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 phys4.11 Page 1 X-Ray transitions to low lying empty states - N series - M series - L series - K series (highest energy) phys4.11 Page 2
K α -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 n f = 1 and n i = 2 for the K α line - x-ray photon energy - using this formula Moseley has determined atomic numbers Z of most of the elements known at the time (1913) phys4.11 Page 3 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) phys4.11 Page 4
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 phys4.11 Page 5 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 H 2 - 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 phys4.11 Page 6
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) phys4.11 Page 7 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 phys4.11 Page 8
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 ~ 10 15 Hz for R ~ 0.1 nm or Γ ~ 1 Hz for R ~ 1 nm phys4.11 Page 9 Why is a bond formed in the H 2 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 H 2 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) phys4.11 Page 10
- w.f. close to the nuclei are similar to Hydrogen w.f. - in s-state the extent of the w.f. is a 0 - ψ a is w.f. of nucleus a, ψ b 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? phys4.11 Page 11 Anti Symmetric H 2 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 H 2 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 phys4.11 Page 12
- for R = 0 and a symmetric w.f. the electron energy would be that of a He ion, i.e. Z 2 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 E b = 2.65 ev at a nuclear separation of 1.06 10-10 m phys4.11 Page 13 - 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 H 2 case (5.3 ev < 2 x 2.65 ev) due to the mutual repulsion between the electrons - as in the H 2 case the symmetric spatial wave functions lead to bonding states whereas the anti symmetric ones lead to anti-bonding states phys4.11 Page 14