Chain-growth polymerization

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Polymer science

Chain-growth polymerization (AE) or chain-growth polymerisation (BE) is a polymerization technique where monomer molecules add onto the active site on a growing polymer chain one at a time. There are a limited number of these active sites at any moment during the polymerization which gives this method its key characteristics.

Chain-growth polymerization involves 3 types of reactions :

1. Initiation: An active species I* is formed by some decomposition of an initiator molecule I
2. Propagation: The initiator fragment reacts with a monomer M to begin the conversion to the polymer; the center of activity is retained in the adduct. Monomers continue to add in the same way until polymers P_i* are formed with the degree of polymerization i
3. Termination: By some reaction generally involving two polymers containing active centers, the growth center is deactivated, resulting in dead polymer

Introduction

IUPAC definition

chain polymerization: A chain reaction in which the growth of a polymer chain proceeds exclusively by reaction(s) between monomer and reactive site(s) on the polymer chain with regeneration of the reactive site(s) at the end of each growth step. (See Gold Book entry for note.)

An example of chain-growth polymerization by ring opening to polycaprolactone

In 1953, Paul Flory first classified polymerization as "step-growth polymerization" and "chain-growth polymerization". IUPAC recommends to further simplify "chain-growth polymerization" to "chain polymerization". It is a kind of polymerization where an active center (free radical or ion) is formed, and a plurality of monomers can be polymerized together in a short period of time to form a macromolecule having a large molecular weight. In addition to the regenerated active sites of each monomer unit, polymer growth will only occur at one (or possibly more) endpoint.

Many common polymers can be obtained by chain polymerization such as polyethylene (PE), polypropylene (PP), polyvinyl chloride (PVC), poly(methyl methacrylate) (PMMA), polyacrylonitrile (PAN), polyvinyl acetate (PVA).

Typically, chain-growth polymerization can be understood with the chemical equation:

${\displaystyle P_x^{\ast }+M\to P_{x+1}^{\ast }+L(x=1,2,\mathrm{3...})}$

In this equation, P is the polymer while x represents degree of polymerization, * means active center of chain-growth polymerization, M is the monomer which will react with active center, and L may be a low-molar-mass by-product obtained during chain propagation. For most chain-growth polymerizations, there is no by-product L formed. However there are some exceptions, such as the polymerization of amino acid N-carboxyanhydrides to oxazolidine-2,5-diones.

This type of polymerization is described as "chain" or "chain-growth" because the reaction mechanism is a chemical chain reaction with an initiation step in which an active center is formed, followed by a rapid sequence of chain propagation steps in which the polymer molecule grows by addition of one monomer molecule to the active center in each step. The word "chain" here does not refer to the fact that polymer molecules form long chains. Some polymers are formed instead by a second type of mechanism known as step-growth polymerization without rapid chain propagation steps.

Reaction steps

All chain-growth polymerization reactions must include chain initiation and chain propagation. Chain transfer and chain termination steps also occur in many but not all chain-growth polymerizations.

Chain initiation

Chain initiation is the initial generation of a chain carrier, which is an intermediate such as a radical or an ion which can continue the reaction by chain propagation. Initiation steps are classified according to the way that energy is provided: thermal initiation, high energy initiation, and chemical initiation, etc. Thermal initiation uses molecular thermal motion to dissociate a molecule and form active centers. High energy initiation refers to the generation of chain carriers by radiation. Chemical initiation is due to a chemical initiator.

For the case of radical polymerization as an example, chain initiation involves the dissociation of a radical initiator molecule (I) which is easily dissociated by heat or light into two free radicals (2 R°). Each radical R° then adds a first monomer molecule (M) to start a chain which terminates with a monomer activated by the presence of an unpaired electron (RM₁°).

• I → 2 R°
• R° + M → RM₁°

Chain propagation

IUPAC defines chain propagation as a reaction of an active center on the growing polymer molecule, which adds one monomer molecule to form a new polymer molecule (RM₁°) one repeat unit longer.

For radical polymerization, the active center remains an atom with an unpaired electron. The addition of the second monomer and a typical later addition step are

• RM₁° + M → RM₂°
• ...............
• RM_n° + M → RM_n+1°

For some polymers, chains of over 1000 monomer units can be formed in milliseconds.

Chain termination

In a chain termination step, the active center disappears, resulting in the termination of chain propagation. This is different from chain transfer in which the active center only shifts to another molecule but does not disappear.

For radical polymerization, termination involves a reaction of two growing polymer chains to eliminate the unpaired electrons of both chains. There are two possibilities.

1. Recombination is the reaction of the unpaired electrons of two chains to form a covalent bond between them. The product is a single polymer molecule with the combined length of the two reactant chains:

• RM_n° + RM_m° → P_n+m

2. Disproportionation is the transfer of a hydrogen atom from one chain to the other, so that the two product chain molecules are unchanged in length but are no longer free radicals:

• RM_n° + RM_m° → P_n + P_m

Initiation, propagation and termination steps also occur in chain reactions of smaller molecules. This is not true of the chain transfer and branching steps considered next.

Chain transfer

An example of chain transfer in styrene polymerization. Here X = Cl and Y = CCl₃.

In some chain-growth polymerizations there is also a chain transfer step, in which the growing polymer chain RM_n° takes an atom X from an inactive molecule XY, terminating the growth of the polymer chain: RM_n° + XY → RM_nX + Y°. The Y fragment ls a new active center which adds more monomer M to form a new growing chain YM_n°. This can happen in free radical polymerization for chains RM_n°, in ionic polymerization for chains RM_n⁺ or RM_n^–, or in coordination polymerization. In most cases chain transfer will generate a by-product and decrease the molar mass of the final polymer.

