Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 1999-2000 Transmission MATTER Scattering Compton Thomson Photoelectric

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 1999-2000

hTransmissionMATTER

hScattering

Compton

h h '

Thomson

Photoelectric absorption

Pair productionh > 1M eV

X-rays

Interaction X-rays - Matter

h

Decay processes

Fluorescence Auger electrons

h f

Primary competing processes and some radiative and non-radiative decay processes


Thomson Observed data

Electron positron

pairs

Compton


Photonuclear

absorptionCro

ss s

ecti

on (

barn

s/at

om)

1

103

106

10 eV 1 KeV 1 GeV1 MeV

Cu Z=29

Energy

Li Z=3 Ge Z=32 Gd Z=64

Energy (KeV)100 102 104104100 102 104

100

102

104

100 102

(

Bar

ns/a

tom

)

100

102

104

100

102

104

X-ray attenuation: atomic cross section

h h

sample


Interactions between x-rays and matter

The fundamental interactions between matter and x-rays having energy less than 1 MeV are: elastic, coherent or Rayleigh scattering inelastic, incoherent or Compton scattering photoelectric absorption

I(t) = I0 exp (- t)

They all contribute to the attenuation of intensity when an x-ray beam crosses matter. The phenomenonis described by the Beer-Lambert law

is the linear absorption coefficient. It is dependent on the energy of the radiation and specific for acertain element and it is the sum of the coefficients representing the phenomena mentioned above

elastic inelastic

the photoelectric linear absorption coefficient, is by far the most important at low energies


1 10 1000.0001

0.001

0.01

0.1

1

10

100

1000

10000

Z=57 Lanthanum

elastic

inelastic

+ elastic + inelasticM

ass

Abso

rptio

n C

oeffi

cien

ts [

cm

2 \ g]

Photon Energy [keV]

Absorption coefficient

Photoelectric mass absorption, coherent scattering, incoherent scattering coefficients of Lanthanum(Z=57) versus energy of the incident photon.

In literature the mass absorption coefficient m= is more likely to be found.

The coefficient for a compound C is calculated taking into account the weight fractions Wi of theelements in the sample considered

C

ii i

ii i

el

i

in

i

W W


Elastic scatteringIt represents the collision of a photon with a tightly bound electron leading to an outgoing photon witha different direction but the same energy.

Rayleigh scattering

Rayleigh scattering occurs mostly at the low energies and for high Z materials.The elastic scattering cross section is given by:

d

d

d

dF x Zel T

,

2

where d

dT is the Thomson scattering cross section for one electron

d

d

rT

0

22

21( cos )


FxZ, is the atomic form factor and xsin

2.

If qk22sin is the momentum transfer and

rn the vector from the nucleus of the atom to the nth electron

we have

F q Z iq r r iq r drnn

Z , exp exp

0 0

1

0 is the ground-state wave function of the atom

Hence with our former notation

F x Z r rrx

rxdr,

sin

44

40

in which (r) is the total electron density.

1.5

1.0

0.5

0.0

0.5

1.0

1.5

0

30

60

90

120

150

180

210

240

270

300

330

d /

d c

oher

ent

[c

m 2

/ g

sr]

Incident Photon Energy = 30 keV Z=14 Si Z=28 Ni Z=42 Mo Z=56 Ba Z=70 Yb

Elastic scattering differential cross section for differentelements as a function of the scattering angle and for a fixedincident photon energy equal to 30 keV.


Inelastic scattering

A loosely bound or free electron hit by a photon undergoes Compton effect and the scattered photonhas different direction and lower energy

Compton scattering

The energy of the outgoing photon is

E

EE

m c1 1

02 cos


The differential inelastic cross section for a single free electron is given by Klein-Nishina formula

d

d

rP EKN

02

2,

where PE, is called the polarisation factor;

P E

,

coscos

cos

cos

1

1 11

1

1 122

2 2

and E

mc02. If

is very small, that is if the photon energy is much smaller than the electron rest

mass, then the polarisation factor reduces to 1 2cos and the Klein-Nishina differential crosssection to the Thomson one.

In order to account for the interference between the waves scattered by each electron in the atom theincoherent scattering function SxZ, is introduced. It is a generalisation of the atomic form factor toinclude excited states.

Indicating with the energy of an excited stationary state as measured from the ground level we canwrite

F q Z iq rnn

Z

, exp

0

1


The incoherent scattering function is then

S q Z F q Z, ,

2

0

0.020

0.015

0.010

0.005

0.000

0.005

0.010

0.015

0.020

0

30

60

90

120

150

180

210

240

270

300

330

d /

d in

cohe

rent

[c

m 2

/ g

sr]

Incident Photon Energy = 30 keV Z=14 Si Z=28 Ni Z=42 Mo Z=56 Ba Z=70 Yb

Inelastic scattering differential cross section for different elements as a function of the scattering angleand for a fixed incident photon energy equal to 30 keV.

