Chapter 40 All About Atoms

In this chapter we continue with a primary goal of physics―discovering and understanding the properties of atoms. 100 years ago researchers struggled to find experiments that would prove the existence of atoms. Today, thanks to scientific and technological progress, we can manipulate atoms in amazing ways: we can image individual atoms using scanning tunneling microscopy; we can drag them on surfaces to make quantum corrals, and even hold an individual atom indefinitely in a trap in order to study its properties when isolated.

(40-1)

Basic Properties

Atoms are stable. Essentially all atoms have remained unchanged for billions of years.

Atoms combine with each other. Atoms stick together to form molecules and stack up to form rigid solids. Even though atoms are mostly empty space, their interactions allow you to stand on a floor without falling through!

These basic properties can be explained by quantum mechanics.

40-2 Some Properties of Atoms

(40-2)

Some Properties of Atoms

Fig. 40-2

Subtler Properties

Atoms Are Put Together Systematically. There are repetitive (periodic) patterns in the properties of different atoms that allow them to be organized into a periodic table.

Ionization energy vs. atomic number (number of protons in nucleus)

Six periods with 2, 8, 8, 18, 18, and 32 atoms in each period, respectively. These numbers are predicted by quantum mechanics.

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Some Properties of Atoms

Subtler Properties, cont’d

Atoms Emit and Absorb Light:

Atoms Have Angular Momentum and Magnetism:

high lowhf E E

Fig. 40-3

“Orbit” of each electron (more correct to think in terms of angular momentum of electronic state) can produce a magnetic moment.

(40-4)

Some Properties of Atoms

Subtler Properties, cont’d

Einstein-de Haas Experiment:

Fig. 40-4

Angular momentum and magnetic moment of atoms are coupled.

Aligning magnetic moments of iron atoms using an external magnetic field causes the iron cylinder to rotate in a direction opposite to the now-aligned angular momenta of the iron atoms (conservation of angular momentum).

(40-5)

40-3 Electron SpinTrapped or free, electrons have intrinsic spin angular momentum S (spin). This is a basic characteristic like the electron’s mass or charge. This leads to two additional quantum numbers that are required to fully specify the electronic state: s (magnitude of the spin, which is always ½ for electrons) and ms (the component of spin along the z-axis).

Electron States for an Atom

Quantum Number Symbol Allowed Value Related to

Principal n 1, 2, 3, … Distance from nucleus

Orbital l 0, 1, 2, …, (n-1) Orbital angular momentum

Orbital magnetic ml -l, -(l-1), …+(l-1), +l Orb. ang. mom. (z-component)

Spin s ½ Spin angular momentum

Spin magnetic ms ± ½ Spin ang. mom. (z-component)

Table 40-1

States with same n form a shell. States with same value for n and l form a subshell.

(40-6)

Orbital Angular Momentum and Magnetism

40-4 Angular Momenta and Magnetic Dipole Moments

1L Orbital Angular Momentum:

Orbital Magnetic Dipole Moment:orb 2

eL

m

orb 12

e

m

orb

orb

Neither nor can be measured experimentally, but their

components along a given axis be measured. Applying

a magnetic field along the axis allows the components of

and to be

L

B z z

L

can

measured.

(40-7)

Orbital Angular Momentum and Magnetic Dipole Moments

Fig. 40-5

B

24

4 2

9.274 10 J/T (Bohr magneton)

eh e

m m

orb, Bz m

Bohr magneton

zL m

: semi-classical angle

between and the axis

cos z

L z

L

L

(40-8)

1 12 2

, B

Neither nor can be measured, but their

components along a given axis (say the -axis) be measured.

where (spin up) or - (spin down)

2

s

z s s

s z s

S

z

S m m

m

can

Fig. 40-6

Spin Angular Momentum and Spin Magnetic Dipole Moment

S, the magnitude of the spin angular momentum, has a single value for any electron, whether free or trapped:

where s (=½) is spin quantum number of the electron.

The spin magnetic dipole moment s is related to S

and is given by:

1 12 21 1 0.866S s s

s

eS

m

1s

es s

m

(40-9)

represents the total angular momentum of atoms containing more

than one electron. is the vector sum of all the orbital and spin

angular momenta of all the electrons.

