- Email: [email protected]
A compound parabolic concentrator as an ultracold neutron spectrometer
Nuclear Instruments and Methods in Physics Research A 721 (2013) 60–64
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
A compound parabolic concentrator as an ultracold neutron spectrometer K.P. Hickerson n, B.W. Filippone California Institute of Technology, United States
art ic l e i nf o
a b s t r a c t
Article history: Received 7 January 2013 Received in revised form 22 March 2013 Accepted 23 March 2013 Available online 3 April 2013
The design principles of nonimaging optics are applied to ultracold neutrons (UCN). In particular a vertical compound parabolic concentrator (CPC) that efficiently redirects UCN vertically into a bounded spatial volume where they have a maximum energy mga that depends only on the initial phase space cross sectional area πa2 creates a spectrometer which can be applied to neutron lifetime and gravitational quantum state experiments. & 2013 Elsevier B.V. All rights reserved.
Keywords: Ultracold neutrons Nonimaging optics
1. Introduction Compound parabolic and elliptical concentrators, designed using the edge ray principle familiar from nonimaging optics [1], have been used with success to collimate, focus and concentrate cold neutrons onto a distant target using the approximation that the neutrons travel in a straight line, as is the case with light [2,3]. In the ultracold limit of the neutron, around a few hundred neV, these straight line reflectors suffer from chromatic aberration as the kinetic energy and gravitational potential approach equal magnitude. In previous uses, this chromatic aberration has been detrimental, particularly for grazing-angle, sideways pointing concentrators as only a narrow portion of the available phase space reaches the target, and the energy spectrum is heavily distorted. While UCN microscopes have been designed using imaging optics to eliminate chromatic aberration [4,5], to date nonimaging concentrators have not been used to measure UCN spectra because as the scale of the concentrator approaches the scale of the curvature of the flight paths, a standard concentrator on its side destroys the spectral information. In this paper we will show that certain cases of these concentrators can be designed for spectroscopy in the UCN limit if the gravitational curvature of the neutrons is taken into account. While previous gravitational spectrometers have achieved remarkable energy resolutions of peV [6], they have done so at the cost of UCN number efficiency by removing unused phase space volume or waiting for UCN to defuse into the measured state from a storage vessel. We show that the principles familiar from nonimaging optics can be applied to UCN optics to design an efficient vertical spectrometer, which
utilizes much of this unused phase space on a first pass, in a time much shorter than the neutron lifetime, or more appropriately, the vessel storage time. As a prime example, we investigate a unique case of a vertical compound parabolic concentrator (CPC) which can quickly isolate UCN in bands as narrow as 13mga FWHM for a guide radius a. For 6 cm diameter guides, this gives 1 neV resolution, achievable after one pass through the optical system in the apogee time of the UCN, v0 =g. Such a CPC spectrometer can be used for a number of new experiments such as measuring the neutron lifetime using UCN. 2. Nonimaging UCN optics In imaging neutron optics, each imaged neutron path is determined by Fermat's principle which also coincides with the classical action principle Z b Z tb mv2 δ dt ¼ δ L dt ¼ 0; ð1Þ ℏ a ta so that the advancing wavefronts and the classical paths also coincide [7,8]. We may consider all potentials, gravitational, magnetic and the Fermi potential, as affecting UCN via an effective index of refraction λ2 ! ! n2 ð r Þ ¼ 1− Vð r Þ: 2mℏ2
For a rotationally symmetric system with angular momentum ℓ, we have Vðr; zÞ ¼ mgz 7 μn Bðr; zÞ þ
n
Corresponding author. Tel.: +1 6266441976. E-mail addresses: [email protected], [email protected] (K.P. Hickerson), [email protected] (B.W. Filippone). 0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.03.049
ð2Þ
ℓ2 þ V F ðr; zÞ: 2mr 2
ð3Þ
For an imaging system, the integral from Fermat's principle is stationary for all neutron paths from object aperture to the image
K.P. Hickerson, B.W. Filippone / Nuclear Instruments and Methods in Physics Research A 721 (2013) 60–64
so we can solve for imaging optical surfaces using Z b δ n2 dt ¼ 0: a
61
ð4Þ
To design a nonimaging optical system, we relax the requirement that this is satisfied for all paths emanating from the input aperture. Not every point in the input aperture must have a conjugate point in the target space. Instead we rely on the edge ray principle familiar from nonimaging photon optics [1] that states that imaged paths serve only as the boundary for the phase space volume of all other enclosed paths. We use Fermat's principle only to solve for the classical edge paths and pick a reflector surface or potential geometry parameters that map the input aperture extrema to the extrema of the output region. All paths within that imaged path boundary will be guaranteed to arrive at the target region, regardless of the path taken, the number of reflections, or the path complexity.
3. Compound parabolic concentrators
Fig. 1. A CPC with a slice cut out.
