Giant dipole resonances in hot nuclear matter in the model of self-relaxing mean field

Nuclear Physics A501 (1989) 289-300 North-Holland, Amsterdam

G I A N T D I P O L E R E S O N A N C E S IN H O T N U C L E A R M A ' I T E R IN T H E

M O D E L OF S E L F - R E L A X I N G M E A N F I E L D J. OKOLOWICZt, M. PLOSZAJCZAKI, S. DROZDZ l and E. CAURIER Centre de Recherches Nucl~aires et Universitd, Louis Pasteur de Strasbourg, Groupe de Physique Nucldaire Th~orique, BP 20, 67037 Strasbourg Cedex, France

Received 15 March 1988 (Revised 6 February 1989) Abstract: The extended time-dependent Hartree-Fock approach is applied for the description of the isovector giant dipole resonance in 4°Ca at finite temperatures. The thermalization process is described using the relaxation-time ansatz for the collision integral. Strong inhibition of the giant-dipole-resonance y-decay is found due to the fast vaporization of the nuclear surface for thermal excitation energies above E * / A ~4.5 MeV. This pre-equilibrium emission of particles in the vapor phase is associated with the radial expansion of nucleus and with the vanishing particle binding energies mainly for protons.

I. Introduction C o n s i d e r a b l e interest has b e e n devoted recently to the studies of giant dipole r e s o n a n c e (L = 1, S = 0, T = 1) in highly excited nuclei. N u m e r o u s e x p e r i m e n t a l groups have tried to make hot nuclei in energetic collisions of heavy ions a n d m e a s u r e the g i a n t - d i p o l e p h o t o n s . These experiments are very difficult a n d until n o w n o c o n s e n s u s has b e e n reached a m o n g different groups on the b e h a v i o u r of g i a n t - d i p o l e m o d e at higher energies 1,2). The theoretical analysis of this p h e n o m e n o n have c o n c e n t r a t e d o n c a l c u l a t i n g the r e s o n a n c e energy a n d the r e s o n a n c e width as a f u n c t i o n of the total a n g u l a r m o m e n t u m for relatively low t h e r m a l excitation energies. A c c o r d i n g to those calculations both the a n g u l a r m o m e n t u m a n d thermal excitations reached in e x p e r i m e n t s are not yet sufficient to see significant deviations from the r e s o n a n c e properties in the cold nucleus. Treating the m e a n field as a d y n a m i c q u a n t i t y one can describe the n u c l e a r v i b r a t i o n a n d the escape width associated with the particle decay 3). The collective vibrations m a y couple to the states outside of the collective subspace a n d to the collective surface oscillations. Softening of the n u c l e a r surface at high t e m p e r a t u r e s may therefore lead to the increase of the particle decay probability. I n all previous calculations the heated vibrating n u c l e u s has b e e n described a s s u m i n g the thermal e q u i l i b r i u m a n d the s m a l l - a m p l i t u d e oscillation of the m e a n field. Both a p p r o x i m a t i o n s c a n n o t be justified at higher excitations. It is believed that the n u c l e u s thermalizes in a few n u c l e o n traversal times, i.e. in the time which is c o m p a r a b l e with the period of one Permanent address: Institute of Nuclear Physics, PL-31-342 Krak6w, Poland. 0375-9474/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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J. Okotowicz et al. / Giant dipole resonances

dipole oscillation. For intermediate-energy heavy-ion collisions the lifetime of the composite system may be of the same order of magnitude. Hence, in this energy range the giant dipole resonance (GDR) is present as a thermal excitation and the detected y-rays are emitted predominantly in the non-equilibrium phase. Heating of the nucleus leads to the softening of the surface modes and hence to the increased anharmonicities in the coupling of the dipole mode with collective surface oscillations and with non-collective states. These two features of G D R in hot nuclei can be included in the description using extended time-dependent Hartree-Fock ( E T D H F ) approach for the non-equilibrated, self-relaxing mean field. In this work we study G D R in 4°Ca at high excitation energies and in the absence of statistical equilibrium, i.e. in the early stage of existence of the hot nucleus. The E T D H F approach is expected to provide an accurate description of the nuclear dynamics associated with both isovector dipole resonance and the heat excitations in that phase of the nuclear reaction.

