X-Ray Spectra:
- continuous part of the spectrum is due to decelerated electrons
- the maximum frequency (minimum wavelength) of the photons generated is determined by
the maximum kinetic energy of the incident electrons through the inverse photo effect
- the discrete lines in the spectrum are due to transitions between inner shells of the atom
- ionization from outer shells takes only a few eV of energy
- the incident electron may ionize an electron from an inner shell of the atom
- the inner shell electrons in high Z atoms have larger binding energies due to the larger
unscreened charge of the nucleus
- fox example in sodium (Na) to ionize a 1s electron takes 1041 eV, the 2s electron takes 63 eV
and the 3s electron 5 eV
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X-Ray transitions to low lying empty states
- N series
- M series
- L series
- K series (highest energy)
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Kα-line energies
- consider refilling of K shell electron hole with a L shell electron
- charge of nucleus is reduced (screened) to Z-1 by one remaining K shell electron
- the L shell electron is bound by that charge, thus the transition frequency is given by
the usual formula with nf = 1 and ni = 2 for the Kα line
- x-ray photon energy
- using this formula Moseley has determined atomic numbers Z of most of the elements
known at the time (1913)
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Auger Effect
- discovered by French scientist
Pierre Auger
- emission of outer shell electron upon
drop of another outer shell electron to
an inner shell vacancy
- excess energy carried by emitted
electron
- effect in competition with X-Ray
generation
- Auger electrons generated within a material may be reabsorbed
- observable Auger electrons are predominantly generated at surfaces of the emitter
- reveals information of electronic states on surfaces (important in the semiconductor
industry)
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Molecules:
- only a small number of elements naturally occur as individual atoms
- most elements occur in bound forms, as individual molecules, in liquids and solids
- how are individual atoms bound to molecules, liquids or solids?
- quantum theory of atoms provides the relevant answers in the context of physics and
chemistry
Molecular Bonds:
- molecules behave as individual particles
- molecules appear in specific composition
- when a stable molecule is formed it lowers the total energy of the system
- if that is not the case the molecule is not formed
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Covalent Bonding:
- attractive force between atoms mediated
by electrons that have a high probability
density in between the two atoms
- the attractive force overcomes the repulsive
force between the nuclei in a certain range
of distances between nuclei
- covalent bonds are formed by electrons
shared between atoms
- example: the hydrogen molecule H2
- total binding energy 4.5 eV is large
- compare with ionization energy 13.6 eV of single hydrogen atom
- typical covalent molecular bonding energies are smaller than ionization energies
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Ionic Bonds:
- ions are formed when one or more electrons are
transferred from one atom to another
- the ions then attract each other forming a ionic bond
- example: NaCl, sodium chloride, salt
- forms a solid in a regular lattice at ambient conditions
- only at high temperatures in the gaseous phase ionic molecules
may be formed
- ionic bonds will be discussed further in the context of solids
later in the lecture
- mixed covalent/ionic bonds can be formed when the electrons are shared unequally
between atoms, as e.g. in HCl (hydrochloric acid)
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Bonding does not occur when:
- the Pauli exclusion principle forbids sharing electrons in the same quantum state
- electrons are forced into excited states of the atom (energy needs to be supplied) to allow for
the bond to be formed
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Sharing of Electrons in a Molecule:
- consider the simplest example of a molecule with only a
single electron: the hydrogen molecular ion
- electron moves in the potential well formed by the two
hydrogen nuclei (just single protons)
- classically the binding energy localizes the electron with
one of the nuclei (extent of w.f. ~ 0.05 nm)
- quantum mechanically the electron can tunnel from one
potential well to the other
- compare with particle in a box , now there is two nearby
boxes with a non-infinite potential barrier between them
- the height of the tunnel barrier depends on the inter
nuclear separation R
- tunneling rate Γ ~ 1015 Hz for R ~ 0.1 nm or Γ ~ 1 Hz for R ~ 1 nm
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Why is a bond formed in the H2+ molecular ion?
- kinetic energy of electron can be lowered if it moves in potential of two hydrogen nuclei
(consider uncertainty principle)
- lowering of kinetic energy must be larger than repulsive energy between two nuclei at a
given inter-nuclear distance
- electron provides some screening of positive charges between two nuclei
Find wave function ψ and total energy E of the
single electron in the H2+ ion as a function of
nuclear distance R
- a minimum in E(R) determines the stable
configuration of the molecule
- consider individual wave functions of two
hydrogen atoms at positions a and b
- compose molecular wave functions from atomic
wave functions (good approximation)
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- w.f. close to the nuclei are similar to Hydrogen w.f.
- in s-state the extent of the w.f. is a0
- ψa is w.f. of nucleus a, ψb is w.f. of nucleus b
- symmetric combination of two w.f.
- w.f. in dependence on R: for small R more overlap,
i.e. larger probability density for electron to be
between the two nuclei
- for R = 0 (if one could force the protons to join to a
single nucleus) the electronic w.f. would be that of a
He+ ion, i.e. similar to that of the H atom but with a
nuclear charge of Z= 2e and thus more localized
- is the electron charge accumulation between the nuclei enough to overcome their repulsion
and form a stable molecule?
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Anti Symmetric H2+ wave function
- consider the anti-symmetric combination of w.f.
- results in a node in the electron probability density
between the nuclei, i.e. a repulsive force between the two
nuclei (no bonding)
- in the limit R-> 0, the combined W.f. is similar to the
He+ 2p excited state w.f., which is of higher energy also
indicating that this state will not form a bond
Total Energy of H2+ molecule:
- consider the symmetric state at large R: the total energy
is that of bound electron in a hydrogen atom -13.6 eV
added to the repulsive interaction between the two nuclei
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- for R = 0 and a symmetric w.f. the
electron energy would be that of a He+
ion, i.e. Z2 Ry = - 4 13.6 eV = - 54.4
eV
- thus the asymptotic energies of the
electrons in bound states are known
for R=0 and R=∞ and the Coulomb
repulsion between the nuclei is known
as well
- the total energy is
- it has a minimum for the symmetric
w.f. forming a bound molecular state
- the binding energy is Eb = 2.65 eV at a nuclear separation of 1.06 10-10 m
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- for the anti-symmetric w.f. the electronic energies for R=0 (first excited energy in a He+
atom) and R=∞ (binding energy of a single electron in the ground state of H) are equal
- the energy only varies weakly with nuclear separation not leading to a bound molecular
state
The Hydrogen Molecule:
- has two electrons that can share the same spatial w.f. (orbital) only if they have different
spin states (exclusion principle)
- the binding is expected to be stronger, because two electrons can contribute to the attractive
forces between the nuclei
- the binding energy turns out to be not quite twice as strong as in the H2+ case (5.3 eV
< 2 x 2.65 eV) due to the mutual repulsion between the electrons
- as in the H2+ case the symmetric spatial wave functions lead to bonding states whereas
the anti symmetric ones lead to anti-bonding states
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