wave behaviour of particles 07

WAVE BEHAVIOUR OF PARTICLES

12 SACE PHYSICS-STAGE 2SECTION 3 TOPIC 4

PRINCE ALFRED COLLEGE


In the previous topics, it was shown that in some circumstances, light exhibits certain behaviours characteristic of waves. In other circumstances, light behaves as particles. Could the reverse be true, namely that particles can behave as waves? This topic investigates this question.


DE BROGLIE’S HYPOTHESISCount Louis de Broglie (1892 - 1970) believed in the symmetry of nature. In 1923 he reasoned that if a photon could behave like a particle, then a particle could behave as a wave.


He turned Compton’s relationship to make wavelength the subject of the equation. Compton- “a photon has momentum”

De Broglie- “An electron has a wavelength”

h

p

p

h


This is called the de Broglie wavelength of a particle.All particles (electrons, protons, bullets, even humans) have a wavelength. They must be moving.They are called “matter waves”.


We cannot see light. We can only make inferences about the nature of light by looking at its properties.Its properties indicate that it is both wave like and particle like in nature.


We also cannot see atoms. We often think of them as exhibiting the properties of particles.But, because we have never seen them, could they be waves pretending to be particles?De Broglie suggested that particles, in some instances could be wave like.

EXAMPLE 1

Calculate the de Broglie wavelength associated with a 1.0 kg mass fired through the air at 100 km/hr.

EXAMPLE 1 SOLUTION

EXAMPLE 1 SOLUTION

Note the wavelength is so small that it cannot be detected and measured. We cannot create slits capable of diffracting such small wavelengths. Can a microscopic object give a more realistic wavelength?

EXAMPLE 2

Calculate the de Broglie wavelength that would be associated with an electron accelerated from rest by a P.D. of 9.0V

EXAMPLE 2 SOLUTION


An electron creates a larger wavelength than a macroscopic object due to the fact that it has a very small mass.The wavelength of an electron is very similar to the wavelength of an x-ray.A beam of electrons should then be able to be diffracted, proving that they have wave like properties.


This wavelength can be measured using a crystal diffraction grating as mentioned previously as the spacing of the atoms in the crystal is in the order of 10-10m. These waves are not caused by the particle but are connected with its motion.


The wavelengths are 1000 x smaller than visible light.Electron beams in electron microscopes are used as they have greater resolving powers and hence greater magnification.

EXAMPLE 3

Calculate the de Broglie of a H atom moving at 158 m s-1 (interstellar space)

EXAMPLE 3 SOLUTION

h

mv

6 626. x 10

1.672 x 10 x 158

-34

-27

= 2.50 x 10-9 m

These are X Rays which do not penetrate the atmosphere

DAVISSON-GERMER EXPERIMENT

C.J. Davisson and L.H. Germer performed an experiment to verify de Broglie’s hypothesis.


Electrons were allowed to strike a nickel crystal. The intensity of the scattered electrons is measured for various angles for a range of accelerating voltages.


It was found that a strong ‘reflection’ was found at θ = 50° when V = 54V. This appeared to be a place of constructive interference, suggesting that the “matter waves” from the electrons were striking the crystal lattice and diffracting into an interference pattern.


The interatomic spacing of Nickel is close to the ‘wavelength’ of an electron. Therefore it would seem possible that electron matter waves could be diffracted.Davisson and Germer set out to verify that the electrons were behaving like a wave using the following calculations.


Theoretical Result (according to de Broglie’s calculation)

The kinetic energy of the electrons is 1/2 mv2 = VeSo mv =

Vem2


The de Broglie wavelength is given by:

For this experiment:

Vem

h

mv

h

2

m10 x 67.1

)10 x (9.11 x )10 x (1.6 x 54 x 2

10 x 625.6

10-

31-19-

-34


This is de Broglie’s theoretical calculaton of what the wavelength should be if a particle were to behave like a wave.


