Table Of ContentInelastic form factors to alpha particle condensate states
in 12C and 16O: what can we learn ?
Y. Funaki1, A. Tohsaki2, H. Horiuchi1, P. Schuck3, and G. Ro¨pke4
1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2 Suzuki Corporation, 46-22 Kamishima-cho Kadoma, Osaka 571-0071 Japan
3 Institut de Physique Nucl´eaire, 91406 Orsay Cedex, France
4 Institut fu¨r Physik, Universit¨at Rostock, D-18051 Rostock, Germany
(Dated: February 9, 2008)
In order to discuss the spatial extention of the 0+ state of 12C (Hoyle state), we analyze the
2
inelasticformfactorofelectronscatteringtotheHoylestate,whichour3αcondensatewavefunction
reproducesverywelllikeprevious3αRGM/GCMmodels. Theanalysisismadebyvaryingthesize
of theHoyle state artificially. As a result, we find that only the maximum valueof the form factor
sensitivelydependsonitssize,whilethepositionsofmaximumandminimumarealmostunchanged.
6 This size dependence is found to come from a characteristic feature of the transition density from
0
the ground state to the Hoyle state. We further show the theoretical predictions of the inelastic
0 form factor to the 2+ state of 12C, which was recently observed above the Hoyle state, and of the
2 inelastic form factor2to the calculated 0+ state of 16O, which was conjectured to correspond to the
3
n 4α condensed state in previous theoretical work bythe present authors.
a
J
1 In this short communicationwe want to reporton our We here use the same notation for the alpha condensate
1 calculation of the elastic and inelastic form factors for wave function, ΦN,J=0(β) as was done in Ref. [2]. The
3α
groundandHoylestatesin12C.Wealsomakepredictions Hamiltonian H is the same as used in Ref. [2, 3]. The
1 for the inelastic form factors, 0+ →2+ in 12C and 0+ → groundandHoyblestatescorrespondtothecasesofλ=1
v α-condensate state in 16O. 1 2 1 and λ=2 in Eq. (4), respectively.
5
As we have shown previously, our alpha particle wave Our results are shown in FIG. 1 and we give our nu-
3
0 function, published in [1, 2], very nicely reproduces on merical values in Table I. Reflecting the fact that our
1 the one hand some experimental data in 12C and on the wave functions of the ground state and the 0+ state are
2
0 other hand it is in close agreement with calculated re- almost equivalent to those given in Ref. [3] using res-
6 sults of Kamimura et al. [2, 3]. It is therefore not so onating group method (RGM), our elastic and inelastic
0 surprising that our wave function also reproduces very (0+ → 0+) form factors almost completely agree with
h/ well the elastic and inelastic (0+ → 0+) form factors in th1ose giv2en in Ref. [3]. In FIG. 2, we predict the in-
1 2
t 12C as this was the case in [3]. This gives quite definite elastic form factor to the 2+ state which is obtained in
- 2
l support to our interpretationof the Hoyle state as being Ref.[7]byusing the 3αcondensatewavefunctions. This
c
u acondensateofalmostindependentalphaparticles[4,5]. state was recently observed at 2.6±0.3 MeV above the
n The form factor is obtainedby performing the Fourier three alpha threshold [8], though the exsistence of this
: transformation of the transition density as follows, state has been suggested for a long time from the theo-
v
i 4π ∞ 2 1 reticalpointofview[9]. Recentlythisstatewascarefully
X |F(q)|2 = ρ(J) (r)j (qr)r2dr exp − a2q2 . investigated by the present authors [7] showing that the
ar 122(cid:12)(cid:12)(cid:12)Z0 J,01 J (cid:12)(cid:12)(cid:12) (cid:16) 2 p ((cid:17)1) 2in+2tesrtparteeteids ainsttimheat3eαlyBroeslea-tceodndtoentshaete0s+2tatseta.te which is
Here a2 = 0.43 fm2 is taken as the finite proton size,
p Wenowmakeastudyofthesensitivityoftheinelastic
which is the same as adopted in Ref. [3] and j (qr) is
J form factor with respect to some theoretical ingredients
the J-th order spherical Bessel function. The transition
of our theory. A quantity of prime interest is the spatial
density ρJ(J,0)1(r) is given as, extention of the Hoyle state which is predicted from our
studies to have a volume 3 to 4 times as large as the
ρ(J) (r)=hΨJM | 12 δ(r−r )|ΨJ=0i/Y∗ (r), (2) one of the ground state of 12C. We therefore repeated
J,01 λ=k i λ=1 JM the calculationofthe inelastic formfactor invaryingthe
X
i=1 b sizeparameteroftheHoylestate. Thecalculationcanbe
where the ground and Hoyle states can be obtained by done as shown in Ref. [2] by adopting as the Hoyle state
solving the following Hill-Wheeler equation, the wave function, Ψ⊥(β) ≡ P⊥ΦN3α,J=0(β), where P⊥ is
defined as the projection operator onto the orthogonal
ΦN3α,J=0(β) (H −E) ΦN3α,J=0(β′) fλJ=0(β′)=0. space to the ground state: b b b
Xβ′ (cid:10) (cid:12) (cid:12) (cid:11)
b (cid:12) (cid:12)b
(3)
P⊥ ≡1−|ΨJλ==10ihΨJλ==10|. (5)
ΨJλ=0 = fλJ=0(β)ΦN3α,J=0(β). (4) Heretheparambeterβcorrespondstothespatialexten-
Xβ tionofthe alphacondensate. InFIG.3,the formfactors
b
2
TABLE I: Numerical values of elastic (upperrow) and inelastic (lower row) form factors in 12C.
