Nuclear pairing and dynamical breakdown of isospin symmetry

ANNALS

OF PHYSICS

159,

Nuclear

184-198 (1985)

Pairing

and Dynamical Breakdown lsospin Symmetry D.

MICHAEL

of

SCADRON

Department of Physics, University of Arizona, Tucson, Arizona 85721 February 1, 1984

Received

Viewed from the perspective of dynamical breakdown of isospin symmetry, the scale and nucleon number A dependence of the valence pairing energy gap A are approximately predicted from the nuclear Debye energy. A Goldberger-Treiman type of relation is also obtained for nuclear pairing in analogy with QCD along with correlated bosons resembling a pion and sigma meson. Finally a critical temperature is discussed such that the BCS relation 0 1985 Academic AIke Tc - 1.8 holds for nuclear pairing as well as for superconductivity. Press, Inc.

I.

INTRODUCTION

It has long been recognized from the semi-empirical massformula associatedwith the binding energy per nucleon B/A vs. nucleon number A curve, that for large A in the A - 50-220 range, a small energy “gap” A exists which changes sign between even-even and odd-odd nuclei according to [ 1]

6Bz

1 A

for EE nuclei

0

for OE nuclei

-A

(1)

for 00 nuclei.

The scale of this energy gap follows the odd-even (OE) empirical pattern [l] A z 12 MeV/fi,

(2)

both for even numbers of protons (n odd) and also for even numbers of neutrons @odd), implying A z 1.1 MeV for a typical medium-heavy nucleus with A z 120. While this presumedvalence (pp or nn) pairing correlation somehowparallels [2] the metallic superconducting nonrelativistic BCS [3] energy gap A, we are not aware of a present theory which establishesthis link in detail. Likewise the consequent correlated pair of nucleons form a 0’ boson excitation, degenerate in energy with the O+ ground state, and resembling a pion which is an s-wave quark-antiquark spinless relativistically bound Nambu-Goldstone boson [4]. Yet we do not know of a nuclear model which relies heavily upon this analogy with relativistic quark physics. 184 0003.4916185 $7.50 Copyright All rights

0 1985 by Academic Press, Inc. of reproduction in any form reserved.

ISOSPIN

SYMMETRY

185

In this paper we attempt to relate this valence nuclear pairing problem to both the superconductive BCS model and the quantum chromodynamic (QCD) quark model by exploiting the ideas of dynamical symmetry breakdown as first proposed for nonrelativistic superconductivity [5] and for relativistic quark physics [6, 71. Such a unification [8] has recently been worked out in detail for superconductivity (dynamical breakdown of charge symmetry) and for QCD (dynamical breakdown of chiral symmetry). Here we extend the link to valence nuclear pairing via dynamical breakdown of isospin symmetry. We accomplish this task by first estimating in Section II the disturbance propagation speed and Debye energy associated with (sound) waves traveling on the surface of the nuclear core. Then in Section III we draw the parallel between the pairing mechanisms for superconductivity, valence nuclear pairing and QCD and focus on the nuclear phonons which correspond on the one hand to phonons in metals and on the other to gluons in QCD. Such dynamical “glue” sets up a nuclear pairing potential which in turn generates the nonperturbative energy gap as discussed in Section IV. Alternatively in Section V we introduce a “nuclear decay constant” f, as a spontaneous symmetry breakdown order parameter, similar to the pion decay constant f, in strong interaction physics and f, for superconductivity [8]. Then in Section VI we demonstrate that fN and A bear the same dynamical “Goldberger-Treiman” [9] relationship as dof, and the dynamically generated quark mass mdyn (i.e., -mN /3) in QCD and also f, and A for superconductivity [8]. In Section VII we turn to the dynamically generated Nambu-Goldstone boson for the valence nuclear pairing theory. Like the pion in QCD, it is a zero energy phase excitation relative to the O+ ground state in EE and 00 nuclei. Although it is difficult to detect explicitly, it may be inferred in conjunction with the 0’ excited state boson which is the analog of the Of ljq u meson in QCD [lo] and the Ct electron-hole boson [ 111, respectively at energies 2mdyn and 24. These nuclear “quasi-bound state” bosons are signals of isospin symmetry breakdown and ultimately are due [l] to the neutron excess for large A. Lastly, in Section VIII we discuss the (isospin) symmetry restoration temperature in medium-heavy nuclei, which might correspond to the “critical” temperature determined from nuclear level densities to be [ 121 k, T, N 0.7 MeV at A z 120. We suggest that this temperature and ratio A/k, T, - 1.8 parallels the BCS prediction [3], although it should be associated with a different Debye energy scale. In Section IX we review our findings and stress that a systematic approach to the valence nuclear pairing problem along the lines of nonrelativistic superconductivity and also relativistic QCD appears to confirm its dynamical isospin symmetry breakdown origin. In the Appendix we give a table comparing the relevant physical quantities in these three apparently disparate but actually quite similar theories of dynamical symmetry breakdown.

