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The coupling of three angular momenta in the optical NMR and ESR of atoms: quantum theory
HVSlCl
Physica B 179 (1992) 342-348 North-Holland
The coupling of three angular momenta in the optical NMR and ESR of atoms" quantum theory M.W. E v a n s 1"2 Institute of Physical Chemistry, Universi O, of ZiJrich, Winterthurerstrafie 190, CH-8057 Zt~rich, Switzerland Received 2 October 1991 Final revised 6 March 1992
A rigorous quantum theory of optical NMR in atoms is developed in the coupling scheme J = L + S: F = J + l, where L, S and I are the orbital, spin electronic and nuclear spin quantum numbers, respectively. The optical NMR Hamiltonian is developed for a coupled state [LSJ1FMv) in terms of products of 9 - j symbols, which are evaluated numerically. The optical NMR resonance condition is described in terms of the selection rule AM~ = 0, +-1 on the azimuthal quantum number of F, the net angular momentum. The theory shows clearly that the original NMR resonance line is shifted and split into several lines by the applied circularly polarised laser of optical NMR,
(A)
1. Introduction Optical, or (laser) N M R spectroscopy is the name given to a new spectral technique in which a circularly polarised laser is used in the sample tube of a conventional N M R spectrometer to induce a new resonance fingerprint of the sample under investigation. A non-linear optical property of the laser, called the conjugate product (HIA)), interacts with an atomic or molecular electronic property called the vectorial or antisymmetric polarisability (a'i=) to induce a magj netic dipole mome n t (rni(ind) ). In tensor notation, with the usual Einstein convention of summation over repeated indices, the interaction energy is AEn
=
. ,, n(A)
- - l O l ( ] 1 1 ij
and is a real scalar quantity because l l i i is a purely imaginary quantity defined through the antisymmetric tensor product /(A) =
~(EiE,
_ E;E,*)
(2)
of the electric field strength E i (in volts per metre) of the laser. Here E j* for example denotes the complex conjugate of Ei. The theory of tensors [1] shows that both ~ j and H ~ I can be expressed as axial vectors, positive to parity inversion (P) and negative to time reversal (T) vt
¢!
o~ i = e q k a j k
//IA~
,
,~,~A~ .
Eii k 1 1 jk
(3)
(4)
(1)
Correspondence to: Dr, M.W. Evans, Cornell University, Engineering and Theory Center Building, Hoy Road, Ithaca, NY 14853-3801, USA. Permanent address: 433 Theory Center, Cornell University, Ithaca, NY 14853, USA. 2 Senior Visiting Research Associate. Materials Research Laboratory, Pennsylvania State University, University Park, PA 16802, USA.
Here s~i k is the rank three totally antisymmetric unit tensor, known as the Levi-Civita s y m b o l The vector representations c~'I a n d / / I A) both have the same P and T symmetries as the magnetic dipole moment, rnl ~"d), which in turn has the same negative T and positive P symmetries of the static magnetic flux density BI °) (in tesla). The interaction energy A E n can therefore be expressed as
0921-4526/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
M.W. Evans / The coupling of three angular momenta in the optical NMR and ESR of atoms
A E n = - i a ' : l l l A) .
(5)
The interaction energy of optical NMR spectroscopy is obtained [2] by adding this to the simplest kind of NMR interaction energy to give AE/~ (ONMR, = __ mlN, BI0) _ ice'~//I A)
with azimuthal components summing up arithmetically:
The selection rules AF=0,±I,
(7)
=
where J/ is the total net electronic angular momentum (orbital and spin) and 3', a T and P positive scalar quantity called the gyroptic ratio. Equation (6) can therefore be written as AEn (°NMn) = --YNIiB¢/,) --i?i, J i I I l A) ' Ji : Lg + 2.002S~,
(8)
where YN is the nuclear gyromagnetic ratio and I i the nuclear spin angular momentum. A simple vector model was then developed in ref. [2] using geometrical methods, to show that the laser in general shifts and splits the original NMR resonance line under consideration. This simple semi-classical approach as put on a rigorous basis in ref. [3] using 6 - j symbols, which were worked out algebraically to give the same expression for the interaction energy as derived in ref. [2]: A E n ( O N M R ) = _MFh(gLI_TIlliZrT(A) +gL27NOZ_ o(0)~}.
(9) This result is valid for one- or two-electron atoms in the coupling scheme (10)
in the coupled state IJ 1 F M F ) , so that the total angular momentum quantum number F takes the values F=J+I
....
,IJ-II,
(11)
AMt== 0, ±1 ,
(13)
govern the optical NMR resonance condition. It is quite clear from eq. (9) that there are several possible resonance conditions, depending on the allowed values of F, J and I. It is also clear that the original resonance frequency, which depends only on I, is shifted and split by the laser in a way that depends on the nature of the atom or molecule under consideration. These predictions have been verified qualitatively in an experiment [4] on a medium sized chiral molecule, in which atomic theory can be useful as a guide to what might be expected. In this paper we develop the atomic theory to the final stage, in which the optical NMR spectrum is worked out in the most general case, involving the three angular momenta L, S and I. Here L is the electronic orbital angular momentum, 2.002S is the electronic spin angular momentum, and G N I is the nuclear spin angular momentum, where gN is the nuclear g number [5]. We use the coupling scheme of eq. (14) to work out the optical NMR interaction energy between coupled states of the type J=L+S,
F = J + I
(12)
M F : Mj + M t .
