EEB, DII, S6, 2010

6) Atom spectroscopy: energy levels (see lab experiment no 3 & lecture 11.02.10)

When analysing data it is often necessary to fit a functional expression to data points to derive some characteristic parameter prior to interpretations. This can be performed by using various computer programs. Instructions about how to do such analysis for specific examples by using IGOR are to be found on the internet (see HERE and HERE).

The problem below is related to your laborotory work and analysis of the H atom spectrum. By defining the ground state energy of atoms equal to zero, energy levels for the H atom (EH(n)) depend on the principal quantum number n as

EH(n) = IPH – R/(n)2 (1)

where IPH is the ionization potential for H and R is the Rydberg constant. For atoms (A) larger than the H atom, the energy levels (EA(n))

can be expressed as

EA(n) = IPA – R /(n +(l))2 (2)

Where (l) is an empirical parameter depending on the angular momentum quantum number (l) and IPA is the ionization potential for A

and R is the Rydberg constant.

Below you will find energy levels for s and d orbitals of the sodium atom (Na) derived from spectra analysis of the Na atom. a-b) Fit the analytical expression (2) to the observed energy levels to determine (l) for the s(l=0) and d(l=2) states. NB / hint: Alternatively you could rearrange expression (2) conveniantly and perfom a line fit. c) Explain the difference in the (l) (l = 0,2) values (see lecture notes from 11.02.10).

n E(cm-1) / s orbitals E(cm-1) / d orbitals

3 29 172.889

4 25 739.991 34 548.766

5 33 200.675 37 036.774

6 36 372.620 38 387.270

7 38 012.044 39 200.93

8 38 968.51 39 728.70

9 39 574.85

IPNa = 41449.44 cm-1.

See fit procedure by IGOR at

http://notendur.hi.is/agust/kennsla/ee10/eeb/Daemi/EEB-DIIs6-10.pxp

n E / s4 257405 33200.76 36372.67 380128 38968.59 39574.8

38

36

34

32

30

28

26

x103

987654

Coefficient values ± one standard deviation K1 = -1.3562 ± 0.000842 = (l=0)

a)

38

36

34

32

30

x103

876543

n E / d3 29172.94 34548.85 37036.86 38387.37 39200.98 39728.7

Coefficient values ± one standard deviation K1 = -0.010683 ± 0.000429 (l=2)

b)

(l=2) = -0.010683 << (l=0) = -1.3562: shielding effects:

See next slides

c)

R2

r

Na: 1s22s22p6 3s

Na+: 1s22s22p6

Skermun/ shielding:

3s

Na*: 1s22s22p6 3p

3p

Na**: 1s22s22p6 3d

3d

11