MICHAEL D.~CADRON

186

II. DISTURBANCE PROPAGATION AND DEBYE ENERGY To begin with, we observe that the confining force which keeps the outer-shell nucleons attracted to the nuclear core is due to isospin- and rotation-symmetric strong-interaction pion exchange. If we impact upon such a spherically symmetric nucleus only slightly, surface waves will propagate as on a two-dimensional vibrating drum. The associated surface wave speed c, can be determined by analogy with the linear speed along a string (T/,u)“~, where T is the string tension and ,U the mass per unit length. In particular the attractive nuclear force (tension) per unit length P’ is equivalent to the nuclear surface energy per unit area, ,y - b surf z 0.94 MeV/fm*, 47cri

(3)

with [ 1] bsurr z 17 MeV and r 0 =: 1.2 fm. Also the surface mass per unit surface area J? can be estimated as Jr%

M surf 41zRz=:

psurf6R - 5 2 . w‘*17 (2 fm) z 160 MeV/fm*,

where we have taken the nuclear density near the surface as one-half the interior density m,(O. 17)/fm3 and the thickness of surface nucleons as 6R N 2 fm. Then we deduce in natural units A = c = 1, a surface speed of magnitude c, = &qH

- l/13,

(54

where the square root structure of c, in (5a) is the area analog of (T/p)“‘. The fact that the surface speed c, in (5a) is less than the “sound” speed oF/3 z 0.095 across the nucleus for an equal mass Fermi gas with [ 1] E, = mN u5/2 z 38.5 MeV is satisfying. Alternatively, approximating the nuclear surface as flat so that c, is driven by only those springs (-l/2) below the surface, we find instead of (5a) that c, N” ;2

c,(eq. mass) = -

1 d 2

“; ) =x>

1

reasonably close to (5a). Henceforth we shall refer to (5b) for concreteness because it involves fewer fitted parameters than does (5a). Given this cS, we may estimate the Debye energy wn associated with the surface waves from 0, z c,kn, where k, is the Debye wave number corresponding to the smallest normal mode for the rotationally symmetric nuclear core. In the case of metallic superconductors, the normal modes are translationally symmetric Bloch inter-ion distance d = 112 so that k, = 27LlL = n/d and waves with k z (l/50) eV for d z 3 A and c s =: 3 x 10P5, which is the speed of sound in wI,=cc, D a metal. For the rotationally symmetric nuclear core, the largest wavelength or lowest

ISOSPIN SYMMETRY

187

FIG. 1. Lowest quadrupole I= 2 excitation mode of a valence pair of nucleons rotating outside of the nuclear core.

normal mode fixes the energy scale of o,,. Thus in Fig. 1 the orbital quantum number is I= 2, for which 1= 2nR/2 = xR where the core radius for A z 120 is R zz r,A’13 z 5.9 fm.

(6)

Then we have k, = 27r/1= 2R-‘, with Debye energy found from (5b) and (6): q, z c, k, = 2c,/R z 4.4 MeV.

(7)

By way of comparison, metallic melting temperatures are of order 500’K which is about twice the above metallic Debye temperature TD = coo/k, E 230’K. Thus it should not be surprising that the nuclear Debye energy (7) is also about one-half the observed binding energy per nucleon -8.5 MeV for heavy nuclei, the latter being the “melting energy” in the nuclear case [ 11. Stated in reverse, we may assumethe energy scale (7) and use it to determine the disturbance surface speedc, now for the third time as c, - l/15.

