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### Radioactivity

- Episode 508: Preparation for the radioactivity topic
- Episode 509: Radioactive background and detectors
- Episode 510: Properties of radiations
- Episode 511: Absorption experiments
- Episode 512: Nuclear equations
- Episode 513: Preparation for the exponential decay topic
- Episode 514: Patterns of decay
- Episode 515: The radioactive decay formula
- Episode 516: Exponential and logarithmic equations

## Radioactivity

Lesson for 16-19

This can be a fascinating area for students. Over the age of 16, they can work with certain radioactive sources, provided they are carefully supervised. One of your tasks as a teacher is to show them how to work safely, and to convince them that it is indeed safe to do so. You can do this by thinking first about the level of background radiation that we all live with, and showing how to ensure that exposure during experimental work does not add significantly to this lifelong exposure.

## Episode 508: Preparation for the radioactivity topic

Teaching Guidance for 16-19

- Level Advanced

Over the age of 16, students can work with certain radioactive sources, provided they are carefully supervised. One of your tasks as a teacher is to show them how to work safely, and to convince them that it is indeed safe to do so.

## Main aims of this topic

Students will:

- work safely with laboratory sources of ionising radiation
- understand the processes of radioactive decay
- use standard notation to represent nuclear processes
- solve problems involving half-life and exponential decay equations

## Prior knowledge

Students should have a basic, descriptive knowledge of radioactivity. They should know that there is background radiation; that radioactivity arises from the breakdown of an unstable nucleus; that there are three types of radioactive emission with different penetrating powers; the natures of alpha and beta particles and of gamma radiation; the meaning of the term half-life

.

## Where this leads

An understanding of radioactive processes is important in some optional topics, such as medical physics.

Exponential decay equations are closely related to equations for capacitor discharge, so there is an opportunity here to draw together two widely separated strands of Physics.

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### Radioactive background and detectors

## Episode 509: Radioactive background and detectors

Lesson for 16-19

- Activity time 70 minutes
- Level Advanced

This episode introduces the ubiquitous nature of radioactivity, and considers its detection. It draws on students’ previous knowledge, and emphasizes the importance of technical terminology.

Lesson Summary

- Demonstration: Detecting background radiation (10 minutes)
- Discussion: Sources of background radiation (15 minutes)
- Demonstration: Radioactive dust (10 minutes)
- Discussion and survey: Sources of radiation – should we worry? (15 minutes)
- Demonstration and discussion: Am-241 source, plus use of correct vocabulary (10 minutes)
- Demonstration: Spark detector (10 minutes)

## Demonstration: Detecting background radiation

Use a Geiger counter to reveal the background radiation in the laboratory. What is the signal

like? (It is discrete, erratic / random.) Does it vary from place to place in the room? (No; it may appear to; this is an opportunity to discuss the need to make multiple or longer-term measurements.) Does it vary from time to time? (No, it’s roughly constant.)

Count for 30 s to get a total count *N* ; repeat several times to show random variation. Calculate the average value of *N*.

(Note: a good rule of thumb is that the standard deviation is

√ *N* _{ave} , so roughly two-thirds of values of *N* will be within ± √ *N* _{ave}.)

Which is better: 10 counts of 30 s, averaged to get the activity, or one count of 300 s? (They amount to the same thing. The statistics of this is probably beyond most post-16 level students.)

Calculate the background count rate from the data (typical value is 0.5 counts per second or 30 counts per minute, but this varies a lot geographically.)

## Discussion: Sources of background radiation

Look at charts showing sources of background radiation. Consider how these might vary geographically, with time, occupation etc.

Pie chart for background radiations

Note that pie charts in text books etc showing relative contributions are often calibrated in units of equivalent dose of radiation called sieverts (symbol Sv); sievert is a unit which takes account of the effects of different types of radiation on the human body.

