PHYSICAL REVIEW A VOLUME 40, NUMBER 9 NOVEMBER 1, 1989

Estimates of spin-exchange parameters for alkali-metal-noble-gas pairs

Thad Ci. Walker'Department ofPhysics, Princeton University, Princeton, New Jersey 08544

(Received 7 July 1989)

Estimates of frequency-shift enhancement factors, binary cross sections for spin exchange andspin relaxation, chemical equilibrium constants, and spin-exchange and spin-rotation interactions invan der Waals molecules are presented for alkali-metal atoms interacting with noble-gas atoms.These results are compared to experiments and are also presented for those pairs which have notbeen studied experimentally.

I. INTRODUCTION

In the last few years spin-exchange techniques have be-come popular for transferring angular momentum tonoble-gas nuclei. ' In such experiments optically pumpedalkali-metal atoms polarize the nuclei via binary col-lisions or formation of van der Waals molecules. Thesetechniques have been used for a variety of applications,including measurements of nuclear magnetic moments,searches for permanent electric dipole moments in atomsand nuclei, spin-polarized nuclear targets, and studiesof quadrupolar wall interactions. The physics of spinexchange has been studied intensively in the last fewyears, but for many alkali-metal —noble-gas pairs no ex-perimental information is available. In the following Icompare calculations of the relevant spin-exchange pa-rameters to the experiments done to date, and present es-timates of the relevant spin-exchange parameters forthose species which have not been measured.

The important spin-exchange phenomena which havebeen studied experimentally to date are frequency shifts,binary cross sections for spin relaxation and spin ex-change, and spin relaxation and spin exchange in boundvan der Waals molecules. A11 of these studies measure inone way or another the strength of the magnetic-dipoleinteraction

Hxs =yN-S, (2)

which causes spin relaxation by coupling S to the rota-tional angular momentum N of the alkali-metal-noble-gaspair. Both a and y are functions of the internuclear sep-aration R of the alkali-metal —noble-gas pair.

The magnetic-dipole coupling (I) is dominated by theFermi-contact interaction

(3)

where p~ is the Bohr magneton, pz is the magnetic mo-ment of the noble-gas nucleus, and g(R) is the wave-function of the alkali-metal valence electron evaluated at

H~s =aK-S

which couples the noble-gas nuclear spin K to the alkali-metal electronic spin S, producing spin exchange, or thestrength of the spin-rotation interaction

the position of' the noble-gas nucleus. Herman hasshown that an enhancement of f(R ) occurs, i.e.,

f(R ) = rig(R ), (4)where P(R ) is the alkali-metal valence-electron wavefunction in the absence of the noble gas, and g )) 1 for allthe noble gases. This enhancement comes about from thelarge kinetic energy acquired by the electron as it scattersin the core potential of the noble-gas atom. Walker, Bo-nin, and Happer have calculated g using theorthogonalized-plane-wave (OPW) and phase-shift (PS)techniques.

In the last few years experiments on the spin-rotationinteraction (2) in bound alkali-metal —noble-gas van derWaals molecules have shown convincingly that for theheavy noble gases the dominant contribution to y comesfrom the interaction of the alkali valence electron withthe noble-gas atom. Wu, Walker, and Happer haveshown that y results from the spin-orbit interaction inthe noble-gas core, and found

(R)= mG dig(R)l'(&)

pR dRHere m and p are the electron mass and the reduced massof the alkali-metal —noble-gas pair, and the constant 6 isdependent only on the spin-orbit interaction of the noblegas. Values of G have been calculated in Ref. 7.

In the following the expressions (3) and (5) will be usedto compare to experimental values of the constants g andG as well as to predict the values of spin-exchange pa-rameters which have not yet been measured. In Sec. IIthe potential curves and wave functions used for the cal-culations are discussed. In Sec. III the values of g and 6found by experiments on van der Waals molecules arepresented. Section IV does the same for experimentswhich have measured binary cross sections. Section Vpresents the results of frequency-shift experiments. InSec. VI the estimates for the alkali-metal —noble-gas pairswhich have not been experimentally investigated arepresented. Table V is a compilation of the calculatedvalues of the spin-exchange parameters for the relevantalkali-metal —noble-gas pairs.