III. VALENCE

PAIRING,

PHONONS, AND THE PAIRING POTENTIAL

Just as metallic Cooper pairs of correlated electrons on the Fermi surface have both linear momentum and spin vectors pointing in opposite directions, so in a medium heavy and heavy nucleus correlated pairs of valence protons or neutrons near the Fermi surface have both angular momentum and spin vectors pointing in opposite directions again becauseof the exclusion principle. (A case in point [ 13] is 0’* with the two valence neutrons having oppositely pointing orbital angular momenta and spins.) A further consequence is that the total angular momentum J,, = L,, + S,, of a correlated p-p or n-n pair vanishes becauseL,, = S,, = 0 (but 1, = I, # 0). This is compatible with an overall J-J (but not L-S) coupling pattern and the observation that all even-even nuclei appear to have zero total angular momentum. Lastly, this angular momentum picture for valence nuclear pairing justifies employing for c, the surface disturbance wave velocity as needed in (7) rather than the linear speed of sound, which is what is excited for translationally symmetric metals and superconductivity.

188

MICHAEL

D.

SCADRON

In all three cases (e-e,p-p, n-n) the corresponding pairing potential is generated by an effective fermion-phonon Friihlich interaction of the form [ 141 R=C,y*V.Ay,

(84

with deformation coupling C, = -V(dE/dV) in metals and (for surface area LZY)

= ctMion (with volume V) for electrons

c, = -d(dE/daq

= c~M,“,f

WI

for protons or neutrons near the surface of the nuclear core. Such nuclear phonons are simulating the core-mediated interaction between a valence-nucleon pair according to the scattering graph of Fig. 2, leading to the attractive momentum-space potential [ 141

q2

v(q’)=--$ surf

o;+

IE, -Jq2

where n is the proton or neutron number density near the surface of the core, E,,, are the energies of the paired nucleons and q, wq are the phonon momentum (magnitude) and energy. Since n in (9) is known [ I] to be about -l/2 the number density throughout the nucleus, we should expect the nuclear mass near the surface of the core, Msurf, in (8b) and (9) also to be about -l/2 the total mass of the nucleus Am,. This is in fact what we deduce from (4) by also using the nuclear radius (6). The next step is to assume these nuclear phonons are at low energy and are of the acoustical type with o 4 z c,q. Then on the Fermi surface with E 1 = E, , the momentum-space pairing potential (9) for low q becomes [14], using (8b), VP) z -c,/n,

(10)

independent of wq or q. On the other hand, the density of states per unit energy per unit volume, N(O), for the paired protons or neutrons on their respective Fermi surfaces has the form N(o)=Vx- l dNTl-

Is’-d;,d&,

\

P or n FIG.

2.

Phonon

exchange

dNTI-!yl dV - C, ’

P n force

between

two protons

or two neutrons.

189

ISOSPIN SYMMETRY

where nil is the number density of valence protons or neutrons with oppositely directed spins and we have again employed (Sb). Since (11) has the structure of the reciprocal of (lo), their product is simply w4

I W>l = n 11/n = l/2,

(12)

because the number density of oppositely paired valence p-p or n-n spins is about one-half the total number density of respective nucleon spins, according to the discussion after Eq. (9). It is important to appreciate that the factor of l/2 in (12) gets reduced to about -l/5 in a superconductor due to the repulsive effect of the Debye screening of Coulomb electrons [3,8, 141. No such repulsive potential obscures (12) for the valence nuclear pairing problem; neutrons are not repelled Coulombically and the proton repulsion is weak because the paired particles are typically separated by the long distance 2R - 12 fm.

IV. NONPERTURBATIVE

ENERGY GAP GENERATION

Now we are able to examine the dynamical problem of the generation of the energy gap d for valence nuclear pairing. As in the case of superconductivity, the Lippmann-Schwinger equation in momentum space sums nonperturbatively at low energies to [3]

T(E) = V(0) + V(0)J &

V(0) + **.

= 1 - N(0) V(0) arcsinh(w,/(2E,

- E, - E2)) ’

(13)

because V(0) in (10) is independent of energy, as in the number density per unit energy N(0) = dn’/dE’ near the Fermi surface. Thus (13) becomes singular at the gap energy -d = E, + E, - 2E, < 0, where using (12) gives the gap for either valence neutron or proton pairing d = w,/sinh(l\r(O)

1V(O)l)-’

z w,/sinh2.

