The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.
-
Upload
jaylon-crutcher -
Category
Documents
-
view
219 -
download
6
Transcript of The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.
![Page 1: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/1.jpg)
The role of the isovector monopole state in Coulomb
mixing.N.Auerbach
TAU and MSU
![Page 2: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/2.jpg)
Coulomb interaction
As a good approximation we take the potential of a uniformly charged sphere.
ii
2
C RrR
Ze=rV 22
3 2
3
2
1
itz2
1
![Page 3: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/3.jpg)
Of interest to us here is the isovector part of the potential. Any off-diagonal matrix element between two states of the isovector part is:
n|M|R
Zen|itr|
R
Ze=n|V|
2
izi
2
C1
032
30
20
20
![Page 4: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/4.jpg)
Isovector monopole
denotes the z-component of isovector monopole operator.
It is obvious that if the state is the giant isovector monopole state, that is the state obtained by acting with the operator on the ground state and normalizing, then the above matrix element will be proportional to
and thus will exhaust the isovector part ( ) of the Coulomb sum rule.
10M
n|
)(M 10
0
00 10
10 |MM| )()(
ztr2
![Page 5: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/5.jpg)
Isovector Excitations
![Page 6: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/6.jpg)
![Page 7: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/7.jpg)
![Page 8: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/8.jpg)
Isospin mixing
![Page 9: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/9.jpg)
(a) Hydrodynamical model
Assuming that the sum is exhausted by the isovector monopole, one obtains:
This is for N=Z nuclei.
Bohr-Mottelson model for the isovector monopole
![Page 10: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/10.jpg)
![Page 11: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/11.jpg)
Energy Weighted Sum Rule
![Page 12: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/12.jpg)
The previous formula is for N=Z nuclei. For other nuclei one has to divide by (T+1).
![Page 13: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/13.jpg)
(d) Microscopic model
Obtained from a p-h calculation and fitting the resulting admixtures with the formula below. (The calculation was done in a good isospin basis which requires the inclusion of 2p-2h components.)
![Page 14: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/14.jpg)
Pure isospin states in N>Z nuclei
![Page 15: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/15.jpg)
Isospin impurities
![Page 16: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/16.jpg)
Skyrme HF-RPA (Isovector monopole strength)
Experimentally observed in several reactions, for example in pion single-charge exchange reactions at Los Alamos (25 years ago).
![Page 17: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/17.jpg)
Skyrme HF-RPA (Coulomb strength).
The actual HF Coulomb field was used to probe the distribution of strength.
![Page 18: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/18.jpg)
![Page 19: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/19.jpg)
The core polarization correction to Coulomb displacement energies
We derive the core polarization correction by realizing that the polarization is due to the admixture of the giant isovector monopole IM) into the g.s. of core.
![Page 20: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/20.jpg)
![Page 21: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/21.jpg)
Transition density for the monopole
(This transition density is proportional to the difference of the proton and neutron densities in the ground states of N=Z nuclei.)
![Page 22: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/22.jpg)
![Page 23: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/23.jpg)
![Page 24: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/24.jpg)
Beta-decay
One of the recent activities in nuclear structure are the attempts to determine the corrections one has to introduce in the evaluation of the beta-decay matrix elements for super-allowed transitions.
![Page 25: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/25.jpg)
Relation to the CKM matrix
![Page 26: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/26.jpg)
Shell model approaches
![Page 27: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/27.jpg)
Coulomb corrections
In order to use the experimental ft values to determine one has to introduce corrections. There is a class of important radiative corrections which we will not treat here. Discussions of these can be found abundantly in the literature.
The second type of correction, that is usually termed as the isospin symmetry breaking term, denoted as and defined by the following equation:
cFF δM=M 1202
udV
![Page 28: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/28.jpg)
Where is the physical Fermi matrix element:
and are the parent and daughter physical states. The symbol stands for the Fermi matrix element obtained in the limit when in the Hamiltonian all the charge-dependent parts are put to zero, and the wave functions are eigenstates of the charge-independent Hamiltonian.
MF
21 Ψ|T|Ψ=M +F
1Ψ| 2Ψ|0FM
![Page 29: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/29.jpg)
Experiment
![Page 30: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/30.jpg)
CKM matrix
Using the measured ft values one canrelate these to the u-quark to d-quark
transition matrix element (m.el.) in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. In the Standard Model (SM) this matrix satisfies the unitarity condition, that is the sum of squares of the matrix elements in each row (column) is equal to one:
1222 =V+V+V ubusud
![Page 31: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/31.jpg)
Hardy-Towner
9999995342.0)00359.0()2257.0()97419.0( 222
![Page 32: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/32.jpg)
A call to arms
G. Miller and A. Schwenk (G.A. Miller and A. Schwenk, Phys. Rev. C78, 035501 , (2008))
“With this, we wish to start and stimulate further efforts to systematically improve ISB corrections, based on an accurate understanding of ISB in nuclear forces”.
(ISB- Isospin Symmetry Breaking)(Isospin versus analog spin, W.M. McDonald
and N. Auerbach, Phys. Lett. 53B, 425 (1975)| )
![Page 33: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/33.jpg)
Phys.Rev. C79, 035502 (2009)
In the present approach we start from a charge-independent Hamiltonian so that the matrix element in eq. (1) is exactly and we then treat the Coulomb force in perturbation theory. In the way we approach the problem there is no need to break up the contribution of the Coulomb interaction into various separate components. All the effects of Coulomb mixing (such as isospin mixing, the change in the radial part of the wave functions, etc) are taken into account in a single term. (Some aspects of this approach have already been presented in the past )
T2
![Page 34: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/34.jpg)
The components with different values are degenerate.
