4.1 Simple Collision Parameters (1)

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4.1 Simple Collision Parameters (1)

• There are many different types of collisions
  taking place in a gas. They can be grouped into
  two classes, elastic and inelastic.
• Elastic Collisions
• the particles conserve their masses, and the
  kinetic energy and momentum is conserved.
• Inelastic Collisions
• kinetic energy can be transformed into rotational
  or vibrational energy, or excitation and
  ionization.

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4.1 Simple Collision Parameters (2)

• Collision Time and Frequency
• Assume a molecule with radius r0 moves with
  velocity v through a cloud of electrons (F. 4.1).
  In the time ?t it sweeps out a cylindrical volume
  V sv?t that was previously filled with nV
  electrons. Here s p r02 is the collisional
  cross section of the molecule. If n is the number
  of electrons per unit volume, i.e., the electron
  number density there will be nV nsv?t
  collisions in the time ?t. The mean time t per
  collision is then
• t ?t/ (nsv?t) 1/nsv.
• The inverse is called the collision frequency ?c
  (Greek symbol, not velocity v)
• c ? 1/t nsv.
• Mean Free Path Length lmfp ? vt 1/sn
• Generally, the electrons have thermal (random)
  velocities and the relative velocities must be
  considered.

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4.2 Binary Elastic Collisions (1)

• The collision process between particles of
  species s and t is controlled by their relative
  velocities and the inter-particle force. We want
  to find the differential cross section sst(gst,q)
  required to calculate the Boltzmann collision
  integral (3.9). Here is the
  magnitude of the relative velocity, and q is the
  scattering angle. If the two colliding particles
  have comparable masses ms and mt, it is
  advantageous to perform the calculations in the
  center-of-mass system defined in equations (4.6)
  to (4.13). Using the laws of conservation of
  momentum and energy, it is easy to show that gst
  gst (gst before, gstafter collision) . The
  direction of the relative velocity vector g
  changes changes at the collision, see Fig.4.3.
• We illustrate the collision process for the
  simple case of a Coulomb collision between an ion
  and and electron. Since the ion mass is so much
  larger than the electron mass it barely changes
  its velocity in response to the collision, i.e.,
  the center-of-mass (CM) system is essentially
  anchored in the ion.

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4.2 Binary Elastic Collisions (2)Coulomb
Collision

• The geometry of the electron-ion collision is
  shown in Fig. 4.4. The ion is at rest in the ion
  frame of reference. Far away (before the
  collision) the electron has the momentum mev0. A
  line through the center of the ion parallel to v0
  has the distance b0 from the electron when the
  electron is still far away. This distance is
  called the impact parameter. The Coulomb force is
  a so-called central force, i.e., it acts along
  the line connecting the two charges.
• The form of the Coulomb law suggests the use of
  polar coordinates r,f in the plane through the
  two particles (Fig. 4.4).

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4.2 Binary Elastic Collisions (3) Coulomb
Collision

• (kinetic potential energy)after (kinetic
  potential energy)before

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4.2 Binary Elastic Collisions (4) Coulomb
Collision
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4.2 Binary Elastic Collisions (5) Coulomb
Collision
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4.2 Binary Elastic Collisions (6) Coulomb
Collision
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