(144

(14b)

However, we know that the Debye energy for valence nuclear pairing is predicted by (7) to be approximately one-half of the binding energy per nucleon, so that (14b) becomes at A z 120, A GZ4.4 MeVi3.63

z 1.2 MeV.

(15)

Not only is the scale of (15) compatible with the empirical value A z 1.1 MeV at A z 120, but the A dependence of (2) is roughly predicted by our analysis. That is, c,

190

MICHAELD.SCADRON

in (5a) and (5b) is very weakly dependent upon nucleon number A and then (7) requires wn - R - ’ - A - ‘I3 so that (14) predicts A - A -‘13, close to the empirical dependencein (2), A - A - I’* . We emphasize that this analysis for nuclear pairing is lessmodel dependent than for the BCS theory of superconductivity because there is only one attractive potential for nuclear pairing satisfying (12), but both an attractive and repulsive potential for superconductivity roughly obeying N(0) 1V(O)] N l/5 as determined (but not predicted) from data. In the latter case A z 20, ee5 - 3 x lop4 eV is the order of the observed metallic gap, whereas A z 2o,e-* - 1 MeV is the scale of the nuclear gap.

V. SPONTANEOUS

SYMMETRY

BREAKDOWN

AND NUCLEAR

DECAY CONSTANT

Thus far we have relied entirely upon the analogy between the e-e Cooper pairs and p-p or n-n Cooper pairs on their respective Fermi surfaces to gain insight into the dynamical structure of symmetry breakdown of isospin (nuclear pairing) from that of the breakdown of charge symmetry (superconductivity). Turning instead to particle physics, the “spontaneous” breakdown of chiral symmetry is measuredby a nonvanishing pseudoscalar pion decay constant f, # 0, where f, can be defined through the action of the axial-vector charge operating on the vacuum [4, 151 Q’, )0) = (-i/2)f, 1rci) # 0, the so-called Nambu-Goldstone realization of the symmetry. Equivalently in relativistic quantum field theory language the decay constant is f, = (014 )0), w here 4 is the scalar sigma field. Since f, has the dimensions of mass (empirically f, z 93 MeV), we may define the nonrelativistic quantum mechanical analog for superconductivity as [g] the “Cooper pair decay constant” f, = vKF%i = (l/20) eV, where the wavefunction normalization J”d3r I$&)l’ cc N, measures the number of Cooper pairs and nJ2 is the corresponding Cooper pair number density. In a similar fashion we may define a “nuclear pairing decay constant” as the analog of the wave function normalization for valence nuclear pairing:

fN = \/n/2Msurf z 3.5 MeV,

(16)

for A FZ120, where n z mi/2 is the nucleon number density measured in natural units. Also, Msurf z M,,,,/2 is th e tota1massof nucleons near the surface of the core. It is the natural mass scale replacing Mi,, for superconductivity which is needed to give the “decay constant” fN in (16) the dimensions of mass. We observe that the energy scalesf, and f, are the same order as the Debye energies w,, - (l/50) eV, 4 MeV, respectively, for superconductivity and valence nuclear pairing. Since two correlated protons have isospin projection I, = l/2 + l/2 = 1, the isospin associated with (16) is I = 1 for valence proton or neutron pairing. We need not worry about suchpn pairing and I # 1 becausethese nucleons are in different Fermi seas.

191

ISOSPIN SYMMETRY

FIG.

3.

Second-order

phase transition

graph

of the free energy

density

vs. pairing

wave

function

The “spontaneity” of this symmetry breakdown can be characterized by a Landautype of free energy density, which for a second-order phase transition can be expressedby the valence pairing wave functions as

The free energy minimum then occurs when the derivative of (17a) with respect to I#]’ vanishes, or 0; = -a/2b and

as depicted in Fig. 3, where the correlated wave function & cc n”’ in (17b) is proportional to fN as defined by (16). The significance of& cc #N # 0 in (16) and (17) is the possibility of redefinition of the ground state (i.e., the vacuum in QCD) according to as is the procedure associated with spontaneousbreakdown of (olo)-,(oIo)-fN chiral symmetry for QCD or for the czmodel.

