Topic 12: Quantum and nuclear physics (HL) Flashcards
What is the photoelectric effect?
The phenomena by which electrons are emitted from the surface of some metals when the surface is illuminated with EM radiation (usually UV)
How many electrons are emitted below the threshold frequency, f0?
No electrons are emitted
Which variable affects the maximum kinetic energy of the emitted electrons?
It depends on the frequency of the incident light
Which variable affects the number of the emitted electrons?
Depends on the intensity of the incident light
What is the delay between light hitting the surface and the emission of the electrons as a result?
No noticeable delay
What does the photoelectric effect show?
Why?
Wave-particle duality. Light is acting as a particle in this instance.
This effect could not happen with the view of light as a wave as any frequency would eventually bring enough energy to the metal plate i.e. there would not be a threshold frequency
What is the setup for the photoelectric effect experiment?
- A variable power supply (for accelerating p.d.)
- Voltmeter in parallel, micro-ammeter in series
- Cathode and anode in a vacuum
- (UV) Light transmitted onto the cathode
Explain why a variable p.d. is used?
The p.d. decelerates the electrons. At a certain value of potential, the stopping potential (VS) no more photocurrent is observed.
(Electrons are brought to rest before arriving at the anode)
What variable affects the stopping potential?
The frequency of the light - linear relationship
What can be calculated using the stopping potential?
The maximum kinetic energy of the electrons as p.d. is energy/charge (we know the charge)
Equation for KEmax of electrons
KEmax = VSe
where VS is the stopping potential
e is the electron charge
Equation for the maximum velocity of electrons
Vmax = √ ( 2 Vs e / m)
where Vs is the stopping potential
e is the electron charge
m is the electron mass
What is the work function?
the minimum energy required for an electron to overcome the attractive forces that act on it within the metal, allowing it to be emitted as a photoelectron
What variable sets the energy of a photon?
the frequency
(E = hf)
What variable sets the amount of photons per second?
the intensity
What happens to the energy absorbed by electrons if it is below the work function?
It will still gain the energy but will quickly be shared out to other electrons, so cannot by emitted
What happens to the energy absorbed that is over the work function?
It is converted into kinetic energy for the electron
Equation for energies in the photoelectric effect
KEmax = Ep - ϕ
Ep = ϕ + KEmax
hf = ϕ + Vse
What is the y-intercept on a energy-frequency diagram for the photoelectric effect?
Negative work function (-ϕ)
What proof do we have that light can behave like waves?
- Reflection
- Refraction
- Diffraction
- Interference
What proof do we have that light is a particle?
- Photoelectric effect
If light waves have particle properties, do other particles have wave properties?
Yes.
High speed electrons can be diffracted through polycrystalline graphite,
What happens to the electron diffraction pattern if the acceleration voltage is decreased?
Intensity decreases and separation increases
What happens to the electron diffraction pattern if the acceleration voltage is increased?
Intensity increases and separation decreases.
In the electron diffraction experiment what causes the electrons to be emitted by the filament?
Thermionic emission
Describe the electron diffraction experiment
- Electrons emitted frim heated filament
- Accelerated using high potential difference
- The chamber is evacuated (no air)
- Electrons pass between the gaps between polycrystalline graphite atoms and are diffracted.
- An interference pattern is observed on the screen (with bright maxima rings and dark minima rings)
What is the De Briglie Hypothesis
The model for matter waves.
The probability function associated with a moving particle where the (amplitude)2 at any given point is the probability of finding the particle at the point.
What does the de Broglie wavelength?
the wavelength of “matter” waves given as
λ = h / p
λ is the de Broglie wavelength
h is Planck’s constant
p is momentum
What can the de Broglie wavelength be written in terms of mass and velocity?
λ = h / mv
as momentum = mass x velocity
What can the de Broglie wavelength be written as when particles are accelerated by a potential difference?
λ = h / √(2meV)
(Using electrical potential energy and kinetic energy equations)
What is the relationship between de Broglie wavelength and kinetic energy?
λ ∝ Ekinetic-1/2
What is the de Broglie wavelength for photons?
λ = hc / E
where λ is wavlength
h is Planck’s constant
c is the speed of light
E is photon energy (E = hf)
What are the three assumptions of the Bohr atomic model?
1) Electrons exist in stationary states
2) Electrons can move between stationary states by absorbing or emitting a quantum of EM radiation
3) The angular momentum of an electron in a stationary state is quantised in integral values of (h/2π)
What is angular momentum and its equation?
the product of a particle’s momentum and its orbital radius
L = pr
where L is angular momentum
p is momentum
r is radius
( also written as L = (mv)r )
What is the relationship between total electron energy and the energy level?
E = (-13.6eV/n2)
What is the Principal Quantum Number?
