Download Hydrogen spectral series

UNIT I
HYDROGEN SPECTRA
The spectral series of hydrogen, on a logarithmic scale.
The emission spectrum of atomic hydrogen is divided into a number of spectral series,
with wavelengths given by the Rydberg formula. These observed spectral lines are due to
electrons moving between energy levels in the atom. The spectral series are important in
astronomy for detecting the presence of hydrogen and calculating red shifts. Further
series were discovered as spectroscopy techniques developed.
Physics
In physics, the spectral lines of hydrogen correspond to particular jumps of
the electron between energy levels. The simplest model of the hydrogen atom is given by
the Bohr model. When an electron jumps from a higher energy to a lower, a photon of a
specific wavelength is emitted.
Electron transitions and
wavelengths for
their resulting
hydrogen.
The spectral lines are grouped into
series according to n'. Lines
are named sequentially starting from the longest wavelength/lowest frequency of the
series, using Greek letters within each series. For example, the 2 → 1 line is called
"Lyman-alpha" (Ly-α), while the7 → 3 line is called "Paschen-delta" (Pa-δ). Some
hydrogen spectral lines fall outside these series, such as the 21 cm line; these correspond
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to much rarer atomic events such as hyperfine transitions. The fine structure also results
in single spectral lines appearing as two or more closely grouped thinner lines, due to
relativistic corrections. Typically one can only observe these series from pure hydrogen
samples in a lab. Many of the lines are very faint and additional lines can be caused by
other elements (such as helium if using sunlight, or nitrogen in the air). Lines outside of
the visible spectrum typically cannot be seen in observations of sunlight, as the
atmosphere absorbs most infra-red and ultraviolet wavelengths.
Rydberg formula
The energy differences between levels in the Bohr model, and hence the wavelengths of
emitted/absorbed photons, is given by the Rydberg formula:
where n is the initial energy level, n′ is the final energy level, and R is the Rydberg
constant. Meaningful values are returned only when n is greater than n′ and the limit of
one over infinity is taken to be zero.
Series
All wavelengths are given to 3 significant figures.
Lyman series (n′ = 1)
n λ (nm)
2
122
3
103
4
97.2
5
94.9
6
93.7
91.1
2
The series is named after its discoverer, Theodore Lyman, who discovered the spectral
lines from 1906-1914. All the wavelengths in the Lyman series are in
the ultraviolet band.
Balmer series (n′ = 2)
n λ (nm)
3
656
4
486
5
434
6
410
7
397
365
Named after Johann Balmer, who discovered the Balmer formula, an empirical equation
to predict the Balmer series, in 1885. Balmer lines are historically referred to as "Halpha", "H-beta", "H-gamma" and so on, where H is the element hydrogen. Four of the
Balmer lines are in the technically "visible" part of the spectrum, with wavelengths
longer than 400 nm. Parts of the Balmer series can be seen in the solar spectrum. H-alpha
is an important line used in astronomy to detect the presence of hydrogen.
Paschen series (n′ = 3)
n λ (nm)
4
1870
5
1280
3
6
1090
7
1000
8
954
820
Named after the Austro-German physicist Friedrich Paschen who first observed them in
1908. The Paschen lines all lie in the infrared band.
Brackett series (n′ = 4)
n λ (nm)
5
4050
6
2630
7
2170
8
1940
9
1820
1460
Named after the American physicist Frederick Sumner Brackett who first observed the
spectral lines in 1922.
Pfund series (n′ = 5)
n λ (nm)
4
6
7460
7
4650
8
3740
9
3300
10 3040
2280
Experimentally discovered in 1924 by August Herman Pfund.
Humphreys series (n′ = 6)
n λ (nm)
7
12400
8
7500
9
5910
10 5130
11 4670
3280
Discovered by American physicist Curtis J. Humphreys.
Further (n′ > 6)
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Further series are unnamed, but follow exactly the same pattern as dictated by the
Rydberg equation. Series are increasingly spread out and occur in increasing
wavelengths. The lines are also increasingly faint, corresponding to increasingly rare
atomic events.