VI. GOLDBERGER-TREIMAN

RELATION

FOR VALENCE

PAIRING

By analogy with superconductivity, a free energy density of the form F/V= cd2 + dA4 for small A also expressesspontaneous symmetry breakdown with nonvanishing gap energy A # 0. Comparing this Landau potential with (17), we might expect that there exists a relation fN cc A since both are “order parameters” associatedwith the breakdown of isospin symmetry and both have the dimensionsof mass(in natural units). In QCD such a proportionality exists between the pion decay constant f, and the dynamically generated quark massm,,,(-1/3m,)

.f, &Ill = %wn3 where the dimensionless pion-quark dynamical relation [8, 161

VW

coupling constant approximately obeys the

(18b)

192

MICHAEL

D.

SCADRON

very close to the chiral-limiting value, mdyn/fnM 315 MeV/90 MeV = 3.5. This Goldberger-Treiman relation (18) at the quark level is a measure of the dynamical breakdown of chiral symmetry, with the fi in (18b) due to quark color and the 2n factor due to Feynman rules associated with the quark loop generating f,. In Section V we have argued that fN in (16) and (17) is similar to f,. The pairing mechanism discussed in Section III not only applies to A but also to mdyn. Specifically a color-singlet qq pair with linear momenta and spins both pointing in opposite directions (compatible with a spinless pion) means that both q and 4 have the sume helicity (a . p) which is inconsistent with massless quarks (like neutrinos) being left-handed and massless antiquarks being righthanded. Thus when the quark acquires a mass mdyn # 0, the (color-singlet) pairing mechanism demands that chiral (handedness) symmetry must be manifestly broken, generating a term -mdYn qq in the QCD Lagrangian, analogous to Aylw in the nuclear Hamiltonian. Given that fN is similar to f, and A is similar to mdyn, the valence nuclear Goldberger-Treiman relation analogous to (18a) is of the form

fNgN =A, where g, is a domensionless considerations to be of order

coupling

(19)

which we must obtain from dynamical

1.2 MeVj3.5 MeV =: 0.34.

g,=Alf,=

(20)

The scale of g, is approximately independent of nucleon number A because A - A - ‘I* from (2) and fN - M,$* - A - I” from (16). To verify (20) dynamically, we return to the attractive phonon-induced potential (9), but now taking ]E, - E,I z A as the Hartree-Fock (HF) self-consistent effect of the gap energy even felt in second order [8]. Then this HF potential in momentum space as obtained from (9), v”,(q*)nW

GqZ +P*)’

(214

with p = A/c, from (5b) and (15) can be Fourier transformed to

VHF@) =j $

VH,(q2)eiqer=

--A--j

*pv dpp3 sinp 0 p* + &r)* *

@lb)

Here we have cut off the momentum integral in (21b) at the maximum phonon momentum transfer in much the same way as done for the calculation of phononinduced resistivity [ 14, 171. With the distance between valence protons (or neutrons) fixed for A z 120 to r z R z 6 fm zz (1.8 ,u-‘, we can express (21b) as VHF

Wa)

ISOSPIN

I=

193

SYMMETRY

1.1~~1~

dpp3

0

p2 + 0.29

sinp

z 11.3,

(22b)

where pF z 269 MeV is the Fermi momentum associated with the nuclear Fermi energy E F E 38.5 MeV and p = A/c s E 18 MeV are in the upper limit of the numerically evaluated integral (22b). On the other hand, we expect VHF in (22) to weakly bind the valence pair relative to the Fermi energy. Applying energy conservation relative to the Fermi surface, we have for the pair KE + V,, = -24, where the kinetic energy of the pair is of order KE = (A/2) + (A/2) = A, since each valence nucleon feeling the gap has an average KE of A/2 relative to the Fermi surface. Thus we deduce [8] VHF z -34,

(23)

which, when combined with (8b), (16), (22), and ,u = A/es, reproduces the now dynamical Goldberger-Treiman relation (19) providing the dimensionless coupling is given by gNz

