Stimulated emission

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Laser light is a type of stimulated emission of radiation.

Stimulated emission is the process by which an incoming photon of a specific frequency can interact with an excited atomic electron (or other excited molecular state), causing it to drop to a lower energy level. The liberated energy transfers to the electromagnetic field, creating a new photon with a frequency, polarization, and direction of travel that are all identical to the photons of the incident wave. This is in contrast to spontaneous emission, which occurs at a characteristic rate for each of the atoms/oscillators in the upper energy state regardless of the external electromagnetic field.

According to the American Physical Society, the first person to correctly predict the phenomenon of stimulated emission was Albert Einstein in a series of papers starting in 1916, culminating in what is now called the Einstein B Coefficient. Einstein's work became the theoretical foundation of the maser and the laser. The process is identical in form to atomic absorption in which the energy of an absorbed photon causes an identical but opposite atomic transition: from the lower level to a higher energy level. In normal media at thermal equilibrium, absorption exceeds stimulated emission because there are more electrons in the lower energy states than in the higher energy states. However, when a population inversion is present, the rate of stimulated emission exceeds that of absorption, and a net optical amplification can be achieved. Such a gain medium, along with an optical resonator, is at the heart of a laser or maser. Lacking a feedback mechanism, laser amplifiers and superluminescent sources also function on the basis of stimulated emission.

Overview

Electrons and their interactions with electromagnetic fields are important in our understanding of chemistry and physics. In the classical view, the energy of an electron orbiting an atomic nucleus is larger for orbits further from the nucleus of an atom. However, quantum mechanical effects force electrons to take on discrete positions in orbitals. Thus, electrons are found in specific energy levels of an atom, two of which are shown below:

When an electron absorbs energy either from light (photons) or heat (phonons), it receives that incident quantum of energy. But transitions are only allowed between discrete energy levels such as the two shown above. This leads to emission lines and absorption lines.

When an electron is excited from a lower to a higher energy level, it is unlikely for it to stay that way forever. An electron in an excited state may decay to a lower energy state which is not occupied, according to a particular time constant characterizing that transition. When such an electron decays without external influence, emitting a photon, that is called "spontaneous emission". The phase and direction associated with the photon that is emitted is random. A material with many atoms in such an excited state may thus result in radiation which has a narrow spectrum (centered around one wavelength of light), but the individual photons would have no common phase relationship and would also emanate in random directions. This is the mechanism of fluorescence and thermal emission.

An external electromagnetic field at a frequency associated with a transition can affect the quantum mechanical state of the atom without being absorbed. As the electron in the atom makes a transition between two stationary states (neither of which shows a dipole field), it enters a transition state which does have a dipole field, and which acts like a small electric dipole, and this dipole oscillates at a characteristic frequency. In response to the external electric field at this frequency, the probability of the electron entering this transition state is greatly increased. Thus, the rate of transitions between two stationary states is increased beyond that of spontaneous emission. A transition from the higher to a lower energy state produces an additional photon with the same phase and direction as the incident photon; this is the process of stimulated emission.

History

Stimulated emission was a theoretical discovery by Albert Einstein within the framework of the old quantum theory, wherein the emission is described in terms of photons that are the quanta of the EM field. Stimulated emission can also occur in classical models, without reference to photons or quantum-mechanics. (See also Laser § History.) According to physics professor and director of the MIT-Harvard Center for Ultracold Atoms Daniel Kleppner, Einstein's theory of radiation was ahead of its time and prefigures the modern theory of quantum electrodynamics and quantum optics by several decades.

Mathematical model

Stimulated emission can be modelled mathematically by considering an atom that may be in one of two electronic energy states, a lower level state (possibly the ground state) (1) and an excited state (2), with energies E₁ and E₂ respectively.

If the atom is in the excited state, it may decay into the lower state by the process of spontaneous emission, releasing the difference in energies between the two states as a photon. The photon will have frequency ν₀ and energy hν₀, given by: $${\displaystyle E_2-E_1=h{\nu }_0}$$ where h is the Planck constant.

Alternatively, if the excited-state atom is perturbed by an electric field of frequency ν₀, it may emit an additional photon of the same frequency and in phase, thus augmenting the external field, leaving the atom in the lower energy state. This process is known as stimulated emission.