II. POTENTIAL CURVESAND WAVE FUNCTIONS

All of the data to be analyzed below depends for its in-terpretation on assumed potential curves and alkali-metal

4959 1989 The American Physical Society

4960 THAD G. WALKER

wave functions. The available information on the poten-tial curves is of varying quality. For the heavy noblegases Ar, Kr, and Xe the scattering experiments of Buckand Pauly have determined potential depths and posi-tions for all the pairs except RbAr and RbXe. Pascaleand Vandeplanque' have used the results of Buck andPauly to constrain their semiempirical calculations of thealkali-metal —noble-gas potentials. For NaAr there alsoexists the direct spectroscopic measurements of Tel-linghuisen et al. " For consistency, I have used the po-tentials of Pascale and Vandeplanque for all calculationsinvolving Ar, Kr, and Xe.

For the light noble gases there is little informationwhich does not require theoretical input; the exception isthe spectroscopic investigation of Lapidovitch et al. ' ofNaNe. For He I have chosen the potentials of Pascale'for the calculations which follow. For NaNe the welldepth calculated by Pascale and Vandeplanque' of 0.25meV is much smaller than the measurement of 1.0 meVby Lapidovich et al. , and the equilibrium separation of6.0 A is significantly greater than the measured 5.29 A.Thus for Ne I have assumed that the trend found byBuck and Pauly for the heavier noble gases —that thewell depth is insensitive to the alkali metal —holds for Neas well. I have therefore taken the Morse potential pa-rameters of Lapidovitch et al. for NaNe to be the samefor the other alkali-metal —Ne molecules, but with thepotential minimum chosen to be somewhat larger thanfor the corresponding alkali-metal —Ar molecule. Theinternuclear separations chosen are NaNe, 5.29 A; KNe,5.64 A; RbNe, 5.68 A; CsNe, 5.80 A.

There are no measurements or calculations of thealkali-metal —Rn potentials so I have chosen the parame-ters for the assumed Morse potential form

(6)

as follows: D=18 meV and a=0.8 A for all thealkali-metal —Rn pairs, and R0=5.1, 5.3, 5.4, and 5.5 .~for NaRn, KRn, RbRn, and CsRn, respectively.

The alkali-metal valence-electron wave functions re-quired for calculating a and y need to be known outsidethe core of the alkali metal for the purposes of this paper.For this purpose the wave functions of choice are thesemiempirical Coulomb wave functions of Bates andDamgaard' which use the experimental binding energyto determine the asymptotic behavior of the wave func-tions. Also, as discussed by Seaton, ' the normalizationof the wave functions assumed by Bates and Damgaardshould be multiplied by a small factor [1+2( —E) ~ dp/de] '~, where —E is the binding energyin rydbergs and p is the quantum defect. The values ofdp/dc. used for this paper are —0.066, —0.156, —0.209,and —0.284 for Na, K, Rb, and Cs, respectively.

III. van der WAALS MOLECULES

In most experiments with heavy noble gases at pres-sures below an atmosphere spin exchange is dominatedby the formation of long-lived van der Waals molecules.The relevant spin-exchange and spin-relaxation phenome-na have been discussed in detail by Happer et al. ,

' the

XerfE,„(N) V~(R )—

kT

Then the chemical equilibrium constant is given by

k,„, =f dNf f(R,N)e ' 4 R dR,0 min

the average value of the spin-rotation interaction is

max max(yN)= f NdN f, ,y(R)f(R, N)

k chem 0 min

—V(R) IkT

~4~R 2dR, (9)

and the average value of the Fermi-contact interaction isR,„[W(a) = f dN f a(R)f (R,N)

chem

x e—V(R)rkT

X4~R2dR . (lo)In the above V~(R) = V(R )+A N /2pR, E,„(N) is themaximum energy of a bound molecule of rotationa1 angu-lar momentum N, R;„(N) and R,„(N) are the turningpoints for the classical motion of the bound molecule inthe potential V~(R), and N, „ is the maximum angularmomentum possible for a bound molecule. Some experi-ments measure slightly different quantities, such as theroot-mean-square values of o, and yA'. By direct calcula-tion I have found that the rms values differ from theabove averages by at most a few percent, so for simplicityI will assume the above method of averaging to interpretall the gas-phase experiments.

