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The Darmstadt effect: a new type of nuclear spectroscopy? (*)

R. ALZETTA(1)(**), T. BUBBA(1), G. LIBERTI(1) and G. PREPARATA(2)

(1_{) Dipartimento di Fisica, Università della Calabria}

I-87030 Roges di Rende, Cosenza, Italy INFN, Sezione di Cosenza - Cosenza, Italy

(2_{) Dipartimento di Fisica, Università di Milano - Milano, Italy}

INFN, Sezione di Milano - Milano, Italy

(ricevuto il 2 Gennaio 1997; approvato il 3 Marzo 1997)

Summary. — We analyze the well-known and hitherto unsolved problem of the

Darmstadt effect, through the idea that the observed sharp e1_e2_{-lines reflect the}

interaction of the “coherent photon clouds” that are dragged along by the colliding heavy ions, and have their origin in the interaction of the zero-point fluctuations of the quantized e.m. field with the different collective states of the nuclei. We find such idea adequate to describe the experimental observations which in the future could give access to a new kind of nuclear spectroscopy.

PACS 25.70.Bc – Elastic and quasielastic scatering. PACS 25.70.Ef – Resonances.

PACS 12.20.Ds – Specific calculations.

About ten years ago experiments conducted at the GSI in Darmstadt [1] revealed that in the collisions between heavy ions, such as for instance238_{U and}232_{Th, very sharp} e1_e2_{states were produced with total energies E}

S4 E11 E2 in the MeV region, and with zero total momentum (back-to-back) in the center of mass of the colliding ions. The subsequent, very careful analysis aimed at relating such strange states to some peculiar low-mass (less than 2 MeV) long-lived particles [2] demonstrated beyond any reasonable doubt that the unexpected phenomena—to be called the Darmstadt effect—could by no means have their origin in the decay of hitherto unobserved low-mass, long-lived particles.

If not the decay products of new particles, what else could such sharp e1_e2_-pairs be? In spite of the early frantic theoretical efforts to give a tenable explanation of the GSI observation, we believe that so far no sensible answer has been given to this

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail address: AlzettaHfis.unical.it

remarkable experimental puzzle whose contours, it must be admitted, are to date still quite uncertain [3]. Be that as it may, in this paper we take the attitude that the early observation of sharp e1_e2_{-pairs were indeed correct and that the physical mechanism} responsible for their production must be searched for within the well-known and established physical laws of the Standard Model. In other words, we believe that no new exotic physics is at work in the surprising Darmstadt observations.

Thus the aim of this paper is to demonstrate that the Darmstadt effect is most probably due to a rather simple, though subtle physical mechanism. We shall do this by carrying out a complete, though approximate calculation of the differential cross-section for e1_e2_{-production in the collisions of two heavy ions, and comparing it with} the existing experimental information.

The simple physical mechanism we have in mind is then the following: the colliding heavy nuclei possess several collective states that couple to the electromagnetic (e.m.) field, the giant-dipole resonance being one of the most studied, but there exist other modes with higher multipoles of lower frequencies as well. Each of those modes will resonantly couple to the appropriate zero-point modes of the quantized e.m. field. Such coupling, as it has being shown elsewhere [4], and in particular for various different phenomena, such as superfluidity, ferromagnetism, cold fusion, involving coherent coupling of the e.m. field with the matter field [5], causes the e.m. modes to “align their phases”, i.e. to become coherent even though no amplification of their amplitudes is to be expected, due to the weakness of the coupling, as one can easily ascertain.

These coherent e.m. fields (linear superposition of modes of well-defined frequen-cies, vi, coinciding with those of the collective nuclear modes) are then dragged along by the colliding heavy ions and can cause the transfer of the ions’ energy and momentum that is necessary to create the observed e1_e2_{-pairs. The very steep} dependence in the vi’s that we find in the calculation implies that only the very low frequencies can be detected in this kind of processes; nevertheless, as we point out in the title, if our explanation turns out to be correct, a new type of nuclear spectroscopy may be accessed in these most interesting experiments.

