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

From Wikipedia, the free encyclopedia

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.

Atomic nucleus

From Wikipedia, the free encyclopedia

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

A model of an atomic nucleus showing it as a compact bundle of protons (red) and neutrons (blue), the two types of nucleons. In this diagram, protons and neutrons look like little balls stuck together, but an actual nucleus (as understood by modern nuclear physics) cannot be explained like this, but only by using quantum mechanics. In a nucleus that occupies a certain energy level (for example, the ground state), each nucleon can be said to occupy a range of locations.

The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford at the University of Manchester based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron in 1932, models for a nucleus composed of protons and neutrons were quickly developed by Dmitri Ivanenko and Werner Heisenberg. An atom is composed of a positively charged nucleus, with a cloud of negatively charged electrons surrounding it, bound together by electrostatic force. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. Protons and neutrons are bound together to form a nucleus by the nuclear force.

The diameter of the nucleus is in the range of 1.70 fm (1.70×10⁻¹⁵ m) for hydrogen (the diameter of a single proton) to about 11.7 fm for uranium. These dimensions are much smaller than the diameter of the atom itself (nucleus + electron cloud), by a factor of about 26,634 (uranium atomic radius is about 156 pm (156×10⁻¹² m)) to about 60,250 (hydrogen atomic radius is about 52.92 pm).

The branch of physics involved with the study and understanding of the atomic nucleus, including its composition and the forces that bind it together, is called nuclear physics.

History

Main article: Rutherford model

The nucleus was discovered in 1911, as a result of Ernest Rutherford's efforts to test Thomson's "plum pudding model" of the atom. The electron had already been discovered by J. J. Thomson. Knowing that atoms are electrically neutral, J. J. Thomson postulated that there must be a positive charge as well. In his plum pudding model, Thomson suggested that an atom consisted of negative electrons randomly scattered within a sphere of positive charge. Ernest Rutherford later devised an experiment with his research partner Hans Geiger and with help of Ernest Marsden, that involved the deflection of alpha particles (helium nuclei) directed at a thin sheet of metal foil. He reasoned that if J. J. Thomson's model were correct, the positively charged alpha particles would easily pass through the foil with very little deviation in their paths, as the foil should act as electrically neutral if the negative and positive charges are so intimately mixed as to make it appear neutral. To his surprise, many of the particles were deflected at very large angles. Because the mass of an alpha particle is about 8000 times that of an electron, it became apparent that a very strong force must be present if it could deflect the massive and fast moving alpha particles. He realized that the plum pudding model could not be accurate and that the deflections of the alpha particles could only be explained if the positive and negative charges were separated from each other and that the mass of the atom was a concentrated point of positive charge. This justified the idea of a nuclear atom with a dense center of positive charge and mass.

Etymology

The term nucleus is from the Latin word nucleus, a diminutive of nux ('nut'), meaning 'the kernel' (i.e., the 'small nut') inside a watery type of fruit (like a peach). In 1844, Michael Faraday used the term to refer to the "central point of an atom". The modern atomic meaning was proposed by Ernest Rutherford in 1912. The adoption of the term "nucleus" to atomic theory, however, was not immediate. In 1916, for example, Gilbert N. Lewis stated, in his famous article The Atom and the Molecule, that "the atom is composed of the kernel and an outer atom or shell." Similarly, the term kern meaning kernel is used for nucleus in German and Dutch.

Principles

A figurative depiction of the helium-4 atom with the electron cloud in shades of gray. In the nucleus, the two protons and two neutrons are depicted in red and blue. This depiction shows the particles as separate, whereas in an actual helium atom, the protons are superimposed in space and most likely found at the very center of the nucleus, and the same is true of the two neutrons. Thus, all four particles are most likely found in exactly the same space, at the central point. Classical images of separate particles fail to model known charge distributions in very small nuclei. A more accurate image is that the spatial distribution of nucleons in a helium nucleus is much closer to the helium electron cloud shown here, although on a far smaller scale, than to the fanciful nucleus image. Both the helium atom and its nucleus are spherically symmetric.

