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Nuclear Rayleigh and whispering gallery waves excited in heavy ion collisions
Volume 94B, number 1
PHYSICS LETTERS
14 July 1980
NUCLEAR RAYLEIGH AND WHISPERING GALLERY WAVES EXCITED IN HEAVY ION COLLISIONS Olga DRAGIJN Comisi~)n Nactonal de Energta At~mwa, Buenos Atres, Argentina and
Herbert I]BERALL 1 Department o f Physics, Cathohc Unzverstty, Washington, DC 20064, USA Recewed 20 January 1980
The gross structure oscillations recently observed in heavy-ion elastic or inelastic scattering and fusion cross sections are here analyzed In terms of nuclear surface (creep) wave resonances. Their dispersion curves characterize these waves as being of Raylelgh or of Whispering-Gallery type
Recent experiments in heavy-ion scattering have concentrated on a study of gross structure resonances appearing in the form of regularly spaced peaks whose energies are correlated among various reaction channels, such as elastic and inelastic differential (at fixed forward or backward angles) and total scattering, as well as in fusion cross sections. These peaks may be considered to be evidence for nuclear molecular resonances occurring during the (temporary) attachment of the incident to the target ion whereby a resonant quaslmolecular system is formed [1 ]. An intermediate structure, which incidentally appears inside many of these peaks, may be ascribed to the doorwaystate nature of the resonant levels [e.g. 1] and to their coupling to higher-order configurations, any such fine structure shall be disregarded in the following. The gross structure peaks have most clearly been noticed in the quasimolecular 12C + 12C (1.e. the 24Mg) system [2], the 12C + 160 (i.e. the 28Si) system [3], and the 12C + 28S1 (4°Ca) and 160 + 28Si (44Ti) systems [4], andhave been tentatively fit 1 Supported by the National Science Foundation, Theoretical Physics Section; attendance at the Third Latin American Workshop, Buenos Aires, 1979, supported by the International Programs Sections, National Science Foundation. 24
to rotational bands by some investigators. Hartree-Fock calculations [1 ] of heavy ion colhsions have indicated the possible formation of molecular-bonded, orbiting two-ion systems which m conventlonal scattering theory are describable as wave packets with heavily weighted high-angular momentum components (l/> ten, twenty or more). At a resonance, one of these components dominates and thus determines the spin of the resonance, as can be unambiguously shown by Pl(cos 0) fits of the angular distributions. Phenomenologlcal fits of this nature may be obtained by adding a Regge pole (creep wave) amplitude [5] to that provided by an optical model, thereby furnishing an explanation for the backward rise of the angular distribution at resonance. In the following, we use the fact that the Regge pole, or creep-wave, behavior of resonant scattering amplitudes may be explicitly obtained through application of the Watson transformation, as e.g. shown recently by one of us for the case of nuclear giantresonance excitation [6], or in classical scattering problems [7]. This approach serves to exhibit a series of circumferential (creep) wave components m the scattering amphtude, and explains the resonances of the system by a phase-matching of any one of the creep waves for which I + ~ wavelengths span the
Volume 94B, number 1
PHYSICS LETTERS
circumference of the system. We analyze the heavyion systems studied in refs. [ 2 - 4 ] in this way, and obtain dispersion curves for the phase velocities of those surface waves which cause the successwe gross structure resonances, as well as some of their attenuation properties. The shape of the dispersion curves show that the creep waves possess the characteristics of classical Raylelgh waves (or in other cases, of Whispering Gallery waves) propagating over curved boundaries, well-known from the theory of acoustic scattering [8,9] and of seismic waves [10], so that they may be considered to be nuclear Raylelgh or Whispering Gallery waves propagating over the curved surface of nuclear matter. To substantiate the foregoing statements, we note that the scattered part of the optical-model wave function contains the factor 21,5l @ ccc (1 - e )Pl(cosO) (1) (apart from the radial wave function) at any distance outside the matching radius of R- matrix theory which may be taken close to the nuclear surface r = R. In the one-level approxlmanon, one has a resonant amplitude [ 11 ] iF/ q/]': cr E - c ° l + ½iF l Pl (cos 0),
(2)
which, by using the transformation [12] t
col ~ col E + (l - lE)colE + ...
