Part 8

We use the equation from page 12:

A₀ = Nφσ (1 - e^{-λt})

where N is the number of target atoms. (One microgram of arsenic contains (10^{-6} gram/75 grams per mole) × 6.02 × 10²³ atoms per mole which is 8 × 10¹⁵ atoms of arsenic.)

φ is the neutron flux. (This would be known to the reactor operator. It is usually measured by inserting materials of known composition and measuring their activation. In this case, φ = 10¹³ neutrons per square centimeter per second.)

σ is the activation cross section. (Neutron cross sections have been measured and tabulated by scientists. For the activation of arsenic-75 to arsenic-76, the cross section is known to be 4.2 × 10^{-24} square centimeter.)

λ is the disintegration constant for arsenic-76. (Here, λ = (ln 2/t{½},(in hours); t{½}, the half-life for arsenic-76, is 26.6 hours so λ = (0.693/26.6) = 0.026.)

t is the time of the irradiation. (Here t is 12 hours.)

Therefore: A₀, the activity of arsenic-76,

= 8 × 10¹⁵ × 10¹³ × 4.2 × 10^{-24} × (1 - e^{-0.026 × 12})

(Note: e is a physical constant, 2.71+)

= 9 × 10⁴ disintegrations per second per microgram

2. Determination of activity of arsenic measured in the sample and corrected back to the time of removal from the reactor.

We use the equation:

A₁ = (R)/(E × F) e^{λt}

where R is the measured count rate. (In this case, R is the number of counts per second observed in the 0.559-MeV gamma-ray peak, which is 5300 counts in 20 minutes or 4.4 counts per second.)

E is the efficiency of the detector. (In this case, it is the number of counts observed in the 0.559 peak for each 0.559-MeV gamma ray emitted by a radioactive material at the sample distance. This is known for the detector being used by making other measurements and, for the set-up used here, is 0.010.)

F is the average number of 0.559-MeV gamma rays emitted in each disintegration of arsenic-76. (This can be deduced from the decay scheme of arsenic-76. See the decay scheme for manganese-56 on page 13. In the decay of arsenic-76 the number of 0.559-MeV gamma rays emitted per disintegration is approximately 0.41.)

λ is the disintegration constant for arsenic-76. (0.026, see page 49.)

t is the decay time. (This is the number of hours from the time the sample was removed from the reactor to the time it was counted, or 5 hours.)

Therefore, A₁, the activity of arsenic-76 produced in the sample at the time of removal from the reactor,

= (4.4 counts per second)/(0.010 × 0.41) e^{0.026 × 5 hours}

= 1200 disintegrations per second

3. Calculation of arsenic concentration in the sample.

We use the equation:

Concentration in parts per million = (A₁)/(A₀ × W) 10⁶

where A₁ and A₀ were determined above and W is the weight of sample analyzed or 300 micrograms (0.0003 gram).

Therefore the concentration is

(1200)/(9 × 10⁴ × 300) × 10⁶ = 44 parts per million.