6.0 Graphs. Graphs illustrate a relationship between two quantities or two sets of experimental data.

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1 6 Graphs 37

2 6. Graphs Graphs illustrate a relationship between two quantities or two sets of eperimental data. A graph has two aes: a horizontal -ais and vertical y-ais. Each is labelled with what it represents. In radiation science, these are usually quantities eg time, activity. Normally, the thing that is being measured (dependent variable) is plotted on the y-ais and the thing causing the change the one we can control (independent variable) is plotted on the -ais. Open a radiation physics tetbook and you will find lots of graphs. They are used to eplain lots of things without having to use lots of words. When we look at a graph, there are several things to note: What quantity each ais represents and its unit of measurement The shape of the graph: if it s a straight line or curved shape for eample, there is usually a mathematical relationship or formula relating the two quantities. This tells us the effect one has on the other The slope or gradient, and the direction eg increasing from left to right or decreasing from left to right 6.1 Straight line graphs Here is an eample of a straight line graph: y

3 the aes are labelled and y it s a straight line which passes through the origin of the graph (,) as increases, y increases A straight line like this represents direct proportion between the two quantities and y. y is changing as changes: y is proportional to. We can use the symbol (proportional to) to write y This is not the same as y =, unless they are changing at eactly the same rate. Look at this table of values that shows this relationship. For eample, when =, y = y You should be able to see that as increases by 1 each time, so does y. This shows they are both increasing at the same rate. This rate of change can be calculated from the gradient of the graph. 6.2 Gradient of a straight line You can work out the gradient between two points on the line, by dividing the rise of the line (the vertical height - that s the change in y) by the run (horizontal distance - the change in ) The origin is the point (,) y rise from to 5 = 5 Run from to 5 = 5 39

4 So we can say, the gradient between the two points = rise = 5 = 1 run 5 The gradient shows how y changes with. In this case, as increases by one, so does y assume the 1, and normally write y = y = 1 but we In general, for a straight line passing through the origin y = m where m is the gradient. Work out the gradient of this graph: This point is called the y- intercept y rise = run = gradient = Notice that this line does not pass through the origin (,). It crosses the y-ais at y = 3 (the y-intercept). Complete a table of values for this graph, so that when =, y = 3; when = 1, y = 4 etc: y We can see that to get from an value to its corresponding y value, we have added 3 4

5 rise = 5 = 1 run 5 Hopefully, you worked out the gradient of the graph was 1 So the formula that relates and y in this case is y = or more usually y = + 3 Note that this gradient of 1 is a positive value, so the graph rises from left to right. If the graph was falling rather than rising, the gradient would be a negative value. 6.3 Equation of a straight line In general, for a straight line y = m + c y-intercept gradient If the y-intercept is zero, then y = m This is the same as y and means y is directly proportional to. So a straight line passing through zero can be used to test direct proportion. 6.4 Using Microsoft Office Ecel to plot a straight line graph Radiographers wear a personal radiation monitor, usually a thermoluminescent dosemeter (TLD) such as lithium fluoride (LiF). When eposed to ionising radiation, electrons within the LiF are given energy and become trapped. In this way, the LiF stores the energy. After 3 months wear, the dosemeter is processed by heating the lithium fluoride discs. Heating releases the electrons from their traps and the stored energy is emitted in the form of light. The amount of light emitted is directly proportional to the eposure to ionising radiation. We will use Ecel to plot a graph from a spreadsheet of the light output from a set of TLDs, eposed to different amounts of -ray photons. 41

6 Open Ecel from the Start menu in the bottom left hand corner or from the icon on the desktop When you select a specific cell, it can be identified by its row number and column letter. A workbook is opened, containing rows and columns, identified by numbers and letters respectively. In cell A1 type the title of this spreadsheet: Light output from lithium fluoride TLDs In cell A2 type Absorbed dose (mgy) and in cell B2 type Light output (arbitrary units). Click on the junction between the column heads and drag to widen the columns. Enter the data: Absorbed dose (mgy) Light output (arbitrary units) The first column will automatically form the ais and the second column the y ais Now create a graph: click and drag to select the two columns of data. Click on the Chart Wizard icon on the toolbar and select the Standard type of graph XY scatter (a scattergram). Click net to see the scattergram and net again. Give an appropriate Chart title and names/units for the and y ais. Place your chart as a new sheet and Finish. You should have a scattergram that appears to show increasing light output gives increasing absorbed dose of - ray photons. Add a line of best fit to these data points by clicking on Chart>add trendline. Select type: linear trendline; under Options, select Display equation on chart. You can now view the trendline and its equation 42

7 The equation is in the form y = m + b y = This means the gradient is 1.17 and the y-intercept.13. However, if we look at the data table, the first point was (,). In producing a line of best fit to the data points, the trendline has not passed through this point. This can be remedied by left-clicking on the trendline to select it. Then right-click and select Format trendline. Under options, select Set intercept =. The trendline will now pass through the point (,) and the equation will be y = 1.11 Analysing this graph, we can say light output is directly proportional to absorbed dose because the line passes through the origin. The gradient is the rate of increase: an absorbed dose of 1 mgy gives a light output of 1.1 units. 6.4 Inverse proportion and hyperbolic curves The previous eample looked at direct proportion: when one quantity increased, another quantity also increased at the same rate. Inverse proportion is where one quantity increases and a second quantity decreases. Mathematically, this is written y 1 and can be epressed either as y is inversely proportional to or y is directly proportional to 1/. It can also be written y -1 See section 1.5 An eample of inverse proportion is the relationship of photon energy (E) to wavelength ( ): E 1 Using the formula given in section 1.5 and the method in question 6, we can produce the following table of values: 43

