Chaotic scattering and escape times of marginally trapped ultracold neutrons.
We compute classical trajectories of Ultracold neutrons (UCNs) in a supercon-ducting Ioffe-type magnetic trap Magnetic trap refers to one of three types of traps used for atoms or charged particles:
UCN University College of the North (The Pas, Manitoba, Candad)
UCN Ultra Cold Neutron
UCN Unión del Centro Nacional realizations are numerically well determined, we predict the median escape time as a function of the midpoint mid·point
1. Mathematics The point of a line segment or curvilinear arc that divides it into two parts of the same length.
2. A position midway between two extremes. of the interval.
Key words: chaotic scattering; escape time; Hamiltonian system In classical mechanics, a Hamiltonian system is a physical system in which forces are velocity invariant. Hamiltonian systems are studied in Hamiltonian mechanics.
In mathematics, a Hamiltonian system is a system of differential equations which can be written in the form of ; magnetic trap; symplectic integration; ultracold neutrons.
There are at present several Ultracold neutron (UCN) experiments that seek to measure the neutron lifetime using magnetic confinement techniques (see these proceedings). Experiments are underway and are being proposed that utilize permanent magnets permanent magnets,
n.pl magnets containing the mineral boron and the rare earth metal neodymium; often combined to create a neoprene or ceramic entity. , superconducting magnets, and a combination of the two to create magnetic field geometries that can be used to spatially confine UCNs. Such techniques are advantageous because they minimize (or eliminate) any interactions that a UCN has with the material walls of the container, thus dramatically reducing (or removing) the largest systematic error that has plagued UCN lifetime experiments to date. Reducing the uncertainty of a neutron lifetime measurement beyond roughly a factor of two (relative to the presently accepted value) is a challenge that requires a new approach and/or substantial increases in the number of available UCNs. Since magnetic trapping techniques offer great promise to meet this challenge, they are very popular.
One key systematic effect that arises when using these new magnetic trapping techniques is marginal trapping; neutrons with a total energy greater than the trap depth may be temporarily contained in the trap. For example, a neutron in a circular orbit
In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. (with its plane perpendicular to the trap axis) can gradually turn into an elliptical el·lip·tic or el·lip·ti·cal
1. Of, relating to, or having the shape of an ellipse.
2. Containing or characterized by ellipsis.
a. orbit (due to angular momentum angular momentum: see momentum.
Property that describes the rotary inertia of a system in motion about an axis. It is a vector quantity, having both magnitude and direction. noncon-servation in an angular dependent field) and thus reach a greater distance from the origin. If the initial energy of this neutron is larger than the trap depth, the neutron can then strike the container wall and be ejected from the trap before it decays. Consequently, the neutron lifetime estimated from data consisting of a mixture of permanently trapped and marginally trapped neutrons will be systematically shorter than the actual lifetime of the neutron.
In this work, as part of the UCN lifetime experiment presently underway at National Institute of Standards and Technology National Institute of Standards and Technology, governmental agency within the U.S. Dept. of Commerce with the mission of "working with industry to develop and apply technology, measurements, and standards" in the national interest. (NIST (National Institute of Standards & Technology, Washington, DC, www.nist.gov) The standards-defining agency of the U.S. government, formerly the National Bureau of Standards. It is one of three agencies that fall under the Technology Administration (www.technology. ) [1-3], we present detailed neutron trajectory simulations in an Ioffe-type supercon-ducting magnetic trap. Neutrons with energies higher than the trap depth can escape from the trap. However, the escape of these marginally trapped neutrons is not immediate and we have shown experimentally that it can introduce a systematic error into the measured neutron lifetime. Analytic calculations  (which have been experimentally verified) have shown that it is possible to remove practically all of these marginally trapped neutrons from the trap by lowering the depth of the trap for a short time and then raising it again. However, this procedure also reduces the number of nonmarginally, i.e., permanently, trapped neutrons.