ANGULAR MOMENTUM
The angular momentum of a rigid object is defined as the product of the moment of
inertia and the angular velocity. It is analogous to linear momentum and is subject to the
fundamental constraints of the conservation of angular momentum principle if there is no
external torque on the object. Angular momentum is a vector quantity. It is derivable
from the expression for the angular momentum of a particle
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The angular momentum of a particle of mass m with respect to a chosen origin is given
by
L = mvr sin θ
or more formally by the vector product
L=rxp
The direction is given by the right hand rule which would give L the direction out of the
diagram. For an orbit, angular momentum is conserved, and this leads to one of Kepler's
laws. For a circular orbit, L becomes
L = mvr
LARMOR PRECESSION
When a pure static magnetic field is present and acting on a system of charges, the
Lorentz force of the field on any charge is always normal to the direction of the motion of
the charge, hence the magnetic field is not able to change the energy of any charge. In
other words, the magnitude of the momentum P is constant. In the nonrelativistic case

 
dP
q  
 q(v  B)  ( p  B) . The direction of the momentum of the charges is changing in
dt
m
a plane normal to the magnetic field. In fact the momentum of any charge is precessing
qB
around the direction of the magnetic field by a frequency =
. The magnetic moment
m
of a charge in this case will be in the direction of the magnetic field. The situation
becomes more complex when the charges have originally (before applying the magnetic
field) a net magnetic moment in some other direction maybe due to another source of
magnetic field or a permanent spin as in electrons. The applied magnetic field tends to
make the magnetic moment of the charge aligned in its direction while the angular
momentum associated with the magnetic moment of the charge tends to maintain its
original direction. The net effect is that the magnetic moment precesses around the
direction of the magnetic field. The situation is analogous to the motion of a spinning
gyroscope that undergoes a precession under the force of gravity when its rotational axis
initially is not vertical. The magnetic moment of a system of moving charges is defined
 
1
as: M  qi (ri  vi ) . Hence the magnetic moment is related to the angular momentum by
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the formula: M  L , where is the gyromagnetic ratio defined as the ratio of the
magnetic dipole moment to the mechanical angular momentum of a system. For classical
q
systems it equals  
. In quantum mechanics, the magnetic moment of an electron
2m
due to its spin is related to the spin by a similar relation but the gyromagnetic ratio is
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multiplied by a factor slightly larger than 2. Regardless of the value of  , the torque
exerted by the magnetic field on the magnetic dipole is expressed as: N  M  B . Since
dL
this torque is what causes the change in the angular momentum of the dipole , N 
,
dt
dL
then L  B 
. That means that the frequency of precession of the magnetic dipole
dt
  B .
The most important application of Larmor Precession is Nuclear Magnetic Resonance.
When the permanent magnetic dipoles of the nuclei are subjected to a very strong
magnetic field in a certain direction, the spin of the nuclei precesses around the magnetic
field in either one of two directions, parallel to the field or anti-parallel. These two
orientations are different in energy by 2 z B . Where  z is the projection of the magnetic
moment of the nucleus in the direction of the field. When external electromagnetic
radiation is applied, the anti-parallel nuclei can acquire a photon of energy f exactly
equal to the difference between the two states and change its state to the other direction
(spin flipping). After some time it can relax to its old state and emit a photon of the same
frequency. The value of the magnetic field in the expression is the magnitude of the sum
of the local weak field due to the magnetic moments of other nuclei and the strong
polarizing field ( B  Bext  Blocal ). Nuclei in different environments (different densities of
atoms around it) have different local fields and hence different resonance conditions.
Thus by changing the value of Bext and scanning across the energy of photons emitted
(same as energy of photons absorbed by the nuclei), the different environments of the
atoms can be detected. This is helpful in spectroscopy. For example, in CH3–CH2–OH the
Hydrogen nuclei have three different absorption peaks for the three different
environments, CH3, CH2, and OH. By irradiating the compound by a constant frequency
signal and sweeping the value of the external magnetic field gradually, three different
peaks will be detected in the radiated signal from the compound at different values of
Bext. This is called Magnetic Resonance Spectroscopy. As shown in figure (1), the
resonance conditions for a bare proton is different from the proton in different chemical
compounds. As implied by the figure, another possible scanning method is by fixing the
magnetic field and sweeping the frequency of the irradiation signal.
Figure (1)
Another important application is the Magnetic Resonance Imaging (MRI). In this
application “an image of a cross-section of tissue can be made by producing a wellcalibrated magnetic field gradient across the tissue so that a certain value of magnetic
field can be associated with a given location in the tissue. Since the proton signal
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frequency is proportional to that magnetic field, a given proton signal frequency can be
assigned to a location in the tissue. This provides the information to map the tissue in
terms of the protons present there. Since the proton density varies with the type of tissue,
a certain amount of contrast is achieved to image the organs and other tissue variations in
the subject tissue.” Different frequency components are discerned from the detected
signal by doing Fourier Transform to the radiated signal. A schematic diagram of this
process is shown in figure (2).
