Lecture 4: Ohio University PHYS7501, Fall 2017, Z. Meisel ([email protected])
Lecture 4: Nuclear Structure 2โข Independent vs collective modelsโข Collective Model
โข Rotationโข Vibrationโข Coupled excitations
โข Nilsson model
Nuclear Models
โข No useful fundamental & universal model exists for nucleiโข E.g. based on the nuclear interaction, how do we describe all nuclear properties?โข Promising approaches include โab initioโ methods, such as Greens Function Monte Carlo,
No-core shell model, Coupled cluster model, density functional theories
โข Generally one of two classes of models is used insteadโข Independent particle models:
โข A nucleon exists within a mean-field (maybe has a few interactions)
โข E.g. Shell model, Fermi gas modelโข Collective models:
โข Groups of nucleons act together (maybe involves shell-model aspects)
โข E.g. Liquid drop model, Collective model
2
โThereโs no small choice in rotten apples.โ Shakespeare, The Taming of the Shrew
Collective Modelโข There are compelling reasons to think that our nucleus isnโt a rigid sphere
โข The liquid drop model gives a pretty successful description of some nuclear properties.โฆcanโt liquids slosh around?
โข Many nuclei have non-zero electric quadrupole moments (charge distributions)
โฆthis means thereโs a non-spherical shape.โฆcanโt non-spherical things rotate?
โข Then, we expect nuclei to be able to be excited rotationally & vibrationallyโข We should (and do) see the signature in the nuclear excited states
โข The relative energetics of rotation vs vibrationcan be inferred from geometry
โข The rotational frequency should go as ๐๐๐๐ โ1๐ ๐ 2
(because ๐ผ๐ผ โก ๐ฟ๐ฟ๐๐ and ๐ผ๐ผ โ ๐๐๐ ๐ 2)
โข The vibrational frequency should go as ๐๐๐ฃ๐ฃ โ1โ๐ ๐ 2
(because itโs like an oscillator)โข So ๐๐๐๐ โช ๐๐๐ฃ๐ฃ 3
Rainwaterโs case for deformationโข A non-spherical shape allows for rotation โฆbut why would a nucleus be non-spherical?โข Consider the energetics of a deformed liquid drop (J. Rainwater, Phys. Rev. (1950))
โข ๐ต๐ต๐ต๐ต๐๐๐๐๐๐๐๐ ๐๐,๐ด๐ด = ๐๐๐ฃ๐ฃ๐๐๐๐๐ด๐ด โ ๐๐๐ ๐ ๐ข๐ข๐๐๐ข๐ข๐ด๐ด โ2 3 โ ๐๐๐๐๐๐๐ข๐ข๐๐๐๐(๐๐โ1)
๐ด๐ด ๏ฟฝ1 3โ ๐๐๐๐๐ ๐ ๐ ๐ ๐ ๐
๐๐โ๐ด๐ด22
๐ด๐ดยฑ ๐๐๐๐๐๐๐๐๐๐๐๐ ๐ด๐ด
โข Upon deformation, only the Coulomb and Surface terms will changeโข Increased penalty for enlarged surfaceโข Decreased penalty for Coulomb repulsion because charges move apartโข The volume remains the same because the drop is incompressible
โข To change shape, but maintain the same volume, the spheroidโs axes can be parameterized asโข ๐๐ = ๐ ๐ (1 + ๐๐) ; ๐๐ = ๐ ๐
1+๐๐; where ๐๐ = 4
3๐๐๐ ๐ 3 = 4
3๐๐๐๐๐๐2
โข It turns out (B.R. Martin, Nuclear and Particle Physics), expanding the surface and Coulomb terms in a power series yields:โข ๐ต๐ต๐ ๐ โฒ = ๐๐๐ ๐ ๐ข๐ข๐๐๐ข๐ข๐ด๐ด โ2 3 1 + 2
5๐๐2 + โฏ ; ๐ต๐ต๐๐โฒ = ๐๐๐๐๐๐๐ข๐ข๐๐
๐๐(๐๐โ1)
๐ด๐ด ๏ฟฝ1 31 โ 1
5๐๐2 + โฏ
โข Therefore, the change in energy for deformation is:โข โ๐ต๐ต = ๐ต๐ต๐ ๐ โฒ + ๐ต๐ต๐๐โฒ โ ๐ต๐ต๐ ๐ + ๐ต๐ต๐๐ = ๐๐2
52๐๐๐ ๐ ๐ข๐ข๐๐๐ข๐ข๐ด๐ด โ2 3 โ ๐๐๐๐๐๐๐ข๐ข๐๐
๐๐(๐๐โ1)
๐ด๐ด ๏ฟฝ1 3(โ๐ต๐ต < 0 is an energetically favorable change)
โข Written more simply, โ๐ต๐ต ๐๐,๐ด๐ด = โ๐ผ๐ผ(๐๐,๐ด๐ด)๐๐2 4
Loveland, Morrissey, Seaborg, Modern Nuclear Chemistry (2006)
b
To get ฮE<0, need Z>116, A>270!So we do not expect deformation
from this effect alone.Nonetheless, heavier nuclei are going
to be more susceptible to deformation.