To determine the optimal lowered trap depth for removal of practically all of the marginally trapped UCNs, one must understand the escape time distribution of UCNs. In this work, we simulate realizations of UCNs and compute their escape times assuming that the trajectories are classical. The escape time is defined to be the time it takes the UCN to first cross the boundary of a cylindrical detection volume. As a caveat, this model for escape simplifies quantum mechanical aspects of neutron interactions. In a later work, we shall develop a more realistic model for neutron escape that accounts for such effects as described in ref. . For each realization, we compute a trajectory using a symplectic integration algorithm [6-13]. Symplectic integrators were developed for Hamiltonian systems and are generally regarded as superior to Runge-Kutta methods The Runge-Kutta methods listed on this page are each defined by the Butcher Tableau, which has the general form:
Our primary result is that the computed escape time for a particular set of initial conditions (momentum and position) does not always stabilize. In general, for the cases studied here, computed escape time stabilizes for cases where the neutron promptly escapes (less than approximately 10 s). For longer escape times, stability is generally rare. We interpret the observed instability of the computed escape time as a manifestation of chaotic behavior. For such trajectories, slight differences in computed trajectories due to numerical noise (which depends on the time step size) can be magnified dramatically at advanced times. For more discussion of this sensitivity in related problems, see refs.  and . In general, chaotic behavior of computed trajectories for Hamiltonian systems is called chaotic scattering [14,15]. We also present an empirical model to predict the median escape time of UCNs as a function of the midpoint of the particular energy interval. As a caveat, the escape time distribution for UCNs in an actual experiment may also be affected by mechanical vibrations and slight instabilities in the magnetic field. In this work, we do not include these possible effects.
2. Physical Model
In the ongoing experiment at NIST, UCNs are produced by inelastic scattering inelastic scattering
The scattering of particles resulting from inelastic collision. of cold neutrons from a reactor in superfluid su·per·flu·id
A fluid, such as a liquid form of helium, exhibiting a frictionless flow at temperatures close to absolute zero.
su [.sup.4]He. By creation of a single phonon phonon (fō`nŏn), quantum of vibrational energy. The atoms of any crystal are in a state of vibration, their average kinetic energy being measured by the absolute temperature of the crystal. in the superfluid, a cold neutron with wavelength near 0.89 nm (speed of approximately 400 m/s) is scattered to a state of near rest. (The mean wavelength of a thermal ensemble of neutrons at 12 K is 0.89 nm (8.9 [Angstrom angstrom (ăng`strəm), abbr. Å, unit of length equal to 10−10 meter (0.0000000001 meter); it is used to measure the wavelengths of visible light and of other forms of electromagnetic radiation, such as ultraviolet ]).) The resulting UCNs are confined in a potential field V (x) produced by the interaction of the magnetic moment of a neutron and a spatially varying magnetic field
V(x) = [mu]|B(x)|, (1)
where B is the magnetic field, |B(x)| is the magnitude of B, and [mu] is the magnetic moment of the neutron [1,2]. In cylindrical coordinates, the trapping volume is approximately -18 cm [less than or equal to] z [less than or equal to] 18 cm and r [less than or equal to] 4.3 cm, where r = [square root of ([x.sup.2] + [y.sup.2])] and is shown in Fig. 1. The maximum of |B| on the escape boundary is approximately 2.1 T. The ratio of the minimum to maximum value of |B| on the escape boundary is 0.573. This minimum point (|B| = 1.2 T) is the trap depth. The total energy of a UCN created at position x is E = [p.sup.2]/(2[m.sub.n]) + V(x) where p is the magnitude of the UCN momentum and [m.sub.n] is the mass of the neutron. Here, we focus on the escape times of UCNs with energies less than 1.6 [V.sub.max], where [V.sub.max] is the maximum potential value on the boundary of the trap. If a UCN had kinetic energy kinetic energy: see energy.
Form of energy that an object has by reason of its motion. The kind of motion may be translation (motion along a path from one place to another), rotation about an axis, vibration, or any combination of equal to [V.sub.max], its speed would be approximately 3.6 m/s. For UCN with low kinetic energies ([p.sup.2]/(2[m.sub.n]) [less than or equal to] 4[V.sub.max]), we assume that the UCN velocity direction is uniformly distributed on the unit sphere In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used. A unit ball is the region enclosed by a unit sphere. and that the conditional probability conditional probability
the probability that event A occurs, given that event B has occurred. Written P(AB). density function of p is proportional to [p.sup.2]. We use this conditional probability density function to simulate realizations of p and compute escape times for realizations of UCN initial position and momentum with total energy less than 1.6 [V.sub.max].