1 Sv = 1 J kg^{-1}

(1 J kg^{-1} = 1 m^{2} s^{-2})

Episode 509-1: Doses (Word, 40 KB)

Episode 509-2: Whole body dose equivalents (Word, 40 KB)

## Demonstration: Radioactive dust

Airborne radioactive substances are attracted to traditional computer and TV screens that use a high voltage. Similarly a charged

balloon will also accumulate radioactive dust and have an activity larger than the average background. The fresh dust in vacuum cleaner bags has a noticeably higher activity too.

Set up a Geiger counter to measure the activity of vacuum cleaner dust; don’t forget to measure background rate also.

Episode 509-3: Radiation in dust (Word, 29 KB)

## Discussion and survey: Sources of radiation – should we worry?

So radiation is all around us

. Indeed, most substances, and things, are radioactive. Students are radioactive! Typically 7000 Bq. So it’s dangerous to sleep with somebody

! However, most of the resulting radiation is absorbed within the owners

body.

Introduce *activity* of a sample as a quantity, measured in becquerels (Bq). Mass of a typical student is 70 kg, so

specific activity = 7000 Bq70 kg

specific activity = 100 Bq kg^{-1}.

The Radiation Protection Division of the Health Protection Agency (formerly the National Radiological Protection Board, NRPB) defines a radioactive substance as having specific activity 400 Bq kg^{-1}.

Should we worry about this? Ask students to complete this survey, and then re-visit at the end of the topic. For each statement, indicate whether they think it is true or false, or they don’t know.

S1: Radioactive substances make everything near to them radioactive.

S2: Once something has become radioactive, there is nothing you can do about it.

S3: Some radioactive substances are more dangerous than others.

S4: Radioactive means giving off radio-waves.

S5: Saying that a radioactive substance has a half life of three days means any produced now will all be gone in six days.

## Discussion and demonstrations: Am-241 source, plus use of correct vocabulary

## Radioactive sources

Follow the local rules for using radioactive sources, in particular do not handle radioactive sources without a tool or place them in close proximity to your body.

Place an Am-241 source close to the GM tube and measure the count rate, which will be impressive, compared to background. (Some end window GM tubes will not detect alpha emission from AM-241 but only weak gamma).

From here on, start to use appropriate technical vocabulary, drawing on students’ earlier experience. For example: Substances are *radioactive*, they emit radiation

when they *decay*. Why are some substances radioactive? (They contain *unstable* nuclei inside their respective atoms.) The unstable nucleus is called the *mother* and when it (she?) decays a *daughter* nucleus is produced (it’s not quite like human procreation!).

Eventually the activity of a radioactive substance must cease. However, point out that the decay of the americium doesn’t seem to be getting any less. There are a very large number of nuclei in there!

Am-241 is used in smoke alarms, so it won’t run out

. They are supplied with 33.3 kBq sources. The half-life is 458 year.

What particular property does nuclear radiation have – what does it do to the matter through which it passes? (It is *ionizing* radiation. It creates ions when it interacts with atoms.) What is an ion? (A neutral atom which has lost or gained (at least one) electron.)

## Simplified diagram of a smoke alarm

To knock an electron from an atom, the ionizing radiation transfers energy to the atom – this is how nuclear radiation is detected. It is not difficult to detect the presence of a single ion – electron pair, so it’s easy to detect the decay of *single* radioactive nucleus. Chemists can detect microgrammes or nanogrammes of chemical substances, physicists can detect individual atomic events.

## Demonstration: Spark detector

Show a spark detector responding to the proximity of an alpha source. (NB At first, do not refer to the source as an alpha source.) Move the source away a few centimetres; you do not need much distance in air to absorb this radiation. Ask students to recall which type of radiation is easily absorbed by air. (Alpha.)

Episode 509-4: Rays make ions (Word, 82 KB)

Comment that a GM tube is not dissimilar to a spark counter, but rolled up to have cylindrical symmetry. At this point, you could discuss the sophisticated design of a GM tube and associated counter. The end window is usually made of mica and has a plastic cover, with holes, to protect the mica.