From nonimaging optics for noncurvilinear rays, the general CPC family has reflective walls of a parabola that are tilted by the acceptance angle θA to the axis of rotation. The traditional CPC design [1] with an aperture at z ¼0 radius a, has the parametric form 2a′ sinðφ−θA Þ −a; 1−cos φ 2að1 þ sin θA Þ cosðφ−θA Þ zðφÞ ¼ ; 1−cos φ rðφÞ ¼
ð5Þ
where a′ is the focal length of the parabola. For nonzero θA , we truncate the CPC to a height of ða′ þ aÞcot θA . When we take the limit θA -0 we find the CPC height as defined in [1] diverges and all UCN paths are contained inside the CPC. In this limit, a′-a and the equation for a vertical CPC becomes zðrÞ ¼
1 ðr þ aÞ2 −a: 4a
ð6Þ
The normal to the wall at the point (r,z) is ð−r−a; 2aÞ n^ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : r 2 þ 2ar þ 5a2
ð7Þ
In this limit, all orbits of the same velocity originating from the input aperture are bound between extrema of the “neutron fountain” caustic, independent of their initial conditions. An interesting property that has been exploited in neutron optics is that a gravitational parabolic path originating at the focus of a parabolic surface will reflect to a conjugate parabolic path that also intersects the focus. Steyerl [9] called this property the “neutron fountain.” As a consequence, points in the neighborhood of the focus of a paraboloid are self-conjugate so the focal plane is imaged back on to itself. Steyerl and Frank used this property to design imaging systems and microscopes [10]. We use this property to show that a compound parabola can efficiently redirect UCN upward as with UCN microscopes but rather than rotate the parabola about its own axis as is done with UCN imaging optics, we place the focus of one parabola coincident with a reflected parabola, forming a compound parabola, and then rotate about the axis of symmetry (see Figs. 1 and 2). For 2D massive particles, it is not generally true that all imaged paths take equal time [7,8], but for the case of the “neutron fountain”, the orbit time from the focus to the opposite parabola wall and reflecting back, is given by ∮ dt ¼
2va g
ð8Þ
Fig. 2. The construction of a CPC using the “neutron fountain” property. For each particle with v ¼ v0 , originating from the focus, these edge orbits are stationary to variations from Fermat's principle and to orbital period. Examples shown are (1) a vertical path from the focus, (2) a focus to focus reflection orbit, pffiffiffi(3) the selfconjugate orbit, (4) an orbit with initial velocity with vr ¼ vz ¼ v0 = 2, and (5) the classical limit of an path originating at the aperture at z¼ 0. The compound parabola is formed by (7) and the reflection (8). In region I, with z 4 v20 =g, UCN are classically forbidden. All UCN from z ¼ 0; r∈½0; a will reach region II. And region III is bounded by the extrema of the edge orbits below 6, the ballistic umbrella of the point f.
where v2a ≡v20 þ 2ag, where v0 is the initial velocity. This can be compared to a vertical orbit which has orbit time 2v20 =g. This time depends only on the parabola's focal length a and the UCN initial velocity magnitude v0, and is independent of the initial angle to the vertical axis. For the flight path from the focus (z¼0) with an initial velocity ðvr ; vz Þ ¼ ðv0 sin θ; v0 cos θÞ, the time to reach the parabolic reflector from the focus is 2a : va −vz
ð9Þ
For each initial vertical velocity, vz ¼ v0 cos θ, for a path starting at focus z ¼0 and angle θ from the vertical axis, there is a reflected
62
K.P. Hickerson, B.W. Filippone / Nuclear Instruments and Methods in Physics Research A 721 (2013) 60–64
path with initial conditions v′z ¼ v0 cos θ′ that intersects the same point on the parabola. As we learned from the “neutron fountain” property, the path starting at the focus will reflect back onto the focus, so the orbit time in Eq. (8) is given by the sum of the two paths 2va 2a 2a ¼ þ : va −vz va −v′z g
ð10Þ
An extremum occurs at vz ¼ v′z ¼ v20 =va . These self-conjugate paths intersect the parabola at a height of v20 =2g−a. This reveals the interesting fact that the kinetic energy spread of the UCN orbits at their apogee is ΔE o mga and is independent of v0. Further, by the edge ray principle, all UCN emanating from the aperture necessarily will be directed to enter region II in Fig. 2. Region II is bounded below by the “ballistic umbrella”, the extrema of all monochromatic UCN paths originating at the parabola focus given by zðrÞ ¼ h−
1 ðr þ aÞ2 ; 4h
h≡
v20 ; g
ð11Þ
and bounded above by the classically forbidden height, z ≤v20 =2g. We are interested in directing UCN vertically and the general solution of the intersection of the CPC wall with the UCN path can be solved in three dimensions with radial symmetry. This allows us to model the optical system in only two dimensions using cylindrical coordinates. Without magnetic fields, the radially symmetric Euler–Lagrange equations are separable with vertical and radial solutions zðtÞ ¼ 12gt 2 þ v0 t þ z0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðtÞ ¼ v2ϕ t 2 þ ðvr t þ r 0 Þ2 :
in Eq. (16) and then iteratively solves the cubic form, which removes the t¼0 solution, iteratively and propagates the path with specular reflections off the CPC wall. This simulation demonstrates that all UCN incident on the input aperture located on the interval r∈½0; aÞ and z ¼0 are redirected to the vertical region z∈½v2 =2g−a; v2 =2g and each UCN path passes through region II, bound by z ≤v20 =2g and Eq. (11) as predicted. We validated the simulation algorithm with a calculation of the simple case of a cylindrical wall with radius a instead of the CPC and compared to the analytic result of the probability distribution for a monochromatic random gas with initial velocity v0, which has the distribution sffiffiffiffiffiffiffi v2 g ð17Þ ; 0 ≤z ≤ 0 : PðzÞ ¼ 2 2g v0 z
ð12Þ
5. Results In Fig. 3 we show the results of a small sample of UCN paths projected onto the ðr; zÞ plane. The free part of the orbits is hyperbolic along the r-axis due to conserved angular momentum, and parabolic in the vertical axis due to gravity. The multiple specular reflections off the CPC walls redirect all UCN into region II as predicted. We ran batches of one million UCN in 5 neV increments with energies ranging from 5 to 40 neV. In Fig. 4, each probability distribution shows UCN are restricted to a horizontal band as wide
We need only solve for the intersection with the reflecting surface −12 g′t 2 þ v′z t þ z′0 ¼ 12rðtÞ;
ð13Þ
where we have defined g′≡
v2a −v2z ; 2a
v′z ≡vz −vr χ;
ð14Þ
and z′0 ≡z0 þ aðχ 2 þ 34Þ;
ð15Þ
where χ ¼ r 0 =2a is a dimensionless parameter. Squaring Eq. (13) gives the quartic equation with coefficients A ¼ g ′2 ;
B ¼ −4g′v′z ;
D ¼ 8v′z z′0 −2vr r 0 ;
2 2 C ¼ 4v′2 z −4g′z′0 −vr −vϕ ;
2 E ¼ 4z′2 0 −r 0 :
ð16Þ
Generally, we set the initial conditions so that ðr 0 ; z0 Þ is at the entrance aperture at z0 ¼ 0 and r 0 ∈½0; aÞ, but the full solution is inappropriate if we have already solved for one intersection and are solving for another on the CPC surface. We need to remove the t¼0 solution from the quartic for all intersections ðr n ; zn Þ for later times t 4 0. This reduces computation time and numerical errors. When ðr n ; zn Þ lay on the wall they are dependent through Eq. (6) when z′n -12r 0 and E ¼ 0. The reflections also allow an iterative procedure to compute r nþ1 ; znþ1 from ðr n ; zn Þ. However, the computation is nonlinear, as the root of a cubic, and n can be any integer depending on initial conditions. There is no easy method to describe the state of an ensemble of UCN after an elapsed time without the use of a Monte Carlo simulation that recursively solves the cubic form of Eq. (16). 4. Monte Carlo simulation We use a Monte Carlo simulation in cylindrical coordinates with rotational symmetry of the CPC to analyze corrections to the 2D limits. The simulation first solves the quartic with coefficients
Fig. 3. A 3D Monte Carlo simulation in the ðr; zÞ plane of 1000 neutrons each with energy E ¼25 neV for one half orbit, 0o t o va =g. The solid circles indicate the end points at t ¼ va =g. Each path passes into region II during the first orbit 0 ≤t o 2va =g.