[FIGURE 1 OMITTED]
We assume that the initial position of the UCN is uniformly distributed in the trapping volume. To determine a neutron trajectory based on its initial position and momentum, we solve the classical equations of motion,
p = F(x) = -[nabla]V (2)
[dot.x] = p/[m.sub.n], (3)
using an optimal fourth order symplectic integration scheme designed for separable sep·a·ra·ble
Possible to separate: separable sheets of paper.
sep Hamiltonian systems where the kinetic energy is a quadratic function A quadratic function, in mathematics, is a polynomial function of the form , where . of momentum . A symplectic map M(t) is a canonical transformation In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations, although it might not preserve the Hamiltonian itself. of a point in position-momentum phase space at initial time t = 0 to a point in position-momentum phase space at time t. That is, (x(t), p(t)) = M(t)(x(0), p(0)). In the optimal fourth order scheme presented in , the predicted value of the position and momentum at time t + [DELTA], is ([x.sub.4], [p.sub.4]), where
For i = 1,4
[p.sub.i] = [p.sub.i-1] + b(i)F([x.sub.i-1])[DELTA] (4)
[x.sub.i] = [x.sub.i-1] + [[a(i)]/[m.sub.n]][p.sub.i][DELTA] (5)
Above, [x.sub.0] = x(t), [p.sub.0] = p(t) and
a(1) = 0.5153528374311229364 b(1) = 0.1344961992774310892
a(2) = -0.085782019412973646 b(2) = -0.2248198030794208058
a(3) = 0.4415830236164665242 b(3) = 0.7563200005156682911
a(4) = 0.1288461583653841854 b(4) = 0.3340036032863214255.
The coefficients a(i), b(i) i = 1,..., 4 are numerically determined to minimize an error constant that measures Hamiltonian (total energy) truncation errors. For more details, we refer the reader to .
To determine when a neutron crosses the boundary, we use a quadratic quadratic, mathematical expression of the second degree in one or more unknowns (see polynomial). The general quadratic in one unknown has the form ax2+bx+c, where a, b, and c are constants and x is the variable. interpolation interpolation
In mathematics, estimation of a value between two known data points. A simple example is calculating the mean (see mean, median, and mode) of two population counts made 10 years apart to estimate the population in the fifth year. method to estimate the maximum value of r and |z| for the trajectory based on computed values of r and z at three successive time steps.
In our numerical approach, we predict |B| at arbitrary points in the trapping volume by using a three dimensional tensor-product spline In computer graphics, a smooth curve that runs through a series of given points. The term is often used to refer to any curve, because long before computers, a spline was a flat, pliable strip of wood or metal that was bent into a desired shape for drawing curves on paper. See Bezier and B-spline. interpolant  where the order of the spline is 4 in each direction. We estimate the tensor-product B-spline coefficients from values of |B| computed on a grid by a numerical code that numerically solves the Biot-Savart law Biot-Savart law
A law of physics which states that the magnetic flux density (magnetic induction) near a long, straight conductor is directly proportional to the current in the conductor and inversely proportional to the distance from the conductor. numerically corresponding to the geometry of the solenoid solenoid (sō`lənoid'), device made of a long wire that has been wound many times into a tightly packed coil; it has the shape of a long cylinder. and current bars that produce the magnetic field. The grid spacing in the x, y, and z direction is 0.1 cm, 0.1 cm, and 0.5 cm. The value of the potential and its gradient is evaluated at arbitrary locations given the B-spline coefficients.
3. Simulation Study
In Fig. 2, we plot a sample predicted trajectory. For this example, CASE A, the computed escape time stabilizes as the time step in the integration scheme decreases. The fractional absolute error |([^.E] - E)/E|, where E is the true energy of the UCN and [^.E] is the predicted energy at escape, generally decreases as the time step decreases (Fig. 3). For other cases, even though the fractional energy at escape is comparable to that observed for CASE A, the computed escape time does not stabilize (see CASE B). For chaotic triatomic triatomic /tri·atom·ic/ (-ah-tom´ik) containing three atoms. systems, Schlier and Seiter remarked that good energy conservation does not ensure that the predicted trajectory is close to the "true" trajectory of interest . Our result is consistent with this remark.