As we have just briefly described in the introductory paragraphs, our proposed explanation of the Darmstadt effect is then based on the idea that the zero-point oscillations of the different (low frequency) collective excitations of a nucleus, through their resonant coupling to the corresponding zero-point fluctuations of the quantized e.m. field, are in fact able to “align” the 8 p 4 (4p32) independent modes of a given frequency v, thus producing a state of the system nucleus plus e.m. field of lowest energy. In this state the modes of the e.m. field at the resonant frequency v, instead of a chaotic statistical mixture, comprise a fully coherent state, whose vector potential has the approximate form

A(x , t) 4e2ivt

g

1 2 vVCD

h

1 /2 8 psin vr vr u , (1)

where VCD is the volume of the coherence domain (CD) of the modes of frequency v (VCD4 4 pRCD3 /3, RCD4 p/v C 600 F, for v C 1 MeV), and u is a complex vector field obeying the trasversality conditions x Q u 40 and ˘Qu40 and such that uQux

4 1 /2. Please note the 8 p factor in eq. (1), stemming from the coherence of the 8 p independent modes of the ground state of the free e.m. field. As a result, the coherence induced on the zero-point fluctuations of the e.m. field, resonating with the low-frequency (vi) collective modes of the nucleus, leads us to picture the nucleus as

surrounded by a “coherent photon cloud” comprising the various frequencies vi and extending over a radius Ri_{4 p/v}i much larger than RN, the nuclear radius. We may thus expect that in the overlap of such clouds (which extend typically for about 103_{F) of} the colliding heavy ions, there takes place the energy momentum exchange necessary to create the observed e1_e2_{pairs. The sharp oscillation frequencies v}

iof the coherent e.m. field

(

see eq. (1)

)

of the cloud will thus lead to the creation of e1_e2_{pairs of likewise} sharp energy

ES4 v11 v2, (2)

where v1 is in the spectrum of the collective excitations of one of the nuclei while v2 belongs to the spectrum of the other. As for the momentum of the pair, the softness of the form factor that can be derived from the e.m. field (1) leads us to expect that the momentum of the pair in the c.m. frame is peaked around the zero value, i.e. the pair is mainly produced “back-to-back”. We note that these are precisely the surprising qualitative features of the e1_e2 _{lines observed in the experiments carried out at} Darmstadt.

In order to substantiate our expectations, in the following we briefly describe the calculation of the cross-section for the process

N1(p1) 1N2(p2) KN1(p8 ) 1 N1 2(p28 ) 1 e1(q1) 1e2(q2) , (3)

whose Feynman diagrams are reported in fig. 1, to which there corresponds the Feynman amplitude M 4 (2ie)2A1i(q1) A2 j (q2) Tijrs (4) with Tijrs4 i u–r(q2)

{

gi 1 q O 2 qO12 m g_{j1 g}j 1 q O 2 qO22 m gi

}

vs(q1) (5)

Fig. 2. – The geometry leading to the estimate (7) of the transit time of the heavy-ion collision. and Ai (q) 4 8 p 2 vVCD



VCD d3_{x u}i_e2iq Q x sin vr vr



2TO2 TO2 dt eiq0t_e2ivt (6)

is the Fourier trasform of the “coherent photon cloud” of frequency v. T is the transit time of the heavy-ion collision, which can be (approximately) estimated (see fig. 2) as

vT 4 2 p v1 1 2 p v2 , (7)

v being the relative velocity. In the Darmstadt experiments v is typically one tenth of

the velocity of light. From (6) we can write

Ai(q) 4 8 p 2 vVCD

FS(q) FT(q0) ui (8)

and approximate the square of the time form factor FT(q0) with the Lorentzian NFT(q0) N24

1 (q02 v)21 G2

, (9)

where the width G is given by G 4 1 T 4 v 2 p v1v2 v11 v2 , (10)

which in the typical experimental conditions for viC 1 MeV implies G G 10 keV. As for the spatial form factor FS(q), for NqNcv (which holds in the relevant

kinematical region) we have approximately NFS(q) N2C

( 4 p)2 2 NqN3 (11)

within the width G

(

see eq. (10)

)

, we can thus write the differential cross-section: ds 4 ( 2 p) 4 v d 3 (P 1Q) d

(

E_{11 E}–v12 v2

)

NMN2 d3_{P d}3_k ( 2 p)6 d3_q 1d 3_q 2 4 E₁E₂( 2 p)6 , (12)

where P 4p18 1 p28 is the total momentum of two outgoing ions,

k 4 M2p18 2 M1p28

M11 M2 (13)

their relative momentum, and Q 4q11 q2 the momentum of e1e2-pair. Please note the energy d-function in eq. (12) which just coincides with eq. (2). Upon integration on the relative momentum k, eq. (12) becomes

(14) _{ds 4} 288 a 2 p8 (M1M2)8 (M11 M2)14 (v11 v2)3 (v1v2)5 Q Q v9XF(Q) q₁q₂d (E_{11 E22 v}12 v2) dE1dE2dVe1_e2,

where F(Q), the form factor of the reaction, is expressed as ( m is the reduced mass: 1 /m 41/M11 1 /M2): F(Q) 4 M2 M1

g

m M1

h

2 F1(Q) 1 M1 M2

g

m M2

h

2 F2(Q) (15) with F1 , 2(Q) 4

g

M1 , 2 m

h

2 _G vNQN

{

p 22 arctan

y

g

M1 , 2 m

h

2 _G vNQN

z

}

. (16)