K.P. Hickerson, B.W. Filippone / Nuclear Instruments and Methods in Physics Research A 721 (2013) 60–64
63
Fig. 4. Results from 106 monochromatic UCN simulations for different energy levels. (a) The probability distribution of UCN at time t ¼ va =g for energy levels 5–40 neV in a CPC. (b) The probability distributions of the apogees of UCN orbits during the first orbit t o 2v0 =g in a CPC. (c) The apogees of 40 neV UCN in a cylindrical guide.
as the focal length of the parabola and the diameter of the input aperture, which for this simulation was set to a¼4 cm. However the probability curves still do not integrate to unity as there are some paths that fall back out of the input aperture at z¼0. While theoretically, these also should reach region II, in practice the CPC would require an input guide so these unphysical orbits are removed. The effect is most dramatic for the lowest energies, E0 ≈mga. While our predicted limits for the width of region II are ΔE omga the practical limits in three dimensions are much tighter and can be determined using our simulation to have a FWHM of ΔE o 13mga≈1:3 neV for each initial energy and a guide width of a ¼4 cm. The simulations rely on the assumption that if a material wall is to be used as a reflecting surface, it must be highly specular. To test the importance of this assumption, we counted the number of reflections for each path generated. This allowed us to get the expected number of reflections and thus compute the probability of non-specular transport. For a 25 cm tall CPC for example, the expected number of reflections was 1.6. For a CPC 40 cm tall, the expected number was 1.7. With a 95% chance of having a nonspecular reflection per bounce, this would still result in an over 90% chance of reaching the region of interest resulting in a strong contrast in a CPC spectrometer.
6. Applications 6.1. Measuring the neutron lifetime The lifetime of the neutron is an important component of refining the free parameters of the standard electroweak model particularly Vud in the CKM matrix. Previous UCN trap designs relied on either walls [11] or magnetic multipoles [12]. Recently, interest has risen in high multipolarity Halbach array traps using permanent magnets, e.g. NdFeB magnets, to create a repulsive wall
for low field seeking UCN [13]. The advantage of such a trap is that UCN of one polarization state can be repelled from the trap walls, thus minimizing neutron capture which quenches the storage lifetime. We believe such a trap has the best possibility of holding UCN with the smallest possible systematic errors usually associated with walled traps in which UCN scatter off of the nuclei of the walls themselves, so that even the lowest cross section materials still have a non-negligible absorptive loss during the trapping time. The difficulty of designing a magnetic trap is that UCN with energy above EH ¼ μn BH ≈50 neV [13] will be able to penetrate the magnetic barrier of the Halbach array, and the UCN may make contact with the surface exposing it to possible absorption by the magnet materials leading to an excess loss of UCN and resulting in a shorter measured lifetime. Therefore, we must filter out UCN with an energy greater than EH before the lifetime experiment begins. A CPC can be used as a possible design of a lifetime experiment as it can redirect UCN to an absorber to quickly and efficiently remove overly energetic and marginally trapped neutrons from a permanent magnetic storage vessel. Typically, permanent magnet Halbach arrays have a maximum surface field strength of 0.8 Tesla due to the limitations of NdFeB magnets. This places a limit on the energies of UCN that a trap may hold to be less than μn B≈40 neV [13]. A magnetically walled CPC designed to hold UCN of this energy would have a maximum height of ≈40 cm and thus UCN entering the trap could reach the absorber in the single half orbit time, pffiffiffiffiffiffiffiffiffiffiffiffi t≈ 2 h=g≈0:3 s. The specularity of such a trap would be very good due to smoothness of the magnetic field generated by a Halbach array, thus a very large majority of UCN could be expected to reach the absorber on the first pass. 6.2. Pulsed UCN source using a CPC The CPC creates an effective spectrometer with resolution of better than E¼ mga. This remarkable property leads us to propose
64
K.P. Hickerson, B.W. Filippone / Nuclear Instruments and Methods in Physics Research A 721 (2013) 60–64
a system for reducing the energy of UCN provided they start with a small phase space volume. For UCN created with a pulsed source such as SD2 , the creation and expulsion time is typically less than 100 ms and the energy distribution is boosted to the Fermi potential of deuterium [14]. This gives a relatively short pulse pffiffiffiffiffiffiffiffiffiffi ffi ðτSD2 5 2a=g ≈0:1 sÞ of narrow band ðΔE≈13mgaÞ UCN which can be directed through a CPC to a trapping region. A mechanical shutter, if carefully timed with the pulse, closing va =g after the pulse, can select out a band of UCN that will remain trapped at a lower energy. Typically, these sources are only a few centimeters across so the minimum energy band that may be trapped is as small as 1–5 neV. Experiments such as GRANIT [15–17] have detected quantization of gravitationally bound states by lowering an absorbing detector onto