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### Properties of radiations

## Episode 510: Properties of radiations

Lesson for 16-19

- Activity time 65 minutes
- Level Advanced

The focus of this episode is the properties of ionizing radiation. It is a good idea to introduce these through a consideration of safety.

Lesson Summary

- Discussion: Ionising radiation and health (10 minutes)
- Demonstration: Deflection of beta radiation (10 minutes)
- Student activity: Completing a summary table (10 minutes)
- Student experiment: Inverse square law for gamma radiation (30 minutes)
- Discussion: Safety revisited (5 minutes)

## Discussion: Ionising radiation and health

Why are radioactive substances hazardous? It is the ionising property of the radiation that makes it dangerous to living things. Creating ions can stimulate unwanted chemical reactions. If the radiation transfers sufficient energy it can split molecules. Disrupting the function of cells may give rise to cancer. Absorption of radiation exposes us to the *risk* of developing cancer.

Thus it is prudent to avoid all unnecessary exposure to ionising radiation. All deliberate exposure must have a benefit that outweighs the risk.

Radioactive *contamination* is when you get a radioactive substance on, or inside, your body (by swallowing it or breathing it in or via a flesh wound). The contaminating material then irradiates you.

How can you handle sources safely in the lab? Point out that you will be safe if you follow your local rules which will incorporate the following:

- Always handle sources with tongs
- Point the sources away from your body (and not at any anybody else)
- Fix the source in a holder which is not adjacent to where your body will be when you take measurements
- Replace sources in lead-lined containers as soon as possible
- Wash hands when finished

## Radioactive sources

Follow the local rules for using radioactive sources, in particular do not handle radioactive sources without a tool or place them in close proximity to your body.

## Demonstration: Deflection of beta radiation

Show the deflection of β by a magnetic field. (Make sure you have a small compass to determine which are the N and S poles of the magnet.) Is the deflection consistent with the LH rule? (Yes; need to recall that electron flow is the opposite of conventional current.) This demonstration is no good with the α particles, as they are absorbed too quickly by the air.)

NB The diagram above is common in textbooks, but is *only* illustrative. For the curvature shown of beta particles, the curvature of the alpha tracks would be immeasurably small.

## Student activity: Completing a summary table

Display the table, with only the headings and first column completed. Ask for contributions, or set as a task; compile results.

Episode 510-1: α, β and γ radiation (Word, 39 KB)

name | symbol | nature | electric charge | stopped by | ionising power | What is it? |
---|---|---|---|---|---|---|

Alpha | α | particle | +2 | mm air; paper | very good | He nucleus |

Beta | β | particle | -1 | mm Al | medium | very fast electron |

Gamma | γ | wave | 0 | cm Pb* | relatively poor | electromagnetic radiation |

Can you see any patterns in the table? (Most ionising – the largest electrical charge – is the least penetrating.) Can you explain this? (The most ionising transfer energy the quickest.) How can the electrical charge determined? Deflection in a magnetic field.)

*NB Gamma radiation is never completely absorbed (unlike alpha and beta) it just gets weaker and weaker until it cannot be distinguished form the background.

## Student experiment: Inverse square law for gamma radiation

Note: since you are unlikely to have sufficient gamma sources for several groups to work simultaneously, this experiment can be part of a circus with others in the next episode. Alternatively, it could be a demonstration.

Gamma radiation obeys an inverse square law in air since absorption is negligible. (Radiation spreads out over an increasing sphere. Area of a sphere = 4 π *r*^{ 2} , so as *r* gets larger, intensity will decrease as 1/ *r*^{ 2} . The effect of absorption by the air will be relatively small.

Episode 510-2: Range of gamma radiation (Word, 38 KB)

(Some students could do an analogue experiment with light, with an LDR or solar cell as a detector.)

When detecting g radiation with a Geiger tube you may like to aim the source into the *side* of the tube rather than the window at the end. The metal wall gives rise to greater secondary electron emission

than the window and this increase the detection efficiency.

