Ultracold neutron accumulation in a superfluid-helium converter with magnetic multipole reflector
aa r X i v : . [ phy s i c s . i n s - d e t ] M a r Ultracold neutron accumulation in a superfluid-helium converterwith magnetic multipole reflector
O. Zimmer and R. Golub Institut Laue Langevin, 38042 Grenoble, France Department of Physics, North Carolina State University, Raleigh, NC 27695, USAJune 4, 2018
Abstract
We analyze accumulation of ultracold neutrons (UCN) in a superfluid-helium converter vesselsurrounded by a magnetic multipole reflector. We solved the spin-dependent rate equation,employing formulas valid for adiabatic spin transport of trapped UCN in mechanical equilib-rium. Results for saturation UCN densities are obtained in dependence of order and strengthof the multipolar field. The addition of magnetic storage to neutron optical potentials canincrease the density and energy of the low field seeking UCN produced and serves to mitigatethe effects of wall losses on the source performance. It also can provide a highly polarizedsample of UCN without need to polarize the neutron beam incident on the converter. Thiswork was performed in preparation of the UCN source project SuperSUN at the ILL.Keywords: ultracold neutrons, UCN source, neutron EDM, neutron decay
Mirror reflection of neutrons is an effect of the neutron optical potential which is mainly due tocoherent s-wave scattering of neutrons by atomic nuclei in condensed matter [1]. Ultracold neu-trons (UCN), which were first produced in Dubna [2] and in Munich [3], have energy sufficientlylow to become totally reflected under any angle of incidence. This peculiar property enablesexperimentalists to store them in ”neutron bottles” made of suitable materials with small crosssections for neutron absorption and providing well depths up to about 300 neV [4, 5]. Storagetime constants of many hundreds of seconds and the possibility to employ also magnetic fieldsand gravity for trapping and manipulation have made UCN a versatile tool to investigate var-ious phenomena of fundamental physics complementary to experiments at high-energy particleaccelerators [6, 7, 8].Among recent experiments with UCN feature a first demonstration of gravity resonancespectroscopy with the goal to search for deviations from Newton’s gravity law at the micrometerlength scale [9], searches for “mirror dark matter” [10, 11], a test of Lorentz invariance [12],searches for axion-like particles [13, 14, 15], a demonstration of the effect of accelerated matteron the neutron wave [16] and of the stability of the Berry geometrical phase for spin particlesunder the influence of noise fluctuations [17]. Earlier work with UCN on the geometrical phasewas published in [18, 19], while its first demonstration with cold neutrons can be found in [20].Long standing are efforts to improve the accuracy of the weak axial-vector and vector couplingconstants of the nucleon derived from precise values of the neutron lifetime [21, 22, 23, 24, 25]and the beta-asymmetry [26, 27, 28, 29]. Among other applications these values are crucial1nput for calculations of weak reaction rates in big-bang nucleo-synthesis and stellar fusion[30, 31], and of the efficiency of neutrino detectors [32]. Also long standing is the search for anon-vanishing neutron electric dipole moment (EDM), which would violate the symmetries ofparity (P) and time reversal (T) and thus via the CPT theorem also the combined symmetriesof charge conjugation and parity (CP). This search was proposed already in 1950 by Purcell andRamsey [33] and has become a prominent route to investigate new mechanisms of CP violationbeyond the standard model’s complex phase of the weak quark mixing CKM matrix, and thematter-antimatter asymmetry in the universe [34]. At the present best level of sensitivity, stillno EDM was observed [35]. Several projects are in preparation or underway with the goal togain at least an order of magnitude in sensitivity [36, 37, 38, 39, 40, 41, 42, 43, 44].Most studies with UCN are counting statistics limited and will strongly profit from newUCN sources which are currently being developed in various laboratories around the world[45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. They are all based on the ”superthermal” UCN productionscheme proposed in 1975 by one of the authors together with Mike Pendlebury [55], using eithersuperfluid He or solid deuterium as a medium for neutron conversion. Early milestones in thedevelopment of the latter were published in [56, 57, 58]. Here we focus on UCN production ina converter of superfluid He installed at the end of a neutron guide, wherein neutrons withenergy 1 meV, respectively, wavelength 0 .
89 nm may loose nearly their entire energy in singlescattering events. At low temperature only few excitations are present in the helium thatare able to scatter UCN back to higher energies. With the vanishing absorption cross sectionof He it becomes possible to accumulate UCN within a converter with reflective boundariesbefore releasing them to an experiment at room temperature. While an earlier attempt to bringthis technique to life was hampered by extraction losses (nonetheless producing record UCNdensities for its time) [59], a more efficient method was developed recently by one of the authorstogether with his co-workers [47, 60, 61, 62]. UCN are extracted through a cold UCN valve anda short vertical UCN guide section, superseding lossy separation window, screens and gaps forthermal insulation between the converter and the UCN guide of the earlier scheme. Work is inprogress to bring the technique to maturity for a new user facility at the ILL, and in particularfor performing a neutron lifetime experiment using magnetic trapping [63, 64, 65]. Also othergroups have recognized the potential advantages of a superfluid helium converter feeding UCNto experiments at room temperature [50, 51], and in some experiments this type of converter isemployed in situ [39, 43, 66].The efficiency of a UCN accumulator at an external neutron beam relies on loss rates beingsufficiently low. Most critical are those losses which occur when UCN hit the walls of theconverter vessel. They are proportional to the frequency of collisions and thus depend on thesize and shape of the converter vessel. From transmission measurements through superfluid Heat 1 .
