Radioactivity & Quantum Phenomena

QUANTUM PHENOMENA (return to CONTENTS)


THE PHOTOELECTRIC EFFECT (Hallswach, 1881) (return to start of page)

Observation:

The electroscope is given a negative charge and the gold leaf diverges as indicated. When ultra violet light is then shone on the zinc plate, the gold leaf is seen to collapse, showing that the charge has been lost. This is due to electrons being emitted by the zinc. The phenomenon is called the photoelectric effect, and the electrons are called photoelectrons. Zinc is said to have a ‘photoemissive surface’ when exposed to ultra violet light. Some materials are photoemissive in visible light.

Laws of the photoelectric effect

When electromagnetic (‘em’) radiation of suitable wavelength is shone on the cathode it emits electrons. If the anode is made positive, it collects the electrons and a current flows.

Using a set-up like the above, the following three laws have been discovered:

1. If the frequency of the incident radiation is reduced, there is a minimum frequency, called the threshold frequency, below which no current is detected, therefore no electrons are being emitted, no matter how intense the radiation. The threshold frequency depends on the material of the cathode - for zinc the frequency is in the ultraviolet.

2. For frequencies above the threshold frequency, the current, and therefore the number of electrons emitted per second, is proportional to the intensity of the incident radiation (E.g. - doubling the intensity, doubles the current).

3. For frequencies above the threshold frequency, if the anode is made negative it repels the electrons and, at a certain p.d., called the stopping potential, VS, the current becomes zero. With the current just stopped, it is found that increasing the intensity of the radiation does not restart it. Also, the value of VS is found to increase with the frequency of the radiation. This implies that electrons are emitted with a range of KEs, up to a certain maximum value, a value which depends only upon the frequency of the incident radiation and not upon its intensity.

Planck’s theory (1900)

Planck was working on a topic apparently unrelated to the photoelectric effect, but something he suggested became fundamental to the explanation of the phenomenon. He postulated that light and all other electromagnetic radiation of frequency f, is emitted in ‘packets’ of energy called ‘quanta’, of size E given by:


Einstein’s theory

Einstein extended Planck’s idea to explain the photoelectric effect.


How does this relate to the three laws of photoelectricity?


2. If the frequency is such that electrons are escaping, then since each photon gives its energy to just one
electron, the number of electrons escaping per second is proportional to the number of photons striking
per second, i.e. to the intensity of the radiation. [which is what law (2) says].

Threshold wavelength


Example

Caesium has a work function of 1.9eV. Find its threshold frequency and threshold wavelength.


Stopping potential

Refer back to Diagram (1) and consider law (3). If the anode is made negative it repels the electrons. At a certain p.d., called the stopping potential, VS, the current becomes zero. This means that even the electrons emitted with the highest kinetic energy, KEmax, are being stopped.

Now, when an electron of charge e coulombs is accelerated though a p.d. V volts, it gains kinetic energy e*V joules. Here we have the opposite situation, where and electron of energy KEmax loses all its energy, due to a p.d. VS, which implies that:

Compare this with the equation for a straight-line graph:


Thus, a graph of VS against f should be a straight line with slope h/e:

The above graph has been produced experimentally, verifying Einstein's equation.

Note - an extract from the Encyclopaedia Britannica:


WAVE-PARTICLE DUALITY (return to start of page)

To explain certain phenomena we have to assume that electromagnetic radiation (visible light etc.) has a dual nature.

Louis de Broglie suggested (in 1924) that, not only does light have a particle-like nature, but the converse might also be true, i.e. that particles of matter might have a wave-like nature.

He suggested that the wavelength of an object moving with momentum p = mv, be given by:

A wavelength determined from this equation is known as a de Broglie wavelength.

It is known that when x-rays (which are electromagnetic waves) pass through certain crystals, that the regular crystal structure acts like a diffraction grating and a diffraction pattern can be produced and seen on a photographic plate.

In 1927 the de Broglie equation was confirmed when beams of electrons (previously considered to be as much particle-like as snooker balls) were successfully diffracted using a nickel crystal as the ‘diffraction grating’. Since then neutrons, protons, hydrogen atoms and helium atoms have all been diffracted.

Example

What is the de Broglie wavelength for an electron accelerated though 5000 V?
(electron mass = 9.1*10-31kg; electron charge = 1.6*10-19C; h = 6.6*10-34Js)


This is similar to the wavelength of x-rays.


RUTHERFORD'S EXPERIMENT (1909) (return to start of page)

The following represents a set-up used by Rutherford to investigate the scattering of alpha particles by thin films of gold:

Alpha particles are positively charged particles emitted by some radioactive materials. They are completely absorbed by a sheet of paper or a few cm of air. Hence the need to use an evacuated container and very thin films of gold in the experiment.

The above experiment was performed in a darkened room. Each impact on the glass front of the microscope was seen as a brief flash of light. The microscope was on a turntable, and could be rotated to any desired angle.

It was observed that:

At the time, it was believed that atoms contained a mixture of positive and negative charges, but it was unclear how they were arranged. The ‘plum pudding’ model of the atom suggested that the charges were more or less evenly spread throughout the atom. The above experiment lead Rutherford to a new model of the atom, called the 'nuclear model'.

