Layer thickness determination via theabsorption of X-rays

Max-Planck-Institut fur extraterrestrische Physik

Giessenbachstraße

85748 Garching

October 25, 2007

Aim of the experiment:

With this experiment a basic understanding of the absorption of X-rays shall be achieved. With this knowledgethe thickness of different samples is being determined. Additionally the setup can be used to identify unknownmaterials by their X-ray-Fluorescence lines. Knowledge in the following fields is necessary for carrying out theexperiment: Atomic and Nuclear Physics, formation mechanisms of X-rays, interaction of photons with matter:scattering, photoelectric effect, compton effect. Solid state physics - basics of semiconductor physics: bandmodel, band gap, doping, pn-junctions (diodes).

1

Contents

1 Introduction 4

2 Theory 5

2.1 Generation of X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Continuum radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Characteristic radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Interaction between X-rays and matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 The photo effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 X-ray fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Experimental Setup 11

3.1 The shape of the spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.1 The Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1.2 The pulse shaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 The noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Physical cause of the noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Connection between noise and shaping time . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.3 Width of the distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Spectral analysis 17

4.1 Characterization of the lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Background estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Qualitative Analysis 19

5.1 Identification of elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Negative effects (parasitics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.1 Line generating effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2.2 Background generating effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.3 Line broadening effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Experimental procedure 21

6.1 Operation of the X-ray tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Operation of the detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.3 Energy calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.4 Determination of the energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.5 Identification and layer thickness determination of different samples . . . . . . . . . . . . . . . . 226.6 Measuring the energy dependance of the absorption coefficient . . . . . . . . . . . . . . . . . . . 236.7 Thickness determination of a sample glued on a carrier which can not be transmitted . . . . . . . 23

7 Radiation protection 24

7.1 Dose quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.2 Consequences of high radiation exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.3 Average radiation exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2

8 Further reading 26

3

Chapter 1

Introduction

The best known application of X-rays if found in medical science in order to detect injuries like e.g. fractures.Here the fact is used that X-rays penetrate materials differently well depending on their composition. The resultis mostly an image where the intensity of the radiation which completely went through the sample is presented.

In this experiment, the permeation of X-rays through matter shall be investigated in detail. It will then beused to determine the thickness of different samples.

The experiment will be conducted in the following way:

1. Determination of the absorber material with its characteristic fluorescence radiation.

2. Determination of the thickness of the sample by measering the transmission of X-rays.

3. Measuring of the energy dependance of the absorption coefficient of the material used.

4. Thickness determination of a sample which is glued onto a carrier which can not be transmitted.

4

Chapter 2

Theory

2.1 Generation of X-rays

Light with an energy between 0.1 and 100 keV is called X-ray Radiation. X-rays can be produced by X-ray tubes.In this process continuum radiation ( the so called ”Bremsstrahlung” ) is produced as well as the characteristicradiation. The typical layout of an X-ray tube is shown in Figure 2.1.

2.1.1 Continuum radiation

Free Electrons are produced in the high vaccuum of an X-ray tube by heating a filament (hot cathode) fromwhich thermal electrons emerge. These electrons are accelerated in a strong electric field and hit a target(anode) made from e.g. Mo, Cr, Cu or Rh. When hitting the anode, the electrons are decelerated and emittheir kinetic energy in the form of short wave electromagnetic radiation. The term ”Bremsstrahlung” originatesfrom german and literally means braking radiation. After their acceleration in the X-ray tube’s electric fieldthe electrons have the energy

Ee = eU (2.1)

where e is the electric charge of the electron and U is the difference of potential which the electron travelledthrough on its way from the filament to the anode. The electron’s energy Ee und the wavelength of the X-raysλ are connected via the following equation:

Ee =hc

λ(2.2)

where c is the speed of light and h is Planck’s constant.From the equations 2.1 and 2.2 the smallest possible wavelength is obtained:

λmin =hc

eU(2.3)

When an electron is decelerated in one step, then the equation 2.3 states the shortest wavelength λmin thatcan be produced. Most of the electrons do not lose their energy in one step but in a great number of differentsteps. This results in a continuous spectrum of X-ray photons.

The intensity distribution of the continuous spectrum is described by Kramer’s Rule:

I(λ) ∝ iZ

(

λ

λmin

− 1

) (

1

λ2

)

(2.4)

I = Intensity in cm−2s−1

i = Tube curentZ = Atomic number of the element which is used as the tube’s target.

From equation 2.3 and Kramer’s Rule, three important statements can be derived (see Figure 2.2):

1. An increase of the applied Voltage U with constant current i leads to a decrease of the minumum wave-length λmin.

5

Figure 2.1: Schematic layout of an X-ray tube

2. There is a linear dependence between the atomic number of the anode material and the intensity of theproduced continuum radiation.rahlung.

3. There is also a linear dependence between the currenti at which the X-ray tube is operated and theintensity I of the continuum radiation. With increasing current more electrons leave the filament andthus more electrons release their energy via bremsstrahlung.

2.1.2 Characteristic radiation

Characteristic radiation occurs, if electrons from inner shells are removed from an atom by energetic radationof energetic electrons. The arising gaps or vacancies are then filled by electrons from higher level shells.(seeSection 2.2.3). The energy difference between these two levels is released in the form of characteristic X-rayswith discrete wavelengths.