2. Discussion of the results

The model consists of the time-dependent Hartree-Fock (TDHF) equations for the single-particle (s.p.) states and the simplified master equations for the timedependent occupation numbers of s.p. states with the conservation laws and H theorem properly satisfied4"5). Particle collisions, which are responsible for maintenance of the thermal equilibrium are described in the relaxation-time approximation 6). The s.p. states are evolved by a generalized set of T D H F equations: ( T + U({n/}, @)}Oi(r, t ) = ihO,t~i(r, t) ,

(1)

where indices i, j run over all proton and neutron s.p. states. T and U in (1) denote the kinetic energy operator and the time-dependent mean field which is generated by the occupation of the set of s.p. states #~(r, t) with occupation numbers n,(t). The evolution of occupation numbers is governed by the linear equation: dn~ n~- ~ dt r

,

(2)

where the relaxation time r is chosen to be the same for all s.p. states. Both ni and ~ depend on time, but their physical meaning is obviously different. Whereas n~(t) (and &~(r, t)) describe the dynamics of the system, the coefficients ~ ( t ) denote the occupation numbers towards which n,(t) tend to approach at time t. Actually the exact shape of ~ ( t ) is not known, but it was argued in ref. 4) that one can use the Fermi distribution &(t)=[exp(e,(t)-#~(t))/kT(t)+l]

1,

(3)

where #7 are the parameters of chemical potentials for protons and neutrons to be determined and T in this model specifies the particle distribution over s.p. orbits.

J. Okotowicz et al. / Giant dipole resonances

291

They are instantaneous chemical potentials and temperature only when the limit of complete thermal equilibrium is reached. The initial temperature is arbitrary but then it must be allowed to vary to keep the excitation energy fixed during the evolution. S.p. energies in (3) depend on time

ei = ~ O~*(r, t)(T+ U~)~bi(r, t) dr. d

(4)

The evolution of parameters/x,, T can be determined by the equations resulting from the particle number

dni .~

and the total energy conservation conditions d//i e,(t) -h-T= o

(6)

for protons and neutrons respectively. Hence, the model contains only one adjustable parameter the relaxation time parameter ~- which plays the role of an average thermal collision time. For further details of this model see ref. 5). In this p a p e r we investigate the isovector giant dipole resonance ( I V - G D R ) in the spherical nucleus 4°Ca for the range of temperatures up to 12 MeV and for various assumptions about the relaxation constant ~-.The H a r t r e e - F o c k (HF) calculations are based on the Skyrme hamiitonian with the B K N force. Equations were solved for s.p. states in 7 shells for protons and neutrons respectively in a cylindrical box of length 40 fm and radius 20 fm with a space mesh Ar = Az = 0.5 fm. Time propagation of the s.p. wavefunctions were obtained using the Crank-Nicholson method with a time step At = 2 × 10 24 s. The calculations have been done in three steps. First the T - - 0 ground state wavefunction has been determined by solving the static H F equations using the imaginary time-step technique 7). In the second step, for each initial temperature separately, the instantaneous occupation numbers (3) have been associated to the set of s.p. orbitals. Finally, this wavefunction has been boosted uniformly with a velocity field proportional to the dipole operator and continued the evolution by solving E D T H F equations. During this time evolution, the wave packet describing the giant dipole excitation oscillates and spreads into the states outside of the collective subspace. It also undergoes the thermal relaxation process. Fig. 1 exhibits the time-evolution of the isovector collective kinetic energy

E = I j ~1

/=1 dri

p(rl . . . . ' r A ; t ) [ ~ J f ( r l ' ' ' ' ' r A ; t ) ] 2

(7)

J. Okotowicz et al. / Giant dipole resonance~

292

~Co -

dipole mode

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0

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5

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(10 -~

~ . . . . . . . . . 3

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s)