Cxas phase measurements of the spin-rotation couplingstrength (yN) have been reported for RbAr, ' RbKr, '

CsKr, ' NaXe, KXe, RbXe, ' and CsXe. Therehave also been molecular-beam measurements of y forKAr (Ref. 21) and RbKr (Ref. 22) which do not measurethe average values, but rather give y(RO). The results ofthese experiments are shown in Table I along with thevalues of G deduced assuming the Pascale and Vande-planque' potential curves.

Measurements of x = ( yN ) /( a ) for KXe, RbXe, andCsXe were made by Xeng et al. By using the measure-

important experimental parameters which are deduced inthese experiments are the average value of the spin-rotation interaction ( yN ) and the average value of theFermi-contact interaction (a). Also obtained are themolecular formation and breakup rates which determinethe chemical equilibrium constant k„.„, =[AX]/[X][ 3],where [ AX] is the density of alkali-metal —noble-gas mol-ecules, [ 3] the alkali-metal number density, and [X] thenoble-gas density.

As shown by Schaefer et al. ,' the fraction of bound

alkali-metal —noble-gas pairs at internuclear separation Rand with angular momentum X is

g2f (R N) Ne fi N /—2PR kT

pkTR 2

ESTIMATES OF SPIN-EXCHANGE PARAMETERS FOR. . . 4961

TABLE I. Measured characteristics of alkali-metal —noble-gas van der %aals molecules compared to theory. OPW and PS refer tothe orthogonalized-plane-wave and phase-shift techniques for calculating ~g~ and G. The measured chemical equilibrium constantsare scaled to 100 C by assuming a T ~ dependence, and compared to the values calculated from the Pascale and Vandeplanque(PV) potentials.

Molecule (a)/h (MHz) Expt. OPW PS (yN)/h (MHz) Expt.6 (eVA )

OPW PSk,„, (A )

Expt. PV

KArRbArRb Kr

CsKrNaXe

XeRb '29xe

Cs ' Xe

2.77+0.27

38+5"38+6"49+8"

54

797881

35

505050

737373

3.35+0.14

26.8+0.8'

172+22g132+12'121+14'141%16'

1.8'3.32

12.1'15.617.096587283

1.91.9

12121239393939

0.960.965.95.95.9

24242424

64~4b

110+9'

256*70"244*44"222' 84"

68

166

310313350

'Molecular-beam measurement of 6, Ref. 21.Reference 17.

'Molecular-beam measurement of G, Ref. 22.Reference 23.

'Reference 18.

Reference 19.gReference 20."Deduced from measurements of Refs. 24 and 8, see text.'Reference 8.

ments of & y N ) for these molecules, & a ) is obtained, andvalues for g deduced. These values are shown in Table I.A recent measurement for RbKr by Schaefer is alsoshown in Table I.

Bouchiat, Brossel, and Pottier have measured the for-mation and breakup rates for RbKr, reported in the form~P and TIP, where ~ is the molecular lifetime, P thenoble-gas pressure, and Tf the three-body formationtime. The chemical equilibrium constant can then be de-duced by the relation k,&, =~PkT/Tf P, and is given inTable I. Likewise Xeng et a/. have measured thecharacteristic pressure po and the molecular formationrate constant Z, which, coupled with the values of & yN )from Wu et al. , give

ZjDok,~, 2~km & y»/h

The values so obtained for KXe, RbXe, and CsXe areshown in Table I.

IV. BINARY COI.LISIONS

Binary collisions between the alkali-metal atoms andthe noble-gas atoms can cause both spin relaxation (fromthe spin-rotation interaction) and spin exchange (via theFermi-contact interaction). The duration of these col-lisions (10 ' sec) is so short that the spins precess bynegligible amounts so that first-order time-dependent per-turbation theory is valid. Thus the transition rate due tothe spin-rotation interaction (2) between the two electron-ic sublevels &

1'~

and & l~of the alkali electron is given by

' f uf(v)d ud blXl

X f" & S~y(R(t))S N~y)dt

where v is the relative velocity of the alkali-metal-noble-gas pair, b the impact parameter of the collision, f (v) theMaxwellian velocity distribution, and we have takeo aclassical trajectory R (t) for the collision. The squaredmatrix element in (11)can be written

TABLE II. Measurements of binary cross sections for spin exchange crsE, and comparison of the de-duced values for g to theory. The measurements have been scaled to 100'C according to a T ' depen-dence.