Rainwaterโs case for deformation
5
โข So far weโve only considered the deformation of the coreโข However, we also need to consider any valence nucleons
โข A non-spherical shape breaks the degeneracy in ๐๐ for a given ๐๐,where the level-splitting is linear in the deformation ๐๐.
โข The strength of the splitting is found by solving theSchrรถdinger equation for single-particle levels in a spheroidal(rather than spherical) well and comparing the spheroidal eigenvalue to the spherical one.
โข The total energy change for deformation then becomes: โ๐ต๐ต ๐๐,๐ด๐ด = โ๐ผ๐ผ ๐๐,๐ด๐ด ๐๐2 โ ๐ฝ๐ฝ๐๐โข The core deformation favors a given m, reinforcing the overall deformation
โข i.e. the valence nucleon interacts with the core somewhat like the moon with the earth, inducing โtidesโ
โข Taking the derivative with respect to ๐๐, we find there is a favored deformation: ๐๐๐ ๐ ๐๐๐๐ = โ๐ฝ๐ฝ2๐ผ๐ผ
โข Since valence nucleons are necessary to amplify the effect,this predicts ground-state deformation occurring in between closed shells
โข Note that the value for ๐ฝ๐ฝ is going to depend on the specific nuclear structure,which shell-model calculations are often used to estimate
S.Granger & R.Spence, Phys. Rev. (1951)
Predicted regions of deformation
6
P. Mรถller et al. ADNDT (2016)B. Harvey, Introduction to Nuclear Physics and Chemistry (1962)
๐ฝ๐ฝ2 โก ๐ผ๐ผ2,0 =43
๐๐5๐๐ โ ๐๐๐ ๐ ๐ ๐ ๐๐๐
Loveland, Morrissey, Seaborg, Modern Nuclear Chemistry (2006)
b
Measured regions of deformation
7
Center for Photonuclear Experiments Data
From:Raman, Nestor, & Tikkanen, ADNDT (2001)N.J. Stone, ADNDT (2005)
Rotation: Rigid rotor
โข The energy associated with a rotating object is: ๐ต๐ต๐๐๐๐๐๐ = 12๐ผ๐ผ๐๐
2
โข Weโre working with quantum stuff, so we need angular momentum instead where ๐ฝ๐ฝ = ๐ผ๐ผ๐๐
โข So, ๐ต๐ต๐๐๐๐๐๐ = 12๐ฝ๐ฝ2
๐ผ๐ผ
โข โฆand ๐ฝ๐ฝ is quantized, so ๐ต๐ต๐๐๐๐๐๐ = ั2๐๐(๐๐+1)2๐ผ๐ผ
โข Thus our rotating nucleus will have excited states spaced as ๐๐(๐๐ + 1) corresponding to rotation
โข For a solid constant-density ellipsoid, ๐ผ๐ผ๐๐๐๐๐๐๐๐๐๐ = 25๐๐๐ ๐
2 1 + 0.31๐ฝ๐ฝ + 0.44๐ฝ๐ฝ2 + โฏ
where ฮฒ = 43
๐๐5
๐๐โ๐๐๐ ๐ ๐ ๐ ๐ ๐ ๐
(A.Bohr & B.Mottelson, Dan. Matematisk-fysiske Meddelelser (1955))
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Rotation: Irrotational Motionโข Rather than the whole nucleus rotating, a tide-like effect could produce something like rotationโข Here nucleons just move in and out in a synchronized fashion,
kind of like people doing โthe waveโ in a stadiumโข Since nucleons arenโt orbiting,
but are just bobbing in and out,this type of motion is calledโirrotationalโ
โข Thankfully, Lord Rayleighworked-out the moment of inertiafor continuous, classical fluidwith a sharp surface(also in D.J. Rowe, Nuclear Collective Motion (1970))
โข ๐ผ๐ผ๐๐๐๐๐๐๐๐ = 98๐๐๐๐๐ ๐
2๐ฝ๐ฝ2
9
B. Harvey, Introduction to Nuclear Physics and Chemistry (1962)
N. Andreacci
Moment of inertia comparisonโข As an example, we can calculate the moment of inertia for 238Pu.
