Odd-particle number random phase approximation and extensions: Applications to particle and hole states around $^{16}$O

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by  Mitsuru Tohyama

Odd-particle number random phase approximation and extensions: Applications to
particle and hole states around 16O
Mitsuru Tohyama1 and Peter Schuck2,3
1Kyorin University School of Medicine, Mitaka, Tokyo 181-8611, Japan
2Institut de Physique Nucle´aire, IN2P3-CNRS, Universite´ Paris-Sud, F-91406 Orsay Cedex, France
3Laboratoire de Physique et de Mod´elisation des Milieux Condens´es, CNRS et,
Universit´e Joseph Fourier, 25 Av. des Martyrs, BP 166, F-38042 Grenoble Cedex 9, France
Thehole-staterandomphaseapproximation(hRPA)andtheparticle-staterandomphaseapprox-
imation(pRPA)forsystemslikeoddAnucleiarediscussed. ThesehRPAandpRPAareformulated
based on the Hartree-Fock ground state. An extension of hRPA and pRPA based on a correlated
groundstateisgivenusingtime-dependentdensity-matrixtheory. Applicationstothesingle-particle
states around 16O are presented. It is shown that inclusion of ground-state correlation affects ap-
3
preciablytheresultsofhRPAandpRPA.Thequestionofthecouplingofthecenterofmassmotion
1
of thecore to theparticle (hole) is also discussed.
0
2
PACSnumbers: 21.60.Jz, 21.10.Pc,27.20.+n
n
a
J
I. INTRODUCTION II. FORMULATION

0
1
Letus considera nucleus consistingof A nucleonsand

] The one-particle states and one-hole states are basic assume that the total Hamiltonian H consists of the ki-
h excitation modes of a nucleus and other many body sys- netic energy term and a two-body interaction. Let us
t tems. The experimental data on the properties of the
- assumethat |0iis the groundstate ofthe Anucleonsys-
cl single-particle states have been accumulated using nu- tem with A even and with energy E0 and |µi an exact
clear reactions such as one-nucleon transfer, pickup and
u eigenstate of the Hamiltonian for the A−1 system with
n knock-outreactions[1],and ithas been foundthat there an eigenvalue E (H|µi=E |µi).
µ µ
[ is a substantial depletion of the spectral strength of the

single-particle states. Theoretical studies have shown
1
v that the strong short-range and tensor components of A. Equations of motion for transition amplitudes
6 thenucleon-nucleoninteractionareresponsibleforapart
2 of the depletion [2] and a substantial part of the frag-
In direct reaction theories such as the the distorted
0 mentation of the single-particle strength is due to the
waveimpulseapproximationandthedistortedwaveBorn
2 coupling to low-lying collective modes [3, 4]. The stan-
approximation the differential cross section for one nu-
.
1 dardapproachtostudythesingle-particlepropertiesmay cleontransferreactionsisrelatedtothespectralfunction
0 be the Green’s function method. Various theoretical ap-
3 proaches have been proposed to implement the coupling Sαα′(ω)
:1 to low-lying collective modes into the self-energy of the Sαα′(ω)= h0|a+α′|µihµ|aα|0iδ(ω+Eµ−E0), (1)
v Green’s function: the particle-phonon coupling model Xµ
i [3,5–7],theTamm-Dancoffapproximation(TDA)[8]and
X the more recent Faddeev random-phase approximation whereaαanda+α aretheannihilationandcreationopera-
r (FRPA) [9]. In the present paper we give a formula- torsofanucleoninasingle-particlestateα,respectively.
a
tionofthe hole-stateRPA(hRPA) usingthe equationof We consider the equations of motion for the transition
amplitudes xµ and Xµ from the A nucleon system to
motion approach (EoM) [10], which has often been used α αβ:γ
to derive the standard RPA, and discuss some aspects the A−1 nucleon system. These amplitudes are defined
by
of hRPA such as the relation to the particle-state RPA
(pRPA), whichhavenot been clarifiedso far inthe liter- xµ =h0|a+|µi, (2)
α α
ature[8,11]. WealsopresentanextensionofoddARPA
Xµ =h0|:a+a+a :|µi, (3)
(oRPA)basedonacorrelatedgroundstateobtainedfrom αβ:γ α β γ
the time-dependent density-matrixtheory (TDDM) [12– where : : implies
14]. The influence of the center-of-mass (c.o.m.) motion
:a+a+a :=a+a+a −(n a+−n a+). (4)
oftheevencoreontheoddsystemisalsodiscussed. The α β γ α β γ γβ α γα β
paper is organized as follows: the formulation of hRPA
Here, nαα′ is the occupation matrix given by
and its extension is given in sect. 2, some properties of
theextendedRPAarealsodiscussedinsect.2,theresults nαα′ =h0|a+α′aα|0i. (5)
obtainedforthesingle-particlestatesaround16Oarepre-
From the EoM relation
sented in sect. 3, and sect. 4 is devoted to a discussion
and conclusion section. h0|[H,a+]=h0|a+(E −H), (6)
α α 0
2
we obtain the equation for xµ The matrix elements of the above equation are given
α
in Appendix A. The normalization of the amplitudes is
h0|[H,a+α]|µi=ωµh0|a+α|µi=ωµxµα, (7) given by
wsidheeroefωtµhe=aEbo0v−e eEqµu.atTiohne icnocmlumduestatteorrmosnwtihtehlaef+αt-ah+βaanγd. (cid:16)x˜µ ∗ Y˜µ ∗(cid:17)(cid:18)XYµµ′′ (cid:19)=δµµ′, (14)
Therefore, xµα couples to Xαµβ:γ. In a way analogous to where x˜µ ∗ and X˜µ ∗ are the left eigenvector of Eq.