In a group of such atoms, if the number of atoms in the excited state is given by N₂, the rate at which stimulated emission occurs is given by $${\displaystyle \frac{\mathrm{\partial }{N}_2}{\mathrm{\partial }t}=-\frac{\mathrm{\partial }{N}_1}{\mathrm{\partial }t}=-B_{21}\rho (\nu )N_2}$$ where the proportionality constant B₂₁ is known as the Einstein B coefficient for that particular transition, and ρ(ν) is the radiation density of the incident field at frequency ν. The rate of emission is thus proportional to the number of atoms in the excited state N₂, and to the density of incident photons.

At the same time, there will be a process of atomic absorption which removes energy from the field while raising electrons from the lower state to the upper state. Its rate is precisely the negative of the stimulated emission rate, $${\displaystyle \frac{\mathrm{\partial }{N}_2}{\mathrm{\partial }t}=-\frac{\mathrm{\partial }{N}_1}{\mathrm{\partial }t}=B_{12}\rho (\nu )N_1.}$$

The rate of absorption is thus proportional to the number of atoms in the lower state, N₁. The B coefficients can be calculated using dipole approximation and time dependent perturbation theory in quantum mechanics as: $${\displaystyle B_{ab}=\frac{e^2}{6{ϵ}_0{\hslash }^2}|⟨a|\overrightarrow{r}|b⟩{|}^2}$$ where B corresponds to energy distribution in terms of frequency ν. The B coefficient may vary based on choice of energy distribution function used, however, the product of energy distribution function and its respective B coefficient remains same.

Einstein showed from the form of Planck's law, that the coefficient for this transition must be identical to that for stimulated emission: $${\displaystyle B_{12}=B_{21}.}$$

Thus absorption and stimulated emission are reverse processes proceeding at somewhat different rates. Another way of viewing this is to look at the net stimulated emission or absorption viewing it as a single process. The net rate of transitions from E₂ to E₁ due to this combined process can be found by adding their respective rates, given above: $${\displaystyle \frac{\mathrm{\partial }{N}_1^{\text{net}}}{\mathrm{\partial }t}=-\frac{\mathrm{\partial }{N}_2^{\text{net}}}{\mathrm{\partial }t}=B_{21}\rho (\nu )(N_2-N_1)=B_{21}\rho (\nu )\mathrm{\Delta }N.}$$

Thus a net power is released into the electric field equal to the photon energy hν times this net transition rate. In order for this to be a positive number, indicating net stimulated emission, there must be more atoms in the excited state than in the lower level: ${\displaystyle \mathrm{\Delta }N>0}$. Otherwise there is net absorption and the power of the wave is reduced during passage through the medium. The special condition ${\displaystyle N_2>N_1}$ is known as a population inversion, a rather unusual condition that must be effected in the gain medium of a laser.

The notable characteristic of stimulated emission compared to everyday light sources (which depend on spontaneous emission) is that the emitted photons have the same frequency, phase, polarization, and direction of propagation as the incident photons. The photons involved are thus mutually coherent. When a population inversion (${\displaystyle \mathrm{\Delta }N>0}$) is present, therefore, optical amplification of incident radiation will take place.

Although energy generated by stimulated emission is always at the exact frequency of the field which has stimulated it, the above rate equation refers only to excitation at the particular optical frequency ${\displaystyle {\nu }_0}$ corresponding to the energy of the transition. At frequencies offset from ${\displaystyle {\nu }_0}$ the strength of stimulated (or spontaneous) emission will be decreased according to the so-called line shape. Considering only homogeneous broadening affecting an atomic or molecular resonance, the spectral line shape function is described as a Lorentzian distribution $${\displaystyle g^{\prime }(\nu )=\frac{1}{\pi }\frac{(\mathrm{\Gamma }/2)}{(\nu -{\nu }_0{)}^2+(\mathrm{\Gamma }/2{)}^2}}$$ where ${\displaystyle \mathrm{\Gamma }}$ is the full width at half maximum or FWHM bandwidth.