Fig. 1. The time dependence of the isovector collective kinetic energy E and the separation of proton and neutron centers of gravity rp r,1, for initial temperatures T = 0 (the solid line) and T = 3 MeV (the dashed line) in 4°Ca. For details see the discussion in the text. a n d the s e p a r a t i o n o f p r o t o n a n d n e u t r o n centres o f gravity for the initial temp e r a t u r e s T = 0 (the solid line) a n d T = 3 M e V (the d a s h e d line), p a n d f in eq. (7) d e n o t e the d i a g o n a l d e n s i t y a n d the velocity field o f isovector d i p o l e oscillations. The time variations o f E a n d r p - r n are strongly c o r r e l a t e d a n d can be used to d e t e r m i n e the f r e q u e n c y o f d i p o l e oscillations. C o u p l i n g o f the giant d i p o l e m o d e with the states o u t s i d e o f the collective s u b s p a c e is r e s p o n s i b l e for d a m p i n g o f the oscillations o f the kinetic energy a s s o c i a t e d with a c o h e r e n t isovector c u r r e n t flow. The d a m p i n g function g({A(E*)}, t) is a p p r o x i mately an e x p o n e n t i a l f u n c t i o n o f the e n e r g y - d e p e n d e n t d e c a y c o n s t a n t A = F~ h a n d can be d e t e r m i n e d n u m e r i c a l l y by the least-squares fit to the first few m a x i m a o f E [ref. 3)]. In T D H F or E T D H F a p p r o a c h e s , 1" is p u r e l y o n e - b o d y in nature a n d describes the total e s c a p e width o f the v i b r a t i o n s 3). The initial collective excitation energy in fig. 1 has been chosen so that the average excitation energy /~* over one oscillation p e r i o d ,

~, = -E* f '''~ g ( a (E*), T

,,,

t') dr,

(81

is a p p r o x i m a t e l y equal to the excitation energy as d e d u c e d from the f r e q u e n c y o f the d i p o l e oscillations. In the a b o v e e q u a t i o n E* is equal to the collective kinetic energy at t = to. An increase o f the t h e r m a l excitation as seen in fig. 1 leads to slowly increasing d a m p i n g o f the d i p o l e excitation (see the collective kinetic energy) from

J, Okotowicz et aL / Giant dipole resonances

293

F = 1.57 MeV at T = 0 to F = 1.97 MeV at T = 3 MeV and to the lowering o f the dipole frequency from ha} = 17.96 MeV at T = 0 to hto = 16.88 MeV at T = 3 MeV. Actually, little is k n o w n about the appropriate value o f the relaxation constant r. If not stated differently, we assume in our calculations that the thermal equilibration time is short c o m p a r e d to the period o f dipole oscillations and use the value r=2X

10-24 s.

To investigate in more details the effect o f particle collisions on properties of G D R , we have calculated the time-evolution o f the isovector collective energy, the separation o f proton and neutron centres o f gravity, the temperature leT and the entropy S for the relaxation time constants ranging from 2 x 1 0 -24 to 2 × 10 20 S. In fig. 2 results are shown for r = 2 × 10 24 (the solid line), r = 2 × 10 23s (the longd a s h e d line), r = 2 × 10 22 s (the short-dashed line) and for r = 2 x 10 -21 S (the dotted line). The entropy S(t) is calculated from the grand canonical formula S = -Y~ [n, In n i + ( 1 - n i ) I n

(1-n~)],

(9)