One has to distinquish between the characteristic radation of the anode material and the characteristic radi-ation of the sample: The characteristic radiation of the anode material superimposes the continuous sprectrumof the X-ray tube only at a few energies. It is produced by the removal of inner electrons from the anodematerial by the previously accelerated electrons. The characteristic radiation of the sample is produced by theradiation coming from the X-ray tube which consists of continuous plus characteristic radiation. Because ofthe different binding energies of electrons in the different elements, a line spectrum is produced by the samplewhich superimposes the continuous radiation of the tube at many positions.

The characteristic line spectrum consists of a number of discrete energy lines. It is characteristic for theparticular anode material and shows certain relative intensities according to the composition.

2.2 Interaction between X-rays and matter

The interaction of electromagnetic radiation with matter is very complex and touches many aspects of modernphysics. If X-ray radiation is travelling through matter, its intensity is attenuated. There are different kinds ofinteractions which all lead to an attenuation of the incident X-rays. The amount of these interactions is stronglydependent on:

1. the energy of the incident radation,

2. the spectral composition of the incident radiation and

3. the chemical and crystalline composition of the sample.

6

Figure 2.2: Dependence of the intensity of the continuum radiation on the current (left), on the voltage (middle)and on the atomic number of the anode material (right); from Bertin (1975).

The attenuation increases exponentially with the sample thickness and is described by the following equation(Lambert-Beer law):

I = I0e−µd (2.5)

I = radiation intensity after passing through matterI0 = radiation intensity before passing through matterd = thickness of the absorberµ = attenuation coefficient ( = 1/attenuation length = 1/λ )

The attenuation coefficient depends on the X-ray energy and on the absorber material. The attenuation ofthe incident X-rays as well as effects like heat build-up, scattering and emission of photons with discrete energiesare due to interaction of the X-rays with electrons in the atomic shell. In this connection one distinquishesbetween three mechanisms. With the Photoeffect electrons are removed from the electron sheath. Scatteringleads to intensity loss when the photons travel through matter. One distinguishes betweens coherent Rayleigh

scattering and incoherent Compton scattering.The attenuation coefficient is hence composed of a coefficent for the photoelectric absorption µph, one for

Rayleigh scattering µra and another one for Compton scattering µcom:

µ = µph + µra + µcom (2.6)

As an example, the attenuation length (λ = 1/µ) of silicon and aluminium against the photon energy isshown in Figure 2.3.

2.2.1 Compton scattering

Compton scattering is when an X-ray photon interacts with a weakly bound electron in an outer shell of theatom. The electron is removed from the atom and carries away a part of the photon energy. The sum of theenergy of the scattered photon and that of the electron is equal to the energy of the incident photon. Thisprocess is described by energy- and momentum conservation. For the energy of the scattered photon we havethe following equation:

E′ =E

1 + (E/mec2)(1 − cosφ)(2.7)

where E and E′ are the energies of the incident and the scattered photon, me is the electron mass, c thespeed of light and φ the angle between the incident and the scattered photon.

7

Figure 2.3: Attenuation length of photons in silicon and aluminium against the photon energy.

The fraction of compton scattered photons increases with increasing energy of the X-rays, decreasing atomicnumber of the scattering atom and with increasing scattering angle φ.

2.2.2 The photo effect

In the photoelectric effect, electrons are removed from the atom if the supplied energy is larger than the bindingenergy EB of the respective electron. The rest of the energy is transferred to kinetic energy of the removedelectron:

hν = EB + Ekin (2.8)

h = Planck’s constantν = frequency of the photon

An Atom with a missing inner electron is in an excited state. After about 10−8 seconds it will return to theground state while emitting energy. There are two possible mechanisms for this: X-ray fluorescence and theAuger effect.

2.2.3 X-ray fluorescence

An electron from a higher shell ”drops” to the gap which was produced by the photoelectric effect. The energydifference ∆E can be emitted completely via X-ray radiation:

∆E = hν (2.9)

This process is called X-ray fluorescence. The emitted fluorescence radiation is characteristic for the partic-ular element as well as for the considered energy transition within the electron shell. The relation between theenergy of the emitted radiation, the atomic number of the emitting element and the principle quantum numbersof the participating electron shells is called Moseley’s law:

hν = R h c (Z − σ)2(

1

n21

− 1

n22

)

(2.10)

R = Rydberg constantZ = atomic numberσ = atomic screening constant (Z − σ is the effective atomic number)

8

Figure 2.4: Schematic representation of the energy transitions for characteristic X-rays (shell model)

n1, n2 = principal quantum number, where n2 > n1.For practical applications, equation 2.10 can be reduced to the following simple relation:

ν ∝ Z2 (2.11)

The lines of the emitted spectrum are sorted in K-, L- and M-series, depending on which shell the electronresides in after the transition (see also figure 2.4). The different lines within one series have different intensities,depending on their quantum mechanical transition probabilities. The line energies are usually not calculatedusing Moseley’s law but rather exist with high precision in tabular form.

The α-lines are the strongest lines because the transitions between directly adjacent shells occur mostfrequently. The energies of those lines identify an element. Usually a Kα-line is enough to determine anelement. For an unambiguous determination the Kβ-line should nevertheless be identified as well. Likewise theLα-, Lβ-, and other lines may be used.

