Martin-Luther-University

Halle-Wittenberg Institute of Physics

Advanced Practical Lab Course

V 16: X-ray Diffraction 1) Record the characteristic of an x-ray counter filled with argon and halogen as a function of

the voltage applied to the counter tube. The X-ray tube is to be carried on an effective voltage of 20.5 kV and a current of 1 mA. The counter tube voltage is to be changed in an interval from 200 V to 480 V. The X-ray tube is running with alternating voltage. The primary X-ray beam is to be filtered by a Zr plate. Additionally to the plot determine characteristic data as length and increase of the plateau, the working point, the zero effect and the limit of the pulse rate.

2) Record the diffraction diagram of a rock salt crystal (cleavage plane (110), lattice constant

a = 5.63×10-8 cm) at 4 different tube voltages (rectifying valve; Ueff < 30 kV) and at a tube current of 1 mA with unfiltered radiation using a Bragg-spectrometer.

3) Determine the anode material of the X-ray tube with this set-up (spectral analysis). 4) Determine Planck´s quantum by means of the voltage dependent, short-wave edge of the

Bremsstrahlung according the Duane-Hunt-law. 5) Record the absorption spectrum of Zr and plot it. Specify the absorption edge of Zr. 6) Find out the lattice constant of another ionic crystal with cubic symmetry by means of

monochromatic X-ray radiation (structure analysis). Specify the composition of the 2-atomic single crystal with the help of literature. In the case of ambiguity explain your choice.

Hints Dead time of the counter is about 100 μs. Do not touch the mica window with a diameter of 9 mm! Use for recording the spectra the PC-digitizer Meilhaus DS1M12 and the corresponding software. Save your data file with the extension *.txt. Open the file with Excel and copy the numbers via clipboard into Origin. Permit spaces as delimiters in the ASCII option window of import.

Questions for testing your knowledge

What about the efficiency at the generation of X-rays using a conventional (sealed) X-ray tube? (Conversion of electrical energy to radiation energy)

What is the difference between X-rays and γ-rays? X-rays are attenuated when they transmit through matter. On which interactions does the

attenuation depend? An absorption spectrum is sketched in figure 3. Explain the characteristic profile. The emission lines are also depicted in figure 3, they are in each case on the “right” side of

the “edge”. Why? Give some capabilities of detection and measurement of X-rays and their intensity. The positions of the X-ray reflections are determined by Bragg´s law. Derive the Bragg

law using figure 2! Which factors describe the intensity of the X-ray reflections? How does the crystalline structure of the sample influence the intensity? Explain the term primitiveness of unit cell! How can you determine the primitiveness by

means of X-rays? Calculate the structure factor of NaCl- and CsCl-structure, respectively.

Basics Generation and properties of X-rays X-rays and also γ-rays are electromagnetic waves with short wavelengths. Some phenomena can merely be explained using the model of photons, i.e. X-rays are considered as particles. Although both X-rays and γ-rays are of the same physical nature, as X-rays are described radiation originating by accelerated charged particles. It can be charged particles moving on a circuit as in a synchrotron storage ring or charged particles impinging on a target as in a conventional sealed tube. Both at the conventional (Coolidge) tube and at the rotating anode electrons are generated by thermionic emission and accelerated by a potential difference. When the electrons are impinging on the anode a small part of the kinetic energy

UeE *= (e-elementary charge, U-applied voltage) are transferred in X-ray radiation on the one hand by slowing down of the electrons in the electric field of the atom cores (Bremsstrahlung) and on the other hand by impact ionization in the electron shell (characteristic X-rays). The slowing down process of the electrons in the electric field is implemented in a multiple-stage, i.e. the kinetic energy is converted into X-ray radiation quasi in portions. The result of this process is the so-called retardation spectrum, which is independent on the anode material and shows a maximum energy and a shortest

wavelength, respectively. The maximum of the retardation spectrum is at 1.5*λGrenz . Specify the corresponding factors of your experiments!

Uehc

Grenz ⋅⋅

=λ (1)

(c-light velocity, h-Planck´s quantum), Equation (1) is known as Duane-Hunt-law. The intensity of the X-rays, i.e. the number of particles per area and time, depends on the atomic number, tube current and tube voltage as it is given in equation (2).

nA UIZI ⋅⋅∝ (2)

Figure 1: Scheme of generation of X-rays.

