Nuclear Size and Density

Transcription

1 Nucear Size and Density How does the imited range of the nucear force affect the size and density of the nucei? Assume a Vecro ba mode, each having radius r, voume V = 4/3π r 3. Then the voume of the entire nuceus is just A V, or the radius of the nuceus is just r nuc = const A 1/3. This aso impies that the (core) density of nucei is independent of the size (A). Charge density found from scattering experiments to be: ρ(0) ρ ) = with a=1.07a 1/3 F, b=0.55f. ( r ( r a) / b 1 e Mass density found to be ~ kg/m 3! Thursday, January 3,

2 Nucear Size and Density Thursday, January 3,

3 Modeing the Binding Energy What we know: Tota binding energy scaes with A. Interior densities are independent of A. Genera tendency to have Z = N. With increasing A, more ikey that N>Z. Many more stabe nucei if N and Z are even (166), than if N or Z are odd (110), than if both are odd (8). Thursday, January 3, 003 3

4 Modeing the Binding Energy Liquid Drop Mode (Weizsacker Formua, Semi- Empirica Mass Formua) M ( Z, A) = Five parameter fit to Atomic Masses: NM n ZM p ZM e a a v = 15.67MeV/c a s = 17.3MeV/c a c = 0.714MeV/c a a = 93.15MeV/c v A a s A 3 a c Z A 1 3 a a ( N Z) 4A δ A 1 Thursday, January 3, MeV / c for even Z and N δ = 0MeV / c for odd A 11.MeV / c for odd Z and N 33

5 Modeing the Binding Energy Semi-Empirica Mass Formua M ( Z, A) = NM n ZM ZM a A a First three terms are just the masses of the constituents p e v s A 3 a c Z A 1 3 a a ( N Z) 4A δ A 1 Fourth term is voume term. This accounts for most of the binding energy. Fifth term is the surface term. Reduces the binding energy (fewer nearest neighbors on the surface). Sixth term is Couomb term. Repusion between protons reduces binding. Seventh term is asymmetry term. Reduces binding as you go away from N=Z. Last term is pairing term. This accounts for the increased binding in even Z or N nucei. Thursday, January 3,

6 What have we earned? Nucear force is short ranged (saturation of binding energy). Surface effects are important (ike in iquid-drop mode). Couomb effects are important. Why is there a tendency towards N>Z for arger A? Density ~ constant and very high. Radius ~ A 1/3. Tendency for N ~ Z. N > Z for heavier nucei. Tuesday, January 8,

7 Mystery of the Magic Numbers At certain vaues of N (and Z), there is an abrupt change in how we the Semiempirica Mass Formua predicts the binding energy. At these magic numbers, the nucei are particuary stabe. This is reminiscent of the stabiity of the nobe gases when there are cosed shes. However, this aso impies that the nuceons are basicay independent of one another moving in the nucear potentia. Considering the density of nucear matter, how can this be? Tuesday, January 8,

8 Fermi Gas Mode Nuceons behave in a simiar fashion to free eectrons in a meta. Each nuceon moves in an attractive net nucear potentia (three-dimensiona finite square we). In the ground state, the nuceons (fermions) occupy the owest possibe energy eves without vioating the excusion principe. Then, since a avaiabe energy eves are fied, nuceon nuceon scattering is suppressed (except for exchange scattering which sti eaves the nuceus unchanged). So, even though the density is extremey high, the nuceons move as if they were independent partices trapped in a we. Tuesday, January 8,

9 Fermi Energy Each state can contain two protons and two neutrons For Z=N=A/ Tuesday, January 8,

10 Resuts from Fermi Gas Mode In order to be stabe, the Fermi energy eve for neutrons and protons must be the same. Since protons fee an additiona Couomb repusion, the potentia depth is shaower, hence there must be more neutrons: neutron excess for higher A nucei. The Fermi Gas mode aso gives a firm foundation for the idea that the nuceons move independenty of each other. One can now ook into the idea that nuceons fi discreet states (as in the atomic case). Tuesday, January 8,

