X-ray diffraction & absorption

X-ray diffraction & absorption

KTH March 31, 2011

Abstract

This laboration consists of two parts: Studying the X-ray diffraction that happens when reflecting X-rays off a crystalline material and their capability to penetrate thick, hard materials. We will study how the X-ray intensity varies with penetration depth and how it depends on different materials such as aluminium, tin etc. In this laboration we will also learn how to produce X-rays of different ”hardness” and also learn how to use a crystal spectrometer.

1 Introduction

When a charged particle is moving in an accelerated motion, for instance in a bent orbit, it is continuously emitting electromagnetic radiation (Fig. 1). The characteristic energy of the emitted radiation is proportional to the cube of the energy of the particle and inversely proportional to the radius of curvature of the orbit. This process occurs very frequently in nature and gives rise to the well known Electromagnetic Spectrum which extends over sixteen orders of magnitude in energy or wavelength (Fig. 2). Thus at the very low energies of the oscillating electrons in a linear radio antenna, where velocity and acceleration are collinear, the emitted radiation has energies less than 10

eV corresponding to wavelengths λ < 1 meter. Other antenna and generator constructions allow the emission of radar- and microwaves with λ down to the millimeter range. In the Universe an innumerable number of radiowave emitting sources have been found where energetic electrons are orbiting for instance neutron stars and supernova remnants. If the magnetic fields are strong enough close to these celestial sources, the radiation may even occur in the X-ray and gamma region (Fig. 2).

The Electromagnetic Spectrum such as the one displayed in Fig. 2 only describes the continuous radiation emitted from charged particles in an accelerated motion. In addition discrete (monoenergetic) radiation is emitted over the same energy range. In the radio wavelength region such discrete emission may originate for instance from deexcitation of molecular rotational and vibrational levels, while monoenergetic light in the near visible range is emitted at electronic transitions in outer atomic shells. As will be examined in the present laboration, discrete X-rays are emitted at transitions between inner atomic shells.

Production of X-rays using an X-ray tube

The principle of a conventional X-ray tube is shown in Fig. 3. Electrons from a filament

are accelerated across a high voltage U and hit an anode, from which X-rays are emitted

Figure 1: When a charged particle is mov- ing in a bent curve, it emits electromagnetic

radiation. Figure 2: The electromagnetic spectrum.

wavelength λ

is obtained when all the energy of the electron is converted into radiation, i.e.

hν

= hc/λ

= eU. (1)

With inserted constants this gives

λ

(˚ A) = 12398/U (volt) (2)

with 1 ˚ A = 10

m = 0.1 nm. Fig. 4 shows the continuous X-ray spectrum emitted from a tube operating at various voltages U ranging from 5 to 25 kV.

X-rays produced from a tube in this way were first observed by Wilhelm R¨ ontgen in 1896 and have found well known, extremely important applications in various fields during the past century. However, the main drawback of X-ray tubes are their extremely low efficiency, since most of the energy of the accelerated electrons are converted into heat. More than ten orders of magnitude higher intensities may be obtained if the electrons are accelerated and stored in a polygon where electromagnetic radiation is emitted from the bent corners. Such an electron storage ring is called a synchrotron and the ”synchrotron light”, which is continuous from the X-ray region up to the infrared region, has found a widespread use in basic science as well as in a number of applications.

The discussion of the characteristic spectrum of X-rays will be discussed in the Modern

Physics course and in the Quantum Physics course later on. Here we will only use the

apparatus for the production of X-rays. The absorption of X-rays in matter is caused by the

photoelectric effect, the Compton effect and by electron-positron production.

Figure 3: Principles of a conventional X-ray tube.

Figure 4: Continuous X-rays from an X- ray tube equipped with a molybdenum anode.

The tube has been operated at five different voltages. Note the characteristic X-rays su- perimposed on the continuous X-rays in the 25 kV run.

2 Part I: Diffraction of X-rays using a crystal spectrometer

When X-rays hit a crystal, reflections occur at the lattice planes at certain angles (Fig. 5).

