QRPA and MQPM

s

1 + ηz

E_ω, v_z = 1

√2

s

1− η_z E_ω, η_z =_z−µ_z−λ, E_ω =^qη_z²+ ∆²_z, µ_z =−ˆj_z^−2X

J b

v_b²Jˆ²[N_zb(J)]⁻²hzb;J|V |zb;Ji,

∆z =−ˆj_z^−1X

J c

ˆjcucvchzz; 0|V |cc; 0i and n=^X

a

ˆj_a²v_a².

(221)

In practice these equations are solved iteratively by, for example, supplying the initial values of ∆_z andλ_z, calculating the other relevant quantities of equations 221, calculating new values for ∆_z andλ_z and repeating the process until self-consistency is reached[10, Chapter 14]. This is usually measured in the absolute difference between the calculated average particle number and the desired particle number, with self-consistency considered achieved when this number is smaller than a limit set before starting the calculations. Different species of nucleons are also considered separately in practical calculations.

2.2.4 QRPA and MQPM

While the BCS model which was presented earlier is a successful nuclear model by itself, it is not the model that is of primary interest in this thesis. Instead, we will use it as a starting point in deriving a more sophisticated model known as the quasiparticle random-phase approximation (QRPA), which in turn will serve as a basis for the actual model used, the microscopic quasiparticle-phonon model (MQPM). The primary advantage of the QRPA over similar models, such as the quasiparticle Tamm-Dancoff approximation (QTDA)[10, Chapter 16], is the inclusion of ground-state correlations which can lead to transitions between states that are highly collective in nature. The most prominent flaw of the model is that the equations do not arise from a variational principle and they are approximations by nature. Similar to the BCS equations, the QRPA equations can be derived with the

EoM method. The derivation is relatively straightforward but somewhat tedious, and an overview of this is presented below.

The basic excitation of QRPA can be written in terms of the variational parameters X_ab^ω and Y_ab^ω, and the quasiparticle operators defined previously as[10, Page 558]

Q^†_ω =^X

ab

hX_ab^ωN_ab(J)^ha^†_aa^†_bⁱ

J M +Y_ab^ωN_ab(J) [˜a_a˜a_b]_{J M}ⁱ

≡^X

ab

hX_ab^ωA^†_ab(J M)−Y_ab^ωA˜_ab(J M)ⁱ,

(222)

where

A^†_ab(J M)≡ N_ab(J)^ha^†_aa^†_bⁱ

J M, A˜_ab(J M) =−N_ab(J) [˜a_a˜a_b]_{J M} and N_ab(J) =

q1 +δ_ab(−1)^J 1 +δ_ab .

(223)

The variations are thenAab(J M) and ˜A^†_ab(J M). The basic excitation is bosonic so we have for the former variation

hQRPA|[δQ,H,Q^†_ω]−|QRPAi=E_ωhQRPA|[δQ,Q^†_ω]|QRPAi ⇔ hQRPA|

"

Aab(J M),H,^X

cd

hX_cd^ωA^†_cd(J M)−Y_cd^ωA˜cd(J M)ⁱ

#

|QRPAi= E_ωhQRPA|

"

A_ab(J M),^X

cd

hX_cd^ωA^†_cd(J M)−Y_cd^ωA˜_cd(J M)ⁱ

#

|QRPAi

(224)

and for the latter hQRPA|

"

A˜^†_ab(J M),H,^X

cd

hX_cd^ωA^†_cd(J M)−Y_cd^ωA˜_cd(J M)ⁱ

#

|QRPAi= E_ωhQRPA|

"

A˜^†_ab(J M),^X

cd

hX_cd^ωA^†_cd(J M)−Y_cd^ωA˜_cd(J M)ⁱ

#

|QRPAi.

(225)

The derivation essentially consists of determining the expectation values of

hA_ab(J M),H,A^†_cd(J M)ⁱ, ^hA_ab(J M),H,A˜_cd(J M)ⁱ, ^hA_ab(J M),A^†_cd(J M)ⁱ and

hA_ab(J M),A˜_cd(J M)ⁱ

(226) along with their hermitian conjugates with respect to the the QRPA vacuum |QRPAi.