25 K a mean free path of 17 m was derived for the 0 .
89 nm neutrons most effective forUCN production [67]. Hence, the vessel can be made several meters long without significantreduction in UCN density. On the other hand, the lateral dimensions of the converter vesselshould match the size of the available neutron beam and guide it to avoid dilution of the incidentflux. The mean free path of UCN in a vessel with such geometry is therefore only in the orderof 5 −
10 cm, leading to high typical frequencies of UCN wall collisions of 50 −
100 per second.It thus becomes challenging to obtain long UCN storage time constants which however area prerequisite for accumulation of a high saturated UCN density. Values measured for narrowvessels are normally well below 200 s. For instance, in a recent experiment on UCN production, arather short storage time constant of 67 s was obtained for a vessel held at 0 . × tube of BeO with Be windows on each end and included a short pipefrom stainless steel. That, nonetheless, a record UCN density was obtained demonstrates thepotential of the method [60]. To our knowledge the Cryo-EDM collaboration achieved with τ = 160 s the so far highest value for a helium converter enclosed within matter boundaries,2sing a 3 m long tubular vessel with diameter 63 mm, made of Be coated copper and closed offby Be windows [39].Magnetic trapping of UCN offers a viable way for a drastic improvement of the storageproperties of the converter vessel, ultimately limited only by the neutron lifetime τ β ≈ ± µ n B of the neutron magnetic moment µ n ≈ / T in a magnetic field B . Suitable configurations of magnetic field gradients keep the lowfield seeking UCN away from walls where otherwise the collisional losses occur. A group atNIST has demonstrated storage time constants consistent with the neutron lifetime within ahelium converter equipped with a superconducting magnetic quadrupole UCN reflector [68, 66].The apparatus was designed to perform neutron lifetime measurements, for which a completesuppression of UCN wall contacts is mandatory. On the other hand, for the sake of enhancing theoutput of a UCN source, combined magnetic and material trapping turns out to be particularlybeneficial. In addition, the phase space for UCN accumulation can be much increased using ahigher multipole order.In this paper we provide an analytic treatment of the rate equation for UCN production andstorage in a superfluid-helium converter confined by material walls and surrounded by a magneticmirror. We show that, combining a converter vessel possessing good (but not exceptional) storageproperties with a magnetic mirror of high multipole order, one may achieve a saturated UCNdensity close to the theoretical limit defined by an ideal experimental bottle, i.e. a square wellpotential without imaginary part. That the magnet needs to generate only part of the trappingpotential is of great practical value for constructing a device using standard superconductingwire technology. The temporal evolution of the spectral UCN density in a closed converter irradiated with thecold beam is governed by a simple rate equation. UCN production is characterized by a spectralrate density p that depends on the spectral flux of the incident beam, and a loss term is due tofinite lifetime τ of UCN in the converter, dn ( ǫ , t ) dt = p ( ǫ ) − n ( ǫ , t ) τ ( ǫ ) . (1)Here we label stored neutrons by their total energy ǫ , defined as their kinetic energy at thepoint of lowest potential energy in the trap. The quantities n ( ǫ , t ) and p ( ǫ ) denote the realspace density, respectively, the spatial UCN production rate density, each per energy intervalof a group of UCN with total energy in the range ( ǫ , ǫ + dǫ ). The saturated spectral UCNdensity obtains when UCN losses balance UCN production for t ≫ τ after having switched onthe beam. It is given by n ∞ ( ǫ ) = p ( ǫ ) τ ( ǫ ) . (2)If one wants to fill a trap with UCN up to a cutoff energy set by the trapping potential V trap ,what matters is the saturated total UCN density which is obtained from n ∞ = Z V trap n ∞ ( ǫ ) dǫ . (3)Many applications of UCN sources involve filling external traps with as many UCN as possible,followed by a long time for holding or manipulation, during which the density in the source canbe refreshed. Therefore n ∞ is a useful parameter of the converter to be optimized . For experiments involving external traps with poor storage properties it will be better to drain UCN frequentlyfrom a partially charged source. However, also in this case a long UCN storage time constant is an asset as it willraise the time-averaged UCN content of the converter. He converter withmultipole magnet and system for UCN extraction. On the left a cut view is shown for n = 12;filled (open) squares indicate electric current flowing into (out of) the