Lecture 2 Cross section Conservation laws Detector principles Spectrum –Signal to noise...
Transcript of Lecture 2 Cross section Conservation laws Detector principles Spectrum –Signal to noise...
Lecture 2
• Cross section
• Conservation laws
• Detector principles
• Spectrum– Signal to noise– resolution
The concept of a cross section originates in a geometrical consideration, which here is illustrated using an example from Nuclear physics:
Assume that two 12C nuclei collide, and consider the process in a coordinate frame in which one of the nuclei is at rest. Then the geometric cross section corresponds to the area around the nucleus at rest which, if hit by the other nucleus, defines that a collision takes place.
The distance between the two nuclear centres is called the impact parameter b. In the drawingthe impact parameter is 2 times the radius of 12C,and at this impact parameter a peripheral collisionwould take place. A fully head-on collision has b=0.
Cross section
at rest
Cross section - definition
YP
The standard unit for cross section is barnbarn (b), (b), (1 mb), 1 barn = 1 b = 10-28 m2 = 10-24 cm2
i.e. the dimension of an area
Assume that one particle/quantum a passes through a material which contains one particle A per surface area Y. Let P be the probability that this causes a particular reaction.Then the cross section for that particular reaction is defined via:
Cross section
Y
d
n
A thin foil (thickness d, area Y) of particle A is irradiated by n particles/time.
What is the reaction rate if the cross section is ?
Cross section
R nP nN d
P Y
NYd Nd
Probability for reaction with one a-particle
and total no of reactions/unit time
Y
d
n
Let N be number of A-particles/volumen the number of incident a-particles
volume = Yd,
Total no of A-particles = NYd
Polar coordinates, r,Θ,φ
0 °Θ, scattering angle
Φ,azimuthal angle
Detector frontarea definesΔΩ
Np
NT
ND
NP; number of projectile particles per second
NT ; number of target particles per cm2
ND ; number of detected particles per second
The result, ND, depends on NP,NT and ΔΩ. Not good for reproducibility
Cross section (σ) normalizes away experiment specific parameters so you get the absolute probability for a given result
σ = ND/ (NP ·NT) which has dimension area i.e unit cm2 or the more useful unit for nuclear dimensions, barn (1barn=10-24cm2)
Differential cross section
σ still depends on the solid angle of the detector. This is normalized away in the differential cross section:
dσ/dΩ = ND/ΔΩ·NP·NT (unit: barn/steradian) or if differentiated both in angle and energy the doubly differential cross section
d2σ/dΩdE = ND/ΔΩ·ΔE·NP·NT (unit: barn/steradian/eV).
A result is often an energy distribution measured in a given angle. It should normally be expressed by this doubly differential cross section
Differential cross section
Differential cross section
NP can be obtained from the beam current (if picoamperes or larger).
If too low (<106 particles per sec), the particles can be counted directly with a detector in the beam. A monitor reaction with known cross section can be used to determine the product NP·NT. Elastic scattering is often used as monitor.
NT can be determined by measuring the thickness of the sample and using the density to calculate the number of nuclei per cm2. More convenient is to measure the area of the target foil and measure its weight. The unit gram/cm2 is a commonly used unit for thickness.
Different types of partial cross sections
• Elastic scattering:Kinetic energy conserved
cross section: s,el, example: d + 39K –> 39K + d
• Inelastic scattering:Kinetic energy not conserved(excitation energy)
cross section: s,inel, example: d + 39K –> 39K* + d
• Absorption reaction:
cross section: a, example: d + 39K –> B + b d ≠ b
• Reaction cross section:
r = a + s,inel
Total photon cross sections
Material: carbon coh: total photon cross section
: atomic photo-effect
coh: coherent scattering (Rayleigh)
incoh: incoherent scattering (Compton)
n: pair production, nuclear field
e: pair production, electron field
ph: photonuclear absorption
From: Thompson and Vaughan (Eds.), X-ray Data Booklet, 2nd edition, Lawrence Berkely National Laboratory 2001Available from http://xdb.lbl.gov
From: Sakurai, AdvanvedQuantum Mechanics, Addison-Wesley, Reading 1967
Rayleighscattering
Elastic scatteringof photons by atoms
From: Moroi, Phys. Rev. 123, 167 (1961)
Photoelectric effect
Ejection of an atomic electronby the absorption of a photon
From: Bjorken and Drell,Relativstic Quantum Mechanics, Mc Graw-Hill, New York 1964
Comptonscattering
Scattering of photonsby free (or quasi-free)
electrons
Photon processes
Pair production(in nuclear field)
From: Bjorken and Drell,Relativstic Quantum Mechanics, Mc Graw-Hill, New York 1964
Production of electron/positronpair on the field of a nucleus
or an electron
Pair production(in electron field)
Photonuclearabsorption
Absorption of a photonby a nucleus
Photon processes
Attenuation
totaltotal = = reactionreaction + + elasticelastic
A beam of particles that passes through a thick target is attenuated (intensity is degraded).