The energy of the radiated X-rays is by the way independent from the binding state of the element becauseprimarily only transitions between the inner shells occur. Exceptions can arise for small energies and lightweightelements. If an electron near the valence band is participating in the emission process, the X-ray energy can beinfluenced by the chemical binding state of the atom. The caracteristic peak can then be displaced to differentenergies for the same element but different binding states. These displacements can only be observed fromlightweight elements which can not be detected within the scope of this lab course experiment.

Selection rules

Electron transitions can not occur from every higher to every lower shell. Only some transitions are permitted.The selection rules given in table 2.2.3 apply.

Auger effect und fluorescence yield

When an electron moves to the ground state the released energy can also be transferred to another electron.This electron from a higher shell leaves the atomic shell - the atom is now doubly ionized. This process is calledAuger effect, radiationless transition or internal conversion. The kinetic energy of the Auger electron is givenby

Ekin = ∆E − EB (2.12)

9

Table 2.1: Selection rules for electron transitionsSymbol Name permitted values selection rules

n principal quantum number 1, 2, . . . , n ∆n 6= 0l azimuthal quantum number 0, 1, . . . , (n − 1) ∆l = ±1m magn. quantum number −l, . . . , 0, . . . , +l -s spin quantum number ±1/2 -j vector sum s + l l ± 1/2 ∆j = ±1 oder 0

fur l = 0 : j 6= −1/2

Table 2.2: Values of the constants A, B, C for equation 2.14.series

constant K L MA −3, 795 · 10−2 −1, 111 · 10−1 −3.60 · 10−4

B 3, 426 · 10−2 1, 368 · 10−2 −3.86 · 10−3

C −1, 163 · 10−6 2, 177 · 10−7 −2.01 · 10−7

where ∆E is the liberated energy from the transition of the first electron to the ground state. The kineticenergy of the Auger electron is again characteristic for the corresponding transition and element. The Augereffect occurs especially for elements with low atomic number because their inner electrons are bound less strongly.

A result of the Auger effect is, that the lines of a particular series is not observed as intensely as one wouldexpect from the number of vacancies in the respective atomic orbital. The K-fluorescence yield WK is definedby the ratio of the number of emitted X-ray photons nγ of the K-series divided by the number of at the sametime produced gaps n in the K shell:

WK =nγ

n(2.13)

The L- and M-fluorescence yields WL und WM are defined analogous. The fluorescence yields are differentfor the different electron shells of an atom. In general the fluorescence yield increases with increasing atomicnumber. The fluorescence zield can be approximated by:

(

W

1 − W

)1/4

= A + BZ + CZ3, (2.14)

where Z is the atomic number and A, B and C are constants for the respective shell. For the detection ofelements with small atomic numbers the fluorescence yield is the limiting factor.

As explained above, a consequence of the Auger effect is the production of doubly ionized atoms. Forexample a K shell vacancy is produced by a primary X-ray photon, an L shell electron ”drops” to the K shellemitting a Kα photon, this photon experiences the Auger effect (which means that it hits another electron inthe same atom) and produces another vacancy in the L shell. The atom is now in the LL-state.

In such doubly ionized atoms the transition of electrons from one shell to another one leads to slightlydifferent energies than in singly ionized atoms. Such lines are called satellite lines. Especially in elements withlow atomic numbers, those lines can be very intensive. E.g. for aluminium, the intensity of the line Al Kα3

(LKLL-Transition) is about 10% of the intensity of Al Kα1,2.

10

Chapter 3

Experimental Setup

The setup which is used in this experiment is presented in figure 3.1. The X-ray radiation from the sourcehits the sample at an angle α and the radiation which is scattered, transmitted or produced there is detectedby a detector at an angle β. The whole setup sits in a housing which screens the X-ray radiation to theoutside. Depending on the angles α and β the transmission or the X-ray fluorescence radiation can primarilybe measured.

A silicon detector is used in this experiment, whose operation is based on semiconductor technology. Forevery incident X-ray photon a charge pulse is generated with its height beeing proportional to the energy of theX-ray photon.

Unlike with frequently used gas flow- or scintillation counters an energy sensitive semiconductor detector asthe Silicon Drift Detector (SDD), which is used here, can do without a special dispersing element because theSDD possesses an intrinsic energy resolution. In order to improve the energy resolution, the detector is cooledby a peltier element. For a further description of the SDD’s operation principles please see the description of theExperiment ”Rontgenfluoreszenzanalyse mit einem Silizium-Driftdetektor” (”X-ray fluorescence analysis witha silicon drift detector”).

The processing of the SDD’s output signal is sketched in figure 3.2. Initially it goes to a charge sensitivepre-amplifier, which resides on the circuit board providing the supply voltages for the SDD. In order to have assmall as possible noise from outside in the signal lines, this circuit board is situated in the direct vicinity of theSDD (about 5cm distance).

The output signal from the first amplifier stage is then led into an amplifier-/pulse shaping unit (”preamp/shaper”, see section 3.1.2). There the signal is amplified again and the step like output signal is transformedinto a gaussian pulse. Now the signal is led to a discriminator-/ADC-unit. The discriminator lets pass onlysignals above a certain voltage (threshold). In this way troublesome noise can be removed. The ADC convertsthe analog signals to digital ones. The Multichannel Analyzer (MCA) collects the signals in different channelsaccording to their amplitudes. So initially one obtains a pulse height spectrum which is not sorted by energybut by channel numbers. In order to attribute energies to the corresponding channel numbers, a calibration hasto be performed. The collected spectrum is then beeing read into a computer for further processing.

