To identify the role of the total collision cross-section in a scattering experi- ment consider the situation depicted schematically in Figure 10.2. In such an experiment, a beam of projectiles, labelled A, with an intensity I (Note: the intensity of the beam is the number of particles per unit area per unit time) passes through a cell of length l containing the target particles, B, at a number density nB.
As the beam of projectiles passes through the cell the number of incident particles that are scattered or lost from the beam must be proportional to the number of target particles with which the beam interacts, i.e.,
dI
I ∝ −nBdz. (10.1)
This can also be expressed as dI
I = −σTnBdz, (10.2)
v
Beam of particles A
Target particles B
- Intensity I
Length l
z
x
Figure 10.2: Schematic diagram of a particle scattering experiment in which a beam of projectiles, A, passes through a cell of length l filled with target particles, B. The intensity of the beam is denoted I, while the number
density of target particles in the cell is nB.
Integrating Equation 10.2 over the length of the target cell leads to Z I(l) I0 dI I = −σTnB Z l 0 dz, (10.3) ln(I) I(l) I0 = −σTnBz l 0 (10.4) ln[I(l)] − ln[I0] = −σTnB(l − 0) (10.5) ln" I(l) I0 # = −σTnBl, (10.6) and so I(l) = I0 exp(−σTnBl ). (10.7)
This expression, Equation 10.7, for the projectile intensity after travelling a distance l through a cell filled with target particles is known as the Beer- Lambert Law. By performing measurements with an apparatus of the kind depicted in Figure 10.2 at different relative collision energies the energy de- pendence ofσTcan be determined. Contributions from the collision energy
to the total collision cross-section arise as a result of (1) the corresponding change in the interaction time, i.e., in collisions at higher energy the inter- action times between the projectiles and the target are small while at lower
energy, the interaction times are longer; (2) effects of the wave nature of the particles when colliding with low energies, since the de Broglie wavelength λdB= h/p depends on the particles momenta the increase in this wavelength
at low collision energies must be considered; and (3) resonances associated with the energy level structure of the particles.
As noted above, σT does not represent the physical size of the target
atom or molecule because of the effects of long range interactions: Coulomb interactions, dipole-dipole interactions, or van der Waals interactions.
The Coulomb interaction, VC, occurs between two charged particles and
and scales as
VC ∝ 1
R, (10.8)
where R is the distance between the two particles. As a result if the pro- jectile and target particles are both chargedσT is significantly greater than
the physical size of the particles because of the effects of the long-range Coulomb interactions.
Dipole-dipole interactions, Vdd, occur between pairs of particles with in-
trinsic electric dipole moments, i.e., ionically bound ground state molecules, and scale as
Vdd ∝
1
R3. (10.9)
The dipole-dipole interaction does not extend over as large a range of inter- particle distances as the Coulomb interaction, so while dipole-dipole in- teractions do increaseσT compared to the physical size of the particles, in
general it is not increased as significantly as in the case of the Coulomb interaction.
The van der Waals interaction, VvdW, occurs between pairs of polarizable
particles that do not possess intrinsic electric dipole moments. This inter- action is therefore more strongly dependent on the inter-particle distance and scales as
VvdW ∝
1
R6. (10.10)
Such interactions do also increase the value ofσT when compared to the
than that associated with either Coulomb interactions or dipole-dipole in- teractions.
These interactions are often said to be ‘soft’ in that they soften the phys- ical edges of two colliding hard spheres. In this sense the Coulomb in- teraction between two charged particles is ‘softer’ than the van der Waals interaction between two neutral particles.
10.4
Cross-sections
Collisions between pairs of particles can be categorised as elastic or inelastic. In elastic scattering processes only the momenta of the particles change but not the internal states, while in inelastic scattering processes the internal energy can change together with the particle momenta. For example, in the case of an electron scattering from a particle A,
Elastic scattering: e−+ A −→ e−+ A (10.11) Inelastic scattering: e−+ A −→ e−+ A∗ (10.12) −→ e−+ A++ e− (10.13) −→ A−+ hν (10.14) where A∗
and hν represent the particle A in an excited state, and a photon, respectively.
Because inelastic scattering processes lead to changes in the internal quan- tum states of the target a total inelastic cross-section,σi f, can be associated
with any pair of initial, i, and final, f , states of the target. This represents the effective area associated with a scattering event in which the target atom is transferred from state i to state f . In these processes the momenta of the colliding particles can also change.
I Hg e- V- V+ Grid V+ - δV (a) Accelerating potential (V) Electron current, I (arb. units) (b)
Figure 10.3:(a) Schematic diagram of the experimental apparatus used by Franck and Hertz to study inelastic scattering of electrons from Hg vapour. (b) Observed changes in electron current recorded at the anode in (a) as the
accelerating potential, V+−V−, was increased. The sharp drops in current
at multiples of 4.9 V indicate an inelastic scattering resonance close to this energy.