Chain transfer to polymer: Branching

Another possibility is chain transfer to a second polymer molecule, result in the formation of a product macromolecule with a branched structure. In this case the growing chain takes an atom X from a second polymer chain whose growth had been completed. The growth of the first polymer chain is completed by the transfer of atom X. However the second molecule loses an atom X from the interior of its polymer chain to form a reactive radical (or ion) which can add more monomer molecules. This results in the addition of a branch or side chain and the formation of a product macromolecule with a branched structure.

Classes of chain-growth polymerization

The International Union of Pure and Applied Chemistry (IUPAC) recommends definitions for several classes of chain-growth polymerization.

Radical polymerization

Based on the IUPAC definition, radical polymerization is a chain polymerization in which the kinetic-chain carriers are radicals. Usually, the growing chain end bears an unpaired electron. Free radicals can be initiated by many methods such as heating, redox reactions, ultraviolet radiation, high energy irradiation, electrolysis, sonication, and plasma. Free radical polymerization is very important in polymer chemistry. It is one of the most developed methods in chain-growth polymerization. Currently, most polymers in our daily life are synthesized by free radical polymerization, including polyethylene, polystyrene, polyvinyl chloride, polymethyl methacrylate, polyacrylonitrile, polyvinyl acetate, styrene butadiene rubber, nitrile rubber, neoprene, etc.

Ionic polymerization

Ionic polymerization is a chain polymerization in which the kinetic-chain carriers are ions or ion pairs. It can be further divided into anionic polymerization and cationic polymerization. Ionic polymerization generates many polymers used in daily life, such as butyl rubber, polyisobutylene, polyphenylene, polyoxymethylene, polysiloxane, polyethylene oxide, high density polyethylene, isotactic polypropylene, butadiene rubber, etc. Living anionic polymerization was developed in the 1950s. The chain will remain active indefinitely unless the reaction is transferred or terminated deliberately, which allows the control of molar weight and dispersity (or polydispersity index, PDI).

Coordination polymerization

Coordination polymerization is a chain polymerization that involves the preliminary coordination of a monomer molecule with a chain carrier. The monomer is first coordinated with the transition metal active center, and then the activated monomer is inserted into the transition metal-carbon bond for chain growth. In some cases, coordination polymerization is also called insertion polymerization or complexing polymerization. Advanced coordination polymerizations can control the tacticity, molecular weight and PDI of the polymer effectively. In addition, the racemic mixture of the chiral metallocene can be separated into its enantiomers. The oligomerization reaction produces an optically active branched olefin using an optically active catalyst.

Living polymerization

Living polymerization was first described by Michael Szwarc in 1956. It is defined as a chain polymerization from which chain transfer and chain termination are absent. In the absence of chain-transfer and chain termination, the monomer in the system is consumed and the polymerization stops but the polymer chain remains active. If new monomer is added, the polymerization can proceed.

Due to the low PDI and predictable molecular weight, living polymerization is at the forefront of polymer research. It can be further divided into living free radical polymerization, living ionic polymerization and living ring-opening metathesis polymerization, etc.

Ring-opening polymerization

Ring-opening polymerization is defined as a polymerization in which a cyclic monomer yields a monomeric unit which is acyclic or contains fewer cycles than the monomer. Generally, the ring-opening polymerization is carried out under mild conditions, and the by-product is less than in the polycondensation reaction. A high molecular weight polymer is easily obtained. Common ring-opening polymerization products includes polypropylene oxide, polytetrahydrofuran, polyepichlorohydrin, polyoxymethylene, polycaprolactam and polysiloxane.

Reversible-deactivation polymerization

Reversible-deactivation polymerization is defined as a chain polymerization propagated by chain carriers that are deactivated reversibly, bringing them into one or more active-dormant equilibria. An example of a reversible-deactivation polymerization is group-transfer polymerization.

Comparison with step-growth polymerization

Polymers were first classified according to polymerization method by Wallace Carothers in 1929, who introduced the terms addition polymer and condensation polymer to describe polymers made by addition reactions and condensation reactions respectively. However this classification is inadequate to describe a polymer which can be made by either type of reaction, for example nylon 6 which can be made either by addition of a cyclic monomer or by condensation of a linear monomer.

Flory revised the classification to chain-growth polymerization and step-growth polymerization, based on polymerization mechanisms rather than polymer structures. IUPAC now recommends that the names of step-growth polymerization and chain-growth polymerization be further simplified to polycondensation (or polyaddition if no low-molar-mass by-product is formed when a monomer is added) and chain polymerization.

Most polymerizations are either chain-growth or step-growth reactions. Chain-growth includes both initiation and propagation steps (at least), and the propagation of chain-growth polymers proceeds by the addition of monomers to a growing polymer with an active centre. In contrast step-growth polymerization involves only one type of step, and macromolecules can grow by reaction steps between any two molecular species: two monomers, a monomer and a growing chain, or two growing chains. In step growth, the monomers will initially form dimers, trimers, etc. which later react to form long chain polymers.

In chain-growth polymerization, a growing macromolecule increases in size rapidly once its growth is initiated. When a macromolecule stops growing it generally will add no more monomers. In step-growth polymerization on the other hand, a single polymer molecule can grow over the course of the whole reaction.

In chain-growth polymerization, long macromolecules with high molecular weight are formed when only a small fraction of monomer has reacted. Monomers are consumed steadily over the course of the whole reaction, but the degree of polymerization can increase very quickly after chain initiation. However in step-growth polymerization the monomer is consumed very quickly to dimer, trimer and oligomer. The degree of polymerization increases steadily during the whole polymerization process.