The peak value of the Lorentzian line shape occurs at the line center, ${\displaystyle \nu ={\nu }_0}$. A line shape function can be normalized so that its value at ${\displaystyle {\nu }_0}$ is unity; in the case of a Lorentzian we obtain $${\displaystyle g(\nu )=\frac{g^{\prime }(\nu )}{g^{\prime }({\nu }_0)}=\frac{(\mathrm{\Gamma }/2{)}^2}{(\nu -{\nu }_0{)}^2+(\mathrm{\Gamma }/2{)}^2}.}$$

Thus stimulated emission at frequencies away from ${\displaystyle {\nu }_0}$ is reduced by this factor. In practice there may also be broadening of the line shape due to inhomogeneous broadening, most notably due to the Doppler effect resulting from the distribution of velocities in a gas at a certain temperature. This has a Gaussian shape and reduces the peak strength of the line shape function. In a practical problem the full line shape function can be computed through a convolution of the individual line shape functions involved. Therefore, optical amplification will add power to an incident optical field at frequency ${\displaystyle \nu }$ at a rate given by $${\displaystyle P=h\nu g(\nu )B_{21}\rho (\nu )\mathrm{\Delta }N.}$$

Stimulated emission cross section

The stimulated emission cross section is $${\displaystyle {\sigma }_{21}(\nu )=A_{21}\frac{{\lambda }^2}{8\pi n^2}{g}^{\prime }(\nu )}$$ where

• A₂₁ is the Einstein A coefficient,
• λ is the wavelength in vacuum,
• n is the refractive index of the medium (dimensionless), and
• g′(ν) is the spectral line shape function.

Optical amplification

Stimulated emission can provide a physical mechanism for optical amplification. If an external source of energy stimulates more than 50% of the atoms in the ground state to transition into the excited state, then what is called a population inversion is created. When light of the appropriate frequency passes through the inverted medium, the photons are either absorbed by the atoms that remain in the ground state or the photons stimulate the excited atoms to emit additional photons of the same frequency, phase, and direction. Since more atoms are in the excited state than in the ground state then an amplification of the input intensity results.

The population inversion, in units of atoms per cubic metre, is

${\displaystyle \mathrm{\Delta }{N}_{21}=N_2-\frac{g_2}{g_1}{N}_1}$

where g₁ and g₂ are the degeneracies of energy levels 1 and 2, respectively.

Small signal gain equation

The intensity (in watts per square metre) of the stimulated emission is governed by the following differential equation:

${\displaystyle \frac{dI}{dz}={\sigma }_{21}(\nu )\cdot \mathrm{\Delta }{N}_{21}\cdot I(z)}$

as long as the intensity I(z) is small enough so that it does not have a significant effect on the magnitude of the population inversion. Grouping the first two factors together, this equation simplifies as

${\displaystyle \frac{dI}{dz}={\gamma }_0(\nu )\cdot I(z)}$

where

${\displaystyle {\gamma }_0(\nu )={\sigma }_{21}(\nu )\cdot \mathrm{\Delta }{N}_{21}}$

is the small-signal gain coefficient (in units of radians per metre). We can solve the differential equation using separation of variables:

${\displaystyle \frac{dI}{I(z)}={\gamma }_0(\nu )\cdot dz}$

Integrating, we find:

${\displaystyle \ln \left(\frac{I(z)}{I_{in}}\right)={\gamma }_0(\nu )\cdot z}$

or

${\displaystyle I(z)=I_{in}{e}^{{\gamma }_0(\nu )z}}$

where

${\displaystyle I_{in}=I(z=0)}$ is the optical intensity of the input signal (in watts per square metre).

Saturation intensity

The saturation intensity I_S is defined as the input intensity at which the gain of the optical amplifier drops to exactly half of the small-signal gain. We can compute the saturation intensity as

${\displaystyle I_S=\frac{h\nu }{\sigma (\nu )\cdot {\tau }_S}}$

where

${\displaystyle h}$ is the Planck constant, and

${\displaystyle {\tau }_{\text{S}}}$ is the saturation time constant, which depends on the spontaneous emission lifetimes of the various transitions between the energy levels related to the amplification.

${\displaystyle \nu }$ is the frequency in Hz

The minimum value of ${\displaystyle I_{\text{S}}(\nu )}$ occurs on resonance, where the cross section ${\displaystyle \sigma (\nu )}$ is the largest. This minimum value is:

${\displaystyle I_{\text{sat}}=\frac{\pi }{3}\frac{hc}{{\lambda }^3{\tau }_S}}$

For a simple two-level atom with a natural linewidth ${\displaystyle \mathrm{\Gamma }}$, the saturation time constant ${\displaystyle {\tau }_{\text{S}}={\mathrm{\Gamma }}^{-1}}$.