that used in deriving Eq. (7), we obtain the equation for αα′ αβα′β′
(13). The occupationmatrix andthe correlationmatrix,
Xµ
αβ:γ whichenterEq. (13)andwhichdescribetheground-state
correlationsin the A nucleon system, can be determined
h0|[H,:a+a+a :]|µi=ω h0|:a+a+a :|µi
α β γ µ α β γ in the framework of Time Dependent Density Matrix
=ωµXαµβ:γ. (8) (TDDM) theory: the TDDM equations [12, 14] consist
of the coupled equations motion for nαα′ and Cαβα′β′,
Onthe left-handside ofthe aboveequationthereappear
expectation values of the terms consisting of three cre- i~n˙αα′ =h0|[a+α′aα,H]0i, (15)
ation operators and two annihilation operators such as
h0|a+λ1a+λ2a+λ3aλ4aλ5|µi, which implies the coupling to a i~C˙αβα′β′ =h0|[:a+α′a+β′aβaα :,H]|0i. (16)
higher-levelamplitude h0|:a+αa+βa+γaβ′aα′ :|µi. Toclose The right-handside of Eq.(16) contains the expectation
the chain of the coupled equations, we factorize these values of three-body operators, which are approximated
terms using xµα and Xαµβ:γ as bythe productsofnαα′ andCαβα′β′ to closethe coupled
chain of the equations of motion. The ground state in
h0|a+λ1a+λ2a+λ3aλ4aλ5|µi+≈CAS(nλ4λ3xXµλµ1)λ,2:λ5 (9) TeqDuDatMionisswghiviecnh saastiasfisetsatn˙iαoαn′ar=y0soalnudtioC˙nαβoαf′βth′ e=T0D. TDhMe
λ5λ4λ2λ3 λ1 stationary solution can be obtained using the gradient
where the correlation matrix Cαβα′β′ is defined by method [15]. This method will be used in our numerical
Cαβα′β′ = h0| : a+α′a+β′aβaα : |0i and AS( ) means that application given later.
the terms in the parentheses are properly antisymmter- In the Hartree-Fock approximation (HF), nαα′ =δαα′
ized [12]. The obtained coupled equations are written for hole states and nαα′ = 0 for particle states, and
as Cαβα′β′ = 0. Keeping in Eq. (13) only the amplitudes
xµ and Xµ , where p and h refer to a particle state
(ǫα−ωµ)xµα+ hλ1λ2|v|αλ3iXλµ1λ2:λ3 =0,(10) anhd a holehhs′t:pate, respectively, corresponds to the TDA
λ1Xλ2λ3 equation of odd particle systems [16]. However, even
within the HF-ground state, Eq. (13) can have all com-
ponentsofXµ ; Xµ ,Xµ ,Xµ ,Xµ ,Xµ
(ǫ +ǫ −ǫ −ω )Xµ + hλ λ |v|λ λ i αβ:γ hh′:p pp′:h hh′:h′′ pp′:p′′ hp:p′
α β γ µ αβ:γ 1 2 3 4 A and Xµ . Such equations have been proposed for the
λ1λX2λ3λ4 hp:h′
firsttime in Ref. [11]andthey havebeen appliedin Ref.
×[(δ ((δ −n )n −C )
λ3α λ4β λ4β γλ2 γλ4λ2β [8]. This very much extended configuration space actu-
+δλ3β(nλ4αnγλ2 +Cγλ4λ2α) ally leads to some difficulties which have been discussed
1 in Ref. [8]. We will take up this discussion again below.
+δ (n n + C ))xµ
λ2γ λ3α λ4β 2 λ3λ4αβ λ1 Inthefollowingwediscusstherelationofxµα withnαα.
+δλ3αnγλ1Xλµ2β:λ4 +δλ3βnγλ2Xλµ1α:λ4 U(1s0i)ngfoErqx.µ(,1w0)efocarnxµαe′limanidnattheeωcomopblteaxincionngjuagnaetequoaftEioqn.
−δλ1γ(nλ4αXβµλ2:λ3 +nλ3βXαµλ2:λ4) for µxµαα′(xµα)∗ µ
1
+ 2(δλ3αδλ4β −δλ3αnλ4β +δλ3βnλ4α)Xλµ1λ2:γ] P(ǫα−ǫα′) xµα′(xµα)∗
=0, (11) Xµ
+ [hαλ |v|λ λ i xµ (Xµ )∗
wherethesubscriptAmeansthatthecorrespondingma- 3 1 2 α′ λ1λ2:λ3
trix is antisymmetrizedand the single-particle states are λ1Xλ2λ3 Xµ
chosen as the eigenstates of the matrix −hλ1λ2|v|α′λ3i Xλµ1λ2:λ3(xµα)∗]=0. (17)
Xµ
′ ′ ′
hα|t|αi+ hαλ|v|αλiAnλ′λ. (12) On the other hand the stationary condition n˙αα′ =0 for
Xλλ′ Eq. (15) gives [15]
Heretisthekineticenergyoperator. Equations(10)and (ǫα−ǫα′)nαα′
(11) are written in matrix form:
+ [hαλ3|v|λ1λ2iCλ1λ2α′λ3
a c xµ xµ λ1Xλ2λ3
(cid:18) b d(cid:19)(cid:18)Xµ (cid:19)=ωµ(cid:18)Xµ (cid:19). (13) −hλ1λ2|v|α′λ3iCαλ3λ1λ2]=0. (18)
3
Equations (17) and (18) suggest that xµ (xµ)∗ and
µ α′ α
µXαµ′β′:β(xµα)∗correspondtonαα′ andPCαβα′β′,respec-
Ptively, though the symmetry of Cαβα′β′ under the ex-
change of α and β is lost in Xµ (xµ)∗. We will
µ α′β′:β α
showbelowthatnαα = µxµαP(xµα)∗approximatelyholds
in the applications to 16PO.