The type of polymerization of a given monomer usually depends on the functional groups present, and sometimes also on whether the monomer is linear or cyclic. Chain-growth polymers are usually addition polymers by Carothers' definition. They are typically formed by addition reactions of C=C bonds in the monomer backbone, which contains only carbon-carbon bonds. Another possibility is ring-opening polymerization, as for the chain-growth polymerization of tetrahydrofuran or of polycaprolactone (see Introduction above).

Step-growth polymers are typically condensation polymers in which an elimination product as such as H₂O are formed. Examples are polyamides, polycarbonates, polyesters, polyimides, polysiloxanes and polysulfones. If no elimination product is formed, then the polymer is an addition polymer, such as a polyurethane or a poly(phenylene oxide). Chain-growth polymerization with a low-molar-mass by-product during chain growth is described by IUPAC as "condensative chain polymerization".

Compared to step-growth polymerization, living chain-growth polymerization shows low molar mass dispersity (or PDI), predictable molar mass distribution and controllable conformation. Generally, polycondensation proceeds in a step-growth polymerization mode.

Application

Chain polymerization products are widely used in many aspects of life, including electronic devices, food packaging, catalyst carriers, medical materials, etc. At present, the world's highest yielding polymers such as polyethylene (PE), polyvinyl chloride (PVC), polypropylene (PP), etc. can be obtained by chain polymerization. In addition, some carbon nanotube polymer is used for electronical devices. Controlled living chain-growth conjugated polymerization will also enable the synthesis of well-defined advanced structures, including block copolymers. Their industrial applications extend to water purification, biomedical devices and sensors.

Inorganic polymer

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

The inorganic polymer (SN)_x

In polymer chemistry, an inorganic polymer is a polymer with a skeletal structure that does not include carbon atoms in the backbone. Polymers containing inorganic and organic components are sometimes called hybrid polymers, and most so-called inorganic polymers are hybrid polymers. One of the best known examples is polydimethylsiloxane, otherwise known commonly as silicone rubber. Inorganic polymers offer some properties not found in organic materials including low-temperature flexibility, electrical conductivity, and nonflammability. The term inorganic polymer refers generally to one-dimensional polymers, rather than to heavily crosslinked materials such as silicate minerals. Inorganic polymers with tunable or responsive properties are sometimes called smart inorganic polymers. A special class of inorganic polymers are geopolymers, which may be anthropogenic or naturally occurring.

Main group backbone

Traditionally, the area of inorganic polymers focuses on materials in which the backbone is composed exclusively of main-group elements.

Homochain polymers

Homochain polymers have only one kind of atom in the main chain. One member is polymeric sulfur, which forms reversibly upon melting any of the cyclic allotropes, such as S₈. Organic polysulfides and polysulfanes feature short chains of sulfur atoms, capped respectively with alkyl and H. Elemental tellurium and the gray allotrope of elemental selenium also are polymers, although they are not processable.

The gray allotrope of selenium consists of helical chains of Se atoms.

Polymeric forms of the group IV elements are well known. The premier materials are polysilanes, which are analogous to polyethylene and related organic polymers. They are more fragile than the organic analogues and, because of the longer Si−Si bonds, carry larger substituents. Poly(dimethylsilane) is prepared by reduction of dimethyldichlorosilane. Pyrolysis of poly(dimethylsilane) gives SiC fibers.

Heavier analogues of polysilanes are also known to some extent. These include polygermanes, [R₂Ge]_n, and polystannanes, [R₂Sn]_n.

Heterochain polymers

Si-based

Heterochain polymers have more than one type of atom in the main chain. Typically two types of atoms alternate along the main chain. Of great commercial interest are the polysiloxanes, where the main chain features Si and O centers: −Si−O−Si−O−. Each Si center has two substituents, usually methyl or phenyl. Examples include polydimethylsiloxane (PDMS, [Me₂SiO]_n), polymethylhydrosiloxane (PMHS, [MeSi(H)O]_n) and polydiphenylsiloxane [Ph₂SiO]_n). Related to the siloxanes are the polysilazanes. These materials have the backbone formula −Si−N−Si−N−. One example is perhydridopolysilazane PHPS. Such materials are of academic interest.

P-based

A related family of well studied inorganic polymers are the polyphosphazenes. They feature the backbone −P−N−P−N−. With two substituents on phosphorus, they are structurally similar related to the polysiloxanes. Such materials are generated by ring-opening polymerization of hexachlorophosphazene followed by substitution of the P−Cl groups by alkoxide. Such materials find specialized applications as elastomers.

General structure of polyphosphazenes. Gray spheres represent any organic or inorganic group.

B-based

Boron–nitrogen polymers feature −B−N−B−N− backbones. Examples are polyborazylenes, polyaminoboranes.

S-based

The polythiazyls have the backbone −S−N−S−N−. Unlike most inorganic polymers, these materials lack substituents on the main chain atoms. Such materials exhibit high electrical conductivity, a finding that attracted much attention during the era when polyacetylene was discovered. It is superconducting below 0.26 K.

Ionomers

Usually not classified with charge-neutral inorganic polymers are ionomers. Phosphorus–oxygen and boron-oxide polymers include the polyphosphates and polyborates.

Transition-metal-containing polymers

Further information: Coordination polymer

Inorganic polymers also include materials with transition metals in the backbone. Examples are Polyferrocenes, Krogmann's salt and Magnus's green salt.

Magnus's green salt is a salt that features a one-dimension chain of weak Pt–Pt bonds.

Polymerization methods

Inorganic polymers are formed, like organic polymers, by:

• Step-growth polymerization: Polysiloxanes;
• Chain-growth polymerization: Polysilanes;
• Ring-opening polymerization: Poly(dichlorophosphazene).

