Depolarization of Spin-Polarized Hydrogen via Collisions with Chlorine Atoms at Ultrahigh Density
Alexandros K. Spiliotis, Michail Xygkis, Michail E. Koutrakis, Dimitrios Sofikitis, T. Peter Rakitzis
DDepolarization of Spin-Polarized hydrogen via Spin-Exchange Collisions with chlorineAtoms at Ultrahigh Density
Alexandros K. Spiliotis,
1, 2
Michail Xygkis,
1, 2
Michael Koutrakis, Dimitrios Sofikitis, and T. Peter Rakitzis
1, 2 Foundation for Research and Technology Hellas, Institute of Electronic Structure and Laser,N. Plastira 100, Heraklion, Crete, Greece, GR-71110 University of Crete, Department of Physics, Herakleio, Greece University of Ioannina, Department of Physics, Ioannina, Greece
Recently, the production of ultrahigh-density ( ∼ cm − ) spin-polarized deuterium (SPD)atoms was demonstrated, from the photodissociation of deuterium iodide, but the upper densitylimit was not determined. Here, we present studies of spin-polarized hydrogen (SPH) densities upto 10 cm − , by photodissociating 5 bar of hydrogen chloride with a focused 213 nm, 150 ps laserpulse. We extract the depolarization cross-section of hydrogen and chlorine atom collisions, which isthe main depolarization mechanism at this high-density regime, to be σ H − Cl = 7(2) × − cm . Wediscuss the conditions under which SPH and SPD densities of ∼ − cm − can be reached, andthe potential applications to ultrafast magnetometry, laser-ion acceleration, and tests of polarizednuclear fusion. PACS numbers:
I. INTRODUCTION
Polarized atomic gases typically have low densities, of ∼ cm − and below, because of depolarizing effects ofcollisions in the production methods of Stern-Gerlach spin separation [1], or optical pumping[2, 3]. The only exceptionshave been the noble gases with nuclear spin (e.g. He and Xe ), for which the nuclear spins can be polarized athigh pressure (via spin-exchange optical pumping), as the depolarization rate of these closed-shell atoms can be verysmall[4]. However, this inability to produce high-density open-shell spin-polarized gases precludes several potentialapplications.Recently, Sofikitis et al. demonstrated the production of ultrahigh-density ( ∼ cm − ), highly spin-polarizedhydrogen (SPH) and deuterium (SPD), from the photodissociation of hydrogen halides with a circularly polarized UVlaser pulse, and a pickup coil to monitor the electron polarization[5].These densities surpass the current state-of-the-artby at least 7 orders of magnitude. To achieve such high densities, the hydrogen halides were photodissociated with a150 ps laser pulse, so that the SPH/SPD are produced nearly instantaneously, on the 10 − s timescale, several ordersof magnitude faster than the production time for conventional polarization methods. This rapid production allowsthe high-density SPH/SPD to survive for 5-100 ns at these high densities[6], before depolarization from collisions.The aim of this paper is to investigate the main depolarization mechanism in detail, and to determine the limits ofSPH density and lifetime.The main SPH depolarization mechanism at ultrahigh densities is expected to be spin-exchange collisions betweenSPH and halogen radicals Y, occurring at a rate k H − Y : H ↑ + Y → H + Y ↑ as opposed to the low-density regime ( ∼ − cm − ), where only ∼ .