(6)
where rnl •) is the nuclear magnetic dipole moment. In ref. [2], this interaction energy was developed using o~'~ ",/i,J,,
343
F=J+I,
(14)
] L S J I F M F ) , to give the optical NMR resonance condition as a function of the three individual quantum numbers L, S and I. This theory is rigorously valid for one- or two-electron atoms and in certain closed shell atoms [5]. For a clear discussion of the validity of the theory, see Silver
[6]. Section 1 introduces the 9 - j formalism [5] required for the problem and derives the most general expression for the interaction energy. Section 2 gives the method of reducing the very complicated 9 - j symbols to simple numbers, which are tabulated for selected values of L, S,
344
M.W. Evans / The coupling o f three angular momenta in the optical N M R and ESR o f atoms
J, 1 and F. Finally a discussion is given of optimised experimental conditions.
2.
The
9-j
ILSJIFMF). To find the expectation value of L in the coupled state we must use an operator which acts only on the coordinate set (1). This is achieved in the general case [6] by setting
formalism
U k: = 1, The problem of evaluating matrix elements of the interaction energy (8) between coupled eigenstates ILSJIFMF) is a special case of the general problem considered by Curl and Kinsey [7] of evaluating matrix elements where there are three kinds of commuting angular momenta. In our case these are L, S and I of an atom. As discussed by Silver [6] any tensor operator of rank k built by coupling of these angular m o m e n t a will have the form
L-
in eq. (16). Here 1 is the unit tensor operator. Similarly ' ,
(18)
I= [{l®l}°®Vkq k .
tl9)
S = [{l®U'e}®l]
From
T kl =
where the assignment of L, S, and I to T, U and V depends on the coupling scheme. In this case we consider the scheme described by eq. (14), i.e. spin rotation interaction with hyperfine coupling to I, Following Silver [6] the coupled angular m o m e n t u m states are written generally as IJlJ2Jl2J3jmj). The reduced matrix elements of X within these states are gwen by eq. (13.8) of Silver [6]: .
(17)
[{Tk' ® l } k ' ® l ] k
eq.
(16)
we
can
therefore
find
( LSJIFI ILI IL'S'J'I'F') using
X k : [{T k, ® Uk2} k,2 ®V%] k ,
k
V k' = 1 ,
L,
k~=k~=0,
J3=j;=l,
U ke -
1,
V k~ =
1 ,
kle=kl=k=l, jl=L,
j,2=J,
Similarly, we can find using
j~=j'z=S, j=F.
(LSJIF I Isl IL'S'J'I'F ')
•
T kl = 1,
U k~ = S ,
k12=k~=kl,
k L= k 3 = 0 ,
(J,J2J12J311XkllJ;J2J;2J;J ')
V k' = ! ,
j:~ = j ; = 1,
Jiz = J,
j = F.
= [(2j + 1)(2j' + 1)(2k + 1) Finally, we can find using
X (2jl 2 + 1)(2j',2 + 1)(2kl, + 1)1 '/-'
J.12 1.1,2 kl2
Jl
Jl
Jl2 ]12
kl k12
T kl
(16)
x (j, llT"lllj'l)(j2llu"211j2,)(j.~liv~',llj;), where the braces denote 9 - j symbols [6-10[. In this paper we reduce these to simple numbers using the F O R T R A N code given by Zare [10]. To implement eq. (16) and to realise why it is necessary, it is important to understand the following. The orbital electronic angular momentum L is a simple operator Tk'(1) of rank k I = 1 acting in space 1 on only one set of variables (Silver [6], chapter 9) of the coupled state
=1,
U k2=l,
k t=k~=kte=0, .t
Jz = ] 2 = S,
(LSJIFIIIIIL'S'J'I'F') Vk'=l,
k~=k=l,
J3 = I,
j=j'=L,
112 =
Jt--J'l-L,
J" j = F.
The interaction energy (8) becomes [11] AEn (°NMR) = _ -
THLilI~ A~ - 2 . 0 0 2 y u S J / l AI gNVNI, BI °~ ,
(20)
where Yu is the gyroptic ratio [12], H I A) the laser's conjugate product, gN the nuclear g number and "?N the nuclear gyromagnetic ratio. This
M.W. Evans / The coupling o f three angular momenta in the optical NMR and ESR of atoms
is evaluated as the matrix elements AEn(ONMR)
~,nn(A)
llz
& =
=
- y n ( L S J I F M F I Li I L ' S ' J ' I ' F ' M F ) I I I
(gl + 2.002g2) + gNYNB zoo)g3 ( F ( F + 1)(2F + 1 ) ) ' / 2
345
,
(26)
where
A~
- 2.002y n
g, = ( L S J I F I ILl I L S J I F )
× ( L S J I F M F I Si I L ' S ' J ' I ' F ' M ' F ) I I I A'
= 3h(2F + 1)(2J + 1)(L(L + 1)(2L + 1)) l/z × (2S + 1)'/2(21+ 1) '/2
-- gNTN
× ( L S J I F M F I Ii I L ' S ' J ' I ' F ' M ' F ) B I °) .
(21) x
I F
Applying the W i g n e r - E c k a r t theorem [6-10], A E n ( O N M R ) = _ ( _ I ) F Mr --MF
0
S J
S J
,
& = (LSJIFIISlILSJIF)
M~
= 3h(2F + 1)(2J + 1)(S(S + 1)(2S + 1)) '/2
× ( y n ( L S J I F I ILl[ I L ' S ' J ' I ' F ' )
× (2L + 1)'/2(21+ 1) l/2
+ 2.002y n ( L S J I F [ ] S i l I L ' S ' J ' I ' F ' ) x
+ gNyN(LSJIF[ I/i[ [ L ' S ' J ' I ' F ' ) ) . (22) At this point we narrow consideration to the diagonal elements. The 3 - j symbol becomes [6-10]
F 1 -M F 0
AEn (°NMR) = _ M v g L ,
S J
S J
,
g3 = ( LSJIFI IIII L S J I F ) = V-3h(ZV + 1)(2J + 1)(1(1 + 1)(21 + 1)) 1/2 x (2L + 1)'/2(2S + 1) '/2
ho~ = [ - ( M F - l) -
x
(23)
(24)
where gc is a function to be determined of the quantum numbers L, S and I. The resonance condition of optical NMR is given in this three angular momentum theory by
IgLI
I F
F)
(--1)F-MFMF M r = (F(F + I ) ( Z J + I)) '/2 '
and the interaction energy reduces to
¢.o-
I F
(--MF)II&], (25)
h
The positive part, or modulus, of gL is used in this expression because gL can be either positive or negative, depending on the state of the atom and on the configuration of//}A) relative to B~°). Defining these latter quantities in the Z axis,
I F
I F
S J
S J
.