A comparison between the elastic and inelastic processes shows that the first ones are more importantfor small energies and high Z elements while for the upper part of the spectrum and the lighter elementsthe other ones become dominant.


10 20 30 40 50 60 70 80 90 1000.00001

0.0001

0.001

0.01

0.1

1

d /

d in

cohe

rent

d /

d c

oher

ent

[cm

2 /

g s

r]

Scattering Angle = 90 degrees Z=14 Si Z=28 Ni Z=42 Mo Z=56 Ba Z=70 Yb

Incoherent Scattering

Coherent Scattering

Incident Photon Energy [keV]

Elastic and inelastic scattering differential cross section for different elements as a function of the incident photonenergy for a fixed scattering angle equal to 90 degrees. (Calculated using the data from reference

The scattering coefficients elastic and inelastic which contribute to the total linear absorption coefficientare the total scattering cross sections, obtained by integration over the solid angle d.

el in

el ind

dd,

,

They are therefore angle independent.



It is the interaction which leads to the fluorescence photons. Fortunately it is dominant in the energyrange we are interested in. A photon incident on an atom and having energy greater than the extractionpotential of an electron in a certain shell may remove the electron and leave a vacancy.

The extraction potentials are also known as absorption edges. At this energy the photoelectric absorption coefficient presents a discontinuity.

20 40 60 800

20

40

60

80

100

120 K-edge L1-edge L2-edge L3-edge

Ene

rgy

of A

bsor

ptio

n E

dges

[ ke

V ]

atomic number Z

Energy of photoelectric absorption edges versus atomic number of the elements.


The total photoelectric absorption coefficient consists of the sum of the coefficients for the single shells.

= K + L1 + L2 + L3 + M1 + . . .

1 10 1000.001

0.01

0.1

1

10

100

1000

10000

Z=57 Lanthanum

-K

-K-L1

-K-L1-L2

-K-L1-L2-L3

Mas

s A

bsor

ptio

n C

oeffi

cien

t [cm

2 / g

]

Energy [keV]

Mass absorption coefficient of Lanthanum (Z=57) versus energy of incident photons split in its components.

When the photons reach an energy equal to an absorption edge, increases abruptly because the photons are suddenly able to remove some more electrons.


One of the components of the single shell photoelectric absorption coefficient may be calculated from if the jump ratios are known. At an energy above the k-edge we have:

K

r

r

( )1

j jkZ

E

3

8

3

k is a constant which changes with the element and the shell.

Between two absorption edges decreases with the photon energy approximately following Bragg-Pierce law

At the edge energy the photoelectric absorption coefficient presents a discontinuity. Such discontinuity is described through the jump ratio r

The edge discontinuity is described through the jump ratio r

E

Er

edgeE

edgeE

lim

lim


De-excitation mechanisms

Radiative transition: Fluorescence

Non radiative transition:Auger effect

Cu K

Cu29

NM

KL

De excitation

Cu29

NM

KL

Cu K photon

Auger electron

Primary X-ray photon

Excited system

Cu29

NM

K L

e- photoelectron

Competing decay processes following the presence of a core hole


0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

Flu

ore

sce

nt

Yie

ld

Atomic Number Z

K

L

2 p3/2

2 p1/22 s

1 s

Kfluorescent

x-ray

Fluorescence

2 p3/2

2 p1/2

2 s

1 s

Auger electron

Auger effect

Prevails for heavvy atoms

Prevails for light atoms

Xs = probability of emissionof a fluorescence photon;

s denotes the edge.

As = probability of emissionof a fluorescence photon from an s edge.

Fluorescence yield:ss

ss AX

X

Auger yield: s1

s


Because of Auger effect the lines of a givel series (for instance the K lines: K1, K2,, K1, …) are not as intense as the would be predicted from the number of vacancies which are present at the associated energy level (tke k level for the K lines).The K fluorescence yield is defined also as the number of photons of all K lines emitted in the unit time divide by the number of K vacancies formed in the same time, i.e.;

K

K K K

K

n n n

N

1 2 1

...

nK ibgRepresets the number of photons of the spectral line i emipptted in unit time

N K is the rate of production of vacancies in the K shell

The fluorescence yields characteristic of the the shell L, L or the shell M, M,… are defined similarly.

The Auger yield is defined as the ratio of the Auger electrons prosuced in the unit time and the vacancies created in the same time interval.