3.1 The shape of the spectrum

As an example, the spectrum of a 55Fe-source is described here, see figure 3.3. 55Fe has a half-life of 2.73 yearsand decays by electron capture into an excited state of 55Mn. During the transition to the ground state X-rayphotons with energies of 5.895 keV (24% probability) and 6.492 keV (2,9% probability) are emitted (amongstothers) - the Mn Kα- and the Kβ-lines. Besides these lines also the socalled espace peak at ≈ 4.2 keV can be seenin the displayed spectrum. The latter is produced in the following way: If a (primary) X-ray photon removesan electron from the K-shell of a silicon atom, electrons from higher shells will fill the gap in the K-shell. Theenergy difference ∆E is released as another X-ray photon. If this X-ray photon leaves the sensitive detectorvolume, not the complete energy Eγ of the primary X-ray photon can be detected but only Eγ − ∆E. The SiKα-line lies at 1.74 keV, the escape-peak therefore has an energy of 4.155 keV.

11

Figure 3.1: Schematic experimental setup

Figure 3.2: The SDD’s output signal is amplified, shaped, discriminated and digitalized. The MCA enters themaximum of each pulse into a histogram. This pulse height spectrum is beeing read by a computer and can befurther processed there.

3.1.1 The Background

Besides these three peaks one also recognizes several fluorescence lines between 4 and 6 keV and a backgroundextending over the whole energy range. The physical origin of those events can be explained by partial chargecollection (”partial events”, see figure 3.4). On the one hand it is possible that the X-ray photon is absorbed inthe thin aluminium coating or at the interface between Al and the p+-Si (case a) Only a part of the charge cannow drift to the active volume and is beeing registered. In the spectrum those events are distributed equally upto the total energy of the X-ray photon. If an X-ray photon is absorbed in the vicinity of the interface betweenthe p+-implantation and the n−-substrate, a partial charge collection can occur as well because the holes inthe p+-implantation can capture a fraction of the electrons. The low energy peak in the spectrum in figure3.4 (at about ADC-channel number 15) is caused by noise in the amplifier electronics. With a little higherdiscriminator threshold this peak is suppressed.

Before a physical justification for the causes of the peak width σ can be given in section 3.2.3, the termshaping time has to be explained first.

12

Figure 3.3: A typical Mn spectrum recorded with an SDD and a Fe-55-source in semi logarithmic representation.One can see both of the Mn-K-peaks, the escape-peak and the pile-up.

3.1.2 The pulse shaper

A pulse shaping is achieved by series connecting of differentiators (CR) and integrators (RC). Differentiatorsattenuate signals with low frequency (high-pass filter), while integrators act as low-pass filters. Thereby effec-tively a higher signal-to-noise ratio is achieved because the signal processing is limited to the frequency rangewhere meaningful detector signals occur. Differentiators and integrators are characterized by the time constant

τ = RC, (3.1)

the product of resistance and capacity (pulse forming constant = shaping time). A CR circuit transforms astep like input signal

Ein(t) =

{

E (t ≥ 0)0 (t < 0)

into an exponentially decaying output signal

Eout(t) = Ee−t/τ .

Accordingly an RC integrator circuit converts it into an exponentially increasing output signal:

Eout(t) = E(

1 − e−tτ)

.

A gaussian pulse shape is achieved in practice by a simple CR differentiator circuit, followed by several RCintegrator circuits. The pulse shaper used in this experiment generates an asymmetric so-called semi-gaussianpulse shape. This means that the rise time of the output signal (≈ 2τ) is shorter than the falling edge (≈ 5τ).So the total pulse duration amounts to 7τ . From this the maximum count rate fmax, without having pile-up(see sections 3.2.2 and 5.2) can be easily estimated for a given shaping time:

fmax =1

7τ(3.2)

The smallest possible shaping time for the pules shaper used here is 0,25 µs, so at most 5, 7 · 105 pulses persecond can be counted. The maximum count rate is still lower because the pulses do not occur regularly butrather randomly.

13

3.2 The noise

The different components contributing to the noise when measuring with semiconductor detectors are listed inthe following equation:

nENC = A12kBT

gm

C2

τ + serial noise (3.3)

+A2

(

2πafC2 +bf

2π

)

+ low frequency noise = 1/f−noise (3.4)

+A3

(

qIl + 2kBTRf

)

τ parallel noise (3.5)

Here nENC is the number of electrons of the equivalent noise charge; gm the transconductance of the JFET;A1, A2, A3 are constants depending on the filter function of the shaper; T is the temperature; C the capacityof the anode; af and bf are constants parametrizing the low frequency noise; Il is the leakage current; Rf is theresistance of the feedback-circuit in the pre-amplifier; τ is the shaping time (see section 3.1.2).

This equation shows that in order to minimize the noise predominantly the capacity C, the temperatureT and the leakage current Il have to be small. Furthermore the different noise components depend differentlyon the shaping time: nENC is for serial noise ∝ τ−1/2, for parallel noise ∝ τ1/2 und the low frequency noise(1/f -noise) independent of the sphapoing time.

3.2.1 Physical cause of the noise

The serial noise (equation 3.3) is thermal noise of a resistance R = 1/gm and is generated within the transistor.The thermal noise arises through brownian motion of the electrons in an electric conductor which leads againto different potentials at both ends of the conductor.