Clearly, gL needs special consideration in the case F = 0, given later. For F ~ 0 it is equally clear that the optical NMR resonance frequency depends on the values allocated to L, S and I, and in general there are many allowed permutations and combinations of these. We arrive at the important conclusion that the circularly polarised laser used in the O N M R technique in atoms can split, as well as shift, the original N M R resonance due to changes in M t alone. The original N M R resonance is recovered properly from eq. (25) as follows. By considering H~ A)= 0 (no laser), L = 0, S = 0 (no consideration of electronic states coupled to IIMI) ) we have g,
--g2 =0
,
M.W. Evans / The coupling o f three angular momenta in the optical N M R and ESR o f atoms'
346
g3
=
2V~h
2
0 0
0 0
.
(27)
U s i n g Z a r e ' s code [10] for 9 - j following section),
i 0yl
=0.288675,
0 0
I
0 0
symbols (see I F 0 0
B z(°)
I F
(28)
i
I F
1 F
1 = 0.00000, i F.
(30)
(29)
'
for all allowed J and I. T h e case F = 0 can occur, for example, for I = J. T h e r e f o r e , there is no O N M R resonance possible in the case F = 0. r%0) For an arbitrary gN'YNDZ = 1 (in units of 10 s Hz) the resonance f r e q u e n c y from eq. (25) is p l o t t e d against the dimensionless
the usual N M R r e s o n a n c e expression [5]. 2.1.
= /.o
= 1.000000,
which implies that g3 = h to within seven decimal places. This gives ¢o=gN'YN
angular m o m e n t u m in the coupled state and f r o m Z a r e ' s code [10],
"Yl I I ( A ]
x -
The case F = O
In the particular case F = 0 there is no net
I
g ~ YN
Z
B(0)
(31)
z
in figs, 1-3 for different L , with S = I = ~.
o~,c~
OPT'~AL NMR OF ATOMS, I • S - 0.5, L • 2, J - 2.5
~
oF ,,To,,s., • s . o~. t.= ~ j .
~5.
INDUCEDF'BEQUENCYSFIFTS,F " t AND2.
LASI~ I~DUCED FREQUB~CY SHIFTS, F = 2 AND 3. FREQUF-NCY.6 .
0.6 O,5
0.,5
.""
0.4-
0.4
, "'
0.3-
0.2
,/""
°)1
0.1 /
Od) , 0.0
,
,
,
,
,
,
, • , " ,
(11
0.2
0.3
0.4
0.5
0.8
0.7
O,B
0.9
,
0.0
1,0
0,1
0,2
0.3
Fig. j=
1 • Plot ~-
5,--:F=2;
of resonance ---:F=3.
frequency
0,4
0.5
0,6
0.7
021
0,9
tO
X
x
vs. sec for 1= S =
2,
Fig. j=
) 3.
Plot
~,--:F=I;
of resonance ---: F=
frequency 2.
vs. s e c f o r 1 = S =
M,W. Evans / The coupling o f three angular momenta in the optical NMR and ESR of atoms OPTICAL NMR OF ATOMS, I • S • n 5, L • 3, J = 3..5. LASER ~
FREQLII~CY ,~,'FTS, F • 3, 4
~[6.
0.5
O.4
O,3
0.2
tl 1
• o d ~ ' o h o h ~ , - ' s s~'~, d~'o~ ~0 x
Fig. 3. Plot of resonance frequency vs. sec for I = S = ~, J= ~;--:F=3;---: F = 4.