Furthermore the kinematical factor X, in the equal-mass case and for v14 v24 v, is given by the simple expression

X 48

g

v v

h

2

[

E₁E_{21 m}e21 q1Q q2

]

. (17)

Setting now q_{64 Q/2 6 d, and integrating over d, one finally gets for the cross-section} at the peak ES4 2 v (18) ds d3 Q 4 4 9 a 2 8 p8 M2 v9 v 11_F(Q) m 2 2 ( 2 O3 )(Q2O4 v2)( (Q2 O4 ) 1 m22 v2) 1 2Q2O4 v2

o

v2 2 m22 Q2O4 v2_{2 Q}2_O4 ,

Fig. 3. – The cross-section at the peak ES4 2 v as a function of the frequency v (v14 v2), in the

equal-mass case (M14 M24 MU).

where now one has

F(Q) 4 v

pQ

g

p 22 arctan v pQ

h

.

(19)

In the final expression, eq. (18), we wish to call the attention to three important aspects:

1) the steep dependence (like v29_{) on the frequency v (v Fm}

e), which strongly enhances the lowest part of the spectrum of collective excitations;

2) the form factor F(Q) that peaks the e1_e2_{-pair cross-section for zero total c.m.} momentum;

3) the large power (v11_{) in the relative velocity that strongly suppresses the effect} in low-energy collisions.

In fig. 3 we plot the cross-section

g

ds

dV

h

e1_e2 as a function of the common frequency

v and in the equal mass case. One can see very clearly the steep decrease for v F1 MeV and the threshold behaviour. In fig. 4 for illustration purposes we plot the

distribution in the angle between e1_e2_{in the c.m. for U 1Th, taking v}

Fig. 4. – The angular distribution between e1_e2_{in the c.m. for U 1Th, with v}

14 0.721 MeV and

v24 1.061 MeV.

and v24 1.061 MeV. Please note the not very dramatic dominance of the “back-to-back” configuration. Finally in table I we give tentative assignments for the frequencies of different heavy ions that may give rise to the observed e1_e2_{lines, and in} table II we compare such assignments with the observed e1_e2_{-lines, and our estimated} cross-sections with those reported by the EPOS and ORANGE groups. Though not perfect, the agreement with experiments that we find is, we believe, far from insignificant. In particular we note that in the U 1U entry of table II the e1_e2 _line which is observed at 555 68 keV is found at 420 keV; this discrepancy, the only serious TABLEI. – Tentative assignments for the frequencies of different heavy ions.

Nucleus v1(keV) v2(keV)

U 721 921

Th 911 1061

Pb 881

TABLE II. – The sum-energy lines with their cross-sections observed in the Orange and EPOS

experiments compared with the sum of possible frequencies of the colliding systems and the calculated cross-sections. Collision system Sum energy (keV) Cross-section (mbOsr ) Sum energy (keV) Cross-section (mbOsr ) Sum energy (keV) Cross-section (mbOsr ) U 1U 555 68 — 630 68 — 815 68 — Orange — — — — — — EPOS 420 2.17 620 1.96 820 1.24 our calculation U 1Th — — — — — — Orange 608 68 — 760 620 — 809 68 — EPOS 610 1.95 760 1.91 810 1.26 our calculation U 1Pb 57666 0.07 787 68 0.11 — — Orange — — 773 610 1–2 — — EPOS 580 1.30 780 0.93 — — our calculation U 1Ta 634 65 1.4 748 68 3.6 805 68 — Orange 625 68 5.8 740 610 3.2 — — EPOS 620 1.87 740 1.71 810 1.36 our calculation

one we find, may well be due to the fact that the relative cross-section has been largely overestimated (this being possible in view of its closeness to threshold) and that in line a 555 68 keV may well arise from the combination of other modes.

In conclusion we would like to point out that the idea that the collective low-frequency excitations organize around heavy nuclei the zero-point fluctuations of the quantized e.m. field in well-defined, coherent e.m. fields, seems remarkably adequate to describe the surprising findings of very sharp e1_e2 _{lines in heavy-ion collisions.} Though quite rudimentary and approximate, the calculation we have presented in this paper gives us a rather clear physical picture of a possible basic mechanism for the production of the sharp e1_e2_{-lines and, at the same time, as we here emphasized, opens} a window on a new type of nuclear spectroscopy that might give us relevant information upon the dynamics of heavy nuclei.

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