310 7.9  Summary Points

Answers

a

Since the spatial displacement x =​ c.δt/​2 the spatial precision Δx =​ c.Δ(δt)/​2. The

error in the arrival time measurement Δ(δt) here is 10% of the sampling time (1/​f)

where f is the sampling frequency so:

Δx =​ 3 × 108 × 0.1 × (1/​(2.7 × 109)/​2 =​ 5.6 × 10–​3 m or 5.6 mm

b

This question illustrates the importance of properly calculating errors

from the known precision of multiple measurement parameters. The SNR =​

ks/​√(ks+​nk2) where ks is the signal rate of the coincidence detection, n =​ 2 for

a delayed confidence method and k2 =​ 2k1. Δt is the rate of random detected

coincidences given a random detection rate k1 from a single detector. Under

normal applications, k2 is much smaller than ks and so SNR is ~ √ks; how­

ever, here ks and k2 have similar orders of magnitude and so both need to be

accounted for. So

SNR =​ 1 × 106/​√(1 × 106 +​ (2 × 3 × 105)) =​ 790.568 counts/​s to three decimal places.

But we need to quote this properly considering the precision by which we

can measure the SNR,

Δ(SNR), which is given by the error on its measurement considering the error on

both ks and k2. Using the general method of differentials to estimate errors,

this indicates that:

Δ(SNR) =​ √((Δks.∂(SNR)/​∂ks)2+​(Δk2.∂(SNR)/​∂k2)2)

The partial derivatives can be worked through as:

∂(SNR)/​∂ks) =​ (1/​√(ks+​nk2)) –​ (1/​2)ks/​(ks+​nk2)3/​2

=​ (1/​√(1 × 106 +​ (2 × 3 × 105)) –​(0.5 × 1 × 106)/​((1 × 106 +​ (2 × 3 × 105))1.5

=​ 5.4 × 10–​4 s per count

∂(SNR)/​∂k2 =​ –​(1/​2)nks/​(ks+​nk2)3/​2 =​ –​(0.5 × 2 × 1 × 106)/​((1 × 106 +​ (2 × 3 × 105))1.5

=​ –​4.9 × 10–​4 s per count. So:

Δ(SNR) =​ √((0.1 × 106 × 5.4 × 10-​4)2 +​ (80 × 103 × (–​4.9 × 10–​4))2)

=​ 67 counts/​s

So:

SNR =​ 791 ± 67 counts/​s

7.9  SUMMARY POINTS

◾There is a plethora of well-​characterized chemical methods to specifically con­

jugate biomolecules to other biomolecules or to nonliving substrates with high

affinity.

◾Thin model organisms of nematode flatworms and zebrafish have proved par­

ticularly useful in generating in vivo biological insight from light microscopy

techniques.

◾Molecular cloning tools have developed mainly around model microbial organisms

and can be used to genetically modify DNA and insert it into foreign cells.

◾High-​quality crystal formation is in general the bottleneck in crystallography.