De Broglie’s Equation and The Wave Nature of Electron
De Broglie’s Equation
❒de Broglie had arrived at his hypothesis with the help of Planck’s
Quantum Theory and Einstein’s Theory of Relativity.
❒He derived a relationship between the magnitude of the wavelength
associated with the mass (m) of a moving body and its velocity.
❒According to Planck, the photon energy (E) is given by the equation:
(h):
Planck’s constant
(v):
the frequency of radiation.
❒By applying Einstein’s mass energy relationship, the energy
associated with photon of mass (m) is given as
where
(c) is the velocity of radiation
❒Comparing equations (1) and (2):
❒The equation (3)
is called de Broglie’s equation and may be put in words as : The
momentum of a particle in motion is inversely proportional to wavelength,
Planck’s constant (h) being the constant of proportionality.
❒The wavelength
of waves associated with a moving material particle (matter waves) is called de
Broglie’s wavelength.
❒The de
Broglie’s equation is true for all particles, but it is only with very small particles,
such as electrons, that the wave-like aspect is of any significance. Large particles
in motion though possess wavelength, but it is not measurable or observable.
❒Let us, for instance
consider de Broglie’s wavelengths associated with two bodies and compare their
values:
(a) For a large mass
Let us consider a stone of mass 100 g moving with a velocity of
1000 cm/sec. The de Broglie’s wavelength λ will be given as follows :
This is too small to be measurable by any instrument and hence no
significance.
(b) For a small mass
Let us now consider an electron in a hydrogen atom. It has a mass =
9.1091 × 10–28 g and moves with a velocity 2.188 × 10–8
cm/sec. The de Broglie’s wavelength (λ) is given as:
❒This value is
quite comparable to the wavelength of X-rays and hence detectable. It is, therefore,
reasonable to expect from the above discussion that:
Everything in nature
possesses both the properties of particles (or discrete units) and also the properties
of waves (or continuity).
The properties of large objects are best described by considering
the particulate aspect while properties of waves are utilized in describing the
essential characteristics of extremely small objects beyond the realm of our
perception, such as electrons.
The Wave Nature of Electron
❒de Broglie’s
revolutionary suggestion that moving electrons had waves of definite wavelength
associated with them, was put to the acid test by Davison and Germer (1927).
❒They
demonstrated the physical reality of the wave nature of electrons by showing
that a beam of electrons could also be diffracted by crystals just like light
or X-rays.
❒They observed
that the diffraction patterns thus obtained were just similar to those in case
of X-rays. It was possible that electrons by their passage through crystals may
produce secondary X-rays, which would show diffraction effects on the screen.
❒Thomson ruled
out this possibility, showing that the electron beam as it emerged from the crystals,
underwent deflection in the electric field towards the positively charged
plate.
Davison and Germers Experiment
❒In their actual
experiment, Davison and Germer studied the scattering of slow moving electrons by
reflection from the surface of nickel crystal.
❒They obtained electrons
from a heated filament and passed the stream of electrons through charged
plates kept at a potential difference of (V) esu. Due to the electric
field of strength (V × e) acting on the electron of charge (e), the electrons
emerge out with a uniform velocity (v) units.
❒The kinetic
energy ½ mv2 acquired by an electron due to the electric
field shall be equal to the electrical force. Thus,
Multiplying by m on both sides,
But according to de Broglie’s relationship
Comparing (1) and (2):
Substituting for
h = 6.6256 × 10–27
erg-sec,
m = 9.1091 × 10–28 g,
e = 4.803 × 10–10 esu,
and changing V esu to V volts by using the conversion factor ⅓× 10−2
, we have:
❒If a potential
difference of 150 volts be applied, the wavelength of electrons emerging out is
λ = 1 Å. Similarly if a potential difference of 1500 volts be created, the
electrons coming out shall have a wavelength 0.1 Å. It is clear, therefore,
that electrons of different wavelengths can be obtained by changing the
potential drop. These wavelengths are comparable with those of X-rays and can undergo
diffraction.
Apparatus used by Davison and Germer
❒The electrons
when they fall upon the nickel crystal, get diffracted. Electrons of a definite
wavelength get diffracted along definite directions.
❒The electron
detector measures the angle of diffraction (say θ) on the graduated circular
scale.
❒According to
Bragg’s diffraction equation, the wavelength λ of the diffracted radiation is
given by ( λ = d sin θ), where d is a constant (= 2.15 for Ni crystal) and θ
the angle of diffraction.
❒By substituting
the experimental value of (θ) in Bragg’s equation (λ = d sin θ), the wavelength
of electrons may be determined. This wavelength would be found to agree with
the value of (λ), as obtained from equation (3).
❒Since
diffraction is a property exclusively of wave motion, Davison and Germer’s (electron
diffraction) experiment established beyond doubt the wave nature of electrons.
❒We have described
earlier that electrons behave like particles and cause mechanical motion in a
paddle wheel placed in their path in the discharge tube. This proves,
therefore, that electrons not only behave like (particles) in motion but also
have ‘wave properties’ associated with them. It is not easy at this stage to
obtain a pictorial idea of this new conception of the motion of an electron.
❒But the application
of de Broglie’s equation to Bohr’s theory produces an important result. The
quantum restriction of Bohr’s theory for an electron in motion in the circular
orbit is that the angular momentum (mvr) is an integral multiple (n) of (h/2π).
That is,
On rearranging, we get:
Putting the value of (h/mv) from equation (1), we
have:
❒Now the
electron wave of wavelength (λ) can be accommodated in Bohr’s orbit only if the
circumference of the orbit, (2πr), is an integral multiple of its wavelength.
Thus de Broglie’s idea of standing electron waves stands vindicated. However,
if the circumference is bigger, or smaller than (nλ), the wave train will go
out of phase and the destructive interference of waves causes radiation of energy.
Solved Problem
Problem (1): Calculate the wavelength of an electron having kinetic
energy equal to
4.55 × 10–25 J. (h = 6.6 × 10–34 kg m2
sec–1 and mass of electron = 9.1 × 10– 31 kg).
Solution:
Problem (2): Calculate the wavelength of an α particle having mass
6.6 × 10–27 kg moving with a speed of 105 cm sec– 1
(h = 6.6 × 10–34 kg m2 sec–1).
Solution:
Reference: Essentials of Physical Chemistry /Arun Bahl, B.S Bahl and G.D. Tuli / multicolour edition.
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