(3)
with
colE = E,
(4)
(5)
A(E) ~ 17r(2/zr(lp + ~)sin 0) 1/2 X ~ ~ exp(--~lCrr -- imTr) e=±l m=0 X exp {i(lp + ½)[e0r - 0) + 2mzr] },
(9)
which represents two creep waves propagating over the surface in opposite directions, plus other such waves which have already encircled the system m times. Each of these waves undergoes a phase jump of a quarter-wavelength at either of the two focal points where the surface waves converge The orbiting quaslmolecular system may be viewed, instead of as a wave packet in the components 4l, alternatively as a wave packet composed of many creep wave components which form a complete set [13], and which will resonate one at a time [14] as the energy is varied. The resonance (of multipolarity l) is generated when the Regge pole, moving along its trajectory, passes the integer l (1.e. l E = l), so that l + g1 wavelengths of the creep wave span the surface of the system, and lead to constructive interference due to phase matching. This is seen from eq. (9) which shows that the surface waves have a wavelength (10)
(6)
(7)
(11)
and an amplitude of the form exp ( - 0 / 0 e ) with a decay angle 0 e = 2/FIE.
where
(~l = Pl/col '
(8)
or asymptotically for l sin 0 >> 1 :
c(E) = R E / ( I E + ½),
where f(/) designates less essennal factors. A Regge pole in this amphtude is located at a position/p(E) in the complex/-plane, lp(E) = l E + ½ lI'lE ,
A(K) = f(lp) rrP/p(-COS 0)/cos 7r(lp + ½),
a phase velocity
el(cos0) A l ( E ) = r i d l - lv(E ) '
and moves along a trajectory m this plane as E (which may be taken as the center-of-mass energy of the colhsIon) increases. Re-evaluating the total wave amphtude at the Regge pole by the use of the Watson transfonnation leads to [ 6 - 8 ]
X = 2rrR/(l E + ½),
may be converted into a resonant expression in the l-variable, i.e. an amphtude factor
14 July 1980
(12)
From eq. (11) we obtain the dispersion curves c(E) of the nuclear surface waves. This is done for the nuclear systems studied in refs. [ 2 - 4 ] where resonance peaks have been observed at a series of resonance energies col, widths Pl of the gross structures have been determined, and the angular momenta l 25
Volume 94B, number 1
PHYSICS LETTERS
of the resonances have been obtained b y P1 (cos 0) fits to the angular distributions. Using these experimental results, we find the dispersion curves shown in the following figures, as well as the decay angles 0 e given b y eq. (12). Fig. 1 presents the quantity c(E)/R for the surface wave that gives rise to the resonances observed [3] in 28S1, as well as Oe(E), plotted versus the centerof-mass energy of the 12 C -- 160 system. The dispersion curve c(E) is seen to start at the origin for E = 0, and to reach a constant value c(oo) at E --, oo. This is the typical behavior of a Raylmgh wave [8,9] propagating over the curved surface of (nuclear) matter, in contrast to higher-order ("Whispering Gallery") modes of surface wave which exist at energies above a certain cutoff value E c that is different for each mode, and for which c(E) descends from an infinite value at E c and approaches (for E -* oo) the bulk wave speed m nuclear matter. The limiting speed c(~o) o f the Rayleigh wave shown in fig. 1, however, represents the speed o f a Raylmgh wave on the flat boundary o f semi-infinite nuclear matter which in general hes below the bulk-speed value. We may state, therefore, that our analysis o f the heav3~-ion scattering in question indicates the existence of Rayleigh waves on the curved surface o f nuclear matter, while higher order or Whispering-Gallery modes that may exist for such a system will be discerned in one o f the following cases. The plot Of0e(E ) in fig. 1 indicates that a t E =
T, t
T,2c6o
14 July 1980
,io I , ~ ,| I ee(E)' rod 5 O"
I '
.
.
.
L2C+12C c(O3)
~ ~
k
20
>
.
~
~
]
_
R
30
~E E
L
i
h
I0
L
J
20
t
i
30
/
i
40
50
Ecm ,MeV
Fig 2 Dispersion curve of Raylelgh-type wave on the surface of 24Mg, excited m 12C + 1 2 C scattering, as well as graph of 0e(E). Error bars show the experimental [2] width.