The 1/f -noise (equation 3.4) originates from electrically active traps in the transistor channel which capturecharge carriers and re-emit them. This causes an increase of the electric field in the transistor channel whichagain influences the current. The interferences in the electric field can be described by the traps’ densityand their capture- and emission-rate. The parallel noise (equation 3.5) includes all currents flowing throughthe electric input. This is primarily the surface leakage current which originates from mobile charge carriersat the surface, caused by defects. Those defects predominantly consist of unavoidable impurities and latticedefects. The leakage current arises from thermal generation of electron-hole pairs in the semiconductor, whichare enabled by energy levels within the band gap. These energy levels come from defects in the crystal latticeand contamination with (predominantly metallic) impurity atoms in the silicon. If those energy levels (traps)lie in the middle of the band gap, the leakage current is halved each time the temperature drops by 7 K.Contributions to the parallel noise are also the leakage current through the gate as well as the feedback currentthrough the feedback resistance in the charge sensitive pre-amplifier.

3.2.2 Connection between noise and shaping time

If the leakage current of the detector could be made infinitely small (e.g. by cooling of the SDD and theelectronics), one would have to adjust the shaping time τ as large as possible, in order to minimize the noise,until the 1/f -noise is the upper limit for the noise. This however is contradictory to the need for high countrates because long shaping times lead to pile-up effects. Pile-up occurs if not only one put two or more signalsare summed up during one shaping time and thus lead to a wrong entry in the spectrum. Hence C has to bedecreased in order to avoid pile-up and have a low noise despite a short shaping time.

3.2.3 Width of the distribution

In order to be able to separate fluorescence lines which lie close to each other, the width of the peaks (the energyresolution) has to be as small as possible. For an SDD the achievable energy resolution σ is:

σ = w

√

n2ENC +

FEγ

w(3.6)

Here F is the Fano factor (for silicon, F = 0, 115), Eγ is the energy of the X-ray photon and w is the creationenergy for an electron-hole-pair. Even with disappearing nENC the width can not be lower than the Fano-limit

σ =√

wFEγ . (3.7)

14

The Fano factor is defined as the mean squared error of the number of created charge carriers divided by themean number of created charge carriers < n >:

F =< n2 > − < n >2

< n >(3.8)

The Fano factor F can also be interpreted as the observed variance of the charge carrier distribution σ2obs divided

by the variance σ2Pois which one would expect from a Poisson distribution:

F =σ2

beob

σ2Pois

(3.9)

Because the processes leading to the generation of charge carriers are not mutually independent, the observedvariance in number of charge carriers is much smaller than expected from a Poisson distribution, where

σPois

w=

√< n >. (3.10)

(< n > is the number of generated charge carriers.) If one inserts equation 3.10 into equation 3.9, solves it forσobs and takes into account that

< n >=Eγ

w(3.11)

(w is the electron-hole-pair creation energy), one obtains the smallest possible width (equation 3.7). BecauseEγ ∝< n > and σ ∝

√< n >, one obtains a smaller relative width

σ

Eγ∝ 1√

< n >,

the smaller the number of generated charge carriers is..

15

Figure 3.4: The different processes which are responsible for events with partial events on the back side ofthe detector: a) Charge generation at the interface between aluminium and silicon. b) Recombination of signalcharges in the vicinity of the interface p+-implantation-n−-silicon. c) Complete charge collection. d) High energyX-ray radiation leaves the detector without interacting with it. The upper plot shows a typical monoenergeticenergy spectrum, the middle one a schematic cross section through the detector, the lower plot shows theCCE-function (charge collection efficiency) up to a depth of 0,8 µm, so just under 0,3% of the complete waferthickness.

16

Chapter 4

Spectral analysis

4.1 Characterization of the lines

Photons with different energies arrive at the detector. There they generate electrons whose number is pro-portional to the respective photon energy. An MCA acquires the distribution of the individual energies andpresents it as a spectrum. Such a counting process is described by Poisson statistics which for large numberscan be approximated by a Gauss distribution (normal distribution) though.

Firstly the positions of the lines are determined approximately. This can be done by hand or with thecomputer and its peak-search algorithm. In order to obtain the exact position, height and FWHM (Full Widthat Half Maximum) of the lines, they are approximated by a least-squares or maximum-likelihood method.Having an energy sensitive detector, one uses the area under the peak as a measure for the intensity which isequal to the number of events within the fitted Gauss peak (Background already subtracted - see next section).

4.2 Background estimation

For all measurements a background correction has to be applied, in order to obtain correct values for theintensities. There are different approaches for the correction of the intensity. They range from a simplesubtraction of an adjacent measuring point up to a least-square fit with second order polynomials. The followingprocedure has proven to be reasonable.

The spectrum is integrated to the left and to the right side of the peak in equal distance to the peak’s centerin two equally wide bands without the peak (Width ηB/2). Two sums NB1 and NB2 arise from that. Also thepeak itself is integrated over a width P , one obtains the value NT = P + B (see figure 4.1). The backgroundamounts to

B =ηP

ηB(NB1 + NB2) (4.1)

From this one obtains the corrected intensity P . The measurement software used in this experiment alreadyperforms a background correction. Nevertheless one has to make sure that the result is reasonable. The straightline displayed by the program should be a good approximation to the background to the right and to the leftof the peak. If this is not the case, the fit has to be repeated.

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Figure 4.1: Background correction of the peak’s area in order to determine the intensity.

