Method for deducing anisotropic spin-exchange rates
aa r X i v : . [ phy s i c s . a t o m - ph ] D ec Method for deducing anisotropic spin-exchange rates
T. G. Walker, I. A. Nelson, and S. Kadlecek Department of Physics, University of Wisconsin-Madison, Madison, WI 53706 Columbia–St. Mary’s Hospital, Milwaukee, WI 53211 Department of Radiology, University of Pennsylvania, Philadelphia, PA 19104 (Dated: October 31, 2018)Using measured spin-transfer rates from alkali atoms to He, combined with spin-relaxation ratesof the alkali atoms due to He and He, it should be possible to differentiate between isotropic andanisotropic spin-exchange. This would give a fundamental limit on the He polarization attainablein spin-exchange optical pumping. For K-He, we find the limit to be 0.90 ± PACS numbers: 32.80.Xx,33.25.+k,34.80.Nz
I. INTRODUCTION AND MOTIVATION
The spin-dependent interactions governing spin-exchange collisions between alkali-metal atoms andnoble-gas atoms are [1] V = α ( R ) S · K + β ( R )(3 S · ˆR ˆR · K − S · K ) (1)where α is the strength of the Fermi-contact or isotropichyperfine interaction between the alkali-metal electronspin S and the noble gas nuclear spin K , and β is thestrength of the anisotropic hyperfine interaction. Both α and β depend on the interatomic separation R .Anisotropic spin-exchange was recently considered byWalter et al. [2] and, on the basis of theoretical argumentsthat have generally been successful in explaining the sizeof various alkali–noble-gas spin-interactions, was foundto be a small effect. If present, anisotropic spin-exchangewould modify the dynamics of polarization transfer froman alkali vapor of number density [A] to the helium nu-cleus to dP He dt = k α [A] ( P A − P He ) + k β [A] (cid:18) − P A − P He (cid:19) − Γ w P He (2)where k α and k β are the rate coefficients arising fromthe two interactions and Γ w represents depolarization atthe wall of the gas enclosure. Note that anisotropic spin-exchange tends to polarize the He nuclei in the directionopposite that of the alkali polarization. In the presence ofcompletely polarized alkali vapor and non-relaxing walls,nearly achievable in practice, the anisotropic interactionwould limit the maximum attainable polarization to P max = k α − k β / k α + k β (3)Walter et al. [2] predicted P max = 0 .
96 for Rb- He and0 .
95 for K- He. Extensive experiments at Wisconsin andNIST [3, 4, 5] have shown that some unknown spin-relaxation mechanism limits the He polarization, evenunder supposedly ideal conditions, to less than 80% for both Rb and K-Rb mixtures. Could one source of this re-laxation be anisotropic spin-exchange? Here we presenta method for experimentally answering this question, bydeducing k β from spin-exchange and alkali-metal spin-relaxation measurements. II. LIMITS FROM WALL RELAXATIONSTUDIES
The approach of P He to saturation in the presence ofa polarized alkali vapor can be experimentally character-ized by its saturation level P ∞ He and rate of approach tosaturation Γ. From Eq. 2, P ∞ He = P A ( k α − k β / A ]Γ w + ( k α + k β )[ A ] (4)Γ = Γ w + ( k α + k β )[ A ] (5)For any given measurement P He ( t ) performed at constant P A [A], Γ w can be eliminated, leaving k α − k β / P ∞ He Γ[A] P A (6)The quantity k SE = k α − k β / P ∞ He or Γ as a function of [A]would allow determination of k α + k β . But it is nowwell-established [3, 5] that Γ w depends strongly on [A],making this approach not feasible.The latest wall studies [5], surveying many cells havinga range of surface to volume ratios S/V , found that theobserved polarization is well-described by P ∞ He P A = 11 + X (7)where X is of the form X = X + X SV (8)If we assume that X represents the fundamental (wall-independent) effects of the anisotropic hyperfine interac-tion, comparison to Eq. 3 yields X = 3 k β k α − k β (9)The factor X , which would represent a limit on P He from collisions in the gas, could be as small as 0 andas large as 0 .
