Reconstruction of Energy and Position in the Antineutrino Detectors

Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Version 11, July 25, 2011)

Abstract

A χ^{2} algorithm is proposed for simultaneous reconstruction of energy and position of a localized event in the antineutrino detectors.

We consider a localized deposition of energy E at a position x = (x, y, z) inside one of the antineutrino detectors. Each of the 192 photomultiplier tubes (PMTs) detects n_{i} photoelectrons at a representative position (x_{i}, y_{i}, z_{i}), along the axis of the PMT, some 2 cm from its front face [1].

We suppose that the number of detected photoelectrons is related to the energy and position of the source by,

n_{i} = \frac{k_{i}NAE}{4πR_{i}^{2}} ,  where R_{i}^{2} = (x − x_{i})^{2} + (y − y_{i})^{2} + (z − z_{i})^{2},  (1)

where k_{i} is a (possibly time-dependent) coefficient that reflects quantum efficiency of PMT i, N is the number of scintillation photons per unit of deposited energy, A = πr_{1}^{2} is the area of the PMT photocathode, r_{1} = 9.5 cm, and the geometric acceptance of photons by a PMT is assumed to fall off with the square of the distance to the source. In the first approximation, all k_{i} are the same, about 0.2.

We form the χ^{2} for the observed data {n_{i}} and the parameters x, y, z and E that are to be determined,

χ^{2} = ∑_{i=1}^{192} \frac{(n_{i} − \frac{k_{i}NAE}{4πR_{i}^{2}} )^{2}}{n_{i}} ,  (2)

where the standard error for the observation of n_{i} photoelectrons is taken to be \sqrt{n_{i}}, and terms for which n_{i} = 0 are omitted from the summation.

The best-fit parameters minimize the χ^{2}, which could be performed by a program such as MINUIT. For a less compute-intensive approach we might consider an iterative procedure, first noting that minimizing the χ^{2} implies that,

0 = −\frac{2π}{NA} \frac{∂χ^{2}}{∂E} = ∑_{i} \frac{\frac{k_{i}}{R_{i}^{2}} (n_{i} − \frac{k_{i}NAE}{4πR_{i}^{2}} )}{n_{i}} ,  (3)

E = \frac{4π}{NA} \begin{matrix}∑ & i & \frac{k_{i}}{R_{i}^{2}} \\ ∑ & i & \frac{k_{i}^{2}}{n_{i}R_{i}^{4}} \end{matrix} ,  (4)

0 = \frac{π}{NAE} \frac{∂χ^{2}}{∂x} = ∑_{i} \frac{\frac{k_{i}(x−x_{i})}{R_{i}^{4}} (n_{i} − \frac{k_{i}NAE}{4πR_{i}^{2}} )}{n_{i}} ,  (5)

x = \frac{4π}{NA} \begin{matrix}∑ & i & \frac{k_{i}x_{i}}{R_{i}^{4}} & (1 − \frac{k_{i}NAE}{4πn_{i}R_{i}^{2}} ) \\ ∑ & i & \frac{k_{i}}{R_{i}^{4}} & (1 − \frac{k_{i}NAE}{4πn_{i}R_{i}^{2}} ) \end{matrix} ,  (6)

etc. Since R_{i} is a function of x, eqs. (4) and (6) are not closed-form solutions. However, we could start with an initial hypothesis as to (x, y, z), perhaps at the center of the detector, or based on a quick estimate such as x = ∑_{i} n_{i}x_{i}/ ∑_{i} n_{i}, and iterate for E and x. Clearly, Monte Carlo studies are be needed to validate this approach.

This model can be augmented in various ways.

If we suppose that the photoelectrons are attenuated with distance R_{i} according to e^{−R_{i}/λ}, for a known attenuation length λ, the χ^{2} would be modified to read,

χ^{2} = ∑_{i=1}^{192} \frac{(n_{i} − \frac{k_{i}NAEe^{−R_{i}/λ}}{4πR_{i}^{2}} )^{2}}{n_{i}} .  (7)

Reflectors at height z_{+} and z_{−} with reflectivity ϵ could be accounted for by introducing,

R_{i±}^{2} = (x − x_{i})^{2} + (y − y_{i})^{2} + (2z_{±} − z − z_{i})^{2},  (8)

and,

χ^{2} = ∑_{i=1}^{192} \frac{[n_{i} − k_{i}E (\frac{e^{−R_{i}/λ}}{R_{i}^{2}} + ϵ\frac{e^{−R_{i+}/λ}}{R_{i+}^{2}} + ϵ\frac{e^{−R_{i−}/λ}}{R_{i−}^{2}} )]^{2}}{n_{i}} .  (9)

The possibility of multiple reflections could be included by defining additional distances R_{i+−}, R_{i−+}, R_{i+−+}, etc.