The nucleus of an atom consists of neutrons and protons, which in turn are the manifestation of more elementary particles, called quarks, that are held in association by the nuclear strong force in certain stable combinations of hadrons, called baryons. The nuclear strong force extends far enough from each baryon so as to bind the neutrons and protons together against the repulsive electrical force between the positively charged protons. The nuclear strong force has a very short range, and essentially drops to zero just beyond the edge of the nucleus. The collective action of the positively charged nucleus is to hold the electrically negative charged electrons in their orbits about the nucleus. The collection of negatively charged electrons orbiting the nucleus display an affinity for certain configurations and numbers of electrons that make their orbits stable. Which chemical element an atom represents is determined by the number of protons in the nucleus; the neutral atom will have an equal number of electrons orbiting that nucleus. Individual chemical elements can create more stable electron configurations by combining to share their electrons. It is that sharing of electrons to create stable electronic orbits about the nuclei that appears to us as the chemistry of our macro world.

Protons define the entire charge of a nucleus, and hence its chemical identity. Neutrons are electrically neutral, but contribute to the mass of a nucleus to nearly the same extent as the protons. Neutrons can explain the phenomenon of isotopes (same atomic number with different atomic mass). The main role of neutrons is to reduce electrostatic repulsion inside the nucleus.

Composition and shape

Protons and neutrons are fermions, with different values of the strong isospin quantum number, so two protons and two neutrons can share the same space wave function since they are not identical quantum entities. They are sometimes viewed as two different quantum states of the same particle, the nucleon. Two fermions, such as two protons, or two neutrons, or a proton + neutron (the deuteron) can exhibit bosonic behavior when they become loosely bound in pairs, which have integer spin.

In the rare case of a hypernucleus, a third baryon called a hyperon, containing one or more strange quarks and/or other unusual quark(s), can also share the wave function. However, this type of nucleus is extremely unstable and not found on Earth except in high-energy physics experiments.

The neutron has a positively charged core of radius ≈ 0.3 fm surrounded by a compensating negative charge of radius between 0.3 fm and 2 fm. The proton has an approximately exponentially decaying positive charge distribution with a mean square radius of about 0.8 fm.

The shape of the atomic nucleus can be spherical, rugby ball-shaped (prolate deformation), discus-shaped (oblate deformation), triaxial (a combination of oblate and prolate deformation) or pear-shaped.

Forces

Nuclei are bound together by the residual strong force (nuclear force). The residual strong force is a minor residuum of the strong interaction which binds quarks together to form protons and neutrons. This force is much weaker between neutrons and protons because it is mostly neutralized within them, in the same way that electromagnetic forces between neutral atoms (such as van der Waals forces that act between two inert gas atoms) are much weaker than the electromagnetic forces that hold the parts of the atoms together internally (for example, the forces that hold the electrons in an inert gas atom bound to its nucleus).

The nuclear force is highly attractive at the distance of typical nucleon separation, and this overwhelms the repulsion between protons due to the electromagnetic force, thus allowing nuclei to exist. However, the residual strong force has a limited range because it decays quickly with distance (see Yukawa potential); thus only nuclei smaller than a certain size can be completely stable. The largest known completely stable nucleus (i.e. stable to alpha, beta, and gamma decay) is lead-208 which contains a total of 208 nucleons (126 neutrons and 82 protons). Nuclei larger than this maximum are unstable and tend to be increasingly short-lived with larger numbers of nucleons. However, bismuth-209 is also stable to beta decay and has the longest half-life to alpha decay of any known isotope, estimated at a billion times longer than the age of the universe.

The residual strong force is effective over a very short range (usually only a few femtometres (fm); roughly one or two nucleon diameters) and causes an attraction between any pair of nucleons. For example, between a proton and a neutron to form a deuteron [NP], and also between protons and protons, and neutrons and neutrons.