15 [5]. Variability in measured X limitsthe certainty of the results, presumably due to its verysensitive dependence on the exact physical and chemicalnature of the wall. III. METHOD
Our basic idea is to determine k α + k β by comparingspin-relaxation measurements of alkali-metal atoms in He and He. The spin-relaxation rate of the alkali-metalatoms due to He is, at low polarization and low enoughtemperatures that the alkali-alkali spin-relaxation ratescan be ignored, γ A = k [ He] = (cid:0) k SR + k α + k β (cid:1) [ He] (10)where k SR is the relaxation produced by the spin-rotationinteraction. The spin-relaxation rate due to He is simply γ A = k SR [ He] (11)since there is no spin-exchange for He. Thus we can usethe relaxation of the alkali atoms in He gas to isolatethe spin-exchange and spin-relaxation contributions. Weargue below that the spin-relaxation rates for the twoisotopes scale linearly with the collision velocities, so that k SR = r µ µ k SR (12)where µ is the reduced mass of the He-alkali pair. Thisscaling should allow us to separate the spin-exchange andspin-rotation contributions to the alkali spin-relaxationrate: k α + k β = k − r µ µ k SR (13)Thus subtracting the scaled He spin-relaxation ratefrom the He spin-relaxation rate isolates the sum of theisotropic and anisotropic spin-exchange rates.Experimentally, the challenge is to measure the alkalispin-relaxation rates carefully enough to preserve signif-icance for the subtraction in the numerator of Eq. 13.The Rb- He spin-exchange rate has now been measuredby two different groups [3, 6] to be 6 . × − cm /s.The relaxation rates for Rb-He are unfortunately about16-50 times bigger (depending on temperature) than thespin-exchange rates. Thus very high precision measure-ments would need to be made.The situation is much better for potassium, where themeasured efficiencies suggest a factor of 10 more favorableratio of spin-exchange to spin-relaxation rates.We now turn to the scaling relation for the two iso-topes. The spin-rotation coupling γ ( R ), R being inter-atomic separation, is inversely proportional to the re-duced mass µ of the colliding pair. This is because the rotation frequency of the atoms about each other is ω = ¯ h N µR (14)which give rise to a Coriolis interaction V ω = − ¯ h ω · L (15)where L is the electronic angular momentum[7]. Thespin-rotation coupling then arises due to the responseof the electron to the effective magnetic field B =¯ hω/ ( g S µ B ). Thus one expects on very general groundsthat γ ( R ) ∝ /µ .The spin-relaxation rate coefficient is an average overthe possible collision trajectories [8] k SR = 8 πvµ h Z ∞ we − w dwb db × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ r o γ ( R ) dR p (1 − b /R ) − V ( R ) /wkT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (16)where w is a dimensionless variable and b the impactparameter of the collision. V ( R ) is the Rb-He potential,which should be very insensitive to the mass of the He nu-cleus. The inverse scaling of γ with reduced mass cancelsthe µ factor in front of the integrals, so that the mass-dependence of the spin-relaxation rate coefficient arisesentirely from the relative velocity factor v ∝ / √ µ . IV. EXPERIMENT
The K-He spin-relaxation measurements were madeat Amersham Health using a 7.1 cm diameter sphericalvalved cell containing K metal with a very small amountof Rb metal dissolved in it. The Rb vapor density wasmeasured to be 2 ± . × − that of the K. The Rbatoms were polarized to typically 20% polarization (par-allel to a 20 G magnetic field) by optical pumping with a60 W diode laser. The polarized Rb atoms then polarizedthe K atoms by spin-exchange collisions. A mechanicalshutter periodically blocked the laser light to allow thealkali polarization to decay due to spin-relaxation.A single-frequency tunable diode laser, operating attypically 3 nm or more from the potassium D1 line at770 nm, was used to monitor the spin-polarization of thealkali atoms by Faraday rotation. The spin-relaxationtransients were then analyzed to extract the slowest de-cay mode of the relaxing atoms. This procedure wasrepeated a number of times as the pressure and com-position of the cell was varied. Two decays were takenat each pressure, with different probe laser intensities. Alinear extrapolation to zero probe laser intensity was per-formed to remove the effect of the probe laser (at most a5% correction).Three gases were used for the experiments. The “ He”gas was actually a 0.9922:0.0078 He-N mixture thatwas the standard Amersham gas mixture. Pure nitrogengas was also used so that the nitrogen contribution to the He relaxation could be corrected for. The third gas was He. The cell was filled with the gas of interest at highpressure. Immediately after filling with the fresh gas, thealkali vapor pressure would suddenly drop, then slowlyrecover over the period of about an hour. The drop inpressure was presumably due to chemical reactions withimpurities in the gases. To vary the gas pressure, hot gaswas pumped out through the cell valve. Since this wasdone with the cell hot, the gas density was determinedfrom the pressure using the ideal gas law at the 150 ◦ Ccell temperature. R e l a x a ti on R a t e ( / s ) He] (amagat)
FIG. 1: Spin-relaxation data for K in two gas mixtures. Thetop data is the He-N mixture, the bottom for pure He.
The measured spin-relaxation decay rates are shownin Fig. 1. On a given day, the data vary smoothly withpressure; however we found some systematic day-to-daychanges that are outside the normal statistical fluctua-tions. For example, the He data points at 6.9 amagatand 2.9 amagat were taken on different days than mostof the other data. The size of these unexplained fluctua-tions is about 4%.
V. ANALYSIS
The data for the two gases were fit to the followingfunction:Γ = D r µ µ G (cid:16) πR (cid:17) + k [G] (17)with the first term representing diffusion, the second K-K relaxation, and the third spin-relaxation due to K-Gcollisions. Based on S. Kadlecek’s thesis[9], we expectΓ < . µ of the K-G pair. Thus only 3 parameters, D , k ( He), and k ( He), were used to fit the entire dataset. The results are: D = 0 . ± .
04 cm /s k ( He) = 0 . ± . k ( He) = 0 . ± . ◦ C temperature, the K and Rb atoms arewell into the regime where the spin-exchange rates be-tween the alkali-metal atoms greatly exceed the spin-relaxation rates for the atoms. Thus the atoms shouldbe well-described by a spin-temperature. The presence ofthe Rb vapor at a concentration of 1/500 slightly modifiesthe usual slowing down factor of 6 for a nuclear spin-3/2atom like K to s = 6 + 10 . /
500 = 6 .