The model (1) assumes that the solid angle of the photocathode depends only on the distance R_{i} and not on the direction. However, for a give R_{i} the solid angle actually varies by a (calculable) factor of ≈ 5 with direction, being smallest when the light travels nearly vertically in the detector [1]. Denoting this “obliquity factor” by α(cos θ_{i}), and noting that distance R_{i} is not necessarily large compared to the radius r_{1} = 9.5 cm of the edge of the photocathode, eq. (1) could be modified as [1],

n_{i} = \frac{k_{i}NE e^{−R_{i}/λ}}{2} (1 − \frac{R_{i}}{\sqrt{R_{i}^{2} + αr_{1}^{2}}} ) + · · · .  (10)

As an example, consider the case that we include single reflections and an obliquity factor. Then,

n_{i} = \frac{k_{i}NE}{2} ∑_{j=1}^{3} ϵ_{j} e^{−R_{ij}/λ} (1 − \frac{R_{ij}}{\sqrt{R_{ij}^{2} + α_{ij}r_{1}^{2}}} ) ,  (11)

where the reflectivity ϵ_{j} is 1 for j = 1 and less for j = 2 and 3 which correspond to light having a single bounce at the top or bottom reflectors, R_{i1} = R_{i}, R_{i2} = R_{i+}, R_{i3} = R_{i−}, α_{ij} = α(cos θ_{ij}),

cos θ_{ij} = \frac{e_{ix}(x − x_{i}) + e_{iy}(y − y_{i})}{R_{ij}} ,  (12)

with (e_{ix}, e_{iy}, 0) being a unit vector along the axis of PMT i. Following [1], the obliquity factor can be approximated as,

α(cos θ) = ∑_{m=0}^{3} a_{m} cos^{m} θ,  (13)

where,

a_{0} = 0,  a_{1} = 1,  a_{2} = 0,  a_{3} = 0 (cos θ > 0.69),  (14)
a_{0} = 0.19,  a_{1} = 0,  a_{2} = 1.71,  a_{3} = −0.95 (cos θ < 0.69).  (15)

A cruder approximation is simply α = cos θ, as would hold if the face of the PMT were flat.

Minimizing the χ^{2} based on eq. (11) leads to,

0 = −\frac{1}{N} \frac{∂χ^{2}}{∂E}  (16)
= ∑_{i} k_{i} ∑_{j=1}^{3} ϵ_{j} e^{−R_{ij}/λ} (1 − \frac{R_{ij}}{\sqrt{R_{ij}^{2} + α_{ij}r_{1}^{2}}} ) [1 − \frac{k_{i}NE}{2n_{i}} ∑_{k=1}^{3} ϵ_{k} e^{−R_{ik}/λ} (1 − \frac{R_{ik}}{\sqrt{R_{ik}^{2} + α_{ik}r_{1}^{2}}} )] ,

E = \frac{\frac{2}{N} ∑_{i} k_{i} ∑_{j=1}^{3} ϵ_{j} e^{−R_{ij}/λ} (1 − \frac{R_{ij}}{\sqrt{R_{ij}^{2}+α_{ij}r_{1}^{2}}} )}{∑_{i} \frac{k_{i}^{2}}{n_{i}} [∑_{j=1}^{3} ϵ_{j} e^{−R_{ij}/λ} (1 − \frac{R_{ij}}{\sqrt{R_{ij}^{2}+α_{ij}r_{1}^{2}}} )]^{2}} ,  (17)

0 = \frac{1}{NE} \frac{∂χ^{2}}{∂x} = ∑_{i} k_{i} ∑_{j=1}^{3} ϵ_{j}e^{−R_{ij}/λ} \{(x − x_{i}) [\frac{1}{λR_{ij}} (1 − \frac{R_{ij}}{\sqrt{R_{ij}^{2} + α_{ij}r_{1}^{2}}} ) + \frac{α_{ij}r_{1}^{2}}{R_{ij}(R_{ij}^{2} + α_{ij}r_{1}^{2})^{3/2}} ]

+\frac{α′_{ij}r_{1}^{2}}{2(R_{ij}^{2} + α_{ij}r_{1}^{2})^{3/2}} (\frac{cos θ_{ij}(x − x_{i})}{R_{ij}} − e_{ix})\}

