Effective Center, Obliquity Factor and Solid Angle of R5912 Photomultiplier Tubes

Kirk T. McDonald

Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Version 4, July 23, 2011)

Abstract

The average location of a photoelectron detected by an R5912 photomultiplier tube (PMT) is at a distance ≈ 2 cm in from the intercept of the axis and the front face of the PMT, and the area of the photocathode as projected onto a plane perpendicular to the direction of an incident photon can be described by an obliquity factor,

α(θ) = \{ \begin{matrix}cos θ & (cos θ > cos θ_{0}), \\ 0.19 + 1.71 cos^{2} θ − 0.95 cos^{2} θ & (cos θ < cos θ_{0}), \end{matrix}  (1)

where θ is the polar angle between the axis of the PMT and the direction of the photon and cos θ_{0} ≈ 0.69. The solid angle subtended by the photocathode with respect to a point at distance r from the average location of the PMT is,

Ω ≈ 2π (1 − \frac{r}{\sqrt{r^{2} + αr_{1}^{2}}} ) ,  (2)

where r_{1} = 9.5 cm.

1 Introduction

As reported by Hamamatsu, the photocathode of their R5912 photomultiplier tubes (PMTs) has a radius of curvature of r_{0} = 13.1 cm and constitutes of spherical cap of chord 2 r_{1} > ∼ 19 cm, as shown on the next page. The polar angle of the edge of the photocathode is θ_{0} > ∼ sin^{−1}(r_{1}/r_{0}) = 46.5^{∘} = 0.81 rad. The distance from the center of curvature to the plane of the cap is r_{2} = r_{0} cos θ_{0} < ∼ 9 cm.

2 Average x, y and z of Photoelectrons

It will often be convenient to represent the coordinates of the photoelectrons j generated at the photocathode of PMT of index i by a single point (⟨x_{j}⟩ , ⟨y_{j}⟩ , ⟨z_{j}⟩) = (x_{i}, y_{i}, z_{i}).

It is appealing to take this point as lying along the symmetry axis of the PMT, although this is strictly correct only for photons with polar angle θ = 0 relative to that axis.

For photons with θ = 0 in a cylindrical coordinate system ρ, ϕ, z′ with z′-axis along that of the PMT and origin at the center of curvature of the photocathode, the average z′ is,

⟨z′⟩ = \frac{∫_{0}^{r_{1}} ρ dρ z′(ρ)}{∫_{0}^{r_{1}} ρ dρ} = \frac{2}{r_{1}^{2}} ∫_{0}^{r_{1}} ρ dρ \sqrt{r_{0}^{2} − ρ^{2}} = \frac{2}{3r_{0}^{2} sin^{2} θ_{0}} [r_{0}^{3} − (r_{0}^{2} − r_{1}^{2})^{3/2}]

= \frac{2r_{0}}{3 sin^{2} θ_{0}} (1 − cos^{3} θ_{0}) = 11.2 cm,  (3)

noting that the surface of the photocathode is given by z′(ρ) = \sqrt{r_{0}^{2} − ρ^{2}}. At the other extreme, θ = 90^{∘}, the average z′ is,

⟨z′⟩ = \frac{∫_{0}^{r_{1}} dρ ∫_{r_{2}}^{z′(ρ)} dz′ z′}{∫_{0}^{r_{1}} dρ ∫_{r_{2}}^{z′(ρ)} dz′} = \frac{∫_{0}^{r_{1}} dρ(r_{0}^{2} − ρ^{2} − r_{2}^{2})/2}{∫_{0}^{r_{1}} dρ (\sqrt{r_{0}^{2} − ρ^{2}} − r_{2})} = \frac{r_{0}^{2}r_{1}/2 − r_{1}^{3}/6 − r_{1}r_{2}^{2}/2}{(r_{0}^{2}θ_{0} − r_{1}r_{2})/2}

= r_{0} \frac{2 sin^{3} θ_{0}}{3(θ_{0} − cos θ_{0} sin θ_{0})} = 10.1 cm.  (4)

I propose taking ⟨z⟩ for the photoelectrons to be on the axis of the PMT and 2 cm in from the front face (rather than distance r_{0} = 13.1 cm in, as perhaps presently used in the analysis).