(IA-tube current, n = 2 for unfiltered radiation, up to 5 for filtered radiation). Besides the retardation spectrum some peaks arise, so called characteristic radiation. The origin of them is the interaction of the impinging electrons with the atomic shell. The incident electrons can knock out electrons from the atomic shell. These vacancies are re-filled by electrons from higher energy levels and finally the energy difference between the levels is emitted as electromagnetic radiation. In the case of electron transitions to energy levels near the atomic core the radiation is in the X-ray range because of the high binding energy. The corresponding wavelengths are described by Moseley´s law (equation (3)).

⎟⎠⎞

⎜⎝⎛ −−⋅= 22

2 11)(1mn

ZR σλ

(3)

(R-Rydberg constant, Z-atomic number, σ-screening constant, σ=1 for K-line and σ=7.4 for L-line, n, m-principal quantum number) According Jönsson und Bergen-Davis the intensity of the characteristic peak is proportional to

( )20UUI −∝ (4)

(U0 - critical excitation voltage of the anode material). Verify this relation and specify the critical excitation voltage! Interaction of X-rays with matter When an X-ray beam interacts with matter (elastically or inelastically) scattering can occur. In the case of elastic scattering the outgoing X-rays have the same energy as the incoming X-rays, only with altered direction. In contrast, inelastic scattering occurs when the energy is transferred from the incoming X-ray to the matter, e.g. by exciting phonons or electrons inside the sample. According the classical scattering theory an electron is excited by an incoming, unpolarized X-ray and emits because of its oscillations an electromagnetic wave with the same wavelength. The intensity of the scattered wave is described by the Thomson formula:

0

2

2

20

2cos1 I

rrIe ⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=

θ (5)

(ro - classic electron radius, θ- scattering angle, r- distance, I0-incident intensity) Most atoms host more than one electron. Considering the scattering at an atom, all partial waves of the electrons of the atomic shell have to sum in phase. The result is the atomic scattering factor fA.

∫ •⋅⋅= rderQf rQiA

rrr rr

)()( ρ (6)

(ρ(r) - electron density distribution, Qr

- scattering vector) In the limiting case of a scattering vector 0→Q the scattered waves of all volume elements are in phase and therefore the atomic factor is equal to Z. The next step is to consider the scattered intensity of a crystal. A crystal is a solid in which the constituent atoms, molecules or ions are packed in a regularly ordered, repeating pattern extending in all three dimensions. One way to derive Bragg´s law is to consider the diffraction of the X-rays on the atoms as a reflection at the net planes. Net planes are specified by Miller´s indices. Bragg´s law (equation (7)) determines the condition of a constructive interference (see figure 2).

λθ nd =⋅⋅ sin2 (7)

For calculating the scattered intensity it is necessary to determine the scattering amplitude:

∑∑ ⋅⋅⋅⋅=n

n

j

j

R

RQi

r

rQijKristall eeQfF

rrrr)( (8)

(Rn- lattice vector, rj-position of atoms in the unit cell, Rn+rj position of atoms in the crystal) The first factor is the structure amplitude, i.e. a summation over all atoms in the unit cell. The other gives the lattice amplitude, i.e. a summation over all unit cells in the crystal. Thus, the scattered intensity is given by

eKristall IFI ⋅= 2 (9) With the help of calculated structure factors the selection rules for the allowed reflection of the corresponding types of structure can be determined. As a consequence the Bravais type and also the primitiveness can be specified. In a quantum mechanical sense X-rays are described as photons, i.e. as an elementary particle. Therefore, it carries momentum and energy. As a particle the photon can only interact with matter by transferring momentum and energy, see for instance the Compton effect. Contrary to Thomson´s scattering it is an inelastic phenomena. In the case of absorption the energy of the photon is transferred to an electron which can be emitted. The rest of energy, i.e. the energy difference between the photon energy and the working function of the emitted electron, contributes to the kinetic energy of the electron. All processes of absorption can be summarized in the linear absorption coefficient, thus, the intensity of the transmitted beam is given by the absorption formula

zeII ⋅−= μ0 (10)

(µ- linear absorption coefficient, z- transmitted thickness) After ionization the atom returns into the ground state by filling up the vacancy with an electron from a higher energy level and emitting a photon. The particular spectral lines depend on the target element and thus are called characteristic lines. Usually these transitions from upper shells into K-shell are called K-lines, into L-shell L-lines and so on. With the help of these lines the element can be determined, e.g. at X-ray spectroscopy or at electron microscopy (element mapping). The emitting photon can also knock out an electron from a higher energy level. This electron is called Auger electron and can be used by the so called Auger spectroscopy, a surface sensitive method. The absorption process reveals a distinctive dependence of the absorption coefficient on the photon energy of the incident X-ray. The absorption spectrum of Pt is depicted in figure 3. Below the bonding energy of the K-electron of 78.395 keV the incoming photon can impact L-and M-electrons only. With increasing energy the absorption coefficient jumps rapidly at the K edge and decreases as E-3. The edge structure at the other edges is analogue.