A neutrally charged atom with a

J

J

1 2 3 1 2 3

eff

eff 1 2 3

1 2 3 1 2 3

tomic number

will have electrons and protons.

Total magnetic dipole moment is:

since

2

Z Z

Z

Z Z

Z

Z Z

J L L L L S S S S

e eL L L L S S S S

m m

eff is weighted more than , is not parallel to .S L J

Fig. 40-7

Orbital and Spin Angular Momentum Combined

(40-10)

is the magnetic field

gradient along the -direction.

The -projection of determines

the direction and magnitude of

the deflecting force.

z

z z

U B

U B

dU dBF

dz dz

dB

dzz

z

40-5 Stern-Gerlach Experiment

Fig. 40-8

Magnetic Deflecting Force on Silver Atom

Stronger B

Weaker B

z

zF

(40-11)

1 1, B B , B B2 2

B B

2 and 2

and

s z s z

z z

dB dBF F

dz dz

Stern-Gerlach Experiment, cont’d

Experimental surprise

Fig. 40-9

when

magnet ON

B

Silver atoms

Meaning of Experiment: is quantized with two possible values

with opposite signs

is also quantized the same way.

The dipole moments of all the electrons

in a silver atom vectorially cancel out

except for the moment of a s

z

zL

ingle

electron.

(40-12)

A proton has a spin magnetic moment

that is associated with the proton's

intrinsic spin angular momentum .

In a magnetic field , the two spin states of

the proton will lead to two orientations of

S

B

, which in turn will have two different

energies since .

2

photon absorption (nuclear magnetic

resonance) at 2 spin flip

radio frequency

z z z

z

U B

E B B B

hf B

f

40-6 Magnetic Resonance

Fig. 40-10

(40-13)

For fixed radio frequency light, when Bext = hf/2mz - Bint→ absorption occurs.

Bint is different for protons in different molecules, so the resonance Bext will be different for protons in different molecules (local environment).

Resonances provide a fingerprint of what (and where in the case of Magnetic Resonance imaging) different proton-containing molecules are present in the material studied.

The net magnetic field that a proton experiences consists of the vector sum of the externally applied magnetic field Bext and internal fields Bint

Magnetic Resonance, cont’d

Fig. 40-11

magnetic dipole moments of atoms and nuclei near the proton→ Bint

ext int ext int absorption when 2 zB B B hf B B

(40-14)

40-7 Pauli Exclusion Principle

No two electrons confined to the same trap (or atom) can have the same set of values for their quantum numbers.

40-8 Multiple Electrons in Rectangular Traps

1. One-dimensional trap. Two quantum numbers n=1, 2, 3… (wavefunction state along L) and ms= +½ or -½.

2. Rectangular corral. Three quantum numbers nx = 1, 2, 3… (wavefunction state along Lx) , ny = 1, 2, 3… (wavefunction state along Ly), and ms= +½ or -½.

3. Rectangular box. Four quantum numbers nx = 1, 2, 3… (wavefunction state along Lx) , ny = 1, 2, 3… (wavefunction state along Ly), nz = 1, 2, 3… (wavefunction state along Lz), and ms= +½ or -½.

(40-15)

Adding electrons to a rectangular trap:

Use energy level diagram.

Start at lowest energy level and move up as lower levels become filled.

Finding the Total Energy

Fig. 40-13

Filled levels

Partially filled level

Empty (unoccupied) level

(40-16)

Four quantum numbers n, l, ml, and ms identify the quantum states of individual electrons in a multi-electron atom.

Subshells are labeled by letters:

l= 0 1 2 3 4 5 . . .

s p d f g h . . .