3. Results and discussion
Figures 1 to 3 show that the laser induced frequency shift from eq. (25) is a linear function of the ratio x, and therefore of the intensity of the laser (in watts per unit area). This is also the result obtained from our original semi-classical theory [2] and the rigorous quantum theory [3] in the simpler coupling scheme F = J + I with J regarded as one quantum number with no separate consideration given to its components L and S. In this paper we give full consideration to the role of L, S and I, and show that the circularly polarized laser can split resonance lines as determined by the structure of the 9 - j symbols. The latter depend on all three quantum numbers, leading to the possibility of a rich optical N M R or optical ESR spectrum. This work is more rigorous than previous models of the O N M R and O E S R spectra, and introduces standard numerical evaluation of the 9 - j symbols to
347
minimise algebraic complexity and maximise the physical content of the treatment. The discontinuity in some of these figures simply reflects the fact that we are using the modulus [gL[ in eq. (25) to calculate the resonance frequency, and the latter cannot be negative, since such a result is physically meaningless. For example, in fig. 1, for I = S = ½ , L = 2 , J = ~, F = 3 , gL is always positive, and a simple straight line is generated with a value at x = 0 which gives the NMR (magnet only) resonance frequency in this coup l e d state. Note carefully that this x = 0 value is quite different from its equivalent for the uncoup l e d nuclear state I = ½, i.e. when the electronic state of the atom is considered to be uncoupled from the nuclear state. In fig. 2 for 1 = S -- ½, L = 2, J = 3, F = 2, gL is negative after x - 0 . 1 , and IgL[ therefore shows a discontinuity at this point. This means that the sign of the interaction energy terms due to the laser is opposite to that due to the magnet, which leads at x - 0 . 1 to a change in sign of the interaction energy. The latter is, however, always a difference in energy, and the resonance frequency in O N M R depends only on the magnitude of the energy difference induced by the combined influence of circularly polarised laser and magnet. The straight line dependence in these figures is as expected from earlier work [2, 3, 11] and is another check on the correctness of our evaluation of the complicated 9 - j symbols based on Zare's code [10]. It is important to note that from figs. 1-3 for a given laser intensity there are several possible O N M R resonance frequencies, because for a given L, S and I there are several possible J and F from the C l e b s c h - G o r d a n series [6-10]. Physically, this means that a circularly polarised laser is capable of splitting as well as shifting the original N M R line due to A M I = +1 ,
(32)
a result that if corroborated in atoms leads to the expectation that O N M R for molecules will, with care, develop into a precise new fingerprint technique of widespread utility, simply because each original N M R feature will be split and shifted in
348
M.W. Evans / The coupling o f three angular momenta in the optical N M R and ESR o f atoms
a characteristic way. ONMR has the additional advantage [12] that the gyroptic dhl is considerably amplified near optical resonances, when photon absorption from a circularly polarised laser generates a magnetic dipole moment as in laser polarisation techniques [13-15]. When a right or left photon is absorbed, however, there is generally a change in J and F as well as M F, and off-diagonal elements of (21) have to be used. The theory in this paper has been restricted to diagonal elements and is valid when J and F do not change, i.e. is valid only for changes in M F. To extend the theory of this paper to paramagnetic molecules requires 3 - j, 6 - j and 9 - j algebra in the double point groups of paramagnetic molecules in which there is a finite electronic angular momentum. In this case it is known that the 3 - j symbols for atoms go over into Harnung's extension [16] of the Griffith V coefficients [17], but nothing is known, apparently, about the way in which the 6 - j and 9 - j symbols must be developed. To extend this atomic theory to the point groups of diamagnetic molecules requires consideration of a magnetic electronic hyperpolarisability, which we refer to elsewhere [18] as the/3 tensor theory. In diamagnetic molecules there is no antisymmetric polarisability [19, 20], but there is a non-vanishing /3, as first discussed by Manakov et al. [21]. In the /3 theory of diamagnetic molecules the 3 - j symbols are replaced by the Griffith V coefficients, but again there appears to be no knowledge of how to develop the 6 - j and 9 - j symbols for the point groups of diamagnetic molecules. It would clearly be useful to develop a F O R T R A N code for such a problem, and we hope that this will be the subject of future work.
Acknowledgements The Swiss N.S.F. is acknowledged for funding this work, and Dr. L.J. Evans for invaluable help with the SAS plotting routine of the Irchel mainframe. ETH Zurich is acknowledged for a major grant of IBM 3090-6S computer time, and the Cornell Theory Center for a CNSF grant of
3090 computer time and other support. The Cotnell Theory Center is recipient of major funding from the US NSF, New York State, and IBM Inc. (USA). Last but not least, many interesting discussions are acknowledged with Professor Warren S. Warren of Princeton University, in whose group the first ONMR experiment is underway, with Professor Jack L. Freed of Cornell and Professor A Lakhtakia of Penn State.
References [1] L.D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University Press. Cambridge, 1982). [2] M.W. Evans, J. Phys. Chem. 95 (1991) 2256. [3] M.W. Evans, J. Mol. Spectrosc.. in press. [4] D. Goswami, C. Hillegas, Q. He, H. Tull and W.S. Warren, Laser NMR Spectroscopy, Proc. Exp. NMR Conf., St. Louis, Missouri, 8-12 April 1991 (contribution from Frick Chem. Lab., Princeton Univ.), [5] P.W. Atkins, Molecular Quantum Mechanics (Oxford University Press, Oxford, 1983, 2nd Ed.). [6] B.L. Silver, Irreducible Tensor Methods (Academic Press, New York, 1976). [7] R.F. Curl, Jr. and J.L. Kinscy, J. Chem. Phys. 35 (1961) 1758. [8] A.R. Edmunds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, N J, 1960). [9] L.C. Biedenham and J.D. Louck, Angular Momentum in Quantum Physics, Theory and Application (Cambridge University Press, Cambridge, 1989). [1(I] R. Zare, Angular Momentum (Wiley, New York, 1988). [11] M.W. Evans, Int. J. Modern Phys. B 5 (1991) 1263, [12] M.W. Evans, Int. J. Modern Phys. B 5 (1991) 1961. [13] C. Lhuillier and F. Lalo~, J. de Phys. 43 (1982) 197, 225. [14] N.R. Bigelow, J.H. Freed and D.M. Lee, Phys. Rev. Lett. 63 (1989) 1609. [15] J.L. Freed and F, LaloE, Sci. Am. 256 (1988) 94. [16] S.E. Harnung, Mol. Phys. 26 (1973) 473. [17] J.S. Griffith, The Irreducible Tensor Method for Molecular Symmetry Groups (Prentice-Hall, Englewood Cliffs, 1962). [18] M.W. Evans, Chem. Phys. 157 (1991) 1. [19] J.P. van der Ziel, P.S. Pershan and L.D. Malmotran, Phys. Rev. Lett. 15 (1965) 190. [20] J.P. van der Ziel, P.S. Pershan and L.D. Malmotran, Phys. Rev. 143 (1966) 574. [21] N.L. Manakov, V.D, Ovsiannikov and S. Kielich, Acta Phys. Polon. A 53 (1978) 581, 595.