Reactions

Inorganic polymers are precursors to inorganic solids. This type of reaction is illustrated by the stepwise conversion of ammonia borane to discrete rings and oligomers, which upon pyrolysis give boron nitrides.

Tunable laser

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CW dye laser based on Rhodamine 6G. The dye laser is considered to be the first broadly tunable laser.

A tunable laser is a laser whose wavelength of operation can be altered in a controlled manner. While all laser gain media allow small shifts in output wavelength, only a few types of lasers allow continuous tuning over a significant wavelength range.

There are many types and categories of tunable lasers. They exist in the gas, liquid, and solid states. Among the types of tunable lasers are excimer lasers, gas lasers (such as CO₂ and He-Ne lasers), dye lasers (liquid and solid state), transition-metal solid-state lasers, semiconductor crystal and diode lasers, and free-electron lasers. Tunable lasers find applications in spectroscopy, photochemistry, atomic vapor laser isotope separation, and optical communications.

Types of tunability

Single line tuning

No real laser is truly monochromatic; all lasers can emit light over some range of frequencies, known as the linewidth of the laser transition. In most lasers, this linewidth is quite narrow (for example, the 1,064 nm wavelength transition of a Nd:YAG laser has a linewidth of approximately 120 GHz, or 0.45 nm). Tuning of the laser output across this range can be achieved by placing wavelength-selective optical elements (such as an etalon) into the laser's optical cavity, to provide selection of a particular longitudinal mode of the cavity.

Multi-line tuning

Most laser gain media have a number of transition wavelengths on which laser operation can be achieved. For example, as well as the principal 1,064 nm output line, Nd:YAG has weaker transitions at wavelengths of 1,052 nm, 1,074 nm, 1,112 nm, 1,319 nm, and a number of other lines. Usually, these lines do not operate unless the gain of the strongest transition is suppressed, such as by use of wavelength-selective dielectric mirrors. If a dispersive element, such as a prism, is introduced into the optical cavity, tilting the cavity's mirrors can cause tuning of the laser as it "hops" between different laser lines. Such schemes are common in argon-ion lasers, allowing tuning of the laser to a number of lines from the ultraviolet and blue through to green wavelengths.

Narrowband tuning

For some types of lasers, the laser's cavity length can be modified, and thus they can be continuously tuned over a significant wavelength range. Distributed feedback (DFB) semiconductor lasers and vertical-cavity surface-emitting lasers (VCSELs) use periodic distributed Bragg reflector (DBR) structures to form the mirrors of the optical cavity. If the temperature of the laser is changed, then the index change of the DBR structure causes a shift in its peak reflective wavelength and thus the wavelength of the laser. The tuning range of such lasers is typically a few nanometres, up to a maximum of approximately 6 nm, as the laser temperature is changed over ~50 K. As a rule of thumb, the wavelength is tuned by 0.08 nm/K for DFB lasers operating in the 1,550 nm wavelength regime. Such lasers are commonly used in optical communications applications, such as DWDM-systems, to allow adjustment of the signal wavelength. To get wideband tuning using this technique, some such as Santur Corporation or Nippon Telegraph and Telephone (NTT Corporation) contain an array of such lasers on a single chip and concatenate the tuning ranges.

Widely tunable lasers

A typical laser diode. When mounted with external optics, these lasers can be tuned mainly in the red and near-infrared.

Sample Grating Distributed Bragg Reflector lasers (SG-DBR) have a much larger tunable range; by the use of vernier-tunable Bragg mirrors and a phase section, a single-mode output range of > 50 nm can be selected. Other technologies to achieve wide tuning ranges for DWDM-systems are:

• External cavity lasers using a MEMS structure for tuning the cavity length, such as devices commercialized by Iolon.
• External cavity lasers using multiple-prism grating arrangements for wide-range tunability.
• DFB laser arrays based on several thermal tuned DFB lasers, in which coarse tuning is achieved by selecting the correct laser bar. Fine tuning is then done thermally, such as in devices commercialized by Santur Corporation.
• Tunable VCSELs, in which one of the two mirror stacks is movable. To achieve sufficient output power out of a VCSEL structure, lasers in the 1,550 nm domain are usually either optically pumped or have an additional optical amplifier built into the device.

Rather than placing the resonator mirrors at the edges of the device, the mirrors in a VCSEL are located on the top and bottom of the semiconductor material. Somewhat confusingly, these mirrors are typically DBR devices. This arrangement causes light to "bounce" vertically in a laser chip, so that the light emerges through the top of the device, rather than through the edge. As a result, VCSELs produce beams of a more circular nature than their cousins and beams that do not diverge as rapidly.

As of December 2008, there is no widely tunable VCSEL commercially available any more for DWDM-system application.

It is claimed that the first infrared laser with a tunability of more than one octave was a germanium crystal laser.

Applications

The range of applications of tunable lasers is extremely wide. When coupled to the right filter, a tunable source can be tuned over a few hundreds of nanometers with a spectral resolution that can go from 4 nm to 0.3 nm, depending on the wavelength range. With a good enough isolation (>OD4), tunable sources can be used for basic absorption and photoluminescence studies. They can be used for solar cells characterisation in a light-beam-induced current (LBIC) experiment, from which the external quantum efficiency (EQE) of a device can be mapped. They can also be used for the characterisation of gold nanoparticles and single-walled carbon nanotube thermopiles, where a wide tunable range from 400 nm to 1,000 nm is essential. Tunable sources were recently used for the development of hyperspectral imaging for early detection of retinal diseases where a wide range of wavelengths, a small bandwidth, and outstanding isolation is needed to achieve efficient illumination of the entire retina. Tunable sources can be a powerful tool for reflection and transmission spectroscopy, photobiology, detector calibration, hyperspectral imaging, and steady-state pump probe experiments, to name only a few.

