UNIT IHYDROGEN 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 their resulting wavelengths for 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

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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

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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 "H-alpha", "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

6 1090

3

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 = r x p

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

. The direction of the momentum of the charges is changing in

a plane normal to the magnetic field. In fact the momentum of any charge is precessing

around the direction of the magnetic field by a frequency = . The magnetic moment 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 as:

. Hence the magnetic moment is related to the angular momentum by the

formula: , where is the gyromagnetic ratio defined as the ratio of the magnetic dipole moment to the mechanical angular momentum of a system.

For classical systems it equals . In quantum mechanics, the magnetic moment of

an electron due to its spin is related to the spin by a similar relation but the gyromagnetic ratio is 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: . Since this torque is what causes the change in the angular momentum of the dipole ,

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, then . That means that the frequency of precession of the magnetic

dipole .

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 . Where 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 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 ( ). 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 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 well-calibrated 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 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

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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.

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.

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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.

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

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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 equally-spaced triplet, in other atoms the magnetic field split the lines into four, six, or even more 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.

"Normal" Zeeman effect

This type of splitting is observed with hydrogen and the zinc singlet.

"Anomalous" Zeeman effect

When electron spin is included, there is a greater variety of splitting patterns.

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This type of splitting is observed for spin 0 states since the spin does not contribute to the angular momentum.

PASCHEN-BACK EFFECT

In the presence of an external magnetic field, the energy levels of atoms are split. This splitting is described well by the Zeeman effect if the splitting is small compared to the energy difference between the unperturbed levels, i.e., for sufficiently weak magnetic fields. This can be visualized with the help of a vector model of total angular momentum. If the magnetic field is large enough, it disrupts the coupling between the orbital and spin angular momenta, resulting in a different pattern of splitting. This effect is called the Paschen-Back effect.

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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.

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.

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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 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).

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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 topsA 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 topsA 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)

MICROWAVE SPECTROMETERThere are two types of microwave spectrometer in use. In the conventional Stark-modulated spectrometer, the sample is contained in a long (1- to 3-metre, or 3.3- to 9.8-foot) 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 Fourier-transformed to an absorption versus frequency spectrum. In both instruments the energy

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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 APPLICATIONSThe 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 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 the12CH2

81Br12C14N 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

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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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