A position-sensitive particle detector turns a true hit at position into a reading…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Fourier analysis and integral transforms

Mathematical Methods for Physics, Ch Fourier transforms, spectra and convolution

Problem

A position-sensitive particle detector turns a true hit at position

u

into a reading spread uniformly over a plateau of width

a

centred on

u

: a hit at

u

contributes

g(x-u)

to the trace at position

x

, where

g(x) = \begin{cases}1/a, & |x| \le a/2\\ 0, & |x| > a/2\end{cases}

(height

1/a

, so that

\int g(x)\,\mathrm{d}x = 1

: one hit always registers as one count, however it is smeared). The trace is then passed through a second, identical stage with the same smearing window, so the overall response relating a true hit at

u=0

to the final reading at

x

is the self-convolution

h(x) = (g*g)(x) = \int_{-\infty}^{\infty} g(v)\,g(x-v)\,\mathrm{d}v.

Both stages are perfectly uniform over their window and respond identically at every position (a translation-invariant response), and edge effects at the ends of the detector are ignored. It has already been shown that a rectangular pulse of height

1

and width

a

, centred at the origin, has Fourier transform

\tilde f(k) = 2\sin(ka/2)/k

under the convention

\tilde f(k) = \int f(x)e^{-ikx}\,\mathrm{d}x

. ((a)) Evaluate the convolution integral directly, by finding the length of overlap between the two width-

a

windows as a function of

x

, and so find

h(x)

for every

x

. ((b)) Without integrating

h(x)

itself, use the convolution theorem together with the rectangle transform quoted above to write down

\tilde h(k)

. ((c)) Take

a = 2.00\,\mathrm{mm}

. Evaluate

\tilde h(k)

k = 0.785\,\mathrm{rad\,mm^{-1}}

(chosen so that

ka/2 = \pi/4

), and state the full base-to-base width of

h(x)

compared with the width

a

of a single stage.

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