1% HCl is dissociated, and the prevailingdepolarization mechanism has been shown to be the three-step reaction of SPH with high-density HY molecules [6]: H ↑ + HY → HY − H ↑ (1) HY − H ↑ → HY − H (2) HY − H ↑ + HY → H ↑ + 2 HY (3)To our knowledge, the depolarization cross-section of SPH by halogen radicals is not cited in the literature. If weassume that σ H − Y is comparable to the depolarization cross-section of SPH by alkali atoms, σ H − Rb ∼ × − cm[7], then, for a density of 10 cm − , the polarization lifetime would be ∼ . et al. [5] used a simple method to estimate σ H − Y . They focused thebeam lenses, one with f=25 mm and one with f=50 mm, placed at a distance l ∼ mm from the center of thecoil. The f=50 mm lens focused the beam ∼ ∼ cm − inside the coil, whereas the f=25 mm lens focused the beam at the center of the coil, creating an a r X i v : . [ phy s i c s . a t o m - ph ] F e b effective SPH density of ∼ cm − near the focus. This way, they expected to differentiate between the low-and high-density depolarization rates. However, they observed lifetimes of ∼ −
20 ns under both density regimes,yielding no evidence of SPD depolarization from I ( P / ) in the high-density regime. The deuterium iodide (DI)density could not be increased above ∼ cm − , to increase the depolarization rate, because the absorption crosssection of DI at 266 nm, 2 × − cm , is large enough to prevent sufficient photolysis laser light reaching the laserfocus, for the geometry of the experiments. The absorption cross section of HCl at 213 nm is 2 orders of magnitudelower, at 2 × − cm . Therefore, the HCl density can be increased to 10 cm − , and the photolysis laser can reachthe laser focus in the cell, without too much loss from absorption, as the optical depth is ∼ σ H − Cl . II. HALOGEN RECOMBINATION
The time dependent halogen-atom density is determined by the recombination rate, so this recombination is firstdiscussed in detail in this section. The recombination reactions are three-body reactions, of the form:2 Y + M → Y + M (4)Depending on which third body M participates in the reaction, the reaction rate is different. As shown in table II,the three-body Cl recombination reaction rates are of order 10 − cm s − at room temperature, when the third body isa chlorine molecule, and somewhat lower in the presence of N . Nonetheless, immediately after the photodissociation,no halogen molecules are present; recombination reactions with M=Y or M=HY should take place to create Y , tomake the three-body recombination reactions faster. Reactions with M=Y are significantly slower than those withM= Y , having a rate of 3 . × − e /T cm s − [8]. Furthermore, to our knowledge, there are no references ofthe rate of recombination reactions with a HY third-body. These rate constants do not justify the existence of asignificant Y density immediately after the photodissociation, therefore, we can assume that the Y atom densityremains constant for the few ns after photodissociation that are of interest to this study. P o p u l a t i o n ( n o r m . ) Time (ns) a) 2 Bar HClc) 2 Bar HCl + 0.5 Bar SF b) 5 Bar HCld) 2 Bar HCl + 2 Bar SF SPHCl ClHCl
FIG. 1: Evolution of populations of SPH, Cl , Cl, and HCl after the photodissociation at various conditions: a) 2 bar HCl b)5 bar HCl c) 2 bar HCl + 0.5 bar SF d) 2 bar HCl + 2 bar C F . Here, we have set σ H − Y =0, to highlight the effect of theCl recombination to the depolarization of SPH. A number of inert gases, such as N , SF , or C F are efficient third bodies for halogen recombination. For example,recombination of Cl radicals in the presence of SF molecules happens at a rate k Cl − SF = 6 . × − cm s − [9],while for Iodine recombination, the rate is k I − SF = 5 . × − cm s − [9]. k Cl − SF is enough to create a populationof Cl comparable to the radical population in a few tens of ns at pressures of 2 bar each of HCl and SF (1d). Sucha high Cl concentration, though, would trigger the efficient reaction: H ↑ + Cl → HCl + Cl (5)which removes SPH, while also creating Cl radicals, an effective SPH depolarizer. Thus, the fast recombination thatremoves Cl, triggers another reaction that replenishes it and simultaneously depolarizes SPH. Taking these reactionsinto account, the rate equations for the system after the UV pulse photodissociation are: d [ SP H ] t dt = − k H − Y [ SP H ] t [ Y ] t − k H − HY [ SP H ] t [ HY ] t − k H − Y [ SP H ( t )][ Y ] t (6) d [ Y ] t dt = k H − Y [ SP H ] t [ Y ] t (7) d [ Y ] t dt = k Y − Y − M [ Y ] t [ M ] t − k H − Y [ SP H ] t [ Y ] t (8) d [ HY ] t dt = k H − Y [ SP H ] t [ Y ] t (9)The rate constants for 5 are shown in table I. All are of order 10 − cm s − , implying that, for Y densities of 10 cm − (similar to the corresponding SPH densities after photodissociation), SPH is removed in less than a ns. Wecan therefore safely conclude that, if the conditions of the experiment favored a very rapid recombination of the Yradicals, a short SPH lifetime would reflect this.Concluding this section, we can say that the observed SPH and SPD lifetimes in [5] are shorter than the limitsimposed by the halide recombination dynamics described above. Therefore, any depolarization observed should beattributed to spin-exchange reactions of SPH with the Y radical. Reaction k H + Cl . × − D + Cl . × − H + Br . × − D + Br . × − H + I × − TABLE I: Rates of the reaction H + Y → HY + Y in (cm s − )[10, 11] (cid:72)(cid:72)(cid:72)(cid:72)(cid:72) Y M Y Ar N Pressure (bar)Cl 5 . × −
