Lecture VI

Many - electron atoms

dr hab. Ewa Popko

S-states probability

P-states probability

The Zeeman effect

The Zeeman effect is the splitting of atomic energy and the associated spectrum lines when the atoms are placed in a magnetic field. This effect confirms experimentally the quantization of angular momentum.

N

S

NS

potential energy

The potential energy of an object in a magnetic field depends on the magnetic moment of the object and the magnetic field at its location

U B

magnetic moment of a current loop

4321 00BnBn

ˆI2

ˆI2

ba

ba

BA

I

The magnetic moment of a wire loop carrying current depends on the current I in the loop and the area A of the loop.

IA

The Zeeman effect

Lme

rmme

rr

eA

Te

iAe

ee 2

v22

v 2

e

The orbiting electron is equivalent to a current loop with radius r and area . 2r

The average current I is the average charge per unit time T for one revolution, given by T=2r/v.

Suppose B is directed towards z-axis. The interaction energy of the atom magnetic moment with the field is:

BU z

where z is the z-component of the vector . On the other hand:

The Zeeman effect

zz Lm

e

2

and Lz=ml with . Thus .....3,2,1,0 lm

BmBm

emBU Bllz

2

ohr magneton

Zeeman effect

The values of ml range from –l to +l in steps of one, an energy level with a particlular value of the orbital quantum number l contains (2l+1) diffrent orbital states. Without a magnetic field these states all have the same energy; that is they are degenerate. The magnetic field removes this degeneracy. In the presence of a magnetic field thy are split into (2l+1) distinct energy levels:

BBmeU B )2/(

,...2,1,0 lBl mwithBmU

Adjacent levels differ in energy by

TeV5

eB 1079.5

2me

μ

Energy diagram forhydrogen, showing the splitting of energy levels resulting from the interaction of the magnetic moment of the electron’s orbital motion with an external magnetic field.

The Zeeman effect

The Zeeman effect

Splitting of the energy levels of a d state caused by an applied magnetic field, assuming only an orbital magnetic moment.

Selection rulesThe photon carries one unit ( ) of angular momentum. Therefore theallowed transitions: l must change by 1 and ml must change by 0 or 1

Solid lines – allowed transitions;dashed - forbidden Nine solid lines give only three energies:Ei-Ef ;Ei-Ef +B;Ei-Ef -B

The Zeeman effect

Conclusions: spectrum lines corresponding to transitions from one set of levels to another set are correspondingly split and appear as a series of three closely spaced spectrum lines replacing a single line.

Anomalous Zeeman effect

Spin angular momentum and magnetic moment

Electron posseses spin angular momentum Ls. With this momentum magnetic momentum is connected:

se

es Lme

g

2

where ge is the gyromagnetic ratio

For free electron ge=2

se

s Lme

Allowed values of the spin angular momentum are quantized :

)1( ssLs

spin quantum number s = ½ 2

3sL

Własny moment pędu - spin

The z – component of the spin angular momentum:

ssz mL

2

12

1

sm

Spin angular momentum and magnetic moment

Be

sz

esz

esz

me

me

Lme

2

21

21sm

21sm

Ls sz

The z- component of the spin magnetic moment

Electron in a magnetic field

BEE sz 0

21sm

21sm

To label completely the state of the electron in a hydrogen atom, 4 quantum numbers are need:

name label magnitude

Principal quantum number

n 1, 2, 3, ...

Orbital quantum number

l 0, 1, 2, ... n-1

magnetic quantum number

ml od –l do +l

Spin quantum number

ms ± 1/2

Many – electron atoms and the exclusion principle

Central field approximation:

- Electron is moving in the total electric field due to the nucleus and averaged – out cloud of all the other electrons.

- There is a corresponding spherically symmetric potential – energy function U( r).

Solving the Schrodinger equation the same 4 quantum numbers are obtained. However wave functions are different. Energy levels depend on both n and l.

• In the ground state of a complex atom the electrons cannot all be in the lowest energy state.