Figure (2)
ZEEMAN EFFECT
When an external magnetic field is applied, sharp spectral lines like the n=3→ 2transition
of hydrogen split into multiple closely spaced lines. First observed by Pieter Zeeman, this
splitting is attributed to the interaction between the magnetic field and the magnetic
dipole moment associated with the orbital angular momentum. In the absence of the
magnetic field, the hydrogen energies depend only upon the principal quantum number n,
and the emissions occur at a single wavelength.
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Note that the transitions shown follow the selection rule which does not allow a change
of more than one unit in the quantum number ml.
Zeeman Interaction
An external magnetic field will exert a torque on a magnetic dipole and the magnetic
potential energy which results is
The magnetic dipole moment associated with the orbital angular momentum is given by
For magnetic field in the z-direction this gives
Considering the quantization of angular momentum, this gives equally spaced energy
levels displaced from the zero field level by
This displacement of the energy levels gives the uniformly spaced multiplet splitting of
the spectral lines which is called the Zeeman effect.
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The magnetic field also interacts with the electron spin magnetic moment, so it
contributes to the Zeeman effect in many cases. The electron spin had not been
discovered at the time of Zeeman's original experiments, so the cases where it contributed
were considered to be anomalous. The term "anomalous Zeeman effect" has persisted for
the cases where spin contributes. In general, both orbital and spin moments are involved,
and the Zeeman interaction takes the form
The factor of two multiplying the electron spin angular momentum comes from the fact
that it is twice as effective in producing magnetic moment. This factor is called the spin
g-factor or gyromagnetic ratio. The evaluation of the scalar product between the angular
momenta and the magnetic field here is complicated by the fact that the S and L vectors
are both precessing around the magnetic field and are not in general in the same direction.
The persistent early spectroscopists worked out a way to calculate the effect of the
directions. The resulting geometric factor gL in the final expression above is called
the Lande g factor. It allowed them to express the resultant splittings of the spectral lines
in terms of the z-component of the total angular momentum, mj.
The above treatment of the Zeeman effect describes the phenomenon when the magnetic
fields are small enough that the orbital and spin angular momenta can be considered to be
coupled. For extremely strong magnetic fields this coupling is broken and another
approach must be taken. The strong field effect is called the Paschen-Back effect.
Hydrogen Zeeman Example
The Zeeman effect for the hydrogen atom offered experimental support for the
quantization of angular momentum which arose from the solution of the Schrodinger
equation.
While the Zeeman effect in some atoms (e.g., hydrogen) showed the expected equallyspaced triplet, in other atoms the magnetic field split the lines into four, six, or even more
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lines and some triplets showed wider spacings than expected. These deviations were
labeled the "anomalous Zeeman effect" and were very puzzling to early researchers. The
explanation of these different patterns of splitting gave additional insight into the effects
of electron spin. With the inclusion of electron spin in the total angular momentum, the
other types of multiplets formed part of a consistent picture. So what has been historically
called the "anomalous" Zeeman effect is really the normal Zeeman effect when electron
spin is included.
"Anomalous" Zeeman effect
"Normal" Zeeman effect
When electron spin is included, there is a greater
This type of splitting is observed variety of splitting patterns.
with hydrogen and the zinc singlet.
This type of splitting is observed for
spin 0 states since the spin does not
contribute to the angular momentum.
PASCHENIn the presence of an
field, the energy levels
This splitting is described
the splitting is small
between the unperturbed
magnetic fields. This can be
model of
total
angular
enough, it disrupts the coupling
momenta, resulting in a different pattern
Paschen-Back effect.
BACK EFFECT
external
magnetic
of atoms are split.
well by the Zeeman effect if
compared to the energy difference
levels, i.e., for sufficiently weak
visualized with the help of a vector
momentum. If the magnetic field is large
between the orbital and spin angular
of splitting. This effect is called the
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In the weak field case the vector model at left implies that the coupling of the orbital
angular momentum L to the spin angular momentum S is stronger than their coupling to
the external field. In this case where spin-orbit coupling is dominant, they can be
visualized as combining to form a total angular momentum J which then precesses about
the magnetic field direction.
In the strong-field case, S and L couple more strongly to the external magnetic field than
to each other, and can be visualized as independently precessing about the external field
direction.
For reference, the sodium Zeeman effect is reproduced below to show the nature of the
magnetic interaction for weak external magnetic fields.
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The following is a model of the changes in the pattern if the magnetic field were strong
enough to decouple L and S. The resulting spectrum would be a triplet with the center
line twice the intensity of the outer lines.
To create this pattern, the projections of L and S in the z-direction have been treated
independently and the ms multiplied by the spin g-factor. The energy shift is expressed as
a multiple of the Bohr magneton mB. The selection rules explain why the transitions
shown are allowed and others not.