19.7 MeV, the surface wave amplitude decreases to 1/e of its original value after moving through an angle 0 e = 790 °, while at E = 25.2 MeV the same decay takes place after 0 e = 126 °. Thus especially in the first case, a substantial clrcumnagivation o f the system by the resonant creep wave is seen to take place. Figs. 2 and 3 present corresponding curves for the systems [2] of 24Mg, and [4] of 44Ti. While c(E) for 12C + 12C (fig. 2) again indicates the presence of a Raylelgh wave on the surface o f the 24Mg system, with the Rayleigh speed limit on a flat nuclear surface agreeing with that from 28Si (fig. 1) to with,
r-
,
,
,
£8Sl
R >
>
2
O"
2O
2
3O
E
w u
v
L
c(~)
R
o,
'
Ib
'
' 20
'
3' 0
'
40'
'
50
Ecm, MeV
Fig. 1. Dispersion curve of Raylelgh-type wave on the surface of 2BSI, excited m 12C + 1 6 0 scattering, as well as graph of Oe(E).Error bars show the experimental [3] width. 26
i
0
l
I0
'
z'o
'
30'
'
4b
'
50
Ecm,MeV
Fig 3. Dispersion curve of Whispering-Gallery-type wave on the surface of 44T1, excited in 160 + 28Si scattering. Error bars show the experimental [4] width.
Volume 94B, number 1
PHYSICS LFTTERS
m 5%, the character of the dispersion curve for 44T! (fig. 3) indicates the excitation of (presumably the lowest-order branch of) nuclear Whispering Gallery waves on this system in the corresponding heavy-ion reaction. The fact that Raylelgh wave resonances are observed at EcM ~ 1 5 - 3 5 MeV for the lighter nuclei under consideration, and Whispering Gallery wave resonances at EcM ~ 20--35 MeV for a heavier nucleus, may be simply due to the fact that heavier nuclei are larger and thus have a flatter surface, so that at a given energy we are here closer to the flat-surface limit, and thus the pattern of the dispersion curves shifts to lower energies as the nucleus becomes larger. It therefore remains to search for the WhisperingGallery wave resonance in the 12C + 12C and 12C + 160 reactions at higher energies, and for the Raylelghwave resonance in the 160 + 28S1 system at lower enerDes, as well as for the higher-order WhisperingGallery resonances in all of these reactions at higher energies yet In conclusmn, it may be said that our analysis of the resonances observed in heavy-ion scattering in terms of nuclear surface waves on the intermediate (nuclear-molecular) systems seems to provide the first indication for the existence, as well as their classification, of nuclear Rayleigh and Whispering Gallery waves on the curved surface of nuclear matter. These waves agree with their classical counterparts insofar as they constitute a solution of the wave equation (in the nuclear case, the SchriSdinger equation), propagate over a boundary of matter and show an equivalent dispersion behavior. They differ from the classical waves as these are actual density waves, while the nuclear creep waves are only the probability amphtudes described by an optical model wave function of two colliding heavy ions. Therefore, rather higher angular momenta, or short wavelengths, are completely acceptable here even for a relatively light intermediate nuclear system. We wish to acknowledge the support of the U.S. National Science Foundation, International Programs
14 July 1980
(Latin America) Section permitting the attendance of one of us (H.U.) at the Third Latin American Condensed Matter workshop in Buenos Aires, Argentina, July 2 - 1 3 , 1979, as well as our support at his workshop by CONICET and CNEA of the Repubhc of Argentina.