B. Equation of motion approach with excitation
operator
Equation (13) lacks some effects such as self-energy
contributions in the configuration Xµ , which should
αβ:γ
be included when a correlated ground state is used. In
order to take account of such effects, we present another
formulationwhichisbasedonEoM[10]. Introducingthe
excitation operator q+ FIG. 1. (a) Self-energy contribution to a particle state and
µ
(b) that to a hole state. The ellipses denoteCαβα′β′ and the
dots theresidual interaction.
q+ = yµ a + Yµ :a+a a : (19)
µ α α αβ:γ γ β α
Xα αXβγ
in [H,:a+a+a :]:
and assuming, as usual, q+|0i = |µi and q |0i = 0 (for α β γ
µ µ
the existence of such a relation, see below), we obtain
[ H,:a+a+a :]= c(αβγ :λ)a+
from Eqs. (7) and (8) α β γ α′
Xα′
(cid:18)BA DC (cid:19)(cid:18)Yyµµ (cid:19)=ωµ(cid:18)NN2111 NN2122 (cid:19)(cid:18)Yyµµ (cid:19),(20) +αX′β′γ′d(αβγ :α′β′γ′):a+α′a+β′aγ′ :
where the matrices are defined as + e(αβγ :λ1λ2λ3λ4λ5)
λ1λ2Xλ3λ4λ5
B(αAβ(αγ ::αα′′))==hh00||{{[[HH,,a:+αa+α],aa+βαa′}γ|0:]i,,aα′}|0i, ((2221)) ×:a+λ1a+λ2a+λ3aλ5aλ4 :. (30)
Using
C(α:α′β′γ′)=h0|{[H,a+α :],:a+γ′aβ′aα′}|0i,
D(αβγ :α′β′γ′ ) N32 ( λ1λ2λ3λ4λ5 :αβγ)
=h0|{[H , :a+αa+βaγ :],:a+γ′aβ′aα′ :}|0i, (23) =h0|{:a+λ1a+λ2a+λ3aλ5aλ4 :,:a+γaβaα :}|0i, (31)
D canbeexpressedase×N . Thesetermsinclude,for
2 32
N11(α:α′)=h0|{a+α,aα′}|0i=δαα′, (24) example,
NN1221((αα:βαγ′:βα′γ′))==hh00||{{:a+αa+α,:aa+β+γa′γaβ:,′aaαα′′}:}|0|0ii==00,,(25) −21δαα′δββ′λ1Xλ2λ3hλ1λ2|v|γ′λ3iACγλ3λ1λ2,
′ ′ ′
N (αβγ :αβ γ )
22 which is a self-energy contribution to the state γ. The
=h0|{:a+αa+βaγ , :a+γ′aβ′aα′ :}|0i. (26) self-energycontributionsareschematicallyshowninFig.
1.
Here { } implies the anticommutator, {A,B} = AB +
The normalization of the amplitudes is given by
BA. The norm matrix N is given in Appendix A. The
22
minaEtrqi.x(e1l3em) seuncthsiansEq.(20)canbeexpressedusingthose (yµ ∗ Yµ ∗)(cid:18)NN2111 NN2122 (cid:19)(cid:18)Yyµµ′′ (cid:19)=δµµ′. (32)
A=a×N (27)
11 The closure relation is written as
B =b×N , (28)
11
C =c×N22. (29) Xµ (cid:18)Yyµµ (cid:19)(yµ ∗ Yµ ∗)(cid:18)NN2111 NN2122 (cid:19)=I, (33)
The matrix D consists of the two types of terms, one
expressed by D = d×N and the other given by D , where I is the unit matrix. We refer to the formulation
1 22 2
whichoriginatesfromthetermswith:a+ a+ a+ a a : Eq. (20) as the extended odd-RPA (EoRPA).
λ5 λ4 λ3 λ2 λ1
4
In the following we discuss the relation between Eqs.
(13) and (20). The transition amplitudes xµ and Xµ
α αβ:γ
are given by yµ and Yµ as
α αβ:γ
xµ N N yµ
= 11 12 . (34)
(cid:18)Xµ (cid:19) (cid:18)N21 N22 (cid:19)(cid:18)Yµ (cid:19)
Inserting this expression into Eq. (13), we obtain
A C yµ N N yµ
=ω 11 12 .(35)
(cid:18)B D1 (cid:19)(cid:18)Yµ (cid:19) µ(cid:18)N21 N22 (cid:19)(cid:18)Yµ (cid:19)
The difference between Eqs. (20) and (35) and thus be-
tween Eqs. (20) and (13) resides in the matrix D. Some
effects of the ground-state correlations such as the self-
energy contributions are missing in Eq. (35) and thus in
Eq. (13) as mentioned above. The importance of these
missing terms will be discussed below in the application FIG. 2. (a) Mass operator for a particle state described by
section. Ypµp′:h and (b) that for a hole state described by Yhµh′:p. The
circles mean the propagators given by Ypµp′:h ((a)) and Yhµh′:p
((b)),and thedots the residual interaction.
1. Symmetry properties
FirstweshowthattheHamiltonianmatrixofEq.(20) a violation of the hermiticity of Eq. (20), though it will
is hermitian. We use the operator identity turn out to be small.
Next we discuss the relationbetween the formulations
h0|{[H,Aˆ],Bˆ}|0i+h0|{[H,Bˆ],Aˆ}|0i for a hole state and a particle state. We can obtain a
=h0|[H,{Aˆ,Bˆ}]|0i. (36) formulation for a particle state using the excitation op-
erator
In Eq. (20), in the matrix A, the operators Aˆ and Bˆ are
q+ = zµ a++ Zµ :a+a+a :. (40)
identified with a+α and aα′, respectively. Since {Aˆ,Bˆ} µ Xα α α αXβγ αβ:γ α β γ
is unity, the right-hand side of Eq. (36) vanishes, which
means h0|{[H,Aˆ],Bˆ}|0i=−h0|{[H,Bˆ],Aˆ}|0i and Since this operator is the conjugate of the hole-state ex-
citation operator Eq. (19), it is easily shown that the
A(α:α′)∗ =−h0|{[H,aα],a+α′}|0i, formulation for a particle state is given as
=h0|{[H,a+],a }|0i=A(α′ :α). (37)
α′ α A C t zµ N N t zµ
=ω 11 12 ,(41)
In the case of the matrix B in Eq. (20) Aˆ is : a+αa+βaγ : (cid:18)B D (cid:19) (cid:18)Zµ (cid:19) µ(cid:18)N21 N22 (cid:19) (cid:18)Zµ (cid:19)
and Bˆ is aα′, and {Aˆ,Bˆ} is reduced to a one-body where the superscript t means the transposition of the
operator. Due to the ground-state condition Eq. (15) correspondingmatrixandω isdefinedbyω =E −E .