Example: n = 3, l = 2→ 3d subshell

40-9 Building the Periodic Table

n l = 0 (s) l = 1 (p) l = 2 (d) ml = 0 -1 0 +1 -2 -1 0 +1 +2

3 __ __ __ __ __ __ __ __ __2 __ __ __ __1 __En

erg

y

Neon: Z = 10→10 electrons

1s2 2s2 2p6

(40-17)

Building the Periodic Table, cont’d

n l =0 (s) l =1 (p) l =2 (d) ml = 0 -1 0 +1 -2 -1 0 +1 +2

3 __ __ __ __ __ __ __ __ __2 __ __ __ __1 __E

ner

gy

Sodium: Z = 11→11 electrons

1s2 2s2 2p6 3s1

n l =0 (s) l =1 (p) l =2 (d) ml = 0 -1 0 +1 -2 -1 0 +1 +2

3 __ __ __ __ __ __ __ __ __2 __ __ __ __1 __E

ner

gy

Chlorine: Z = 17→17 electrons

1s2 2s2 2p6 3s2 3p6

For smaller atoms such as these, one can assume that the energy only depends on n.

degenerate

(40-18)

Building the Periodic Table , cont’d

Iron: Z = 26→26 electronsFor atoms with a larger number of electrons, the interactions among the electrons causes shells with the same n but different l to have different energies (degeneracy lifted).

1s2 2s2 2p6 3s2 3p6 3d6 4s2

Due to interactions, it takes less energy to start filling the 4s subshell before completing the filling of the 3d subshell, which can accommodate 10 electrons.

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X rays are short-wavelength (10-10 m), high-energy (~keV ) photons. Photons in the visible range: ~ 10-6 m; ~eV.

Useful for probing atoms

40-10 X Rays and Ordering of Elements

Fig. 40-14

Fig. 40-15

0min

hcK hf

min

0

(cutoff wavelength)hc

K

Independent of target material

(40-20)

1. Energetic electron strikes atom in target, knocks out deep-lying (low n value). If deep-lying electron in n = 1 (K-shell), it leaves a vacancy (hole) behind.

2. Another electron from a higher energy shell in the atom jumps down to the K-shell to fill this hole, emitting an x-ray photon in the process.

Characteristic X-Ray Spectrum

Fig. 40-16

If the electron that jumps into the hole starts from the n = 2 (L-shell), the emitted radiation is the K line. If it jumps from the n = 3 (M-shell), the emitted radiation is the K line. The hole left in the n = 2 or n = 3 shells is filled by still higher lying electrons, which relax by emitting lower energy photons (higher lying energy levels are more closely spaced).

(40-21)

Moseley (1913) bombarded different elements with x rays. Nuclear charge, not mass, is the critical parameter for ordering elements.

Ordering Elements

Fig. 40-17

(40-22)

12

2215

15

7

(10.2 eV) 12.46 10 Hz 1

4.14 10 eV s

where constant 4.96 10 Hz

ZEf Z

h

f CZ C C

Accounting for the Moseley Plot

Ordering Elements, cont’d

4

2 2 2 20

1 13.60 eV= , for 1, 2,3,

8n

meE n

h n n Energy levels in hydrogen:

2

2

13.60 1n

eV ZE

n

Approximate effective energy levels in multi-electron atom with Z protons (replace e2 x e2 with e2 x (e(Z - 1))2:

2 2

2 1 2 2

2

13.60 eV 1 13.60 eV 1

2 1

(10.2 eV) 1

Z ZE E E

Z

K energy:

K frequency:

(40-23)

Lasers have many uses: Small: voice/data transmission over optic fibers, CDs, DVDs, scannersMedium: medical, cutting (from cloth to steel), weldingLarge: nuclear fusion research, astronomical measurements, military applications

4. Laser light can be sharply focused: Can be focused into very small spot so that all the power is concentrated into a tiny area. Can reach intensities of 1017 W/cm2, compared to 103 W/cm2 for oxyacetylene torch.

40-11 Lasers and Laser Light

1. Laser light is highly monochromatic: Its spread in wavelength is as small as 1 part in 1015.

2. Laser light is highly coherent: Single uninterrupted wave train up to 100 km long. Can interfere one part of beam, with another part that is very far away.

3. Laser light is highly directional: Beam spreads very little. Beam from Earth to Moon only spreads a few meters after traveling 4 x 108 m.

(40-24)

40-12 How Lasers Work

Fig. 40-19

0xhf E E

0

0xE E kT

xN N e

Thermal distribution (Boltzmann):

To get more stimulated emission than absorption, x > N0 → population inversion → not in thermal equilibrium

(40-25)

Fig. 40-22

Helium-Neon Gas Laser

Fig. 40-20

Thermal Equilibrium Population Inversion

Fig. 40-21

(40-26)