CHAPTER 40 : INTRODUCTION TO QUANTUM PHYSICS 40.2) The Photoelectric Effect Light incident on certain metal surfaces caused electrons to be emitted from the surfaces = photoelectric effect The emitted electrons = photoelectrons Figure (40.6) A diagram of an apparatus in which the photoelectric effect can occur An evacuated glass or quartz tube contains a metallic plate E connected to the negative terminal of a battery and another metallic plate C is connected to the positive terminal of the battery When the tube is kept in the dark – the ammeter reads zero – no current in the circuit When plate E is illuminated by light by light having a wavelength shorter than some particular wavelength that depends on the metal used to make plate E – a current is detected by the ammeter – a flow of charges across the gap between plates E and C This current arises from photoelectrons emitted from the negative plat (the emitter) and collected at the positive plate (the collector)
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CHAPTER 40 : INTRODUCTION TO QUANTUM PHYSICS 40.2) The Photoelectric Effect Light incident on certain metal surfaces caused electrons to be emitted from.
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CHAPTER 40 : INTRODUCTION TO QUANTUM PHYSICS
40.2) The Photoelectric Effect
Light incident on certain metal surfaces caused electrons to be emitted from the
surfaces = photoelectric effect
The emitted electrons = photoelectrons
Figure (40.6)
A diagram of an apparatus in which the photoelectric effect can occur
An evacuated glass or quartz tube contains a metallic plate E connected to the negative terminal of a battery and another metallic plate C is
connected to the positive terminal of the battery
When the tube is kept in the dark – the ammeter reads zero – no current in the circuit
When plate E is illuminated by light by light having a wavelength shorter than some particular wavelength that depends on the metal used
to make plate E – a current is detected by the ammeter – a flow of charges across the gap between plates E and C
This current arises from photoelectrons emitted from the negative plat (the emitter) and collected at the positive plate (the collector)
Figure (40.7)
A plot of photoelectric current versus potential difference V between plates E and
C for two light intensities
At large values of V – the current reaches a maximum value – the current increases as the intensity of the incident light increases
When V is negative (when battery in the circuit is reversed to make plate E positive
and plate C negative) – the current drops to a very low value because most of the emitted
photoelectrons are repelled by the now negative plate C
Only those photoelectrons having a kinetic energy greater than the magnitude of eV reach plate C – where e = the charge on the electron
When V is equal to or more negative than - V s (the stopping potential) – no
photoelectrons reach C and the current is zero
The stopping potential – independent of the radiation intensity
The maximum kinetic energy of the photoelectrons is related to the stopping potential through the relationship:
smax VeK (40.7)
Features of the photoelectric effect – could not be explained by classical physics or by the wave theory of light
No photoelectrons are emitted if the frequency of the incident light falls below some cutoff frequency fc (is
characteristic of the material being illuminated) (wave theory – predicts that the photoelectric effect should occur at any frequency, provided the light intensity is suffiently high)
The maximum kinetic energy of the photo-electrons is
independent of light intensity (wave theory – light of higher intensity should carry more energy into the metal per unit time and eject photoelectrons having higher kinetic energies)
The maximum kinetic energy of the photo-electrons
increses with increasing light frequency (wave theory – prdicts no relationship between photoelectron energy and incident light frequency)
Photoelectrons are emitted from the surface almost instantaneously (less that 10-9 s after the surface is illuminated) – even at low light intensities (the photoelectrons are expected to require some time to absorb the incident radiation before they acquire enough kinetic energy to escape from the metal)
Successful explanation by Einstein
Assumed that light (or any other electromagnetic wave) of frequency f can be
considered a stream of photons
Each photons has an energy E = hf
Figure (40.8)
In Einstein’s model – a photon is so localized that it gives all its energy hf to a single
electron in the metal
According to Einstein – the maximum kinetic energy for these liberated photoelectrons is :
hfKmax (40.8) Photoelectric effect equation
Where = work function of metal ( the minimum energy with which an electron is bound in the metal and is on the order of a
few electron volts) – Table (40.1)
Photon theory of light – explain the features of the photoelectric effect that cannot be understood using the concepts of classical physics :
• The effect is not observed below a cutoff frequency – the energy of the photon must be greater than or equal to .
• If the energy of the incoming photon does not satisfy this condition – the electrons are never ejected from the surface, regardless of the light intensity.
• Kmax is independent of light intensity – If the light intensity is doubled, the number of photons is doubled – doubles the number of photoelectrons emitted.
• Their maximum kinetic energy (= hf – ) – depends only on the light frequency and the work function, not on the light intensity
• Kmax increases with increasing frequency is easily understood with Equation (40.8)
• Photoelectrons are emitted almost instantaneously – the incident energy arrives at the surface in small packets and there is a one-to-one interaction between photons and photoelectrons.
