PHYSICS DEPARTMENT

Wave-Particle Dualism

In 1925, Louis de-Broglie suggested that if radiations can behave as waves in some experiments and as particles in others then one must expect the particles like protons and electrons to exhibit the wave nature under some suitable conditions. It means that it should be possible to device certain experiments in which the particles like electrons, protons etc. can be made to exhibit characteristics of waves. Two such experiments, one by Davisson and Germer in 1927 and other by G.P. Thomson in 1928, were performed and de-Broglie's suggestion was found to be true.

Davisson and Germer's

experiment

In 1927, two American Physicists Davisson and Germer determined the de- Broglie wavelength for slow electrons accelerated under small potential difference and gave the first experimental evidence for existence of matter waves. The experimental set up used by them is as shown in fig.

1. Electron Gun: it consists of a tungsten filament F that produces electrons by process of thermionic emission.

2. High tension Battery B2 : it provides suitable potential difference to accelerate the electrons.

3. Cylinder A: it provides a fine beam of electrons.4. Ni crystal :it acts as scatterer. It is capable of

rotation about an axis perpendicular to plane of paper.

5. Electron collector: it is capable of rotating about the same axis as the Ni crystal target. So it collects the scattered electron beam in a given direction.

6. Galvanometer: it is connected to the electron collector. The deflection in galvanometer is proportional to the intensity of electron beam.

7. Evacuated chamber C: it encloses the entire apparatus.

Initially the electrons are accelerated under a potential difference of about 40V and this potential difference is increased in small steps to perform the experiment at different fixed values of the potential difference. The beam of accelerated electrons is made to fall normally on the crystal. For each value of potential difference, polar graphs are drawn between angle θ and the collector current. Angle θ known as co-latitude is the angle between beams of electrons on the Ni crystal and that which enters the electron-collector. The curves obtained corresponding to different accelerating voltages are as shown in fig.

Theory

Experimental verification

A smooth curve is obtained for potential difference of 40 volt .A bump begins to appear on the curve. This bump moves upwards with increase of accelerating potential difference and becomes most prominent at accelerating voltage of 54 volt and for θ=500. Beyond 54 volt. the bump decreases and is lost completely at about 68 volt. Hence the most prominent maxima observed corresponding to accelerating potential difference of 54 volt at angleθ=500 with the incident electron beam gives a strong evidence that electrons are associated with waves, which after reflection from the regularly spaced atoms in Ni-crystal give constructive interference as in case of Bragg's X-ray diffraction from crystals.

The collector current is maximum at co-latitude angle θ=500

corresponding to accelerating voltage V=54 volt when accelerated electron beam is made to fall normally on the crystal.

As shown in fig, the incident and scattered beams of electrons make an angle of φ=650 with the Bragg's planes. The separation 'd' between Bragg's planes can be determined by using X-rays of known wavelength λ and applying Bragg's equation 2d sin φ=nλ

The value of 'd’ for Ni crystal used in Davisson and Germer experiment was determined to be 0.91A0.

Now the wavelength ‘λ’ of matter waves associated with electrons can be calculated from Bragg's equation by using φ=650.

For d=0.91A0 and n=1 λ=2×0.91×sin650

or λ=1.65 A0

But from de-Broglie equation, the wavelength of accelerated electrons is given by λ= A0

when accelerating voltage is V=54 volt,

λ= A0 =1.66 A0

As there is an excellent agreement between the two values of wavelength'λ', therefore, Davisson and Germer's experiment verifies de-Broglie hypothesis of wave nature of particles in motion.

12.27V

12.2754

G.P. Thomson's method

Experimental Setup

Filament F is heated by passing electric current through it using low tension battery B1. The electrons are emitted by filament due to thermionic emission. These electron are then accelerated under a high potential difference of 10KV to 50KV applied to cylinderical anode A by battery B2. A fine beam of electron coming out of the anode A is made to fall on a thin foil(thickness=10-6cm) of a metal like gold, silver etc. The electrons after diffraction are received on a photographic plate P placed with its plane parallel to the foil. The photographic plate shows a pattern of a number of concentric rings around a central spot as shown in fig.

Theory of G.P. Thomson’s experiment

In this method, the waves of a given wavelength(de-Broglie) under go selective reflections by those micro crystals(of metal foil) whose planes are suitably oriented. According to Bragg's equation

2dsinθ=nλWhen large umber of micro crystals with all possible orientations are present

then some of them may have appropriate glancing angle θ to yield a Bragg reflection from one or more sets of planes.

Corresponding to angle of scattering α=2θ, the diffracted waves of electron

will follow a cone whose axis lies along the direction of incident beam. As a result, concentric rings are formed on the photographic plate as shown in fig(b)

Let D be diameter of nth dark ring and L the distance of photographic plate from plane of foil then

=2θ

D=4θ LFor nth order dark ring obtained at angle θ, Bragg's equation gives nλ=2d sinθ

nλ=2d θ nλ=2d. or nλ=

4DL

2dDL

/ 2DL

de-Broglie wavelength of an electron accelerated through potential difference V is given by λ= A0

But when potential difference V is high due to relativistic effects the de-Broglie wavelength of accelerated electron is given by

In this equation, α is the relativistic correction factor is given by

where m0 is rest mass of electron

150

(1 )2

V

20300

eVm c

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On using value of λ as given above and solving for D, we get

or DV1/2=constant.(neglecting small correction factor) This equation was experimentally verified by G.P Thompson. Further he also determined the value of lattice spacing 'd' for foils of different metals by using the expression

The value of 'd 'for different metals as given by this equation are very close to the values as determined from X-ray diffraction method.

Thus, G. P. Thompson method confirms the existence of de-Broglie's matter waves and proves the correction de-Broglie's wave equation.

2 150

(1 )2

nLdD V

2 150

(1 )2

nLDd V

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