NMD-06

Well-counter geometry

ε = ½ (1 − h/√(h²+R²)). On-axis geometric efficiency of a circular well.

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CountingWell counter

Governing equation

\varepsilon=\tfrac12\left(1-h/\sqrt{h^{2}+R^{2}}\right)

$h \in \left[0.1, 10\right]$

where

h: Source height (cm)
R: Well radius (cm)
\varepsilon: Geometric efficiency (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\varepsilon=\tfrac12\left(1-h/\sqrt{h^{2}+R^{2}}\right)

$h \in \left[0.1, 10\right]$

$h = 0.1$ $\varepsilon = 0.467$

Along the curve

②

Domain of the variable — blue

h_{\min} = 0.1,\quad h_{\max} = 10

$h \in \left[0.1, 10\right]$

$h = 0.1$ $h = 0.1$

Along the curve

③

Response curve — green

\varepsilon=\tfrac12\left(1-h/\sqrt{h^{2}+R^{2}}\right)

$h \in \left[0.1, 10\right]$

$h = 0.1$ $\varepsilon = 0.467$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\varepsilon}{\mathrm{d}h}

$h \in \left[0.1, 10\right]$

$h = 0.162$ $\varepsilon^{\prime} = -0.327$

Along the curve

Lecture brief

Historical brief

MIRD schema, in-vivo decay, uptake fractions and PET coincidence turned unsealed sources into organ dose. The sheets compute activity, residence and well-counter geometry. This sheet (NMD-06 — Well-counter geometry) is the form associated with Well counter. Working symbols: $h$ , $R$ $\rightarrow$ $\varepsilon$ . A NaI well almost 4π-wraps a small tube. Solid angle Ω/4π is the geometric ε.

Purpose

Purpose: compute

\varepsilon

from

h

R

in Nuclear medicine via

\varepsilon=\tfrac12\left(1-h/\sqrt{h^{2}+R^{2}}\right)

ε = ½ (1 − h/√(h²+R²)). On-axis geometric efficiency of a circular well. Use it when a real nuclear medicine question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

h = 2.000\,\mathrm{cm}

R = 1.500\,\mathrm{cm}

, the governing relation

\varepsilon=\tfrac12\left(1-h/\sqrt{h^{2}+R^{2}}\right)

yields

\varepsilon = 0.1000\,\mathrm{—}

. A crystal cup and a point source on axis. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Source height h2 cmWell radius R1.5 cm

Outputs

Geometric efficiency \varepsilon0.1000 —

Reading speed

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Narration of this film

A crystal cup and a point source on axis.

A NaI well almost 4π-wraps a small tube. Solid angle Ω/4π is the geometric ε.

Reading speed

Watch on YouTube