Molecule

Na HeRb 'HeRb "NeRb Kr

Ref.

25262723

0

osE (A )

(1.26+0.22) X 10(1.68+0.24) X 10-'(5.6+0.2) x 10{1.3+0.3) X 10

Expt.

9.610.11952

OPW

9.59.5

1535

PS

5.35.39.4

44

4962 THAD G. WALKER

TABLE III. Measurements of spin-relaxation cross sections (o.sR) for alkali-metal —noble-gas pairs,and deduced values of G compared to theory.

Molecule

RbArRbKrCsKrNaXeRbXe

Ref.

1818192818

0 2asR (A )

(6.08+0.20) X 10(2.66+0. 12)X 10-'(2.55+0.36) X 10

1.09 X 10-'(1.64+0.9)X 10

Expt.

2.013115235

G (eVA )

opw

1.912123939

ps

0.965.95.9

2424

P2V 2P V

2 y(R)dR2

(1—cos 8) y(R (t))dt = (1—cos 8)4A 'o [u (1 b /R—

) —2V(R)/p]'(12)

where ro is the classical turning point of the trajectoryand 0 is the angle between v and the axis of quantization.

The integral over angles in (11) is now possible, givinga spin-relaxation cross section de6ned via the relation

= 2

[X]u

of

a = we dub db-8m. p

SR

y(R)dR"o [(1 b /R ) ——V(R)lwkT]'

(13)where w =pu l2kT and u=&8kT/vrp, . This is insensi-tive to temperature. A similar result is obtained for thespin-exchange cross section:

crsE= 2 Je dw b db2' pg2 kT

a (R)dR"o [1 b /R —V(R—)lwkT]'

(14)

should be taken into account when analyzingtemperature-dependent data on spin-exchange cross sec-tions. It should also be noted that both (13) and (14) de-pend sensitively on the repulsive wall of the potential,where the factor [1 b /R ——V(R)lwkT]' is smallest.The analogous cross sections for free-electron spin relaxa-tion and spin exchange have been calculated by Walker,Bonin, and Happer.

Table II shows the binary spin-exchange cross sectionmeasurements which have been made to date, and valuesof g deduced from these cross sections. Soboll reportsthe Na relaxation rate for the slowest time constant.Thus his reported cross section is o.sE/(2I+1), whereI=—,

' is the nuclear spin of Na. To account for o.sR Ihave also subtracted his He cross section from his Hecross section. Chupp and Coulter report the cross sec-tion for transitions between magnetic sublevels of the'Ne nucleus, so o.sE is twice this value.

Table III shows some of the measurements of osRwhich have been reported. Here much of the older datahas been incorrectly interpreted due to problems with op-tically thick vapors, neglect of multiple time constants,and neglect of the efT'ects of the alkali-metal nuclearspin. ' Thus only reported results which include theseeft'ects are presented in Table III.

With the above de6nitions, the basic equations for spinexchange via binary collisions are V. FREQUENCY SHIFTS

(s, ) =2uo'sE[X]((s' —s,') &I(., ) —&K' —K,') &s, ))dt

UcrsR[X](si )

and

As discussed by Schaefer et al. ,' under virtually all

practical conditions the relevant parameter for describingthe shifts of alkali-metal EPR frequencies and noble-gas

(IC, ) =2ucrsE[ A]((E K, ) (S,)—(16)

TABLE IV. Measurements of the frequency-shift enhance-ment factor Kp from Ref. 16 and the deduced values of g com-pared to theory.

A point which is sometimes overlooked is the Tdependence of o.sE, which arises because the spin-exchange probability is proportional to the square of theinteraction time and therefore proportional to theinverse-square of the relative velocity. This dependence

Molecule

RbKrRbXe

ICp

270+95643+260

Expt.