J

2c n2 s z 0.34. I

We observe that (24) is precisely the required value (20). The caveat is that g, in (24) is moderately dependent upon the distance r separating the valence pair of nucleons needed to convert the integral (21b) to (22b). While r varies between 0 and the core diameter 2R, it is satisfying that our initial (and obvious) choice r z R turns out to match the coupling (24) with (20). Note that V,, - A3 in (22a) but V,, -A in (23). Thus one power of A cancels out when combining (22) and (23) providing A # 0. Such a cancellation is characteristic of all nonperturbative dynamical symmetry breakdown problems as for (i) ferromagnetic magnetization M cc tanh M - M + M3/3; (ii) superconductivity [8] with A3 cc Vnet cc A, and likewise for the temperature-dependent gap discussed in Section VIII; (iii) four-fermion field theory [6] with 6m = m CCm j d4p(p2 - m2) - ’ generating a gap equation; (iv) the Goldberger-Treiman relation f, = mdyn/gnqq combined with a QCD quark loop giving [8] f, ozg,,,mdyn; (v) the U(1) gap equation in QCD with [ 181 quark annihilation graph strength &, cc rnt, = q2. In all these cases the nonzero nonperturbative mass scale cancels out of the calculation and the resulting dimensionless coupling has a mean field theory square root (analogous to the critical exponent l/2) structure as in (24) or leads to a gap equation. Thus we conclude that the spontaneous symmetry breakdown parameter fN in Section V is linked to the gap energy order parameter A via the dynamical Goldberger-Treiman relation (19) and (24). One final observation concerning the derivation of this Goldberger-Treiman relation for nuclear pairing: the attractive Hartree-Fock potential (21) is generated by phonon exchange alone. For superconductivity the net Hartree-Fock potential is

194

MICHAELD.

SCADRON

the difference between (attractive) phonon exchange and (repulsive) photon exchange, the latter setting up a (Debye) screening effect which is absent for nuclear pairing. In this sensenuclear pairing is closer to QCD, since in the latter color-singlet case gluon exchange also sets up an attractive interaction between the bound quark and antiquark, and repulsive forces can be neglected. For QCD, however, there would appear to be no analog of the Debye energy in the color-singlet channel. VII.

BOSON EXCITATIONS

AND ISOSPM SYMMETRY

BREAKDOWN

It is a general thorem of field theory that symmetry breakdown (leading to A - 1 MeV and f, - 3 MeV for valence nuclear pairing) must be accompanied by zero-energy Nambu-Goldstone (pp or nn) boson “phase modes” [4]. Although such superconductivity C- electron-hole zero-energy phase modes [19] are “eaten up” while coupling to the (C- photon) plasma frequency, their presence is reflected by the Meissner effect and also by the Cooper pair boson condenstate effect of magnetic flux quantization [20]. For QCD and also for the o model, the pion represents the zero-energy phase mode. The fact that it materializes as an observed O- qq s-wave bound state enjoying all the zero-mass properties of partially conserved axial currents [ 151 (PCAC), but is not “eaten up,” is due to the absence of a repulsive color-singlet qq potential. While the latter is also true for valence nuclear pairing as stressedat the end of Sections III and VI, one further obstacle blocks the observance of the valence pairing Nambu-Goldstone modes. Such phase modes are characterized by a constant amplitude magnitude ]$ -f, with exponentially varying phase for wavelengths large relative to the pairing distance R - 6 fm. More quantitatively the associatedcoherence length l is of order [20] (N u,/A - 0.2/l MeV - 40 fm,

(25)

which is much greater than the size of a single nucleus, thus inhibiting the observance of the Nambu-Goldstone zero-energy phase modes. Put another way, the translational symmetry in a superconductor supports L + co and k + 0, E, -+ 0, whereas the rotational symmetry in a nucleus requires quantized wavelengths to be finite, so it is difftcult to support E, + 0 in the latter case. An alternative picture is that the coherence length (25) can be lowered to within nuclear dimensionsof -10 fm if the Nambu-Goldstone phase mode has energy above the ground state. We are not aware that such phase modes have been as yet identified. The interacting boson model [21] (IBM) is one such place where they may occur. In either case we must try to identify the second nuclear-pairing boson quanta associatedwith dynamical symmetry breakdown: the O+ analog of the Cf electronhole “amplitude mode” in superconductivity [ 111 at energy 24 and the O+ p-wave cjq CJmeson with chiral-limiting mass [6, lo] 2mdyn- 630 MeV, long inferred from the intermediate range of the 3S, isoscalar NN force. That its energy should also be 24 above the (s boson) ground state was suggestedin Ref. [22].