General gain equation

The general form of the gain equation, which applies regardless of the input intensity, derives from the general differential equation for the intensity I as a function of position z in the gain medium:

${\displaystyle \frac{dI}{dz}=\frac{{\gamma }_0(\nu )}{1+\overline{g}(\nu )\frac{I(z)}{I_S}}\cdot I(z)}$

where ${\displaystyle I_S}$ is saturation intensity. To solve, we first rearrange the equation in order to separate the variables, intensity I and position z:

${\displaystyle \frac{dI}{I(z)}\left[1+\overline{g}(\nu )\frac{I(z)}{I_S}\right]={\gamma }_0(\nu )\cdot dz}$

Integrating both sides, we obtain

${\displaystyle \ln \left(\frac{I(z)}{I_{in}}\right)+\overline{g}(\nu )\frac{I(z)-I_{in}}{I_S}={\gamma }_0(\nu )\cdot z}$

or

${\displaystyle \ln \left(\frac{I(z)}{I_{in}}\right)+\overline{g}(\nu )\frac{I_{in}}{I_S}\left(\frac{I(z)}{I_{in}}-1\right)={\gamma }_0(\nu )\cdot z}$

The gain G of the amplifier is defined as the optical intensity I at position z divided by the input intensity:

${\displaystyle G=G(z)=\frac{I(z)}{I_{in}}}$

Substituting this definition into the prior equation, we find the general gain equation:

${\displaystyle \ln \left(G\right)+\overline{g}(\nu )\frac{I_{in}}{I_S}\left(G-1\right)={\gamma }_0(\nu )\cdot z}$

Small signal approximation

In the special case where the input signal is small compared to the saturation intensity, in other words,

${\displaystyle I_{in}\ll I_S}$

then the general gain equation gives the small signal gain as

${\displaystyle \ln (G)=\ln (G_0)={\gamma }_0(\nu )\cdot z}$

or

${\displaystyle G=G_0=e^{{\gamma }_0(\nu )z}}$

which is identical to the small signal gain equation (see above).

Large signal asymptotic behaviour

For large input signals, where

${\displaystyle I_{in}\gg I_S}$

the gain approaches unity

${\displaystyle G\to 1}$

and the general gain equation approaches a linear asymptote:

${\displaystyle I(z)=I_{in}+\frac{{\gamma }_0(\nu )\cdot z}{\overline{g}(\nu )}{I}_S}$

Structural isomer

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Structural_isomer 

In chemistry, a structural isomer (or constitutional isomer in the IUPAC nomenclature) of a compound is a compound that contains the same number and type of atoms, but with a different connectivity (i.e. arrangement of bonds) between them. The term metamer was formerly used for the same concept.

For example, butanol H₃C−(CH₂)₃−OH, methyl propyl ether H₃C−(CH₂)₂−O−CH₃, and diethyl ether (H₃CCH₂−)₂O have the same molecular formula C₄H₁₀O but are three distinct structural isomers.

The concept applies also to polyatomic ions with the same total charge. A classical example is the cyanate ion O=C=N⁻ and the fulminate ion C⁻≡N⁺−O⁻. It is also extended to ionic compounds, so that (for example) ammonium cyanate [NH₄]⁺[O=C=N]⁻ and urea (H₂N−)₂C=O are considered structural isomers, and so are methylammonium formate [H₃C−NH₃]⁺[HCO₂]⁻ and ammonium acetate [NH₄]⁺[H₃C−CO₂]⁻.

Structural isomerism is the most radical type of isomerism. It is opposed to stereoisomerism, in which the atoms and bonding scheme are the same, but only the relative spatial arrangement of the atoms is different.^[5][6] Examples of the latter are the enantiomers, whose molecules are mirror images of each other, and the cis and trans versions of 2-butene.

Among the structural isomers, one can distinguish several classes including skeletal isomers, positional isomers (or regioisomers), functional isomers, tautomers, and structural isotopomers.

Skeletal isomerism

A skeletal isomer of a compound is a structural isomer that differs from it in the atoms and bonds that are considered to comprise the "skeleton" of the molecule. For organic compounds, such as alkanes, that usually means the carbon atoms and the bonds between them.

For example, there are three skeletal isomers of pentane: n-pentane (often called simply "pentane"), isopentane (2-methylbutane) and neopentane (dimethylpropane).