Correct readings for background.

How can we get a straight line graph? We expect

*I* ∝ *r*^{ -2},

so a plot of *I* versus *r*^{ –2} should be direct proportion (i.e. a straight line through the origin). It is much easier to see if a graph is a straight line, rather than a particular curve. Lift the graph and look along the line – it’s easy to spot a trend away from linear. However, two points are worth noting:

- Sealed γ sources do not radiate in all directions, so do not expect perfect 1
*r*^{ 2}behaviour - You do not know exactly where in the Geiger tube the detection is taking place, so plotting
*I*^{ - ½ }against*r*gives an intercept, the systematic error in the measurement.

## Discussion: Safety revisited

Return briefly to the subject of safe working. Background radiation is, say, 30 counts per minute. How far from a gamma source do you have to be for the radiation level to be twice this? Would this be a safe working distance? (Probably.) How much has your lifetime dose of radiation been increased by an experiment like the above? (Perhaps one hour at double the background radiation level – a tiny increase. It will be safe enough to carry out a few more experiments.)

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### Absorption experiments

## Episode 511: Absorption experiments

Lesson for 16-19

- Activity time 40 minutes
- Level Advanced

This gives students the opportunity to work with radioactive sources.

Lesson Summary

- Student experiments: Absorption of radiation and report back (40 minutes)
- Demonstration: Absorption of radiation by living matter
- Student experiment (optional)

## Student experiments

Groups could work in parallel and report back to a plenary session.

Remind them to correct for the background count (taken at least twice – at the start and end of the main experiments and the two results averaged).

Range of alpha radiation

Episode 511-1: Use a spark counter (Word, 62 KB)

Range of beta radiation

Episode 511-2: The range of beta particles in aluminium and lead (Word, 38 KB)

Range of gamma radiation

An optical analogue for the absorption of γ particles by lead is the absorption of light by successive microscope slides.

Episode 511-3: Absorption in a liquid (Word, 54 KB)

Absorption of γ particles is an example of exponential decrease – check the data for a constant half thickness

, thus suggesting the type of physics involved. (Each mm of absorber is reducing the intensity by the same fraction.)

Episode 511-4: Absorbing radiations (Word, 38 KB)

## Demonstration: Absorption of radiation by living matter

To simulate the absorption of radiation by living matter use slices of different vegetables as absorbers, or a slice of bacon to represent human flesh.

Episode 511-5: Absorption in biological materials (Word, 53 KB)

## Student experiments: Optional

The first requires a sealed radium-226 source. Because Ra-226 is the parent to a chain of radioactive daughters, granddaughters and so on, you get a mixture of αs, βs and γs emitted. Challenge students to use absorbers to establish that all three radiation types are being emitted. (The maximum energies are:
*E*_{ α } = 7.7 MeV
*E*_{ β } = 3.3 MeV
*E*_{ γ } = 2.4 MeV)

The second is an extension of the β absorption experiment. You could speculate that some β particles might be back-scattered

(like Rutherford’s α particle scattering that first demonstrated the existence of the nucleus). A quick try shows that some β particles are indeed back-scattered.

## Radioactive sources

Follow the local rules for using radioactive sources, in particular do not handle radioactive sources without a tool or place them in close proximity to your body. Deliberately placing a radioactive source in contact with the skin would increase your dose of ionising radiation unnecessarily and increase the risks to your health. This is a criminal offence.

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### Nuclear equations

## Episode 512: Nuclear equations

Lesson for 16-19

- Activity time 70 minutes
- Level Advanced

Now that your students are familiar with different types of radiation, you can look at the processes by which they are emitted.