plane. The neutron opticalpotentials are: V for the cylindrical inner surface over the length L , e V at the beam window andat the UCN valve, ≥ e V for the UCN extraction system, and V He ≈ . R is situated within a multipole magnet and illuminated homogeneously by a coldneutron beam passing along the r = 0 axis. UCN accumulation takes place over a length L ≫ R between a beam window and a UCN valve fully immersed in the helium as in the apparatusdescribed in [69]. Shown is a butterfly valve but also different types may be envisaged, suchas an iris diaphragm valve. For experiments at room temperature UCN are released into awindow-less extraction system with a short vertical guide section as described in [61] . In thesection for UCN accumulation the cylindrical wall possesses a neutron optical potential V + iW [1, 4, 5], with V = 2 π ~ m n X l N l b l , W = ~ X l N l vσ l ( v ) , (4)where m n is the neutron mass, N l is the atomic number density of the nuclear species l withcoherent bound neutron scattering length b l , and σ l ( v ) is the loss cross section (sum of crosssections for neutron capture and upscattering; v is the UCN velocity in the medium and vσ l ( v )usually constant over the whole UCN spectrum). The beam window and the UCN valve aremade of (or coated with) a material with neutron optical potential e V + i f W .A radial n -polar magnetic field with modulus B ( r ) = B R (cid:16) rR (cid:17) n − (5)can be generated as shown, using a regular arrangement of an even number of n straight currentbars on the outer cylinder surface, with opposite current in adjacent bars (in practice one employslong racetrack coils). A neutron moving in such a field has a magnetic potential energy of V m ( r ) = ± V m R (cid:16) rR (cid:17) n − , V m R = | µ n | B R , (6) It is also conceivable to place the UCN valve closer to (or within) the extraction guide. This would howeverconsiderably increase the surface of wall material exposed to the UCN during accumulation. Here we analyse thesystem as shown in fig. 1. V trap = min (cid:16) ( V ± V m R − V He ) Θ ( V ± V m R − V He ) , (cid:16) e V − V He (cid:17) Θ (cid:16) e V − V He (cid:17)(cid:17) , (7)i.e. the minimum of the total potential (neutron optical wall potential and magnetic interactionpotential, reduced by the neutron optical potential of the converter medium). The first argumentdescribes the situation at the cylindrical wall (with potential V ), the second one at the end disks(potential e V ≥ V ). The expression employs the step function Θ ( x ) = 1 for x > x ) = 0for x ≤
0, and V He ≈ . He as calculated using eq. 4 with N He ≈ . × cm − and b He ≈ . × − cm. For the lfs neutrons the lowest potential energyprevails on the line r = 0, whereas for the hfs neutrons it has its minimum close to the cylindricalwall (see Fig. 2). Note however that the hfs neutrons will only be able to leave the magneticfield if they still possess kinetic energy at r = 0 and hence only for ǫ > V m R according to ourprevious definition of ǫ . Since we are not further interested in the fate of hfs neutrons unableto escape the multipolar field after opening the UCN valve, and for simplicity of the equationsto follow, we define ǫ in the sequel commonly for both spin states as the kinetic energy in thepoint of lowest magnetic field (i.e. on the line r = 0). Equation 3 thus describes, individually forboth spin states, UCN densities of useable UCN (i.e. those with ǫ > V trap as definedin eq. 7. We define the polarization of the saturated ensemble of useable UCN as P ∞ = n ∞ , lfs − n ∞ , hfs n ∞ , lfs + n ∞ , hfs . (9)As shown in Fig. 2 for two situations it will depend on the relative strengths of magnetic andneutron optical potentials. The inverse of the time constant appearing in the loss term in eq. 1 is comprized of severalcontributions, τ − ( ǫ , T ) = τ − ( ǫ ) + τ − ( ǫ ) + τ − ( T ) + τ − + τ − ( ǫ ) + τ − β , (10)where the argument indicates now also the dependence on the temperature T of the converter.From left to right, they describe UCN loss at collisions with the walls of the converter vessel,escape of UCN through an imperfectly closed UCN valve and through slits caused by manufac-turing imperfections of the vessel assembly, upscattering by thermal excitations in the helium,absorption by He impurities, UCN depolarization at wall collisions, and neutron beta decay.5igure 2: Schematic of the magnetic and neutron optical potentials in the closed UCN accumu-lator shown in Fig. 1. In the upper figure, V − V He > V m R . Low field seeking (lfs) UCN withtotal energy ǫ < e V − V He are trapped (solid line shade). High field seeking (hfs) UCN with0 < ǫ < V − V He − V m R are trapped, too (dashed-line shade), leading to polarization P ∞ < V − V He < V m R where hfs UCN with ǫ > P ∞ = 1. Only hfs UCN with ǫ < τ − maybecome relevant only if trapping is at least partly magnetic. In an experimental study a de-polarization probability per wall collision of 7 × − was measured for a bottle made of Be[72]. Hence, τ − < τ − β even for the fastest neutrons in the narrow trap geometry discussedhere. Like the first two rate constants in eq. 10, τ − will be further suppressed due to themultipole magnet (provided that spin transport is adiabatic), and we therefore neglect it. Fortemperatures below 1 K [69, 73], τ − ( T ) ≈ ( T [K])