i

where the sum goes over all s.p. orbitals. The excitation energy is E * / A ~ 1.9 MeV ( T = 3 MeV). Results presented in fig. 2 suggest that particle collisions at the thermal energies o f a r o u n d E * / A = 1.25 MeV have little influence on the dynamics o f the dipole mode. The energy o f the collective dipole oscillations is used partially to heat the nucleus. The whole process is to a very g o o d a p p r o x i m a t i o n an isentropic one. Results for the dipole frequency and the escape width at various values o f r are summarized in table 1. For r ~> 2 × 10-22s both the dipole frequency and the escape width are u n c h a n g e d . It is worth noticing that the change of the dipole frequency with the temperature is smaller for longer relaxation times. An opposite t e n d e n c y can be seen for the escape width. The main reason for the growth o f the particle-emission rates from the G D R in hot nuclei is a gradual b u i l d u p of a hot zone of weakly b o u n d particles at the nuclear surface. These particles o c c u p y s.p. states in the c o n t i n u u m or near to it and are essentially u n b o u n d . This can be seen in fig. 3 which presents the particle density in 4°Ca as a function o f the distance from the center o f the nucleus at T = 0 (the solid line), T = 2 MeV (the dashed line) and T = 4 MeV (the dotted line). These curves have been obtained by solving the static self-consistent H F equations at finite temperatures using the imaginary time-step technique 7). It is seen that a noticeable change in the shape o f density p(r), signifying vaporization of the nuclear surface, begins first for temperatures T ~> 3 MeV i.e. for the thermal excitation energies E * / A > 1.25 MeV. The nucleons in the density tail are weakly b o u n d or u n b o u n d and, hence, the collective dipole boost applied to them leads to their separation from the rest of the nucleus. This observation correlates well with the change o f F from 1 . 5 7 M e V at T = 0 to 1 . 6 9 M e V at T = 2 M e V , 1 . 9 7 M e V at T = 3 M e V and 2.38 MeV at T = 4 MeV. In this range o f thermal excitations the average binding energy decreases from 6.59 to 4.35 MeV for protons and from 10.46 to 7.67 MeV

294

J. Okotowicz et aL / Giant dipole resonances 'WCa

-

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mode

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2

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n e u t r o n c e n t e r s o f g r a v i t y r p - r n, t h e t e m p e r a t u r e

i

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of proton and

k T and the entropy for the initial temperature

3 MeV

in 4°Ca. E T D H F calculations are performed for the broad range of relaxation time constants: 7= 2 x 1 0 24 s ( t h e s o l i d l i n e ) , "r = 2 x 10 23 s ( t h e d a s h e d l i n e ) , ~ " - 2 x 10 .22 s ( t h e s h o r t - d a s h e d l i n e ) a n d r-2

x 10 -2~ s ( t h e d o t t e d l i n e ) . F o r d e t a i l s s e e t h e d i s c u s s i o n in t h e text.

for neutrons. The fast vaporization of the nuclear surface and hence the strong enhancement of particle decay width in the pre-equilibrium phase becomes at E*/A> 2.85 MeV ( T > 4 MeV) a dominant mechanism of attenuating the dipole gamma ray strength. Fig. 4 presents the time dependence of the isovector collective energy, the separation of proton and neutron centres of gravity, the temperature and the entropy for the excitation energy E*/A ~ 5 MeV which corresponds to the initial temperature of 6 MeV. The evolution of the nucleus at this temperature is qualitatively different from those seen at lower excitations because the dipole boost

295

J. Okotowicz et al. / Giant dipole resonances TABLE 1 Giant dipole frequency and the total escape width for different relaxation parameters in 4°Ca at the excitation energy E * = 1.9 MeV ( T = 3 MeV) r [s]

hw [MeV]

F [MeV]

2 × 10 24

16.88

2 × 10 2 x 10 2×10 2×10

16.99 17.28 17.36 17.37

1.97 2.09 2.15 2.13 2.13

23 22 2t 2o

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1 0 -~

1 0 -2

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v

-t.B

(¢1 t"0

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1 0 -4

1 . 1 0 -s 0

2

4

6

8

10

12

14

16

18

r (fro) Fig. 3. Nucleon density for various distances from the center of the nucleus is plotted for temperatures T = 0 (the solid line), T = 2 MeV (the dashed line) and T = 4 MeV (the dotted line).