µ µ µ 0
the right-hand side of Eq. (36) vanishes, which means Equation (41) implies that (zµ, Zµ) is the left-hand
h0|{[H,Aˆ],Bˆ}|0i=−h0|{[H,Bˆ],Aˆ}|0i and eigenvector of Eq. (20). Thus Eq. (20) gives simulta-
neously the particle states and the hole states. This is
B(αβγ :α′)∗ =−h0|{[H,:a+a a :],a+}|0i,
γ β α α′ completely analogous to pp(hh)RPA (see Ref. [16].)
=h0|{[H,a+],:a+a a :}|0i
α′ γ β α
′
=C(α :αβγ). (38)
2. Hartree-Fock approximation for the ground state
Similarly, for the matrix D in Eq. (20) Aˆ is : a+a+a :
α β γ
If we make the usual approximation to take for the
and Bˆ is :a+γ′aβ′aα′, and {Aˆ,Bˆ} is reduced to at most a ground state |0i the HF one, N in Eq. (26) becomes
two-body operator. Due to the ground-state conditions 22
Eqs. (15) and (16) the right-hand side of Eq. (36) van- N22(αβγ : α′β′γ′)=(δαα′δββ′ −δαβ′δβα′)δγ′γ
ishes, which implies
×(n0 +n0 n0 −n0 n0 −n0 n0 ),(42)
γγ αα ββ γγ αα γγ ββ
′ ′ ′ ∗ ′ ′ ′
D(αβγ :αβ γ ) =D(αβ γ :αβγ). (39)
wheren0 isequalto1or0. InHF,N isnon-vanishing
αα 22
Therefore, the Hamiltonian matrix in Eq. (20) is hermi- only for Ypµp′:h and Yhµh′:p. These amplitudes Ypµp:h and
tian. Intheapplications,shownbelow,wedonottakeall Yµ correspond to the backward amplitudes of yµ and
hh′:p h
the matrix elements of nαα′ and Cαβα′β′, which causes ypµ, respectively. Hereafter we refer to this formulation
5
of N are nonvanishing for all configurations due to the
22
ground state correlations and, therefore, all other Yµ
αβ:γ
could be included, in principle. We investigate the effect
ofinclusionofthoseother amplitudes inEoRPAinsome
limited cases.
3. The RPA ground state wave function
The choice of the subspace spanned by the afore men-
tionedfouramplitudesyµ,yµ,Yµ ,Yµ maybegiven
h p pp′:h hh′:p
adifferentrationale. Weconsiderthefollowingtwoquasi-
particle operators which consist only of the forward and
backwardamplitudes
1
qα+ = ypαa+p − 2 Yhαh′:pa+ha+h′ap, (43)
Xp hXh′p
FIG. 3. (a) Mass operator for a particle state described by
1
Ycihrµhc′l:epsamnedan(bt)htehpartofpoargaathoorslegsitvaetnebdyesYchrµihb′:epd((bay))Yapµpn′d:h.YpTµp′h:he qρ+ =Xh yhρah− 2pXp′hYpρp′:ha+hapap′ (44)
((b)),and the dotsthe residual interaction. and neglect the coupling of yα to Yα and that of yρ
p pp′:h h
to Yρ . This oRPA scheme actually corresponds to
hh′:p
the standard RPA for even nucleon systems. This can
consistingofthefouramplitudes,yµ,yµ,Yµ andYµ for example be seen in the following way. It can easily
h p pp:h hh′:p
be shown that the operators q and q kill the following
as odd-RPA (oRPA). The mass operators of the one- α ρ
RPA vacuum, i.e. q|Zi=0 with
body Green’s function derived from oRPA are schemati-
cally shown in Figs. 2 and 3. Since oRPA describes the |Zi=e41Pzpp′hh′a+paha+p′ah′|HFi (45)
holestatesandparticlestatessimultaneously,the single-
particle strength can be spread over both positive and under the conditions
negative energy regions. We consider that the strength ypα∗zpp′hh′ =Yhαh∗′:p′ (46)
below the Fermi energy ǫF of the core nucleus belongs Xp
to the states in the A−1 system, while that above ǫ
to the states in the A+1 system. The TDA hole-statFe yhρ∗zpp′hh′ =Ypρp∗′:h′, (47)
equationisobtainedbykeepingonlyyµ andYµ : since Xh
h hh′:p
the couplingofyµ toYµ is includedinadditiontothe where |HFi is the HF ground state of an even A system.
h hh′:p
couplingtothebackwardamplitudeYµ ,ouroRPAac- These two quasiparticles (one for the particle addition
pp′:h (α)andonefortheparticleremoval(ρ))span,asseen,ex-
tually corresponds to some sort of second RPA for even
actlythespaceofthefouramplitudesdiscussedinII.B.2.
nucleonsystems. Onemaythinkthatinadditiontheam-
plitudes Yµ and Yµ should be included in oRPA However, the single equation for the four amplitudes is
hp:h′ pp′:p′′ now split into two independent 2 × 2 equations corre-
because they respectively express the backward propa-
sponding to the two operators introduced in Eqs. (43)
gations of the particle - hole pair and the hole-hole pair
in Yµ . However,these amplitudes cannot be included and (44), respectively. Using in these equations the HF
hh′:p ground state as in II.B.2, we see, that we have one type
because the norm of these amplitudes is not defined in
HF (the matrix elements N for Yµ and Yµ van- of ’forward’goingamplitudes and one type of ’backward
22 hp:h′ pp′:p′′ going’ amplitudes in analogy with what we know from
ish in HF). As mentioned above, the formulation Eq.
