Properties and Absorption

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Take a metal sheet three millimetres thick, which lets exactly half of an X-ray beam through. Now put two such sheets together, six millimetres in all. The first sheet stops half, the second sheet stops half, and half plus half makes one whole, so the beam is stopped completely, right? It sounds so convincing, doesn't it? By the end of this video you will see exactly why it is wrong. Today we learn the properties of X-rays and how they are absorbed, and questions on half-value thickness come up often in JEE, so let's begin.

0:36Properties of X-rays

So, first, what exactly are X-rays? They are electromagnetic waves, the same family as light. So in vacuum they travel at c, the speed of light, the small c you see on screen. Okay? Now let me ask you this. We send a beam of X-rays between the plates of a charged capacitor. Is the beam deflected, and what does that show? No, it is not deflected at all, and that shows that X-rays carry no charge. So neither an electric field nor a magnetic field can bend them. Right? Next, what can they do? They ionise gases, which means they knock electrons out of the gas atoms, and they affect photographic plates, just like light does. Clear? They also cause fluorescence, which makes some materials glow, and they eject photoelectrons from metals, just as in the photoelectric effect. Got it? Now here is a lovely question. To diffract light we can use a ruled grating, but to diffract X-rays we use crystals. Why crystals, and not ordinary ruled gratings? Yes, it is all about size. Lambda, the wavelength of X-rays, is about zero point one nanometre; the wavy sign means about. That is comparable with the spacing of atoms in a crystal. The lines of a ruled grating are thousands of times further apart, so for X-rays a crystal makes a far better grating. With me?

2:04Absorption: the beam fades exponentially

Now the main part, absorption. An X-ray beam enters a material with intensity I naught, the I with a small zero below. After a thickness x, the intensity that comes out is given by I equals I naught, times e to the power minus mu times x. Okay? Here mu, the Greek letter that looks like a u with a tail, is the absorption coefficient, and x is the thickness. The bigger the mu, the faster the beam fades. Right? Now tell me, what thickness lets exactly half of the intensity through? Set I equal to I naught over two, and solve for x. So, e to the power mu times x must equal two, which means mu times x equals the natural log of two, written l n two. This thickness is the half-value thickness, x with a small one over two below. It equals l n two over mu, that is, zero point six nine three over mu. Clear? Now back to our puzzle. One half-value thickness stops half the beam. So two of them stop the whole beam, don't they? No, that is the trap, because the second layer never sees the full beam! It receives only half, and it stops half of that. So each half-value thickness halves whatever is left, and after n of them, I equals I naught over two to the power n. Got it? So two half-value thicknesses leave I naught over four, a quarter of the beam, and not zero. Half plus half making one whole was our mistake, because the two halves are halves of different amounts. And that solves our puzzle, okay?

3:49What absorbs X-rays more strongly?

Alright, now let us see which materials absorb more. Absorption rises steeply with Z, the atomic number of the material, so heavier atoms absorb much more. Right? It also rises with lambda, the wavelength. The longer the wavelength, the more easily the X-rays are absorbed. Clear? Now use this in a hospital. In an X-ray photograph, why do the bones stand out so clearly against the flesh around them? Yes, exactly, bone absorbs much more than flesh, so fewer X-rays reach the plate behind the bone, and its shadow shows clearly. And for the same reason, lead, with its very high atomic number, is used for shielding, to protect people from X-rays. Okay? Now two words you will often hear, soft and hard X-rays. Hard X-rays have shorter wavelengths and greater penetrating power, while soft X-rays, with longer wavelengths, are absorbed more easily. Got it? And in a given tube, raising the tube voltage makes the beam harder, so the X-rays go deeper into the material. With me?

4:55Example 1: fraction through 9 mm when half-value is 3 mm

Now, example one. The half-value thickness of a material for certain X-rays is three millimetres. What fraction of the intensity passes through nine millimetres? Pause the video and try it first. Done? Okay, let's check. Nine millimetres over three millimetres gives three, so the beam crosses three half-value thicknesses. Okay? Now, nine is three times three, so some students quickly write one third. Is one third correct? No, because each half-value thickness halves what is left. So I over I naught is one half, times one half, times one half, that is, one half to the power three, which is one eighth. So one eighth of the intensity gets through. Clear?

5:43Example 2: absorption coefficient and thickness for 1%

Now a harder one, example two, on the same material, whose half-value thickness is three millimetres. Find the absorption coefficient, and the thickness that cuts the intensity to one percent. Pause the video and try it first. Done? Let's go. Mu equals l n two over the half-value thickness, so zero point six nine three over three, which gives zero point two three one per millimetre. The m m to the power minus one on screen means per millimetre. Right? Now, one percent means I over I naught is one over one hundred. So e to the power mu times x equals one hundred, and x equals l n one hundred over mu; the double arrow means it follows. Okay? Now careful here. Many students say l n one hundred is two, since ten squared is one hundred. Is that right? No, and this is the trap, because two is the log of one hundred to base ten! The natural log of one hundred is two times the natural log of ten, which is about four point six zero five, and not two; the wavy equals sign means approximately. Clear? So x equals four point six zero five over zero point two three one, which is about nineteen point nine millimetres. Got it? And here is a topper's quick check with half-value thicknesses. Two to the power seven is one hundred and twenty eight, which is greater than one hundred, so we need just under seven half-value thicknesses. Seven times three is twenty one millimetres, and nineteen point nine is just under that, so our answer makes sense. With me?

7:23Quick revision: properties, absorption law, Z and wavelength

So, quick revision. X-rays are electromagnetic waves with no charge, so fields do not deflect them, and crystals diffract them. The intensity falls as I equals I naught, times e to the power minus mu times x, and each half-value thickness, l n two over mu, halves the intensity. And absorption rises with atomic number and with wavelength, so bone absorbs more than flesh, and lead is used for shielding. Now go and try the practice problems on this page. You will find them easy. All the best!