UCN passing over a reflective horizontal plate. Currently, these experiments are limited by the poor statistics due to the low occupancy of the lowest ground states. A pulsed source coupled to a CPC with a trapping region could generate large numbers of UCN in these states as well as increase the experiment time due to the low lateral velocity of the trapped UCN. If combined with a pulsed UCN source and a mechanical shutter, the CPC can be used to collect high densities of UCN with reduced energy at a higher gravitational potential above the source. These can be guided to an experimental chamber to make large statistics quantized gravitational states measurements. UCN with such low lateral velocity can also be dropped into unperturbed vertical paths for efficient and compact nn oscillation searches. 7. Conclusion We have applied the design principles of nonimaging optics to UCN transport optics. We use these principles to design an apparatus, a special case of a vertical CPC, that redirects monochromatic UCN (with velocity v0) from a Lambertian horizontal disk source, upward into a bound region. The region is strictly bound below the classically forbidden upper half-plane, and above the ballistic umbrella of the focus, h−ðr þ aÞ2 =4a≥z≥h, where h ¼ v20 =g. We have showed, using Monte Carlo simulation, that this bound holds empirically in three dimensions for a range of initial velocities and analyzed the resultant probability distribution of the UCN both after one-half average orbit time va =g and at the apogee of each individual orbit. We discussed the possibility of using these data to construct an apparatus that can have a shutter to trap UCN in the lower kinetic energy phase of the orbit. Such a device could trap UCN in the target region with energies E o mga. Such a device could be used to preserve the phase space of a
spallation UCN source to more efficiently gravitationally cool UCN for use in experiments that require population of the lowest energy UCN such as gravitational states experiments and nn searches. We also discussed how to construct an efficient spectrometer with effective resolution of E≈mga=3. Such a spectrometer could be used to remove marginally trapped UCN from a magnetic trap for a precision neutron lifetime experiment. We present the CPC as an interesting use of UCN nonimaging optics, but there are many new experimental designs that can also use the same design principles. We hope that this will motivate many novel and efficient geometries in future experiments that require efficient UCN transport. References [1] R. Winston, J.C. Mi nano, P. Benítez, Nonimaging Optics, Elsevier Academic Press, San Diego, 2005. [2] M. Baldo-Ceolin, P. Benetti, T. Bitter, F. Bobisut, E. Calligarich, et al., Zeitschrift für Physik C 63 (1994) 409. [3] R. Golub, D.J. Richardson, S.K. Lamoreaux, Ultra-cold Neutrons, Adam Hilger, New York, 1991. [4] S.S. Arzumanov, S.V. Masalovich, A.N. Strepetov, A.I. Frank, JETP Letters 39 (1984) 486. [5] S.S. Arzumanov, S.V. Masalovich, A.N. Strepetov, A.I. Frank, JETP Letters 44 (1986) 213. [6] V.V. Nesvizhevsky, H.G. Börner, A.K. Petukhov, H. Abele, S. Baeler, F.J. Rue, T. Stöferle, A. Westphal, A.M. Gagarski, G.A. Petrov, A.V. Strelkov, Nature 415 (2001) 297. [7] A. Steyerl, J. Phys. 45 Coll. C3, Suppl. No. 3: C3-255, 1984. [8] A.I. Frank, Soviet Physics Uspekhi 30 (1988) 110. [9] A. Steyerl, W. Drexal, S.S. Malik, E. Gutsmeidl, Physica B 151 (1988) 36. [10] A. Steyerl, W. Drexel, T. Ebisawa, E. Gutsmiedl, R.G.K.A. Steinhauser, W. Mampe, P. Ageron, Revue de Physique Appliquée 23 (1988) 171. [11] A. Serebrov, V. Varlamov, A. Kharitonov, A. Fomin, Y. Pokotilovski, P. Geltenbort, J. Butterworth, I. Krasnoschekov, M. Lasakov, R. Tal'daev, A. Vassiljev, O. Zherebtsov, Physics Letters B 605 (2005) 72. [12] C.R. Brome, J.S. Butterworth, K.J. Coakley, M.S. Dewey, S.N. Dzhosyuk, R. Golub, G.L. Greene, K. Habicht, P.R. Huffman, S.K. Lamoreaux, C.E.H. Mattoni, D. N. McKinsey, F.E. Wietfeldt, J.M. Doyle, Physical Review C 63 (2001) 055502. [13] P.L. Walstrom, J.D. Bowman, S.I. Penttila, C. Morris, A. Saunders, Nuclear Instruments and Methods in Physics Research Section A 599 (2008) 82. [14] A. Saunders, J.M. Anaya, T.J. Bowles, B.W. Filippone, P. Geltenbort, R.E. Hill, M. Hino, S. Hoedl, G.E. Hogan, T. Ito, K.W. Jones, T. Kawai, K. Kirch, S. K. Lamoreaux, C.Y. Liu, M. Makela, L.J. Marek, J.W. Martin, C.L. Morris, R. N. Mortensen, A. Pichlmaier, S.J. Seestrom, A. Serebrov, D. Smith, W. Teasdale, B. Tipton, R.B. Vogelaar, A.R. Young, J. Yuan, Physics Letters B 593 (2004) 55. [15] P. Schmidt-Wellenburg, P. Geltenbort, V. Nesvizhevsky, C. Plonka, T. Soldner, F. Vezzu, O. Zimmer, A source of ultra-cold neutrons for the gravitational spectrometer GRANIT, 2008. [16] G. Pignol, K.V. Protasov, D. Rebreyend, F. Vezzu, V.V. Nesvizhevsky, A.K. Petukhov, H.G. Börner, T. Soldner, P. Schmidt-Wellenburg, M. Kreuz, P.G.D. Forest, J.M. Mackowski, J.L.M.C. Michel, N. Morgado, L. Pinard, A. Remillieux, A. M. Gagarski, G.A. Petrov, A.M. Kusmina, A.V. Strelkov, H. Abele, S. Baeler, A.Y. Voronin, GRANIT project: a trap for gravitational quantum states of UCN, 〈arXiv:0708.2541v1〉, 2007. [17] S. Baessler, M. Beau, M. Kreuz, V.N. Kurlov, V.V. Nesvizhevsky, G. Pignol, K. V. Protasov, F. Vezzu, A.Y. Voronin, Comptes Rendus Physique 12 (2011) 707.