History

The first true broadly tunable laser was the dye laser in 1966. Hänsch introduced the first narrow-linewidth tunable laser in 1972. Dye lasers and some vibronic solid-state lasers have extremely large bandwidths, allowing tuning over a range of tens to hundreds of nanometres. Titanium-doped sapphire is the most common tunable solid-state laser, capable of laser operation from 670 nm to 1,100 nm wavelengths. Typically these laser systems incorporate a Lyot filter into the laser cavity, which is rotated to tune the laser. Other tuning techniques involve diffraction gratings, prisms, etalons, and combinations of these. Multiple-prism grating arrangements, in several configurations, as described by Duarte, are used in diode, dye, gas, and other tunable lasers.

Nuclear shell model

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

In nuclear physics, atomic physics, and nuclear chemistry, the nuclear shell model utilizes the Pauli exclusion principle to model the structure of atomic nuclei in terms of energy levels. The first shell model was proposed by Dmitri Ivanenko (together with E. Gapon) in 1932. The model was developed in 1949 following independent work by several physicists, most notably Maria Goeppert Mayer and J. Hans D. Jensen, who received the 1963 Nobel Prize in Physics for their contributions to this model, and Eugene Wigner, who received the Nobel Prize alongside them for his earlier groundlaying work on the atomic nuclei.

The nuclear shell model is partly analogous to the atomic shell model, which describes the arrangement of electrons in an atom, in that a filled shell results in better stability. When adding nucleons (protons and neutrons) to a nucleus, there are certain points where the binding energy of the next nucleon is significantly less than the last one. This observation that there are specific magic quantum numbers of nucleons (2, 8, 20, 28, 50, 82, and 126) that are more tightly bound than the following higher number is the origin of the shell model.

The shells for protons and neutrons are independent of each other. Therefore, there can exist both "magic nuclei", in which one nucleon type or the other is at a magic number, and "doubly magic quantum nuclei", where both are. Due to variations in orbital filling, the upper magic numbers are 126 and, speculatively, 184 for neutrons, but only 114 for protons, playing a role in the search for the so-called island of stability. Some semi-magic numbers have been found, notably Z = 40, which gives the nuclear shell filling for the various elements; 16 may also be a magic number.

To get these numbers, the nuclear shell model starts with an average potential with a shape somewhere between the square well and the harmonic oscillator. To this potential, a spin-orbit term is added. Even so, the total perturbation does not coincide with the experiment, and an empirical spin-orbit coupling must be added with at least two or three different values of its coupling constant, depending on the nuclei being studied.

The empirical proton and neutron shell gaps are numerically obtained from observed binding energies. Distinct shell gaps are shown at labeled magic numbers, and at ${\displaystyle N=Z}$.

The magic numbers of nuclei, as well as other properties, can be arrived at by approximating the model with a three-dimensional harmonic oscillator plus a spin–orbit interaction. A more realistic but complicated potential is known as the Woods–Saxon potential.

Modified harmonic oscillator model

Consider a three-dimensional harmonic oscillator. This would give, for example, in the first three levels ("ℓ" is the angular momentum quantum number):

level n	ℓ	mℓ	ms
0	0	0	+⁠1/2⁠
			−⁠1/2⁠
1	1	+1	+⁠1/2⁠
			−⁠1/2⁠
		0	+⁠1/2⁠
			−⁠1/2⁠
		−1	+⁠1/2⁠
			−⁠1/2⁠
2	0	0	+⁠1/2⁠
			−⁠1/2⁠
	2	+2	+⁠1/2⁠
			−⁠1/2⁠
		+1	+⁠1/2⁠
			−⁠1/2⁠
		0	+⁠1/2⁠
			−⁠1/2⁠
		−1	+⁠1/2⁠
			−⁠1/2⁠
		−2	+⁠1/2⁠
			−⁠1/2⁠

Nuclei are built by adding protons and neutrons. These will always fill the lowest available level, with the first two protons filling level zero, the next six protons filling level one, and so on. As with electrons in the periodic table, protons in the outermost shell will be relatively loosely bound to the nucleus if there are only a few protons in that shell because they are farthest from the center of the nucleus. Therefore, nuclei with a full outer proton shell will have a higher nuclear binding energy than other nuclei with a similar total number of protons. The same is true for neutrons.

This means that the magic numbers are expected to be those in which all occupied shells are full. In accordance with the experiment, we get 2 (level 0 full) and 8 (levels 0 and 1 full) for the first two numbers. However, the full set of magic numbers does not turn out correctly. These can be computed as follows:

• In a three-dimensional harmonic oscillator the total degeneracy of states at level n is ${\displaystyle \frac{(n+1)(n+2)}{2}}$.
• Due to the spin, the degeneracy is doubled and is ${\displaystyle (n+1)(n+2)}$.
• Thus, the magic numbers would be$${\displaystyle \sum _{n=0}^k(n+1)(n+2)=\frac{(k+1)(k+2)(k+3)}{3}}$$for all integer k. This gives the following magic numbers: 2, 8, 20, 40, 70, 112, ..., which agree with experiment only in the first three entries. These numbers are twice the tetrahedral numbers (1, 4, 10, 20, 35, 56, ...) from the Pascal Triangle.

