3. The isotropic values are the same as in fig. 2, and uniaxial anisotropies are set to 10% of their isotropic values. In the upper frame, the two states I - t t ) and I- t - ) are omitted since they are too distant and behave a lot like in fig. 2. Obviously, anisotropy has lifted the quartet degeneracy in zero
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E[
1
o.oJ 0
1
I-+t>.l-+-> 1
1
200
I
600 400 H [Gauss]
600
4
1000
Fig. 3. Upper: Energy levels for an uniaxial e-p-k system with isotropic couplings A,,=4000 MHz, &=I500 MHz, &= - 2000 MHz, and anisotropies DI set to 10% of the corresponding isotropic constants. Lower: Lifetime curves for the system.
field to the extent that there are no four doublets in total. Accordingly, there are also four pairs of lifetimes. They show a quite dramatic behaviour as a function of field. As in fig. 2, the states 1+ - - ) and I- - + ) undergo an avoided crossing at about 500 G due to an isotropic coupling term, obeying the selection rule AM= 0. A second avoided crossing is observed between the states 1t t - ) and I- - - ) at about 600 G, obeying the selection rule Id!!1 =2 which, as seen in section 2, has to be ascribed to the anisotropic interaction. In our simulations we kept the isotropic e-p coupling constant at 400 MHz. All variations are therefore ascribed to anisotropy, and to the additional nucleus involved. In summary, the conclusions are the following. Whereas in high fields the nuclei are decoupled and the lifetimes are always the same, the
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influence of additional nuclei is noticeable in low fields even for relatively small couplings, in particular in the presence of anisotropy. For large couplings the average lifetime is greatly reduced in zero field and changes little as the field increases. However, resonances due to avoided level crossings may appear, with their positions depending on the size of the couplings. Similar effects are routinely observed in spectra of muon substituted organic radicals [ 131. Experimentally, it will be difficult to pick up the distribution of lifetimes of so many different states especially to fix in the fitting procedure the contributions of long-lived states, since they are field dependent in the presence of anisotropy. The present work does not invalidate the analysis of previous experimental work in this area. Rather, we wanted to point out possible pitfalls. Indeed, depending on the size of the coupling constants the consequences of further nuclei and anisotropy can be serious. It is difficult to estimate the isotropic nuclear couplings involved, since they are highly influenced by spin polarization effects. So far, there has been no experimental determination of these terms, and theoretical work in this direction would therefore be highly desirable. With the greatly improved ab initio techniques available today [141 this should not be impossible, although the inclusion of a positron represents of course a particular problem. Anisotropies are of dipolar origin and easier to estimate since they depend mostly on the magnitude of the magnetic moments and on their distance. The contribution between electron and positron will thus be by far the largest and should be similar to electron-electron anisotropies in triplet molecules of comparable size. For triplet naphthalene these amount to D= 3000 MHz and E = 4 11 MHz [ 15 ] and are thus even larger than assumed in fig. 3. It is not impossible that anisotropy effects contributed to the discrepancies noted for the r] value of Ps in the observations with naphthalene [6] since it is easy to get preferential alignment of the platelets in polycrystalline samples.
Acknowledgement We thank O.E. Mogensen for his encouragement 449
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to look into this problem and for providing
documentation
us with
of his work.
References [I ] M. Deulsch, Progress in nuclear physics, Vol. 3, ed. O.R. Frlsch (Pergamon Press, Oxford, 1953). [2]A. Rich,Rev. Mod. Phys. 53 (1981) 127. (3lA.P. Mills. J. Chem. Phys. 62 (1975) 2646. (411. Billard, J.Ch. .4bbC and G. Duplltre, Chem. Phys. 127 (1988) 273. [ 5] A. Bisi, G. Consolati and L. Zappa, Hyperline Interaction 36 (1987) 29. [6] W. G6rniak and T. Goworek, Chem. Phys Letters I77 (1991) 23. [ 71 S. Rochanakij and D.M. Schrader, Radiation Phys. Chem. 32 (1988) 557.
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[R] 1. Billard, J.Ch. Abbe and CT.DuplPtre, J. Chem. Phys. 91 (1989) 1579. [9] O.E. Mogensen, Chem. Phys. Letters 163 (1989) 145. [IO] D.M. Schrader, in: Advances in chemistry series, Vol. 175. Positronium and muonium chemistry, ed. H.J. ,4che (American Chemical Society, Washmgton, 1979). [ 1I] A. Carrington and A.D. McLachlan, Introduction LO magnetic resonance (Harper, New York, 1967 ). [ 121 I.J. Billard, Influence du Champ MagncQue sur la PhysicoChlmie du Posltromum dans la Matlere Condens&, Thtse, CNRS, Strasbourg ( 1989). [ 131 E. Roduner, Lecture notes in chemistry, Vol. 49. The positive muon as a probe in free radxal chemistry. Potential and limitations ofthe pSR techniques (Springer, Berlin, 1988). [ 141 D.M. Chipman, Theorct. Chim. Acta, in press. [ 151J.E. Wertz and J.R. Bolton, Electron spm resonance (McGraw-Hill, New York, 1972).