Interaction Between Ionizing Radiation And Matter, Part 2 ... Interaction Between Ionizing Radiation

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  • Interaction Between Ionizing Radiation And Matter, Part 2

    Charged-Particles

    Audun Sanderud

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  • • Incoming charged particle interact with atom/molecule:

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    o Excitation / ionization

    Ionization

    Excitation

    • Ion pair created from ionization

  • • Interaction between two particles with conservation of kinetic energy ( and momentum):

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    o Elastic collision

    m1, v m2 m1, v1

    m2, v2 χ θ

    • Classic mechanics give: 2 2 2

    0 1 1 1 2 2

    1 1 1 2 2

    1 1 2 2

    1 1 1T m v m v m v 2 2 2

    m v m v cos m v cos 0 m v sin m v sin

    θ χ θ χ

    = = +

    = + = +

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    o Elastic collision(2)

    ( )

    2 1 1 2

    2 1 2 1 2 1 2

    1

    2

    2m v cos 4m m cosv , v v 1 m m m m

    sin 2tan m cos 2 m

    χ χ

    χθ χ

    ⇒ = = − + +

    = −

    • These equations gives the maximum transferred energy:

    ( ) 2 1 2

    max 2 2,max 02 1 2

    m m1E m v 4 T 2 m m

    = = +

  • • Proton(#1)-electron(#2): θmax=0.03

    o, Emax=0.2 % T0

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    o Elastic collisions(3)

    a) m1>>m2 a) m1=m2 a) m1

  • • Rutherford proved that the cross section of elastic scattering is:

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    o Elastic collisions-cross section

    ( )4 d 1 d sin 2

    σ θ

    ∝ Ω

    → Small scattering angels most probable • Differentiated by the energy

    2

    d 1 dE E σ ∝

    → Small energy transferred most probable

  • • Stopping power, (dT/dx): the expectation value of the rate of energy loss per unit of pathlength. Dependent on: -type of charged particle

    -its kinetic energy -the atomic number of the medium

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    o Stopping power

    T0 T0-dT

    dx

    nv targets per volume unit max max

    min min

    max

    min

    E E A

    V v E E

    E A

    E

    N Zd ddT En dx n dx EdT dx EdT dT A dT

    N ZS dT d EdT dx A dT

    σ σσ ρ

    σ ρ ρ

    ⎛ ⎞⎟⎜= = = ⎟⎜ ⎟⎜⎝ ⎠

    ⎛ ⎞ ⎛ ⎞⎟⎜ ⎟⎜= ⎟= ⎟⎜ ⎜⎟ ⎟⎜⎜ ⎟⎜ ⎝ ⎠⎝ ⎠

    ∫ ∫

  • • The charged particle collision is a Coulomb-force interaction

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    o Impact parameter

    • The impact parameter b useful versus the classic atomic radius a

    • Most important: the interaction with electrons

  • • b>>a: particle passes an atom in a large distance

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    o Soft collisions

    • The result is excitations (dominant) and ionization; amount energy transferred range from Emin to a certain energy H

    • Small energy transitions to the atom

    • Hans Bethe did quantum mechanical calculations on the stopping power of soft collision in the 1930

    • We shall look at the results from particles with much larger mass then the electron

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    o Soft collisions(2)

    r0: classic electron radius = e2/4πε0mec2 I: mean excitation potential β: v/c z: charge of the incoming particle ρ: Density of the medium

    NAZ/A: Number of electrons per gram in medium H: Maximum transferred energy at soft

    collision

    ( ) 2 2 2 2 2

    c,soft 2soft 0 e eA 2 2 2

    c

    S dT 2 r m c z 2m c HN Z ln dx A I 1

    π β β ρ ρ β β

    ⎡ ⎤⎛ ⎞⎛ ⎞ ⎟⎜⎢ ⎥⎟ ⎟⎜ ⎜= ⎟ = −⎟⎜ ⎢ ⎥⎜⎟ ⎟⎜ ⎟⎜ ⎜ ⎟⎝ ⎠ − ⎟⎢ ⎥⎜⎝ ⎠⎣ ⎦

  • • The quantum mechanic effects are specially seen in the excitation potential I

