phys4.10 Page 1 When filling electrons in any subshell spins first remain unpaired in every quantum state and have parallel spin.

Transcription

1 Filling of Electronic States - usual filling sequence: 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2 4d 10 5p 6 6s 2 4f 14 5d 10 6p 6 7s 2 6d 10 5f 14 - consider potassium (K): the 4s shell is filled before the 3d shell because of the stronger binding of the 4s electrons - the transition elements have electrons in the 3d or 4d subshells respectively - the lanthanides (rare earths) have identical 5s 2 5p 6 6s 2 configurations and differ by the number of electrons in the in the 4f shell - the actinides have identical 6s 2 6p 6 7s 2 configurations and differ only in the 5f and 6d electrons - for both lanthanides and actinides the chemical properties are dominated by the outer s and p electrons and thus result in very similar properties for all of them phys4.10 Page 1 Hund's Rule When filling electrons in any subshell spins first remain unpaired in every quantum state and have parallel spin. examples: Z configuration spin of p electrons Boron 5 1s 2 2s 2 2p 1 Carbon 6 1s 2 2s 2 2p 2 Nitrogen 7 1s 2 2s 2 2p 3 Oxygen 8 1s 2 2s 2 2p 4 Fluorine 9 1s 2 2s 2 2p 5 Neon 10 1s 2 2s 2 2p 6 note: - The Coulomb repulsion between electrons gives rise to Hund's rule. Electrons with the same spin in the same subshell must have different magnetic quantum numbers m l which have different spatial distributions minimizing the overlap and thus the Coulomb energy. - ferromagnetism of the 3d metals iron (Fe 26 ), cobalt (Co 27 ) and nickel (Ni 28 ) is a consequence of Hund's rule Fe 26 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 6 phys4.10 Page 2

2 phys4.10 Page 3 Spin Orbit Coupling The interaction between the spin magnetic moment of an electron with its orbital angular momentum is called spin-orbit coupling. - this effect gives rise to fine structure doubling of spectral lines classical model: - consider electron moving around nucleus with charge +Ze - in the reference frame of the electron the positively charged nucleus generates a magnetic field B - the spin magnetic moment μ B of the electron interacts with that field giving rise to a potential energy of the electron spin - every electron state with orbital angular momentum will be split into two substates phys4.10 Page 4

3 Spin orbit coupling in hydrogen - nuclear field - mean radius - orbit frequency - for an electron in a 2p state phys4.10 Page 5 Total Angular Momentum Each electron in an atom contributes with its orbital angular momentum L and its spin angular momentum S to the total angular momentum J of the atom. - consider the simple case of atoms with a single electron in the outer shell, i.e. all group I elements (H, Li, Na, K,... ) and ions such as He +, Be +, Mg +, B 2+, Al 2+ etc. - in these atoms the total angular momentum is given by - with the magnitude - and z-component - the simultaneous quantization of L, S and J allows only for certain relative orientations to be allowed - for one electron atoms there are only two relative positions possible phys4.10 Page 6

4 Relative orientation of L and S in one electron atoms - two possibilities: quantization of total angular momentum J for the two possibilities of coupling between L and S phys4.10 Page 7 Precession of angular moments without external field: - the magnitude and direction of J is conserved - L and S precess around J with external magnetic field B: - J precesses around B - J z is quantized - L and S precess around J - this gives rise to the anomalous Zeeman effect phys4.10 Page 8

5 LS coupling - in many electron atoms the total angular momentum J is the vector sum of the individual orbital angular moments L i and spin angular moments S i in most of the many electron atoms - the individual orbital angular moments L i are observed to couple to a resultant orbital angular moment L - the individual spin angular moments S i couple to the resultant spin angular momentum S - the moments then interact via the spin orbit coupling to form the total angular momentum J the angular momentum magnitudes L, S, J and their z-components L z, S z, J z are quantized in the usual way with their respective quantum numbers phys4.10 Page 9 quantization rules: note that the resultant angular moment quantum numbers are indicated by capital letters, whereas the individual electron quantum numbers are not capitalized note: - L and M L quantum numbers are always integer or 0 - all other quantum numbers are half integer for an odd number of electrons - and integer or 0 for an even number of electrons - for L > S the total angular momentum J can have 2S + 1 different values - for L < S the total angular momentum J can have 2L + 1 different values phys4.10 Page 10