Physica B 179 (1992) 342-348 North-Holland
The coupling of three angular momenta in the optical NMR and ESR of atoms" quantum theory M.W. E v a n s 1"2 Institute of Physical Chemistry, Universi O, of ZiJrich, Winterthurerstrafie 190, CH-8057 Zt~rich, Switzerland Received 2 October 1991 Final revised 6 March 1992
A rigorous quantum theory of optical NMR in atoms is developed in the coupling scheme J = L + S: F = J + l, where L, S and I are the orbital, spin electronic and nuclear spin quantum numbers, respectively. The optical NMR Hamiltonian is developed for a coupled state [LSJ1FMv) in terms of products of 9 - j symbols, which are evaluated numerically. The optical NMR resonance condition is described in terms of the selection rule AM~ = 0, +-1 on the azimuthal quantum number of F, the net angular momentum. The theory shows clearly that the original NMR resonance line is shifted and split into several lines by the applied circularly polarised laser of optical NMR,
(A)
1. Introduction Optical, or (laser) N M R spectroscopy is the name given to a new spectral technique in which a circularly polarised laser is used in the sample tube of a conventional N M R spectrometer to induce a new resonance fingerprint of the sample under investigation. A non-linear optical property of the laser, called the conjugate product (HIA)), interacts with an atomic or molecular electronic property called the vectorial or antisymmetric polarisability (a'i=) to induce a magj netic dipole mome n t (rni(ind) ). In tensor notation, with the usual Einstein convention of summation over repeated indices, the interaction energy is AEn
=
. ,, n(A)
- - l O l ( ] 1 1 ij
and is a real scalar quantity because l l i i is a purely imaginary quantity defined through the antisymmetric tensor product /(A) =
~(EiE,
_ E;E,*)
(2)
of the electric field strength E i (in volts per metre) of the laser. Here E j* for example denotes the complex conjugate of Ei. The theory of tensors [1] shows that both ~ j and H ~ I can be expressed as axial vectors, positive to parity inversion (P) and negative to time reversal (T) vt
¢!
o~ i = e q k a j k
//IA~
,
,~,~A~ .
Eii k 1 1 jk
(3)
(4)
(1)
Correspondence to: Dr, M.W. Evans, Cornell University, Engineering and Theory Center Building, Hoy Road, Ithaca, NY 14853-3801, USA. Permanent address: 433 Theory Center, Cornell University, Ithaca, NY 14853, USA. 2 Senior Visiting Research Associate. Materials Research Laboratory, Pennsylvania State University, University Park, PA 16802, USA.
Here s~i k is the rank three totally antisymmetric unit tensor, known as the Levi-Civita s y m b o l The vector representations c~'I a n d / / I A) both have the same P and T symmetries as the magnetic dipole moment, rnl ~"d), which in turn has the same negative T and positive P symmetries of the static magnetic flux density BI °) (in tesla). The interaction energy A E n can therefore be expressed as
0921-4526/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
M.W. Evans / The coupling of three angular momenta in the optical NMR and ESR of atoms
A E n = - i a ' : l l l A) .
(5)
The interaction energy of optical NMR spectroscopy is obtained [2] by adding this to the simplest kind of NMR interaction energy to give AE/~ (ONMR, = __ mlN, BI0) _ ice'~//I A)
with azimuthal components summing up arithmetically:
The selection rules AF=0,±I,
(7)
=
where J/ is the total net electronic angular momentum (orbital and spin) and 3', a T and P positive scalar quantity called the gyroptic ratio. Equation (6) can therefore be written as AEn (°NMn) = --YNIiB¢/,) --i?i, J i I I l A) ' Ji : Lg + 2.002S~,
(8)
where YN is the nuclear gyromagnetic ratio and I i the nuclear spin angular momentum. A simple vector model was then developed in ref. [2] using geometrical methods, to show that the laser in general shifts and splits the original NMR resonance line under consideration. This simple semi-classical approach as put on a rigorous basis in ref. [3] using 6 - j symbols, which were worked out algebraically to give the same expression for the interaction energy as derived in ref. [2]: A E n ( O N M R ) = _MFh(gLI_TIlliZrT(A) +gL27NOZ_ o(0)~}.
(9) This result is valid for one- or two-electron atoms in the coupling scheme (10)
in the coupled state IJ 1 F M F ) , so that the total angular momentum quantum number F takes the values F=J+I
....
,IJ-II,
(11)
AMt== 0, ±1 ,
(13)
govern the optical NMR resonance condition. It is quite clear from eq. (9) that there are several possible resonance conditions, depending on the allowed values of F, J and I. It is also clear that the original resonance frequency, which depends only on I, is shifted and split by the laser in a way that depends on the nature of the atom or molecule under consideration. These predictions have been verified qualitatively in an experiment [4] on a medium sized chiral molecule, in which atomic theory can be useful as a guide to what might be expected. In this paper we develop the atomic theory to the final stage, in which the optical NMR spectrum is worked out in the most general case, involving the three angular momenta L, S and I. Here L is the electronic orbital angular momentum, 2.002S is the electronic spin angular momentum, and G N I is the nuclear spin angular momentum, where gN is the nuclear g number [5]. We use the coupling scheme of eq. (14) to work out the optical NMR interaction energy between coupled states of the type J=L+S,
F = J + I
(12)
M F : Mj + M t .