Sodium was used as the basis of the model for convenience, but the fields required to
create Paschen-Back conditions for sodium are unrealistically high. Lithium, on the other
hand, has a spin-orbit splitting of only 0.00004 eV compared to 0.0021 eV for sodium.
Such small energy values are sometimes expressed in "wavenumbers", or 1/l in cm-1. In
these units the lithium separation is about 0.3 cm-1and the sodium separation is about 17
cm-1 .The Paschen-Back conditions are met in some lithium spectra observed on the Sun,
so this effect does have astronomical significance.
STARK EFFECT
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The splitting of atomic spectral lines as a result of an externally applied electric field was
discovered by Stark, and is called the Stark effect. As the splitting of a line of the helium
spectrum shows, the splitting is not symmetric like that of the Zeeman effect.
The splitting of the energy levels by an electric field first requires that the field polarize
the atom and then interact with the resulting electric dipole moment. That dipole moment
depends upon the magnitude of Mj, but not its sign, so that the energy levels are split into
J+1 or J+1/2 levels, for integer and half-integer spins respectively.
The Stark effect has been of marginal benefit in the analysis of atomic spectra, but has
been a major tool for molecular rotational spectra.
CHARACTERISTIC X-RAY SPECTRUM
An X is a very high energy photon with a wavelength in the range of 0.01 to 10
nanometers, and is also referred to as a roentgen ray. A stream of such photons is called
an X-ray or X-ray beam, and has remarkably high penetration power in radiography,
radiology, radiotherapy, and scientific research.
X-rays, like other wave-based things, are defined by the range of their wavelength
spectrum. When any substance is used in an X-ray tube, a set of characteristic X-ray
frequencies or wavelengths are produced. All elements have a characteristic X-ray
spectrum produced from this activity, and there is a very strong correlation between
atomic number and the frequency of certain lines in the X-ray spectrum. Through
processes like this, the X-ray spectrum can be used to identify elements in a specific
sample.
Typically, the X-ray spectrum will be measured using the x-ray diffraction pattern of the
elements involved. A diffractometer records the diffraction thus created, and a scientist
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can interpret the resulting data to match it to known elements or substances, or to
determine that the substance being tested is something unfamiliar. X-ray spectrum
analysis allows for simple identification of substances without destroying samples.
LINEAR, SYMMETRIC AND ASYMMETRIC TOP MOLECULES
Linear molecules
As mentioned earlier, for a linear molecule
. For most of the
purposes, IA is taken to be zero. For a linear molecule, the separation of lines in the
rotational spectrum can be related directly to the moment of inertia of the molecule, and,
for a molecule of known atomic masses, can be used to determine the bond
lengths (structure) directly. For diatomic molecules, this process is trivial, and can be
made from a single measurement of the rotational spectrum. For linear molecules with
more atoms, rather more work is required, and it is necessary to measure molecules in
which more than one isotope of each atom have been substituted (effectively this gives
rise to a set of simultaneous equations that can be solved for the bond lengths).
Examples of linear molecules: dioxygen (O=O), carbon monoxide (O≡C*), hydroxy
radical (OH), carbon dioxide (O=C=O), hydrogen cyanide (HC≡N), carbonyl sulfide
(O=C=S), chloroethyne (HC≡CCl), acetylene (HC≡CH)
Symmetric tops
A symmetric top is a molecule in which two moments of inertia are the same. As a matter
of historical convenience, spectroscopists divide molecules into two classes of symmetric
tops, Oblate symmetric
tops (saucer
or
disc
shaped)
with IA = IB < IC and Prolate symmetric tops(rugby football, or cigar shaped)
with IA < IB = IC. The spectra look rather different, and are instantly recognizable. As for
linear molecules, the structure of symmetric tops (bond lengths and bond angles) can be
deduced from their spectra.
Examples of symmetric tops:
Oblate: benzene (C6H6), cyclobutadiene (C4H4), ammonia (NH3)
Prolate: chloromethane (CH3Cl), propyne (CH3C≡CH)
Asymmetric tops
A molecule is termed an asymmetric top if all three moments of inertia are different.
Most of the larger molecules are asymmetric tops, even when they have a high degree of
symmetry. In general, for such molecules, a simple interpretation of the spectrum is not
normally possible. Sometimes asymmetric tops have spectra that are similar to those of a
linear molecule or a symmetric top, in which case the molecular structure must also be
similar to that of a linear molecule or a symmetric top. For the most general case,
however, all that can be done is to fit the spectra to three different moments of inertia. If
the molecular formula is known, then educated guesses can be made of the possible
structure, and, from this guessed structure, the moments of inertia can be calculated. If
the calculated moments of inertia agree well with the measured moments of inertia, then
the structure can be said to have been determined. For this approach to determining
molecular structure, isotopic substitution is invaluable.