Experimental Result (according to Davisson-Germer)

X-ray diffraction had already shown the interatomic distance was 0.215 nm for nickel. Since θ = 50°, the angle of incidence to the reflecting crystal planes in the nickel crystal is 25°as shown below:


dsin θ = mλ

For the first order reinforcement…

λ = dsinθ = (.215 x 10-9)(sin50°)

= 1.65 x 10-10 m


The close correspondence between the theoretical prediction for the wavelength by de Broglie (1.67 x 10-10 m) and the experimental results of Davisson-Germer (1.65 x 10-10 m) provided a strong argument for the de Broglie hypothesis.

APPLICATION – ELECTRON MICROSCOPES

LIGHT MICROSCOPESA normal light microscope is based on at least two converging lenses, the objective and the eyepiece. There is a limit to how much the conventional microscope can magnify the image. This is due to diffraction.


This determines the minimum distance between two points on the object that can be distinguished as separate. Instead of coming to a focus at a point, the light focuses to a small disc. Any attempt to increase the magnification just magnifies the diffraction disc.


For light microscopy, the minimum distance, using light of wavelength of about 5 x 10-7 m, is about 2 x 10-7 m. This corresponds to a magnification of about 1000. Using ultraviolet light, the magnification can be increased

to 3000 x.


X-rays have smaller wavelengths and so could be considered for use. The problem is that they do not refract significantly and are unsuitable for conventional microscopes, as they cannot be focused easily.


ELECTRON MICROSCOPES:


Once the wavelike properties of electrons were discovered, people realised that they had the properties that were required for high magnification; 1) they have a small wavelength and 2) they can be focused using electric or magnetic fields.


Just as an X-ray tube can produce electrons, electrons can be produced for an electron microscope in the same manner by accelerating of electrons across a large P.D. This takes place in an electron gun with P.D.’s in the range of 40 kV to 100 kV.


The work done by the electric field and assuming the electrons start from rest, their kinetic energy is given by qV. In the case where the accelerating potential is 60 KV, the kinetic energy is: K = qV = (1.6 x 10-19) x (60 x 103) = 9.60 x 10-15 J.


To determine the wavelength of the electrons, the de Broglie relationship is used, = h/p. The momentum must first be determined from the kinetic energy:

K = ½mv2 = ½m2v2/m = p2/2m


And so the momentum can be determined by:

P =

=

P = 1.32 x 10-22 kgms-1

mK2

)10 x (9.60 x )10 x (9.11 x 2 15-31-


= h/p = 6.63 x 10-34/1.32 x 10-22 = = 5.01 x 10-12m

This value is about 100 000 times smaller than visible light.

This makes it easier to distinguish between two points that are separated by only 1 x 10-10 m and have useful magnifications of over 1 million. The problem remains how to focus them.


In a Transmission Electron Microscope (TEM), the electron gun replaces the lamp and electrostatic lenses (usually magnetic lenses) replace the optical lens. The electron image is converted to visible light on a fluorescent screen. The electrostatic lens is shown below:


A parallel beam of electrons that enters the lens along any line except the central vertical axis, experiences a force due to the electric field that deflects them toward the central axis. Their paths are such that all electrons reach this central axis at the same distance from the lens. This is the focal length of the lens.


A magnetic lens is shown below:


The result of using this lens is the same as the electrostatic lens but the path of the electrons is a little more complicated. At any instant, the motion of the electron can be resolved into components parallel and perpendicular to the field.


For a field that is correctly shaped with the appropriate magnitude, the original parallel beam can come together along the central axis at a fixed distance that is the focal length of the lens. In present day electron microscopes, magnetic lenses have virtually replaced electrostatic lenses.


To recap… electron microscopes can focus on smaller objects due to the fact that an electron has a smaller wavelength than visible light.The electron can also be focused using electric and magnetic fields.We use the ability of an electron (particle) to behave like a wave in the use this technology.


Scanning Electron Microscope


Hard Disc


Ant holding a microchip


DNA

wave behaviour of particles 07

Technology

wave behaviour of particles

particles electrons

properties of particles

larger wavelength

realistic wavelength

broglie hypothesis

electron matter waves

electron beams