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Chapter 5

Qualitative Analysis

5.1 Identification of elements

All elements show a characteristic X-ray spectrum where the intensities of the different lines exist in fixed ratios.In order to identify an element unambiguously, one usually uses not a single line in a spectrum but rather agroup of lines belonging together and their relative intensities.

For example if one discovers the Kα-line of an element, also the Kβ-line of the same element has to exist withthe correct intensity ratio. If this is not found, the Kα-line is questionable. If the intensity ration is not correct,a second line can be superimposed or effects like absorption or amplification can be involved. Alternatively alsotransitions to the L-shell can be considered in the same way if the resolution and energy range of the spectrometerpermit this. The energies of the K- and the L-lines of different elements and their relative intensities are foundin data reference books or in the periodic table which is available at the experimental setup. The qualitativeanalysis generally precedes any further measurement in order to determine the elements contained in a sample.In doing so it can be determined in a simplified way, which elements are a main constituent of the sampleand which ones are only contained in traces. If the line intensity from the sample is known as well as the lineintensity for the pure element one can estimate the concentration by

cA ≈ Isample

Ipure

. (5.1)

5.2 Negative effects (parasitics)

5.2.1 Line generating effects

Several effects generate further lines which have nothing to do with the fluorescence spectrum and can easilylead to misinterpretations. Others lead to a displacement of the fluorescence lines.

Escape-Peaks arise from secondarily generated photons in the detector which leave the detector volume.Additional lines are produced which are displaced with respect to the characteristic lines by the energy of theSi-Kα-line (1.74 keV).

Rayleigh scattering The primary X-ray radiation, consisting of the continuous Bremsstrahlung and thecharacteristic lines of the X-ray tube’s anode is scattered elasticly by the atoms of the sample. The scatteredtube spectrum is superimposed on the characteristic sample spectrum in the detector. As a remedy filters canbe used which block the anode lines, e.g. Ni for a Cu-target (see the absorption edges of the different elementsin figure 2.3).

Compton scattering When scattering inelasticly at shell electrons, X-ray photons transfer energy to theelectrons. Afterwards the photons have a lower energy by ∆E. The value of ∆E depends primarily on thescattering angle and on the energy (see equation 2.7). Hence this effect leads to additional lines at E −∆E andoccurs predominantly for lightweight atoms and high X-ray energies.

Pile-up If two or more photons hit the detector at the same time and create electron-hole-pairs, additionallines are created with double or multiple energy of that of the actual characteristic lines (see figure 3.3).

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Figure 5.1: Quantum efficiency of the SDD

5.2.2 Background generating effects

If the background increases the recognition of weak lines (e.g. L, M) is hampered significantly.Partial events Because of the spacial distribution of the charge cloud within the detector, charges can

leave the detector volume without beeing registered. This is essentially a geometric effect which depends onthe position where the photon hits the detector and manifests itself in a continuous distribution below thecharacteristic lines.

5.2.3 Line broadening effects

If the measured peaks are strongly broadened, problems will arise during the data analysis in distinguishing thelines and determining their position and height.

Overlapping peaks from two closely adjacent lines can be difficult to distinquish. Often only one singlebroadened Gaussian peak can be recognized. The position of that line can then not be determined preciselyany more.

Energy resolution The energy resolution of the detector and the electronics limits the quality of the wholesystem. The broader the lines get, the worse the quality of the line characterization will be.

5.3 Limits

With the available X-ray source only transitions below a limiting energy can be excited. The limiting energy isdetermined by the acceleration voltage of the X-ray tube (see equation 2.3).

The silicon drift detector can detect only photons in a certain energy range. Here the limits are governedby the entrance window, the detector thickness and the detector material. Absorption data can e.g. be foundon the homepage of the Center for X-ray Optics (see further reading in chapter 8).

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Chapter 6

Experimental procedure

6.1 Operation of the X-ray tube

Before the X-ray tube is activated, chapter 7, radiation protection, should be read. Inside the X-ray apparatusa dose output of more than 10 Sv/h can be generated in the radiation cone. This dose output can damage livingtissue already at short exposure times. Outside the apparatus the dose rate is reduced to less than 1µSv/h, avalue which lies in the order of magnitude of natural radiation exposure. Befor each operation the housing andespecially the lead glass screens and sliding doors as well as the lead glass tube around the X-ray tube have tobe checked for intactness.

Operation of the X-ray tube:

• Connect to the power supply and switch on the X-ray apparatus (Switch at the left side of the housing).

• Push button U and adjust the acceleration voltage to 20kV (turning knob ADJUST).

• Push button I and adjust the heating current to 0,01 mA.

• Check, whether the lead glass doors are closed properly and push HV ON/OFF: The high voltage controllamp blinks and the hot cathode of the X-ray tube shines. A safety switch prevents X-ray radiation frombeeing generated while the doors are open.

6.2 Operation of the detector

• Switch on the supply voltage ±12V (is needed for the operation of the amplifier chips on the SDD circuitboard) at the lower Rohde & Schwarz power supply pack. The voltages are adjusted already and mustnot be changes. The output to the very right is for the peltier elemtent. The maximal voltage here is 12V.

• Switch on the high voltage (-150 V) at the upper Rohde & Schwarz power supply pack.