Lesson Summary

- Discussion: Nuclide notation and
*N*–*Z*plot (10 minutes) - Student Questions: Practice with notation (10 minutes)
- Worked Examples: Equations for alpha, beta and gamma decay (20 minutes)
- Student Questions: Practice with nuclear equations (30 minutes)

## Discussion: Nuclide notation

Revise nuclide notation: AZX

Discuss how *A*(mass or nucleon number), *Z* (charge or atomic number) and *N*(neutron number are related):

*A* = *Z* + *N*.

Discuss isotopes (common examples: H, D and T, U-235 and U-238, C-14 and C-12).

Set the task of finding out the name for nuclides having the same *A* but different *Z* (isobars), and the same *N* but different *Z* (isotones).

Show an *N*-*Z* plot (Segrè plot).

## Student questions: Practice with notation

Set some simple questions involving nuclide notation.

Episode 512-1: Nuclide notation (Word, 36 KB)

Grid showing change in *A* and *Z* with different emissions.

## Worked examples: Equations for alpha, beta and gamma decay

Nuclear decay processes can be represented by nuclear equations. The word *equation* implies that the two sides of the equation must balance

in some way.

Episode 512-2: Decay processes (Word, 53 KB)

You could give examples of equations for the sources used in school and college labs.

α sources are americium-241, 24195Am

24195Am → 23793Np + 42He

β ^{-} sources are strontium-90, 9038Sr

9038Sr → 9039Np + 0-1e

The underlying process is:

n → p + e^{-} + n

Here, n is an antineutrino. Your specification may require you to explain why this is needed to balance the equation.

You can translate n → p + e^{-} + n into the AZ notation:

10n → 11H + 0-1e

γ sources are cobalt-60, 6027Co. The γ radiation comes from the radioactive daughter 6028Ni of the β decay of the 6027Co.

The 6028Ni is formed in an excited state

and so almost immediately emits a γ ray. They are only emitted after an α or β decay, and all such γ rays have a well-defined energy. (So a cobalt-60 source which is a pure gamma emitter must be designed so that betas are not emitted. How? – (by encasing in metal which is thick enough to absorb the betas but which still allows gammas to escape.)

## Student questions: Practice with nuclear equations

Episode 512-3: Practice with nuclear equations (Word, 66 KB)

The more unusual decay processes (positron emission, neutron emission, electron capture) could be included, and students challenged to write them as nuclear equations.

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### Preparation for the exponential decay topic

## Episode 513: Preparation for the exponential decay topic

Lesson for 16-19

- Level Advanced

Students will:

- Define the term half-life
- Make calculations involving numbers of half-lives
- Relate half-life to decay probability l
- Measure the half-life of a fast-decaying nuclide
- Use exponential and logarithmic equations for radioactive decay

## Main aims

## Prior knowledge

Basic, descriptive radioactivity should already have been covered. Students will have previously been introduced to the term half-life, but are unlikely to be confident in using the quantity in calculations.

## Where this leads

The mathematics of exponential decay parallels that of capacitor discharge, damped SHM etc, so these would be suitable topics to tackle next. If students have already met these topics, you could usefully spend time drawing the parallels between them.

### Up next

### Patterns of decay

## Episode 514: Patterns of decay

Lesson for 16-19

- Activity time 130 minutes
- Level Advanced

This episode assumes that students already have an idea of half-life, and links it to the empirical decay curve.

Lesson Summary

- Discussion and demonstration: Measuring half life (40 minutes)
- Worked examples: Involving whole numbers of half lives (30 minutes)
- Student example and discussion: Plotting a graph (30 minutes)
- Student questions: Calculations (30 minutes)

## Discussion and demonstrations: Measuring half life

Each radioactive nuclide has its own unique half-life. Values range from millions of years (e.g. uranium-238 at 4.47 × 10^{9} year) to minute fractions of a second

(e.g. beryllium-8 at 7 × 10^{-17} s).