02. We also mustaccount for a slight amount of Rb-He spin-relaxation,measured by Baranga et al [6] to be 41.2/s-amagat forRb He, and, using the mass scaling, 36.1/s-amagat for He. We therefore find k SR = s × . − . /
500 = 2 .
10 /s-amagat= 7 . × − cm /s (19)and, using the mass scaling, k SR = 1 . k SR = 2 .
39 /s-amagat= 8 . ± . × − cm /s (20)These are the first measurements of spin-relaxation of Kby He.Since the He gas is actually a mixture, a correction forN must also be made. From Ref. [9], and confirmed by ameasurement at 28 psig, we find that nitrogen contributes1.24/s-amagat for the 0.78% mixture used. We thereforefind for the total K- He spin-destruction rate coefficient(spin-exchange plus spin-rotation), k = s × . − . − . /
500 = 4 .
04 /s-amagat (21)The spin-exchange contribution is therefore k α + k β = k − k SR = 1 .
65 /s-amagat= 6 . ± . × − cm /s (22)The latest measurements of k SE [10] give k α − k β / . ± . × − cm /s. Therefore the X-factor due toanisotropic spin-exchange is X = 6 . ± . . ± . − . ± .
13 (23)This in turn implies that spin-exchange using K- He col-lisions is fundamentally limited to a He polarization of P max = 11 + X = 0 . ± .
11 (24)This result, though it does not rule out P max = 1, istantalizing since it suggests there may actually be a fun-damental contribution to the X-factor. Higher preci-sion measurements of both the spin-exchange rate coef-ficient and the spin-relaxation measurements are neededto reach a definitive conclusion.We can combine our He spin-relaxation results with k SE to obtain a spin-exchange efficiency η = k SE3 k = 5 . ± . . ± . . ± .
02 (25)This is in slight disagreement (1.25 σ ) with the Baranga et al. result [6] of 0.295 ± He. This issue is not onlyof interest for fundamental reasons, but it has practical importance for maximizing the attainable polarizationin spin-exchange optical pumping. Considerable effort atNIST and Wisconsin [5] has gone into trying to improvethe wall-relaxation performance of He spin-exchange.The best polarization observed to date is 81%. If thisvalue is approaching the fundamental limit for the pro-cess, there is little to be gained through further laboriouswall studies. On the other hand, if the limit is 95% orhigher, there is room to substantially improve the per-formance of spin-exchange pumped targets for applica-tions such as neutron spin-filters [12], magnetic resonanceimaging [13], and electron scattering [14].
Acknowledgments
This work was supported by the National ScienceFoundation, the Department of Energy, and AmershamHealth. The authors benefitted from discussions with T.Gentile and B. Driehuys. [1] T. Walker and W. Happer, Rev. Mod. Phys. , 629(1997).[2] D. K. Walter, W. Happer, and T. G. Walker, Phys. Rev.A , 3642 (1998).[3] B. Chann, E. Babcock, L. W. Anderson, and T. G.Walker, Phys. Rev. A , 032703 (2002).[4] B. Chann, E. Babcock, W. Chen, T. Smith, A. Thomp-son, L. W. Anderson, T. G. Walker, and T. R. Gentile,J. Appl. Phys. , 6908 (2003).[5] E. Babcock, B. Chann, T. G. Walker, W. C. Chen, andT. R. Gentile, Phys. Rev. Lett. , 083003 (2006).[6] A. Ben-Amar Baranga, S. Appelt, M. V. Romalis, C. J.Erickson, A. R. Young, G. D. Cates, and W.Happer,Phys. Rev. Lett. , 2801 (1998).[7] T. G. Walker, J. H. Thywissen, and W. Happer, Phys.Rev. A , 2090 (1997).[8] T. G. Walker, Phys. Rev. A , 4959 (1989).[9] S. Kadlecek, Ph.D. thesis, University of Wisconsin- Madison (1999).[10] E. Babcock, Ph.D. thesis, University of Wisconsin-Madison (2005).[11] E. Babcock, I. Nelson, S. Kadlecek, B. Driehuys, L. W.Anderson, F. W. Hersman, and T. G. Walker, Phys. Rev.Lett. , 123003 (2003).[12] E. Babcock, S. Boag, K. Andersen, M. Becker,C. Beecham, F. Bordenave, J. Chastagnier, W. Chen,R. Chung, T. Chupp, et al., Physica B: Condensed Mat-ter , 2655 (2009).[13] J. H. Holmes, R. L. O’Halloran, E. K. Brodsky, Y. Jung,W. F. Block, and S. B. Fain, Mag. Res. in Med. , 1062(2008).[14] K. Slifer, M. Amarian, L. Auerbach, T. Averett,J. Berthot, P. Bertin, B. Bertozzi, T. Black, E. Brash,D. Brown, et al. (Jefferson Lab E94010 Collaboration),Phys. Rev. Lett.101