• Switch on the pulse shaper and the ADC in the NIM rack. To begin with set the pulse shaper to ashaping time of 1 µs. The chip temperature can be determined via the voltage of the temperature diodein the SDD. A decrease of temperature by 1 K results in an increase of that voltage by 30 mV. At roomtemperature (21.6◦C) the voltage is 4.649 V. Note that with longer operation the environment of thedetector will warm up. Thus, switch on the water cooling in order to stabilize the detector temperature.

6.3 Energy calibration

Position the detector into the beam of the X-ray tube. For all following measurements make sure that the deadtime does not exceed 1.5%. This can be achieved e.g. by inserting an additional absorber into the beam orby rotating the detector by a small angle away from the beam. How would a higher dead time influence themeasurements?

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• Adjust the acceleration voltage to 20 kV, the current to 0.01 mA. Watch the detector’s (amplified) outputsignal on the oscilloscope (BNC Connector at the output of the amplifier). Change the acceleration voltageto 30 kV. Watch the output signal of the pulse shaper on the oscilloscope. How does the signal changefor different shaping times? Set the shaping time to 1 µs for the following measurements.

• Record a spectrum of the X-ray tube at different acceleration voltages (15 to 35 kV) and with cooleddetector. How does the shape of the spectrum change? Determine the voltage from which on charac-teristic lines of molybdenum are excited. Adjust the voltage for the following measurements so that thecharacteristic lines are best visible (in doing so, consider also the signal-to-noise ratio of the characteristiclines).

• Perform an energy calibration by means of the characteristiv lines of the X-ray tube (Mo Kα at 17.44keV and Mo Kβ at 19.63 keV). The measurement software offers an built-in energy calibration: open thewindow calibration and perform a Gauss fit for both characteristic lines. The channel number can betransferred directly to the window for the energy calibration by clicking the button ”Fit”. The energieshave to be entered by hand and the values are adopted with clicking ”Add ≫”. Then a straight line is fittedto the values by clicking ”Calibration”. The fit values are displayed in the same window and should bewritten down for further analyses. The checkbutton ”Use calibration” has to be checked. Furthermore theenergy axis has to be made visible via the window ”Axis Options”. The calibration should be performedin such a way that energies up to ∼28 keV can be measured. To achieve this the amplification factorcan be adjusted at the shaper (”Gain Fine/Coarse”). After each change the calibration has to be redone!Therefore write down the calibration parameters for each shaping time.

• In order to improve the energy calibration, two further lines at known energies shall be used. For thispurpose, the sample holder has to be equipped with a piece of copper and set to an angle of 45◦. Thedetector is set to 90◦ in order to measure the X-ray fluorescence radiation of the copper sample (setupnumber 1, figure 3.1). Record the spectrum of the copper sample and use the Cu-Kα- and Kβ-lines forthe energy calibration.

6.4 Determination of the energy resolution

Position the detector directly in the beam and adjust the X-ray tube’s emission current to 0,01 mA. Measurethe peak width (the energy resolution) of the Mo-Kα-line for different temperatures T and shaping times τ .How does the energy resolution change and why? After changing the shaping time, the calibration has to beredone! It is thus recommended that you first leave the shaping constant, calibrate and measure at differenttemperatures before going to the next shaping time and recalibrate. Which shaping times are optimal for whichtemperatures? For the following measurements set the optimal temperature and shaping time.

6.5 Identification and layer thickness determination of different sam-ples

Four unknown samples are available at the experiment. These shall firstly be identified by their fluorescencelines (Setup 1, figure 3.1) and then their thickness shall be measured.To do this, the detector is set to an angle of 0◦ (compare setup 2, figure 3.1). Set the acceleration voltage to25 kV and the current to 0.01mA. Make sure that the dead time has an acceptable value. If this is not thecase, decrease the intensity by rotating the detector by a few degrees away from the beam.

Now position the sample in the beam and set the sample angle to 90◦. Measure the spectrum transmittedby the sample with sufficient exposure time so that the Mo-Kα-line shows enough statistics. Measure the areaunder its peak by fitting a Gaussian and write down this value. Make sure that the background is fitted correctly.Now remove the sample and measure the spectrum without the sample with the same exposure time. Determinethe area under the Mo-Kα-peak again. From those two values and the material dependent attenuation length(or ”linear attenuation coefficient”) the sample’s thickness can be determined. The attenuation length can befound on the internet at http://www.cxro.lbl.gov/optical_constants/ or athttp://physics.nist.gov/PhysRefData/FFast/html/form.html.

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6.6 Measuring the energy dependance of the absorption coefficient

Use one of the identified samples from the previous step and determine the energy dependance of its absorptioncoefficient. To do this, take a spectrum from the X-ray tube with the sample and make sure that the continuumis recorded with sufficient statistics. Save this spectrum on the computer and measure the spectrum from theX-ray tube without absorber with the same exposure time. Save this spectrum as well. By dividing one curveby the other one obtains µ(E). Explain the properties of this curve and compare it to the theoretical curve.

6.7 Thickness determination of a sample glued on a carrier which

can not be transmitted

Another sample is available consisting of two layers. The carrier material is lead which can not be transmittedby the X-rays. Now the thickness of the copper shall be determined by using the fluorescence radiation of thelead which is produced after the X-rays from the X-ray tube have gone through the copper layer at an angle α.The fluorescence ratiation is measured after it has again gone through the copper at an angle β (see figure 3.1).By measuring at two different detector angles β1 and β2 the thickness of the copper can be determined.