Were it not for the Auger effect the yield fluorescence of a given line series would always be 1.

The fluorescent yield depends on the atomic number Z and the line series; an empirical approximation is given by:

Z

A Z

4

4

Where A is a constant having value 104 for K lines and 108 for L lines.


The Auger EffectThe Auger effect, and the fluorescence photons are secondary interactions: they can only happen after aphotoelectric absorption has occurred.

Fluorescent yield and Auger electron yield versus atomic number of the elements. (Curves calculatedusing fit parameters from reference


It sets the basis for elemental fluorescence analysis.It consists of a series of discrete energies correspondingto the energy difference between two atomic levelsand is therefore characteristic of the emitting element.

The characteristic spectrum

When a K-shell electron is removed the vacancy can befilled by an L-electron, this will result in another vacancyand photons belonging to different lines will be emitted.

n 0

l = 1

j 1, 0

Siegbahn IUPAC Siegbahn IUPAC

K1 K-L3 L1 L3-M5K2 K-L2 L2 L3-M4K1 K-M3 L1 L2-M4K2 K-N2,N3 L2 L3-N5K3 K-M2 L3 L1-M3

L4 L1-M2


Moseley was the first to investigate and find the relationship between the atomicnumber of an element and the energy of its spectral lines and it can be described as

E ( Z ) = kj ( Z - j)2

20 40 60 800

20

40

60

80

100

120 K1

K2

K1

K2

L1

L2

L1

L2

Cha

ract

. Lin

e E

nerg

y [k

eV]

Atomic Number Z

where kj and j are fixed for a certain transition j

Energy of the most important lines versus atomic number Z.


The atomic scattering factors.In the classical theory of dispersion the atom is assumed to scatter radiation as if it was formed bydipole oscillators whose natural frequencies are those of the absorption edges of the electronic shells. Inthis approximation oscillators are formed by bound electrons moving harmonically with respect to afixed nucleus. An electromagnetic wave falling on an atom forces the electrons to oscillate; the dipoleradiates with same frequency of the incident wave. The dipole scattering factor is defined as the ratio ofthe amplitude of the wave scattered by a bound electron to the one scattered by a free electron.

fA

Abound

free

The quantity obtained happens to be complex and is usually separated in its components.

f f if

The expression is generalised by summing the contributions of all the oscillators contained in the atomand we can write

f = f1 + if2

0 5 10 15 20 25 30-10

0

10

20

30

40

50

60

70

Z=73 Ta Z=41 Nb Z=14 Si

Atom

ic s

catte

ring

fact

or f 1

Incident photon energy [keV]

Atomic forward scattering factorf1 as a function of energy


0 5 10 15 20 25 300

10

20

30

40 Z=73 Ta Z=41 Nb Z=14 Si

Atom

ic s

catte

ring

fact

or f 2

Incident photon energy [keV]Atomic forward scattering factor f2 as a function of energy.

The real part of the atomic scattering factor for energies above and far away from the absorption edge is approximately equal to the number of “relevant” electrons: for a photon energy above the K absorption edge f1=Z for, an energy between the L3 and the K absorption edge f1=Z-2 and so on.


Material density [g/cm3] 10-6 10-8

Plexiglas 1.16 0.9 0.055Glassy carbon 1.41 1.0 0.049Quartz glass 2.20 1.5 0.46Silicon 2.33 1.6 0.84Tantalum 16.6 9.1 87.5Platinum 21.45 11.7 138.2Gold 19.3 10.5 129.5

Real and imaginary part of the refractive index calculated for Mo K radiation (17.478 keV)

It can then be shown that the refractive index for a material with N atoms per unit volume is given by

nN r

f if 12

02

1 2

in which

ande is the electron chargec is the velocity of lightm0 is the electron rest mass

N r

Afa 0 2

12

N r

Afa 0 2

22

NN

Aa is the number of atoms per unit volume


Reflection and refraction of x-raysReflection and refraction of x-rays can be described with the same formalism as for visible light.A monoenergetic beam meeting the border surface between two media with different dielectricconstant is considered. The incident, reflected and refracted beam are approximated to plane waves.On the border surface the contour conditions derived from Maxwell laws (Fresnel conditions) areapplied and an analytical form for the reflected and for the refracted beam is obtained.

Reflection and refraction of x-rays on the border surface between two media

A first consequence of the Fresnel conditions is that the frequency of the outgoing beams equal thefrequency of the incoming beam. Moreover the incident angle is equal to the reflection angle

1 1incident reflected

and the refraction angle follows Snell’s law.