For all cases studied, stability is achieved when the computed escape time is less than approximately 10 s. To illustrate the stability problem for longer escape times, we plot the computed escape time for 25 cases. We halt the trajectory after 100 s if there is no escape. For these cases, we vary the time step from [10.sup.-3] s to [10.sup.-7] s. For computed escape times between 10 s and 30 s, stability is sometimes achieved. Beyond 30 s, stability occurs just once for an escape time of 86 s (CASE C).
In a sensitivity analysis, we compute the trajectory of interest for the cases A, B, and C. For each case, we predict nine other trajectories with the same initial velocity the velocity of a moving body at starting; especially, the velocity of a projectile as it leaves the mouth of a firearm from which it is discharged.
See also: Velocity but slightly different initial positions. For cases A and B, after about 8 s, the distance between the reference trajectory and the other trajectories grew by about ten orders of magnitude (Fig. 5). Thus, it is not a surprise that the computed escape time is unstable for these cases. For CASE C, the distance grew only by about two orders of magnitude. Thus, it is plausible that we can compute a long escape time for CASE C even though we cannot for CASE B.
An open question is whether one can consistently estimate the probability density function Probability density function
The function that describes the change of certain realizations for a continuous random variable. of escape times for an ensemble of realizations even though we cannot, in general, determine the escape time for all realizations in the ensemble. This point was raised in refs  and . In other words Adv. 1. in other words - otherwise stated; "in other words, we are broke"
put differently , is the probability density function of computed escape times for an ensemble of trajectories (with random initial positions and velocities) computed for a given time step the same as the probability density function of the true escape times for the ensemble? This is an open question.
If more than half of the computed escape times are well determined numerically for given energy interval, the median is well determined for that interval. Thus, we can still extract useful information from the computed escape times even though some may not be well determined. We bin computed escape time data from simulated realizations of UCNs according to according to
1. As stated or indicated by; on the authority of: according to historians.
2. In keeping with: according to instructions.
3. energy, and empirically model the logarithm logarithm (lŏg`ərĭthəm) [Gr.,=relation number], number associated with a positive number, being the power to which a third number, called the base, must be raised in order to obtain the given positive number. of the median escape time for each bin as a function of the bin midpoint, [bar.E], as follows:
[FIGURE 2 OMITTED]
ln[MED([t.sub.esc]/[t.sub.o])]([bar.E]) = [[gamma].sub.1] + [[gamma].sub.2] ln [[[bar.E] - [V.sub.min]]/[V.sub.max]] + [epsilon]([bar.E]), (6)
where the maximum potential value on the boundary of the trap is [V.sub.max] and the minimum potential on the boundary is [V.sub.min] = 0.573 [V.sub.max] (shown in Fig. 6 as a thick vertical line) [t.sub.o] = 1 s and [epsilon]([bar.E]) is the prediction error for the bin with midpoint [bar.E]. The energy bins had uniform width of 0.01 [V.sub.max]. Above 0.655 [V.sub.max], the computed median for each bin stabilized as a function of time step (the minimum time step was [10.sup.-5] s). We determine the model parameters by the method of ordinary least squares and their associated 1-sigma random uncertainties by a nonparametric bootstrap See boot.
(operating system, compiler) bootstrap - To load and initialise the operating system on a computer. Normally abbreviated to "boot". From the curious expression "to pull oneself up by one's bootstraps", one of the legendary feats of Baron von Munchhausen. resampling scheme [17-19]. The basic idea of the bootstrap is to generate synthetic data based solely on the observed data without making any distributional assumptions. Our observed data can be represented as N data pairs [z.sub.i] = ([x.sub.i], [y.sub.i]) i = 1,..., N, where [x.sub.i] is the ith energy bin midpoint and and [y.sub.i] is the ith median escape time. We draw N realizations of a random integer that is uniformly distributed between 1 and N. Denote this string of random integers as ([j.sub.1], [j.sub.2],..., [j.sub.N]). Our bootstrap replication of the observed data is [z.sub.[j.sub.1]], [z.sub.[j.sub.2]], [z.sub.[j.sub.3]],..., [z.sub.[j.sub.N]]. We refit the model to each bootstrap replication of the data and store the associated parameter estimates. The standard deviation In statistics, the average amount a number varies from the average number in a series of numbers.
