aa r X i v : . [ nu c l - t h ] O c t ON THE PROPERTIES OF THE
Th ISOTOPE
V. I. Isakov Petersburg Nuclear Physics Institute, Gatchina 188300, Russia,
National Research Centre Kurchatov Institute
A b s t r a c t
Electromagnetic properties of the deformed neutron-odd nucleus
Th are investigated in theframework of the unified model, with primary emphasis upon the properties of the low-lying isomericstate.
On the basis of detailed analysis of γ -transitions in Th attendant α -decay of U, it wasestablished the existence in the daughter nuclei
Th of the low-lying level with the excitationenergy of only a few eV. This is the most low-lying state known by now. The next one is thelevel 1/2 + in U, with the excitation energy equal to 76.5 eV. The latest experimental data[1] point to the value of the excitation energy equal to ∼ ∼ ± µ s was measuredin [3]. However, the energy of this state is not yet measured in the direct experiment.Here, we carry out theoretical analysis of the characteristics of Th, and make an attemptto describe decay properties of it‘s low-lying levels, as well as to propose an alternative way forexcitation of the above-mentioned state in the reaction of the Coulomb excitation.In Fig.1 we show experimental scheme of levels and the decay scheme for the low-lyingstates in
Th, that are known by now from the experiment [4, 5]. Here, one can easily observerotational bands characteristic to the deformed nuclei. Thus, we perform theoretical analysisfor this deformed neutron-odd nuclei basing on the ideas of the unified model proposed in thepapers [6]– [9].The wave function of the axially-symmetric odd nuclei in the framework of the unified modelreads as Ψ
JMK = s J + 116 π h D JMK ( θ i ) · χ K + D JM − K ( θ i ) · χ JK i . (1)Second term in (1) provides symmetry of the wave function to reflection relatively the planeorthogonal to the symmetry axis, while χ K = X Nℓ Λ s x K ( N ℓ Λ s ) | N ℓ Λ s i , χ JK = X Nℓ Λ s ( − J − ℓ − / x K ( N ℓ Λ s ) | N ℓ − Λ − s i . (2) E-mail [email protected] /2 + , 97.1 keV7/2 + , 42.4 keV5/2 + , gr.st. (7/2 + ), 71.8 keV(5/2 + ), 29.2 keV(3/2 + , 0.0076 keV) Th Band [N,n z , =633] Band [N,n z , =631] Q=+4.3(9) barn =0.33(7) * B(E2)= 332(8) W.u.** B(E2)=170(30)W.u.** B(M1)=0.0076(12)W.u.* B(E2)=92(4)W.u. ** B(E2)=6.2(8)W.u.** B(M1)=0.0117(14)W.u. . . n.m. Fm ,1 W.u.(M1)=1.79 (n.m.) = 0.24(1)=0.24(1)=0.18(2) ** B(M1)=0.011(4)W.u. . * data from the CE** data from decay ** B(E2)=65(7)W.u.** B(E2)=300(160)W.u. = 0.23(7)=0.20(1) ? = 0.22(1) (9/2 + ), 125.4 keV Figure 1:
Low-lying levels in Th In (1) and (2) χ K are Nilsson orbitals [10] that represent the decomposition of the single-particle functions of the axially-symmetric deformed potential over the spherical-symmetricfunctions, Λ and s are projections of orbital moment and spin on the symmetry axis, K = Λ + s .We define reduced transition matrix elements and reduced transition rates by the relations h J M | ˆ m ( λµ ) | J M i = ( − J − M J λ J − M µ M ! h J k ˆ m ( λ ) k J i , h J k ˆ m ( λ ) k J i = ( − J − J h J k ˆ m ( λ ) k J i . (3) B ( λ ; J → J ) = h J k ˆ m ( λ ) k J i J + 1 , B ( λ ; J → J ) = 2 J + 12 J + 1 B ( λ ; J → J ) . (4)For E m ( E , core ) µ = D µ ( θ i ) · π ZR · β | e | , while ˆ m ( E , s.p. ) µ = X ν D µν ( θ i ) · ˆ m ( E , intr. ) ν . (5)Then, we obtain h Ψ J K k ˆ m ( E , core ) + ˆ m ( E , s.p. ) k Ψ J K i == ( − J − J q (2 J + 1) C J K J K δ ( K , K ) δ ( α , α ) 34 π | e | · ZR · β ++ ( − J − J s J + 1)4 π · X Nℓ Λ(1 , x K ( N ℓ Λ s ) x K ( N ℓ Λ s ) . ( u u − v v ) ×× s ℓ + 12 ℓ + 1 C ℓ ℓ h δ ( s , s ) C J K K − K ) J K C ℓ Λ K − K ) ℓ Λ ++ δ ( s , − s )( − J − ℓ − / C J − K − K − K ) J K C ℓ − Λ − K − K ) ℓ Λ i e eff h | r | i . (6)2n (6) u and v are the coefficients of the Bogoliubov transformation, that accounts thesuperfluid correlations, while e eff is the effective quadrupole charge for the odd particle.Quadrupole moment of state is expressed via the reduced E Q ( J ) = vuut π J (2 J − J + 1)(2 J + 1)(2 J + 3) h J k ˆ m ( E k J i . (7)Consider now M M l -forbidden transitions in spherical nuclei. Inthis way, the M m ( M µ = s π µ N h g R ˆ J + ( g ℓ − g R )ˆ ℓ + ( g s − g R )ˆ s + δ ˆ µ ( M , tens. ) i µ , (8)where δ ˆ µ ( M , tens. ) µ = κr [ Y ⊗ ˆ s ] µ · τ . (9)In (9) τ = +1 for neutrons ( n ) and τ = − p ); κ = − .
031 fm − ; g ℓ ( p ) ≈ . g ℓ ( n ) ≈ . , g s ( p ) = 3 . g s ( n ) = − . g R = Z/A = 90 /
229 = 0 . . The values ofparameters g l , g s and κ were defined by us before [11, 12] from the description of magneticmoments as well as l -allowed and l -forbidden M M h Ψ J K k ˆ m ( M , core ) + ˆ m ( M , s.p. ) k Ψ J K i == g R µ N δ ( K , K ) δ ( J , J ) δ ( α , α ) s J ( J + 1)(2 J + 1)4 π ++ ( − J − J s J + 1)4 π µ N X Nℓ Λ s (1 , x K ( N ℓ Λ s ) x K ( N ℓ Λ s ) . ( u u + v v ) ×× ( ( g ℓ − g R ) q ℓ ( ℓ + 1) δ ( n , n ) δ ( ℓ , ℓ ) (cid:20) δ ( s , s ) C J K K − K ) J K ×× C ℓ Λ K − K ) ℓ Λ + δ ( s − s )( − J − ℓ − / C J − K − K − K ) J K C ℓ − Λ − K − K ) ℓ Λ (cid:21) ++ ( g s − g R ) √ δ ( n , n ) δ ( ℓ ℓ ) (cid:20) δ (Λ , Λ ) C J K K − K ) J K C / s K − K )1 / s ++ δ (Λ , − Λ )( − J − ℓ − / C J − K − K − K ) J K C / − s − K − K )1 / s (cid:21) − (10) − κ h | r | i (cid:20) C J K K − K ) J K h ℓ Λ / s | [ Y ⊗ ˆ s ] K − K ) | ℓ Λ / s i ++( − J − ℓ − / C J − K − K − K ) J K h ℓ − Λ / − s | [ Y ⊗ ˆ s ] − K − K ) | ℓ Λ / s i (cid:21)) . K = Λ + s , K = Λ + s , while h ℓ Λ / s | [ Y ⊗ ˆ s ] µ | ℓ Λ / s ] i = 32 s ℓ + 1)2 π C ℓ ℓ ×× X j j q (2 j + 1) C j K j K µ C j K ℓ Λ / s C j K ℓ Λ / s ℓ / j ℓ / j , µ = Λ + s − Λ − s ; (11)Magnetic moments of states are defined by the relation µ J = s πJ J + 1)(2 J + 1) h J k ˆ m ( M k J i . (12)For the E β and the entering values of J and K : Q ( J, K ) = 3 K − J ( J + 1)( J + 1)(2 J + 3) Q ; Q ( K = J ) = J (2 J − J + 1)(2 J + 3) Q ; Q = 3 √ π | e | ZR · β. (13) B ( E J + 1 , K → J, K ) = 3 K ( J + 1 + K )( J + 1 − K ) J ( J + 1)( J + 2)(2 J + 3) · π Q , (14) B ( E J + 2 , K → J, K ) = 3( J + 2 + K )( J + 1 + K )( J + 2 − K )( J + 1 − K )(2 J + 2)(2 J + 3)( J + 2)(2 J + 5) · π Q . (15)By using experimental data shown in Fig.1 and formulas (13)–(15), one can easily definethe magnitude of the deformation parameter β which average value turns out to be β ≈ . β , that corresponds to maximal value of the binding energy B in Th obtained in calculations [13], which were performed in the Hartree–Fock–Bogoliubovapproach with the Gogny interaction. This value of β was used by us in our calculations thatinvolve the “intrinsic” function χ .Consider now transitions between the states of different bands | ( J , J ′ ) K i → | ( J , J ‘2 ) K i ,where the initial as well as final states have different values of ( J, J ‘ ), but the same values of K .We see from formula (6) that in case of the E u u − v v ), which value is very sensitive to small variation of the single-particle scheme,especially when the entering single-particle orbitals are close to the Fermi level. This is just thecase under consideration. In addition, the value of the effective quadrupole charge e eff is ratherindefinite here, as it is not clear, what part of the quadrupole transition strength should beincluded in the single-particle mode after taking into account rotation of the core in the obviousway. Thus, direct calculations of the E λ satisfies the condition K + K > λ , as it takes place if we consider E M B ( λ ; J ′ K → J ′ K ) = [ C J ′ K λ ( K − K ) J ′ K ] [ C J K λ ( K − K ) J K ] B ( λ ; J K → J K ) . (16)As we know from the experiment the value of B ( E J = 9 / , K = 5 / → J = 5 / , K =3 /
2) = 6.2(8) W.u., we can define in this way all interband E M M u u + v v ) is closeto unity, while the values of g s and κ are known. Thus, calculations of M E M Th areshown in Tables 1–3. As one can see from Table 1, the magnitude B ( M
1; 9 / , / → / , / g s ( n ) ≈ g s ( n ) free , which contradicts generally established conception. Note that if weborrow the value of the M h χ / | ˆ m ( M , s.p. ) | χ / i from the experimental data on the | / , / i → | / , / i and | / , / i → | / , / i M Th, as compared to results of direct calculations.By using data on transition rates shown in Table 1, B ( M
1; 3 / , / → / , /
2) = 0 . µ N and B ( E
2; 3 / , / → / , /
2) = 8 . γ -transition [15] (also the private communication of M.B. Trzhaskovskaya), we findthe half-lives for this transition equal to T / ( M
1) = 5 . · − s and T / ( E
2) = 2 . · − s (in-cluding electron conversion). Here, conversion coefficients are very large: α M tot (0 . ≈ . · and α E tot (0 . ≈ . · . It is important that at such small transition ener-gies, conversion coefficients rapidly grow with decrease of the transition energy (approximately, α M tot ∼ / (∆ E ) − ǫ and α E tot ∼ / (∆ E ) − ǫ , where ǫ ∼ . As a result, the half-life of the stateof interest at such small transition energies in practice does not depend on energy, but only onthe transition matrix element . In [16, 17] one can find other evaluations of the magnitude of T / (3 / , / → / , / .Below, we discuss the problem of population of the above-mentioned isomeric state by themethod different from α and β -decays. In the paper [19], authors proposed the method whichemploys synchrotron radiation, while in [20] the authors suggested pumping m Th by thehollow-cathode discharge. Here, we consider the chance for excitation of the isomeric state in The latest theoretical estimations for transition rates in
Th are in [18] E M N n z (1)Λ ] = [633] and [ N n z (2)Λ ] = [631] in Th. Calculations ofthe E B ( E J = 9 / K = 5 / → J = 5 / K = 3 / . E
2) =83.2 e fm . Results on the B ( M
1) values are shownin the units of µ N (1 W.u.( M
1) = 1 . µ N ), and they were obtained by calculations basing onformula (10) with β = 0 . , g s ( n, eff) = − ,
04 and κ = − .