standard ph RPA for even systems with corresponding
(13) allows us to implement all the Xµ amplitudes
αβ:γ amplitudes X and Y. As a matter of fact, it recently
including Xµ and Xµ because there is no restric-
hp:h′ pp′:p′′ hasbeenshown[17]thatalsoforthe standardph-RPAa
tion of the norm matrix. (If Eq. (35) is used instead generalized operator can be found which annihilates the
of Eq. (13), Xhµp:h′ and Xpµp′:p′′ are projected out, how- state Eq. (45). It is given by the following form
ever.) The inclusion of all the amplitudes of Xµ can
give quite unphysical results because the sumαoβf:γsome Qν = [Xpνh∗a+hap−Ypνh∗a+pah]
unperturbed energies corresponding to Xµ fall near Xph
αβ:γ
ǫ , whichmakesit difficult to distinguishthe hole states 1
F + ην a+a a+a
fromtheparticlestates(seealsoRef. [8]). Fortheserea- 2 p1p2ph p2 p1 p h
sons we mainly present the results in EoRPA calculated phXp1p2
Yuspµipn′:ghoannldyYthhµhe′:fpouinroaRmPpAlit,uadltehsocuogrhretshpeonmdaintrgixtoelyemhµ,eynpµts, − 21phXh1h2ηhν1h2pha+h1ah2a+pah. (48)
6
This destruction operator kills the vacuum Eq. (45), i.e. the ρ part of the mass operator corresponding to Fig.
Q|Zi=0, under the conditions 3(b)). It may certainly be appealing to work with an
approachwhichisbasedonagroundstatewavefunction.
zphp′h′ = (X−1)νphYpν′h′ (49)
Xν GoingbeyondtheuseofaHFgroundstate,wecando
1 as in this work considering EoRPA as described above.
ην = Xν z (50)
p1p2ph 2 p1h1 pp2hh1 However,thereisalsothepossibilitytomixevenandodd
Xh1 RPA’s. For example it has turned out that Self Consis-
1
ην = Xν z . (51) tent RPA (SCRPA) based on the vacuum Eq. (45) gives
h1h2ph 2Xp1 p1h1 pp1hh2 very good results [17]. For instance, it also solves the
two particle case exactly. One thus could use SCRPA
We see that there are additional terms to the standard
tocalculatethe correlationfunctionsappearinginanex-
ph-RPAoperatorwhichcontainspecifictwo-bodyterms.
tended oRPA. To use the ansa¨tze Eqs. (43) and (44)
ThecorrespondingtermsinQ+ canschematicallybeob-
ν directly seems difficult, since they correspond to a non-
tainedinaugmentingtheadditionoperatorEq. (43)bya
linear transformation among the fermion operators.
destructora andtheremovaloperatorEq. (44)byacre-
h
atora+. Theη-termsarealsosmallamplitude(backward
p
going) terms which can be added to the standard RPA, 5. Spurious modes
evaluated with the HF state. They improve the results
of standard RPA [18]. We, therefore, see that complete
First we discuss an RPA-like formulation that can
consistency between RPA in even and odd systems can
bring the c.o.m motion of an odd system at zero excita-
be achieved.
tionenergy. We considerforanA+1systemthe ground
state |Φ i and an excited state |Φ i with excitation en-
0 µ
ergy ω . Using the equation of motion
4. Green’s Function Description µ
hΦ |[a+a ,H]|Φ i=ω hΦ |a+a |Φ i
0 α′ α µ µ 0 α′ α µ
Itmaybeinstructivetocasttheaboveamplitudeequa- =ω xµ (55)
µ αα′
tions into Green’s function language. For this we write
down a Dyson equation and assuming |Φ0i = a+p|HFi, we obtain the following
equation
Gωkk′ =G0kδkk′ +G0kXk1 Mkωk1Gωk1k′, (52) ωµxµαα′ =(ǫα−ǫα′)xµαα′
where −(n0αα−n0α′α′) hαλ′|v|α′λiAxµλλ′
Xλλ′
G0 = 1−n(k0) + n(k0) (53) + [δαphpλ′|v|α′λiA−δα′phαλ′|v|pλiA]xµλλ′
k ω−εk+iη ω−εk−iη Xλλ′
is the free or HF Green’s function with the occupation + [hαp|v|λpiAxµλα′ −hλp|v|α′piAxµαλ]. (56)
numbersn(0) equalto0or1. Themassoperatorisgiven Xλ
k
by The first two lines of the above equation have the same
Yρ Yρ∗ formasthestandardRPAforanevenAsystem,andthe
Mkk′ =αhh′pX1p2p′1p′2hkh|v|p1p2iωp−1pΩ2:hNρ+p1′1p+′2:hiη′ tinhiardpaanrtdicfloeusrttahtetepr.mFsoarrtehdeuteottaoltmheomadednittuiomnaolpneuraclteoorn
+×hp′1p′2|v|h′k′hikp|v|h h iYhα1h2:pYhα′1∗h′2:p′ P =Xαα′hα′|−i~∇|αia+α′aα, (57)
1 2 ω−ΩN−1−iη which satisfies [P,H]=0, we evaluate ω hΦ |P|Φ i as
ρpp′hX1h2h′1h′2 α µ 0 µ
×hh′1h′2|v|p′k′i, (54) ωµhΦ0|P|Φµi= hα′|−i~∇|αiωµxµαα′. (58)
where Yα,ρ and Ω are the TDA 2p-1h and 2h-1p Xαα′
α,ρ
amplitudes and eigenvalues, respectively, obtained from Using the right-hand side of Eq. (56) and the trans-
the corresponding TDA equations [16]. In the case lational invariance of the interaction [19], we can show
where we strictly work with oRPA corresponding to ω hΦ |P|Φ i=0, which implies ω =0. Thus the exci-
µ 0 µ µ
the ground state Eq. (45), the coupled system of tationenergyofthec.o.mmotionofanoddsystemgiven
particle and hole propagation in above Dyson equation by Eq. (56) is zero from the ground state |Φ i and ǫ
0 p
decouples into two separate Dyson equations, one for from |HFi. In order to obtain this conclusion, however,
the particles (with the α part of the mass operator weneedtoinclude allcomponentsofxµ becauseofthe
αα′
corresponding to Fig. 3(a)) and one for the holes (with last two terms on the right-hand side of Eq. (56).