100 s , (11)so that for T < . τ − contributes with less than 10% of τ − β . The rate constant τ − canbe suppressed below any relevant level by purification of the helium from He using a superleak[61, 74, 75] or the heat flush technique [76]. As a result, we are left with τ − and τ − asdominating contributions, and the rate constant due to neutron decay sets an ultimate lowerlimit for a perfect converter.For the losses due to wall collisions we want to apply an analytic description. If we assumethe trapped UCN in mechanical equilibrium we can use formulas derived in the book [4] wherethe authors analyzed the effect of the earth’s gravitational field on neutrons moving in a bottle.We adapt the notation to our case and replace the height parameter h by the radial coordinate r characterizing the multipolar magnetic field strength. We neglect the gravitational field, whichis a good approximation for a horizontal source with less than 10 cm diameter. The kineticneutron energy for the two spin states is then given by E ( r ) = ǫ ∓ | V m ( r ) | . The energy, E := E ( R ) = ǫ ∓ V m R , (12)is positive for the extractable hfs neutrons which can always explore the whole trap. The lowfield seeking neutrons on the other hand may have too low energy to hit the walls. For E > µ ( E ) for UCN loss during a collision with the cylindrical wall of the heliumfilled converter vessel can be written as µ ( E ) = 2 f Re V ′ E arcsin r EV ′ − r V ′ E − ! for E ≤ V ′ , (13)valid for a neutron optical potential V + iW with small losses, i.e. f = W/V ≪
1, and writing V ′ = V − V He , (14)with V He given in eq. 8. For convenience in later calculations we have included a projection ontothe real part of the expression. It offers a handy formulation of the case where neutrons havetoo low energy to hit the wall, ensuring that µ ( E <
0) = 0 without need to specify the rangeof E in advance as positive. The function µ ( E ) rises monotonously with E from µ = 0 for E = 0 to µ = πf for E = V ′ . For E > V ′ we may set µ = 1 since we are not interested here incalculating the dynamics of marginally trapped neutrons. For E = V ′ / µ ≈ . f . Note that,since we consider a long trap for which 2 πRL ≫ πR , we will neglect losses due to f W at theend disks. The magnetic multipole suppresses wall losses of lfs UCN for several reasons. First,only a fraction of them has sufficient energy to hit the lossy wall. Second, those lfs neutronswith ǫ > V m R hit the wall due to magnetic deceleration with a reduced kinetic energy E (eq.12), leading to reduced losses due to µ ( ǫ − V m R ) < µ ( ǫ ). Third, the rate of wall collisions ofthese neutrons is reduced as well, leading to a further suppression in the expressions for τ − and τ − , which we calculate next. 7 / ǫ V m R
110 15 12
43 85
24 0 .
005 0 .
021 0 .
133 0 .
533 0 .
696 0 .
758 0 . .
067 0 .
133 0 .
333 0 .
667 0 .
778 0 .
822 0 . .
159 0 .
252 0 .
465 0 .
739 0 .
824 0 .
859 0 . .
248 0 .
351 0 .
555 0 .
785 0 .
855 0 .
883 0 . .
325 0 .
429 0 .
619 0 .
818 0 .
876 0 . . .
39 0 .
492 0 .
667 0 .
841 0 .
892 0 .
913 0 . γ ′ for low field seeking neutrons, as defined with the upper sign in eq. 18 forvarious values of ǫ /V m R .In mechanical equilibrium, a group of neutrons with total energy in the range ( ǫ , ǫ + dǫ )will occupy the accessible phase space in the trap with uniform density. As a result of phasespace transformation under the influence of a conservative potential (see section 4.3.1 in thebook [4]), the real space spectral UCN densities in different positions are related by n ( ǫ , t, r ) = Re s ǫ ∓ | V m ( r ) | ǫ n ( ǫ , t, , (15)where projection onto the real part ensures n ( ǫ , t, r ) = 0 for the lfs UCN for r > R ∗ defined by ǫ = | V m ( R ∗ ) | . We define an effective volume of the source for neutrons with total energy ǫ as γ ( ǫ ) = 2 πL Re Z R s ǫ ∓ | V m ( r ) | ǫ rdr. (16)The spectral UCN density averaged over the entire volume of the converter is then given by n ( ǫ , t ) = γ ′ ( ǫ ) n ( ǫ , t, , (17)and the reduced quantity, γ ′ ( ǫ ) = γ ( ǫ ) πR L = 2 Re Z r ∓ V m R ǫ r n − rdr, (18)was derived using eq. 6. Values for γ ′ listed in Table 1 show that, the higher the multipolarity,the less significant becomes the reduction of the density of the lfs neutrons with respect to asquare well potential of same depth. The spectral current density of neutrons at any point inthe vessel, per unit area and per energy interval about ǫ , is given by the gas kinetic relation J ( ǫ , t, r ) = 14 n ( ǫ , t, r ) v ( ǫ , r ) . (19)The spectral rate of UCN collisions with the cylindrical wall of the helium container is given by2 πRLJ ( ǫ , t, R ). The speed v ( ǫ , R ) of the neutrons as they hit the wall is related to the speedat r = 0 through v ( ǫ , R ) = Re r ǫ ∓ V m R ǫ v ( ǫ , . (20)With eq. 15 we obtain J ( ǫ , t, R ) = ǫ ∓ V m R ǫ J ( ǫ , t,