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
A compound parabolic concentrator as an ultracold neutron spectrometer K.P. Hickerson n, B.W. Filippone California Institute of Technology, United States
art ic l e i nf o
a b s t r a c t
Article history: Received 7 January 2013 Received in revised form 22 March 2013 Accepted 23 March 2013 Available online 3 April 2013
The design principles of nonimaging optics are applied to ultracold neutrons (UCN). In particular a vertical compound parabolic concentrator (CPC) that efficiently redirects UCN vertically into a bounded spatial volume where they have a maximum energy mga that depends only on the initial phase space cross sectional area πa2 creates a spectrometer which can be applied to neutron lifetime and gravitational quantum state experiments. & 2013 Elsevier B.V. All rights reserved.
Keywords: Ultracold neutrons Nonimaging optics
1. Introduction Compound parabolic and elliptical concentrators, designed using the edge ray principle familiar from nonimaging optics [1], have been used with success to collimate, focus and concentrate cold neutrons onto a distant target using the approximation that the neutrons travel in a straight line, as is the case with light [2,3]. In the ultracold limit of the neutron, around a few hundred neV, these straight line reflectors suffer from chromatic aberration as the kinetic energy and gravitational potential approach equal magnitude. In previous uses, this chromatic aberration has been detrimental, particularly for grazing-angle, sideways pointing concentrators as only a narrow portion of the available phase space reaches the target, and the energy spectrum is heavily distorted. While UCN microscopes have been designed using imaging optics to eliminate chromatic aberration [4,5], to date nonimaging concentrators have not been used to measure UCN spectra because as the scale of the concentrator approaches the scale of the curvature of the flight paths, a standard concentrator on its side destroys the spectral information. In this paper we will show that certain cases of these concentrators can be designed for spectroscopy in the UCN limit if the gravitational curvature of the neutrons is taken into account. While previous gravitational spectrometers have achieved remarkable energy resolutions of peV [6], they have done so at the cost of UCN number efficiency by removing unused phase space volume or waiting for UCN to defuse into the measured state from a storage vessel. We show that the principles familiar from nonimaging optics can be applied to UCN optics to design an efficient vertical spectrometer, which
utilizes much of this unused phase space on a first pass, in a time much shorter than the neutron lifetime, or more appropriately, the vessel storage time. As a prime example, we investigate a unique case of a vertical compound parabolic concentrator (CPC) which can quickly isolate UCN in bands as narrow as 13mga FWHM for a guide radius a. For 6 cm diameter guides, this gives 1 neV resolution, achievable after one pass through the optical system in the apogee time of the UCN, v0 =g. Such a CPC spectrometer can be used for a number of new experiments such as measuring the neutron lifetime using UCN. 2. Nonimaging UCN optics In imaging neutron optics, each imaged neutron path is determined by Fermat's principle which also coincides with the classical action principle Z b Z tb mv2 δ dt ¼ δ L dt ¼ 0; ð1Þ ℏ a ta so that the advancing wavefronts and the classical paths also coincide [7,8]. We may consider all potentials, gravitational, magnetic and the Fermi potential, as affecting UCN via an effective index of refraction λ2 ! ! n2 ð r Þ ¼ 1− Vð r Þ: 2mℏ2
For a rotationally symmetric system with angular momentum ℓ, we have Vðr; zÞ ¼ mgz 7 μn Bðr; zÞ þ
n
Corresponding author. Tel.: +1 6266441976. E-mail addresses: [email protected], [email protected] (K.P. Hickerson), [email protected] (B.W. Filippone). 0168-9002/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2013.03.049
ð2Þ
ℓ2 þ V F ðr; zÞ: 2mr 2
ð3Þ
For an imaging system, the integral from Fermat's principle is stationary for all neutron paths from object aperture to the image
K.P. Hickerson, B.W. Filippone / Nuclear Instruments and Methods in Physics Research A 721 (2013) 60–64
so we can solve for imaging optical surfaces using Z b δ n2 dt ¼ 0: a
61
ð4Þ
To design a nonimaging optical system, we relax the requirement that this is satisfied for all paths emanating from the input aperture. Not every point in the input aperture must have a conjugate point in the target space. Instead we rely on the edge ray principle familiar from nonimaging photon optics [1] that states that imaged paths serve only as the boundary for the phase space volume of all other enclosed paths. We use Fermat's principle only to solve for the classical edge paths and pick a reflector surface or potential geometry parameters that map the input aperture extrema to the extrema of the output region. All paths within that imaged path boundary will be guaranteed to arrive at the target region, regardless of the path taken, the number of reflections, or the path complexity.
3. Compound parabolic concentrators
Fig. 1. A CPC with a slice cut out.
