Chapter 9, Problem 9.14P

A counting system has a background of 360 counts during a 20-minute counting period. What is the lower limit of detection with this system for counting times of
(a) 2 minutes?
(b) 20 minutes?
(c) 200 minutes?

To determine

Lower limit of detection for counting times of 2 min

Answer to Problem 9.14P

Lower limit detection is 24 cpm.

Explanation of Solution

Given:

Time during which gross count was made, $t_g=2\min$

Background count, $n_{\text{b}}=360counts$

Time during which background count was made, $t_{\text{b}}=20\min$

Formula used:

Background counting rate, $r_b=\frac{n_b}{t_b}$

Lower limit of detection, $\text{LLD}=\frac{3.29\sqrt{r_b{t}_g\left(1+\frac{t_g}{t_b}\right)}+3}{t_g}$

Calculation:

Background counting rate, $r_b=\frac{n_b}{t_b}=\frac{360}{20}=18cpm$

Lower limit of detection, $\text{LLD}=\frac{3.29\sqrt{r_b{t}_g\left(1+\frac{t_g}{t_b}\right)}+3}{t_g}$

  $\text{LLD}=\frac{3.29\times \sqrt{\left(18\frac{\text{ counts }}{\min }\right)\times 2\min \times \left(1+\frac{2\min }{20\min }\right)}+3}{2\min }=12\text{cpm}$

Conclusion:

Lower limit detection is 12 cpm.

To determine

Lower limit of detection for counting times of 20 min

Answer to Problem 9.14P

Lower limit detection is 4.6 cpm.

Explanation of Solution

Given:

Time during which gross count was made, $t_g=20\min$

Background count, $n_{\text{b}}=360counts$

Time during which background count was made, $t_{\text{b}}=20\min$

Formula used:

Background counting rate, $r_b=\frac{n_b}{t_b}$

Lower limit of detection, $\text{LLD}=\frac{3.29\sqrt{r_b{t}_g\left(1+\frac{t_g}{t_b}\right)}+3}{t_g}$

Calculation:

Background counting rate, $r_b=\frac{n_b}{t_g}=\frac{360}{20}=18cpm$

Lower limit of detection, $\text{LLD}=\frac{3.29\sqrt{r_b{t}_g\left(1+\frac{t_g}{t_b}\right)}+3}{t_g}$

  $\text{LLD}=\frac{3.29\times \sqrt{\left(18\frac{\text{ counts }}{\min }\right)\times 20\min \times \left(1+\frac{20\min }{20\min }\right)}+3}{20\min }=4.6\text{cpm}$

Conclusion: Lower limit detection is 4.6 cpm.

To determine

Lower limit of detection for counting times of 200 min

Answer to Problem 9.14P

Lower limit detection is 658 cpm.

Explanation of Solution

Given:

Time during which gross count was made, $t_g=200\min$

Background count, $n_{\text{b}}=360counts$

Time during which background count was made, $t_{\text{b}}=20\min$

Formula used:

Background counting rate, $r_b=\frac{n_b}{t_b}$

Lower limit of detection, $\text{LLD}=\frac{3.29\sqrt{r_b{t}_g\left(1+\frac{t_g}{t_b}\right)}+3}{t_g}$

Calculation:

Background counting rate, $r_b=\frac{n_b}{t_b}=\frac{360}{20}=18cpm$

Lower limit of detection, $\text{LLD}=\frac{3.29\sqrt{r_b{t}_g\left(1+\frac{t_g}{t_b}\right)}+3}{t_g}$

  $\text{LLD}=\frac{3.29\times \sqrt{\left(18\frac{\text{ counts }}{\min }\right)\times 200\min \times \left(1+\frac{200\min }{20\min }\right)}+3}{200\min }=3.3\text{cpm}$

Conclusion:

Lower limit detection is 3.3 cpm.