References [1] K.A Erb and D A. Bromley, Physics Today, January 1979, p. 34 [2] T.M Cormler et al., Phys. Rev. Lett 40 (1978) 924 [3] D. Shaplra et al., Phys. Rev. Lett. 40 (1978) 371 [4] J Barrette et al, Phys. Rev Lett. 40 (1978) 445. [5] KW McVoy, Phys Rev. C3(1971) 1104, R.C Fuller and O. Dragfin, Phys Rev Lett 32 (1974) 617 [6] A.R. Farhan, J. George and H i)berall, Nucl Phys A305 (1978) 189 [7] J.D Murphy, J George, A Nagl and H. i~lberall,J. Acoust. Soc. Am. 65 (1979) 368, H Uberall et al., J. Acoust. Soc. Am., to be pubhshed [8] H i~lberall,Surface waves m acoustics, m Physical acoustics, eds W P Mason and R.N Thurston (Academic Press, New York, 1971) Vol 10, p 1, H. i~/berall,m Proc IUTAM Symp on Modern problems m elasnc wave propagation, eds. J Achenbach and J. Mlklowltz (Wiley, New York, 1978) p. 239 [9] G.V. Frisk, J W. Dickey and H i~lberall,J. Acoust. Soc. Am. 58 (1975) 996, G.V Frisk and H. Uberall, J Acoust. Soc. Am. 59 (1976) 46, J W Dickey, G.V. Frisk and H. Uberall, J. Acoust. Soc Am 59 (1976) 1339. [10] W Ewmg, W Jardetzky and F. Press, Elastic waves m layered media (McGraw-Hill,New York, 1957). [11] D. Brink, J, Grabowskl and E. Vogt, Nucl. Phys A309 (1978) 359. [12] K.W McVoy, m: Classical and quantum mechamcal aspects of heavy Ion colhslons, Lecture Notes l~ Physics No. 33, eds. 1 Ehlers and H A. WeldenmfiUer (Springer Verlag, Berhn, 1975) p 127. [13 ] A Sommerfeld, Partlelle Dlfferentlalglmchungen der Physlk (Akademlsche Verlagsgesellschaft, Leipzig, 1948) p 216 [14] J.W. Dickey and H. Uberall, J Acoust Soc. Am 63 (1978) 319.
27
PHYSICS LETTERS
14 July 1980
NUCLEAR RAYLEIGH AND WHISPERING GALLERY WAVES EXCITED IN HEAVY ION COLLISIONS Olga DRAGIJN Comisi~)n Nactonal de Energta At~mwa, Buenos Atres, Argentina and
Herbert I]BERALL 1 Department o f Physics, Cathohc Unzverstty, Washington, DC 20064, USA Recewed 20 January 1980
The gross structure oscillations recently observed in heavy-ion elastic or inelastic scattering and fusion cross sections are here analyzed In terms of nuclear surface (creep) wave resonances. Their dispersion curves characterize these waves as being of Raylelgh or of Whispering-Gallery type
Recent experiments in heavy-ion scattering have concentrated on a study of gross structure resonances appearing in the form of regularly spaced peaks whose energies are correlated among various reaction channels, such as elastic and inelastic differential (at fixed forward or backward angles) and total scattering, as well as in fusion cross sections. These peaks may be considered to be evidence for nuclear molecular resonances occurring during the (temporary) attachment of the incident to the target ion whereby a resonant quaslmolecular system is formed [1 ]. An intermediate structure, which incidentally appears inside many of these peaks, may be ascribed to the doorwaystate nature of the resonant levels [e.g. 1] and to their coupling to higher-order configurations, any such fine structure shall be disregarded in the following. The gross structure peaks have most clearly been noticed in the quasimolecular 12C + 12C (1.e. the 24Mg) system [2], the 12C + 160 (i.e. the 28Si) system [3], and the 12C + 28S1 (4°Ca) and 160 + 28Si (44Ti) systems [4], andhave been tentatively fit 1 Supported by the National Science Foundation, Theoretical Physics Section; attendance at the Third Latin American Workshop, Buenos Aires, 1979, supported by the International Programs Sections, National Science Foundation. 24
to rotational bands by some investigators. Hartree-Fock calculations [1 ] of heavy ion colhsions have indicated the possible formation of molecular-bonded, orbiting two-ion systems which m conventlonal scattering theory are describable as wave packets with heavily weighted high-angular momentum components (l/> ten, twenty or more). At a resonance, one of these components dominates and thus determines the spin of the resonance, as can be unambiguously shown by Pl(cos 0) fits of the angular distributions. Phenomenologlcal fits of this nature may be obtained by adding a Regge pole (creep wave) amplitude [5] to that provided by an optical model, thereby furnishing an explanation for the backward rise of the angular distribution at resonance. In the following, we use the fact that the Regge pole, or creep-wave, behavior of resonant scattering amplitudes may be explicitly obtained through application of the Watson transformation, as e.g. shown recently by one of us for the case of nuclear giantresonance excitation [6], or in classical scattering problems [7]. This approach serves to exhibit a series of circumferential (creep) wave components m the scattering amphtude, and explains the resonances of the system by a phase-matching of any one of the creep waves for which I + ~ wavelengths span the