1

2

2

1

cos

cos

n

n


nc

v

where c is the velocity of light in vacuum and v is the phase velocity of light in a medium.The phase velocity in a medium is energy dependent and is in the x-ray region bigger than c; as aconsequence the refraction index also depends on the energy of the incident photon and it is for x-raysslightly smaller than 1. Usually absorption effects in the media cannot be avoided and the index is acomplex number. Thus it is written as

n = 1- - i

The coefficient is called decrement because it reduces the real part of the refractive index from theunitary value. The coefficients is a measure of the absorption and is related to the linear photoelectricabsorption coefficient by

4

n is called the refractive index and is defined by


Critical angle and total reflection

The refractive index for air can be approximated to the refractive index of vacuum which is 1.Snell’s law becomes hence

1

2

2

1n

cos

cos

n2 is smaller than 1. This means that the refractive angle 2 must be smaller than the incident angle 1.The incident angle for which the refraction angle becomes 0 is called the critical angle

coscrit n 2

Since n2 is very close to one crit must be very close to 0 and the cosine can be approximated with thefirst two terms of the Taylor series in 0.

cos

critcrit n 12

2

2

If we ignore the imaginary component of the refractive index we obtain

crit 2


10 1000

5

10

15

20

25

30

35 Z=14 Si Z=41 Nb Z=73 Ta

Crit

ical

ang

le [m

rad]


10 1000.1

1

10

Crit

ical

ang

le [m

rad]


Theoretical critical angle for different reflector materials as a function of incident photon energy in a linear -logarithmic graph and log-log graph. Far away from the absorption edges the ratio between the critical angle for different materials is constant.

Material critical angle [mrad]17,5keV 60keV 100keV

Silicon 1.8 0.5 0.3Niobium 3.1 0.9 0.6Tantalum 4.2 1.2 0.7Platinum 4.8 1.4 0.8Gold 4.5 1.3 0.8Theoretical critical angle for different materials and different energies of the incident photons.


ReflectivityThe reflectivity of a surface is defined as the ratio

RE

Eincident

reflected

2

whereEincident is the electric field associated to the incident beamEreflected is the electric field associated to the reflected beam

Using Fresnel formulas we get

R

h h

h h

c

c

2 1

2 1

1

2

1

2

in which hc c

2 2 22

1

2

1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.2

0.4

0.6

0.8

1.0E = 17.6 keV

Z=14 Si Z=41 Nb Z=73 Ta

Ref

lect

ivity

/ crit

Theoretical reflectivity different materials as a function of the ratio

c for a fixed incident photon energy.


0 20 40 60 80 1000.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

= 0.8 c

Z=14 Si Z=41 Nb Z=73 Ta

Ref

lect

ivity


Theoretical reflectivity for different reflector materials as a function of incident photon energy at a fixedratio

c.

For the calculation it is assumed that the reflecting surface is „perfectly“ polished for every material.


Penetration Depth.

When a beam coming from vacuum or air hits the surface of an optically denser medium under an angle smaller than thecritical one it is „totally“ reflected. This reflection does not entirely occur in the first layer of atoms but the beam penetratesinto the material and the importance of the phenomenon depends on the reflector material; from the theoretical considerationsalready explained it can be quantified. With z

e1 is indicated the depth at which the beam intensity is reduced of a factor 1/e

z

c

Ee

c c

1

2 2 2 2

1

2 2 2

1

22

2

4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.41

10

100

1000 E = 17.6 keV

Z=14 Si Z=41 Nb Z=73 Ta

Pen

etra

tion

dept

h z 1

/e [n

m]

/ critTheoretical penetration depth for different reflector materials as a function of the ratio

c for a fixed incident photon energy.


0 20 40 60 80 1001

10

= 0.8 c

Z=14 Si Z=41 Nb Z=73 Ta

Pen

etra

tion

dept

h z 1

/e [n

m]


Theoretical penetration depth for different reflector materials as a function of incident photon energy at a fixedratio

c

For the light elements the penetration depth increases, for these elements and for high energies inelasticscattering becomes important and therefore the background „noise“ grows up.


Diffraction

An x-ray beam impinging on an ordered, periodical structure like a crystal or a multilayer undergoes multiplereflections from the different planes of the lattice or the delimitation surfaces between the various layers.

Multiple reflection of an x-ray beam from the different plane s of a crystal.

The reflected waves interfere with each other and diffraction patterns are obtained. In first approximation (that is if we ignore dispersion effects inside the crystal) the phenomenon follows Bragg’s law sin2dn

where n is the order diffraction, d is the interplanar spacing and is the angle between the reflecting planes and the incident beam.

Documents

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 1999-2000 Transmission MATTER Scattering Compton Thomson Photoelectric