195

ISOSPIN SYMMETRY

Consider, for example, the even-even tin nucleus, ‘*‘Sri, with nucleon number A = 120 roughly midway between closed shells of neutrons [23]. Its three excited low-lying 0+ states (lzo Sn*) are 1.87, 2.16, and 2.59 MeV above the ground state [24], all near 6E - 24 for A - 1 MeV. However, only the 1.87 and 2.59 MeV states admit to both two-particle transfer reactions 1231; the 2.16 MeV state is not seen in the pickup reaction [24] ‘**Sn + ‘*‘Sn* while the 1.87 and 2.59 MeV states are seen [24] in both ‘**Sn+ ‘*‘Sn* and in the stripping reaction “*Sn 4 ‘*‘Sn*. Such reactions are analogous to the QCD transitions 71t) CJTLwith the two-neutron pair emitting or absorbing a phonon in the transfer reactions Sn(N f 2) + Sri*(N). To reconfirm that the 1.87 or 2.59 MeV ‘*‘Sn* 0’ states are viable candidates as “isoscalar pairing-vibration amplitude modes,” we note [25] that they must have large radial matrix elements to the ground state ‘*‘Sn as required of a radial amplitude mode, envisioned as a pairing-vibration about the minimum 1$N 1 in Fig. 3. While this is difficult to see in the isoscalar inelastic reaction [23] a+ l*OSn+ a’ + 12’Sn*, both the 1.87 and 2.59 MeV states (but not the 2.16 MeV state) are excited in [24] d + ‘*‘Sn --t d’ + ‘*‘Sn*, where d is the isoscalar deuteron. To distinguish between these two Ot states, we note that the observed binding energies of tin are [26] B(“*Sn) = 1004.984 MeV, B(‘19Sn) = 1011.469 MeV, B( 12’Sn) = 1020.574 MeV, B( ‘*‘Sn) = 1026.754 MeV, then requiring the neutronpairing even-odd energy gap to be [l] A, = f

[B(“‘Sn)

- 3B(r19Sn) + 3B(12’Sn) -B(‘*‘Sn)]

= 1.386 MeV.

(26)

Thus 24, = 2.77 MeV favors the 0 ’ ‘*‘Sn* state at 2.59 MeV as the “c-like” (i.e., Higgs-like) isoscalar boson associated with the dynamical breakdown of isospin symmetry.

VIII.

SYMMETRY

RESTORATION

TEMPERATURE

Up to this point the order parameters of symmetry breakdown, A or mdyn, have been measured at zero temperature. However another way to probe the strength of symmetry breakdown is to identify a critical temperature T,, above which the symmetry breakdown (ferromagnetic) phase melts down to the symmetry (paramagnetic or normal) phase. In superconductivity, BCS showed [3] that a A (at varying temperature) vs. T curve separates the phases as depicted in Fig. 4 with [3,27] A/k, T, z ne-)‘E z 1.76, which is consistent with data. To be more specific, the self-consistent equation for the temperature-dependent gap in a uniform medium of fermions of energy E, is [3,27] d3k A(T) A(T)=k,TIV(0)lI~,:+E:,

where the phonon momentum

integral in (27a) is cut off at the Debye momentum

(274 k, .

196

MICHAEL D. SCADRON

A h I

Normal phase I

T

TC

FIG. 4. Gap energy vs. temperature curve for superconductivity and for valence nuclear pairing.

For d(T) # 0, the gap cancels out of (27a) and then passing to the critical temperature k, T, where A(T,) = 0, one finds near the BCS Fermi surface

W)

1 I WI

w&ksTc dx xtanhx% = 0

5.1,

WI

where the hyperbolic tangent factor in (27b) is due to the statistical distribution of fermions. The numerical value of (27b) is for the superconductivity case where w,/k, - 230°K and T, - 1.5’K means that wD/2k, T, - 75 so that the upper limit of the integral is approaching its asymptote. On the other hand, for valence nuclear pairing the critical temperature T, has been estimated [ 121 from level density models of medium-heavy nuclei as found from neutron scattering data off Co and In combined with Fermi gas and superconductivity models for 58Fe, 5*Ni and “%n. The result is [12] k, T, 5 0.7 MeV

(28)

for A z 120. Combining (28) with the nuclear Debye energy of wu z 4.4 MeV from (7), the upper limit of the integral in (27b) becomes wD/2k, T, z 3.1, signilicantly less than w,/2k, T, z 75 for superconductivity. Applying the former upper limit to the BCS integral in (27b) we obtain for valence nuclear pairing 1

-

tanh x E 1.96,

(29)

which is essentially the estimate (12), now found in an entirely different manner. Alternatively, given (29) we predict d E 1.26 MeV from (14), which when combined with (28) leads to a A(T) vs. T graph as already depicted in Fig. 4, again with A/k, T, z 1.80.

