Interaction X rays  Matter ...

Author
nguyennguyet 
Category
Documents

view
218 
download
0
Embed Size (px)
Transcript of Interaction X rays  Matter ...

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
hνTransmissionMATTER
hνScattering
Compton
h hν ν' <
Thomson
Photoelectric absorption
Pair productionhν > 1M eV
Xrays
Interaction Xrays  Matter
hν
Decay processes
Fluorescence Auger electrons
h fν
Primary competing processes and some radiative and nonradiative decay processes

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
Thomson Observed data
Electron positronpairs
Compton
Photoelectric absorption
Photonuclearabsorption
Cro
ssse
ctio
n (b
arns
/ato
m)
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
σ(B
arns
/ato
m)
100
102
104
100
102
104
Xray attenuation: atomic cross section
hν hν
sample

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
Interactions between xrays and matter
The fundamental interactions between matter and xrays having energy less than 1 MeV are:− elastic, coherent or Rayleigh scattering− inelastic, incoherent or Compton scattering− photoelectric absorption
I(t) = I0 exp (µ t)
They all contribute to the attenuation of intensity when an xray beam crosses matter. The phenomenonis described by the BeerLambert law
µ 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
µ = τ + σelastic + σinelastic
τ the photoelectric linear absorption coefficient, is by far the most important at low energies

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
1 10 1000.0001
0.001
0.01
0.1
1
10
100
1000
10000
Z=57 Lanthanum
τ σelastic σinelastic τ + σelastic + σinelastic
Mas
s A
bsor
ptio
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 µm=µ/ρ 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
µρ
µρ
τρ
σρ
σρ
=
=
+
+
∑ ∑
C
ii i
ii i
el
i
in
i
W W

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
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:
( )dd
d
dF x Zel T
σ σΩ Ω
= ,2
where d
dTσ
Ω is the Thomson scattering cross section for one electrond
d
rTσ θΩ
= +02
2
21( cos )

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
( )F x Z, is the atomic form factor and xsin
=
θ
λ2
.
If r
hq
k=
22sin
θ is the momentum transfer and
rrn the vector from the nucleus of the atom to the nth electron
we have
( ) ( ) ( ) ( )F q Z iq r r iq r drnn
Zr r r r r r r, exp exp= ⋅ = ⋅
=∑ ∫Ψ Ψ0 0
1
ρ
ψ0 is the groundstate wave function of the atom
Hence with our former notation
( ) ( ) ( )F x Z r r rxrx
dr,sin
=∞
∫44
40π ρ
π
π
in which ρ(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 σ
/ d Ω
coh
eren
t
[cm
2 /
g s
r]
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.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
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 cosθ

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
The differential inelastic cross section for a single free electron is given by KleinNishina formula
( )dd
rP EKN
σθ
Ω= 0
2
2,
where ( )P Eθ , is called the polarisation factor;
( )( )( )
( )( )P Eθ α θ
θα θ
α θ,
coscos
cos
cos=
+ −+ +
−+ −
1
1 11
1
1 122
2 2
and α =E
m c02 . If α is very small, that is if the photon energy is much smaller than the electron rest
mass, then the polarisation factor reduces to 1 2+ cos θ and the KleinNishina 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 ( )S x Z, is introduced. It is a generalisation of the atomic form factor toinclude excited states.
Indicating with ε the energy of an excited stationary state as measured from the ground level we canwrite
( ) ( )F q Z iq rnn
Z
ε εr r r, exp= ⋅
=∑ Ψ Ψ0
1

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
The incoherent scattering function is then
( ) ( )S q Z F q Z, ,=>∫ ε
ε
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 σ
/ d Ω
inco
here
nt
[c
m 2 /
g s
r]
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.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
10 20 30 40 50 60 70 80 90 1000.00001
0.0001
0.001
0.01
0.1
1
d σ
/ d Ω
inco
here
nt
d σ /
d Ω 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 σelastic and σinelastic which contribute to the total linear absorption coefficientare the total scattering cross sections, obtained by integration over the solid angle dΩ.
σσ
el in
el ind
dd,
,= ∫ Ω ΩThey are therefore angle independent.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
Photoelectric absorption
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 τ presents a discontinuity.
20 40 60 800
20
40
60
80
100
120 Kedge L1edge L2edge L3edge
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.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
The total photoelectric absorption coefficient τ consists of the sum of the coefficients for the single shells.
ττ = ττK + ττL1 + ττL2 + ττL3 + ττM1 + . . .
1 10 1000.001
0.01
0.1
1
10
100
1000
10000
Z=57 Lanthanum
τ ττK ττKτL1 ττKτL1τL2
ττKτL1τL2τL3
Mas
s A
bsor
ptio
n C
oeffi
cien
t [cm2
/ 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, τ increases abruptly because the photons are suddenly able to remove some more electrons.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
One of the components of the single shell photoelectric absorption coefficient may be calculated from τ if the jump ratios are known. At an energy above the kedge we have:
τ τKr
r=
−( )1
τ j jkZ
E
=3
8
3
k is a constant which changes with the element and the shell.
Between two absorption edges τ decreases with the photon energy approximately following BraggPierce law
At the edge energy the photoelectric absorption coefficient τ presents a discontinuity. Such discontinuity is described through the jump ratio r
The edge discontinuity is described through the jump ratio r
( )( )EE
r
edgeE
edgeE
τ
τ
−
+
→
→=lim
lim

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
Deexcitation mechanisms
Radiative transition: Fluorescence
Non radiative transition:Auger effect
Cu Kα
Cu29
NM
KL
De excitation
Cu29
NM
KL
Cu Kαphoton
Auger electron
Primary Xray photon
Excited system
Cu29
NM
K L
e photoelectron
Competing decay processes following the presence of a core hole

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
0 20 40 60 800.0
0.2
0.4
0.6
0.8
1.0
Flu
ores
cent
Yie
ld
Atomic Number Z
K
L
2 p3/22 p1/22 s
1 s
Kα1fluorescent
xray
Fluorescence
2 p3/22 p1/22 s
1 s
Auger electronAuger 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
+=ω
Auger yield: sω−1
sω

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
Because of Auger effect the lines of a givel series (for instance the K lines: Kα1, Kα2,, Kβ1, …) 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.;
ω α α βKK K K
K
n n n
N=
+ + +1 2 1
...
nK ib g Represets the number of photons of the spectral line i emipptted in unit timeNK is the rate of production of vacancies in the K shell
The fluorescence yields characteristic of the the shell L, ωL or the shell M, ωM,… are defined similarly.
The Auger yield is defined as the ratio of the Auger electrons prosuced in the unit time and the vacancies created inthe 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:
ω =+Z
A Z
4
4
Where A is a constant having value 104 for K lines and 108 for L lines.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
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
Flu
ores
cent
Yie
ld
Aug
er E
lect
ron
Yie
ld
0 20 40 60 800.0
0.2
0.4
0.6
0.8
1.0
ωK 1ωKωL 1ωL
Atomic Number Z

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
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 Kshell electron is removed the vacancy can befilled by an Lelectron, 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
Kα1 KL3 Lα1 L3M5Kα2 KL2 Lα2 L3M4Kβ1 KM3 Lβ1 L2M4Kβ2 KN2,N3 Lβ2 L3N5Kβ3 KM2 Lβ3 L1M3
Lβ4 L1M2

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
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  σ j)2
20 40 60 800
20
40
60
80
100
120 Kα1 Kα2 Kβ1 Kβ2 Lα1 Lα2 Lβ1 Lβ2
Cha
ract
. Lin
e E
nerg
y [k
eV]
Atomic Number Z
where kj and σj are fixed for a certain transition j
Energy of the most important lines versus atomic number Z.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
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 3010
0
10
20
30
40
50
60
70
Z=73 Ta Z=41 Nb Z=14 Si
Ato
mic
sca
tterin
g fa
ctor
f 1
Incident photon energy [keV]
Atomic forward scattering factorf1 as a function of energy