The exact form of the correlated QRPA ground state |QRPAi is not known prior to solving the QRPA equations, so the BCS ground state |BCSi will be used as an approximate vacuum to derive the equations[10, Page 558]. The approximation of replacing the QRPA vacuum expectation value of the commutator of operators Q_ω and Q^†_ω with the corresponding expectation value with respect to the BCS vacuum is known as the quasiboson approximation (QBA). The replacement of the QRPA ground state with the BCS ground state as an effective vacuum is the reason that the QRPA does not satisfy a variational principle.

With the BCS ground state as the approximate vacuum, it can then be shown that[10, Page 558]

hBCS|^hA_ab(J M),A^†_cd(J⁰M⁰)ⁱ|BCSi=hBCS|^hA_ab(J M),A˜_cd(J⁰M⁰)ⁱ|BCSi= δ_acδ_bdδ_{J J}⁰δ_{M M}⁰.

(227)

This reduces the right sides of equations 224 and 225 toEωX_ab^ω andEωY_ab^ω respectively.

For the left sides, we can define matrices Aand B with elements[10, Page 559]

A_ab,cd=hBCS|[A_ab(J M),H,A^†_cd(J M)] and B_ab,cd =− hBCS|[A_ab(J M),H,A˜_cd(J M)]

(228) when the Hamiltonian H is expressed in terms of the BCS quasiparticle operators as[10, Chapter 13.3.2]

H = 1 2

X

b

ˆj_b² E_b

(E_b−η_b)

η_b+1 2µ_b

−1 2∆²_b

+^X

b

ˆj_bE_b[a^†_b˜a_b]₀₀+V_RES, (229)

where

V_RES= 1 4

X

αβγδ

v_αβγδN[c^†_αc^†_βc_δc_γ], (230) with the normal ordering N[· · ·] taken with respect to the BCS vacuum. By substi-tuting the particle-hole operators by their expressions in terms of the quasiparticle operators (equations 188) in the above expression, the residual interaction V_RES can be shown to consist of a sum of parts[10, Chapter 16.1]

H₄₀ = 1 2

X

abcdJ

(−1)^JV_abcd⁽⁴⁰⁾(J)^ha^†_aa^†_bⁱ

J ·^ha^†_ca^†_dⁱ

J + h.c., (231)

H₃₁= ^X

abcdJ

(−1)^JV_abcd⁽³¹⁾(J)^ha^†_aa^†_bⁱ

J ·^ha^†_ca˜_dⁱ

J + h.c., (232) and

H₂₂ = 1 2

X

abcdJ

(−1)^JV_abcd⁽²²⁾(J)^ha^†_aa^†_bⁱ

J ·[˜a_ca˜_d]_J, (233) where

V_abcd⁽⁴⁰⁾(J) = −1

2[N_ab(J)N_cd(J)]⁻¹u_au_bv_cv_dhab;J|V |cd;Ji, (234)

V_abcd⁽³¹⁾(J) = −1

2[N_ab(J)N_cd(J)]⁻¹(u_au_bv_cu_d−v_av_bu_cv_d)hab;J|V |cd;Ji (235) and

V_abcd⁽²²⁾(J) =−1

2[N_ab(J)N_cd(J)]⁻¹(u_au_bu_cu_d−v_av_bv_cv_d)hab;J|V |cd;Ji+ 2u_av_bu_cv_d^X

J⁰

[N_ad(J⁰)N_cb(J⁰)]⁻¹Jˆ⁰²







j_a j_b J j_c j_d J⁰







had;J⁰|V |cb;J⁰i.