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    o Soft collisions(3)

    • High Z – small transferred energy less likely

    Atomic number, Z

    M ea

    n ex

    ci ta

    tio n

    po te

    nt ia

    l, I/Z

    [e V

    ]

  • • b

  • • The total collision stopping power is then (soft + hard):

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    o Collisions stopping power

    • Important: increase with z2, decrease with v2, not dependent on particle mass

    ( ) 2 2 2 2 2

    c,soft c,hard 2c 0 e eA 2 2

    S SS 4 r m c z 2m cN Z ln A 1 I

    π β β ρ ρ ρ β β

    ⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥⎟⎜= + = −⎟⎢ ⎥⎜ ⎟⎜ ⎟− ⎟⎢ ⎥⎜⎝ ⎠⎣ ⎦

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    o Sc/ρ in different media

    • I and electron density (ZNA/A) gives the variation

  • • Electron-electron scattering more complicated; interaction between identical particles

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    o SC for electrons/positrons

    • Sc,hard/ρ: electron-elektron; Møller cross section positron-electron; Bhabha cross section

    The characteristics similar to that of heavy particles

    • Sc,soft/ρ: Bethe’s soft coll. formula

    ( ) ( )

    ( ) 22 2 2

    2c 0 eA e22 2

    e

    2S 2 r m c zN Z Cln F 2 , T / m c A Z2 I / m c

    τ τπ τ δ τ ρ β

    ±

    ⎡ ⎤⎛ ⎞⎟⎜ +⎢ ⎥⎟⎜ ⎟= + − − ≡⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

    ( ) ( ) ( )

    2 2

    2

    / 8 2 1 ln2 F 1

    1 − τ − τ+τ = −β +

    τ+

    ( ) ( ) ( )

    2

    2 3 14 10 4F 2ln 2 23

    12 2 2 2 +

    ⎧ ⎫⎪ ⎪β ⎪ ⎪⎪ ⎪τ = − + + +⎨ ⎬⎪ ⎪τ+ τ+ τ+⎪ ⎪⎪ ⎪⎩ ⎭

  • • The approximation used in the calculations of SC assume v>>vatomic electron

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    o Shell correction

    • C/Z depend on particle velocity and medium

    • When v~vatomic electron no ionizations will occur

    • Shell correction C/Z handles this, and reduce SC/ρ

    • Occur first in the K-shell - highest atomic electron speed

  • • Charged particles polarizes the medium

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    o Density-effect correction

    • Weaker interaction with distant atoms because of the reduction of the Coulomb force field

    Charged (+z) particle

    eff eff pol

    eff pol

    E E E

    E E

    = +

    <

    • Polarization increase with (relativistic) speed

    • Most important for electrons / positrons • But: polarization not important at low ρ

  • • Density-effect correction δ reduces Sc/ρ in solid and liquid elements

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    o Density-effect correction(2)

    • Sc/ρ (water vapor) > Sc/ρ (water)

    Dashed curves: Sc without δ

  • • When charged particles are accelerated by the Coulomb force from atomic electrons or nucleus photons can be emitted; Bremsstrahlung D

    ep ar

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    Radiative stopping power

    • The Lamor equation (classic el.mag.) denote the radiation power from an acceleration, a, of a charged particle:

    ε0: Permittivity of a vacuum

    Charged particle atomic

    electron

    2 2

    3 0

    (ze) aP 6 cπε

    =

  • • The case of a particle accelerated in nucleus field:

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    o Radiative stopping power(2)

    • Comparison of proton and electron as incoming:

    22 2 2 2 2

    2 2 0 0

    zZe zZe ZzF ma a P a z 4 r 4 mr m

    ⎛ ⎞⎟⎜ ⎟= = ⇒ = ⇒ ∝ ∝⎜ ⎟⎜ ⎟⎜πε πε