11 Fermi Gas Mode and Neutron Stars When the core of a star burns out and coapses under gravitationa pressure, the Fermi energy of the eectrons increase unti the reaction: p e n ν e dominates the inverse reaction: n p e ν e since the eectron wi be Paui bocked. Eventuay, a of the protons wi be converted to neutrons, the couomb repusion between nucei disappears and you are eft with neutron matter. Tuesday, January 8, 003 4

12 Radius of Neutron Stars 1km Tuesday, January 8,

13 Hyperon Spectroscopy In the Fermi mode, a energy eves of a nuceus are fied up to the Fermi energy and interactions between nuceons are suppressed, other than (unobservabe) swaps between indistinguishabe nuceons. How then to mark nuceons in a particuar energy eve? Repace a nuceon with a hyperon (hadron with strange quark content), e.g. a Lambda Λ. K K A Λ A π n Λ π B Λ = B n E π E K ( M Λ M n ) c recoi energy Tuesday, January 8,

14 Hyperon Spectroscopy The Lambda can go into any of the energy eves, since it is not affected by the Paui excusion of the other nuceons. The pion spectra then refects the binding energy of the Lambda. For heavier nucei, the potentia depth stays approximatey constant, but the radius of the we increases as A 1/3, and hence the energy of a particuar state decreases approximatey ineary with the radius squared. Tuesday, January 8,

15 She Mode (Back to Magic Number Probem) Thursday, January 3,

16 She Mode With the Fermi Gas Mode, we can now understand the nuceus as a coection of independent nuceons, each of which are moving in a spherica potentia we whose depth is determined by the net potentia from the rest of the nuceons and whose extent is given by the nucear radius (proportiona to A 1/3 ). Next step is to sove Schrödinger's equation for this probem. The wave functions can be separated into two parts, radia and spherica: R n ( r), Y m with : = ( θ, ϕ) n = 1,,3,4, L s, p, d, f, g, L number of nodes 1 orbita anguar momentum Energy is independent of m (=integer between /-) Since nuceons have spin ½, the n eves are (1) degenerate. Tuesday, January 8,

17 She Mode First, must assume some form of the potentia. For a but the ightest nucei, start with Woods-Saxon Potentia to fit the density distribution: V0 V centra ( r) = ( r R) / a 1 e With soutions to Schrödinger's equation, can construct the energy eve diagram based on the degeneracy of each eve: Remember, that magic numbers were:,8,0,8,50,8,16 One Can see that for higher magic numbers, this doesn t work. Many different forms for the potentia were tried with no success. Something missing?! Tuesday, January 8,

18 She Mode with S L Remember from atomic physics, that LS couping spits degenerate eves into fine structure. ~10-4 effect on the energy (order α ). higher j (=±½) has higher energy (ess bound) Magnetic in origin (eectron s spin coupes with magnetic fied created by its orbita motion) Spin-Orbit couping in nucei is different in two important respects: It is strong (on the same order as E n ) It is inverted from the atomic case. (ower j has higher energy). Both are indications that this couping is not magnetic in origin. Thursday, January 30,

19 She Mode with S L ) ( ) ( ) ( h s r V r V r V s centra = New term in potentia: Then, ) ( 1 1 for 1) ( 1 for 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( r V E j j s s j j s s s s j j s s s j s s h h h h = = = = = = = = Thursday, January 30,

20 Thursday, January 30,

21 Predictions Tota anguar momentum of onepartice and one-hoe states For nucei with one too many nuceons to fi a she (or one too few), the tota nucear spin is determined by the spin of the extra nuceon (or hoe). Exampes: 7 N 15 is douby magic except for a proton hoe in the 1p / subshe. So it shoud have a spin i equa to the vaue j = 1/ for that subshe. This prediction agrees with measurement. 8 O 17 is douby magic except for an extra neutron in the 1d 5/ subshe. So it shoud have i = j = 5/, in agreement with measurement. 19 K 39 is predicted to be douby magic except for a proton hoe in the d 3/ subshe, so it shoud have i = j = 3/. It does. 83 Bi O9 is douby magic except for an extra proton in the 1h 9/ subshe. So its spin shoud be i = j = 9/. This agrees with measurement. Thursday, January 30, 003 5

