Give the theory of normal zeeman effect and show how you can determine the value of specific charge [e / m) of electron with its help.
Answers
Answer:
In 1896 Pieter Zeeman discovered that spectral lines emitted by atoms were split into
three closely spaced lines when the atoms were placed in an external (to the atoms)
magnetic field (see Figure ZE-1 and the More section The Zeeman Effect). Of the
three lines, one had the frequency (and wavelength) of the original, no-field spectral
line, one was at a slightly lower frequency (longer wavelength), and the third was at a
slightly higher frequency (shorter wavelength). The frequency differences Df of the
two new lines from the frequency of the original line were equal. This observation
was explained by H. A. Lorentz using classical mechanics and classical electromag-
netic theory.1
He treated the additional motion of the electron in the atom due to the
external magnetic field as a simple harmonic vibration resulting from an elastic
restoring force acting to return the electron to some equilibrium position. The vibra-
tion frequency of electron was, by electromagnetic theory, that given to the emitted
electromagnetic wave resulting from the harmonic acceleration of the charged elec-
tron and was equal to
f = A
a
m ZE-1
where m is the electron mass and a is a positive constant dependent on the properties
of the particular atom.
If the external magnetic field H (H = B>m, where m permeability) is applied
in the 1z direction, then a force is introduced given by
F = q
c
v * H ZE-2
The components of F are
Fx = qH
c
dx
dt
Fy = qH
c
dy
dt
Fz = 0 ZE-3
and the equations of motion of the charge become
for x: md2
x
dt
2 = -ax + qH
c
dx
dt
1
H. A. Lorentz, The Theory of Electrons (London: Macmillan & Co., 1909). This book records the
lectures delivered by Lorentz at Columbia University during the spring 1906 term.
Explanation:
for y: md2
y
dt
2 = -ay - qH
c
dy
dt
for z: md2
z
dt
2 = -az ZE-4
As Lorentz described, solving the z equation yields the original no-field frequency f0.
Solving the x and y equations and noting that qH>mc V f0 leads to the two approxi-
mate solutions
f+ = f0 + qH
mc
and f- = f0 - qH
mc
ZE-5
Thinking about this situation a bit more, you may recall that the magnetic force given
by Equation ZE-2 is always perpendicular to both v and H and, therefore, does no
work on the charge q. Hence, it does not change the energy (i.e., frequency of rota-
tion) of the charge. However, as the H field is applied, there is a dH>dt, which, via
Maxwell’s second law, does result is work being done on the charge as H increases
from zero to its final value. That work is the energy acquired by the magnetic dipole
moment associated with the charge’s orbital motion in the
magnetic field H. A complete classical solution yields, on sub-
stituting the electron’s charge e for the general charge q,
Df = { eH
4pmc
ZE-6
The classical solution “explains” the normal Zeeman effect,
which is exhibited by relatively few atoms, but gives no sug-
gestion for an explanation of the anomalous Zeeman effect,
which requires electron spin, unknown to Lorentz at the time,
for its explanation (see the More section The Zeeman Effect).
Measurements of the Zeeman effect have provided a
wealth of information on such topics ranging from atomic
structure to the magnetic fields of the Sun. However,
Zeeman’s original application of his discovery, using
Lorentz’s theoretical explanation and the known values of the
speed of light c and his external magnetic field H, was the
determination of the charge-to-mass ratio e>m for the elec-
tron. This was a spectroscopic measurement of the wave-
length (i.e., frequency) differences Dl between
the new spectral lines that appeared with the field “on” and
the original line l0 with the field “off.” His was the first such
measurement, preceding Thomson’s by about a year.
Zeeman’s measured value of e>m, about 1.6 * 1011 C>kg,
compares well with the currently accepted value of
1.759 * 1011 C>kg.
Weak H field
No H field
Weak H field
No H field
He singlet
(a)
(b)
D1 D2
Normal Zeeman
Effect
Na D doublet
Anomalous Zeeman
Effect
ZE-1 (a) The red line in the He singlet spectrum at
667.8 nm exhibits the normal Zeeman effect when the
excited He atoms are placed in a weak magnetic field.
(b) The yellow Na D-lines at 589.0 nm (D1) and
589.6 nm (D2) provide an example of the anomalous
Zeeman effect when the excit
Answer:
The Zeeman effect is the effect of splitting the energy levels of an atom when it is placed in an external magnetic field. The normal Zeeman effect: ... In this situation, the electron moving in a magnetic field experiences a Lorentz force that slightly changes the orbit of the electron and hence its energy.
Explanation:
The splitting between the two energy states is called electron Zeeman interaction (EZI) and is proportional to the magnitude of B0, as illustrated in Figure 1. The energy difference between the two Zeeman states is given by ΔE = E(mS = +1/2) - E(mS = -1/2) = geβeB0/h (in Hz).