q [fm−1] 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
|F(q)0+→0+|(×10−1) 9.4 7.7 5.5 3.3 1.6 0.56 0.030 0.15 0.16 0.11 0.065 0.033
1 1
|F(q)0+→0+|(×10−2) 0.98 3.3 5.2 5.5 4.3 2.5 0.96 0.069 0.27 0.29 0.20 0.11
1 2
1.0 elastic form factor 10−2
10−1 RGM 10−3 01+ → 22+
10−2 condensed w.f.
10−4
10−3
2 10−4 2 10−5
)| )|
q q
(F10−5 (F10−6
| |
10−6
10−7
10−7
10−8
10−9
0 5 10 15 0 5 10 15
q2 [fm−2] q2 [fm−2]
(a) (a)
10−2 (×10−2)
01+ → 02+(RGM) 3.5 01+ → 22+
10−3 01+ → 02+
3.0
10−4
2.5
10−5
2
)q| )q| 2.0
(F10−6 (F
| | 1.5
10−7 1.0
10−8 0.5
10−9
0 5 10 15 0 1 2 3 4
q2 [fm−2] q [fm−1]
(b) (b)
FIG. 1: (Color online) (a): Experimental values of elastic FIG.2: (a): Theoreticalpredictionofinelasticformfactorto
form factor in 12Carecompared with ourvaluesobtainedby the 2+ state of 12C using the wave function of [7]. (b): The
2
solving the Hill-Wheeler equation Eq. (3) for ΨJ=0. The re- formfactorshownin(a)isplottedasafunctionofq inlinear
λ=1
sult given in Ref. [3] using resonating group method (RGM) scale for ordinate.
is also shown. (b): Experimentalvalues ofinelastic form fac-
tor in 12C to the Hoyle state are compared with our values
obtained by using the Hoyle state wave function ΨJ=0 and
λ=2 from the ground state to the Hoyle state calculated at
those given in Ref. [3] (RGM). The experimental values are
taken from Ref[6]. several β = (βx,βy,βz) values are shown. Short-hand
notations β = β = β = β and β := (β = β ,β ) =
x y z 1 x y z
(5.27fm,1.37fm)[10]areusedhereandinthefollowing.
The corresponding r.m.s. radii are shown in parenthe-
3
2.5 R0=3.78 fm
HW ) δ=(R−R0)/R0
10−2 ββ==23..6796((33..0586)) 2()q|exp 2.0 max(|F(q)|2exp)=3.β0=×21.06−93(R=3.08fm)
10−3 β→1 (3.78) (x|F 1.5 β=3.76(R=3.56fm)
a
10−4 β=5.27(4.35) m β=5.27(R=4.35fm)
)/
2()|Fq| 1100−−65 2(()x|Fq| 01..50 β=7.3β→81((RR==35..7683ffmm))
a
10−7 m
0
10−8
−0.2 0 0.2 0.4
10−9 δ
0 5 10 15
q2 [fm−2]
FIG. 4: The ratio of the value of maximum height, the-
(a) ory versus experiment, for the inelastic form factor, i.e.
max|F(q)|2/max|F(q)|2 ,isplottedasafunctionofδ,which
exp
is defined as δ = (R−R )/R . R and R are the r.m.s.
0 0 0
(×10−2) HW radii corresponding to Ψ⊥(β = βx = βy = βz) and Ψ⊥(β1),
8.0 β=2.69(3.08) respectively. Here R0 =3.78 fm. Unit of β is in fm.