Sealed school/college sources have half-lives chosen to ensure that they will remain radioactive over a period of years (though Co-60 will become significantly less active year-by-year):

Radiation | Source | Nuclide | Notes | T_{1/2} |
---|---|---|---|---|

α | americium-241 | 24195Am | Also emits gammas | 458 year |

β | strontium-90 | 9038Sr | The energetic betas come from the daughter Ys-90 | 28.1 year |

γ | cobalt-60 | 6027Co | Also emits betas, but these may be absorbed internally | 5.26 year |

You need something else if you are to measure half-life in the course of a lesson. A short half-life source commonly available in schools and colleges is protactinium-234,
*T*_{ ½ } = 72 s.
Another is radon-220,
*T*_{ ½ } = 55 s.

Demonstrate the measurement of half-life for one of these. (There is a data logging opportunity here.) It is vital to correct for background radiation. Explain how to find *T*_{ ½ } from the graph.

(Students could repeat this experiment for themselves later.)

## Demonstration of half life of protactinium: Measuring half life

Episode 514-1: Half life of protactinium (Word, 64 KB)

## Worked examples: Involving whole numbers of half-lives

Here is a set of questions involving integral numbers of half lives. You could set them as examples for your students, or work through them to explain some basic ideas.

Question: The nuclear industry considers that after 20 half lives, any radioactive substance will no longer present a significant radiological hazard. The half life of the fission product from a nuclear reactor, caesium-137, is 30 years. What fraction will still be active after 20 half lives?

Answer: Use the long method. Calculate ½ of ½ and so on for twenty steps:

1/contents4 after 10 half lives

1/2048, …, 1/1048576 after 20 half lives

i.e. less than one millionth of the original quantity remains radioactive.

A quicker method is to calculate ½ ^{20}. Show how this is done with a calculator, using the y^{x} key.

Question: How many years into the future will Cs-137 be safe

?

Answer: 20 × 30 years = 600 years

Question: If after ten half lives the activity of a substance is reduced to one thousandth of its original value, how many more half lives must elapse so that the original activity is reduced to one millionth of its original value?

Answer: Ten half lives reduces activity by a factor of 11000. One millionth is 11000 × 11000, so ten more half lives are needed.

## Student example and discussion: Plotting a graph

Provide your students with the following data.

time | activity / Bq |
---|---|

0 | 10057 |

5 | 6253 |

10 | 3648 |

15 | 2296 |

20 | 1403 |

25 | 798 |

30 | 508 |

35 | 307 |

40 | 201 |

Ask them to inspect the data and estimate the half-life. (It lies between 5 and 10 seconds.)

Now ask them to plot a graph and thereby determine the half life of the substance. They should deduce a total time of 35 seconds for five halvings of the original activity, gives a half life of 7 seconds.

Emphasise the definition of half life *T*_{ ½ }. The half life of a radioactive substance is the time taken *on average* for half of *any quantity* of the substance to have decayed. (Check the precise wording of your specification.)

Introduce the term *exponential* to describe this behaviour, in which a quantity decreases by a constant factor in equal intervals of time.

Episode 514-2: Half life (Word, 25 KB)

Exponential behaviour in nature is very common. It appears in the post-16 specification several times, so this may not be the first time your students have met it. (The equalisation of electric charge on a capacitor whose plates are connected via a resistor; the reduction in amplitude of a damped simple harmonic oscillator; the absorption of electromagnetic radiation passing through matter (e.g. γ rays by lead, visible light by glass); Newton’s Law of Cooling. So either refer back to previous situations and make the analogy explicit, or flag forward to other examples to come.

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### The radioactive decay formula

## Episode 515: The radioactive decay formula

Lesson for 16-19

- Activity time 50 minutes
- Level Advanced

Here, the key idea is the random nature of the decay. Avoid simply pulling pull equations out of the air – at least make them plausible.

Lesson Summary

- Discussion: The meaning of the decay constant l (15 minutes)
- Discussion: The link with half-life (15 minutes)
- Student experiments: Analogue experiments linking probability with decay rates (20 minutes)

## Discussion: The meaning of the decay constant l

Start from the definition of the decay constant λ: the probability or chance that an *individual* nucleus will decay per second. (You may like to comment on the problem with notation in physics. λ is used for wavelength as well as the decay constant. The context should make it unambiguous.)