• In this interaction the photon’s energy is imparted to an electron that then has enough energy to leave the metal – constrast to the wave theory, in which the incident energy is distributed uniformly over a large area of the metal surface
Final confirmation of Einstein’s theory
Experiment observation of a linear relationship between Kmax and f –Figure (40.9)
fc
Kmax
f
Figure (40.9)
The intercept on the horizontal axis – the cutoff frequency below which no photoelectrons are emitted, regardless of light intensity
The frequency is related to the work function through the relationship fc = / h
The cutoff frequency corresponds to a cutoff wavelength of :
hc
h/
c
f
c
cc (40.9) c = speed of light
Wavelengths greater than c incident on a material having a work function do not result in the emission of photoelectrons
40.3) The Compton Effect
Compton and his co-workers – the classical wave theory of light failed to explain the
scattering of x-rays from electrons
Classical theory
Electromagnetic waves of frequency fo incident on electrons should have two effects (Figure (40.10a)) :
Radiation pressure should caused the
electrons to accelerate in the direction of
progagation of the waves
The oscillating electric field of the incident
radiation should set the electrons into oscillation at the apparent frequency f’
f’ = the frequency in the frame of the moving electrons
Frequency f’ is different from the frequency fo of the incident
radiation because of the Doppler effect : Each electron first absorbs
as a moving particle and then reradiates as a moving particle –
exhibiting two Doppler shifts in the frequency of radiation
Because different electrons will move at different speeds after the interaction –
depending on the amount of energy absorbed from the electromagnetic waves – the
scattered wave frequency at a given angle should show a distribution of Doppler-shifted
values
Compton’s experiment
At a given angle – only one frequency of radiation was observed
Could explain these experiment by treating photons not as waves but as point-like particles having enegy hf and momentum hf/c and by
assuming that the energy and momentum of any colliding photo-electron pair are conserved
Compton effect – adopting a particle model for wave (a scattering phenomenon)
Figure (40.10b) – the quantum picture of the exchange of momentum and energy between an individual x-ray photon and an electron
Figure (40.10b) :
• In the classical model – the electron is pushed along the direction of prpagation of the incident x-ray by radiation pressure.
• In the quantum model – the electron is scattered through an angle with respect to this direction – a billiard-ball type collision.
Figure (40.11a)
A schematic diagram of the apparatuse used by Compton
The x-rays, scattered from a graphite target – were analyzed with a rotating crystal
spectrometer
The intensity was measured with an ionization chamber that generated a current proportional
to the intensity
The incident beam consisted of monochromatic x-rays of wavelength
o = 0.071 nm.
Figure (40.11b) – the experimental intensity-versus-wavelength plots observed by Compton for four
scattering angles (corresponding to in Fig. (40.10)
The graphs for the three nonzero angles show two peaks
At o At ’ > o
The shifted peak at ’ is caused by the scattering of x-rays from free electrons, and it
was predicted by Compton to depend on scattering angle as :
cos1cm
h'
eo (40.10)
Compton shift equation
Where me = the mass of the electron
cm
h
e
= Compton wavelength c of the electron
nm 00243.0cm
h
ec
The unshifted peak at o (Figure (40.11b)) – is caused by x-rays scattered from electrons tightly bound to the
target atoms
This unshifted peak also is predicted by Eq. (40.10) if the electron mass is replaced with the mass of a
carbon atom, which is about 23 000 times the mass of the electron
There is a wavelength shift for scattering from an electron bound to an atom – but it is so small that it was
undetectable in Compton’s experiment
Compton’s measurements were in excellent agreement with the predictions of Equation (40.10)
Derivation of the Compton Shift Equation
By assuming that the photon behaves like a particle and collides elastically with a free electron initially at rest – Figure (40.12a)
The photon is treated as a particle having energy E = hf = hc/ and mass zero
In the scattering process – the total energy and total linear momentum of the system must be
conserved
Applying the principle of conservation of energy to this process gives :
eo
K'
hchc
hc/o = the energy of the incident photon, hc/’ = the energy of the
scattered photon, and Ke = the kinetic energy of the recoiling electron
Because the electron may recoil at speeds comparable to the speed of light – use the relativistic expression Ke = mec2 – mec2
2e
2e
o
cmcm'
hchc
(40.11)
22 c/v1/1 where
Apply the law of conservation of momentum to this collision – noting that both the x and y components of
momentum are conserved
The momentum of a photon has a magnitude p = E/c and E = hf
p = hf/c
Substituting f for c gives p = h/
Because the relativistic expression for the momentum of the recoiling electron is pe = mev : we obtain the following expression for the x and y components of
linear momentum, where the angles are as described in Fig. (40.12b) :
cosvmcos'
hh :component x e
o(40.12)
sinvmsin'
h0 :component y e (40.13)
Eliminating v and from Eq. (40.11) to (40.13) – a single expression that relates the remaining three
variables (’, o, and ) – the Compton shift equation :
cos1cm
h'
eo
40.5) Bohr’s Quantum Model of the Atom
The basic ideas of the Bohr theory as it applies to the hydrogen atom:• The electron moves in circular orbits around the proton under the
influence of the Coulomb force of attraction – Figure (40.15)
• Only certain electron orbits are stable – the electron does not emit energy in the form of radiation – the total energy of the atom remains constant – and classical mechanics can be used to describe the electron’s motion
• Radiation is emitted by the atom when the electron “jumps” from a more energetic initial orbit to a lower-energy orbit.
