The hyperfine quadrupole moment of muonium in the ground state

1.B : I.IB.3

1Vuclear Physics A305 (1978) 411-417 ; © North-flollamd Pr~blti/tlmy

Co., Amsterdam

Not to be reproduced by photoprlnt or mkxo9lm without welttm p~~ $om the publisher

THE HYPERFINE QUADRUPOLE MOMENT OF MUONIUM IN THE GROUND STATE DOUG BEDER

Departnurrrt ojPhysies, University ojBritish

Cohonbia,

Vancouver, BC, Canada

Received 13 February 1978 (Revised l0 April 1978) Ahetract : The quadrupole moment of the ground state of the p + e - atom is calculated to be (3m,m~ - ' ; effects in a crystal are briefly discussed.

1. Calcalatioo of gaadrapole moment Due to the interaction of the magnetic dipole moments of the electron and muon (i.e. the hyperfine interaction); the p-e atom in the triplet-spin ground state is not purely S-wave, but contains a small D-wave admixture. (This is amusingly similar to the physics of the deuterium nucleus.) Consequently, this ground state acquires a quadrupole moment, albeit a rather small one. The present work is intended to ascertain this moment in order that one might judge whether the resultant level splitting in the electric field gradients of available crystal lattices is feasibly observable . This was motivated by experimental work in muon spin resonance being conducted t) at the TRIUMF facility in Vancouver. One might first be tempted to employ Rayleigh-Schrödinger perturbation theory to evaluate the hyperfine effect . Indeed, one can straightforwardly obtain analytic expressions for the effect of the 1 = 2 discrete hydrogenic states which are admixed into the ground state - this is relegated to the appendix as an exercise for the reader . Unfortunately, this "bound state" contribution to the quadrupole moment is only a small fraction of the correct answer, as was only evident in hindsight; this point was not realized in a recent note 4) in the literature. We therefore turn to a simple approach analogous to one commonly used in discussing the Stark effect t). We write the wave function for the triplet-spin ground state as ~ar = ~a°r + ~~,

where Js = M, ~~°r is the 1 = 0 "unperturbedH ground state, and ~,~ results from the hyperfine interaction. Inserting this in the Schrôdinger equation, we then have, to t

An equivalent method which hes been verified to give the same result here is discussed in ref.').

412

D . BEDER

lowest order inthe hyperfine interaction, (Ho - Eo)~~ _

(dE-H~o~~~"

Here the unperturbed ground state wave function is exp (-r/ao)XM~ ao = ~m lan~~a - (~ô)} The unperturbed Coulomb Hamiltonian and ground state energy are denoted by Ho and Eo respectively, and the hyperfine interaction is given by ~M

C3(s~ Pxs P)-sa sw -4ns~ " sub3(r)J rs . mume . s e _ m~,m~ V~Cr-3Y10(~)~~p~~N

Hi~~ _ -

+(dS: ~ 0) terms+(1= 0 b-function).

Eq. (5) is obtained by explicit decomposition of the spin~ependent term; here ~lar =

-1,~

~+ Z,

M=0 M = f 1.

In order to proceed further we observe that we need only evaluate the M = 0 component of the quadrupole tensor - that this suffices follows from the WignerEckart theorem. We define our quadrupole moment operator : ~o = ~Z YZO(~) = rz(3 cost B-1).

From our above description of the wave function we see that Now, although ¢~ contains both l = 0 and 1= 2 parts, we are guaranteed from eqs. (~ and (8) that we only need to know the part of ~M proportional to YZO(Sl)XM. Consequently, we isolate contributions to eq . (2) with this angular dependence, and thus only need the first term in our decomposition of Ham` as given in eq. (5). Now, if . we write ~er(1 = 2. is = 0) = Xér YZO(~) ~P ( - r/ao)R(r)~ then the radial part of the wave function satisfies [from eq . (2)]

HYPERFINE QUADRUPOLE MOMENT

41 3

Thus our problem reduces finally to the solution of a simple differential equation for R (just as with the Stark effect calculation, except that we needed to exert care in dealing with the angle and spin dependence). The result is simple : 4e ea~

1 ~

3

. 0/

The quadrupole moment is then trivially evaluated using eqs. (8), (9) and (11) : At this point we should note : (a) The quadrupole moment is concentrated at moderate (but not small) radii ; we shall assess this below. (b) No trace of the Coulomb interaction is seen in the result (12). This could have been anticipated had dimensional analysis of the result been undertaken! (c) The M-dependence nM which we obtained by explicitly decomposing Hi"` is indeed consistent with the Wigner-Eckart theorem, as it must be. (d) Contrary to what one expects with uncoupled partial waves, here the D-wave amplitude is not proportional to rz. By examining eq. (10) (which determines ~) for small r, one sees that ~ would be proportional to rz only if H'"` were so proportional . (e) The hyperfine-induced quadrupole moment is very small. Here ~z. (13) Qo = -2.4 x 10-za In order to appreciate what energy level shifts should result at a crystal lattice site, we present a crude estimate assuming an electric field E = E(z)~= : ro

where e is the electron charge and ro = 1 A. In this configuration the quadrupole interaction energy is âE: dEQ = ~e~¢oi

ôz

(15)

= 0.28 x 106(Hz) x qx. A more worrisome feature is that the quadrupole moment arises from radii which are significant compared to typical lattice spacings, so that expressing the interaction energy in terms of local field gradients may be a poor approximation. Let us consider the quadrupole moment density weighted by rz ; pQ(r) oc r This density is maximum at

Cl + ~~ exp (-2r/ao). 0

r = 6ao(1 +~) = 0.77ao.

