RELATIVISTIC CONTRIBUTIONS TO THE QUADRUPOLE INTERACTION IN ATOMIC NITROGEN

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RELATIVISTIC CONTRIBUTIONS TO THE QUADRUPOLE INTERACTION IN ATOMIC

NITROGEN

P. Sandars

To cite this version:

P. Sandars. RELATIVISTIC CONTRIBUTIONS TO THE QUADRUPOLE INTERACTION IN ATOMIC NITROGEN. Journal de Physique Colloques, 1970, 31 (C4), pp.C4-225-C4-227.

�10.1051/jphyscol:1970436�. �jpa-00213891�

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J O U R N A L DE PHYSIQUE Colloque C4, supplément au n° 11-12, Tome 31, Nov.-Déc. 1970, page C4-225

RELATIVISTIC CONTRIBUTIONS TO THE QUADRUPOLE INTERACTION IN ATOMIC NITROGEN

P. G. H . S A N D A R S

The Clarendon Laboratory, Oxford University

Résumé. — L'interaction quadrupolaire dans l'état fondamental de l'azote atomique a été mesurée récemment. Sa faible valeur, b = 1,32 Hz, a suscité une recherche théorique des différents effets relativistes qui contribuent à b. On trouve qu'un traitement simpliste des effets relativistes à l'intérieur de la couche 2 p aboutit à un résultat qui est plus grand que la valeur expérimentale par un ordre de grandeur. On est par conséquent en présence, soit d'une compensation remar- quable entre différents termes, soit d'une divergence sérieuse entre théorie et expérience.

Abstract. — The quadrupole interaction in the ground state of atomic nitrogen has recently been measured. Its small size, b = 1.32 Hz, has stimulated a theoretical investigation of the various relativistic effects which can contribute to the b. We find that a simple minded treatment of rela- tivistic effects within the 2 p shell yields a result which is an order of magnitude larger than the experimental value. There is, therefore, either a remarkable cancellation between terms or a serious discrepancy exists between theory and experiment.

1. Introduction. — C r a m p t o n et al. [1] have recently reported a measurement of the q u a d r u p o l e interaction in t h e ground state of atomic nitrogen. In this com- m u n i c a t i o n we r e p o r t a calculation which yields a result in complete disagreement with the experiment.

Several years ago, Holloway, Luscher and Novick [2]

investigated t h e hyperfine structure of the N ground state using an optical p u m p i n g technique. They found for the quadrupole coupling constant for N1 4

Very recently, C r a m p t o n et al. [1], using a very sen- sitive method based on the hydrogen maser, found the following value for t h e interaction constant :

T h e interest in this result comes from its small size. It must be compared to e2 Q < r~3 >/h which is the n o r m a l order of magnitude of the quadrupole constant. F o r nitrogen with < r~3 > = 3.1 a0 [2]

and O = + 0.015 barns [3] we find :

The obvious explanation for the factor of 107 between the experimental measurement and t h e order of magnitude quoted above is that the ground state is essentially a pure S state which has no quadrupole interaction on account of its spherical symmetry.

We show in this paper t h a t this argument is n o t correct ; there are interaction mechanisms which lead to contributions to the quadrupole constant greatly in excess of the experimental value. Either a remarkable cancellation must occur or there is a serious discrepancy between theory and experiment.

2. Second order dipole. — In their paper Holloway and al. [2] pointed out that an appreciable contribution to the quadrupole constant comes from the magnetic dipole hyperfine structure acting in second order within the 2 p3 shell. This pseudo-quadrupole interaction comes from an energy term which in an obvious notation takes the form

Holloway and al. found that this leads to a contribution to the quadrupole constant :

The numerical result has been verified by Bessis and Ambry [4].

3. Breakdown of LS coupling within the 2 p3 shell.

It is straight-forward to show t h a t t h e d o m i n a n t contributions from the breakdown of LS coupling within t h e 2 p3 shell are

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l.V

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970436

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C4-226 P. G. H. SANDARS

and

where H ,,,. ,,, H,,. and HQ are respectively the spin- orbit, the spin-spin and the quadrupole Hamiltonians.