โข The NNDC chart says for this nucleus ๐ฝ๐ฝ = 0.285โข ๐ผ๐ผ๐๐๐๐๐๐๐๐๐๐ = 2
5๐๐๐ ๐ 2 1 + 0.31๐ฝ๐ฝ + 0.44๐ฝ๐ฝ2 + โฏ โ 2
5๐ด๐ด ๐๐0๐ด๐ด1/3 2 1 + 0.31๐ฝ๐ฝ + 0.44๐ฝ๐ฝ2
= 25(1.2๐๐๐๐)2 ๐ด๐ด5/3๐๐๐๐๐๐ 1 + 0.31๐ฝ๐ฝ + 0.44๐ฝ๐ฝ2 = 5874 ๐๐๐๐๐๐ ๐๐๐๐2
โข ๐ผ๐ผ๐๐๐๐๐๐๐๐ = 98๐๐๐๐๐ ๐
2๐ฝ๐ฝ2 โ 98๐๐๐ด๐ด ๐๐0๐ด๐ด1/3 2๐ฝ๐ฝ2
= 98๐๐(1.2๐๐๐๐)2 ๐ด๐ด5/3๐๐๐๐๐๐ ๐ฝ๐ฝ2 = 3778 ๐๐๐๐๐๐ ๐๐๐๐2
โข We can obtain an empirical rotation constant for 238Puโข The energy associated with excitation from the 1st 2+ excited state to the 1st 4+ state is:โ๐ต๐ต = ๐ต๐ต๐๐๐๐๐๐4+ โ ๐ต๐ต๐๐๐๐๐๐2+ = ั2
2๐ผ๐ผ4 4 + 1 โ ั2
2๐ผ๐ผ2 2 + 1 = 7 ั2
๐ผ๐ผโข From NNDC, ๐ต๐ต 21+ = 44๐๐๐๐๐๐ & ๐ต๐ต 41+ = 146๐๐๐๐๐๐ , so ๐ผ๐ผ๐๐๐๐๐๐๐๐ = 7
102ั2๐๐๐๐๐๐โ1
โข Take advantage of fact that ั๐๐ โ 197๐๐๐๐๐๐ ๐๐๐๐ and 1๐๐๐๐๐๐ โ 931.5๐๐๐๐๐๐/๐๐2,๐ผ๐ผ๐๐๐๐๐๐๐๐ = 14
0.102๐๐๐๐๐๐ั2 1๐๐๐๐๐๐
931.5๐๐๐๐๐๐/๐๐2= 0.074 ั2๐๐2
๐๐๐๐๐๐2๐๐๐๐๐๐ = 0.074 197๐๐๐๐๐๐ ๐ข๐ข๐ ๐ 2
๐๐๐๐๐๐2๐๐๐๐๐๐
= 2859 ๐๐๐๐๐๐ ๐๐๐๐2
10โฆcloser to irrotational
Empirical moment of inertiaโข ๐ต๐ต๐๐๐๐๐๐ = ั2๐๐(๐๐+1)
2๐ผ๐ผ, so measuring โ๐ต๐ต between levels should give us ๐ผ๐ผ
โข It turns out, generally: ๐ผ๐ผ๐๐๐๐๐๐๐๐ < ๐ผ๐ผ๐๐๐๐๐๐๐๐ < ๐ผ๐ผ๐๐๐๐๐๐๐๐๐๐
11
D.J. Rowe, Nuclear Collective Motion (1970))
A.Bohr & B.Mottelson, Dan. Matematisk-fysiske Meddelelser (1955))
Rotational bands: sequences of excited states
โข ๐ต๐ต๐๐๐๐๐๐ = ั2๐๐(๐๐+1)2๐ผ๐ผ
, so for a given ๐ผ๐ผ, โ๐ต๐ต โ ๐๐(๐๐ + 1)
โข Note that parity needs to be maintained because rotation issymmetric upon reflection and so 0+ ground-states canonly have j=0,2,4,โฆ (because ๐๐ = (โ1)๐ฝ๐ฝ)
โข Without observing the decay scheme, picking-out associated rotational states could be pretty difficult
โข Experimentally, coincidence measurements allow schemes to be mapped
12
B. Harvey, Introduction to Nuclear Physics and Chemistry (1962)
L. van Dommelen, Quantum Mechanics for Engineers (2012) T.Dinoko et al. EPJ Web Conf. (2013)
158Er
Rotational bandsโข Rotation can exist on top of other excitationsโข As such, a nucleus can have several different
rotational bands and the moment of inertia๐ผ๐ผ is often different for different bands
โข The different ๐ผ๐ผ lead to different energyspacings for the different bands
13
T.Dinoko et al. EPJ Web Conf. (2013)
Loveland, Morrissey, Seaborg, Modern Nuclear Chemistry (2006)
Rotational bands: Backbendโข The different ๐ผ๐ผ for different rotational bands
creates the so-called โbackbendโโข This is when we follow the lowest-energy
state for a given spin-parity (the โyrastโ state)belonging to a given rotational bandand plot the moment of inertia andsquare of the rotational frequency
14
160YF.Beck et al. PRL (1979)
Loveland, Morrissey, Seaborg, Modern Nuclear Chemistry (2006)
๐ป๐ป ๐ต๐ต 21+โก ๐ต๐ต 41+ /๐ต๐ต 21+โ ๐ฝ๐ฝ2
I.Angeli et al. J.Phys.G (2009)
Zr
Inferring structure from rotational bandsโข Since ๐ต๐ต๐๐๐๐๐๐ = ั2๐๐(๐๐+1)
2๐ผ๐ผand ๐ผ๐ผ โ ๐ฝ๐ฝ , we can use rotational bands to probe deformation
โข More deformed nuclei have larger ๐ฝ๐ฝ, so excited state energies for the band should be lowโข The 1st 2+ excited state energy is often used to probe this
โข The rotor model for rotational bands isvalidated by the comparison of bandexcited state energy ratios to the rotorprediction