• The frequency f of the photon emitted in the jump is related to the change in the atom’s energy and is independent of the frequency of the electron’s orbital motion
• The frequency of the emitted radiation is found from the energy-conservation expression :
Ei – Ef = hf (40.18) where Ei = the energy of the initial state, Ef = the energy of the final
state, and Ei > Ef
• The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum : The allowed orbits are those for which the electron’s orbital angular momentum about thenucleus is an integral multiple of ħ = h/2 :
mevr = nħ n = 1, 2, 3, … (40.19)
Using these four assumptions
Calculate the allowed energy
levels
emission wavelengths of the
hydrogen atom
Electric potential energy of the system (Fig. (40.15)) : U = keq1q2/r = – kee2/r
where ke is the Coulomb constant and the negative sign arises from the charge – e on the electron
The total energy of the atom which contains both kinetic and potential energy terms :
r
ekvm
2
1UKE
2
e2
e (40.20)
Newton’s second law :
r
vm
r
ek 2e
2
2e
The Coulomb attractive force kee2/r2 exerted on the electron must equal the
mass times the centripetal acceleration (a = v2/r) of the electron
The kinetic energy of the electron is :
r2
ek
2
vmK
2e
2e (40.21)
Substituting this value of K into Eq. (40.20) – the total energy of the atom is :
r2
ekE
2e (40.22)
The total energy is negative – indicationg a bound electron-
proton system
Energy in the amount of kee2/2r must be added to the atom to remove the electron and make the total energy
of the system zero
Obtain an expression for r, the radius of the allowed orbits – by solving Equations (40.19) and (40.21) for v and equationg the results :
rm
ek
rm
hnv
e
2e
22e
222
2ee
22
n ekm
hnr n = 1, 2, 3, … (40.23)
The radii have discrete values – they are quantized
The result is based on the assumption that the electron can exist only in certain allowed
orbits determined by the integer n
The orbit with the smallest radius = Bohr radius ao (corresponds to n = 1) :
nm 0529.0ekm
ha
2ee
2
o (40.24)
A general expression for the radius of any orbit in the hydrogen atom by substituting Equation (40.24) into Equation (40.23) :
nm) 0529.0(nanr 2o
2n (40.25)
Radii of Bohr orbits in hydrogen
Figure (40.16) – the first three circular Bohr orbits of the hydrogen atom
The quantization of orbit radii immediately leads to energy quantization
Substituting rn = n2ao into Equation (40.22) – obtained the allowed energy levels of hydrogen
atom :
2o
2e
n n
1
a2
ekE (40.26)n = 1, 2, 3, …
Inserting numerical values :
eV n
606.13E
2n n = 1, 2, 3, … (40.27)
Energy levelsOnly energies satisfying this
equation are permitted
Ground state = the lowest allowed energy level (n = 1, energy E1 = – 13.606 eV)
First excited state = the next energy level (n = 2, energy E2 = E1/22 = – 3.401 eV)
Figure (40.17) – an energy level diagram showing the energies of these discrete energy
states and the corresponding quantum numbers n
The uppermost level – corresponding to n = (or r = ) and E = 0 : represent the state for which the electron is removed from the
atomIonization energy = the minimum energy required to ionize the atom (to completely
remove an electron in the ground state from the proton’s influence)
Figure (40.17) – the ionization energy for hydrogen in the ground state (based on
Bohr’s calculation = 13.6 eV
Eqs. (40.18) and (40.26) – to calculate the frequency of the photon emitted when the
electron jumps form an outer orbit to an inner orbit :
2i
2fo
2efi
n
1
n
1
ha2
ek
h
EEf (40.28)
Frequency of a photon emitted from hydrogen
Because the quantity measured experimentaly is wavelength – use c = f
to convert frequency to wavelength :
2i
2fo
2e
n
1
n
1
hca2
ek
c
f1(40.29)
2i
2f
H n
1
n
1R
1(40.30)
Identical to relationships discovered by Balmer and Rydberg (Eqs. (40.14) to (40.17)
The constant kee2/2aohc = Rydberg constant, RH = 1.097 373 2 x 107m-1
Bohr extended his model for hydrogen to other elements
In general – to describe a singel electron orbiting a fixed nucleus of charge +Ze (where Z = the atomic number of the