β=3.76(3.56)
6.0 β→1 (3.78) on the r.m.s. radius of the 0+2 state, we plot on FIG. 4
β=5.27(4.35) thevariationofthisheightwithrespecttothe sizeofthe
()Fq|4.0 Hwhoyenletshtaeter.man.sdrwaedisueseothfattheth0is+hsetiagtheticshcahnagnegsesdtr.onItgliys
| 2
therefore allowed to say that the measurement of the in-
2.0 elastic form factor of α-particle condensate states allows
viaourmodelwavefunctiontodeducetheradiusofsuch
astate. Weshouldnotethatthe wavefunctionofthe0+
2
0 1 2 3 4 state ΨJλ==20 canbe describedratherwellby Ψ⊥(β) as far
asreasonableβ valuesareadopted. Thesquaredoverlap
q [fm−1]
between ΨJλ==20 and Ψ⊥(β) amounts to 99.2% at β =β1.
Due to this almost complete equivalence between both
(b) wave functions, ΨJλ==20 and Ψ⊥(β1), we understand that
the corresponding inelastic form factors obtained by us-
ing both wave functions, i.e. denoted by HW and β in
1
FIG. 3: (Color online) (a): The inelastic form factors to the FIG.3,almostcompletelyagreewithoneanother. Asfor
Hoylestateareplottedasafunctionofq2. Thewavefunction theotherchoicesofβ,Ψ⊥(β)alsohasareasonablylarge
Ψ⊥(β) is adopted as the Hoyle state, the size of which is amountofsquaredoverlapwithΨJ=0,i.e. 64.4%,90.1%,
λ=2
artificially changedbyvaryingthevaluesoftheparameterβ. and 81.8% at β =2.69 fm, 3.76 fm, and 5.27 fm, respec-
β is defined as β ≡ βx = βy = βz and β1 denotes (βx = tively. It should be emphasized that the fact that the
βy,βz) = (5.27 fm,1.37 fm) [10]. The result using the wave wavefunctionΨ⊥(β)whichisparametrizedbyβ ismore
functions of ground and Hoyle states which are obtained by or less a good approximation of the 0+ state guarantees
solvingtheHill-WheelerequationisdenotedbyHW.(b): The 2
the validity of the above discussion of size dependence.
form factors shown in (a) are replotted as a function of q in
We can analyze the reason for these features of the
linear scale for ordinate. The r.m.s. radii corresponding to
Ψ⊥(β) are shown in parentheses. Units of all numbersare in form factor in the following simple way. In FIG. 5 we
fm. show the transition density r2ρ(0) (r) for different val-
0201
ues of β. Due to the orthogonality between ΨJ=0 and
λ=1
ses. We see that the overall structure of the form factor P⊥ΦN3α,J=0(β) one has the relation 0∞r2ρ(002)01(r)dr = 0.
is not very much affected by artificially changing the ra- Wbebnote that the position of the nRode of r2ρ(002)01(r) at
dius of the Hoyle state, i.e. for instance the position of r ≈ 2.5 fm and the point where this transition density
the minimum is only changed in very slight proportions. drops approximately to zero, i.e. at r ≈ 6.0 fm, do not
However,we can see that the amplitude decreasesas the dependonthe variousvaluesofβ. Alsothe featureofan
r.m.s radius of the 0+ state increases. In order to see approximate odd function around the nodal point holds
2
how sensitively the height of the firstmaximum depends forallβ-values. Itclearlycanbe concludedthat one can
4
approximately write
10−2
ρ(002)01(r)≈f(β)ρ˜(002)01(r), (6) 10−3 (01+)th → (03+)th
where f(β) ≥ 0 is independent of r and a decreasing 10−4
functionofβ, whereasρ˜(0) (r) isindependent ofβ. This
means that the form fac0t2o0r1F(q) just changes amplitude 2 10−5
)|
but not shape when the size of the Hoyle state is varied. (Fq 10−6
This analysis is completely consistent with the features |
seen in FIG. 3. 10−7
0.30 β=3.76(3.56) 10−8
β→ (3.78) 10−9
1
0 5 10 15
0.15 β=5.27(4.35) q2 [fm−2]
1) β=7.38(5.63)
−
m
(a)
f
(
) 0
r
(
ρ(0)00,21 (×10−2)
2r−0.15 2.5 (01+)th → (03+)th
2.0
−0.30 1.5
0 2 4 6 )q|
(F
r (fm) | 1.0
0.5
FIG.5: (Coloronline)TransitiondensitiesdefinedinEq.(2)
multiplied byr2,i.e. r2ρ(0) (r),correspondingtowavefunc-
gtiiovnens Ψby⊥((ββ)xw=itβhyd,βiffze)r=en02t(051β.2(7=fβmx,1=.3β7yfm=)β[z1)0]v.alTuhese.rβ.m1.sis. 0 1 q 2[fm−1] 3 4
radiicorrespondingtoΨ⊥(β)areshowninparentheses. Units
of all numbersare in fm.