J. Okotowicz et al. / Giant dipole resonances

296

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T=6

MeV

and

the

relaxation

time

constant

24S.

of the wavefunction initiates both the isovector oscillations and the isoscalar expansion of protons and neutrons in the vapor phase. Hence, r p - r n does not change the sign during the time-evolution as it should be in the isovector mode. Also the subsequent maxima of rp - r, grow even though the isovector kinetic energy is strongly damped. The collective oscillatory motion associated with excitation of I V - G D R is superimposed on the isoscalar expansion of particles in the vapor phase. The ratio of collective energies for isoscalar and isovector dipole modes becomes a quantitative measure of the maintenance of the initial coherence in the motion of protons against neutrons excited through the isovector field. Fig. 5 exhibits the isoscalar collective kinetic energy of the expansion and the nuclear mean square

297

J. Okotowicz et al. / Giant dipole resonances

30

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Fig. 5. The time dependence of the isoscalar dipole expansion energy and the nucleur r.m.s, radius in 4°Ca with the initial temperature T - 6 MeV. For details see the discussion in the text. radius as functions of the time. The isoscalar kinetic energy of the expansion grows from zero at t = 0 to its maximal value - 3 0 MeV at t--~ 2.5 × 10 -22 s. Note that the expansion time scale is comparable with the oscillation period of GDR. During this short time the ratio of isoscalar and isovector collective energies grows rapidly from 0 to - 5 . 5 . This rapid loss of collectivity of the isovector dipole mode which obviously is related to the growing disorder in the motion of protons against neutrons, should strongly attenuate the E1 strength from the studied excitation energy range. Note also that the r.m.s, radius of the nucleus grows gradually (see fig. 5). The box in which we allow the nucleons to move is too small to see a clear separation of the particles in the vapor and liquid phases, but the phenomenon of rapid emission of vapor particles during the first period of isovector dipole oscillations is clearly seen. It should be stressed that the large increase of the nuclear r.m.s, radius (see fig. 5) is predominantly due to the evaporation of weakly bound or unbound particles which decouple fast from the rest of the nucleons participating in the isovector dipole mode. Hence, the calculated increase in the r.m.s, radius is not accompanied by any significant lowering of the dipole frequency. The dipole frequency at E * / A = 5 MeV ( T = 6 MeV) is hto = 16.1 MeV. The escape width at this temperature F = 4.76 MeV is significantly larger than seen at T = 4 MeV, emphasizing again the enhancement of the particle emission from the resonance. Going to still higher temperatures, one observes a gradual disappearance of the collective dipole oscillations of the nuclear liquid in the background of the isoscalar expansion of the vapor

298

J. Okotowicz et aL / G i a n t dipole resonances

particles. Fig. 6 summarizes E T D H F results for T = 6, 8, 10 and 12 MeV showing the isovector kinetic energy and rv-r, 1 as functions for time. G D R is weakly pronounced at E * / A = 8 MeV ( T = 8 MeV) and disappears at E * / A = 1075 MeV ( T = 10 MeV) and E * / A = 13 MeV ( T = 12 MeV). The dipole frequency at E * / A = 8 MeV is hw = 15.6 MeV wheras the escape width is 6.9 MeV. At this temperature on the average, protons are already unbound and neutrons are only weakly bound ( B / N = - 2 . 1 8 M e V ) . Protons become unbound at the temperature 7.3MeV. Obviously, this does not mean that the liquid phase for protons disappears completely. However, one might expect that around this limiting temperature, the dipole motion of protons against neutrons will loose its coherent features and dissolves completely in the background of the isoscalar expansion mode. Indeed, the kinetic energy of the expansion which at T = 6 MeV reaches the value - 3 0 MeV grows rapidly to - 6 0 MeV at T = 8 MeV and ~90 MeV at T = 10 MeV. This can be read out from the fig. 7 which shows the collective kinetic energy of the radial expansion, the r.m.s, radius of the nucleus, the instantaneous temperature and entropy for the range of temperatures T = 6, 8, 10 and 12 MeV. Note that the expansion time scale is approximately independent of the thermal excitation of the nucleus and equals ~2.5 x 1022 S. The same time takes the nucleus to emit hot particles from the vapor phase. This time scale is much shorter than the particle decay time from the equilibrated system. It should be noticed that the nucleus is significantly cooled while expanding for T~>8 MeV. In particular the limiting temperature for the expansion at E * / A = 13 MeV ( T = 12 MeV) is only ~10.5 MeV. Note, that the evolution of the nucleus