These angles θ are found from Huygens principle and we find from Fig. 5 that constructive interferences between two adjacent beams occur if the optical path difference is equal to an integer multiple of the wavelength λ of the incident X-rays or

AB + BC − AE = 2AB − AE = 2d/ sin θ − 2d cos θ/ tan θ = nλ (3) i.e.

nλ = 2d sin θ (Bragg’s law) (4)

Figure 5: When X-rays hit a crystal under an angle θ each atom acts as a source for a new elementary

wave (left figure). These elementary waves cause constructive interferences (right figure) at the condi-

tions given by Braggs law (Eq. 4) which gives the scattering angle for incident X-rays with wavelength

λ in a material with lattice constant d.

if known characteristic X-rays from a tube are applied. In the present laboration, both possibilities will be applied.

Figure 6: Equipment for measurements of diffrac- tion / reflection of X-rays used in the present lab- oration. X-rays from a tube with a copper an- ode is reflected from a crystal under the angle θ, which is measured from the pointer arm on the scale (0 − 90

), and detected by a Geiger-Muller (GM) tube on a second rotatable arm which moves twice as fast as the pointer arm, i.e it is located at the angle 2θ. The Bragg spectrum is recorded by measuring the reflected intensity as a function of θ.

Determination of X-ray energies

The spectrum produced by an X-ray tube can look as in Fig.7 where one can see the charac- teristic peaks K

and K

as well as the ”bremsstrahlung” curve.

Figure 7: X-ray diffraction spectrum from a LiF crystal and a tube with a copper anode measured using the equipment displayed in Fig. 6. The con- tinuous X-rays start at some minimum angle θ

which depends on the voltage U applied to the tube.

The discrete peaks originate from K

and K

radi- ation in copper. The complete spectrum is recorded at a tube voltage of 25 kV, while spectra close to θ

run at 22 and 19 kV are included for a com- parison.

A typical X-ray spectrum recorded with the present equipment with applied voltages

U = 25, 22 and 19 kV to the the X-ray tube and using a LiF crystal is shown in Fig. 7. At the

minimum angle θ

the continuous radiation is already detectable and a broad distribution

is observed as well as the K

and K

radiation from the copper anode of the tube. These

transitions occur at θ = 22

.6 and 200

.3, respectively, which correspond to λ = 1.54 ˚ A and

1.39 ˚ A from Bragg’s law (Eq. 4) with d(LiF) = 2.01 ˚ A. The weak peak at the right end of

the spectrum stems from the second order diffraction of K

.

Tasks during Part I

• Use the spectrometer and the LiF crystal and generate spectra at 10 kV, 20 kV and 25 kV. Are the spectra different? If so, why? Measure the peak angles and estimate uncertainties in your peak measurements.

• Since we know d for the LiF crystal, we can calculate the energy (or wavelenght) of the peaks in the specta. This should be presented in the report, with uncertainties.

• Switch to the KBr crystal and generate new spectra at the same voltage. Using Bragg’s law, calculate d for the KBr crystal. This should also be presented in the report, with uncertainties.

3 Part II: Absorption of X-rays by matter

The absorption of X-rays in matter increases exponentially with the thickness x of the ab- sorbing layer and the mass absorption coefficient µ. The remaining intensity I(x) after a passage through a layer with thickness x can be described by the following formula:

I(x) = I(0)e

(5)

where µ is the linear absorption coefficient. For carbon, aluminum, iron and tin the mass absorption coefficient, µ/ρ (cm

/gram), and density, ρ (g/cm

), are: (0.18, 2.2), (0.38, 2.7), (2.0, 7.9) and (11.0, 7.3) respectively for 50 keV X-rays.

Question to answer

Calculate how thick layers (in mm) of these elements are needed to stop 99 % of incident 50 keV X-rays.

Tasks during Part II

• Adjust the voltage to 25 kV and put the spectrometer arm to an angle where the K

peak has its maximum.

• Measure the intensity of the X-rays without an absorber, then redo the measurements with aluminum of varying thickness. Produce graphs of the intensity as a function of the thickness x and fit an exponential function to the data to determine the linear absorption coefficient µ for aluminum, with uncertainty.

• Redo the measurements but with different materials of the same thickness. Produce a

graph of the intensity as a function of Z, the atomic number of the material. Explain

the graph.