In particular, the first six shells are:

• level 0: 2 states (ℓ = 0) = 2.
• level 1: 6 states (ℓ = 1) = 6.
• level 2: 2 states (ℓ = 0) + 10 states (ℓ = 2) = 12.
• level 3: 6 states (ℓ = 1) + 14 states (ℓ = 3) = 20.
• level 4: 2 states (ℓ = 0) + 10 states (ℓ = 2) + 18 states (ℓ = 4) = 30.
• level 5: 6 states (ℓ = 1) + 14 states (ℓ = 3) + 22 states (ℓ = 5) = 42.

where for every ℓ there are 2ℓ+1 different values of m_l and 2 values of m_s, giving a total of 4ℓ+2 states for every specific level.

These numbers are twice the values of triangular numbers from the Pascal Triangle: 1, 3, 6, 10, 15, 21, ....

Including a spin-orbit interaction

We next include a spin–orbit interaction. First, we have to describe the system by the quantum numbers j, m_j and parity instead of ℓ, m_l and m_s, as in the hydrogen–like atom. Since every even level includes only even values of ℓ, it includes only states of even (positive) parity. Similarly, every odd level includes only states of odd (negative) parity. Thus we can ignore parity in counting states. The first six shells, described by the new quantum numbers, are

• level 0 (n = 0): 2 states (j = ⁠1/2⁠). Even parity.
• level 1 (n = 1): 2 states (j = ⁠1/2⁠) + 4 states (j = ⁠3/2⁠) = 6. Odd parity.
• level 2 (n = 2): 2 states (j = ⁠1/2⁠) + 4 states (j = ⁠3/2⁠) + 6 states (j = ⁠5/2⁠) = 12. Even parity.
• level 3 (n = 3): 2 states (j = ⁠1/2⁠) + 4 states (j = ⁠3/2⁠) + 6 states (j = ⁠5/2⁠) + 8 states (j = ⁠7/2⁠) = 20. Odd parity.
• level 4 (n = 4): 2 states (j = ⁠1/2⁠) + 4 states (j = ⁠3/2⁠) + 6 states (j = ⁠5/2⁠) + 8 states (j = ⁠7/2⁠) + 10 states (j = ⁠9/2⁠) = 30. Even parity.
• level 5 (n = 5): 2 states (j = ⁠1/2⁠) + 4 states (j = ⁠3/2⁠) + 6 states (j = ⁠5/2⁠) + 8 states (j = ⁠7/2⁠) + 10 states (j = ⁠9/2⁠) + 12 states (j = ⁠11/2⁠) = 42. Odd parity.

where for every j there are 2j+1 different states from different values of m_j.

Due to the spin–orbit interaction, the energies of states of the same level but with different j will no longer be identical. This is because in the original quantum numbers, when ${\displaystyle \overrightarrow{s}}$ is parallel to ${\displaystyle \overrightarrow{l}}$, the interaction energy is positive, and in this case j = ℓ + s = ℓ + ⁠1/2⁠. When ${\displaystyle \overrightarrow{s}}$ is anti-parallel to ${\displaystyle \overrightarrow{l}}$ (i.e. aligned oppositely), the interaction energy is negative, and in this case j=ℓ−s=ℓ−⁠1/2⁠. Furthermore, the strength of the interaction is roughly proportional to ℓ.

For example, consider the states at level 4:

• The 10 states with j = ⁠9/2⁠ come from ℓ = 4 and s parallel to ℓ. Thus they have a positive spin–orbit interaction energy.
• The 8 states with j = ⁠7/2⁠ came from ℓ = 4 and s anti-parallel to ℓ. Thus they have a negative spin–orbit interaction energy.
• The 6 states with j = ⁠5/2⁠ came from ℓ = 2 and s parallel to ℓ. Thus they have a positive spin–orbit interaction energy. However, its magnitude is half compared to the states with j = ⁠9/2⁠.
• The 4 states with j = ⁠3/2⁠ came from ℓ = 2 and s anti-parallel to ℓ. Thus they have a negative spin–orbit interaction energy. However, its magnitude is half compared to the states with j = ⁠7/2⁠.
• The 2 states with j = ⁠1/2⁠ came from ℓ = 0 and thus have zero spin–orbit interaction energy.

Changing the profile of the potential

The harmonic oscillator potential ${\displaystyle V(r)=\mu {\omega }^2{r}^2/2}$ grows infinitely as the distance from the center r goes to infinity. A more realistic potential, such as the Woods–Saxon potential, would approach a constant at this limit. One main consequence is that the average radius of nucleons' orbits would be larger in a realistic potential. This leads to a reduced term ${\displaystyle {\hslash }^{2}l(l+1)/2m{r}^2}$ in the Laplace operator of the Hamiltonian operator. Another main difference is that orbits with high average radii, such as those with high n or high ℓ, will have a lower energy than in a harmonic oscillator potential. Both effects lead to a reduction in the energy levels of high ℓ orbits.

Predicted magic numbers

Low-lying energy levels in a single-particle shell model with an oscillator potential (with a small negative l² term) without spin–orbit (left) and with spin–orbit (right) interaction. The number to the right of a level indicates its degeneracy, (2j+1). The boxed integers indicate the magic numbers.

Together with the spin–orbit interaction, and for appropriate magnitudes of both effects, one is led to the following qualitative picture: at all levels, the highest j states have their energies shifted downwards, especially for high n (where the highest j is high). This is both due to the negative spin–orbit interaction energy and to the reduction in energy resulting from deforming the potential into a more realistic one. The second-to-highest j states, on the contrary, have their energy shifted up by the first effect and down by the second effect, leading to a small overall shift. The shifts in the energy of the highest j states can thus bring the energy of states of one level closer to the energy of states of a lower level. The "shells" of the shell model are then no longer identical to the levels denoted by n, and the magic numbers are changed.