Figure 2: Derivation of Bragg´s law.

Detection of X-rays Photographic film was the first detector for X-rays. In the last years image plates and CCD chips replace this technique. X-ray detectors other than the two-dimensional detectors may be classified into three types: ionization counters, scintillation counters and semiconductor detectors. In this experiment the first type is used, i.e. a so called halogen counter, that is a cylindrical envelope with two electrodes, a central wire maintained at an appropriate potential and a grounded coaxial conducting cylinder and filled with argon and halogen gas as quenching gas. X-rays enter the counter through a 3 µm thick window of mica Near the anode the electric field strength depending on the distance r from the centre (see equation (11)) is very high.

⎟⎠⎞⎜

⎝⎛

⋅=

A

K

ZR

rrr

UrEln

1)( Anode: 0→r (11).

(UZ – applied voltage, rK – radius of the cylinder, rA – radius of the central wire). According to the applied voltage different interaction phenomena appear and result in a typical characteristic as shown in figure 4.

In the voltage range up to UZ < 100 V free electrons (primary electrons) are generated by primary photoionization process. As a consequence a current is flowing to the anode, obeying Ohm´s law because of recombination of a part of the electrons with the ions in the gas-filled chamber.

Between 100 V and 200 V (see figure 4) the recombination is suppressed and the current reaches a saturation value. In this range ionization chambers are running.

At voltages UZ > 200 V the interaction of the primary electrons with the atoms start up the secondary ionization process. In this range the gain factor is about 100 and does not depend on the energy of the incoming radiation. The proportional or gas gain factor depends on the type of the gas, the gas pressure and the applied voltage. The signal is a function of the energy of the detected particles and not of the number, thus, the counter is used as a proportional counter to analyze energy with an accuracy of about some percent.

Figure 3: Absorption spectrum of Pt.

A further rise of the voltage above the proportional range leads to a higher detection sensitivity. One primary electron can initiate a gas discharge and a signal proportional to the number of the detected particle independent on their energy (Geiger-Müller-counter) is recorded.

When the applied voltage is too high a Townsend discharge can occur and as a consequence the counter can be destroyed.

Dead time correction: Immediately after a discharge the central wire is environed with positive ions preventing to count further incoming particles. The cathode is electrically screened by the space charge effect. As a consequence the electric field is decreased and a discharge can not take place, the counter is “dead”. Only when the positive ions migrate to the cathode the electric field is rebuilt and further incoming particles can be detected. Usually the dead time is determined experimentally. The intensity is proportional to the tube current and governed by

θ00

1 nn

n+

= ,

( n – measured pulse rate, n0 – actual pulse rate, θ - dead time). Gas fillings of ionization counters Because of its high gas gain factor inert gases are used either with an additive or without one. Alkanes like methan and ethan or halogens like Br and Cl are used as quenching gases to reduce the dead time.

Figure 4: Scheme of the current-voltage characteristic of a gas-filled counter. The intensity of the incident ionizing radiation is constant.

Experimental devices Bragg´s spectrometer Shortly after the discovery of the X-rays in 1895 by Conrad Wilhelm Röntgen, besides the most common use of X-rays in medicine, two main fields in natural science have emerged,

spectrometric applications and investigations of the structure of solids. The first spectrometer was constructed by W.H. and W.L. Bragg and consists of an X-ray tube, a crystal as sample and a detector. The sample and the detector are rotated around the goniometer axis in a ratio of 1 : 2. The orientation and the adjustment of the single crystal determine the reflecting net planes, thus only the different orders of the corresponding reflection can one be recorded. The intensities of the various reflections decrease as the square of the order. Verify this correlation! In the case of a powder sample an orientation of the sample is not necessary, the reflection are measured in any case. Point out the differences between single crystal, mosaic crystal and powder crystal! References:

Nielsen: Elements of Modern X-Ray Physics Kittel: Introduction to Solid State Physics

Figure 5: Scheme of a Bragg spectrometer.