13.6eV
It is the minimum required energy required for an electron to move from the ground state for a hydrogen atom
What does the wave function give?
The wave function squared is the probability that a particle can be observed as a function of time
What is the particle in a box model?
It is a potential well depicting how much energy is required in order for the electron to shift energy states
What is the wave function collapse?
According to the wave function, a particle’s position is just a probability.
However, once observed, this means that it can no longer be a probability and thus the wave function collapses
What is the Heisenberg uncertainty principle?
A quantum system can only be predicted probabilistically, the Heisenberg uncertainty principle shows that the uncertainty in a quantum’s position is related to the uncertainty is its momentum.
What is the electron’s wave function shape?
it is a sine wave
If you can measure wavelength precisely, what else can you determine accurately?
(formula)
The momentum
p = h / λ
You know the momentum of a particle precisely but…?
all information about its position is lost.
(unlimited uncertainty in its position)
How can you know the uncertainty in position for electron diffraction?
It would be ± gap size
If the graph for the wave function shows the largest uncertainty in momentum, then what does it look like and what does it show?
High amplitude focused towards one point
i.e. smallest uncertainty in position
Electron in a box model: what is the smallest possible value for the momentum?
smallest momentum = smallest wavelength
p = h / 2L
What is the type of interactions in the photoelectric effect?
instantaneous 1:1 interactions
Points to remember for the photoelectric effect
(4)
- Light is a stream of photons
- Each photon’s energy is E = hf
- there are instantaneous 1:1 interactions between incident photons and the surface electrons
- If the photon energy is greater than or equal to the work function, electrons are emitted. If not, none are emitted.
What impact will an increase of intensity have on the maximum kinetic energy of electrons?
No impact.
KE depends on photon energy, determined by frequency.
Equation for standing waves (for electron in a box model)
λn = 2L / n
How do you know how many peaks and troughs to draw for a given confined length and wavelength?
nλ = L
n = L/λ
n is number of peaks and troughs
Conditions for pair production
(2)
- Strong electric field
- Minimum energy photon turns into a matter and antimatter pair
(typically gamma to electron + positron)
What quantities must be conserved during both pair production and annihilation?
- Lepton number
- Baryon number
- Charge
- Strangeness
Why is the minimum photon energy needed to create a particle-antiparticle pair?
The energy of a photon is transferred into mass (E=mc2)
As two particles are created, it is multiplied by 2.
Why is it far less common for pair production to produce a proton-antiproton pair rather than electron-positron pairs?
Protons/anti-protons have far more mass than electrons/positrons. From E=mc2, it requires much more energy,
What is annihilation?
When electrons and positrons destroy each other (they release two photons (gamma)
What are conjugate variables?
A pair of variables which when the uncertainty of one decreases, the uncertainty of the other increases
What is the relationship between the uncertainty in the energy of a particle and the time it exists for?
ΔE Δt ≥ h/4π
Why must low energy photons that produce electron-positron pairs annihilate a short time after?
Due to the relationship between uncertainties in time and energy ()
t is very small, then E is very big so the range of energy is large, so could be below (E=mc2) value, therefore annihilating quickly.
[ASK SIR FOR EXPLANATION]
Why is it possible for a ground state electron to escape from the atom with less than the principal quantum number?
Particles have an infinitesimally small probability of being anywhere in the universe.
This allows the electron to “borrow” energy from its surroundings and pass the energy barrier, later “repaying” this energy back into the surroundings.
How could you show quantum tunnelling on a graph?
A probability (wave function)2 vs displacement graph with a maximum at zero as a bell curve.
A physical barrier with a small part of the graph beyond the barrier.
Explain how quantum tunnelling explains why fusion occurs in the Sun at lower temperatures than the laws of electrostatics would predict
Quantum tunnelling allows for the slight chance that hydrogen will fuse at lower temperatures. As there are a lot of atoms, it will happen very often, therefore causing fusion at lower temperatures.
Explain how a scanning, tunnelling microscope forms images using quantum tunnelling
When the particle tunnels through the board, the current produced is detected, which can be used to map an image.
What were the main results of the Rutherford alpha scattering experiment?
(3)
- most of the alpha particles passed through the gold leaf undeflected
- some alpha particles were deflected through very wide angles
- some alpha particles were rebounded in the opposite direction (e.g. didn’t go past the leaf)
What were the main interpretations of the results of the Rutherford alpha scattering experiment?
(3)
- most of the atom is empty space
- the atom contains small dense regions of electrical charge
- these small dense regions are positively charged
(he essentially found the nucleus)
Explain how the method of closest approach works
- as an alpha particle goes head-on with a gold nucleus, its kinetic energy falls whilst its electric potential energy increases.
- when the alpha particle is at its closest to the nucleus, KE is zero while EPE is maximum, and momentarily stops moving.