Halo nuclei and nuclear force range limits

The effective absolute limit of the range of the nuclear force (also known as residual strong force) is represented by halo nuclei such as lithium-11 or boron-14, in which dineutrons, or other collections of neutrons, orbit at distances of about 10 fm (roughly similar to the 8 fm radius of the nucleus of uranium-238). These nuclei are not maximally dense. Halo nuclei form at the extreme edges of the chart of the nuclides—the neutron drip line and proton drip line—and are all unstable with short half-lives, measured in milliseconds; for example, lithium-11 has a half-life of 8.8 ms.

Halos in effect represent an excited state with nucleons in an outer quantum shell which has unfilled energy levels "below" it (both in terms of radius and energy). The halo may be made of either neutrons [NN, NNN] or protons [PP, PPP]. Nuclei which have a single neutron halo include ¹¹Be and ¹⁹C. A two-neutron halo is exhibited by ⁶He, ¹¹Li, ¹⁷B, ¹⁹B and ²²C. Two-neutron halo nuclei break into three fragments, never two, and are called Borromean nuclei because of this behavior (referring to a system of three interlocked rings in which breaking any ring frees both of the others). ⁸He and ¹⁴Be both exhibit a four-neutron halo. Nuclei which have a proton halo include ⁸B and ²⁶P. A two-proton halo is exhibited by ¹⁷Ne and ²⁷S. Proton halos are expected to be more rare and unstable than the neutron examples, because of the repulsive electromagnetic forces of the halo proton(s).

Nuclear models

Main article: Nuclear structure

Although the standard model of physics is widely believed to completely describe the composition and behavior of the nucleus, generating predictions from theory is much more difficult than for most other areas of particle physics. This is due to two reasons:

• In principle, the physics within a nucleus can be derived entirely from quantum chromodynamics (QCD). In practice however, current computational and mathematical approaches for solving QCD in low-energy systems such as the nuclei are extremely limited. This is due to the phase transition that occurs between high-energy quark matter and low-energy hadronic matter, which renders perturbative techniques unusable, making it difficult to construct an accurate QCD-derived model of the forces between nucleons. Current approaches are limited to either phenomenological models such as the Argonne v18 potential or chiral effective field theory.
• Even if the nuclear force is well constrained, a significant amount of computational power is required to accurately compute the properties of nuclei ab initio. Developments in many-body theory have made this possible for many low mass and relatively stable nuclei, but further improvements in both computational power and mathematical approaches are required before heavy nuclei or highly unstable nuclei can be tackled.

Historically, experiments have been compared to relatively crude models that are necessarily imperfect. None of these models can completely explain experimental data on nuclear structure.

The nuclear radius (R) is considered to be one of the basic quantities that any model must predict. For stable nuclei (not halo nuclei or other unstable distorted nuclei) the nuclear radius is roughly proportional to the cube root of the mass number (A) of the nucleus, and particularly in nuclei containing many nucleons, as they arrange in more spherical configurations:

The stable nucleus has approximately a constant density and therefore the nuclear radius R can be approximated by the following formula,

${\displaystyle R=r_0{A}^{1/3}}$

where A = Atomic mass number (the number of protons Z, plus the number of neutrons N) and r₀ = 1.25 fm = 1.25 × 10⁻¹⁵ m. In this equation, the "constant" r₀ varies by 0.2 fm, depending on the nucleus in question, but this is less than 20% change from a constant.

In other words, packing protons and neutrons in the nucleus gives approximately the same total size result as packing hard spheres of a constant size (like marbles) into a tight spherical or almost spherical bag (some stable nuclei are not quite spherical, but are known to be prolate).

Models of nuclear structure include:

Cluster model

The cluster model describes the nucleus as a molecule-like collection of proton-neutron groups (e.g., alpha particles) with one or more valence neutrons occupying molecular orbitals.

Liquid drop model

Main article: Semi-empirical mass formula

Early models of the nucleus viewed the nucleus as a rotating liquid drop. In this model, the trade-off of long-range electromagnetic forces and relatively short-range nuclear forces, together cause behavior which resembled surface tension forces in liquid drops of different sizes. This formula is successful at explaining many important phenomena of nuclei, such as their changing amounts of binding energy as their size and composition changes (see semi-empirical mass formula), but it does not explain the special stability which occurs when nuclei have special "magic numbers" of protons or neutrons.