031 fm − . E, M ( λ ) J K J K B (1 → E, M ( λ ) J K J K B (1 → E E E M E M E M E M E M E M E M E M E For the E dσ E ( ξ, ϑ ) d Ω = Z e ¯ hv ! a B ( E ↑ ) df E ( ξ, ϑ ) d Ω ,a ≈ . Z Z E (MeV) (1 + A /A ) · − cm , ξ = Z Z A / (1 + A /A )∆ E . E − / · ∆ E ) / . (17)Here, A , Z and E refer to the projectile, E and ∆ E are energy in the laboratory systemand the excitation energy in MeV, a is half the distance of the closest drawing in the backwardscattering. Functions f E ( ξ, ϑ ) are expressed [22] via integrals over trajectories. If ∆ E/E = 0,then we obtain df E ( ξ = 0 , ϑ ) d Ω = π (h − π − ϑ ϑ i · ϑ/ ) . (18)In a general case, we have [22] σ E ( ξ ) = 4 . A E kin ( A , MeV) B ( E ↑ , barn ) Z (1 + A /A ) f E ( ξ ) barn , (19)where f E ( ξ = 0) = 0.895. 6able 2: Reduced E M K = 5 / K = 3 / B ( E
2) values are in the Weisskopf units and were calculated byusing β = 0 .
22. Numbers in square brackets show experimental results [4, 5]. The M1 rates arein the units of µ N , and they were calculated by using g s ( n, eff) = − .
04 and κ = − .
031 fm − . E, M ( λ ) J K J K B (1 → E, M ( λ ) J K J K B (1 → E E E M E M E M E M E M E Table 3: Electric quadrupole and magnetic dipole moments of the lowest states of
Th. Here,by calculation of quadrupole moments we used averaged value of β =0.22, while by calculationof magnetic moments we used β = 0 . , g s ( n, eff) = − .
04 and κ = − .
031 fm − . Quantity(
J , K ) Exp. Calc. Quantity(
J , K ) Exp. Calc. Q (5 / , /
2) +4.3(9) barn +2.9 barn Q (3 / , /
2) – +1.6 barn µ (5 / , /
2) +0.46(4) µ N +0 . µ N µ (3 / , /
2) – +0 . µ N For the M dσ M ( ξ, ϑ ) d Ω = Z e ¯ hc ! λ − c ( p ) B ( M ↑ ) df M ( ξ, ϑ ) d Ω , λ − c ( p ) = ¯ hm p c . (20)For ξ = 0 we obtain df M ( ξ = 0 , ϑ ) d Ω = 16 π − ( π − ϑ ) / · tan ϑ/ sin ϑ . (21)We see from Eq.(21), that by ∆ E → ϑ → M ξ = 0, P ( M , ξ = 0 , ϑ ) = dσ ( M , ϑ ) /dσ (Coul , ϑ ) ∼ ϑ by ϑ →
0. Thus, the divergence of the M ϑ → ϑ →
0, inthis case the the colliding nuclei are far from each other, and the Coulomb interaction betweennuclei is really screened by the electron clouds. Really, almost all electron charge of atom islocated at distances less than the Bohr radius R B = ¯ h / ( m e e ). In this way, we should excludeintervals more than R max , i.e. exclude scattering angles less than ϑ min , where ϑ min = 2 arcsin (cid:16) R max /a − (cid:17) ≈ aR max , R max ≈ R B . (22)7able 4: Comparison between the cross sections σ and the “effective” cross sections σ eff for the Coulomb excitation of the Th levels by protons and α -particles.Energy Protons, 6 MeV Protons, 10 MeV He, 10 MeVLevel keV σ , barn σ eff , barn σ , barn σ eff , barn σ , barn σ eff , barn3 / +1 / +2 / +1 / +2 / +1 / +2 σ M = 0 . · − Z B ( M ↑ ) f M ( ξ = 0 , ϑ min ) barn . (23)Here, B ( M
1) is in the units of µ N and f M ( ξ = 0 , ϑ min ) = 32 π π Z ϑ min [1 − ( π − ϑ ) / · tan ϑ/ sin ϑ dϑ . (24)For ϑ min 1 , less than 1 we have f M ( ξ = 0 , ϑ min 1 ) ≈ f M ( ξ = 0 , ϑ min 2 ) + 32 π (cid:18) ϑ min 2 ϑ min 1 (cid:19) . (25)For protons and α -particles with energies 10 MeV bombarding Th, ϑ min ∼ . and f M ( ξ = 0 , ϑ min = 0 . ) = 186. The corresponding cross section is negligible as comparedto the E f M rapidly decreases, see also [23]. Thus, all levels considered by ushere, are populated in the Coulomb excitation by means of the E / + level may happen not onlydue to the direct Coulomb excitation from the ground state, but also due to the dischargingof the excited higher-lying states. This process is very important as many of these statesare actively excited due to large B ( E
