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)!