We may then suppose that the highest j states for n = 3 have an intermediate energy between the average energies of n = 2 and n = 3, and suppose that the highest j states for larger n (at least up to n = 7) have an energy closer to the average energy of n−1. Then we get the following shells (see the figure)

• 1st shell: 2 states (n = 0, j = ⁠1/2⁠).
• 2nd shell: 6 states (n = 1, j = ⁠1/2⁠ or ⁠3/2⁠).
• 3rd shell: 12 states (n = 2, j = ⁠1/2⁠, ⁠3/2⁠ or ⁠5/2⁠).
• 4th shell: 8 states (n = 3, j = ⁠7/2⁠).
• 5th shell: 22 states (n = 3, j = ⁠1/2⁠, ⁠3/2⁠ or ⁠5/2⁠; n = 4, j = ⁠9/2⁠).
• 6th shell: 32 states (n = 4, j = ⁠1/2⁠, ⁠3/2⁠, ⁠5/2⁠ or ⁠7/2⁠; n = 5, j = ⁠11/2⁠).
• 7th shell: 44 states (n = 5, j = ⁠1/2⁠, ⁠3/2⁠, ⁠5/2⁠, ⁠7/2⁠ or ⁠9/2⁠; n = 6, j = ⁠13/2⁠).
• 8th shell: 58 states (n = 6, j = ⁠1/2⁠, ⁠3/2⁠, ⁠5/2⁠, ⁠7/2⁠, ⁠9/2⁠ or ⁠11/2⁠; n = 7, j = ⁠15/2⁠).

and so on.

Note that the numbers of states after the 4th shell are doubled triangular numbers plus two. Spin–orbit coupling causes so-called 'intruder levels' to drop down from the next higher shell into the structure of the previous shell. The sizes of the intruders are such that the resulting shell sizes are themselves increased to the next higher doubled triangular numbers from those of the harmonic oscillator. For example, 1f2p has 20 nucleons, and spin–orbit coupling adds 1g9/2 (10 nucleons), leading to a new shell with 30 nucleons. 1g2d3s has 30 nucleons, and adding intruder 1h11/2 (12 nucleons) yields a new shell size of 42, and so on.

The magic numbers are then

•   2
•   8=2+6
•  20=2+6+12
•  28=2+6+12+8
•  50=2+6+12+8+22
•  82=2+6+12+8+22+32
• 126=2+6+12+8+22+32+44
• 184=2+6+12+8+22+32+44+58

and so on. This gives all the observed magic numbers and also predicts a new one (the so-called island of stability) at the value of 184 (for protons, the magic number 126 has not been observed yet, and more complicated theoretical considerations predict the magic number to be 114 instead).

Another way to predict magic (and semi-magic) numbers is by laying out the idealized filling order (with spin–orbit splitting but energy levels not overlapping). For consistency, s is split into j = ⁠1/2⁠ and j = −⁠1/2⁠ components with 2 and 0 members respectively. Taking the leftmost and rightmost total counts within sequences bounded by / here gives the magic and semi-magic numbers.

• s(2,0)/p(4,2) > 2,2/6,8, so (semi)magic numbers 2,2/6,8
• d(6,4):s(2,0)/f(8,6):p(4,2) > 14,18:20,20/28,34:38,40, so 14,20/28,40
• g(10,8):d(6,4):s(2,0)/h(12,10):f(8,6):p(4,2) > 50,58,64,68,70,70/82,92,100,106,110,112, so 50,70/82,112
• i(14,12):g(10,8):d(6,4):s(2,0)/j(16,14):h(12,10):f(8,6):p(4,2) > 126,138,148,156,162,166,168,168/184,198,210,220,228,234,238,240, so 126,168/184,240

The rightmost predicted magic numbers of each pair within the quartets bisected by / are double tetrahedral numbers from the Pascal Triangle: 2, 8, 20, 40, 70, 112, 168, 240 are 2x 1, 4, 10, 20, 35, 56, 84, 120, ..., and the leftmost members of the pairs differ from the rightmost by double triangular numbers: 2 − 2 = 0, 8 − 6 = 2, 20 − 14 = 6, 40 − 28 = 12, 70 − 50 = 20, 112 − 82 = 30, 168 − 126 = 42, 240 − 184 = 56, where 0, 2, 6, 12, 20, 30, 42, 56, ... are 2 × 0, 1, 3, 6, 10, 15, 21, 28, ... .

Other properties of nuclei

This model also predicts or explains with some success other properties of nuclei, in particular spin and parity of nuclei ground states, and to some extent their excited nuclear states as well. Take ¹⁷
₈O (oxygen-17) as an example: Its nucleus has eight protons filling the first three proton "shells", eight neutrons filling the first three neutron "shells", and one extra neutron. All protons in a complete proton shell have zero total angular momentum, since their angular momenta cancel each other. The same is true for neutrons. All protons in the same level (n) have the same parity (either +1 or −1), and since the parity of a pair of particles is the product of their parities, an even number of protons from the same level (n) will have +1 parity. Thus, the total angular momentum of the eight protons and the first eight neutrons is zero, and their total parity is +1. This means that the spin (i.e. angular momentum) of the nucleus, as well as its parity, are fully determined by that of the ninth neutron. This one is in the first (i.e. lowest energy) state of the 4th shell, which is a d-shell (ℓ = 2), and since p = (−1)^ℓ, this gives the nucleus an overall parity of +1. This 4th d-shell has a j = ⁠5/2⁠, thus the nucleus of ¹⁷
₈O is expected to have positive parity and total angular momentum ⁠5/2⁠, which indeed it has.