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0 40

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o

t0 0 o.o

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0.2

0.3

0.4

time (10 -21 s)

Fig. 6. The time dependence of isovector collective energy and separation of proton and neutron centers of gravity. Expansion of 4°C was initialized by the large-amplitude dipole boost and thermal excitation of T = 6 MeV (the solid line) and 8, 10, 12 MeV (shorter dashes correspond to the higher energy).

299

J. Okolowicz et al. / Giant dipole resonances

100

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Fig. 7. The time dependence of the isoscalar collective expansion energy, the r.m.s, radius of the nucleus, the instantaneous temperature and entropy. The expansion o f 4°Ca was initialized as in fig. 6. For details see the caption of fig. 6 and the discussion in the text. f o l l o w i n g the d i p o l e e x c i t a t i o n is i s e n t r o p i c b o t h at these very high t e m p e r a t u r e s 8) a n d at l o w e r t e m p e r a t u r e s a l t h o u g h this was not i m p o s e d in calculations. The d i s a p p e a r a n c e o f the collectivity in the i s o v e c t o r d i p o l e m o d e might be o b s e r v a b l e t h r o u g h the r a p i d increase o f the a n i s o t r o p y o f the e m i t t e d p r o t o n s a n d neutrons. T h e r e m a y be f o u n d two c o m p o n e n t s to the energy d i s t r i b u t i o n o f emitted p a r t i c l e s , a high t e m p e r a t u r e strongly a n i s t r o p i c c o m p o n e n t to be a s s o c i a t e d with the i m m e d i a t e v a p o r i z a t i o n a n d the p r e e q u i l i b r i u m particle e m i s s i o n a n d a l o w e r t e m p e r a t u r e c o m p o n e n t f r o m the c o m p o u n d nucleus e v a p o r a t i o n . O u r c a l c u l a t i o n s p r e d i c t the c h a n g e in the b e h a v i o r o f the G D R at the t h e r m a l e x c i t a t i o n energies p e r p a r t i c l e b e t w e e n 4.5 a n d 5.9 MeV ( T ~ 6-7 MeV) d u e to the strongly e n h a n c e d

300

J. Okotowicz et al. / Giant dipole resonances

particle d ecay width a n d related with that d e g r a d a t i o n o f the o r d e r e d m o t i o n o f nucleons. T h es e effects s h o u l d have a strong influence on the spectral distribution and multiplicity o f the d i p o l e p h o t o n s le a d in g to the strong a t t e n u a t i o n o f the d i p o l e strength from the region o f excitation energies a b o v e E * / A ~ - 4 . 5

MeV. It should

be n o t i c e d that in this range o f th e r m a l excitations both the p r o t o n s e p a r a t i o n energy Sp an d the particle b i n d i n g energy Bp are a lr ead y very small. O n e may speculate that the limit o f the collective i s o v e c to r d i p o l e m o t i o n in nuclei is scaled by the values o f S~, and B v at high te m p e r a t u r e s . Recently, a strong inhibition o f the G D R y - d e c a y was o b s e r v e d at the excitation energies ab o v e E * / A ~ 2 . 7

MeV in ~'~Sn

[ref. 2)]. Th e e x p e r i m e n t a l values o f Sp (S,1) and Bp in ~°Sn are l o w er than those c a l c u l a t e d for 4°Ca a n d hence, one might expect that the ab o v e given limits o f collective m o t i o n s h o u l d be strongly r e d u c e d for ~°Sn. On e o f us (M.P.) wishes to thank S. O g a z a for useful discussions. Note added in proof: The omission o f the n e u t r o n and p r o t o n n u m b e r in the definition o f the quantity q , - r n

causes that the o r d i n a t e scale in all the figures

presenting this q u a n t i t y s h o u l d be d i v i d e d by 20.

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