Skeletal isomers of pentane

n-Pentane	Isopentane	Neopentane

If the skeleton is acyclic, as in the above example, one may use the term chain isomerism.

Position isomerism (regioisomerism)

See also: Arene substitution pattern § Ortho, meta, and para substitution

Position isomers (also positional isomers or regioisomers) are structural isomers that can be viewed as differing only on the position of a functional group, substituent, or some other feature on the same "parent" structure.

For example, replacing one of the 12 hydrogen atoms –H by a hydroxyl group –OH on the n-pentane parent molecule can give any of three different position isomers:

Pentan-1-ol	Pentan-2-ol	Pentan-3-ol

Another example of regioisomers are α-linolenic and γ-linolenic acids, both octadecatrienoic acids, each of which has three double bonds, but on different positions along the chain.

Functional isomerism

Functional isomers are structural isomers which have different functional groups, resulting in significantly different chemical and physical properties.

An example is the pair propanal H₃C–CH₂–C(=O)-H and acetone H₃C–C(=O)–CH₃: the first has a –C(=O)H functional group, which makes it an aldehyde, whereas the second has a C–C(=O)–C group, that makes it a ketone.

Another example is the pair ethanol H₃C–CH₂–OH (an alcohol) and dimethyl ether H₃C–O–CH₂H (an ether). In contrast, 1-propanol and 2-propanol are structural isomers, but not functional isomers, since they have the same significant functional group (the hydroxyl –OH) and are both alcohols.

Besides the different chemistry, functional isomers typically have very different infrared spectra. The infrared spectrum is largely determined by the vibration modes of the molecule, and functional groups like hydroxyl and esters have very different vibration modes. Thus 1-propanol and 2-propanol have relatively similar infrared spectra because of the hydroxyl group, which are fairly different from that of methyl ethyl ether.

Structural isotopomers

Main article: Isotopomer

In chemistry, one usually ignores distinctions between isotopes of the same element. However, in some situations (for instance in Raman, NMR, or microwave spectroscopy) one may treat different isotopes of the same element as different elements. In the second case, two molecules with the same number of atoms of each isotope but distinct bonding schemes are said to be structural isotopomers.

Thus, for example, ethene would have no structural isomers under the first interpretation; but replacing two of the hydrogen atoms (¹H) by deuterium atoms (²H) may yield any of two structural isotopomers (1,1-dideuteroethene and 1,2-dideuteroethene), if both carbon atoms are the same isotope. If, in addition, the two carbons are different isotopes (say, ¹²C and ¹³C), there would be three distinct structural isotopomers, since 1-¹³C-1,1-dideuteroethene would be different from 1-¹³C-2,2-dideuteroethene. And, in both cases, the 1,2-dideutero structural isotopomer would occur as two stereoisotopomers, cis and trans.

Structural equivalence and symmetry

Structural equivalence

Two molecules (including polyatomic ions) A and B have the same structure if each atom of A can be paired with an atom of B of the same element, in a one-to-one way, so that for every bond in A there is a bond in B, of the same type, between corresponding atoms; and vice versa. This requirement applies also to complex bonds that involve three or more atoms, such as the delocalized bonding in the benzene molecule and other aromatic compounds.

Depending on the context, one may require that each atom be paired with an atom of the same isotope, not just of the same element.

Two molecules then can be said to be structural isomers (or, if isotopes matter, structural isotopomers) if they have the same molecular formula but do not have the same structure.

Structural symmetry and equivalent atoms

Structural symmetry of a molecule can be defined mathematically as a permutation of the atoms that exchanges at least two atoms but does not change the molecule's structure. Two atoms then can be said to be structurally equivalent if there is a structural symmetry that takes one to the other.

Thus, for example, all four hydrogen atoms of methane are structurally equivalent, because any permutation of them will preserve all the bonds of the molecule.

Likewise, all six hydrogens of ethane (C
₂H
₆) are structurally equivalent to each other, as are the two carbons; because any hydrogen can be switched with any other, either by a permutation that swaps just those two atoms, or by a permutation that swaps the two carbons and each hydrogen in one methyl group with a different hydrogen on the other methyl. Either operation preserves the structure of the molecule. That is the case also for the hydrogen atoms in cyclopentane, allene, 2-butyne, hexamethylenetetramine, prismane, cubane, dodecahedrane, etc.