22 Predictions Tota anguar momentum of one-partice and onehoe states 8 Pb 07 is douby magic except for a neutron hoe in the 1i 13/ subshe. So the excusion principe predicts that it shoud have a spin i = j = 13/. However, the measured spin is i = 1/. Need correction for pairing term, and aso sma couomb corrections for the proton spectra. This drives the 1i 13/ eve beow the 3p 1/ eve. Thursday, January 30,

23 Summary of Pairing Each nuceon coupes L with S to form tota anguar momentum, J. Nuceons interact to form pairs with tota anguar momentum = 0. Nuceons want to be cose (spatiay) because the nuceon-nuceon interaction is strong and attractive. Odd-A nucei have tota spin of the odd (unpaired) nuceon. Tuesday, February 4, 003 6

24 Parity Parity = 1 (nucear eigenfunction is even function of space variabes) Parity = -1 (nucear eigenfunction is odd function of space variabes) Since the nuceons move basicay independenty, nucear eigenfunctions are just the product of the nuceon eigenfunctions Tota parity of the nucei = parity of the eigenfunction of the odd nuceon = (-1) Tuesday, February 4,

25 Magnetic moments Predictions Can be determined from the properties of the extra (or missing) nuceon in one-partice and one-hoe states. E.g., 16 O is douby magic and therefore each nuceon is paired with another to produce zero tota anguar momentum. It s nucear magnetic moment is then zero. This works for ighter nucei, but in heavier nucei, breaks down because of nuceon-nuceus interactions. Again, compicated systems! Tuesday, February 4,

26 Magnetic Moments Tuesday, February 4,

27 Coective Mode A bend between the she mode and the iquid drop mode. Nuceons in unfied subshes move in a net nucear potentia produced by the core of fied subshes. That potentia is not the sphericay symmetric potentia of the she mode, but can undergo deformations in shape simiar to what one woud expect from the iquid drop mode. Even in the ground state the core of nuceons are affected by the nuceons in the unfied she. Their (non sphericay symmetric) presence can distort the core and create a tida effect. Tuesday, February 4,

28 Coective Mode This tida effect can create a net charge current with its own anguar momentum, adding to the net magnetic moment of the unpaired nuceon from the she mode. This corrects the she mode predictions and accounts for the magnetic moments of the nucei. This can aso account for another effect that the she mode does not correcty predict: eectric quadrupoe moments. Tuesday, February 4, 003 Extra proton Negative eectric quadrupoe moment Proton hoe Positive eectric quadrupoe moment 67

29 Deformations and Pairing Force The pairing force aso tends to keep nuceons in cose spatia proximity = neighboring vaues of m. This effect adds to the coective distortions from spherica symmetry and can create arge quadrupoe moments in haf-fied shes of arge nucei. Tuesday, February 4,

30 Coective Mode The coective mode does an exceent job of predicting the eectric quadrupoe moments of most nucei. Notice the change in sign of the moments at the magic numbers. The coective mode can aso be used to understand other coective deformations of nucei, incuding dipoe and quadrupoe resonances and fission. Tuesday, February 4,

31 Modes Summary Name Assumptions Theory Used Properties Predicted Liquid Drop Mode Simiar mass densities, binding energies proportiona to masses Cassica (asymmetry and pairing terms introduced with no justification) Accurate average masses and binding energies (SEMF) Fermi Gas Mode Nuceons move independenty in a net nucear potentia Quantum statistics of Fermi gas of nuceons Depth of net nucear potentia, asymmetry term She Mode Nuceons move in net nucear potentia with strong and inverted spin-orbit term Schrödinger equation soved for potentia Magic numbers, nucear spins, nucear parities, pairing term Coective Mode Net nucear potentia undergoes deformations Schrödinger equation soved for nonspherica potentia Magnetic dipoe moments, eectric quadrupoe moments Tuesday, February 4,