Inelastic electron scattering was first demonstrated in a classic exper- iment of Franck and Hertz1. In their experiment, depicted schematically in Figure 10.3(a), electrons from a heated cathode operated at a potential V−
were accelerated toward a metal grid through a potential difference V+−V−
before the resulting current, I, was measured on an anode. As the accelerat- ing potential was increased from 0 V the electron current initially increased as expected, however, for an accelerating potential of ∼ 4.9 V an abrupt drop in current was observed. This reduction in electron current indicated that as the electrons passed through the Hg vapor lost kinetic energy. The resonant behaviour of this loss suggested that the energy at which it oc- cured corresponded to the energy difference between two electronic states in Hg, and that in the electron scattering process the Hg atoms were being excited. This was later confirmed by determining the wavelength of the light,λ = 254 nm, emitted by the Hg vapor at this resonant collision energy 1J. Franck and G. Hertz, Über Zusammenstöße zwischen Elektronen und Molekülen
des Quecksilberdampfes und die Ionisierungsspannung desselben [On the collisions be- tween electrons and molecules of mercury vapor and the ionization potential of the same], Verhandlungen der Deutschen Physikalischen Gesellschaft 16 457 (1914).
of E/e = 4.9 eV.
10.5
X-ray spectra
In the experiments of Franck and Hertz electrons were accelerated to ki- netic energies of a few electron volts. As a result the relevant inelastic scattering processes involved the excitation of atomic electrons in the outer, most weakly bound, orbitals of Hg. Transitions involving these outer- shell electrons typically occur at photon energies corresponding to 1–10 eV, or wavelengths in the visible or ultraviolet regions of the electromagnetic spectrum.
However, if accelerated to higher energies, e.g., 1–100 keV, inelastic elec- tron scattering can be exploited to excite tightly-bound inner-shell electrons in atoms. For example, in the Ne atom with the ground state configuration 1s22s22p6, the inner-shell electrons are those in the 1s orbital. Transitions involving these electrons typically occur at wavelengths from 0.1 Å to 10 Å, in the extreme ultraviolet or X-ray regions of the electromagnetic spectrum. It is difficult to generate laser radiation at these wavelengths, however, high energy inelastic electron scattering can be exploited to study transitions in- volving inner-shell electrons in atoms and molecules, and to generate X-ray radiation.
Electron beam
Cathode Electron acceleration Metal target Anode e- e-Figure 10.4:Schematic diagram of apparatus used for high energy electron bombardment of metal targets, and X-ray generation.
X-ray wavelength
Intensity
λ
0e
-e
- X-ray photone
-e
- X-ray photone
-(a)
(b)
(c)
Figure 10.5: (a) The electron acceleration in the Coulomb field of the nucleus that gives rise to the broad background continuum in an X-ray spectrum. (b) The inner-shell excitation process that gives rise the sharp features in an X-ray spectrum. (c) A schematic X-ray spectrum.
gets, experimental arrangements of the kind depicted schematically in Fig- ure 10.4 are typically employed. The excitation of inner-shell electrons by inelastic electron scattering leads to the emission of X-ray photons. The resulting X-ray spectra have two components (1) a broad background con- tinuum, and (2) sharp features characteristic of the target atom or molecule. These components can be seen in Figure 10.5(c).
Continuous spectrum: The broad background continuum in an X-ray spec- trum arises as a result of the acceleration of an incoming electron in the Coulomb field of the positively charged nucleus [see Figure 10.5(a)]. In general if accelerated a charged particle emits radiation. This emis- sion is not quantised and appears as a continuous component of the complete X-ray spectrum [see Figure 10.5(c)]. The maximum X-ray
photon energy corresponds to the incoming electron being deceler- ated to a stand-still as it passes the nucleus of the target atom. When this occurs the wavelength of the emitted photon,λ0, is
λ0 =
hc Ei,
(10.15) where Eiis the initial kinetic energy of the incident electron.
Discrete spectrum: The discrete (sharp) components of an X-ray spectrum result from an incident electron colliding with an inner-shell electron in the atom leaving a hole [see Figure 10.5(b)]. When this occurs the electronic configuration of the atom is no longer the lowest in energy for the remaining electrons. Consequently, the electrons rearrange and an electron from a higher orbital can decay to fill the hole. In this process an X-ray photon is emitted. This photon has an energy equal to the energy difference between the excited state from which the atomic electron decayed, and the state containing the hole. This process results in the sharp features in Figure 10.5(c). The photon energies at which these sharp features appear are dependent on the energy level structure of the target atom.
A first approximation to the energies of the sharp features in X-ray spectra can be obtained by assuming that they can be described by an equation similar to the Rydberg formula. In this case, the energy,∆E, associated with each sharp line corresponds to the difference in energy between an initial state, Ei, and a final state, Ef, such that
∆E hc = Ei−Ef hc (10.16) ' R∞Z2eff 1 n2f − 1 n2i , (10.17)
where Zeff = Znucl−S, with Znucl the nuclear charge, and S a screening
constant.
X-ray transitions therefore form series each of which is associated with the decay to a particular final state. Transitions for which nf = 1 form a