(6)
where rnl •) is the nuclear magnetic dipole moment. In ref. [2], this interaction energy was developed using o~'~ ",/i,J,,
343
F=J+I,
(14)
] L S J I F M F ) , to give the optical NMR resonance condition as a function of the three individual quantum numbers L, S and I. This theory is rigorously valid for one- or two-electron atoms and in certain closed shell atoms [5]. For a clear discussion of the validity of the theory, see Silver
[6]. Section 1 introduces the 9 - j formalism [5] required for the problem and derives the most general expression for the interaction energy. Section 2 gives the method of reducing the very complicated 9 - j symbols to simple numbers, which are tabulated for selected values of L, S,
344
M.W. Evans / The coupling o f three angular momenta in the optical N M R and ESR o f atoms
J, 1 and F. Finally a discussion is given of optimised experimental conditions.
2.
The
9-j
ILSJIFMF). To find the expectation value of L in the coupled state we must use an operator which acts only on the coordinate set (1). This is achieved in the general case [6] by setting
formalism
U k: = 1, The problem of evaluating matrix elements of the interaction energy (8) between coupled eigenstates ILSJIFMF) is a special case of the general problem considered by Curl and Kinsey [7] of evaluating matrix elements where there are three kinds of commuting angular momenta. In our case these are L, S and I of an atom. As discussed by Silver [6] any tensor operator of rank k built by coupling of these angular m o m e n t a will have the form
L-
in eq. (16). Here 1 is the unit tensor operator. Similarly ' ,
(18)
I= [{l®l}°®Vkq k .
tl9)
S = [{l®U'e}®l]
From
T kl =
where the assignment of L, S, and I to T, U and V depends on the coupling scheme. In this case we consider the scheme described by eq. (14), i.e. spin rotation interaction with hyperfine coupling to I, Following Silver [6] the coupled angular m o m e n t u m states are written generally as IJlJ2Jl2J3jmj). The reduced matrix elements of X within these states are gwen by eq. (13.8) of Silver [6]: .
(17)
[{Tk' ® l } k ' ® l ] k
eq.
(16)
we
can
therefore
find
( LSJIFI ILI IL'S'J'I'F') using
X k : [{T k, ® Uk2} k,2 ®V%] k ,
k
V k' = 1 ,
L,
k~=k~=0,
J3=j;=l,
U ke -
1,
V k~ =
1 ,
kle=kl=k=l, jl=L,
j,2=J,
Similarly, we can find using
j~=j'z=S, j=F.
(LSJIF I Isl IL'S'J'I'F ')
•
T kl = 1,
U k~ = S ,
k12=k~=kl,
k L= k 3 = 0 ,
(J,J2J12J311XkllJ;J2J;2J;J ')
V k' = ! ,
j:~ = j ; = 1,
Jiz = J,
j = F.
= [(2j + 1)(2j' + 1)(2k + 1) Finally, we can find using
X (2jl 2 + 1)(2j',2 + 1)(2kl, + 1)1 '/-'
J.12 1.1,2 kl2
Jl
Jl
Jl2 ]12
kl k12
T kl
(16)
x (j, llT"lllj'l)(j2llu"211j2,)(j.~liv~',llj;), where the braces denote 9 - j symbols [6-10[. In this paper we reduce these to simple numbers using the F O R T R A N code given by Zare [10]. To implement eq. (16) and to realise why it is necessary, it is important to understand the following. The orbital electronic angular momentum L is a simple operator Tk'(1) of rank k I = 1 acting in space 1 on only one set of variables (Silver [6], chapter 9) of the coupled state
=1,
U k2=l,
k t=k~=kte=0, .t
Jz = ] 2 = S,
(LSJIFIIIIIL'S'J'I'F') Vk'=l,
k~=k=l,
J3 = I,
j=j'=L,
112 =
Jt--J'l-L,
J" j = F.
The interaction energy (8) becomes [11] AEn (°NMR) = _ -
THLilI~ A~ - 2 . 0 0 2 y u S J / l AI gNVNI, BI °~ ,
(20)
where Yu is the gyroptic ratio [12], H I A) the laser's conjugate product, gN the nuclear g number and "?N the nuclear gyromagnetic ratio. This
M.W. Evans / The coupling o f three angular momenta in the optical NMR and ESR of atoms
is evaluated as the matrix elements AEn(ONMR)
~,nn(A)
llz
& =
=
- y n ( L S J I F M F I Li I L ' S ' J ' I ' F ' M F ) I I I
(gl + 2.002g2) + gNYNB zoo)g3 ( F ( F + 1)(2F + 1 ) ) ' / 2
345
,
(26)
where
A~
- 2.002y n
g, = ( L S J I F I ILl I L S J I F )
× ( L S J I F M F I Si I L ' S ' J ' I ' F ' M ' F ) I I I A'
= 3h(2F + 1)(2J + 1)(L(L + 1)(2L + 1)) l/z × (2S + 1)'/2(21+ 1) '/2
-- gNTN
× ( L S J I F M F I Ii I L ' S ' J ' I ' F ' M ' F ) B I °) .
(21) x
I F
Applying the W i g n e r - E c k a r t theorem [6-10], A E n ( O N M R ) = _ ( _ I ) F Mr --MF
0
S J
S J
,
& = (LSJIFIISlILSJIF)
M~
= 3h(2F + 1)(2J + 1)(S(S + 1)(2S + 1)) '/2
× ( y n ( L S J I F I ILl[ I L ' S ' J ' I ' F ' )
× (2L + 1)'/2(21+ 1) l/2
+ 2.002y n ( L S J I F [ ] S i l I L ' S ' J ' I ' F ' ) x
+ gNyN(LSJIF[ I/i[ [ L ' S ' J ' I ' F ' ) ) . (22) At this point we narrow consideration to the diagonal elements. The 3 - j symbol becomes [6-10]
F 1 -M F 0
AEn (°NMR) = _ M v g L ,
S J
S J
,
g3 = ( LSJIFI IIII L S J I F ) = V-3h(ZV + 1)(2J + 1)(1(1 + 1)(21 + 1)) 1/2 x (2L + 1)'/2(2S + 1) '/2
ho~ = [ - ( M F - l) -
x
(23)
(24)
where gc is a function to be determined of the quantum numbers L, S and I. The resonance condition of optical NMR is given in this three angular momentum theory by
IgLI
I F
F)
(--1)F-MFMF M r = (F(F + I ) ( Z J + I)) '/2 '
and the interaction energy reduces to
¢.o-
I F
(--MF)II&], (25)
h
The positive part, or modulus, of gL is used in this expression because gL can be either positive or negative, depending on the state of the atom and on the configuration of//}A) relative to B~°). Defining these latter quantities in the Z axis,
I F
I F
S J
S J
.