0) Θ ( ǫ ∓ V m R ) , (21)8ith the step function Θ ( x ) as already used in eq. 7. For the loss term in eq. 1 due to collisionswith the cylindrical wall we can thus write n ( ǫ , t ) τ wall ( ǫ ) = 2 R µ ( ǫ ∓ V m R ) J ( ǫ , t, R ) . (22)Further evaluation using eq. 21, eq. 19 for r = 0, eq. 17, and inserting µ from eq. 13 leads us to τ − ( ǫ ) = v ( ǫ , Rγ ′ ( ǫ ) f V ′ ǫ Re arcsin r ǫ ∓ V m R V ′ − s ǫ ∓ V m R V ′ (cid:18) − ǫ ∓ V m R V ′ (cid:19)! . (23)For the calculation of the corresponding expression for losses through slits it is reasonableto assume them to be situated at r = R , e.g. at the seam of the tube or at its connections tothe circular windows for the cold beam. Assuming that any UCN hitting a slit will be lost anddenoting the total surface of all slits by A , their contribution to the loss term in eq. 1 is givenby n ( ǫ , t ) τ slit ( ǫ ) = 2 R A πRL J ( ǫ , t, R ) , (24)neglecting the small surface of the disks at the ends. Hence, τ − ( ǫ ) = Av ( ǫ , γ ( ǫ ) ǫ ∓ V m R ǫ Θ ( ǫ ∓ V m R ) , (25)and we see that, for the lfs neutrons, the ordinary gas kinetic expression represented in the firstfraction on the right side becomes reduced for V m R > We first consider UCN production in absence of the magnetic multipole field. For homogeneousirradiation with the cold neutron beam guided through the converter, neglecting decrease ofintensity due to reflection losses and neutron scattering in the helium, the UCN production ratedensity is position independent and given by p = Z V trap p ( ǫ ) dǫ = KV / . (26)The V / dependence follows for a homogeneous population of states within a sphere in momen-tum space with spectral UCN production rate density p ( ǫ ) = 32 K √ ǫ . (27)The factor K due to single phonon emission has been calculated on the basis of neutron scat-tering data and confirmed in several experiments [77, 62, 78, 79, 80, 81, 82], albeit with modestexperimental accuracy limited by detection efficiency and other corrections. For UCN withmaximum energy determined by V trap = V − V He ≈
233 neV for Be or Ni with natural isotopiccomposition, it is given by K ≈ − cm − Φ . (cid:2) cm − s − nm − (cid:3) / (233neV) / , (28)where Φ . is the differential unpolarized neutron flux density at a neutron wavelength of0 .
89 nm. The flux unit is chosen numerically close to values available at existing facilities, e.g.9 / V m R V trap
34 58 12 .
229 0 .
37 0 .
456 0 . . .
517 0 .
585 0 . .
512 0 .
609 0 .
665 0 . .
589 0 .
672 0 .
72 0 . .
645 0 .
718 0 .
759 0 . .
688 0 .
752 0 .
788 0 . κ as defined in eq. 32 for various values of n and V m R /V trap .the monochromatic beam H172A at the ILL [47]. An additional, usually smaller contributionto UCN production is due to multi-phonon processes.When adding the multipolar magnetic field, the spectral UCN production rate density be-comes dependent on position and spin state, p lfs(hfs) ( ǫ , r ) = 34 K ( r ) Re p ǫ ∓ | V m ( r ) | , (29)where the r dependence of K accounts for a spatially varying flux density of the neutron beam.The factor 1 / K ( r ) = K , and the spatially averagedspectral UCN production rate density for the two spin states can be expressed in terms of thenormalized effective volume from eq. 18, i.e. p lfs(hfs) ( ǫ ) = 34 Kγ ′ ( ǫ ) √ ǫ . (30)Without magnetic field, p , lfs = p , hfs = p / p given in eq. 26, whereas with field, p lfs(hfs) = Z V trap p lfs(hfs) ( ǫ ) dǫ (31)are different due to the spin-dependent γ ′ ( ǫ ) and V trap from eq. 7. We note particularly thatthe ratio of total production rates for lfs UCN with the magnetic multipole switched on andswitched off, κ = p lfs p , lfs , (32)is smaller than unity due to the phase space reduction by the magnetic multipole. The valuesquoted in Table 2 demonstrate a positive effect of high multipolar order n on κ and hence onthe saturated UCN density calculated in the next section. There are however practical limits.First, thermal insulation between the magnet and the much colder helium container necessitatesan annular gap over which the field would drop too strongly if n is chosen too large. Second,the maximum field strength achievable with a given maximum current density in the currentbars around the converter of given diameter decreases with n . For R = 5 cm, and taking intoaccount the results given in the next section, n ≈
12 turns out to be a reasonable choice.