The rules for the ordering of the nucleus shells are similar to Hund's Rules of the atomic shells, however, unlike its use in atomic physics, the completion of a shell is not signified by reaching the next n, as such the shell model cannot accurately predict the order of excited nuclei states, though it is very successful in predicting the ground states. The order of the first few terms are listed as follows: 1s, 1p⁠3/2⁠, 1p⁠1/2⁠, 1d⁠5/2⁠, 2s, 1d⁠3/2⁠... For further clarification on the notation refer to the article on the Russell–Saunders term symbol.

For nuclei farther from the magic quantum numbers one must add the assumption that due to the relation between the strong nuclear force and total angular momentum, protons or neutrons with the same n tend to form pairs of opposite angular momentum. Therefore, a nucleus with an even number of protons and an even number of neutrons has 0 spin and positive parity. A nucleus with an even number of protons and an odd number of neutrons (or vice versa) has the parity of the last neutron (or proton), and the spin equal to the total angular momentum of this neutron (or proton). By "last" we mean the properties coming from the highest energy level.

In the case of a nucleus with an odd number of protons and an odd number of neutrons, one must consider the total angular momentum and parity of both the last neutron and the last proton. The nucleus parity will be a product of theirs, while the nucleus spin will be one of the possible results of the sum of their angular momenta (with other possible results being excited states of the nucleus).

The ordering of angular momentum levels within each shell is according to the principles described above – due to spin–orbit interaction, with high angular momentum states having their energies shifted downwards due to the deformation of the potential (i.e. moving from a harmonic oscillator potential to a more realistic one). For nucleon pairs, however, it is often energetically favourable to be at high angular momentum, even if its energy level for a single nucleon would be higher. This is due to the relation between angular momentum and the strong nuclear force.

The nuclear magnetic moment of neutrons and protons is partly predicted by this simple version of the shell model. The magnetic moment is calculated through j, ℓ and s of the "last" nucleon, but nuclei are not in states of well-defined ℓ and s. Furthermore, for odd-odd nuclei, one has to consider the two "last" nucleons, as in deuterium. Therefore, one gets several possible answers for the nuclear magnetic moment, one for each possible combined ℓ and s state, and the real state of the nucleus is a superposition of them. Thus the real (measured) nuclear magnetic moment is somewhere in between the possible answers.

The electric dipole of a nucleus is always zero, because its ground state has a definite parity. The matter density (ψ², where ψ is the wavefunction) is always invariant under parity. This is usually the situation with the atomic electric dipole.

Higher electric and magnetic multipole moments cannot be predicted by this simple version of the shell model for reasons similar to those in the case of deuterium.

Including residual interactions

Residual interactions among valence nucleons are included by diagonalizing an effective Hamiltonian in a valence space outside an inert core. As indicated, only single-particle states lying in the valence space are active in the basis used.

For nuclei having two or more valence nucleons (i.e. nucleons outside a closed shell), a residual two-body interaction must be added. This residual term comes from the part of the inter-nucleon interaction not included in the approximative average potential. Through this inclusion, different shell configurations are mixed, and the energy degeneracy of states corresponding to the same configuration is broken.

These residual interactions are incorporated through shell model calculations in a truncated model space (or valence space). This space is spanned by a basis of many-particle states where only single-particle states in the model space are active. The Schrödinger equation is solved on this basis, using an effective Hamiltonian specifically suited for the model space. This Hamiltonian is different from the one of free nucleons as, among other things, it has to compensate for excluded configurations.

One can do away with the average potential approximation entirely by extending the model space to the previously inert core and treating all single-particle states up to the model space truncation as active. This forms the basis of the no-core shell model, which is an ab initio method. It is necessary to include a three-body interaction in such calculations to achieve agreement with experiments.

Collective rotation and the deformed potential

In 1953 the first experimental examples were found of rotational bands in nuclei, with their energy levels following the same J(J+1) pattern of energies as in rotating molecules. Quantum mechanically, it is impossible to have a collective rotation of a sphere, so this implied that the shape of these nuclei was non-spherical. In principle, these rotational states could have been described as coherent superpositions of particle-hole excitations in the basis consisting of single-particle states of the spherical potential. But in reality, the description of these states in this manner is intractable, due to a large number of valence particles—and this intractability was even greater in the 1950s when computing power was extremely rudimentary. For these reasons, Aage Bohr, Ben Mottelson, and Sven Gösta Nilsson constructed models in which the potential was deformed into an ellipsoidal shape. The first successful model of this type is now known as the Nilsson model. It is essentially the harmonic oscillator model described in this article, but with anisotropy added, so the oscillator frequencies along the three Cartesian axes are not all the same. Typically the shape is a prolate ellipsoid, with the axis of symmetry taken to be z. Because the potential is not spherically symmetric, the single-particle states are not states of good angular momentum J. However, a Lagrange multiplier ${\displaystyle -\omega \cdot J}$, known as a "cranking" term, can be added to the Hamiltonian. Usually the angular frequency vector ω is taken to be perpendicular to the symmetry axis, although tilted-axis cranking can also be considered. Filling the single-particle states up to the Fermi level produces states whose expected angular momentum along the cranking axis ${\displaystyle ⟨J_x⟩}$ is the desired value.

Related models

Igal Talmi developed a method to obtain the information from experimental data and use it to calculate and predict energies which have not been measured. This method has been successfully used by many nuclear physicists and has led to a deeper understanding of nuclear structure. The theory which gives a good description of these properties was developed. This description turned out to furnish the shell model basis of the elegant and successful interacting boson model.

A model derived from the nuclear shell model is the alpha particle model developed by Henry Margenau, Edward Teller, J. K. Pering, T. H. Skyrme, also sometimes called the Skyrme model. Note, however, that the Skyrme model is usually taken to be a model of the nucleon itself, as a "cloud" of mesons (pions), rather than as a model of the nucleus as a "cloud" of alpha particles.