On the other hand, the hydrogen atoms of propane are not all structurally equivalent. The six hydrogens attached to the first and third carbons are equivalent, as in ethane, and the two attached to the middle carbon are equivalent to each other; but there is no equivalence between these two equivalence classes.

Symmetry and positional isomerism

Structural equivalences between atoms of a parent molecule reduce the number of positional isomers that can be obtained by replacing those atoms for a different element or group. Thus, for example, the structural equivalence between the six hydrogens of ethane C
₂H
₆ means that there is just one structural isomer of ethanol C
₂H
₅OH, not 6. The eight hydrogens of propane C
₃H
₈ are partitioned into two structural equivalence classes (the six on the methyl groups, and the two on the central carbon); therefore there are only two positional isomers of propanol (1-propanol and 2-propanol). Likewise there are only two positional isomers of butanol, and three of pentanol or hexanol.

Symmetry breaking by substitutions

Once a substitution is made on a parent molecule, its structural symmetry is usually reduced, meaning that atoms that were formerly equivalent may no longer be so. Thus substitution of two or more equivalent atoms by the same element may generate more than one positional isomer.

The classical example is the derivatives of benzene. Its six hydrogens are all structurally equivalent, and so are the six carbons; because the structure is not changed if the atoms are permuted in ways that correspond to flipping the molecule over or rotating it by multiples of 60 degrees. Therefore, replacing any hydrogen by chlorine yields only one chlorobenzene. However, with that replacement, the atom permutations that moved that hydrogen are no longer valid. Only one permutation remains, that corresponds to flipping the molecule over while keeping the chlorine fixed. The five remaining hydrogens then fall into three different equivalence classes: the one opposite to the chlorine is a class by itself (called the para position), the two closest to the chlorine form another class (ortho), and the remaining two are the third class (meta). Thus a second substitution of hydrogen by chlorine can yield three positional isomers: 1,2- or ortho-, 1,3- or meta-, and 1,4- or para-dichlorobenzene.

ortho-Dichlorobenzene	meta-Dichlorobenzene	para-Dichlorobenzene
1,2-Dichlorobenzene	1,3-Dichlorobenzene	1,4-Dichlorobenzene

For the same reason, there is only one phenol (hydroxybenzene), but three benzenediols; and one toluene (methylbenzene), but three toluols, and three xylenes.

On the other hand, the second replacement (by the same substituent) may preserve or even increase the symmetry of the molecule, and thus may preserve or reduce the number of equivalence classes for the next replacement. Thus, the four remaining hydrogens in meta-dichlorobenzene still fall into three classes, while those of ortho- fall into two, and those of para- are all equivalent again. Still, some of these 3 + 2 + 1 = 6 substitutions end up yielding the same structure, so there are only three structurally distinct trichlorobenzenes: 1,2,3-, 1,2,4-, and 1,3,5-.

1,2,3-Trichlorobenzene	1,2,4-Trichlorobenzene	1,3,5-Trichlorobenzene

If the substituents at each step are different, there will usually be more structural isomers. Xylenol, which is benzene with one hydroxyl substituent and two methyl substituents, has a total of 6 isomers:

2,3-Xylenol	2,4-Xylenol	2,5-Xylenol
2,6-Xylenol	3,4-Xylenol	3,5-Xylenol

Isomer enumeration and counting

Enumerating or counting structural isomers in general is a difficult problem, since one must take into account several bond types (including delocalized ones), cyclic structures, and structures that cannot possibly be realized due to valence or geometric constraints, and non-separable tautomers.

For example, there are nine structural isomers with molecular formula C₃H₆O having different bond connectivities. Seven of them are air-stable at room temperature, and these are given in the table below.

Compounds with molecular formula C₃H₆O

Name	Molecular structure	Melting point (°C)	Boiling point (°C)	Comment
Allyl alcohol		–129	97
Cyclopropanol			101–102
Propionaldehyde		–81	48	Tautomeric with prop-1-en-1-ol, which has both cis and trans stereoisomeric forms
Acetone		–94.9	56.53	Tautomeric with propen-2-ol
Oxetane		–97	48
Propylene oxide		–112	34	Has two enantiomeric forms
Methyl vinyl ether		–122	6

Two structural isomers are the enol tautomers of the carbonyl isomers (propionaldehyde and acetone), but these are not stable.