[1 − \frac{k_{i}NE}{2n_{i}} ∑_{k=1}^{3} ϵ_{k} e^{−R_{ik}/λ} (1 − \frac{R_{ik}}{\sqrt{R_{ik}^{2} + α_{ik}r_{1}^{2}}} )] ,  (18)

where α′ = dα/d cos θ = ∑_{m=1}^{3} m a_{m} cos^{m−1} θ. Hence, we can write,

x = \frac{A}{B} ,  (19)

where,

A = ∑_{i} k_{i} ∑_{j=1}^{3} ϵ_{j}e^{−R_{ij}/λ} \{x_{i} [\frac{1}{λR_{ij}} (1 − \frac{R_{ij}}{\sqrt{R_{ij}^{2} + α_{ij}r_{1}^{2}}} ) + \frac{[α_{ij} + (α′_{ij}/2) cos θ_{ij}]r_{1}^{2}}{R_{ij}(R_{ij}^{2} + α_{ij}r_{1}^{2})^{3/2}} ]

−\frac{e_{ix}α′_{ij}r_{1}^{2}}{2(R_{ij}^{2} + α_{ij}r_{1}^{2})^{3/2}} \} [1 − \frac{k_{i}NE}{2n_{i}} ∑_{k=1}^{3} ϵ_{k} e^{−R_{ik}/λ} (1 − \frac{R_{ik}}{\sqrt{R_{ik}^{2} + α_{ik}r_{1}^{2}}} )] ,  (20)

and,

B = ∑_{i} k_{i} ∑_{j=1}^{3} ϵ_{j}e^{−R_{ij}/λ} [\frac{1}{λR_{ij}} (1 − \frac{R_{ij}}{\sqrt{R_{ij}^{2} + α_{ij}r_{1}^{2}}} ) + \frac{[α_{ij} + (α′_{ij}/2) cos θ_{ij}]r_{1}^{2}}{R_{ij}(R_{ij}^{2} + α_{ij}r_{1}^{2})^{3/2}} ]

[1 − \frac{k_{i}NE}{2n_{i}} ∑_{k=1}^{3} ϵ_{k} e^{−R_{ik}/λ} (1 − \frac{R_{ik}}{\sqrt{R_{ik}^{2} + α_{ik}r_{1}^{2}}} )] .  (21)

Similar expressions hold for y, but the z-dependence in R_{ij} and cos θ_{ij} leads to the substitutions e_{ix} → 0, and x_{i} → 2z_{±} − z_{i} (rather than x_{i} → z_{i}) for j, k = 2, 3 in the expressions for A and B.

For a maximum-likelihood approach to this issue, see [2]. An alternative to the analytic approach considered here is to use a Monte Carlo simulation of the detector, and to fit the results for the number of photoelectrons to some form n_{i} = Ef(x, y, z, x_{i}, y_{i}, z_{i}) whose derivatives with respect to x, y and z are reasonably simple. Then, an iterative χ^{2} method could be constructed similar to that given in this note. A step in this direction has been made in [3].

Comments:

I wrote a simplified Monte Carlo event generator that assumes the faces of the PMTs are flat, and that there are no reflections. Using the actual E and x for the (point) source of scintillation, the reconstruction of energy E according to eq. (17) is reasonable, but the reconstruction of source position x according to eqs. (19)-(21) is poor.

I note that the form of eq. (20) is peculiar in that it weights PMT coordinate x_{i} according to [n_{i}(meas) − n_{i}(pred)]/n_{i}(meas), which gives more significance to PMTs in which the observed signal differs more from the expectation. This may be a hint that the iterative algorithm considered here has poor convergence. Instead, it may be more appropriate to find the best-fit E and x by numerical minimization of the χ^{2} (or by numerical maximization of the closely related likelihood function).

References

[1] K.T. McDonald, Effective Center and Obliquity Factor of R5912 Photomultiplier Tubes

    (July 11, 2011), http://kirkmcd.princeton.edu/dayabay/pmt_center.pdf

[2] L. Wen et al., Maximum likelihood reconstruction of a detector with reflective panels, Nucl. Instr. and Meth. A629, 296 (2011),

   http://kirkmcd.princeton.edu/examples/detectors/wen_nim_a629_296_11.pdf

[3] S. Ryland Ely, A Deterministic Code for Calculating the AD Light Response (Apr. 29,

   2009), http://dayabay.ihep.ac.cn/DocDB/0035/003525/001/lightsims.pdf