The terms in the semi-empirical mass formula, which can be used to approximate the binding energy of many nuclei, are considered as the sum of five types of energies (see below). Then the picture of a nucleus as a drop of incompressible liquid roughly accounts for the observed variation of binding energy of the nucleus:

Volume energy. When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. So, this nuclear energy is proportional to the volume.

Surface energy. A nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.

Coulomb energy. The electric repulsion between each pair of protons in a nucleus contributes toward decreasing its binding energy.

Asymmetry energy (also called Pauli Energy). An energy associated with the Pauli exclusion principle. Were it not for the Coulomb energy, the most stable form of nuclear matter would have the same number of neutrons as protons, since unequal numbers of neutrons and protons imply filling higher energy levels for one type of particle, while leaving lower energy levels vacant for the other type.

Pairing energy. An energy which is a correction term that arises from the tendency of proton pairs and neutron pairs to occur. An even number of particles is more stable than an odd number.

Shell models and other quantum models

Main article: Nuclear shell model

A number of models for the nucleus have also been proposed in which nucleons occupy orbitals, much like the atomic orbitals in atomic physics theory. These wave models imagine nucleons to be either sizeless point particles in potential wells, or else probability waves as in the "optical model", frictionlessly orbiting at high speed in potential wells.

In the above models, the nucleons may occupy orbitals in pairs, due to being fermions, which allows explanation of even/odd Z and N effects well known from experiments. The exact nature and capacity of nuclear shells differs from those of electrons in atomic orbitals, primarily because the potential well in which the nucleons move (especially in larger nuclei) is quite different from the central electromagnetic potential well which binds electrons in atoms. Some resemblance to atomic orbital models may be seen in a small atomic nucleus like that of helium-4, in which the two protons and two neutrons separately occupy 1s orbitals analogous to the 1s orbital for the two electrons in the helium atom, and achieve unusual stability for the same reason. Nuclei with 5 nucleons are all extremely unstable and short-lived, yet, helium-3, with 3 nucleons, is very stable even with lack of a closed 1s orbital shell. Another nucleus with 3 nucleons, the triton hydrogen-3 is unstable and will decay into helium-3 when isolated. Weak nuclear stability with 2 nucleons {NP} in the 1s orbital is found in the deuteron hydrogen-2, with only one nucleon in each of the proton and neutron potential wells. While each nucleon is a fermion, the {NP} deuteron is a boson and thus does not follow Pauli Exclusion for close packing within shells. Lithium-6 with 6 nucleons is highly stable without a closed second 1p shell orbital. For light nuclei with total nucleon numbers 1 to 6 only those with 5 do not show some evidence of stability. Observations of beta-stability of light nuclei outside closed shells indicate that nuclear stability is much more complex than simple closure of shell orbitals with magic numbers of protons and neutrons.

For larger nuclei, the shells occupied by nucleons begin to differ significantly from electron shells, but nevertheless, present nuclear theory does predict the magic numbers of filled nuclear shells for both protons and neutrons. The closure of the stable shells predicts unusually stable configurations, analogous to the noble group of nearly-inert gases in chemistry. An example is the stability of the closed shell of 50 protons, which allows tin to have 10 stable isotopes, more than any other element. Similarly, the distance from shell-closure explains the unusual instability of isotopes which have far from stable numbers of these particles, such as the radioactive elements 43 (technetium) and 61 (promethium), each of which is preceded and followed by 17 or more stable elements.

There are however problems with the shell model when an attempt is made to account for nuclear properties well away from closed shells. This has led to complex post hoc distortions of the shape of the potential well to fit experimental data, but the question remains whether these mathematical manipulations actually correspond to the spatial deformations in real nuclei. Problems with the shell model have led some to propose realistic two-body and three-body nuclear force effects involving nucleon clusters and then build the nucleus on this basis. Three such cluster models are the 1936 Resonating Group Structure model of John Wheeler, Close-Packed Spheron Model of Linus Pauling and the 2D Ising Model of MacGregor.