(30) Thus to summarize, there are three energy scales wn, A, k, T,. For superconductivity (SC) vs. nuclear pairing (NP) we have (w,/kT,),, $+ (wD/k, T,),, and (N(0) 1V(O)]);c’ > (N(0) ( V(O)()$ but (A/k, T,),, z 1.76 is almost the same as Wa Tc),, = 1.80. In QCD, computer computations on the lattice deduce a chiral symmetry restoration temperature of order*’ k, T, - 200 MeV. Moreover a finite temperature calculation of the various nonperturbative “order parameters” in QCD, the dynamically generated quark mass and the quark condensate and the scalar 0 mass, shows that all three “melt” at temperature2’ k, T, E 200 MeV. Thus once again mdyn/k, T, z 1.6, close indeed to A/k, T, for the above nonrelativistic theories.

197

ISOSPIN SYMMETRY

IX. CONCLUSION We have attempted to explore the consequences of valence nuclear pairing viewed as dynamical breakdown of isospin symmetry and have compared this picture with that of superconductivity and QCD, respectively viewed as dynamical breakdown of charge symmetry and of chiral symmetry. The nonrelativistic binding force for Q = 2 Cooper pairs of electrons and also for I= 1 pairs of nucleons is mediated by phonons with wg = c,q in the low q acoustical region. Such phonons are the nonrelativistic analog of spin 1 massless gluons, also with wq = cq. In all four cases the Cooper pairing of e-e, p-p, n-n, q--q requires the spins and linear or angular momenta to be aligned in opposite directions. The three theories have gap energies (mdrn for QCD), “decay constants,” Goldberger-Treiman relations and boson excitations both at zero energy and energy 24 relative to the ground state, all signatures of dynamical symmetry breakdown. Lastly although superconductivity and valence nuclear pairing have vastly different Debye energy scales relative to the energy gaps, both are destroyed by symmetry restoration energies kBTc such that the BCS ratio A/k, Tc - 1.76 to 1.80 always holds. We conclude that valence nuclear pairing is indeed governed by dynamical breakdown of isospin symmetry and displays many striking analogies with both nonrelativistic superconductivity and relativistic QCD. APPENDIX:

SUMMARY

TABLE FOR DYNAMICAL SYMMETRY BREAKDOWN Superconductivity

Fermions Confining energy scale Mediator of force Disturbance speed Debye energy Pairing potential Symmetry breakdown Pairing gap Many-body order parameter Dimensionless GT coupling constant Massless boson collective modes Massive boson collective modes

e E, - 5 eV Phonon c, - 10-5 c % - l/50 eV v sttq vre, Charge sym. A - 3 x 10e4 eV f, - l/20 eV

Valence nuclear pairing

Quantum chromodynamics

P? n

49 4

E, - 39 MeV Phonon c,- l/15 c wn-4MeV V att

Isospin sym. A-1MeV

mhad.

1 GeV Gluon c, - c -

V att Chiral sym. mdyn - 3 15 MeV

f, - 3.5 MeV

f, - 90 MeV

t?N- l/3

57nqq- 3.5

C- e-e hole Eaten by photon

o+pporn-n

0- qqNGB n, K seen

C + eehole seen m,+ -24 -

O+pporn-n m(‘?3n*) =: 24

g,-

Symmetry restoration temperature Afk,

10-z

T, - 2OK T, E 1.76

Seen?

k, T, - 0.7 A/kB T, z

MeV 1.8

0+ qqseen m, = 2mdyn MeV %yn/ks Tc - 1.6

k, T, - 200

MICHAEL D.SCADRON

198

ACKNOWLEDGMENTS The author is grateful for illuminating conversations with D. Bailin, G. Bertsch, F. Iachello, J. Vary and especially to J. D. Garcia and R. A. Young for continual advice on how to contrast valence nuclear pairing with superconductivity. This work is supported in part by the U.S. Department of Energy under Contract DE-AC02-80ER10663.

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