Units: λ is measured in s^{-1} (or h^{-1} , year^{-1} , etc).

If you have a sample of *N* undecayed nuclei, what will its activity *A* be? In other words, how many of the *N* will decay in a second?

*A* = λ × *N* (because the probability for each of the *N* is λ).

As time passes, *N* will get smaller, so *A* represents a *decrease* in *N* . To make the formula reflect this, it needs a minus sign.

*A* = − λ × *N*

Plausibility: The more undecayed nuclei you have, and the greater the probability that an individual one will decay, the greater the activity of the sample.

In calculus notation, this is

d *N* d *t* = − λ × *N*

Ask your students to put this equation into words. (The rate of decay of undecayed nuclei *N* is proportional to the number N of undecayed nuclei present.) This is the underlying relationship in any process that follows *exponential decay* .

More generally: if the rate of change is proportional to what is left to change, then exponential decay follows. Conversely if data gives an exponential decay graph, then you know something about the underlying process.

Episode 515-1: Smoothed out radioactive decay (Word, 43 KB)

Episode 515-2: Half-life and time constant (Word, 35 KB)

## Discussion: The link with half-life

How are λ and *T*_{ ½ } linked?

If a nuclide has a large value of λ, will it have a long or short *T*_{ ½ } ? (A high probability of decay implies it will decay quite quickly whihc leads to a short half life, and vice versa.)

What type of proportionality does this suggest? (Inverse proportion.)

Thus λ ~ 1*T*_{ ½ }.

In fact,

*T*_{ ½ } = ln(2)λ

*T*_{ ½ } = 0.693λ

This is a very important and useful formula. Depending which of l or *T*_{ ½ } is easiest to measure experimentally, the other can be determined. In particular it allows very long half lives (sometimes millions of years) to be determined. How could we measure the 4.5 billion year half-life of uranium-238? (Take a sample of a known number of U-238 atoms; count how many decay in one second. This gives l, and we can calculate half-life.)

## Student experiments (or demonstrations): Analogue experiments linking probability with decay rates.

Explore some analogue systems to reinforce the way in which decay probability is related to half-life. Each gives a good exponential graph. For example:

Throw a large number of dice. A 6 represents decayed

, and this dice is removed. Each throw represents the same time interval. Each face has a probability of 16 of being upwards on each throw. (A quick way to find out how many remain to decay is to weigh them.)

Episode 515-3: Modelling radioactive decay (Word, 36 KB)

Cubes with differently coloured faces instead of dice can incorporate 3 different decay constants

e.g. colouring three faces red gives a chance of a red face being uppermost representing decay has a probability of decay of ½; colour two faces blue (chance of decay = 2 × 16, so 13), and one face yellow (chance of decay 16). Sugar cubes can be used, painted with food colouring.

Throw drawing pins: decay → point upwards. Safety: beware sharp points!

Is the chance of decay 12?

The drop in height of the head on a glass of beer usually shows exponential behaviour – opportunity for a field trip?

Water flowing out through a restriction: at any instant the water remaining represents the undecayed nuclei; the water that has flowed out represents the decayed nuclei. Flow rate depends upon the pressure head ~ the height of water remaining ~ quantity of water remaining.

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### Exponential and logarithmic equations

## Episode 516: Exponential and logarithmic equations

Lesson for 16-19

- Activity time 145 minutes
- Level Advanced

Students may find this mathematical section difficult. It is worth pointing out that they have already covered the basic ideas of radioactive decay in the earlier episodes.