(statistics) standard deviation - (SD) A measure of the range of values in a set of numbers. of the model parameter estimates computed from the bootstrap data is our estimate of the 1-sigma uncertainty of the model parameter estimated from the observed data. In this study, we simulate [10.sup.4] bootstrap replications of the data.
For 0.655 [V.sub.max] < [bar.E] < 0.8 [V.sub.max], the estimated parameters are [^.[gamma].sub.1] = -8.16 and [^.[gamma].sub.2] = -3.53. The bootstrap estimates of the 1-sigma random uncertainties for these estimates are 1.19 and 0.64 respectively. For E > 0.8 [V.sub.max], the estimated parameters are [^.[gamma].sub.1] = -4.74 and [^.[gamma].sub.2] = -1.00. The associated bootstrap estimates of the 1-sigma random uncertainties are respectively 0.04 and 0.09. For clarity, we write the predicted median escape time for the bin with midpoint [bar.E] as
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
MED([t.sub.esc]/[t.sub.o])([bar.E]) = exp([[gamma].sub.1]([[bar.E] - [V.sub.min]]/[V.sub.max])[.sup.[[gamma].sub.2]]. (7)
We modeled the trajectories of UCNs in a magnetic trap classically and predicted the trajectory of the neutron by integrating the laws of motion laws of motion
See Newton's laws of motion. using a symplectic integration scheme. In our approach, we modeled the potential produced by the spatially varying field in the trap using a spline interpolation In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. scheme. Since this interpolation scheme is an approximate method, the computed escape times may differ from those computed by use of an exact model for the potential. Since each component of the magnetic field in the trap satisfies Laplace's equation Laplace's equation
In mathematics, a partial differential equation whose solutions (harmonic functions) are useful in investigating physical problems in three dimensions involving gravitational, electrical, and magnetic fields, and certain types of fluid motion. , it might be possible to develop a special function expansion approximation for the potential. As a caveat, the escape time distribution for UCNs in an actual experiment may be affected by mechanical vibrations and slight instabilities in the magnetic field. In this work, we did not include these possible effects.
We defined the escape time of a neutron to be the first time that the classical neutron trajectory crosses the boundary of the trapping volume. The trapping volume boundary is the union of the endcaps and the surface of a cylinder. For all cases considered, we computed numerically stable escape times for trajectories that promptly escaped the trap in about 10 s or less. However, in general, for longer escape times, stability is rare. How to interpret a computed escape for a particular time step for such unstable cases is not clear. Other researchers [8,10] have remarked that one does not expect to compute the escape time for a particular trajectory in a chaotic set in a stable manner. However, they suggested that it may be possible to estimate the probability density function of the escape times for the chaotic set. We are not aware of any proof of this conjecture.
We also predicted the median escape time for simulated UCNs based on escape time data at energies where the median was well determined. The accuracy of this model for lower energies is an open issue. In future work, we plan to study a more realistic model for neutron escape that accounts for additional quantum effects. In this more realistic model, neutrons would escape or scatter off the boundary according to stochastic laws.
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
We thank B. Alpert of NIST for many useful conversations. This work was supported in part by the National Science Foundation under grant numbers PHY-0354263 and PHY-0354264.
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K. J. Coakley
National Institute of Standards and Technology, Boulder, CO 80305 USA
J. M. Doyle, S. N. Dzhosyuk, and L. Yang
Harvard University, Cambridge, MA 02138 USA
P. R. Huffman
North Carolina State University History
National Institute of Standards and Technology, Gaithersburg, MD 20899 USA
Accepted: August 11, 2004
Available online: http://www.nist.gov/jres