Clearly, gL needs special consideration in the case F = 0, given later. For F ~ 0 it is equally clear that the optical NMR resonance frequency depends on the values allocated to L, S and I, and in general there are many allowed permutations and combinations of these. We arrive at the important conclusion that the circularly polarised laser used in the O N M R technique in atoms can split, as well as shift, the original N M R resonance due to changes in M t alone. The original N M R resonance is recovered properly from eq. (25) as follows. By considering H~ A)= 0 (no laser), L = 0, S = 0 (no consideration of electronic states coupled to IIMI) ) we have g,
--g2 =0
,
M.W. Evans / The coupling o f three angular momenta in the optical N M R and ESR o f atoms'
346
g3
=
2V~h
2
0 0
0 0
.
(27)
U s i n g Z a r e ' s code [10] for 9 - j following section),
i 0yl
=0.288675,
0 0
I
0 0
symbols (see I F 0 0
B z(°)
I F
(28)
i
I F
1 F
1 = 0.00000, i F.
(30)
(29)
'
for all allowed J and I. T h e case F = 0 can occur, for example, for I = J. T h e r e f o r e , there is no O N M R resonance possible in the case F = 0. r%0) For an arbitrary gN'YNDZ = 1 (in units of 10 s Hz) the resonance f r e q u e n c y from eq. (25) is p l o t t e d against the dimensionless
the usual N M R r e s o n a n c e expression [5]. 2.1.
= /.o
= 1.000000,
which implies that g3 = h to within seven decimal places. This gives ¢o=gN'YN
angular m o m e n t u m in the coupled state and f r o m Z a r e ' s code [10],
"Yl I I ( A ]
x -
The case F = O
In the particular case F = 0 there is no net
I
g ~ YN
Z
B(0)
(31)
z
in figs, 1-3 for different L , with S = I = ~.
o~,c~
OPT'~AL NMR OF ATOMS, I • S - 0.5, L • 2, J - 2.5
~
oF ,,To,,s., • s . o~. t.= ~ j .
~5.
INDUCEDF'BEQUENCYSFIFTS,F " t AND2.
LASI~ I~DUCED FREQUB~CY SHIFTS, F = 2 AND 3. FREQUF-NCY.6 .
0.6 O,5
0.,5
.""
0.4-
0.4
, "'
0.3-
0.2
,/""
°)1
0.1 /
Od) , 0.0
,
,
,
,
,
,
, • , " ,
(11
0.2
0.3
0.4
0.5
0.8
0.7
O,B
0.9
,
0.0
1,0
0,1
0,2
0.3
Fig. j=
1 • Plot ~-
5,--:F=2;
of resonance ---:F=3.
frequency
0,4
0.5
0,6
0.7
021
0,9
tO
X
x
vs. sec for 1= S =
2,
Fig. j=
) 3.
Plot
~,--:F=I;
of resonance ---: F=
frequency 2.
vs. s e c f o r 1 = S =
M,W. Evans / The coupling o f three angular momenta in the optical NMR and ESR of atoms OPTICAL NMR OF ATOMS, I • S • n 5, L • 3, J = 3..5. LASER ~
FREQLII~CY ,~,'FTS, F • 3, 4
~[6.
0.5
O.4
O,3
0.2
tl 1
• o d ~ ' o h o h ~ , - ' s s~'~, d~'o~ ~0 x
Fig. 3. Plot of resonance frequency vs. sec for I = S = ~, J= ~;--:F=3;---: F = 4.