The spatially averaged saturated spectral densities for lfs and hfs UCN follow from eq. 2 witheq. 30, i.e. n ∞ , lfs(hfs) ( ǫ ) = 34 Kγ ′ ( ǫ ) √ ǫ τ ( ǫ ) , (33)10 ,0 0,2 0,4 0,6 0,8 1,0050010001500 n , h f s , n , l f s ( c m - ) V mR /(V V He ) f (s) 3x10 -6
711 3x10 -5
319 5x10 -5
227 1x10 -4
131 2x10 -4 lfshfs Figure 3: Saturated densities of low field and high field seeking UCN in a converter vesselwith diameter 10 cm, held at 0 . e V = V = 252 neV (e.g. for a converter vessel made entirelyof Be), and for various values of f = W/V . The solid lines show results for the best valueof f previously achieved for Be, while the upmost curves show the situation for unrealisticallylow f for illustration. Values are given for an unpolarized differential neutron flux density ofΦ . = 10 cm − s − nm − at λ = 0 .
89 nm, as available at the monochromatic beam positionH172A at the ILL. A characteristic time constant τ is calculated for neutrons with velocity v = v max , including in eq. 10 the rates τ − β , τ − and the wall collisional losses calculated usingeq. 23 for V m R = 0 and ǫ = V trap .using the corresponding sign in eqs. 18 and in the expression for τ ( ǫ ). Hence, using eq. 3and writing out all arguments relevant for characterizing the multipolar magnetic field and theconverter, the saturated total mean UCN densities in the converter are given by n ∞ , lfs(hfs) (cid:16) R, V m R , n, V, f, e V , T (cid:17) = 34 K Z V trap γ ′ ( ǫ , V m R , n ) √ ǫ τ − ( ǫ , R, V m R , n, V, f, T ) dǫ . (34)The dependence on e V is contained in the upper limit of integration, see eq. 7. From the variouscontributions to the rate constant τ − (see eq. 10) we retain the terms due to wall collisions,upscattering (eq. 11) and neutron beta decay, assuming that the wall losses can entirely bedescribed by eq. 23 and that there is no He in the converter and no slit in the vessel.A first calculation of n ∞ , lfs and n ∞ , hfs was performed for a vessel featuring the neutron opticalpotential of Be at all walls (i.e. e V = V = 252 neV). Beryllium has become a standard materialfor UCN trapping, with a best reported experimental value of f = 3 × − in the low temper-ature limit [83, 84], despite a much smaller theoretical value (the finding that this was never11eached was termed ”anomalous losses” and has triggered many experimental investigations andspeculations). However, it might be more realistic to consider also worse values for f , assumingthat efficient cleaning procedures cannot be applied in situ (e.g. baking is excluded in presenceof indium seals). Figure 3 shows results exemplary for multipole order n = 8 and n = 12, as afunction of the magnetic trapping potential at the cylindrical wall of the vessel, normalized tothe trapping depth without magnetic field. The density of lfs UCN increases with n as expecteddue to the increase in trapping phase space, while that of the hfs neutrons decreases. Hence,for partial magnetic trapping of the lfs neutrons (characterized by V m R < V − V He , see Fig. 2),higher multipole order leads to higher UCN polarization defined in eq. 9. For instance, for a12-pole with B R = 2 . P ∞ = 86% for f = 3 × − while for a worse f = 2 × − it improves to P ∞ = 97%. As also obvious from the curves, the poorer the neutron optical UCNstorage performance, the larger will be the improvement of lfs UCN density due to the multipolemagnet. For the experimental cases reported in the introduction, with measured loss rate ratiosas high as τ − /τ − β ≃ . n ∞ , lfs exhibit a maximum for values V m R /V trap <
1. This can beunderstood as resulting from the competition of storage time constant τ and the effective trapvolume γ ′ entering eq. 34. For bad values of f the optimum obtains for V m R /V trap close to 1,while for a trap with excellent storage properties the multipole field reduces the UCN densityeven at low field values because the factor γ ′ < τ .For illustration of this behaviour we also added a curve for an unrealistic converter vessel withhypothetical f = 3 × − .Next we consider an interesting further opportunity for buildup of a high lfs UCN densitywhich takes advantage of the fact that the multipole magnet increases not only storage timeconstants but also the potential energy of the neutron at the cylindrical wall. As a result, thetrapping depth of the converter vessel becomes larger if the disks providing axial confinement aremade of a material with larger neutron optical potential e V > V (remember eq. 7 and see Fig. 1).Since the surface of the disks is small, one may even employ materials which would be unsuitablefor the entire vessel, for an unfavorably large e f = f W / e V or because coating the tubular sectionwith sufficient quality might be unavailable. While diamondlike carbon has already been studiedin some detail [85, 86, 87, 88], further candidate materials able to extend the spectrum for UCNtrapping beyond the Be cutoff have been the scope of recent investigations [89]. Particularlypromising is boron nitride in the cubic phase (cBN). Its neutron optical potential of 324 neV iseven larger than that of diamond (304 neV) but due to the large absorption cross section of theisotope B, e f = 1 . × − is also excessively large. Enrichment of the weakly absorbing Bhowever may reduce e f down to 3 . × − , along with a further increase of e V to a theoretical valueof 351 neV. Using experiments on transmission with time-of-flight analysis and cold neutronreflectometry, the authors of ref. [89] have demonstrated a value of 305 ±