Remarks concerning the lab report

Present the measurements and analyses graphically and/or in table form. Comment on the results and performa detailed calculation of errors where this is possible.

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Chapter 7

Radiation protection

A considerable amount of human radiation exposure is caused by X-ray radiation. Therefore the causes andconsequences of radioactive exposure will be described in this chapter.

The X-ray apparatus used in this experiment has a qualification approval and special safety precautions.The experimental chamber and the X-ray tube reside behind lead glass screens. As soon as they are opened, theX-ray tube will switch off immediately. The does output during operation is below 1 µSv/h (see next section).

7.1 Dose quantities

The biological effect of ionizing radiation is essentially a consequence of the ionization taking place within thebiologic material (tissue, bones, . . .). In cells, few ionizations can be enough to affect individual functionalitiesor destroy the ability for cell division. Quantitatively the radiation exposure is described by the absorbed dose

D =dE

dm.

The SI-unit of the absorbed dose is the Gray: 1 Gy = 1 J/kg (=100 rad). The biological effect of a radiationexposure can not be sufficiently described by the absorbed dose alone. E.g. the irradiation with α particlesis more harmful than with electrons even with the same absorbed dose because the ionization is distributedgeometrically different (the ionizations are more dense along the particle path for α-particles than they are forfast electrons).

For the purpose of radiation protection the absorbed dose is multiplied by the radiation weighting factor(RBE-factor, relative biological effect) in order to consider the different biological effect of the different radiationtypes. The product of radiation weighting factor and absorbed dose is called equivalent dose:

H = qD

The unit of the equivalent dose is the Sievert: 1 Sv = 1 J/kg (= 100 rem). Photons (and thus also X-rays) andelectrons have a radiation weighting factor of 1, for slow neutrons q = 2 . . . 5, for fast neutrons q = 10 . . . 15,and for α-particles and heavy ions q = 20.

7.2 Consequences of high radiation exposure

When considering the biological effect of a radiation exposure, one distinguishes between the acute radiationeffects and the long-term damages. While exposures on the whole body below 0,25 Sv do not seem to cause anyacute effects, doses over 1 Sv affect e.g. the hematopoietic organs. Doses of more than 5 Sv are lethal within ashort time.

The most important long-term damage is the formation of tumors. With equivalent doses of about 1 to 2Sv the probability of tumor diseases doubles compared to the value without irradiation. Also smaller doses cancause cancer diseases with correspondingly lower probability.

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radiation source < H > (mSv/a)cosmic radiation 0,45terrestrial gamma radiation 0,6air 1,8global fallout 0,04endogenous radioactive nuclides 0,35X-ray examinations 1,8TV / monitors 0,01nuclear power plants 0,00003

Table 7.1: Average equivalent doses per year

7.3 Average radiation exposure

Natural radiation exposure of humans is subdivided into cosmic, terrestrial and endogenous (= produced insidethe body) radiation. In table 7.1 average equivalent doses are given for the radiation exposure of the public.

Cosmic radiation is mainly made up of protons and α-particles which interact with the nuclei of the moleculesof the air. By this process secondary radiation is produced, consisting predominantly of protons, neutrons, pions,muons, kaons, electrons and photons. The average dose output of cosmic radiation at sea level is about 400µSv/a. The yearly dose increases with the height above sea level by about 10 µSv/a per 30m.

The terrestrial radiation is the radiation of the natural radioactive nuclides with very long half-life timesand their decay products. It is regionally strongly variable and reaches e.g. in Germany values of 0.1 mSv/a(northern Germany) or 1.5 mSv/a (Bavarian Forest).

The uranium contained in the soil or building materials causes a relatively high radiation exposure by one ofits decay products, 222Rn, when it is inhaled. Radon is a noble gas and hence diffuses through matter withoutinteraction. It decays via α-emission with a half-life of 3.8 days. Especially in closed rooms it can accumulatequickly. Also the global fallout contributes to the terrestrial radiation. This is an increase of radioactivity(mainly 3H, 90Sr and 137Cs) on the earth’s surface as a consequence of atmospheric nuclear testing in the sixtiesand of the reactor accident of Chernobyl. The endogenous radiation exposure arises from radioactive nuclei like40K and uranium inside the human body.

The largest source of artificial radiation exposure is given by medical diognostics. The X-ray tube usedin this experiment produced a dose output of 1 µSv/h at 10cm distance from the lead glass screening. For acomputer monitor it is about 5 µSv/h.

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Chapter 8

Further reading

• E. Bertin, Principles and Practice of X-ray Spectrometric Analysis, Plenum Press, 1975

• G. Lutz, Semiconductor Radiation Detectors, Springer, 1999

• R. Jenkins, An Introduction to X-ray Spectrometry, Heyden, 1974

• R. Jenkins, Quantitative X-ray Spectrometry, Dekker, 1995

• Strahlenschutz, Radioaktivitat und Gesundheit, Bayerisches Staatsministerium fur Landesentwicklungund Umweltfragen

• www.hll.mpg.de und www.ketek.net - Publications

• www-cxro.lbl.gov - Center for X-ray Optics

Imprint

Setup of the experiment: Michael Bauer and Nico Cappelluti 2006

Manual List of Versions:

• Version 1.0: Michael Bauer 11.10.2006First Version

• Version 1.1: Martin Muhlegger 25.10.2007Small corrections and translation into English

Experiment supervisor: Elena Orlando

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