โข The ratio of the yrast 4+ and 2+ excited stateenergies isgenerally closeto the rotorprediction fornuclei far fromclosed shells
15
Asaro & Perlman, Phys. Rev. (1953)
Rotor(4+/2+)
Rotor(6+/2+)
Inferring structure from rotational bandsโข To keep life interesting, ๐ผ๐ผ can change for a single band, indicating a change in structure,
e.g. how particular nucleons or groups of nucleons are interacting
16
32MgH.Crawford et al. PRC(R) (2016)
Vibrational modesโข Considering the nucleus as a liquid drop, the nuclear volume should be able to vibrateโข Several multipoles are possible
โข Monopole: in & out motion (ฮป = 0)โขa.k.a the breathing mode
โข Dipole: sloshing back & forth (ฮป = 1)โข If all nucleons are moving together, this is just CM motion
โข Quadrupole: alternately compressing & stretching (ฮป = 2)
โข Octupole: alternately pinching on one end & then the other (ฮป = 3)โข + Higher
โข Protons and neutrons can oscillate separately (โisovectorโ vibrations)โข All nucleons need not move together
โข e.g. the โpygmy dipoleโ is the neutron skin oscillation17
Isoscalar IsovectorH.J. Wollersheim
The gifs to the right are for giant resonances,
when all protons/neutrons act collectively
Vibrational modesโข Additionally, oscillations can be grouped by spin
โฆleaving a pretty dizzying range of possibilities
18
The myriad of possible nuclear vibrations are discussed in a friendly manner here: Vibrations of the Atomic Nucleus, G. Bertsch, Scientific American (1983)
TAMU
G.Bertsch, Sci.Am. (1983)
Scattering experiments are key to identifying the vibrational properties of particular excited states
because one obtains characteristic diffraction patterns.
ฮป = 0 ฮป = 2
๐ ๐ ๐ ๐ ๐๐๐ ๐ , ฮป = 1
Rough energetics of vibrational excitationsโข In essence, a nuclear vibration is like a harmonic oscillatorโข There is some oscillating deviation from a default shape and a restoring force attempts to
return the situation to the default shapeโข The restoring force differs for each mode and so therefore do the characteristic frequencies ๐๐,
which have a corresponding energy ั๐๐โข Nuclear matter is nearly incompressible, so the monopole oscillation takes a good bit of
energy to excite.For even-even nuclei, the monopole oscillation creates a 0+ state at โ 80๐ด๐ดโ1/3๐๐๐๐๐๐
โข Neutrons and protons are relatively strongly bound together, so exciting an isovector dipole also takes a good bit of energy
For even-even nuclei, the dipole oscillation creates a 1- state at โ 77๐ด๐ดโ1/3๐๐๐๐๐๐โข The squishiness of the liquid drop is more amenable to quadrupole excitations,
so these are the lowest-energy excitationsFor even-even nuclei, the quadrupole oscillation creates a 2+ state at ~1-2MeV.The giant quadrupole oscillation is at โ 63๐ด๐ดโ1/3๐๐๐๐๐๐
โข Similarly, octupolar shapes can also be accommodatedFor even-even nuclei, the octupole oscillation creates a 3- state at ~4MeV
19
Vibrational energy levelsโข Just as the quantum harmonic oscillator eigenvalues
are quantized, so too will the energy levels for differentquanta (phonons) of a vibrational mode.