7
Now we discuss the c.o.m of a core nucleus in odd A would arise in cold fermionic atom systems where one
nuclei whose treatment is of particular relevance. In the could ask the question what happens to an odd fermion
standardparticlevibrationcouplingmodel[3,5]thespu- whichiscoupledtotheso-calledKohnmode,i.e. acoher-
riousmodeissimplydiscarded,first,forthetranslational ent c.o.m. motion of the underlying even system, in the
mode, on physical grounds but also because the RPA external harmonic container. Since the mass of the core
amplitudes of a zero mode cannot be normalized. On can be very large, e.g. with a million of atoms, the fac-
the other hand, e.g. in the case of rotations, it would torization can become quite valid and also the ph-TDA
be very important to find a way to include the rota- for the Kohn mode will become very collective. It could
tional mode, since it is a physical state. In order to be interesting to investigate this question in more detail
learnsomethingaboutthecouplingofsingle-particlemo- theoretically and experimentally because the treatment
tion and recoil of the core nucleus, we first show that of Goldstone modes in single-particle mass operators is,
ω hµ|Pa |0i = ǫ hµ|a |0i holds in the mean-field ap- to the best of our knowledge, an unsolved problem.
µ α α α
proximation. Using the complex conjugate of Eq. (8),
we evaluate ω hµ|Pa |0i such that
µ α
ω hµ|Pa |0i=hµ|[Pa ,H]|0i C. The A=2 case
µ α α
=hµ|[P,H]a |0i+hµ|P[a ,H]|0i
α α
=hµ|P[a ,H]|0i, (59) We show that our formulation is exact for an A = 2
α
system. InthecaseofanA=2system,TDDMgivesthe
where we use [P,H] = 0. If we use the mean-field coupled equations of motion for nαα′ and the two-body
approximation for [aα,H], that is, [aα,H] = ǫαaα, then density matrix ραβα′β′, which are defined as
we obtain ω hµ|Pa |0i = ǫ hµ|a |0i, which means
µ α α α
that the strength |hµ|Paα|0i|2 is concentrated at the nαα′(t)=hΦ(t)|a+α′aα|Φ(t)i, (60)
state with ω = ǫ . In the general case the mean-field
approximatioµn is nαot valid as Eq. (10) indicates. In the ραβα′β′(t)=hΦ(t)|a+α′a+β′aβaα|Φ(t)i, (61)
realistic applications ofour oRPAor EoRPAapproaches
where |Φ(t)i is the time-dependent total wavefunction
shown below, we will, therefore, see that a large portion
|Φ(t)i = exp[−iHt]|Φ(t = 0)i. The equations in TDDM
of the strength is distributed to an energy region lower
are written as [20]
than ǫ , which can be interpreted as a recoil effect of
α
the core nucleus.
i~n˙αα′ = (hα|t|λinλα′ −hλ|t|α′inαλ)
In the past, the question of the spurious modes Xλ
appeared essentially in the particle-vibration coupling + [hαλ1|v|λ2λ3iρλ2λ3α′λ1
model [5] which is derived from the Green’s function λ1Xλ2λ3
method factorizing in the mass operator the 2p-1h (2h- ′
−ρ hλ λ |v|αλ i], (62)
1p) propagator into an ph-RPA propagator and a HF αλ1λ2λ3 2 3 1
singleparticlepropagator. Inthespectralrepresentation
of the RPA propagator the spurious mode is then dis-
cardedbecause of the zero energy mode and the ensuing i~ρ˙αβα′β′ = (hα|t|λiρλβα′β′ +hβ|t|λiραλα′β′
diverging amplitudes. On the other hand, if one could Xλ
′ ′
solve the 2p-1h (2h-1p) propagator in the mass operator −hλ|t|αiραβλβ′ −hλ|t|β iραβα′λ)
ethxeaocrtyly, s(eu.rge.lyinnaomproodbelle)morwwitithhaascpounrsiiostuesnmthoitgiohneroofrtdheer + [hαβ|v|λ1λ2iρλ1λ2α′β′
λX1λ2
core nucleus wouldbe present. From our analysis above, ′ ′
−hλ λ |v|αβ iρ ]. (63)
it appears that the mass operator should be calculated 1 2 αβλ1λ2
with 2h-1p (2p-1h) TDA amplitudes. It could very well
Here the single-particle states are arbitrary. Since there
be that this approach gives more realistic results than
are no higher-levelreduced density matrices in an A=2
the particlevibrationcouplingmodelwherethe spurious
system, these two equations are exact. When the two-
mode is discarded. That is what our derivationseems to
bodydensitymatrixinEq. (62)isapproximatedbyanti-
indicate.
symmetrized products of the occupation matrices, Eq.
In any case, e.g. in the case of rotations, it would be
(62) is equivalent to the equation in the time-dependent
necessary to include this mode, since it is physical. One
HF theory. The ground state is given as a stationary
could push the argument even further and assume that,
solution of these equations.
since,e.g. therotationisverycollective,thefactorization
The equation for the transition amplitude xµ is
of the 2h-1p (2p-1h) TDA into a ph-TDA + plus a hole α
(particle) is a good approximation (the neglected terms
coming only from exchange). Because of its strong col- (hλ|t|αi−δαλωµ)xµλ + hλ1λ2|v|αλ3iX˜λµ1λ2:λ3
lectivity,eventuallyalltheothercouplingstointrinsicph Xλ λ1Xλ2λ3
modes could be neglected. Actually analogous questions =0, (64)
8
where X˜µ =h0|a+a+a |µi. The equation for X˜µ is
αβ:γ α β γ αβ:γ TABLE I. Single-particle energies ǫ and occupation proba-
given as α
bilitiesn calculatedinTDDM.Thesingle-particleenergies
αα
((hλ|t|αi−δ ω )X˜µ in HF are given in parentheses.