3. Results and discussion
Figures 1 to 3 show that the laser induced frequency shift from eq. (25) is a linear function of the ratio x, and therefore of the intensity of the laser (in watts per unit area). This is also the result obtained from our original semi-classical theory [2] and the rigorous quantum theory [3] in the simpler coupling scheme F = J + I with J regarded as one quantum number with no separate consideration given to its components L and S. In this paper we give full consideration to the role of L, S and I, and show that the circularly polarized laser can split resonance lines as determined by the structure of the 9 - j symbols. The latter depend on all three quantum numbers, leading to the possibility of a rich optical N M R or optical ESR spectrum. This work is more rigorous than previous models of the O N M R and O E S R spectra, and introduces standard numerical evaluation of the 9 - j symbols to
347
minimise algebraic complexity and maximise the physical content of the treatment. The discontinuity in some of these figures simply reflects the fact that we are using the modulus [gL[ in eq. (25) to calculate the resonance frequency, and the latter cannot be negative, since such a result is physically meaningless. For example, in fig. 1, for I = S = ½ , L = 2 , J = ~, F = 3 , gL is always positive, and a simple straight line is generated with a value at x = 0 which gives the NMR (magnet only) resonance frequency in this coup l e d state. Note carefully that this x = 0 value is quite different from its equivalent for the uncoup l e d nuclear state I = ½, i.e. when the electronic state of the atom is considered to be uncoupled from the nuclear state. In fig. 2 for 1 = S -- ½, L = 2, J = 3, F = 2, gL is negative after x - 0 . 1 , and IgL[ therefore shows a discontinuity at this point. This means that the sign of the interaction energy terms due to the laser is opposite to that due to the magnet, which leads at x - 0 . 1 to a change in sign of the interaction energy. The latter is, however, always a difference in energy, and the resonance frequency in O N M R depends only on the magnitude of the energy difference induced by the combined influence of circularly polarised laser and magnet. The straight line dependence in these figures is as expected from earlier work [2, 3, 11] and is another check on the correctness of our evaluation of the complicated 9 - j symbols based on Zare's code [10]. It is important to note that from figs. 1-3 for a given laser intensity there are several possible O N M R resonance frequencies, because for a given L, S and I there are several possible J and F from the C l e b s c h - G o r d a n series [6-10]. Physically, this means that a circularly polarised laser is capable of splitting as well as shifting the original N M R line due to A M I = +1 ,
(32)
a result that if corroborated in atoms leads to the expectation that O N M R for molecules will, with care, develop into a precise new fingerprint technique of widespread utility, simply because each original N M R feature will be split and shifted in
348
M.W. Evans / The coupling o f three angular momenta in the optical N M R and ESR o f atoms
a characteristic way. ONMR has the additional advantage [12] that the gyroptic dhl is considerably amplified near optical resonances, when photon absorption from a circularly polarised laser generates a magnetic dipole moment as in laser polarisation techniques [13-15]. When a right or left photon is absorbed, however, there is generally a change in J and F as well as M F, and off-diagonal elements of (21) have to be used. The theory in this paper has been restricted to diagonal elements and is valid when J and F do not change, i.e. is valid only for changes in M F. To extend the theory of this paper to paramagnetic molecules requires 3 - j, 6 - j and 9 - j algebra in the double point groups of paramagnetic molecules in which there is a finite electronic angular momentum. In this case it is known that the 3 - j symbols for atoms go over into Harnung's extension [16] of the Griffith V coefficients [17], but nothing is known, apparently, about the way in which the 6 - j and 9 - j symbols must be developed. To extend this atomic theory to the point groups of diamagnetic molecules requires consideration of a magnetic electronic hyperpolarisability, which we refer to elsewhere [18] as the/3 tensor theory. In diamagnetic molecules there is no antisymmetric polarisability [19, 20], but there is a non-vanishing /3, as first discussed by Manakov et al. [21]. In the /3 theory of diamagnetic molecules the 3 - j symbols are replaced by the Griffith V coefficients, but again there appears to be no knowledge of how to develop the 6 - j and 9 - j symbols for the point groups of diamagnetic molecules. It would clearly be useful to develop a F O R T R A N code for such a problem, and we hope that this will be the subject of future work.
Acknowledgements The Swiss N.S.F. is acknowledged for funding this work, and Dr. L.J. Evans for invaluable help with the SAS plotting routine of the Irchel mainframe. ETH Zurich is acknowledged for a major grant of IBM 3090-6S computer time, and the Cornell Theory Center for a CNSF grant of
3090 computer time and other support. The Cotnell Theory Center is recipient of major funding from the US NSF, New York State, and IBM Inc. (USA). Last but not least, many interesting discussions are acknowledged with Professor Warren S. Warren of Princeton University, in whose group the first ONMR experiment is underway, with Professor Jack L. Freed of Cornell and Professor A Lakhtakia of Penn State.
References [1] L.D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University Press. Cambridge, 1982). [2] M.W. Evans, J. Phys. Chem. 95 (1991) 2256. [3] M.W. Evans, J. Mol. Spectrosc.. in press. [4] D. Goswami, C. Hillegas, Q. He, H. Tull and W.S. Warren, Laser NMR Spectroscopy, Proc. Exp. NMR Conf., St. Louis, Missouri, 8-12 April 1991 (contribution from Frick Chem. Lab., Princeton Univ.), [5] P.W. Atkins, Molecular Quantum Mechanics (Oxford University Press, Oxford, 1983, 2nd Ed.). [6] B.L. Silver, Irreducible Tensor Methods (Academic Press, New York, 1976). [7] R.F. Curl, Jr. and J.L. Kinscy, J. Chem. Phys. 35 (1961) 1758. [8] A.R. Edmunds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, N J, 1960). [9] L.C. Biedenham and J.D. Louck, Angular Momentum in Quantum Physics, Theory and Application (Cambridge University Press, Cambridge, 1989). [1(I] R. Zare, Angular Momentum (Wiley, New York, 1988). [11] M.W. Evans, Int. J. Modern Phys. B 5 (1991) 1263, [12] M.W. Evans, Int. J. Modern Phys. B 5 (1991) 1961. [13] C. Lhuillier and F. Lalo~, J. de Phys. 43 (1982) 197, 225. [14] N.R. Bigelow, J.H. Freed and D.M. Lee, Phys. Rev. Lett. 63 (1989) 1609. [15] J.L. Freed and F, LaloE, Sci. Am. 256 (1988) 94. [16] S.E. Harnung, Mol. Phys. 26 (1973) 473. [17] J.S. Griffith, The Irreducible Tensor Method for Molecular Symmetry Groups (Prentice-Hall, Englewood Cliffs, 1962). [18] M.W. Evans, Chem. Phys. 157 (1991) 1. [19] J.P. van der Ziel, P.S. Pershan and L.D. Malmotran, Phys. Rev. Lett. 15 (1965) 190. [20] J.P. van der Ziel, P.S. Pershan and L.D. Malmotran, Phys. Rev. 143 (1966) 574. [21] N.L. Manakov, V.D, Ovsiannikov and S. Kielich, Acta Phys. Polon. A 53 (1978) 581, 595.