LECTURE 22

More Atom Building

PHYSICS 420SPRING 2006Dennis Papadopoulos

0,1,2, 1n l K0,1,2, 1n l K

1,2,3,n K1,2,3,n K

There are a total of n subshells, each specified by an angular momentum quantum number , and having an angular momentum of

l

1L v

l l h

The allowed energy levels are quantized much like or particle in a box. Since the energy level decreases a the square of n, these levels get closer together as n gets larger.

Fig. 8-6, p.267

Fig. 8-7, p.272

Magnetic moment of a current loop:

iA

currentarea

enclosed by current

loop

Orbiting electrons form a current loop which give rise to a magnetic field.

Since the current is defined as the direction of flow of positive charge, the orientation of the magnetic moment will be antiparallel to the

angular momentume of the electron and can be found using the right hand rule.

The current loop of the orbiting electron sets up a magnetic dipole which behaves like a bar magnet with the north-south axis directed along .

Kepler’s Law of areas: A line that connects a planet to the sun sweeps out

equal areas in equal times.

2

LA

T m

v

22 22

r m AL mvr m r r m

T T T

v

vA

T

qi

2 2

q eiA L L

m m

v vv

e

Remember that the z component of angular momentum is quantized in units of so the magnetic dipole

moment is quantized as well: h

249.274 10 J/T2B

e

m

h

The Bohr magneton

2 2z z Be e

e eL m m

m m l l

h

where

2

22

dm

d

l

B

L

t

v v

Torque exerted:

Magnetic dipole tends to want to align itself with the magnetic field but it can never align due to the

uncertainty principle!

m ll lm ll lThere are a total of orbitals within each subshell, these can be thought of as projections of the angular momentum on the z axis.

2 1l

2

1s 2

1s

3( 1)

2S s s v

h h

The electron has an intrinsic magnetic moment called “spin”. The orientation of the angular momentum vector of this apparent rotation motion can only have a manitude of ½.

Chemical properties of an atom are determined by the least tightly bound electrons. VALENCE ELECTRONS

Factors:

•Occupancy of subshell

•Energy separation between the subshell and the next higher subshell.

Pauli Exclusion Principle• To understand atomic spectroscopic data for optical frequencies,

Pauli proposed an exclusion principle:

No two electrons in an atom may have the same set of quantum numbers (n, ℓ, mℓ, ms).

• It applies to all particles of half-integer spin, which are called fermions, and particles in the nucleus are fermions.

The periodic table can be understood by two rules: 1) The electrons in an atom tend to occupy the lowest energy levels

available to them.

2) Pauli exclusion principle.

s shell l=0

p shell l=1

Beryllium and magnesium not inert even though their outer subshell is closed…why??

Helium and Neon and Argon are inert…their outer subshell is closed.

Alright, what do we add next???

Uh-oh…3d doesn’t come next…why???

Fig. 9-16, p.324

nucleus: charge +3e

closed s “shell”: total charge -2e

orbiting electron “sees” net charge of +e…like hydrogen

Note that for higher orbital angular momentum, the energy more nearly equals to the corresponding levels in hydrogen. The more nuclear charge an electron “sees”,

2s state more tightly bound since it has a nonzero probability density inside the 1s shell and “sees” more of the nuclear charge.

Note the “pulling down” of the energy of the low angular momentum states with respect to hydrogen due to the penetration of the electric shielding.

The energy separating shells becomes smaller with increasing n.

Electrons in lower angular momentum states penetrate shielding more, and thus are more tightly bound. As the energy levels become closer together, some lower angular momentum states of higher n may actually have a lower energy.

Excited state 4s lies lower than 3d!!

Order of filling, then, is not what we naively expect. The degree of charge screening plays

a BIG role.

4s comes BEFORE 3d…

…and apparently 5s comes before 4d, and 6s comes before 5d…

The term with the maximum multiplicity lies lowest in energy.

Also, mutual repulsion forces electrons to higher energy states.

Note the filling of aligned spins

before “doubling up”.

Hund’s rule

Lanthanide series (or “rare earths” due to low abundance)

Actinide series

Stick them at the bottom to keep things from getting too awkward.

The Periodic TableInert Gases:

• Last group of the periodic table

• Closed p subshell except helium

• Zero net spin and large ionization energy

• Their atoms interact weakly with each other

Alkalis:

• Single s electron outside an inner core

• Easily form positive ions with a charge +1e

• Lowest ionization energies

• Electrical conductivity is relatively good

Alkaline Earths:

• Two s electrons in outer subshell

• Largest atomic radii

• High electrical conductivity

The Periodic TableHalogens:• Need one more electron to fill outermost subshell• Form strong ionic bonds with the alkalis• More stable configurations occur as the p subshell is filled

Transition Metals:• Three rows of elements in which the 3d, 4d, and 5d are being filled• Properties primarily determined by the s electrons, rather than by

the d subshell being filled• Have d-shell electrons with unpaired spins• As the d subshell is filled, the magnetic moments, and the

tendency for neighboring atoms to align spins are reduced

The Periodic TableLanthanides (rare earths):

• Have the outside 6s2 subshell completed

• As occurs in the 3d subshell, the electrons in the 4f subshell have unpaired electrons that align themselves

• The large orbital angular momentum contributes to the large ferromagnetic effects

Actinides:

• Inner subshells are being filled while the 7s2 subshell is complete

• Difficult to obtain chemical data because they are all radioactive

• Have longer half-lives

The ionization energies are periodic…

If you are filling shells in order of successively higher energies, why does this happen??

The atomic radius is surprisingly constant. Why does is not scale with the number of electrons?

eVn

ZEn 2

2

6.13

Fig. 9-17, p.325

2 21 ( / 2 )( /1) 24 KeVoE ke a Z ;

KE[

Continuous + line spectrum,

Cut-off at maximum exciting electron energy