The angular integrals in (3.1) and (3.2) can be worked out in the conventional manner and one finds

and

where

<

and 17 are the usual spin-orbit and spin-spin parameters. The expressions (3.3) and (3.4) differ in sign from tlie equivalent results obtained by Hollo- way and al. [2] because these authors incorporated an incorrect phase in one of their matrix elements.

Using the values

5

⁼70 cm-', ^q⁼-71 cm-',

(WAS - W2,,) = 28,840 cm-' and Q =

+

.015 barns, we obtain :

b2 =

+

^{.3 H z ,} b, ⁼

+

2.9 HZ.

These results are similar to the calculated value of Bessis and Ambry [4] who found

4. Relativistic Radial Integrals. ^-It is well known from other work [5, 61 that the existence of the diffe- rent radial integrals

i n the Dirac theory leads to important contribution to the quadrupole interaction in half-filled sliell atoms.

Perhaps the simplest way of treating this effect is to use the equivalent operator technique of Sandars and Beck [7].

I t is straight-forward to show that the dominant contribution to the quadrupole constant is given by :

H;" is the effective quadrupole operator defined in reference (7). The angular integral can be readily evaluated to yield

where

It is important to note that this result is entirely equivalent to a ^JJ coupling treatment and equation (4.3) may be obtained in a straight-forward way by using the well-known transformations between LS and jj coupling for the 2 p3 sliell.

We have calculated the value of the radial parameter by solving the Dirac equation in the following poten- tials :

Thomas-Fermi, Hartree-Fock-Slater, non-relativistic Hartree and relativistic Hartree. The results agree closely and may be expressed in the form

We have also calculated A from the radial functions which are solutions of a relativistic Hartree-Fock calculation of Mayers [8] and obtain

Using the latter value of A we find

The total contribution in tlie central field approxi- mation, in which we restrict our attention to the 2 p sliell. is therefore

5. Discussion. ^-The order of magnitude discrepancy between the experimental result and our simple minded central field calculation has made as tliorough theoretical investigation essential. This is now under way and the results will be reported elsewhere. We can, however, comment on one rather obvious effect which we have neglected, namely configuration interaction by the magnetic spin-spin operator. This is likely to be important because in LS coupling it is a tensor of rank 2 in orbital space and hence can contribute to the quadrupole interaction directly througli terms of the form :

<

2 p3 4 S 3 / 2

I

^ff,.,.

1

2 p2 n' 1' 4 ~ 3 / z

>

x

<

2 ^P2II' 1' 4 ~ 3 1 2

I

H Q 12 p3 4 ~ 3 1 2

>

hb, = 8

C-

22'1' - W,,'I' JD)

(5.1) We have made approximate calculations of the contribution from (5.1) using the differential equation methods of Sternheimer [9] to carry out tlie sums over single particle states. We find 6,

-

³^Hz ^{which is}

inadequate to resolve the discrepancy.

Acknowledgements. - The author is indebted to Dr. D. F. Mayers for the use of his programmes and for his unpublished data on nitrogen, and to Mr. M . Barret for his skill arid patience in programming.

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RELATlVISTlC CONTRIBUTIONS TO THE QU.4DIIUPOLE INTERACTION

References

[I] CRAMPTON (S. B.), BERG (H. C.), ROBINSON (H. G.) and [5] EVANS (L.), SANDARS (P. G. H.), WOODGATE (G. K.), RAMEY (N. F.), P11)'s. Rev. Letters, 1970, 24, 195. Proc. Roy. Soc., 1965, A 229, 108.

121 HoLLoWAy (W. W.), L U S C H ~ R (Jr. E.) andNOvlcK (R.), [6] EVANS (L.), SANDARS (P. G . He), WOODGATE ( G . K.), Phys. Rev., 1962, 126, 2109. Proc. Roy. Soc., 1965, A 229, 114.

[31 OYKoNsK1 1968, 49, 5354. (C. T.) and HA (T. K.), J . Chem. P h ~ s . Rev., [71

sAm-

(P. G . H.) and B~~~ (J.), hC. R ~sot., ~ . [4] BESSIS (N.) and AMBRY (C.), La Structure Hyperfine 1965, A 229, 97.

MagnCtique des atomes et des molCcules, Editions [81 MAYERS (D. F.1, Rivate communication, 1970.

du C. N - R. S., Paris 1966, p. 95. [9] STERNHEIMER (R. M.), Phys. Rev., 1954, %, 951.