3 Obliquity Factor

Photons with polar angle θ less than 90^{∘} −θ_{0} = 43.5^{∘} can strike the photocathode anywhere, and the projection of the area of the photocathode onto a plane perpendicular to the photon’s trajectory is,

A_{proj}(θ) = A cos θ,  (5)

where A = π r_{1}^{2} = 284 cm^{2} is the area of the plane of the cap. For θ greater than 90^{∘} − θ_{0} = 43.5^{∘} the projected area has a complicated behavior, reaching a minimum for θ = 90^{∘},

A_{proj}(θ = 90^{∘}) = 2 ∫_{0}^{r_{1}} dρ ∫_{r_{2}}^{z′(ρ)} dz′ = r_{0}^{2}θ_{0} − r_{1}r_{2} = A\frac{θ_{0} − cos θ_{0} sin θ_{0}}{π sin^{2} θ_{0}} = 0.19A ≡ aA. (6)

For θ near 90^{∘} the projected area varies slowly with angle, so I suppose that it can be represented for 0 < cos θ < cos θ_{0} with the form,

A_{proj}(0 < cos θ < cos θ_{0}) = A(a + b cos^{2} θ + c cos^{3} θ),  (7)

where b and chosen such that A_{proj}(cos θ_{0}) = A cos θ_{0} and dA_{proj}(cos θ_{0})/d cos θ = A. Hence,

b = \frac{2 cos θ_{0} − 3a}{cos^{2} θ_{0}} = 1.71,  c = \frac{2a − cos θ_{0}}{cos^{3} θ_{0}} = −0.95.  (8)

Introducing the obliquity factor α = A_{proj}/A we have,

α(θ) = \{ \begin{matrix}cos θ & (cos θ > cos θ_{0}), \\ a + b cos^{2} θ + c cos^{2} θ & (cos θ < cos θ_{0}). \end{matrix}  (9)

For completeness, we note that if we represent the average location of PMT i by (x_{i}, y_{i}, z_{i}) and a unit vector along its axis by (e_{ix}, e_{iy}, 0), then a photon that originates at point (x, y, z) has polar angle with respect to this PMT given by,

cos θ_{i} = \frac{e_{ix}(x − x_{i}) + e_{iy}(y − y_{i})}{\sqrt{(x − x_{i})^{2} + (y − y_{i})^{2} + (z − z_{i})^{2}}} .  (10)

4 Solid Angle

The number of photoelectrons generated in a PMT by a point source at distance r from the effective center of the PMT is proportional to the solid angle subtended by the PMT. The PMT has projected area παr_{1}^{2} which is equivalent to a circle of radius r_{eff} = r_{1}\sqrt{α}. The equivalent circle subtends a cone of half angle θ_{eff} related by tan θ_{eff} = r_{eff/r} The solid angle Ω subtended by the equivalent circle is,

Ω = 2π(1 − cos θ_{eff}) = 2π (1 − \frac{r}{\sqrt{r^{2} + r_{eff}^{2}}} ) = 2π (1 − \frac{r}{\sqrt{r^{2} + αr_{1}^{2}}} ) .  (11)

Only when r_{1} ≪ r does the solid angle have the simple form παr_{1}^{2}/r^{2}.

Approximating the projected area of the photocathode by an equivalent circle would lead to some error in the solid angle Ω for θ ≈ 90^{∘} and r ≈ r_{1}. However, such a source region is in the outer LAB buffer, where there is no scintillation. For scintillation sources close to a photomultiplier, θ < ∼ θ_{0}, the use of an equivalent circle is a reasonable approximation.