โข Similarly, the energy levels have an even spacing,๐ต๐ต๐๐ = (๐ ๐ + 1
2)ั๐๐โข Even-even nuclides have 0+ ground states, and thus,
for a ฮป = 2 vibration, ๐ ๐ = 2 excitations will maintain the symmetry of the wave-function (i.e. ๐ ๐ = 1 excitations would violate parity)
โข Therefore, the 1st vibrational state will be 2+
โข We can excite an independent quadrupole vibration by adding a second phononโข The second phonon will build excitations on the first, coupling to either 0+,2+, or 4+
โข Employing a nuclear potential instead winds up breaking the degeneracy for states associated with a given number of phonons
20
Loveland, Morrissey, Seaborg, Modern Nuclear Chemistry (2006)
Vibrational energy levels are pretty obvious for spherical nuclei
21
G. Harvey, Introduction to Nuclear Physics and Chemistry (1962)
1-ph
onon
3-ph
onon
s2-
phon
ons
Spejewski, Hopke, & Loeser, Phys. Rev. (1969)
โข The typical signature for vibrating spherical nucleus is ๐ต๐ต(41+)/๐ต๐ต(21+) โ 2โฆthough that obviously wonโt be the case for a deformed nucleus
Matin, Church, & Mitchell Phys. Rev. (1966)
22
D. Inglis, Phys. Rev. (1955)
radware
Rotational bands can build on vibrational statesโข Deformed nuclei can simultaneously vibrate and rotateโข The coupling depends on whether the vibration maintains
axial symmetry or notโข The two types, ๐ฝ๐ฝ and ๐พ๐พ, are in reference to how the
vibration deforms the shape in terms of Hill-Wheelercoordinates (Hill & Wheeler, Phys. Rev. (1953))
โข Exemplary spectra:
23
Side view Top view Side view Top view
Bohr & Mottelson, Nuclear Structure Volume II (1969)
B. Harvey, Introduction to Nuclear Physics and Chemistry (1962) N. Blasi et al. Phys.Rev.C (2014)
ฮฒ
gs
ฮณ ฮฒ
gs
ฮณ
24
Single-particle states can build on vibrational states
B. Harvey, Introduction to Nuclear Physics and Chemistry (1962)
โข For some odd-A nuclei, excited states appear to result from the unpaired nucleon to a vibrational phonon
โข For 63Cu, the ground state has an unpaired p3/2 nucleon
โข Coupling this to a 2+ state allows 2 โ 32 โค ๐๐ โค 2 + 3
2 , i.e. 12
โ, 32โ, 52
โ, 72โ
โข Another example:
Canada, Ellegaard, & Barnes, Phys.Rev.C (1971)
Recap of basic structure models discussed thus farโข Schematic shell model
โข Great job for ground-state ๐ฝ๐ฝ๐๐โข Decent job of low-lying excited states for spherical nuclei, particularly near closed shellsโข Miss collective behavior that arises away from shell closures
โข Collective modelโข Rotational excitations explain several ๐ฝ๐ฝ๐๐ for deformed, even-even nuclei
These are โmid shellโ nuclei, because theyโre not near a shell closureโข Vibrational excitations explain several ๐ฝ๐ฝ๐๐ for spherical, even-even nuclei
These are โnear shellโ nuclei, because theyโre near a shell closureโข Miss single-particle behavior that can couple to collective excitations
What do we do for collective behavior for odd-A nuclei?the Nilsson model (a.k.a. deformed shell model)
25
Nilsson Model: combining collective & single-particle approachesโข Our schematic shell model was working perfectly fine
until you threw it away like a cheap suit because of a little deformation! Luckily, Mottelson & Nilsson (Phys. Rev. 1955) werenโt so hasty.