Examples of asymmetric tops: anthracene (C14H10), water (H2O), nitrogen dioxide (NO2)
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MICROWAVE SPECTROMETER
There are two types of microwave spectrometer in use. In the conventional Starkmodulated spectrometer, the sample is contained in a long (1- to 3-metre, or 3.3- to 9.8foot) section of a rectangular waveguide, sealed at each end with a microwave
transmitting window (e.g., mica or Mylar), and connected to a vacuum line for
evacuation and sample introduction. The radiation from the source passes through a
gaseous sample and is detected by a crystal diode detector that is followed by an
amplifier and display system (chart recorder). In order to increase the sensitivity of the
instrument, signal modulation by application of a high-voltage square wave across the
sample is used. The second type is the Fourier-transform spectrometer, in which the
radiation is confined in an evacuated cavity between a pair of spherical mirrors and the
sample is introduced by a pulsed nozzle that lowers the temperature of the sample to less
than 10 K. The sample is subjected to rotational energy excitation by application of a
pulsed microwave signal, and the resulting emission signal is detected and Fouriertransformed to an absorption versus frequency spectrum. In both instruments the energy
absorbed or emitted as the molecules undergo transitions from one quantized rotational
state to another is observed. The Fourier-transform instrument has the advantage of
providing higher resolution (1 kilohertz [kHz] relative to 30 kHz) and of exhibiting a
much simpler spectrum due to the low sample temperature that insures that the majority
of the molecules are in the few lowest energy states.
For observation of its rotational spectrum, a molecule must possess a permanent electric
dipole moment and have a vapour pressure such that it can be introduced into a sample
cell at extremely low pressures (5–50 millitorr; one millitorr equals’ 1 × 10−3 millimetre
of mercury or 1.93 × 10−5 pound per square inch). The spectra of molecules with
structures containing up to 15 atoms can be routinely analyzed, but the density and
overlapping of spectral lines in the spectra of larger molecules severely restricts analysis.
MOLECULAR APPLICATIONS
The relationship between the observed microwave transition frequency and the rotational
constant of a diatomic molecule can provide a value for the internuclear distance. The
quantitative geometric structures of molecules can also be obtained from the measured
transitions in its microwave spectrum. In addition to geometric structures, other
properties related to molecular structure can be investigated, including electric dipole
moments, energy barriers to internal rotation, centrifugal distortion parameters, magnetic
moments, nuclear electric quadrupole moments, vibration-rotation interaction parameters,
low-frequency vibrational transitions, molecular electric quadrupole moments, and
information relative to electron distribution and bonding. Microwave spectroscopy has
provided the detailed structure and associated parameters for several thousand molecules.
The use of Fourier-transform spectrometers has provided a method for studying many
short-lived species such as free radicals (i.e., OH, CN, NO, CF, CCH), molecular ions
(i.e., CO+, HCO+, HCS+), and Van der Waals complexes (i.e., C6H6−HCl, H2O−H2O,
Kr−HF, SO2−SO2). There is a special relationship between microwave spectroscopy
and radio astronomy. Much of the impetus for the investigation of the microwave spectra
of radical and molecular ions stems from the need for identifying the microwave
emission signals emanating from extraterrestrial sources. This collaboration has resulted
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in the identification in outer space of several dozen species, including the hydroxyl
radical, methanol, formaldehyde, ammonia, and methyl cyanide.
For a polyatomic molecule, which is characterized by three moments of inertia, the
microwave spectrum of a single molecular species provides insufficient information for
making a complete structure assignment and calculating the magnitude of all bond angles
and interatomic distances in the molecule. For example, the values of the three moments
of inertia of the12CH281Br12C14N molecule will depend on eight bond parameters (four
angles and four distances), hence it is not possible to obtain discrete values of these eight
unknowns from three moments. This problem can be circumvented by introducing the
assumption that the structure of the molecule will not significantly change if one or more
atoms are substituted with a different isotopic species. The three moments of an
isotopically substituted molecule are then derived from its microwave spectrum and,
since they depend on the same set of molecular parameters, provide three additional
pieces of data from which to obtain the eight bond parameters. By determining the
moments of inertia of a sufficient number of isotopically substituted species, it is possible
to obtain sufficient data from which to completely determine the structure. The best
structural information is obtained when an isotopic species resulting from substitution at
each atom site in the molecule can be studied.
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