180 [12]5 . × − . × − . × − . × − . × − − . × − [14]Br 4 . × −
100 [15]1 . × − . × − . × − . × − [16]I 3 . × − × − × −
100 [17]TABLE II: halogen three-body recombination rates for various chaperons M in cm s − III. DETERMINATION OF σ H − Y The rate equations for SPH depolarization from collisions with Y radicals read as
FIG. 2: Plot of the spin polarized Hydrogen density inside the coil, after the photodissociation of 2 bar HCl. Note that thelower limit of the plot densities is 0.1% d [ SP H ] t dt = − k H − Y [ SP H ] t [ Y ] t − k H − HY [ SP H ] t [ HY ] t (10) d [ Y ] t dt = 0 (11) d [ HY ] t dt = 0 (12)where we neglect all Y recombination reactions, since they are significantly slower than the depolarization lifetime.It follows that, to calculate k H − Y , the SPH (and Y) density immediately after the photodissociation must be known.If the photodissociation is 100% efficient, HY (0) = 0, and the second term of 10 vanishes. Otherwise, the secondterm is non-negligible, and can be calculated using the value of k H − HY from [6] (7 . × − cm ).Note that, as mentioned in [5], and was confirmed in this study, no evidence of spin-polarized halogens (i.e., aFourier peak at 150 MHz, the hyperfine frequency of Cl P / [18]) is found in the data. The absence of such evidencemeans that halogen atoms are quickly ( (cid:28) A. Experiment
A simple experiment was designed to estimate k H − Y , by measuring the SPH polarization lifetime for various SPHdensities. The experimental setup is the same as the one used in [5], with hydrogen chloride(HCl) as the SPH sourcegas. Briefly, 1-5 bar HCl are introduced in a gas cell, and a circularly polarized, 213 nm, 150 ps laser pulse is used tophotodissociate HCl. The 4 mm diameter beam is focused by a 5 cm lens, placed on a one-dimensional translationstage, by which we control the distance of the lens from the cell, and therefore the position of the focus inside thecell. A pickup coil with a length of 4 mm, 4.5 turns, and a 2 mm diameter, is used to detect the time evolution ofthe SPH produced by the photodissociation of HCl. The time-dependent signal that the coil picks up is of the form: I ( t ) = e − t/τ cos 2 πf t (13)Where t is the time, τ the polarization lifetime, and f is the hyperfine beating frequency, which is equal to 1.420415GHz for the unperturbed hyperfine hydrogen Hamiltonian H = A HF I · J (14)with A HF the hyperfine constant, and I , J the quantum operators for the nuclear and electron spin, respectively.With this setup, we can scan the position of the focus through the coil length, effectively varying the density of theSPH produced inside the coil.The dimension of the beam was confirmed to follow the distribution of a Gaussian beam using a razor-edge scheme.The waist at focus was estimated to be 6 ± µ m, close to the diffraction limit of d = 5 µ m. A theoretical model basedon the saturated Beer-Lambert law is used to simulate the densities created by a laser beam of known dimensions andenergy. The produced SPH density through the length pickup coil length, is shown in figure 2, for an HCl pressureP=2 bar, corresponding to an initial number density of 5 . × cm − . The model shows that the SPH density withinthe Rayleigh range of the beam ( l ∼ . mm ) is close to the initial HCl density for pressures of up to 5 bar. However,due to the large beam divergence and the low absorption cross-section of hydrogen chloride, the density outside ofthe Rayleigh range is much lower, of order 10 − cm − . If SPH depolarization at a high density environment is fast,this variation in density within a few mm would result in polarization lifetime gradients in space. Since the pickupcoil is longer than this length, the different SPH polarization lifetimes would be imprinted in the pickup coil signal(13) as more than one exponential: I ( t ) = (cid:90) (cid:90) V ρ SP H ( r, θ ) e − t/τ ( r,θ ) cos (2 πf t ) dr dθ (15)where ρ SP H and τ depend on r and θ .A shorter coil would, of course, offer better spatial resolution, but at the detriment of the signal-to-noise ratio(SNR).For this reason, in this experiment, we chose to work with a 4.5 turns, 4 mm long pickup coil. Further investigationon the optimization of the coil is needed, to achieve shorter length while retaining a high SNR. B. Calculation of the Expected SPH Density