โข Consider a deformed nucleus with axial symmetry that has asingle unpaired nucleon orbiting the nucleus
โข Weโre sticking to axial symmetry becauseโขMost deformed nuclei have this property (mostly prolate)โขThe math, diagrams, and arguments are easier
โข The nucleon has some spin, ๐๐, which has a projection ontothe axis of symmetry, ๐พ๐พ
โข ๐๐ has 2๐๐+12 possible projections ๐พ๐พโข Therefore, each single particle state from our shell model
now splits into multiple states, identified by their ๐พ๐พ, each of which can containtwo particles (spin up and spin down) for that given projection
26
Nuclear Rotation Axis
R.Casten, Nuclear Structure from a Simple Perspective (1990)
Loveland, Morrissey, Seaborg, Modern Nuclear Chemistry (2006)
Nilsson model: single-particle level splitting
27
R.Casten, Nuclear Structure from a Simple Perspective (1990)
โข Consider the options for our nucleonโs orbit around the nucleusโข Orbits with the same principle quantum number will have the same radiusโข Notice that the orbit with the smaller projection of ๐๐ (๐พ๐พ1)
sticks closer to the bulk of the nucleus during its orbitโข Since the nuclear force is attractive,
the ๐พ๐พ1 orbit will be more bound (i.e. lower energy) than the ๐พ๐พ2 orbitโข The opposite would be true if the nucleus in our picture was oblate,
squishing out toward the ๐พ๐พ2 orbitโข Therefore, for prolate nuclei, lower ๐พ๐พ single-particle levels
will be more bound (lower-energy),whereas larger ๐พ๐พ states will be more bound for oblate nuclei
๐๐
KK
โข Continuing with our schematic picture, we see that the proximity of theorbiting nucleon to the nucleus isnโt linear with ๐พ๐พ, since sin๐๐~๐พ๐พ
๐๐
โข So the difference in binding for โ๐พ๐พ = 1 increases as ๐พ๐พ increasesโข Now, considering the fact that single particle levels of different ๐๐
can have the same projection ๐พ๐พ,we arrive at a situation that isessentially the two-state mixingof degenerate perturbation theory,where it turns out the perturbationbreaks the degeneracy and causesthe states to repel each otherin a quadratic fashion where the strength of the deflectiondepends on the proximity of the states in energy
28
Nilsson model: single-particle level splitting
R.Casten, Nuclear Structure from a Simple Perspective (1990)
R.Casten, Nuclear Structure from a Simple Perspective (1990)
D.Griffiths, Introduction to Quantum Mechanics (1995)
R.Casten, Nuclear Structure from a Simple Perspective (1990)
Nilsson Model: single particle levels vs ๐ฝ๐ฝ
29
Nucleons: 8 < ๐๐ < 20 or 8 < ๐๐ < 20Protons: Z>82
Neutrons: N>126
Protons: 50<Z<82
Neutrons: 82<Z<126
B. Harvey, Introduction to Nuclear Physics and Chemistry (1962)
Nilsson Model: Example
30
B. Harvey, Introduction to Nuclear Physics and Chemistry (1962)
โข Consider 25Al, for which we expect ๐ฝ๐ฝ2 โ 0.2, like 27Alโข There are 13 protons and 12 neutrons,
so the unpaired proton will be responsible for ๐ฝ๐ฝ๐๐โข Filling the single-particle levels,
โข We place two protons in the 1s1/2 level, which isnโt shownโข Then two more in 1/2-, two more in 3/2-, two more in 1/2-,
two more in the 1/2+, two more in the 3/2+
โข And the last one winds up in the 5/2+ levelโข So, we predict ๐ฝ๐ฝ๐๐.๐ ๐
๐๐ = โ5 2+
โข For the first excited sate,it seems likely the proton will hopup to the nearby 1/2+ level
โข Agrees with dataโข Since 25Al is deformed,
we should see rotational bandswith states that have (integer)+๐๐and โ ๐๐(๐๐ + 1) spacing
Further Readingโข Chapter 6: Modern Nuclear Chemistry (Loveland, Morrissey, Seaborg)โข Chapter 7: Nuclear & Particle Physics (B.R. Martin)โข Chapter 14, Section 13: Quantum Mechanics for Engineers (L. van Dommelen)โข Chapter 5, Section G: Introduction to Nuclear Physics & Chemistry (B. Harvey)โข Chapter 8: Nuclear Structure from a Simple Perspective (R. Casten)
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