αλ µ λβ:γ
Xλ
ǫ [MeV] n
+hλ|t|βiX˜µ −hγ|t|λiX˜µ ) α αα
αλ:γ αβ:λ orbit proton neutron proton neutron
+ hλ λ |v|αβiX˜µ 1s1/2 −32.5 (−32.1) −36.2 (−35.9) 0.98 0.98
λX1λ2 1 2 λ1λ2:γ 11pp31//22 −−1182..33 ((−−1182..20)) −−2115..88 ((−−2115..86)) 00..9931 00..9931
=0. (65) 1d5/2 −3.8 (−3.8) −7.1 (−7.2) 0.08 0.08
Since there are no higher-level transition amplitudes in 2s1/2 1.2 (1.5) −1.5 (−1.2) 0.02 0.02
an A = 2 system, these two equations are also exact.
From Eq. (64) we obtain
1
2 (hα′|t|αi+δαα′ωµ)xµα′(xµα)∗ B. Ground state
µXαα′
′ The occupation probabilities calculated in TDDM are
= hα|t|αinαα′ shown in Table I. The largest deviation from the HF
Xαα′ values(n0 =1or0)is about10%,whichmeans thatthe
1 αα
+ hλ λ |v|λ λ iρ ground state of 16O is a strongly correlated state. A re-
2 1 2 3 4 λ3λ4λ1λ2
λ1λX2λ3λ4 cent shell-model calculation by Utsuno and Chiba [22]
also gives a similar result for the ground state of 16O.
=h0|H|0i, (66)
The correlation energy E in the ground state, which is
c
where nαα′ and ραβα′β′ are exactly given by defined by Ec = αβα′β′hαβ|v|α′β′iCα′β′αβ/2, is −19.6
nαα′ = xµα′(xµα)∗, (67) MpeenVsa.tAedlabrygethpeoPritnicorneaosfetihnetchoerrmelaeatino-nfieelnderegnyerigsycodmue-
Xµ
to the fractional occupation of the single-particle states.
ραβα′β′ = X˜αµ′β′:β(xµα)∗. (68) The resulting energy gain due to the ground-state corre-
Xµ lations, which is given by the total energy difference be-
tween HF and TDDM, is with 5.2 MeV relatively small.
Equation (66) corresponds to the relation between the
Such kind of scenario is similar to the one well known
totalground-stateenergyandthe single-particleGreen’s
from BCS theory [16].
function [21].
III. APPLICATIONS TO 16O C. Spectral functions
A. Calculational details In Figs. 4 and 5 the spectral functions of the proton
1p and 1p hole states in 15N calculated in EoRPA
1/2 3/2
(Eq. (20)) (solid line) are shown, respectively, and com-
In this paper, we make a firstschematic applicationof
our theory to proton hole states in 15N and proton par- pared with the results in TDA (dotted line). Since the
results in oRPA are similar to the EoRPA results, they
ticle states in 17F. We do not consider the correspond-
arenotshown. Asalreadymentioned,intheEoRPAcal-
ing neutron states because there are less experimental
culations we consider only the same Yµ amplitudes as
data. We consider the 1s , 1p , 1p , 1d , 2s , αβ:γ
1/2 3/2 1/2 5/2 1/2 thoseusedinoRPA.Tofacilitateacomparisonofvarious
1d , 2p , 2p 1f and 1f states for both pro-
3/2 3/2 1/2 7/2 5/2
calculations, we smooth the distributions using an arti-
tons and neutrons. The continuumstates are discretized
ficial width Γ = 0.5 MeV. As shown in Table I the HF
byconfiningthesingle-particlewavefunctionsinasphere
energies of the proton 1p and 1p states are −12.0
of radius 12 fm. We use a simplified residual interaction 1/2 3/2
MeV and −18.2 MeV, respectively. In TDA the main
which consists only of the t and t terms of the Skyrme
0 3
peak is shifted upwards from the HF position due to the
III force. Its strength is reduced by 20% to put the spu-
coupling to the configurations Yµ whose unperturbed
rious c.o.m motion of 16O at approximately zero energy hh′:p
energiesaredistributedbelow−40MeV.InEoRPA(and
in the standard RPA. For the ground-state calculation
of 16O in TDDM, we only use the bound single-particle oRPA)the mainpeakis slightlyshifted downwardsfrom
the HF position due to the coupling to both Yµ and
states,the1s1/2,1p3/2,1p1/2,1d5/2and2s1/2states,and hh′:p
consideronlythetwoparticle-twoholetypecorrelations the backward amplitudes Ypµp′:h whose unperturbed en-
in Cαβα′β′. We also neglect the off-diagonal elements of ergies are located above 0 MeV (see Fig. 4 and Fig.
nαα′ between the 1s1/2 and 2s1/2 states. The ypµ am- 5). The strength distribution in the positive energy re-
plitudes for the proton 2s , 2p and 2p states are gion corresponds to the states in 17F. The strengths of
1/2 1/2 1/2
neglected because their contributions are negligible. the main peak of the proton 1p state calculated in
1/2
9
1.2 0.4
1.0
0.3
1] 0.8
-V 0.015 -1]
Me 0.6 0.03 eV 0.2
S [ 0.4 0.010 0.02 S [M
0.005 0.01 0.1
0.2
0.000 0.00
-60 -50 -40 -30 0 10 20 30
0.0 0.0
-50 -40 -30 -20 -10 0 10 20 -60 -50 -40 -30 -20 -10
E [MeV] E [MeV]
FIG. 4. Spectral function of the proton 1p1/2 state in 15N FIG. 6. Same as Fig. 4 but for theproton 1s1/2 state.
calculated in EoRPA (solid line). The dotted line shows the
result in TDA. The distributions are smoothed with an ar-
tificial width Γ = 0.5 MeV. The strength distribution in the
goodapproximation. Thesingle-particlestrengthsofthe
positiveenergyregion isduetothecouplingtothebackward
amplitude Ypµp′:h and indicates the states in 17F. The small mainpeakoftheproton1p3/2statecalculatedinEoRPA,
strength distributions in thepositive and negativeenergy re- RPAandTDAare0.88,0.82and0.93,respectively. The
gions are shown in the insets. sum of the occupation probabilities of the proton 1p3/2
state distributed in the negative energy region is 0.93
in EoRPA, which corresponds to n = 0.93 in TDDM.
