phys 342 - lecture 18 notes - f12

8/11/2019 PHYS 342 - Lecture 18 Notes - F12

1/14

Lecture 18

Motion of the Nucleus

C.M.

R r0Center of mass:

00

00 rM

mRrmR =!=

Newtons law:

20

2

00

0

20

20

2

0

20

20

)(4

1)(

)(4

1

rR

erR

Mm

Mm

rR

e

RMrm

+

=+

+

+==

!"

#

!"##


2/14

Lecture 18

Reduced Mass

Therefore, we have

00

0

2

2

0

2,

4

1

rRrandMm

Mmwhere

r

er

+=

+

=

=

!"

#

The effects of finite mass of the nucleus can be corrected

simply by replacing the mass of the electronby the reduced

mass

.


3/14

Lecture 18

Effective

Rydberg Constant

Rm

0e4

80

2h

3c

e4

80

2h

3c

M m0

M

We know

We getR

M

e4

80

2h

3c R

1

1 m0 /M

For example,

RH R

1

1 m0 /M

H

109677.584 cm1

RD

R 1

1 m0 /M

D

109707.419 cm1


4/14

Lecture 18

Energy Level Diagram

2

6.13

n

eVEn !=

Energy levels of the

hydrogen atom:

Ground state: n=1

eVE 6.131 !=

Excited states: n>1


5/14

Lecture 18

Transitions

Transitions from higher energy levels to lower energy

levels produce emission lines. For instance, the Lymanseries of the H atom originate in transitions from excited

states to the ground state.

Transitions from lower energy levels to higher energy

levels produce absorption lines. In extreme cases, anelectron in the ground state can absorb sufficient energy

to become a free electron. The minimum energy

required for such a process is referred to as the

ionization energy.

An electron can make a transition from one energy level to the

other:


6/14

Lecture 18

Excitation Mechanisms

Photo-excitation: an atom in the ground state absorbs a

photon (of energy less than the ionization energy) from the

radiation field to make a transition to an excited state.

Collisional excitation: an atom in the ground state acquires

energy (which is less than the ionization energy), in the

process of colliding with other energetic particles, to make

a transition to an excited state.


7/14

Lecture 18

Photoionization

Lyman series of the H atom correspond to transitions from

n>1to the ground state n=1.

What is the inverse process?

Photo-excitation: *,1 AhvA n !+

As nincreases, the spacing between adjacent energy levels

decreases, and the energy of the nthlevel approaches that ofionization. Therefore, it is possible for an electron in the

ground state to absorb a photon of energy greater than the

ionization energy to leave the atom. This process is known

as photoionization.


8/14

Lecture 18

Lyman Limit

There is a lower limit to the wavelength of photons associated

with each line series, arising fundamentally from the upper

limit to the energies of quantized states of an atom. The limit

is sometimes referred to as the series limit.

For example, for the Lyman series of the H atom, the limit

(commonly known as the Lyman limit) is given by

$& '=

22

111

mnhc

EI

(

1=n

!=m hc

EI

Ly

=

!

1


9/14

Lecture 18

Radiative Recombination

Recombination and Radiative Cascade:

!+++!+ "+

21 hvhvAeA

If an electron absorbs a photon of energy photon is greaterthanEI, the electron will escape the atom with excess kinetic

energy. Photoionization can be symbolically expressed as

!++

"+

eAhvA

What is the inverse process?

For instance, a proton collides with an free electron to

form an H atom, emitting photons in the process.


10/14

Lecture 18

Collisional Excitation

An atom (A) can also be ionized or excited by acquiring

energy in a collision with another particle (B):

BABA !+"+ *

or

BeABA !++"+ #+

It is through this process that Frank and Hertz revealed the

first experimental evidence for the existence of discrete

energy levels in atoms.


11/14

Lecture 18

Demonstration of Ionization

Only when the electrons

have a certain minimum

energy eVi does the

current appear. Thecorresponding voltage

Vi is the ionization

potential of the atoms,

i.e., eVi

=EI .


12/14

Lecture 18

Franck-Hertz Experiment

The space between the grid andanodeAis filled with Hg vapor.

A braking voltage VB=0.5V is

applied between the grid andA.

As soon as the voltage VGbetween the cathode and the

grid exceeds VB ,the current

increases but drops suddenly at

the integer multiples of about

5 V.

An intense emission line at

2537 A is observed, which

corresponds to a photon energy

of 4.85 eV.


13/14

Lecture 18

Physical Explanation

The energies of the electron in the Hg

atoms are quantized.

The discrete grid voltages at which the

current shows sudden drops corresponds

to the difference in the potential energybetween the ground state and an excited

state.

When the excited atoms return to the

ground state, they emit photons ofthe same energy.

Improvement in the experimental technique led to the discovery

of more closely spaced structures, corresponding to energy levels.


14/14

Lecture 18

Photo-excitation vs. Collisional Excitation

As an example, sodium vapor at low pressure can be excited to

fluorescence by illumination with the yellowNaline (of which

the photons have an energy 2.11 eV). The excitation occurs

only when the light used has exactly the quantum energy 2.11

eV. Both smaller or larger quantum energies are ineffectivein producing an excitation.

On the other hand, sodium vapor can be excited by collisions

with electrons. In this case, the yellow line is emitted whenever

the energy of the electrons is equal toorgreater than2.11 eV.

The difference is a fundamental one. It is due to the fact that

the energy of photons is quantized but that of free electrons

is not.

phys 342 - lecture 18 notes - f12

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