2) values. In this way, we took into account excitationof all levels shown in Fig. 1, as well as all possible E M B ( E
2) and B ( M
1) values were borrowed by us from Tables 1 and 2, while thenecessary conversion coefficients were borrowed from [24]. Results of our calculations of crosssections are demonstrated in Table 4. Here, σ corresponds to the direct excitation, while σ eff is the effective cross section, that includes settlement of the 3 / +1 state by γ -transitions fromthe high-lying levels. One can easily see that the allowance of feeding from the high-lyingstates leads to considerable increase of population of the isomeric state. Note, that taking intoaccount additional excited states leads to further increase of σ eff as compared to σ .8or example, let’s take the foil of Th with thickness d = 10 µm . The density ρ of Th isabout 3 · atoms/cm . Suppose that we have constant in time beam of 10 MeV protons witha beam current j equal to 1 µ A ( ∼ . · atoms/s).Then, the counting rate for transitionsfrom the 0.0076 keV level (allowing also for the settlement of this level from the high-lying statesthat are excited in the process of the Coulomb excitation) is N = j · σ eff · ρ · d ≈ · s − .However, this level decays mainly by the electron conversion ( α M tot ≈ . · ). Thus, thecounting rate for γ -quanta is only N γ ∼ · − s − , i.e. ∼
20 d − . However, one should keepin mind that metallic Th is not transparent for “blue” γ -rays. Thus, it is better to use a targetfrom the radiolucent glassy material containing Th atoms.The author acknowledge M.B. Trzhaskovskaya for discussions and calculations concerningproblems of atomic structure, as well as Yu. N. Novikov and A.V. Popov for useful criticalremarks. References [1] B.R. Beck, J.A. Becker, P. Beiersdorfer, et al., Phys. Rev. Lett. , 142501 (2007).[2] Lars von der Wense, B. Seiferle, M. Laatiaoui et al. , Nature , 47 (2016).[3] B. Seiferle, Lars von der Wense and P.G.Thirolf, Phys. Rev. Lett. , 042501 (2017).[4] S.J. Goldstein, et al. Phys. Rev. C , 2793 (1989).[5] E. Browne, J.K. Tuli, Nucl. Data Sheets , 2657 (2008).[6] A. Bohr, Dan. Mat. Fys. Medd. , No. 14 (1952).[7] A. Bohr and B. Mottelson, Dan. Mat. Fys. Medd. , No. 16 (1953).[8] A. Bohr, Rotational states of atomic nuclei , Copenhagen (1954).[9] A. Bohr and B. Mottelson, Dan. Mat. Fys. Medd. , No. 1 (1955).[10] S.G. Nilsson, Dan. Mat. Fys. Medd. , No. 16 (1955).[11] S.A. Artamonov, V.I. Isakov, S.G. Kadmensky et al. , Sov. J. Nucl. Phys. , 486 (1982).[12] V.I. Isakov, Physics of Atomic Nuclei, , 811 (2016).[13] S. Hilaire and M. Girod, Eur. Phys. J., A , 237 (2007); see alsohttp://phynu.cea.fr/HFB-Gogny eng.htm[14] G. Alaga, K. Alder, A. Bohr, B. Mottelson, Dan. Mat. Fys. Medd. , No 9 (1955).[15] F.F. Karpeshin and M.B. Trzhaskovskaya, Phys. Rev. C , 054313 (2007).[16] V.F. Strizhov and E.V. Tkalya, Sov.Phys. JETP , 387 (1991).917] E.V. Tkalya, C. Schneider, J. Jeet, and E.R. Hudson, Phys. Rev. C , 054324 (2015).[18] Nikolay Minkov and Adriana P´alffy, arXiv:1704.07919v2[nucl.th].[19] J. Jeet, Ch. Schneider, S.T. Sullivan, et al., Phys. Rev. Lett. , 253001 (2015).[20] N.N. Inamura, T. Mitsugashira, et al., Hyperfine Interactions , 115 (2005).[21] K.A. Ter-Martirosyan, Sov. Phys. JETP , 284 (1952).[22] K. Alder, A. Bohr, T. Huus, B. Motetlson, and A. Winther, Rev. Mod. Phys. , 432(1956).[23] K. Alder and A. Winther, Dan. Mat. Fys. Medd., , No. 1 (1956).[24] T. Kibe’di, N.W. Burrows, M.B. Trzhaskovskaya, et al., Nucl. Instr. and Meth., A589