From nonimaging optics for noncurvilinear rays, the general CPC family has reflective walls of a parabola that are tilted by the acceptance angle θA to the axis of rotation. The traditional CPC design [1] with an aperture at z ¼0 radius a, has the parametric form 2a′ sinðφ−θA Þ −a; 1−cos φ 2að1 þ sin θA Þ cosðφ−θA Þ zðφÞ ¼ ; 1−cos φ rðφÞ ¼
ð5Þ
where a′ is the focal length of the parabola. For nonzero θA , we truncate the CPC to a height of ða′ þ aÞcot θA . When we take the limit θA -0 we find the CPC height as defined in [1] diverges and all UCN paths are contained inside the CPC. In this limit, a′-a and the equation for a vertical CPC becomes zðrÞ ¼
1 ðr þ aÞ2 −a: 4a
ð6Þ
The normal to the wall at the point (r,z) is ð−r−a; 2aÞ n^ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : r 2 þ 2ar þ 5a2
ð7Þ
In this limit, all orbits of the same velocity originating from the input aperture are bound between extrema of the “neutron fountain” caustic, independent of their initial conditions. An interesting property that has been exploited in neutron optics is that a gravitational parabolic path originating at the focus of a parabolic surface will reflect to a conjugate parabolic path that also intersects the focus. Steyerl [9] called this property the “neutron fountain.” As a consequence, points in the neighborhood of the focus of a paraboloid are self-conjugate so the focal plane is imaged back on to itself. Steyerl and Frank used this property to design imaging systems and microscopes [10]. We use this property to show that a compound parabola can efficiently redirect UCN upward as with UCN microscopes but rather than rotate the parabola about its own axis as is done with UCN imaging optics, we place the focus of one parabola coincident with a reflected parabola, forming a compound parabola, and then rotate about the axis of symmetry (see Figs. 1 and 2). For 2D massive particles, it is not generally true that all imaged paths take equal time [7,8], but for the case of the “neutron fountain”, the orbit time from the focus to the opposite parabola wall and reflecting back, is given by ∮ dt ¼
2va g
ð8Þ
Fig. 2. The construction of a CPC using the “neutron fountain” property. For each particle with v ¼ v0 , originating from the focus, these edge orbits are stationary to variations from Fermat's principle and to orbital period. Examples shown are (1) a vertical path from the focus, (2) a focus to focus reflection orbit, pffiffiffi(3) the selfconjugate orbit, (4) an orbit with initial velocity with vr ¼ vz ¼ v0 = 2, and (5) the classical limit of an path originating at the aperture at z¼ 0. The compound parabola is formed by (7) and the reflection (8). In region I, with z 4 v20 =g, UCN are classically forbidden. All UCN from z ¼ 0; r∈½0; a will reach region II. And region III is bounded by the extrema of the edge orbits below 6, the ballistic umbrella of the point f.
where v2a ≡v20 þ 2ag, where v0 is the initial velocity. This can be compared to a vertical orbit which has orbit time 2v20 =g. This time depends only on the parabola's focal length a and the UCN initial velocity magnitude v0, and is independent of the initial angle to the vertical axis. For the flight path from the focus (z¼0) with an initial velocity ðvr ; vz Þ ¼ ðv0 sin θ; v0 cos θÞ, the time to reach the parabolic reflector from the focus is 2a : va −vz
ð9Þ
For each initial vertical velocity, vz ¼ v0 cos θ, for a path starting at focus z ¼0 and angle θ from the vertical axis, there is a reflected
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path with initial conditions v′z ¼ v0 cos θ′ that intersects the same point on the parabola. As we learned from the “neutron fountain” property, the path starting at the focus will reflect back onto the focus, so the orbit time in Eq. (8) is given by the sum of the two paths 2va 2a 2a ¼ þ : va −vz va −v′z g
ð10Þ
An extremum occurs at vz ¼ v′z ¼ v20 =va . These self-conjugate paths intersect the parabola at a height of v20 =2g−a. This reveals the interesting fact that the kinetic energy spread of the UCN orbits at their apogee is ΔE o mga and is independent of v0. Further, by the edge ray principle, all UCN emanating from the aperture necessarily will be directed to enter region II in Fig. 2. Region II is bounded below by the “ballistic umbrella”, the extrema of all monochromatic UCN paths originating at the parabola focus given by zðrÞ ¼ h−
1 ðr þ aÞ2 ; 4h
h≡
v20 ; g
ð11Þ
and bounded above by the classically forbidden height, z ≤v20 =2g. We are interested in directing UCN vertically and the general solution of the intersection of the CPC wall with the UCN path can be solved in three dimensions with radial symmetry. This allows us to model the optical system in only two dimensions using cylindrical coordinates. Without magnetic fields, the radially symmetric Euler–Lagrange equations are separable with vertical and radial solutions zðtÞ ¼ 12gt 2 þ v0 t þ z0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðtÞ ¼ v2ϕ t 2 þ ðvr t þ r 0 Þ2 :
in Eq. (16) and then iteratively solves the cubic form, which removes the t¼0 solution, iteratively and propagates the path with specular reflections off the CPC wall. This simulation demonstrates that all UCN incident on the input aperture located on the interval r∈½0; aÞ and z ¼0 are redirected to the vertical region z∈½v2 =2g−a; v2 =2g and each UCN path passes through region II, bound by z ≤v20 =2g and Eq. (11) as predicted. We validated the simulation algorithm with a calculation of the simple case of a cylindrical wall with radius a instead of the CPC and compared to the analytic result of the probability distribution for a monochromatic random gas with initial velocity v0, which has the distribution sffiffiffiffiffiffiffi v2 g ð17Þ ; 0 ≤z ≤ 0 : PðzÞ ¼ 2 2g v0 z
ð12Þ
5. Results In Fig. 3 we show the results of a small sample of UCN paths projected onto the ðr; zÞ plane. The free part of the orbits is hyperbolic along the r-axis due to conserved angular momentum, and parabolic in the vertical axis due to gravity. The multiple specular reflections off the CPC walls redirect all UCN into region II as predicted. We ran batches of one million UCN in 5 neV increments with energies ranging from 5 to 40 neV. In Fig. 4, each probability distribution shows UCN are restricted to a horizontal band as wide
We need only solve for the intersection with the reflecting surface −12 g′t 2 þ v′z t þ z′0 ¼ 12rðtÞ;
ð13Þ
where we have defined g′≡
v2a −v2z ; 2a
v′z ≡vz −vr χ;
ð14Þ
and z′0 ≡z0 þ aðχ 2 þ 34Þ;
ð15Þ
where χ ¼ r 0 =2a is a dimensionless parameter. Squaring Eq. (13) gives the quartic equation with coefficients A ¼ g ′2 ;
B ¼ −4g′v′z ;
D ¼ 8v′z z′0 −2vr r 0 ;
2 2 C ¼ 4v′2 z −4g′z′0 −vr −vϕ ;
2 E ¼ 4z′2 0 −r 0 :
ð16Þ
Generally, we set the initial conditions so that ðr 0 ; z0 Þ is at the entrance aperture at z0 ¼ 0 and r 0 ∈½0; aÞ, but the full solution is inappropriate if we have already solved for one intersection and are solving for another on the CPC surface. We need to remove the t¼0 solution from the quartic for all intersections ðr n ; zn Þ for later times t 4 0. This reduces computation time and numerical errors. When ðr n ; zn Þ lay on the wall they are dependent through Eq. (6) when z′n -12r 0 and E ¼ 0. The reflections also allow an iterative procedure to compute r nþ1 ; znþ1 from ðr n ; zn Þ. However, the computation is nonlinear, as the root of a cubic, and n can be any integer depending on initial conditions. There is no easy method to describe the state of an ensemble of UCN after an elapsed time without the use of a Monte Carlo simulation that recursively solves the cubic form of Eq. (16). 4. Monte Carlo simulation We use a Monte Carlo simulation in cylindrical coordinates with rotational symmetry of the CPC to analyze corrections to the 2D limits. The simulation first solves the quartic with coefficients
Fig. 3. A 3D Monte Carlo simulation in the ðr; zÞ plane of 1000 neutrons each with energy E ¼25 neV for one half orbit, 0o t o va =g. The solid circles indicate the end points at t ¼ va =g. Each path passes into region II during the first orbit 0 ≤t o 2va =g.