6 Hyperfine structure - atomic nuclei also have intrinsic angular moments and magnetic moments - these also contribute to the total angular moments - the angular moments are small and thus make only small (but observable and also useful) contributions to the total energy of electrons - the three sources of angular momentum in an atom phys4.10 Page 11 Term Symbols - as for single electron states symbols are used to indicate the total orbital angular momentum of a many electron atom - in this case capitalized letters are used - a superscript number before the letter indicates the multiplicity of the state, i.e. the number of possibilities to combine the total L and total S to a total angular momentum J - for L > S the total momentum ranges from L - S to L + S, i.e. the multiplicity is 2 S in the situation S = 0 the multiplicity is 1, hence there is only one state with J = L that is called a singlet - for S = 1/2 there is a doublet state (L+ 1/2, L - 1/2) - for S = 1 there is a triplet state (L+ 1, L, L - 1) phys4.10 Page 12

7 - for S > L the multiplicity is given by 2 L the total angular momentum of the state realized is then indicated by a subscript after the letter - example: 'doublet P three halves' - in atoms with a single outer shell electron the principal quantum number is used as a prefix - e.g. in sodium (Na) phys4.10 Page 13 Find the different possible values of J for two electrons in with orbital quantum numbers l 1 = 1 and l 2 = 2 and spin quantum numbers s 1,2 = ± 1/2. - three different possibilities to form L = l 1 - l 2 l 1 + l 2 = 1, 2, 3 - two possibilities to form S = s 1 - s 2 s 1 + s 2 = 0, 1 - orbital and spin angular moments - five possibilities to form J = L - S L + S = 0, 1, 2, 3, 4 phys4.10 Page 14

8 Hydrogen (H, Z = 1) Spectrum: - states labeled by n and l - selection rule: Δl = ± 1 detailed structure of n = 2 and n = 3 states: - splitting of H α line into 7 sub lines - states with same n and different j have different energies - also states with same n and j but different l have different energies Lamb shift (discovered 1947) between 2 2 S 1/2 and 2 2 P 1/2 phys4.10 Page 15 Sodium (Na, Z = 11) Spectrum: - electron configuration 1s 2 2s 2 2p 6 3s 1 - single outer s electron in potential of nucleus screened by closed inner shells - spectrum similar to that of Hydrogen for large l states (3d, 4f, 5f, 6f) - lower angular momentum states have higher probability densities close to the nucleus, resulting in stronger binding and thus lower energies single electron atom radial electron probability densities P(r) dr in dependence on angular quantum number l phys4.10 Page 16

9 Helium (He, Z = 2 ) spectrum: selection rules: - 2 electrons in the 1s ground state - LS coupling is dominant - in single electron transitions ΔL = 0 is forbidden and Δl = ± 1 = ΔL is required - for initial states J = 0, J has to change by ±1 - no radiative spin transitions between para and ortho Helium (ΔS = 0), only transitions by collisions - no n = 1 triplet due to exclusion principle - large binding energies - largest ionization energy of any element phys4.10 Page 17 Mercury (Hg, Z = 80) spectrum: - 2 outer shell electrons, 78 closed subshell inner electrons - spectrum shown for one of the two electrons in lowest energy state - singlet (S=0) and triplet (S=1) states of outer shell electrons are observed - violation of ΔS = 0 selection rule, e.g. in 3 P 1 -> 1 S 0 transitions - violation of ΔS = 0 selection rule and ΔJ = 0 rules in 3 P 2 -> 1 S 0 and 3 P 0 -> 1 S 0 transitions - break-down of LS coupling is observable - in high Z atoms JJ coupling dominates phys4.10 Page 18

10 Experiment Mercury (Hg) Spectrum: * yellow (579.1 nm) visible line 6 1 D 2 -> 6 1 P 1 ** green (546.1 nm) visible line 7 3 S 1 -> 6 3 P 0 *** blue (435.8 nm) visible line 7 3 S 1 -> 6 3 P 1 **** UV line(253.7 nm) visible line 6 3 P 1 -> 6 1 S 0 phys4.10 Page 19