The action of the isospin lowering and raising operators, , gives:
1,2 +T Tz
T +,T
12 TT,|T=TT,|T TT,|T=TT,|T+ 21
![Page 35: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/35.jpg)
We now add to the charge independent Hamiltonian a charge dependent part .
The dominant part in the charge dependent interaction is the charge asymmetric Coulomb force . (While the charge-dependent components of the two-body nuclear force might be important in changing the relative spacing of levels in the analog nucleus, its influence on isospin mixing is expected to be small). In what follows we will deal only with off-diagonal matrix elements of the Coulomb interaction. Because of the long range nature of the Coulomb force, the prevailing part will be in such cases the one-body part.
CDV
VC
![Page 36: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/36.jpg)
Wave functions
We will now find in perturbation theory the effect of the charge-dependent part on the wave functions of the two members of the isomultiplet:
111,11 NM|ε+M|ε+TT,|=Ψ T+T+TTT,T
1211,1111,12 1
NM|η+M|η+M|η+TT,|=Ψ T+T+TTT,TTTT
![Page 37: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/37.jpg)
Admixtures
0
1
EE
M|V|TT,=ε
Ti,+TM
Ti,+T)(
Ci
i 0,1
11
111
EE
M|V|TT,=η
Ti,+TM
Ti,+T)(
Ci
i 1,0,1
Ti,+TT,T,=εi 1,0 (TVi+T )(C /1 )EE
Ti,+TM 0
)E(ETVi+TTi,+TT,T,=ηTi,+TM
)(Ci 11
1 /11,0
![Page 38: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/38.jpg)
![Page 39: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/39.jpg)
where , are the components of the isovector monopole, and where
'z
'T T,M| 'z
' T,T
21
21 1 +TT ε+ε+=N 2
122
12 1 +TTT η+η+η+=N
![Page 40: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/40.jpg)
Symmetry potential
u will denote the reduced matrix element
![Page 41: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/41.jpg)
siT
Mi EE
MeV 41 31A
si
Mi EE
ωAξ
V=κ
12
![Page 42: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/42.jpg)
Admixtures
![Page 43: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/43.jpg)
12
11110021
1212
NN
T
+Tηε+ηε+T=Ψ|T|Ψ +
2121
201 1 ε+ε+=N
2121
20
212 1 η+η+η+=N
![Page 44: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/44.jpg)
Coulomb correction
2
21
1221 1212
ε
ωAξ
V+TT=|ΨT|Ψ +
ωAξ
V)+(T=δc
11421ε 2
1321
4114 ε
ξA
V)+(T=δc
2132
1
418 ε
ξA
V=δc
For T=1 nuclei:
![Page 45: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/45.jpg)
Coulomb correction
1. Hydrodynamical model
2. Non-Energy Weighted Sum Rule (NEWSR)
3. Energy Weighted Sum Rule (EWSR)
4.Microscopic
27106.0 A=δc
3/77100.67 A=δc
27105.7 A=δc
3/571018.0 A=δc
![Page 46: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/46.jpg)
Coulomb corrections
These numbers are considerably smaller (factors 3-8) than the Hardy-Towner results.
![Page 47: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/47.jpg)
Discussion
Why is there this difference between the results of our approach and the ones discussed above? It is difficult to pinpoint exactly the reasons; one possible reason is that in the other works collective effects are not included. In the present work on the contrary, the mixing with IVMS takes into account effects of collectivity. The IVMS is a collective excitation and because of the repulsive nature of the particle-hole interaction in the isovector mode it is shifted to higher energies and its strength is reduced. This leads to reduced Coulomb mixing both, in the proton wave function and in the isopin impurity of the isospin quantum number.
![Page 48: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/48.jpg)
Limitations
It is important to asses the uncertainties in our treatment of the Coulomb correction. As already mentioned the symmetry potential strength is not well determined. This parameter determines the splitting between the various isospin components. There is another factor that influences this splitting, namely the different degree of collectivity of these isospin components. In large neutron excess nuclei this might alter considerably the spacing [8], but in T=1 nuclei the collectivity of the various components of the IVMS is similar and the effect on the spacing is small.
c
![Page 49: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/49.jpg)
Discussion
The assumption of equal reduced matrix elements for the transitions to the various isospin components of the IVMS in expressions (12-13) has a very small effect for the nuclei considered. The IVMS has a spreading width and this could bring some fraction of strength to lower energies and influence the result. Also the centroid energy contains some degree of uncertainty. The use of a simplified charge distribution (homogenous sphere) and the neglect of short-range non-Coulomb charge-dependent interactions might affect the results somewhat. The possibility that the Thomas –Ehrman effect [8] might have some influence on the orbital and binding- energy dependence should not be forgotten. There are possibly a number of other small uncertainties. If we rely on an intuitive estimate that the maximal uncertainty is 50%, still our results will be considerably lower than the ones found in previous studies.
![Page 50: The role of the isovector monopole state in Coulomb mixing. N.Auerbach TAU and MSU.](https://reader037.fdocuments.in/reader037/viewer/2022110116/551b15945503462e578b5d14/html5/thumbnails/50.jpg)
Recent work, Liang et al.Relativistic RPA