The laser used in this experiment emits a λ = 213 nm, τ p = 150 ps, E = 3 mJ, D = 4 mm Gaussian pulse. Thepick-up coil’s diameter and length are d = 2 mm and l = 3.5 mm respectively, and is placed 3 cm away from thewindow of the cell. The coil is relatively short in order to facilitate the acquisition of rapidly changing signal. Thephotodissociation cross section for HCl at 213 nm (room temperature) is σ = 1 . × − cm[19]. The number of SPHfragments created can be calculated as follows. The initial available photons are: N ph = λEhc (16)with h is Planck’s constant, and c the speed of light. Applying the saturated Beer-Lambert law, we find that thephotons that reach the coil are: N ph = N ph e − σln (1 − e − σln ) (17)where n is the particle density and is related to the pressure P (in bar) for HCl as: n = P × . × cm − (18) IV. RESULTS AND DISCUSSION
From figure 2, we can observe the variation in SPH density inside the coil. Inside the Rayleigh range of thebeam, HCl is almost or entirely depleted (70-100%), and there exist only hot spin-polarized hydrogens and chlorinesimmediately after the photodissociation. As discussed earlier, Cl is almost entirely depolarized within less than 1 nsafter the photodissociation, thus making SPH-Cl spin-exchange more efficient. The fast depolarization would then betriggered by spin-exchange collisions between SPH and unpolarized Cl. At positions outside the Rayleigh range, SPHand Cl account for less than 1% of the total number density, and are surrounded by HCl molecules. In that case, themain depolarization mechanism is via the HCl-SPH complex formation (1), and the depolarization rate is more thanan order of magnitude lower. Using this density calculation for H, Cl, and HCl at every position of the beam, we canextract a value for the H-Cl depolarization cross section from eqs. 10. To that end, a Finite Element Analysis (FEA)is employed, where the volume of the beam is separated in sections of { x,y,z } =1 μ m × μ m × μ m, and the timeevolution of the SPH in each section is calculated. SPH /N ν SP H E q ( k m / s ) No inert Gas1:13:15:1
FIG. 3: Thermal velocity distribution vs photodissociation fraction, for various ratios of C F partial pressure to hydrogenhalide partial pressure. To extract σ H − Cl , the distribution of velocities should also be calculated. The velocity of SPH after HY pho-todissociation is ˜ v ≈
20 km/s[20]. However, thermal equilibrium between H, Cl (and HCl, where less than 100% ofHCl molecules have been photodissociated) should be reached quickly after the photodissociation, as the hard-spherecollision rate is 40/ns.Using the Equipartition theorem, the kinetic energy of the H atoms, E H , as a function of the dissociated fractionx of the HCl molecules, is given by: E H = E H K + E EKE × (cid:2) x/ x + C pHCl R (1 − x ) (cid:3) (19)where E H K = 0.038 eV is the thermal energy of hydrogen at room temperature, E EKE = E photon − E bond =1 . eV is the excess kinetic energy from the photodissociation, C p HCl the heat capacity of HCl, and x the fractionof photodissociated molecules, x = n/ N , with n the SPH density produced by the photodissociation, and N thenumber density of HCl prior to the photodissociation.Figure 3 shows the equilibrium SPH velocity as a function of the SPH density. We see that the thermal equilibriumvelocity of hydrogen when HCl is completely photodissociated is reduced to ∼
12 km/s, as a result of the frequentcollisions with the initially much colder( ∼
300 K) Cl atoms, whereas at low SPH densities, SPH velocity is the roomtemperature velocity, ∼ σ H − Cl . Figure 4 show the obtained SPH signal after photodissociation by the laser pulse, anda fit with σ H − Cl = 7 × − cm , for various focusing geometries, and at two initial HCl pressures, 2 and 5 bar.When the beam is focused inside the coil, there is a steep drop observed in the early part of the signal, followed bya lower slope at later times. The steep drop should be attributed to the high density inside the Rayleigh range, wherethe main depolarization mechanism is via spin-exchanging collisions between SPH and Cl. At later times, the slopebecomes gradually lower, approaching the low-density depolarization rate. The value of σ H − Cl = 7 × − cm fitsthe data well at all focusing geometries and at both 2 bar and 5 bar, lending confidence to the determination of theSPH-Cl spin-exchange cross section.Note that an inert gas with a high heat capacity can be used, to cool the spin-polarized hydrogen down to a lowerequilibrium temperature, and as such lower the collision rate, and thus the spin-exchange collision rate(assuming thatthe spin-exchange cross section does not increase too much at low energies).A suitable candidate may be hexafluorethane ( C F ), which has a heat capacity of 105 J mol − K − at roomtemperature, which raises to over 170 J mol − K − at high temperatures. Additionally, C F is transparent at middleand far ultraviolet, reducing the possibility of reacting fragments emerging from dissociation or ionization. By addingan extra factor k E C F , where k = N C F /N HCl the number density ratio of C F : HCl , and E C F the equilibriumthermal energy of C F , eq. 19 becomes: E H a = E H K + E EKE × x/ x + C pHCl C pH (1 − x ) + k C pC F C pH (20)The equilibrium velocity of SPH after HCl photodissociation in the presence of various partial pressures of C F isshown in figure 3. The reduction of the equilibrium velocity by a factor of 3 observed with a 3:1 C F /HCl ratio isexpected to prolong the polarization lifetime by an equal factor at the high-density regime.While the degree of electronic polarization of SPH can be practically 100%, as electronic polarization production isnearly instantaneous, the polarization of the nucleus occurs after a hyperfine half-period, which is ∼