(236)

The above equations can be used to determine the explicit forms of the matrix elements of A and B. After a long, but a rather straightforward derivation these can be shown to be[10, Chapters 16.2 and 18.1.2]

A_ab,cd = (E_a+E_b)δ_acδ_bd+ (u_au_bu_cu_d+v_av_bv_cv_d)hab;J|V |cd;Ji+ Nab(J)Ncd(J)^h(uavbucvd+vaubvcud)hab⁻¹;J|VRES|cd⁻¹;Ji − (−1)^jc+jd+J(u_av_bv_cu_d+v_au_bu_cv_d)hab⁻¹;J|V_RES|dc⁻¹;Jiⁱ

(237)

and

B_ab,cd=−(u_au_bv_cv_d+v_av_bu_cu_d)hab;J|V |cd;Ji+

Nab(J)Ncd(J)^h(uavbvcud+vaubucvd)hab⁻¹;J|VRES|cd⁻¹;Ji − (−1)^jc+jd+J(u_av_bu_cv_d+v_au_bv_cu_d)hab⁻¹;J|V_RES|dc⁻¹;Jiⁱ.

(238)

The matrices can be shown to have the properties

A^† =A and B^T=B, (239)

and they can be used to write equations 224 and 225 as[10, Chapters 11.2.1 and which together constitute the QRPA equations. They can be combined into a single matrix equation

This is a non-Hermitian matrix eigenvalue problem for the excited states and excita-tion energies.

The QRPA is a realistic many-body theory that is widely in use in the present day nuclear physics research, but for the nuclei considered in this thesis it is not applicable by itself. The reason for this is that the QRPA is capable of describing even-even nuclei only (or odd-odd nuclei in the case of pnQRPA[10, Chapter 19]).

To model odd-even nuclei, we will first consider even-even nuclei adjacent to the odd-even nuclei of interest and solve the BCS and QRPA equations for the occupation amplitudesu_a andv_a, and the QRPA parameters X^ω and Y^ω. These will then be used to define the basic excitation Γ^†_k(jm) of the microscopic quasiparticle-phonon model by[8]

The MQPM states thus consist of single-quasiparticle components and quasiparti-cles coupled together with QRPA phonons that are essentially three-quasiparticle components with good angular momentum quantum numbers j and m.

The derivation of the MQPM equations is again similar to the derivation of the equations of the previous two models, and we will only present the key features of it below. A detailed derivation can be found in e.g. [8]. In the EoM method we have the variations a_b and ^ha^†_aQ^†_ω^i†

jm. Running these through the EoM method equations 163 while using the BCS state as the vacuum results in the matrix equation



The explicit forms of the submatricesA, B, A⁰ and N are presented in appendix C.

After solving the above equation for Cⁱ and Dⁱ, the MQPM states can be constructed by operating with Γ^†_k(jm) on the BCS state. In the case of this thesis, the most relevant quantities associated with these states are the reduced neutral-current one-body transition densities between them. To see why, we return to the definitions of the multipole operators of equations 78. These operators Oλµ are spherical tensors, so they can be expressed in the second quantized form of equation 152 as[24]

Oλµ=λ⁻¹





X

proton orbitalsab

(a||O^p_λ||b)^hc^†_ac˜b

i

λµ+ ^X

neutron orbitalsab

(a||O_λⁿ||b)^hc^†_a˜cb

i

λµ



, (245) where we have decomposed the operator O_λµ into partsO^p_λµ and O_λµⁿ that operate on only the protons and neutrons respectively (O_λµ,O^p_λµ and O_λµⁿ are functions of q in the case of the multipole operators of equations 78). The reduced matrix elements of the multipole operators appearing in equation 104 then consist of sums of terms proportional to (again suppressing the quantum numbers of the nuclear states other than the angular momenta J_i andJ_f) (a||O^r_λ||b)(J_f||^hc^†_a˜c_bⁱ

λ||J_i), where r=p orn.

The factor (a||O^r_λ||b) will be considered in the next section while (J_f||^hc^†_a˜c_bⁱ

λ||J_i), the reduced one-body neutral-current transition density, is precisely where the nuclear physics enters into the scattering cross section calculation. It is independent of the scattering process and is determined solely from the nuclear model employed. The explicit forms of these reduced transition densities are somewhat complicated and will not be derived here. They can be found in appendix C.

In document Neutral current supernova neutrino-nucleus scattering off 127I and 133Cs (sivua 65-70)