4469

OPW

3550

ps

4473

ESTIMATES OF SPIN-EXCHANGE PARAMETERS FOR. . . 4963

(yN)/h (MHz) k,h, (A )Molecule

TABLE V. Estimates of spin-exchange parameters of alkali-metal —noble-gas pairs. All numbers are for 100 C. The values of go 5used are He, —9.5; Ne, 15; Ar, —22; Kr, 44; Xe, —73; Rn, 114. The values of G are, in units of eV A; Ne, 0.24; Ar, 1.9; Kr, 12; Xe,39; Rn, 124. The potential curves used to obtain these estimates are explained in Sec. II.

~,„(A') ~sE «) ( a ) /h (MHz)

Na HeK HeRb HeCs HeNa 2'Ne

K 'NeRb 2'Ne

CsNa ArK ArRb ArCs 'ArNa KrK' KrRb 83KrCs' KrNa' Xe

Rb' XeCs' XeNa Rn

Rb RnCs Rn

6.58.58.8

10.02734384429566183

120230280340310660730880730

130014001800

1.8 x10-'1.6x10-'1.8 x10-'2.0x10-"2.0x10-'4.9X 105.3 X 107.9x10-'7.2 x10-'1.9 x10-'2.4x10-'2.9 x10-'6.1x10-'1.8 x10-'2.0x10-'2.3 X 104.7x10-'1.0x10-'1.1x10-'1.3 X 10

1.2X10 '2.1 x 102.1x10-'2.7x 109.8 x10-'1.5 x10-'2.3x10 '2.9x 102.8 x10-'1.3 x10-'2.0X 103.8 x10-'6.1x10-'3.4x10-'7.1x10-'1.2X 101.3 x10-'8.0X 101.6x10-'2.7x 102.7 X 101.2x10-'2.5 x10-'4.9X 10

0.0860.110.110.132.42.61.92.8

2023212070886566

190250190200

0.110.180.180.270.731.061.01.60.981.61.82.1

19333340

7.7151520

5.76.66.77.1

60726886

120150170180230310310350470510530550

NMR frequencies is the frequency-shift enhancement fac-tor Kp which is the ratio of the average alkali-metalvalence electron density at a noble-gas nucleus to thealkali-metal atom density. Schaefer et al. have shownthat Kp is given by the following simple formula:

f ~ y(g ) ~

2 —V(R)/kT4 g 2' (17)

Since the Boltzmann factor is nearly 1 over most of theintegral above, Kp is only weakly temperature dependent.Kp is sensitive to the location of the repulsive wall of thepotential, but insensitive to the well depth. Thus if g hasbeen measured in some other exPerirnent, Kp becomes agood probe of the repulsive wall of the potential. Todate, the experimental measurements of Kp have been lim-ited by systematic errors to accuracies of about 35%, butthis will certainly be improved in the future.

To date, the only quantitative measurements of the fre-quency shifts are for RbKr and RbXe. ' The results ofthese experiments are shown in Table IV.

VI. ESTIMATES OF SPIN-EXCHANGEPARAMETERS

The comparisons in Tables I—IV of the values of g and6 obtained from the OPW and PS methods to those de-duced from experiment and potential curves show thatthe experiments favor larger values of q and G. Table Vgives the calculated values of all the spin-exchange pa-rameters for the alkali-metal —noble-gas pairs using thelarger of the PS and OPW values for g and G. The evi-dence of Tables I—IV suggests that these estimates shouldbe fairly reliable for those alkali-metal —noble-gas pairswhich have not been investigated to date. This will beuseful for future experiments which involve spin ex-change between optically pumped alkali-metal atoms andnoble-gas nuclei.

ACKNOWLEDGMENTS

W. Happer's interest and helpful comments were great-ly appreciated. This work was supported by the U.S. AirForce Oi%ce of Scientific Research under Grant No. 88-0165.

*Present address: Joint Institute for Laboratory Astrophysics,University of Colorado and National Institute for Standardsand Technology, Boulder, CO 80309-0221.

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