8 wavelength 1-13 m wavelength 1-13 m Photon energy (kev) wavelength 1-13 m Using Ecel to plot a scattergram, we get a curve. If we select a power trendline and equation, the graph looks like this: relationship of photon energy and wavelength y = photon energy (kev) The shape of the curve is a hyperbola. It does not pass through the origin. The equation indicates y is inversely proportional to since y -1. It can be seen that as photon energy increases, wavelength decreases. We could also show that y is directly proportional to 1/ by calculating one divided by the photon energy and then plotting the graph. This should give a straight line. wavelength against 1/photon energy 5 4 y = /photon energy 44

9 6.5 Eponential graphs The simplest eponential relationship is y = a where y is a variable, a is a constant value and is a variable power (also called an eponent) You will meet several eponential relationships in radiography, which at first glance look frightening: Used to calculate the activity of a radioactive source at a certain time For eample, radioactive decay: A t = A o e - t Used to calculate potential difference across a capacitor: used in capacitor discharge units in mobile radiography and some eposure timer circuits Used to calculate the intensity of photons transmitted through a thickness of material Capacitor charging and capacitor discharging V t = V o 1-e t/rc V t = V o e t/rc Attenuation of -ray photons in a material I t = I o e - All of these can be considered in the same form as y = a variable power variable constant Let s look at radioactive decay A t = (A o e - ) t variable power: time variable: activity at a specific time constant: relating original activity and fraction of unstable nuclei decaying per second 45

10 And capacitor charging 1 e indicates an increase rather than a decrease: this is called eponential growth variable: pd at a specific time V t = (V o 1-e 1/RC)t variable power: time constant: relating original pd and fraction of pd charging per second And attenuation in a material I t = (I o e - ) variable power: thickness of material variable: intensity transmitted through a specific thickness of material constant: relating original intensity and fraction of photons removed per cm of material If we plot a graph of any of these variable powers against the dependent variable, we will get an eponential curve: This decreasing curve shows eponential decay Equal fractional changes in the dependent variable: 5% in this eample y Equal changes in the variable power: 1 in this eample 46

11 percentage transmission of -ray beam Note, the eponential curve starts on the dependent variable ais (y-ais) at a specific value often represented by 1% Note the eponential decay curve will never theoretically reach the -ais (and shouldn t be drawn doing so) Equal changes in the variable power will give equal fractional changes in the dependent variable In both diagnostic imaging and radiotherapy, a routine quality control check involves measuring the Half Value Thickness (HVT) of the -ray equipment. This is done by measuring the transmission of the -ray beam through thin metal sheets, usually aluminium. Sample data looks like this: mm Al % transmission Graph to show Half Value Thickness Ecel has been used to indicate an eponential trendline. The y-intercept has been fied at y = 1e thickness of aluminium (mm) A line has been drawn across from the 5% transmission to intercept with the trendline, and then dropped vertically to the - ais. This indicates a half value thickness of approimately 2.4 mm of aluminium for this particular -ray unit, meaning this thickness will reduce the intensity of the beam by one half. Notice the equation of the line: 1 indicates the original transmission; is the equal fractional change. 47

12 6.6 Eponential curves and straight lines Logarithm can be shortened to log This is where we meet the logarithm A logarithm converts a number into a power to which a base number is raised a power 1 = 1 1 A number the base number A logarithm converts a number into a power to which a base number is raised So Logs to the base of 1 are called common logs The log of the number 1, is 1 (the power to which the base of 1 is raised) 1 = 1 1 log 1 = 1 1 = 1 2 log 1 = 2 1 = 1 3 log 1 = 3 This works for all the numbers in between 1 and 1 and 1 and 1 Eg 56 = log 56 =1.75 From the table above, we can see that the logs of 1, 2 and 3 increase by one each time (equal changes of 1) but these represent the numbers 1, 1 and 1 that increase by a factor of 1 each time (equal fractional changes). If this sounds familiar, it s because an eponential change is where equal changes in one quantity give equal fractional changes in the other. 48

13 This means, if we plot the log of the fractional change, it will be converted into an equal change and give a straight line graph! Look at this eample y plotted against : and log y plotted against : Here is the table of values used for these graphs: y log y y y The eponential formula and natural logarithms A logarithm converts a number into a power to which a base number is raised e is a mathematical number = Instead of using the base of 1, we could use the base e. This would then give us the natural logarithm, which works in the same way. e = e 1 log e e = 1 See section 6.6 above or we say that the natural log of e to the base of e = 1 in the same way as the log of 1 to the base of 1 = 1 This is useful in eponential relationships, which contain the value e such as I t = I o e - When the log e of the fractional change is plotted, the e term becomes 1 because log e e = 1 and the gradient. This is how: 49

14 y See section 6.5 for meaning of symbols I t = I o e - eponential attenuation These steps make use of certain rules log e I t = log e I o + log e e - taking natural logs of both sides log e I t = log e I o comes down and log e e = 1 log e I t = log e I o - and putting the other way round: log e I t = - + log e I o y = m + c See section 6.3 Comparing this with the equation for a straight line, y = m + c, we can see the gradient is and the y intercept is log e I o y plotted against : 12 1 = -.69 y-intercept = y = 1e log e y plotted against : If the graph is plotted in Ecel, using a log scale on the y ais and a trendline added, the equation of the straight line obtained can be displayed, where the gradient = - = -.69 y-intercept = log e 1 = 4.6 y log e y = By plotting a straight line like this, it proves the eponential relationship and gives a much easier graph for making measurements such as gradient and interpolating between points. 5