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Physics 395 Homework 2: Ionization Potentials and Energy Levels of Atoms, Assignments of Health sciences

The University of Texas at AustinHealth sciences

A physics homework assignment for a university course, specifically for the topic of ionization potentials and energy levels of atoms. The assignment includes problems related to the li atom, the energy ordering of atomic shells in a neutral atom, and the helium atom. Students are required to use units of ev, look up energy levels and quantum numbers from spectroscopic tables, draw energy level diagrams, and solve differential equations using mathematica or other coding tools.

Typology: Assignments

Pre 2010

Uploaded on 08/27/2009

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Physics 395, Spring 2004
Homework #2
Due: Friday, Feb. 20
1. Lithium Atom.
For this problem, use units of eV.
a) What is the ionization potential of the Li++ ion? (Hint: it is hydrogenic).
b) The spectroscopic tables refer to the Li atom as Li I, the Li+ ion as Li II, and the Li++
ion as Li III. Look up energy levels of these species at physics.nist.gov, and identify the
ionization potential (IP) of each of these. The ionization potential is the energy required
to remove the most weakly bound electron, and will be evident as the “series limit” of the
energy levels with singly excited electrons. Also, look up the quantum numbers and the
energies of the first two excited states of each of these species, and the quantum numbers
and energy of the first excited state of each of these species with two excited electrons.
c) Draw an energy level diagram showing the Li ground state energy, and the ionization
thresholds for Li+(ground state) + e−, Li++(ground state) + 2e−, and Li+++ + 3e−. Show the
first two excited levels of Li, and the first two doubly excited levels of Li. Also, show the
ionization threshold for Li+(1st excited state) + e−, Li+(2nd excited state) + e−, Li+(1st
doubly excited state) + e−, Li++(1st excited state) + 2e−, and Li++(2nd excited state) + 2e−.
d) The first energy level of Li with two excited electrons can autoionize. Describe the
possible products of this autoionization. (Can the atom decay into Li+(ground state) + e−?
to Li+(excited state) + e−? to Li++ + 2e−? etc.)
e) The ground configuration of Li is 1s2 2s. Since the two 1s electrons cannot be in the
same state, one of the 1s electrons must have spin projection ms = +1/2, and the other
must have ms = −1/2. Suppose that the 2s electron has ms = +1/2. (There will be another
set of quantum states associated with ms = −1/2 for the 2s electron, which we are
neglecting for the moment.) Write the wavefunction for this state as a Slater determinant.
Expand the Slater determinant out to show the wavefunction explicitly in the form
()
[]
∑−=
P
snsnsn
PmmmP
N3313222111 )()()(1
!
1
321 rrrr lll
ϕϕϕψ
where nili = 1s or 2s. (i.e. actually write out all the terms in the sum.) Can you rewrite
this as a simple product (space wavefunction)×(spin state) like we did for helium?
pf3

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Download Physics 395 Homework 2: Ionization Potentials and Energy Levels of Atoms and more Assignments Health sciences in PDF only on Docsity!

Physics 395, Spring 2004

Homework

Due: Friday, Feb. 20

  1. Lithium Atom.

For this problem, use units of eV.

a) What is the ionization potential of the Li++^ ion? (Hint: it is hydrogenic).

b) The spectroscopic tables refer to the Li atom as Li I, the Li+^ ion as Li II, and the Li++ ion as Li III. Look up energy levels of these species at physics.nist.gov, and identify the ionization potential (IP) of each of these. The ionization potential is the energy required to remove the most weakly bound electron, and will be evident as the “series limit” of the energy levels with singly excited electrons. Also, look up the quantum numbers and the energies of the first two excited states of each of these species, and the quantum numbers and energy of the first excited state of each of these species with two excited electrons.

c) Draw an energy level diagram showing the Li ground state energy, and the ionization thresholds for Li+(ground state) + e−, Li++(ground state) + 2e−, and Li+++^ + 3e−. Show the first two excited levels of Li, and the first two doubly excited levels of Li. Also, show the ionization threshold for Li+(1st^ excited state) + e−, Li+(2nd^ excited state) + e−, Li+(1st doubly excited state) + e−, Li++(1st^ excited state) + 2e−, and Li++(2nd^ excited state) + 2e−.

d) The first energy level of Li with two excited electrons can autoionize. Describe the possible products of this autoionization. (Can the atom decay into Li+(ground state) + e−? to Li+(excited state) + e−? to Li++^ + 2e−? etc.)

e) The ground configuration of Li is 1s^2 2s. Since the two 1s electrons cannot be in the same state, one of the 1s electrons must have spin projection ms = +1/2, and the other must have ms = −1/2. Suppose that the 2s electron has ms = +1/2. (There will be another set of quantum states associated with ms = −1/2 for the 2s electron, which we are neglecting for the moment.) Write the wavefunction for this state as a Slater determinant. Expand the Slater determinant out to show the wavefunction explicitly in the form

= ∑ ( −) [ ]

P

n s n s n s

P (^) P m m m N^1

r ψ ϕ l r 1 ϕ l r 2 ϕ l r 3

where nili = 1s or 2s. (i.e. actually write out all the terms in the sum.) Can you rewrite this as a simple product (space wavefunction)×(spin state) like we did for helium?

  1. Energy ordering of the atomic shells.

a) The Thomas-Fermi model for a neutral atom, which treats the electrons in an atom as a degenerate Fermi gas, provides a simple approximate method to determine the charge distribution in an atom, and for the “average” potential V(r) experienced by an electron in an atom. The result of this model is that

r

Ze V r

2 ( )= − (1)

where χ is the solution to the differential equation

3 / 2 1 / 2 2

2 = x^ − dx

d

and x = r/b is a dimensionless radial coordinate, with

1 / 3

0 2 1 / 3

Z

a meZ

b (^)  ≅ 

= ^ π^ h (3)

The charge density in the atom is ρ = −e n(r), with the electron density

( ) [ ] 3 / 2 2 2

3 / 2 ( ) 3

( ) V r

m n r = −

π h

One boundary condition at the origin is χ(0) =1. The other boundary condition is set by the condition that χ(r) → 0 as r → ∞. This occurs for χ′(0) ≅ 1.589.

a) Adapt your Mathematica or other code from HW#1, problem 2, to solve the differential equation for χ and to plot out the resulting electron density distribution and radial potential V(r). Plot out the density distribution, in atomic units, for the case of the neutral Rb atom, which has N = Z = 37.

b) The Hartree model starts with a guess for the radial potential V(r) experienced by an electron. The resulting eigenfunctions and energies for the single electron orbitals will be found by solving a radial Schrödinger equation in the effective potential Veff = V(r) + Vcent, where Vcent is the centrifugal potential, just as you did for hydrogen. Suppose that you can use the Thomas Fermi potential you found in part (a) as your guess for V(r), and plot out the effective radial potential for a Rb atom in atomic units for angular momentum states l = 0, 1, 2, and 3.

c) The convention for the labeling of the orbitals in a many electron atom is the same as for hydrogen. The l = 0 states are labeled in order of increasing energy as 1s, 2s, 3s, …

The l = 1 states are labeled in order of increasing energy as 2p, 3p, …. For the d-states it