350 ps for SPHand ∼ C F . A notable result isthat ¿90% polarized protons can be produced at a density of about 10 cm − , and the same degree of polarizationfor deuterons is achievable up to ∼ × cm − . A threefold increase in proton densities would yield a polarizationof 70%, equal to that of deuteron densities of × cm − . For these simulations, we assumed σ H − Cl = σ D − I , and σ D − I ≈ σ H − I , with σ H − I = 5 × − cm s − [11]. The rate of chlorine recombination in the presence of C F has been previously measured to be 3 . × − cm s − [21], whereas, in the absence of experimental measurements,we assume the cross section of iodine recombination in the presence of C2F6 to be similar to that in the presence of SF [9].We can then deduce expected values for spin polarized electron, proton and deuteron densities produced via thephotodissociation method. About 2 . × cm − ∼ × cm − for protons and ∼ cm − fordeuterons. V. CONCLUSIONS
To our knowledge, the value of σ H − Cl presented here is the first experimentally measured value of the H-Cl spin-exchange rate, and is consistent with the upper limit of σ D − I that Sofikitis et al. [5] suggested in an experimentconducted at one order of magnitude lower densities for DI photodissociation. The measured value shows thatlifetimes of order of a few ns are possible, for SPH at densities above 10 cm − . Calculation of σ H − Cl , particularlyas a function of collision energy, will be helpful to corroborate the results and conclusions of this work, or to helppoint out issues that still need to be elucidated..The experimental method presented here could be improved to achieve the capability to test higher SPH densitieswith a better spatial resolution. For better spatial resolution, a shorter coil is required, which, however, would reducethe inductance and quality factor, and consequently the signal-to-noise ratio. A potential solution for this would bethe use of a microstrip coil, which offers a high quality factor at GHz frequencies at a very compact size. Furthermore,higher energy lasers would produce higher SPH densities, possibly without the need for tight focusing, thus eliminatingthe SPH distribution gradients that the 3 mJ laser used in this study creates. Ultrahigh density SPH has severalnovel applications, including ns-resolved magnetometry, laser-ion acceleration, and tests of polarized nuclear fusion[22]. Using SPH for magnetometry allows improvement in time resolution of atomic magnetometers by several ordersof magnitude, and with spatial resolution of about 10 μ m [23]. Laser-ion acceleration can produce intense, GeV-scale,electron or proton pulses, from gas targets of density ∼ cm − , which have been unpolarized; only SPH productionof HY photodissociation has demonstrated polarized electron, proton, and deuteron densities of ∼ cm − , sufficientfor laser-ion acceleration [24–27]. It is known that polarized nuclei increase the fusion cross sections of the D+T andD + He reactions by ∼ FIG. 4: Experimental Data (Blue) and fits generated by finite element analysis of the produced SPH inside the pickupcoil(Light Red) for different positions of the beam focus related to the coil center, at an initial HCl pressure of 2 bar(upperpanel) and 5 bar (lower panel). The value for σ H − Cl used is 7 × − cm . x is the distance of the lens relative to the positionwhere the beam is focused at the center of the coil. FIG. 5: Same as fig. 4, with the beam focused inside the coil, and two different values for the H-Cl cross section, σ H − Cl SE =10 − cm (left) and σ H − Cl SE = 5 × − cm (right) FIG. 6: Degree of polarization of protons, 350 ps after complete photodissociation of HCl(upper figure), and deuterons, 1.5 nsafter complete photodissociation of DI(lower figure), at various pressure conditions, and at an initial temperature of T=300 K. [1] D Szczerba, L.D van Buuren, J.F.J van den Brand, H.J Bulten, M Ferro-Luzzi, S Klous, H Kolster, J Lang, F Mul,H.R Poolman, and M.C Simani. A polarized hydrogen/deuterium atomic beam source for internal target experiments. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and AssociatedEquipment , 455(3):769 – 781, 2000.[2] B. Clasie, C. Crawford, J. Seely, W. Xu, D. Dutta, and H. Gao. Laser-driven target of high-density nuclear-polarizedhydrogen gas.
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