(b)
On these grounds we also want to make a prediction
of the inelastic form factor to the α-condensate state in
FIG.6: (a): Inelasticformfactorin16Otothe4αcondensate
16O. In FIG. 6 we show this form factor calculated with
state as obtained from the wave function corresponding to
the α-particle condensate wave function for 16O deter-
the third eigen energy state in Ref. [1]. (b): The form factor
minedpreviously[1]. Thislatterstateisactuallythe3rd
shown in (a) is plotted as a function of q in linear scale for
0+ stateofourcalculationwhoseenergyisatE0+ =14.1 ordinate.
3
MeV.Weseethattheinelasticformfactorfor16Oresem-
bles very much in its structure the one of 12C. The posi-
tions of minimum and maximum are almost unchanged, agreement may be a pure coincidence and more experi-
while the height of the first maximum is relatively sup- mental evidences are needed.
pressed compared to the case of 12C. A candidate of the The experimental determination of the corresponding
4αcondensatestatemayhavebeenobservedat13.5MeV form factor would be highly welcome and an eventual
with an alpha decay width of 0.8 MeV [11]. This new agreement with our calculated result, we think, a clear
state is the 5th 0+ state experimentally, corresponding indication of the dilute α-particle structure of the corre-
tothe 3rd0+ stateofourcalculation. Anargumentthat sponding 0+ state in 16O.
in 16O the α-condensate state is around 13.5 MeV could In conclusion, we showedthat, without adjustable pa-
goasfollows: Itiswellknownthatthesecond0+ statein rameters, our proposed condensate wave function for al-
16Oat6.06MeVhasastructureofanα-particleorbiting pha particles [1] nicely reproduces the experimental in-
inanS-wavearoundan12C-core[9,12,13]. Excitingthis elastic form factor 0+ → 0+ in 12C. Together with its
1 2
12C-coretothe Hoylestate wefindthe excitationenergy highsensitivityonthemagnitudeoftheHoylestatewith
7.65MeV+6.06MeV=13.71MeV.Ofcourse,thisclose respect to its size and the reproduction of other experi-
5
mentaldata[1,14],webelievethatthealmostidealBose terAspectssponsoredbyInstituteofPhysicalandChem-
condensatenatureoftheHoylestateisnowfirmlyestab- icalResearch(RIKEN),andissupportedbytheMinistry
lished. We also made predictions for the inelastic form of Education, Culture, Sports, Science and Technology
factor to the 2+ state in 12C which we interpreted in (MEXT),Grant-in-AidforJSPSFellows,(No.03J05511)
2
[7] as a quadrupole particle-hole excitation of the Hoyle andbytheGrant-in-Aidforthe21stCenturyCOE“Cen-
state. Apredictionoftheformfactor0+ →αcondensate ter for Diversity and Universality in Physics” from the
1
statein16Oisalsopresentedanditisarguedthatanex- MEXT of Japan. This work was done as a part of
perimental confirmation of this form factor undoubtedly the Japan-France Research Cooperative Program under
wouldrevealthecondensatecharacterofthecorrespond- CNRS/JSPS bilateral agreement.
ing state.
Acknowledgements
This work was partially performed in the Research
Projectfor Study ofUnstable Nuclei fromNuclear Clus-
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Rev.Lett. 87, 192501 (2001). and D.Baye, Phys.Rev. C 36 (1987), 54.
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1
R¨opke,Phys.Rev. C 67, 051306 (R) (2003). inFIG.2of[2].Thismayindicateastronglyoblatemin-
[3] Y. Fukushima and M. Kamimura, Proc. Int. Conf. imum in contradiction with the spherical β-values em-
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Kamimura, Nucl. Phys.A 351, 456 (1981). the minimum at β = (βx =βy,βz) =(5.7 fm,1.3 fm) is
[4] T.YamadaandP.Schuck,nucl-th/0506048,appearedin extremelyshallowandconnectedbyanalmostflatvalley
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238 valley is more than 90%. This is also underlined by the
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1
631; A. Nakada, Y. Torizuka and Y. Horikawa, Phys. little off thecontinuousline.
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[8] M. Itoh et al., Proc. of the 8th Int. Conf. on Cluster- 277.
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