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Fig. 4. Results from 106 monochromatic UCN simulations for different energy levels. (a) The probability distribution of UCN at time t ¼ va =g for energy levels 5–40 neV in a CPC. (b) The probability distributions of the apogees of UCN orbits during the first orbit t o 2v0 =g in a CPC. (c) The apogees of 40 neV UCN in a cylindrical guide.
as the focal length of the parabola and the diameter of the input aperture, which for this simulation was set to a¼4 cm. However the probability curves still do not integrate to unity as there are some paths that fall back out of the input aperture at z¼0. While theoretically, these also should reach region II, in practice the CPC would require an input guide so these unphysical orbits are removed. The effect is most dramatic for the lowest energies, E0 ≈mga. While our predicted limits for the width of region II are ΔE omga the practical limits in three dimensions are much tighter and can be determined using our simulation to have a FWHM of ΔE o 13mga≈1:3 neV for each initial energy and a guide width of a ¼4 cm. The simulations rely on the assumption that if a material wall is to be used as a reflecting surface, it must be highly specular. To test the importance of this assumption, we counted the number of reflections for each path generated. This allowed us to get the expected number of reflections and thus compute the probability of non-specular transport. For a 25 cm tall CPC for example, the expected number of reflections was 1.6. For a CPC 40 cm tall, the expected number was 1.7. With a 95% chance of having a nonspecular reflection per bounce, this would still result in an over 90% chance of reaching the region of interest resulting in a strong contrast in a CPC spectrometer.
6. Applications 6.1. Measuring the neutron lifetime The lifetime of the neutron is an important component of refining the free parameters of the standard electroweak model particularly Vud in the CKM matrix. Previous UCN trap designs relied on either walls [11] or magnetic multipoles [12]. Recently, interest has risen in high multipolarity Halbach array traps using permanent magnets, e.g. NdFeB magnets, to create a repulsive wall
for low field seeking UCN [13]. The advantage of such a trap is that UCN of one polarization state can be repelled from the trap walls, thus minimizing neutron capture which quenches the storage lifetime. We believe such a trap has the best possibility of holding UCN with the smallest possible systematic errors usually associated with walled traps in which UCN scatter off of the nuclei of the walls themselves, so that even the lowest cross section materials still have a non-negligible absorptive loss during the trapping time. The difficulty of designing a magnetic trap is that UCN with energy above EH ¼ μn BH ≈50 neV [13] will be able to penetrate the magnetic barrier of the Halbach array, and the UCN may make contact with the surface exposing it to possible absorption by the magnet materials leading to an excess loss of UCN and resulting in a shorter measured lifetime. Therefore, we must filter out UCN with an energy greater than EH before the lifetime experiment begins. A CPC can be used as a possible design of a lifetime experiment as it can redirect UCN to an absorber to quickly and efficiently remove overly energetic and marginally trapped neutrons from a permanent magnetic storage vessel. Typically, permanent magnet Halbach arrays have a maximum surface field strength of 0.8 Tesla due to the limitations of NdFeB magnets. This places a limit on the energies of UCN that a trap may hold to be less than μn B≈40 neV [13]. A magnetically walled CPC designed to hold UCN of this energy would have a maximum height of ≈40 cm and thus UCN entering the trap could reach the absorber in the single half orbit time, pffiffiffiffiffiffiffiffiffiffiffiffi t≈ 2 h=g≈0:3 s. The specularity of such a trap would be very good due to smoothness of the magnetic field generated by a Halbach array, thus a very large majority of UCN could be expected to reach the absorber on the first pass. 6.2. Pulsed UCN source using a CPC The CPC creates an effective spectrometer with resolution of better than E¼ mga. This remarkable property leads us to propose
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a system for reducing the energy of UCN provided they start with a small phase space volume. For UCN created with a pulsed source such as SD2 , the creation and expulsion time is typically less than 100 ms and the energy distribution is boosted to the Fermi potential of deuterium [14]. This gives a relatively short pulse pffiffiffiffiffiffiffiffiffiffi ffi ðτSD2 5 2a=g ≈0:1 sÞ of narrow band ðΔE≈13mgaÞ UCN which can be directed through a CPC to a trapping region. A mechanical shutter, if carefully timed with the pulse, closing va =g after the pulse, can select out a band of UCN that will remain trapped at a lower energy. Typically, these sources are only a few centimeters across so the minimum energy band that may be trapped is as small as 1–5 neV. Experiments such as GRANIT [15–17] have detected quantization of gravitationally bound states by lowering an absorbing detector onto UCN passing over a reflective horizontal plate. Currently, these experiments are limited by the poor statistics due to the low occupancy of the lowest ground states. A pulsed source coupled to a CPC with a trapping region could generate large numbers of UCN in these states as well as increase the experiment time due to the low lateral velocity of the trapped UCN. If combined with a pulsed UCN source and a mechanical shutter, the CPC can be used to collect high densities of UCN with reduced energy at a higher gravitational potential above the source. These can be guided to an experimental chamber to make large statistics quantized gravitational states measurements. UCN with such low lateral velocity can also be dropped into unperturbed vertical paths for efficient and compact nn oscillation searches. 