15 neV for their 2 µ m thick deposit of cBN (with natural isotopic composition) on a circular silicon waver. Thedeviation from the ideal value is due to a cubic phase content of 90%, which was measuredindependently by IR spectroscopy. For highly enriched material, and assuming the same cubicphase content, one may expect a neutron optical potential of about 330 neV.Figure 4 shows saturated UCN densities calculated for a trap with a Be-coated cylindricalwall with f = 3 × − and with the axial UCN confinement provided by Be, BN (90% cubic),or cubic C N . The latter features an extraordinarily large theoretical value of e V = 391 neV.The curves for e V > V for the lfs UCN start with a slope larger than in the case e V = V (dash-dotted). This is due to a B R dependent increase of the integration range, V trap , in eq. 34,as long as V m R < e V − V . Kinks in the curves appear at magnetic field values correspondingto V m R = e V − V where the full trapping depth is reached. While for a high quality Be trapwith e V = V = 252 neV the gain in UCN density is not too impressive (lowest two curves), for e V = 330 neV a magnetic 12-pole with B R = 2 . n , h f s , n , l f s ( c m - ) B R (T) C N , 100% cubic, 391 neV BN, 90% cubic, 330 neV
Be, 252 neV lfs
Figure 4: Saturated densities of low field and high field seeking UCN in converter vessels withdiameter 10 cm, V = 252 neV, f = 3 × − , T = 0 . e V as indicated in the legend. Thethicker of each pair of curves is for multipolarity n = 12, the thinner for n = 8, and all valuesare given for Φ . = 10 cm − s − nm − (unpolarized). The kinks visible for the two upperpairs of curves appear at field values corresponding to V m R = e V − V . Densities of the hfs UCNfor e V ≥ V are independent on e V but are smaller for the higher multipole order.13 -5 f = 2x10 -4 n , h f s , n , l f s ( c m - ) B R (T) Figure 5: Saturated densities of low field and high field seeking UCN in converter vessels withdiameter 10 cm, V = 252 neV, T = 0 . f = 3 × − (solid curves) or f = 2 × − (dashed). Beam window and UCN valve (see Fig. 1) are coated with c BN with e V = 330neV. The magnetic multipole order is varied between n = 6 and 24. Values are given forΦ . = 10 cm − s − nm − (unpolarized). 14 V (neV) B R (T) f × − × − × − × −
330 2 . n ∞ , lfs (cid:0) cm − (cid:1) P ∞ .
89 0 .
949 0 .
969 0 . . n ∞ , lfs (cid:0) cm − (cid:1) P ∞ .
86 0 .
939 0 .
965 0 . ≥
252 0 n ∞ , lfs (cid:0) cm − (cid:1)
820 390 230 130 P ∞ R = 5 cm, T = 0 . f . Values are given for an unpolarized differential flux densityof Φ . = 10 cm − s − nm − .density n ∞ , lfs by a factor 2 . − to 1880 cm − . Hence, if the experiment connectedto the source can use the high-energy UCN it provides, the magnetic multipole is an asset evenfor a vessel with very good storage properties. Note that the saturated high field seeker UCNdensity n ∞ , hfs for e V ≥ V does not depend on e V , since for them the trapping potential is givenby V − V m R (see upper part of Fig. 2). Hence, the larger e V , the larger will be the polarization P ∞ defined in eq. 9 (see also Table 3). Note also that, as sketched in the lower part of Fig. 2, formagnetic fields providing a trapping potential stronger than the neutron optical one ( B R & . P ∞ = 1.Figure 5 shows the dependence of saturated UCN density on the multipole order, for trapswith Be-coated cylindrical wall with f = 3 × − , respectively f = 2 × − , each with endwindows with a potential of e V = 330 neV. Again, one notes the positive influence of highermultipole order on the number of trapped UCN and on the polarization. Table 3 quotes values for n ∞ , lfs and P ∞ for n = 12 and for various values of f . One can see, for instance for f = 2 × − ,that the magnetic field enhances n ∞ , lfs by more than a factor five from 230 cm − to 1210 cm − .One also notes that the multipole field stabilizes the output of the source, by mitigating theinfluence of the loss coefficient of the converter wall surface. This includes possible deteriorationof the wall quality with time, which will then also be much less an issue than without the field.For application of the source for feeding a magnetic trap, e.g. for neutron lifetime experimentswith typical trapping potentials in the range (50 − n ∞ , lfs on the upper bound of the trappedUCN energy spectrum. Figure 6 shows this dependence, for traps with n = 12 and again withthe cylindrical section made of Be with f = 3 × − , respectively f = 2 × − . Values arecalculated using eq. 34 with the potential e V − V He set to different values starting from 60 neVand increased in steps of 20 neV. We see that, the lower e V , the lower will be the magneticfield needed to optimize the UCN density. The reason is that lowering e V reduces wall collisionallosses due to the energy dependence of µ ( E ) defined in eq. 13 and due to wall hits occuring at asmaller rate, whereas the effective volume γ ′ decreases quickly with V m R for a low-energy UCNspectrum (see Table 1). The solid curves in Fig. 6 tell us that, for the converter vessel coatedwith an excellent Be mirror, f = 3 × − , the multipole magnet will offer some advantage onlyfor not too low UCN cutoff energy. However, for a more realistic situation, f = 2 × − , gainsdue to the magnet are rather significant even for low-energy UCN spectra. For example, forfeeding an external trap with trapping depth 60 neV, it will improve n ∞ , lfs by a factor 2 at B R ≈ . . B R ≈ .