Pauli’s exclusion principle states that no two electrons can occupy the same quantum – mechanical state. That is, no two electrons in an atom can have the same values of all four quantum numbers (n, l, ml and ms )

Shells and orbitals

Nmax - maximum number of electrons occupying given orbital

n shell orbital

1 K 0 s

2 L 0 s

L 1 p

3 M 0 s

M 1 p

M 2 d

4 N NNN

01

23

sp

df

Nmax

2

2

2

6

6

6210

1014

Shells K, L, M

n 1 2 3

0 0 1 0 1 2

m 0 0 -1 0 1 0 -1 0 1 -2 -1 0 1 2

ms

N 2 8 18

N : number of allowed states state with ms = +1/2 state with ms = -1/2

1s22s22p2

1s22s22p4

carbon

oxygen

Hund’s rule - electrons occupying given shell initially set up their spins paralelly

The periodic table of elements

Atoms of helium, lithium and sodium

n =1, = 0 n =1, = 0 n =1, = 0

n =2, = 0 n =2, = 0n =2, = 0

n =2, = 1 n =2, = 1

n =3, = 0

Helium (Z = 2) Lithium(Z = 3) Sodium (Z= 11)

1s

2s

2p

3s

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10

6p6 7s2 6d10 5f14

110

25

15

23

22

21

26

162

43:

43:

43:

43:

43:

43:

43:

431:

sdCu

sdMn

sdCr

sdV

sdTi

sdSc

spCa

spsK

Electron configuration – the occupying of orbitals

Example: l = 1, s = ½

1 jjJ

21

21

23

21

21

23

21

21

23

21

, lub,,,

1lub1

jj mm

jj

j = 3/2 j = 1/2

SLLJ

Possible two magnitudes of j : l-sjslj or

jjjjmmJ jjz ,1,,1,,

Total angular momentum - J

TheStern-Gerlach experiment

Diamagnetics

.Diamagnetics Shells totally filled with electrons. Total magnetic moment equals zero. (In a filled orbital, the vectors for both the orbital angular momentum and the spin angular momentun point in all posible directions and thus cancel).

• Noble gas

- He, Ne, Ar…..• diatomic molecule gas

- H2, N2…..

• solid states of ionic bonds

- NaCl(Na+, Cl-)…• solid states of covalent bonds

- C(diamond), Si, Ge…..• most organic materials

. Paramagnetics Shells partially filled with electrons Total magnetic moment different from zero.

Paramagnetics

Bef JJg )1(

BJHef Mg,

The component of the magnetic moment directed towards external magnetic field

Fine and hyperfine structure

Line splittings resulting from magnetic inetractions are called fine structure.

The nucleus of the atom has also magnetic dipole moment that interacts with total magnetic moment of electrons. These effects are called hyperfine structure.

NMR ( nuclear magnetic resonance)

Like electrons, protons also posses magnetic moment due to orbital angular momentum and spin ( they are also spin-1/2 particles) angular momentum.

Spin flip experiment:

Protons, the nuclei of hydrogen atoms in the tissue under study, normally have random spin orientations. In the presence of a strong magnetic field, they become aligned with a component paralell to the field. A brief radio signal flips the spins; as their components reorient paralell to the field, they emit signals that are picked up by sensitive detectors. The differing magnetic environment in various regions permits reconstruction of an image showing the types of tissue present.

An electromagnet used for MRI imaging

Wilhelm Roentgen 1895

Roentgen lamp

2

maxmin

v

2e

AC

m hceV h

Roentgen 1895; X -ray: 10-12m – 10-9m

X-ray continuum spectra

2

maxmin

v

2e

AC

m hceV h

ACeV

hcmin

X-ray spectra and Moseley law

The continous –spectrum radiation is nearly independent of the target material.

Sharp peaks (characteristic spectra) depend on the accelerating voltage and the target element. Frequencies of the peaks as a function of the element’s atomic number Z:

215 )1)(1048.2( ZHzf

Moseley law

ACeV

hcmin

X-ray spectra and Moseley law - explanationCharacteristic x-ray radiation is emitted in transitions involving the inner shells of a complex atom.

Let us assume, that due to electric field one of the two K – electrons is knocked out of the K shell. The vacancy can be filled by another electron falling in from the outer shells. K is the transition from n=2 to n=1. As the electron drops down it is attracted by Z protons in the nucleus screened by the one remaining electron in the K shell. The energy before (Ei) an after (Ef) transition:

22 2/)6.13()1( eVZEi )6.13()1( 2 eVZE f

)2.10()1( 2 eVZEK

215 )1)(1047.2( ZHzh

Ef

X-ray diffraction pattern

X-ray diffraction pattern

Diffraction maxima: md sin2

X