16) (17)

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D . BEDER

However, if we consider the distribution of pQ(r~ we ought to examine ~(r) _ ~drpQ(r)' ~~ drpQ(r) 0

0

We find ~(ao) = 1-0.57,

~(2a o) = 1-0.18.

In other words, about half the quadrupole moment arises for r > ao, and fully 6 arises for r > 2ao; hence our comment above. 2. F~ergy eigemalues with electric wnd magcetk ßdds preseat We consider the case of perpendicular electric and magnetic fields E = E(z~~ +BB~.

(19)

The interaction Hamiltonian is now easily written down ; we omit the interaction of the muon dipole moment with B, as it is much smaller than the electron dipole interaction. We consider the low field limit, retaining terms through O(B 2~ and hence must include the singlet state in our diagonalization of the following Hamiltonian : Hm~ = Mare.=i +~ Q~ . B+E(P,_1-3P,ao).

(20)

Here P is the (triplet, singlet) projection operator

E = hyperfine energy = ~ez(memNaô)-1 . The eigenvalues for "tripletn states are just the usual, with an additive quadrupole eßoct (these expressions are approximations valid for B not too small) : (21)

As estimated in eq. (15~ we feel that a realistic expectation is that a < 1 MHz In this case the M = 0,1 crossover of levels occurs when the Zeeman splitting (eB/?m,)

HYPERFINE QUADRUPOLE MOMENT

41 5

is less than 1 MHz, well below the typical field strength for muon precession experiments (after all, the muon only lives 2.2 peek). The usual M = 0 .-. M = ± 1 precession frequency is now split into two: z (22) ml 2me t C~a+ (2m)'

4E).

Again, for typical experiments, the quadratic (in B) term is negligible compared to ~a. For B parallel to E, diagonalization is trivial (for our simple E): (23)

Now we obtain two (M = 0 "-" M = f 1) precession frequencies z

1 4E

(24) ~ 2me ~ ~~- ~~~~ /~ The quadrupole splitting is twice as big now as for the perpendicular field configuration. Should this be observed, it would constitute sufficient evidence that one was seeing a quadrupole eBect to justify further theoretical work (e.g. for general E, B configurations) ; this, of course, is a problem which has been thoroughly addressed by nuclear-magnetio-resonance experts. In the present case one has two serious complications to consider, however: (a) As alluded to above, the large (radial) extension of the quadrupole moment makes it dubious that the quadrupole interaction energy can be expressed simply in terms of local field gradients. (b) One ought to consider whether lattice eûects could also induce a quadrupole moment. In appendix B we briefly indicate how to treat more general field configurations. Appendix A RAYLEIGH-SCHRODINGER PERTUR$ATION RESULT FOR DISCRETE STATES

Straightforward application of perturbation theory leads to (Qier = ~ 2~1~i°r~~~n : l = 2i E 1 .

.

Eo

~ n~

1 = 2i~°`i,~~i~

One now (a) rewrites Hi°' as in eq. (5); (b) trivially performs angular integrations ; (c) expresseQ radial bound state Coulomb wave functions in terms ofconfluent hypergeometricfunctions (1F1); and (d) performs radial integrals analytically [using results

41 6

D. BEDER

from the Bateman transform tables 3)] in terms of ZFl hypergeometric functions. The result is for discrete hydrogenic states 1 128e2 (n+2)!n 2 ZFl ~Q)rr = ~° (E E 6' 25mem~ "~ "- °) -1(n-3)!(n+1)9 3~ ~' n+1) C

(A.2) The first confluent Z F 1 can be reduced s) to

-s

CCl n/ ~Cl+ n/~ via a Kummer transformation; the second Z Fl is easily calculated numerically. A little arithmetic exertion then demonstrates that the contribution from discrete hydrogenic states is only about 12 ~ of our correct answer . That this is the case should be suspected when one discovers that in the closure approximating the quadrupole moment is several times larger than our result, and twenty times larger thanthe discrete state contribution ! After this work was completed we observed a recent paper ~) presenting the discrete state result of this appendix as the full result ! Appendix B In a general field {E,} we can express the quadrupole interaction energy as follows : If the electrostatic potential ~ is expanded in multipoles then

A --- ~ r~Y,~(Ll)

16x

21+1 Eu",

where the ooe$'cients of interest in the quadrupole case are given by ôx1i

-

~ a~x1 rs Y2~(~)~Es~ _

- ~

a~xl ~a~s~~

Thus if cartegian fie~d gradients are giveq the Es, can be identified. We then have the matrix elements of A between triplet states ofmagnetic quantum numbers M', M:
(B.3)

where we have used the Wigner-Eckart theorem, and C is a Clebsch-Gordan coefl'dent. An interesting special case occurs for the simple E field of our examples, when B, E are inclined at an angle 8. For this case we can show that the frequency splitting is proportioflal') to 3 cost 8-1 ; our special cases illustrate this result .

HYPERFINE QUADRUPOLE MOMENT

41 7

I thank M. H. L. Pryce for a comment which led me to the present solution method, and J. Brewer for preventing total indolence during a sunny summer; I also thank J. Berlinsky for useful conversations.

1) 2) 3) 4) S)

References

J. Brewer, UBC, private communication E. Merzbacher, Quantum mechanics (Whey, NY) pp. 383-4 Tables of integral transforma, Bateman Manuscript Project, vol 2 (McGraw-Hill, NY, 1954) p, 219 V. G. Baryshevslcy and S. A. Kuten, Phys . Lett . 64A (1977) 238 M. H. L. Pryce, private communication