- by knowing the kinetic energy of the alpha particle, you can deduce the radius of closest approach, which would be the maximum radius the nucleus could be.
What is the electrostatic (electric) potential energy equation?
EPE = k x (Qq/r)
k is the Coulomb constant which is equal to 1/(4πƐ0)
EPE = 1/(4πƐ0) x (Qq/r)
where Ɛ0 is the permittivity of free space, Q is the charge of one particle, q is the charge of the other particle and r is the distance between the two.
How can the electrostatic (electric) potential energy equation be applied to the method of closest approach?
Eα = (kZe x 2e)/rc
where k is the coulomb constant, Z is the proton number, e is the elementary charge, rc is the closest distance.
What could the EPE of an alpha particle be assumed as?
Its kinetic energy
What is the radius of a nucleus dependant on?
R = R0 A1/3
It is dependant on the mass number i.e. A
Derive the nuclear density equation
as density = mass/volume
nucleus mass = Au (mass number x unified atomic mass)
nucleus volume = (4/3)πR3 = (4/3)πAR03
nuclear density = Au / ((4/3)πAR03)
= 3u/4πR03
which is a constant
What is one of the problems with the Rutherford scattering method?
At higher energies, the alpha particles were able to approach the target nucleus so closely that the strong nuclear attractive force overcomes the electrostatic repulsion.
What is the equation of light incident on a small circular object of diameter D?
sin θ ≈ λ/D
as θ is close to 0º
What is the wavelength of an electron?
λ = hc/E
Which energy levels are higher: those of electrons or those of the nucleus.
Those of the nucleus, but the principle is still the same
What does the classical model say about alpha decay?
It shouldn’t happen.
How would you draw alpha decay and different energy level?
A high horizontal line on the left, with three low lines on the right, both sides joined with dotted lines which each have an energy value.
There are lines that indicate gamma-ray emission between the three lower lines. (all pointing downwards).
How does quantum mechanics explain why alpha decay occurs?
The wave function exists outside of the nucleus and therefore requires less energy to decay.
(Quantum tunnelling)
What properties does the part of the wave function in the potential well have?
- high kinetic energy
- high probability
- short wavelength
What properties does the part of the wave function outside the potential well have?
- less momentum
- longer wavelength
- small but finite chance
What two properties does radioactive decay have?
- random
- unpredictable
Why is it impossible to predict when an unstable nucleus will decay?
Radioactive decay is a quantum mechanical process.
However, we can work out the probability.
What is a decay constant?
What is its unit?
The probability that an individual nucleus will decay in a given time period.
The symbol given is λ.
The unit is time-1 e.g. s-1, min-1, hour-1 etc.
What is radioactive activity?
What is its unit?
Activity is the number of nuclei decaying in a given time, usually 1 second.
The symbol is A.
The unit is Becquerels (Bq).
What is the equation for activity, in terms of the decay constant?
A = Nλ
or
A = - Nλ (as N will decrease over time)
or
dN/dt = - Nλ
What is activity in differential form?
What does this show us?
A = dN/dt
It shows us that radioactive decay is exponential.
Derive the two equations for radioactive activity.
dN/dt = - Nλ
1/N (dN) = - λ(dt)
N0∫N 1/N dN = - λ 0∫t dt
[ln N]NN0 = - λ [t]t0
ln N - ln N0 = - λt
ln N/N0 = - λt
N/N0 = e-λt
N = N0e-λt
A = A0e-λt
What do the graphs of number of unstable nuclei left vs time and activity vs time look like?
They are negatively exponential curves.
How can you graph the log of N graph?
ln N - ln N0 = -λt
ln N = -λ t + lnN0
y = m x + c
which would be a negative linear graph of gradient -λ
and N0 = ey-intercept
What is a half-life?
The amount of time taken for the number of unable nuclei in a substance to half.
Derive the half-life equation
Since A = A0e-λt, after 1 half like A = A0/2
so A0/2 = A0e-λt1/2
1/2 = e-λt1/2
ln 1/2 = -λt1/2
(ln 1/2) / -λ = t1/2
(same for number of nuclei remaining)
For the radioactive decay equations, what does N signify?
N = number of unstable nuclei remaining
(NOT number of nuclei that have decayed!!)
How can you measure long half-lives?
- First, take a pure sample of the nuclide (i.e. one that hasn’t decayed yet)
- Measure its mass and calculate the number of nuclei in the sample.
- Measure the count rate using a GM tube
- Divide the area of a sphere of radius (distance between source and GM sensor) by the area of the sensor
- Multiply this by the activity measured to give you the total activity of the substance.
- The decay constant can be calculated using A/N.
How do you adjust the activity measured for long half-lives?
Activity adjustment = area of a sphere with a radius equal to the distance between the source and GM sensor / area of GM sensor window.