Lesson Summary

- Discussion: The exponential decay equation (15 minutes)
- Student question: An example using the equation (20 minutes)
- Discussion: The logarithmic form of the equation (15 minutes)
- Worked example: Using the log equation (20 minutes)
- Student questions: Practice calculations (60 minutes)
- Discussion (optional): Using Lilley’s formula (15 minutes)

## Discussion: The exponential decay equation

Explain that the equation

*N* = *N*_{0} × e^{-λ t}

can be used to generate an exponential decay graph. Work through a numerical example, perhaps related to the dice-throwing analogue
(N_{0} = 100 ;
λ = 16). Make sure that your students know how to use the e^{x} key on their calculators.

Emphasise that similar equations apply to activity

*A* = *A*_{0} × e^{-λ t}

and count rate

*C* = *C*_{0} × e^{-λ t}

(Not all the radiation emitted in all directions by a source will collected by a detector lined up in one direction from a source)

## Student question: An example using the equation

Set students the task of drawing a graph for a lab source, e.g. Co-60 ( λ = 0.132 y^{-1}
,
C_{0} = 200 counts s^{-1}
). They should first calculate and tabulate values of C at intervals of 1 year, and then draw a graph. From the graph, deduce half-life. Does this agree with the value from

*T*_{ ½ } = ln(2)λ

or

*T*_{ ½ } = 0.693λ

## Discussion: The logarithmic form of the equation

Point out that a straight line graph is usually more useful than a curve, particularly when dealing with experimental data. Introduce the equation
ln *N* = ln (*N*_{0}) − λ *t*.
Emphasise that this embodies the same relationship as the exponential equation. Use a sketch graph to how its relationship to the straight line equation

*y* = *m**x* + *c*

(
intercept = ln(*N*_{0})
, gradient = −l).

## Worked examples: Using the log equation

Start with some experimental data (e.g. from the decay of protactinium), draw up a table of ln (count rate) against time. Draw the log graph and deduce l (and hence half-life). The experimental scatter should be obvious on the graph, and hence the value of a straight line graph can be pointed out.

You will need to ensure that your students can find natural logs using their calculators.

## Student questions: Practice calculations

Your students should now be able to handle a range of questions involving both e^{x} and ln functions. It is valuable to link them to some of the applications of radioactive materials (e.g. dating of rocks or ancient artefacts, diagnosis and treatment in medicine, etc).

Episode 516-1: Decay in theory and practice (Word, 42 KB)

Episode 516-2: Radioactive decay with exponentials (Word, 77 KB)

Episode 516-3: Radioactive decay used as a clock (Word, 48 KB)

## Radio carbon dating

Episode 516-4: Two important dating techniques (Word, 29 KB)

## Discussion (optional): Using Lilley’s formula

Some students may benefit from a simpler approach to the mathematics of radioactive decay, using Lilley's formula

fraction = ½ ^{n}.

When first introduced at pre-16 level, radioactivity calculations are limited to integral number of half lives. After 1, 2, 3, …, half-lives, 1/2, 1/4, 1/8, … remains. The pattern here is that after n half lives, a fraction = ½ ^{n} remains to decay.

This formula works for non integral values of n ; i.e. it also gives the fraction remaining yet to decay after any non-whole number of half-lives (e.g. 2.4, or 3.794). To use this formula, a little skill with a calculator is all that is required.

For example: The *T*_{ ½ } of 146C is 5730 years. What fraction of a sample of 146C remains after 10 000 years? Answer:

fraction remaining = ½ ^{n}

The number of half lives,

n = 10 0005730

n = 1.745

Thus the

fraction remaining = ½ ^{1.745}

And using the y^{x} button on a calculator gives fraction = 0.298.

If students know how to take logarithms (or can lean the log version of the formula), they can solve other problems:

How many years will it take for 99% of 6027Co to decay if its half life is 5.23 yr?

The fraction remaining is 1%, so

fraction = 0.01.

Hence

0.01 = ½ ^{n}

Taking logs of both sides gives

ln(0.01) = n × ln( ½ )

giving

n = 6.64 half lives,

and so the

number of years = 6.64 × 5.23 years

number of years = 34.7 years.