Rutherford’s interpretation of the observations:

The smallness of the nucleus, relative to the overall size of an atom, is implied by the rarity of large deflections. The diameter of the nucleus being ~10-15m whereas the diameter of an atom is ~10-10m. Thus, much of the volume of an atom is empty space.

Rutherford’s nuclear model explained the observation in the scattering experiment, but raised a problem. An electron in orbit is accelerating, and it was thought that an accelerating charge should emit radiation continuously, and therefore the electron should lose energy and spiral into the nucleus in a very short time, i.e. the atom should collapse in on itself. Clearly this does not happen since atoms are mostly very stable structures.


THE BOHR ATOM (1911) (return to start of page)

Niels Bohr made two suggestions:

  1. An electron can revolve around the nucleus in certain allowed orbits in which it does not emit radiation. Each orbit is associated with a definite amount of energy
  2. An electron can ‘jump’ from one orbit of energy E2 to another of lower energy E1 and the energy difference is emitted as a photon of electromagnetic radiation of frequency f given by:

Thus, each downward ‘jump’ or ‘transition’ produces the emission of a photon of radiation of a definite energy, frequency and wavelength.

Also, if an electron absorbs a photon of energy exactly equal to the energy gap between two levels ( i.e. if  hf = E2- E1) it can jump up to the higher level.

The hydrogen spectrum

An important feature of the Bohr model of the atom is that it gives an explanation for the distinct wavelengths emitted by hydrogen, i.e. the hydrogen spectrum, described below.

If a hydrogen lamp is viewed through a diffraction grating, in an otherwise dark room, the hydrogen spectrum appears as a distinct set of colours, each of a different wavelength. If the hydrogen lamp is arranged so that the light emerges from a vertical slit, the spectrum is seen as a set of distinct coloured lines on a dark background. This is called a line emission spectrum. It can be represented by:

The Bohr ‘picture’ of the hydrogen atom

Each wavelength corresponds to an electron transition between two particular ‘energy levels’.

The circles represent allowed orbits of different energy.
Each orbit or ‘energy level’ is assigned a ‘quantum number’, n, with an energy value En.


ENERGY LEVEL DIAGRAMS (return to start of page)

The possible energy levels of electrons in atoms can be conveniently represented by horizontal lines, each with an assigned energy value.

An electron at rest well away from the nucleus is taken to have zero energy. As it ‘falls’ into the atom it loses energy, this energy being emitted as electromagnetic radiation. Having started with zero energy, and then losing energy, its energy becomes less than zero, i.e. negative.

An atom is said to be in its ground state when every electron in it is in the lowest allowed energy state, otherwise it is in an excited state.

Recall from earlier notes that:

The following is an energy level diagram for hydrogen:
 

The ionisation energy of an atom is the minimum energy needed to completely remove an electron from the atom in its ground state. For hydrogen, the ionisation energy = 13.6eV = 21.8*10-19J

Now, if an electron is accelerated from rest through 13.6V it gains kinetic energy 13.6eV, by definition of the electron volt, so it has exactly the right amount of energy to knock an electron out of a hydrogen atom in its ground state. Hence, we say that the ionisation potential of hydrogen is 13.6V.

Example
From the information in the above diagram calculate the longest wavelength that occurs in each series.
(h = 6.6*10-34Js; c = 3.0*108ms-1)

From this, the smaller the energy difference the longer the wavelength of the emitted photon.

So we use the smallest energy jump for each series:


Note: The visible spectrum has wavelengths from about 4*10-7m (violet) to about 7*10-7m (red). Thus, the above wavelength of 6.6*10-7m in the Balmer series would be visible red light. The Lyman wavelength is less than 4*10-7m, and is in the ultraviolet, and the Paschen wavelength is longer than 7*10-7m, and is in the infrared.


TYPES OF OPTICAL SPECTRA (return to start of page)

Spectra can be classified as ‘emission spectra’ or ‘absorption spectra’, and are further subdivided within these groups:

1. Emission spectra

a) Line spectra

These are emitted by the atoms of luminous gases or vapours at low pressure. The atoms are far enough apart not to interact very much, and each atom emits the same set of wavelengths.

Each element has a different line spectrum, i.e. a line spectrum is characteristic of the element emitting it. The spectrum of hydrogen has already been described.

Yellow sodium lamps are used in certain experiments in the laboratory. Many street lights are sodium lamps. When viewed through a diffraction grating (which disperses the light, i.e. separates the individual wavelengths), the spectrum is seen to contain two yellow lines very close together, which means that they have nearly the same wavelength.

b) Band spectra

These are emitted by the molecules of glowing gases or vapours. The interactions of the atoms within the molecules produce complex spectra. A band spectrum consists of well defined groups or bands of lines.

c) Continuous spectra

These are emitted by hot solids, hot liquids and hot gases at high pressure. The atoms are close together and many interactions occur and a continuous range of wavelengths is emitted (from red to violet, as well as wavelengths outside the visible spectrum).

2. Absorption spectra

These are obtained when part of the light emitted by a source is absorbed.

In general if a substance emits a particular set of wavelengths, then it can absorb those same wavelengths. For example, if white light is passed though sodium vapour, the emerging light has two dark lines in its spectrum where the bright lines in the sodium emission spectrum occur. This is the line absorption spectrum of sodium.

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A Level Physics - Copyright © A C Haynes 1999 & 2004