7. Conclusion We have applied the design principles of nonimaging optics to UCN transport optics. We use these principles to design an apparatus, a special case of a vertical CPC, that redirects monochromatic UCN (with velocity v0) from a Lambertian horizontal disk source, upward into a bound region. The region is strictly bound below the classically forbidden upper half-plane, and above the ballistic umbrella of the focus, h−ðr þ aÞ2 =4a≥z≥h, where h ¼ v20 =g. We have showed, using Monte Carlo simulation, that this bound holds empirically in three dimensions for a range of initial velocities and analyzed the resultant probability distribution of the UCN both after one-half average orbit time va =g and at the apogee of each individual orbit. We discussed the possibility of using these data to construct an apparatus that can have a shutter to trap UCN in the lower kinetic energy phase of the orbit. Such a device could trap UCN in the target region with energies E o mga. Such a device could be used to preserve the phase space of a
spallation UCN source to more efficiently gravitationally cool UCN for use in experiments that require population of the lowest energy UCN such as gravitational states experiments and nn searches. We also discussed how to construct an efficient spectrometer with effective resolution of E≈mga=3. Such a spectrometer could be used to remove marginally trapped UCN from a magnetic trap for a precision neutron lifetime experiment. We present the CPC as an interesting use of UCN nonimaging optics, but there are many new experimental designs that can also use the same design principles. We hope that this will motivate many novel and efficient geometries in future experiments that require efficient UCN transport. References [1] R. Winston, J.C. Mi nano, P. Benítez, Nonimaging Optics, Elsevier Academic Press, San Diego, 2005. [2] M. Baldo-Ceolin, P. Benetti, T. Bitter, F. Bobisut, E. Calligarich, et al., Zeitschrift für Physik C 63 (1994) 409. [3] R. Golub, D.J. Richardson, S.K. Lamoreaux, Ultra-cold Neutrons, Adam Hilger, New York, 1991. [4] S.S. Arzumanov, S.V. Masalovich, A.N. Strepetov, A.I. Frank, JETP Letters 39 (1984) 486. [5] S.S. Arzumanov, S.V. Masalovich, A.N. Strepetov, A.I. Frank, JETP Letters 44 (1986) 213. [6] V.V. Nesvizhevsky, H.G. Börner, A.K. Petukhov, H. Abele, S. Baeler, F.J. Rue, T. Stöferle, A. Westphal, A.M. Gagarski, G.A. Petrov, A.V. Strelkov, Nature 415 (2001) 297. [7] A. Steyerl, J. Phys. 45 Coll. C3, Suppl. No. 3: C3-255, 1984. [8] A.I. Frank, Soviet Physics Uspekhi 30 (1988) 110. [9] A. Steyerl, W. Drexal, S.S. Malik, E. Gutsmeidl, Physica B 151 (1988) 36. [10] A. Steyerl, W. Drexel, T. Ebisawa, E. Gutsmiedl, R.G.K.A. Steinhauser, W. Mampe, P. Ageron, Revue de Physique Appliquée 23 (1988) 171. [11] A. Serebrov, V. Varlamov, A. Kharitonov, A. Fomin, Y. Pokotilovski, P. Geltenbort, J. Butterworth, I. Krasnoschekov, M. Lasakov, R. Tal'daev, A. Vassiljev, O. Zherebtsov, Physics Letters B 605 (2005) 72. [12] C.R. Brome, J.S. Butterworth, K.J. Coakley, M.S. Dewey, S.N. Dzhosyuk, R. Golub, G.L. Greene, K. Habicht, P.R. Huffman, S.K. Lamoreaux, C.E.H. Mattoni, D. N. McKinsey, F.E. Wietfeldt, J.M. Doyle, Physical Review C 63 (2001) 055502. [13] P.L. Walstrom, J.D. Bowman, S.I. Penttila, C. Morris, A. Saunders, Nuclear Instruments and Methods in Physics Research Section A 599 (2008) 82. [14] A. Saunders, J.M. Anaya, T.J. Bowles, B.W. Filippone, P. Geltenbort, R.E. Hill, M. Hino, S. Hoedl, G.E. Hogan, T. Ito, K.W. Jones, T. Kawai, K. Kirch, S. K. Lamoreaux, C.Y. Liu, M. Makela, L.J. Marek, J.W. Martin, C.L. Morris, R. N. Mortensen, A. Pichlmaier, S.J. Seestrom, A. Serebrov, D. Smith, W. Teasdale, B. Tipton, R.B. Vogelaar, A.R. Young, J. Yuan, Physics Letters B 593 (2004) 55. [15] P. Schmidt-Wellenburg, P. Geltenbort, V. Nesvizhevsky, C. Plonka, T. Soldner, F. Vezzu, O. Zimmer, A source of ultra-cold neutrons for the gravitational spectrometer GRANIT, 2008. [16] G. Pignol, K.V. Protasov, D. Rebreyend, F. Vezzu, V.V. Nesvizhevsky, A.K. Petukhov, H.G. Börner, T. Soldner, P. Schmidt-Wellenburg, M. Kreuz, P.G.D. Forest, J.M. Mackowski, J.L.M.C. Michel, N. Morgado, L. Pinard, A. Remillieux, A. M. Gagarski, G.A. Petrov, A.M. Kusmina, A.V. Strelkov, H. Abele, S. Baeler, A.Y. Voronin, GRANIT project: a trap for gravitational quantum states of UCN, 〈arXiv:0708.2541v1〉, 2007. [17] S. Baessler, M. Beau, M. Kreuz, V.N. Kurlov, V.V. Nesvizhevsky, G. Pignol, K. V. Protasov, F. Vezzu, A.Y. Voronin, Comptes Rendus Physique 12 (2011) 707.