8T for V trap = 120 neV. 15 -5 f = 2x10 -4 n , l f s ( c m - ) B R (T) Figure 6: Saturated density of low field seeking UCN in converter vessels with diameter 10 cm, V = 252 neV, T = 0 . f = 3 × − ,dashed ones for f = 2 × − . e V − V He is varied between 60 neV and 311 neV. Values are givenfor Φ . = 10 cm − s − nm − (unpolarized).16 Conclusions
As our analysis shows, a multipole magnet can lead to a large gain in the saturated density oflow field seeking UCN because the presence of the field reduces the number of neutrons hittingthe material walls and reduces the energy and wall collision rate of those that do. In addition, itacts as a source-intrinsic UCN polarizer without need to polarize the incident beam and henceavoiding associated losses. A 12-pole magnet with field B R = 2 . R = 5 cmseems technically feasible using standard NbTi superconducting wire technology, as investigatedin an independent study using a finite element code. Based on results of experimental workdone by other groups, a promissing candidate vessel able to provide a UCN spectrum withexceptionally high cutoff consists of a Be trap closed off by disks coated with c BN. Alternativematerials are diamondlike carbon with V close to 300 neV depending on the abundance of sp chemical bonds, or enriched Ni with V = 346 neV and a theoretical f = 8 . × − , whichhowever is magnetic so that UCN depolarization might be an issue that needs experimentalstudy. We note that, in order to extract the full benefits, the incoming cold beam will needto be transported by a supermirror guide, with a top layer deposit of a good UCN reflectingmaterial with neutron optical potential V . An experimental study of a UCN source prototypeinvolving a converter vessel coated with such type of mirror is currently underway at the ILL, inpreparation of the UCN source project SuperSUN which will include a 12-pole magnet arounda 3 m long cylindrical converter vessel.Our benchmark converter is able to provide a saturated low field seeker UCN density almostas high as an unrealistic, perfect trap with f = 0 and Be cutoff, for which one calculates n ∞ , lfs = 2060 cm − when exposed to a neutron beam with differential flux density Φ . =10 cm − s − nm − . For a pure Be trap equipped with the 12-pole magnet one calculates n ∞ , lfs =1380 cm − for f = 3 × − as previously achieved for this material, while for f ten timesworse, we still obtain n ∞ , lfs = 1110 cm − . Without magnet on the other hand, n ∞ , lfs wouldget suppressed by a factor five, which impressively demonstrates the capability of the magneticmultipole reflector to mitigate the influence of a poor loss coefficient f of the converter wallsurface. Obviously, also if loss of quality of the inner converter surface with time will be anissue, the magnet offers a valuable practical advantage.Including the high field seekers into the discussion, one first notes that without magnetic fieldthey are equally well trapped, so that in this case (and still assuming the usual situation of anunpolarized cold neutron beam for UCN production) the total UCN density in the source will bea factor two higher. However, for experiments requiring polarized UCN such as magnetic trapsfor precise determination of the neutron lifetime, or the neutron EDM experiment, this factor isof no use. Values for polarization of the trapped ensemble of UCN after saturation of the sourcewere quoted in Table 3 and are typically well beyond 90% for the system discussed. As obviousfrom Fig. 2, low-energy UCN may stay poorly polarized whereas for the high-energy part ofthe spectrum the high field seekers, after magnetic acceleration to the cylindrical wall, will havekinetic energy beyond its cutoff potential and quickly get lost. Poor polarization is not a problemfor experiments using magnetic traps which can be designed for quick cleaning out the wrongspin component. For experiments that would profit from a very high initial polarization onemight coat the converter vessel with a low-loss material with potential V < V m R , which wouldlead to nearly 100% UCN polarization since the high field seekers would stay untrapped. For asystem with V > V m R one may still, if needed, increase the polarization by delayed extractionof the UCN after having switched off the saturating neutron beam. The high field seekers willthen quicker leave the trap than the low field seekers due to shorter trapping time constants.One might also cut out the lowest-energy part of the spectrum by a vertical UCN guide sectionwith suitably chosen length.As an important detail not affecting our conclusions we note that, in addition to the multipole17eld, it will be necessary to apply a bias field in the order of some 10 mT along the converteraxis to avoid depolarization in the region around r = 0 , where the multipole field is zero. Inaddition, we can consider using axial magnetic pinch fields to increase V trap and thereby thedensity of the UCN for an extended energy spectrum. For extraction, the field at one end needsto be ramped down, so that an iris type UCN valve might be most appropriate for this case. Anextended UCN spectrum would be interesting if UCN of any velocity were beneficial, such as inUCN transmission experiments, or in combination with a phase space transformation by lettingthe UCN rise against the gravitational field. Note however that low-loss extraction of such aUCN spectrum will be a challenge. On the other hand, some studies might be performed in situusing static pinch fields, such as experiments on UCN upscattering in superfluid He, for whichany increase in UCN density will be very welcome.
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