The strength of attenuation depends on all processes possible for the beam, i,e, the sum of all different cross sections.
Attenuation
dn
dx ntotalN
n(x) n0 e totalNx
n ntotalNx
With the solution:
The change n of the number of particles in the beam in a segment x will be:
Let x -> dx :
Attenuation length
• N has dimension 1/length
• Nx = x/is attenuation length
x/
0 1 2 3 4 5 6 7 8 9 10
Inte
nsity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
10-3
Photoemission Principle
IN h(mono-energetic)
OUT e-
Sample
Schematic experiment
For the outgoing electrons we measure the number of electrons versus their kinetic energy.
In addition the direction of the electrons may be detected (and in some cases their spin).
NOTE the direction of the Binding Energy (BE) scale
From Energy Conservation (Esample is the total energy of the sample
before and after the electron is emitted)
h + Esample(before) = Esample(after) + Ekin(e-)
i.e. a Binding Energy EB (or if you like, BE) can be defined
EB = h Ekin(e-) = Esample(after) - Esample(before)
To beam line
Why do we see a clear signal from the surface layer in photoemission ?
Photon Energy
Surface signal
(The first atomic layer)
Bulk signal
Attenuation length of soft X-ray photons in solids is of the order of 1000 Å.
Is it reasonable that we see a clear signal from the surface atoms when the attenuation length of the exciting radiation is much larger than the distances between layers?
Conservation laws:
• Energy
• Momentum
• Angular momentum
• Charge
• Other quantum numbers
Coulomb scattering• Coulomb interaction - electromagnetic force
between projectile and target. • Normally the interaction is elastic, but both
Coulomb excitation and disintegration can happen.
In elastic Coulomb scatteringthe particle trajectories are bent in the Coulomb field.
Elastic Coulomb scattering• Rutherfords formula
– From classical conservation laws Rutherfords famous formula
• dependence
• dependence
• dependence1
sin4 2
dd
zZe2
4 0
2
1
4Ta
2
1
sin4 2
1T
a
2
Z2
Rutherfords formula
Z2
1T
a
2
1
sin4 2
Shadowing cone
The ideal detector•Sensitivity for radiation
•Cross section•Size (mass)•Transparency
•Response•Energy-signal•Linearity
•Response function for radiation
•Time •Pile-up, dead time
•Resolution•Fwhm
Resolution of spectral features
From: Gerthsen & Vogel, Physik, 17th edition, Springer-Verlag, Berlin and Heidelberg 1993
Separation <
FWHM
Modelling & curve fitting can ”increase”the resolving power to a certain extent:
Improving the resolution is better!(but not always possible!)
From: Beutler et al., Surf. Sci. 396 (1998) 117Smedh et al., Surf. Sci. 491 (2001) 99
Physical limits to the resolving power of an instrument
From: Gerthsen & Vogel, Physik, 17th edition, Springer-Verlag, Berlin and Heidelberg 1993
≈ sin = 1.22 /D
Rayleigh criterion
Lens Optical microscope
dmin = 1.22 / (2n sin ) ≈ 200 nm
for optical microscopy
: wavelength of light (min. 450 nm)n: refractive index of light (often 1.56): collecting angle
Minimum distance that can beresolved:
From: J. Stöhr, NEXAFS Spectroscopy, Springer-Verlag, Berlin and Heidelberg 1992
Signal-to-noise and Signal-to-background ratios
SB = Is / Ib
SN = Is / In =IS
( IS + Ib )1/2=
( )IS1 + 1/SB
1/2
From statistics: for large N the noise scales like N .
What to optimise - SB or SN?
Also depends on what you can optimise!
Noise: statistical phenomenon Background: physical phenomenon!
When all external (systematic) noise has been removed the only way left is to increase thenumber of counts!
Counting time Choice of method
Choice of sample
Choice of method
Choice of geometry
Low count rateGood SB
High count rateBad SB
Intermediate count rateBad SB
Method of choice!
Example: X-ray absorption measured usingdifferent detection methods
From: J. Stöhr, NEXAFS Spectroscopy, Springer-Verlag, Berlin and Heidelberg 1992
Unproblematic backgrounds ...
... and problematic backgrounds:background = background(x)!