αα
Summingthewholespectralweightsinnegativeandpos-
1.2 itive energy regions gives, of course, the sum rule value
of one.
1.0
The results for the proton 1s state are shown in
1/2
-1] 0.8 0.015 0.03 Fisig−.326..1TMheeVH.FTheenesrtgryenogfththies pfrraogtomnen1tse1d/2dhuoeletosttahtee
V
e 0.6 coupling to the configurations Yµ : the unperturbed
S [M 0.4 00..000150 00..0012 eanbeorugty−o2f6tMheeVc.oSnifingceurtahteiobnac(k1wpa1hr/hd2′):cp−o1n(fi1gp3u/r2a)t−io1n1sd5Y/p2µp′:ihs
are energetically well separated, there is no significant
0.2
0.000 0.00
-60 -50 -40 -30 0 10 20 30 difference between the TDA and oRPA results. There-
0.0 fore,the oRPAresultis notshowninFig. 6. Comparing
-50 -40 -30 -20 -10 0 10 20 with the results obtained from Eq. (35), we found that
E [MeV] the D term in the matrix D, which is given by e×N
2 32
and describes the self-energy contributions to the one
particle - two hole configurations, play a role in shifting
FIG. 5. Sameas Fig. 4 but for the proton 1p3/2 state. the strength to lower energy region. The summed occu-
pation probability of the proton 1s state distributed
1/2
in the negative energy region is 0.98 in EoRPA, which
EoRPA,oRPAandTDA are0.88,0.82and 0.95,respec- corresponds to nαα =0.98 in TDDM.
tively. When the forward amplitude Yµ is neglected Thespectralfunctionoftheproton1d statein17Fis
hh′:p 5/2
inoRPA,the mainpeakisfurthershifteddownto−14.9 showninFig. 7. TheHFenergyofthe1d stateis−3.8
5/2
MeV andhas strength 0.89. This indicates that the cou- MeV. The main peak is shifted downwards from the HF
pling to the backwardamplitudes Ypµp′:h plays an impor- position in TDA due to the coupling to Ypµp′:h, while, on
tantroleindepletingthesingle-particlestrength. Theef- thecontrary,itisshiftedupwardinoRPAduetothe ad-
fects ofthe ground-statecorrelationsincludedinEoRPA ditionalcouplingtothebackwardamplitudesYhµh′:p. The
play a role in slightly reducing the correlations in oRPA ground-state correlations included in EoRPA play a role
due to fractionaloccupationof the single-particle states. in slightly reducing correlations in oRPA. The states lo-
The sum of the strength of the proton 1p state dis- catedbelowthesingle-particleenergyoftheproton1p
1/2 1/2
tributed in the negative energy regionis 0.91 in EoRPA, statecorrespondtothestatesin15N.Thesummedoccu-
which corresponds to n = 0.91 in TDDM, (see Table pation probability of the proton 1d state distributed
αα 5/2
I). Thus the relation n = |hµ|a |0i|2 holds to a belowthe proton1p state is0.06inEoRPA,while the
αα µ α 1/2
P
10
1.2 0.6
1.0 0.5
-1V] 0.8 0.03 actor 0.4
S [Me 0.6 0.02 S f 0.3
0.4 0.2
0.01
0.2 0.00 0.1
-50 -40 -30 -20
0.0 0.0
-30 -20 -10 0 10 -60 -50 -40 -30 -20 -10 0
E [MeV] E-E [MeV]
p
FIG.7. SameasFig. 4butfortheproton1d5/2 statein17F. FIG. 9. Spectroscopic factors for the proton 1s1/2 and 2s1/2
The dot-dashed line shows the result in oRPA. The strength statescalculatedinoRPAarecomparedwithexperiment[23]
distribution below −15 MeV is due to the coupling to the (red bars).
backward amplitudeYhµh′:p and shows thestates in 15N.
0.6
3.5
0.5
3.0
or 0.4
or 2.5 act
act 2.0 S f 0.3
S f
1.5 0.2
1.0 0.1
0.5
0.0
0.0 -60 -50 -40 -30 -20 -10 0
-40 -30 -20 -10 0 E-E [MeV]
p
E-E [MeV]
p
FIG. 10. Same as Fig. 9 butfor EoRPA.
FIG. 8. Spectroscopic factors for the proton 1p1/2 and 1p3/2
states calculated in EoRPA are compared with experiment
[23] (red bars). of TDA and RPA-type calculations [8, 9]. Studies on
the effect of short-range correlations have predicted a
strength reduction of about 10% in 16O [24–26]. The
corresponding value for n in TDDM is 0.08.
αα spectroscopic factors for the proton 1s and 2s
1/2 1/2
states calculated in oRPA is compared with experiment
[23] (red bars) in Fig. 9. The EoRPA results are
D. Comparison with experiment
also compared with experiment (red bars) in Fig. 10.
Since the inclusion of ground-state correlations causes a
The spectroscopic factors (defined by (2j + downward shift of the strength, the agreement with the
1)×transition strength) calculated in EoRPA for data becomes somewhat worse in EoRPA. The strong
the proton 1p1/2 and 1p3/2 states are compared with fragmentation below −15 MeV cannot be reproduced in
experiment [23] (red bars) in Fig. 8. The main peak these oRPA and EoRPA calculations. Probably higher
of the proton 1p1/2 state is considered as the ground configurations are needed. The spectroscopic factors for
state of 15N and the hole-state energy is measured the proton 1d and 1d states calculated in oRPA
5/2 3/2
from this threshold in the following. The results in arecomparedwithexperiment[23](redbars)inFig. 11.
EoRPA are reasonable though they overestimate the The EoRPA results are also compared with experiment
experimental data and cannot reproduce the strength in Fig. 12. Due to a downward shift of the strength,
distributionaround−10MeV. This is a commonfeature the agreement with the data is worsened in EoRPA. We