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
0 5 10 15 20 25 300
10
20
30
40 Z=73 Ta Z=41 Nb Z=14 Si
Ato
mic
sca
tterin
g fa
ctor
f2
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=Z2 and so on.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
Material density ρ [g/cm3] δ⋅106 β⋅108
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 Kαα 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
( )n N r f if= − +12
02
1 2
λπ
in which
ande is the electron chargec is the velocity of lightm0 is the electron rest mass
δπ
ρλ=
N r
Afa 0 2 12
βπ
ρλ=
N r
Afa 0 2 22
NN
Aa=ρ
is the number of atoms per unit volume

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
Reflection and refraction of xraysReflection and refraction of xrays can be described with the same formalism as for visible light.A monoenergetic beam meeting the border surface between two media with different dielectricconstant ε 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 xrays 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
θ θ1 1incident reflected=
and the refraction angle follows Snell’s law.
1
2
2
1
θθ
=cos
cos
n
n

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
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 xray region bigger than c; as aconsequence the refraction index also depends on the energy of the incident photon and it is for xraysslightly 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 δ  iβ
The coefficient δ is called decrement because it reduces the real part of the refractive index from theunitary value. The coefficients β is a measure of the absorption and is related to the linear photoelectricabsorption coefficient by
βλµπ
=4
n is called the refractive index and is defined by

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
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
θθ
n2 is smaller than 1. This means that the refractive angle θ2 must be smaller than the incident angle θ1.The incident angle for which the refraction angle becomes 0 is called the critical angle
cosθcrit n= 2
Since n2 is very close to one θcrit must be very close to 0 and the cosine can be approximated with thefirst two terms of the Taylor series in 0.
cosθθ
critcrit n≈ − ≈12
2
2
If we ignore the imaginary component of the refractive index we obtain
θ δcrit ≈ 2

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
10 1000
5
10
15
20
25
30
35 Z=14 Si Z=41 Nb Z=73 Ta
Crit
ical
ang
le [m
rad]
Incident photon energy [keV]
10 1000.1
1
10
Crit
ical
ang
le [m
rad]
Incident photon energy [keV]
Theoretical critical angle for different reflector materials as a function of incident photon energy in a linear logarithmic graph and loglog 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.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
ReflectivityThe reflectivity of a surface is defined as the ratio
RE
Eincident
reflected
=
r
r
2
whererEincident is the electric field associated to the incident beamrEreflected 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
θθθθ
in which hc c
=
+
−
+
θθ
θθ
βδ
2 2 2 2
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
θ / θcrit
Theoretical reflectivity different materials as a function of the ratio θ/θc for a fixed incident photon energy.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
0 20 40 60 80 1000.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
θ = 0.8 θc
Z=14 Si Z=41 Nb Z=73 Ta
Ref
lect
ivity
Incident photon energy [keV]
Theoretical reflectivity for different reflector materials as a function of incident photon energy at a fixedratio θ/θc.
For the calculation it is assumed that the reflecting surface is „perfectly“ polished for every material.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
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
=
− + − −
h
θ θ β θ θ
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 [
nm]
θ / θcritTheoretical penetration depth for different reflector materials as a function of the ratio θ/θc for a fixed incident photon energy.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
0 20 40 60 80 1001
10
θ = 0.8 θc Z=14 Si Z=41 Nb Z=73 Ta
Pen
etra
tion
dept
h z 1/e [
nm]
Incident photon energy [keV]
Theoretical penetration depth for different reflector materials as a function of incident photon energy at a fixedratio θ/θc
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.

Giuseppe Dalba, La Fisica dei Raggi X, Dipartimento di Fisica, Università di Trento, a.a. 19992000
Diffraction
An xray 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 xray 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 θλ sin2dn =
where n is the order diffraction, d is the interplanar spacing and θ is the angle between the reflecting planes and the incident beam.