Volume 94B, number 1
PHYSICS LETTERS
circumference of the system. We analyze the heavyion systems studied in refs. [ 2 - 4 ] in this way, and obtain dispersion curves for the phase velocities of those surface waves which cause the successwe gross structure resonances, as well as some of their attenuation properties. The shape of the dispersion curves show that the creep waves possess the characteristics of classical Raylelgh waves (or in other cases, of Whispering Gallery waves) propagating over curved boundaries, well-known from the theory of acoustic scattering [8,9] and of seismic waves [10], so that they may be considered to be nuclear Raylelgh or Whispering Gallery waves propagating over the curved surface of nuclear matter. To substantiate the foregoing statements, we note that the scattered part of the optical-model wave function contains the factor 21,5l @ ccc (1 - e )Pl(cosO) (1) (apart from the radial wave function) at any distance outside the matching radius of R- matrix theory which may be taken close to the nuclear surface r = R. In the one-level approxlmanon, one has a resonant amplitude [ 11 ] iF/ q/]': cr E - c ° l + ½iF l Pl (cos 0),
(2)
which, by using the transformation [12] t
col ~ col E + (l - lE)colE + ...
(3)
with
colE = E,
(4)
(5)
A(E) ~ 17r(2/zr(lp + ~)sin 0) 1/2 X ~ ~ exp(--~lCrr -- imTr) e=±l m=0 X exp {i(lp + ½)[e0r - 0) + 2mzr] },
(9)
which represents two creep waves propagating over the surface in opposite directions, plus other such waves which have already encircled the system m times. Each of these waves undergoes a phase jump of a quarter-wavelength at either of the two focal points where the surface waves converge The orbiting quaslmolecular system may be viewed, instead of as a wave packet in the components 4l, alternatively as a wave packet composed of many creep wave components which form a complete set [13], and which will resonate one at a time [14] as the energy is varied. The resonance (of multipolarity l) is generated when the Regge pole, moving along its trajectory, passes the integer l (1.e. l E = l), so that l + g1 wavelengths of the creep wave span the surface of the system, and lead to constructive interference due to phase matching. This is seen from eq. (9) which shows that the surface waves have a wavelength (10)
(6)
(7)
(11)
and an amplitude of the form exp ( - 0 / 0 e ) with a decay angle 0 e = 2/FIE.
where
(~l = Pl/col '
(8)
or asymptotically for l sin 0 >> 1 :
c(E) = R E / ( I E + ½),
where f(/) designates less essennal factors. A Regge pole in this amphtude is located at a position/p(E) in the complex/-plane, lp(E) = l E + ½ lI'lE ,
A(K) = f(lp) rrP/p(-COS 0)/cos 7r(lp + ½),
a phase velocity
el(cos0) A l ( E ) = r i d l - lv(E ) '
and moves along a trajectory m this plane as E (which may be taken as the center-of-mass energy of the colhsIon) increases. Re-evaluating the total wave amphtude at the Regge pole by the use of the Watson transfonnation leads to [ 6 - 8 ]
X = 2rrR/(l E + ½),
may be converted into a resonant expression in the l-variable, i.e. an amphtude factor
14 July 1980
(12)
From eq. (11) we obtain the dispersion curves c(E) of the nuclear surface waves. This is done for the nuclear systems studied in refs. [ 2 - 4 ] where resonance peaks have been observed at a series of resonance energies col, widths Pl of the gross structures have been determined, and the angular momenta l 25
Volume 94B, number 1
PHYSICS LETTERS
of the resonances have been obtained b y P1 (cos 0) fits to the angular distributions. Using these experimental results, we find the dispersion curves shown in the following figures, as well as the decay angles 0 e given b y eq. (12). Fig. 1 presents the quantity c(E)/R for the surface wave that gives rise to the resonances observed [3] in 28S1, as well as Oe(E), plotted versus the centerof-mass energy of the 12 C -- 160 system. The dispersion curve c(E) is seen to start at the origin for E = 0, and to reach a constant value c(oo) at E --, oo. This is the typical behavior of a Raylmgh wave [8,9] propagating over the curved surface of (nuclear) matter, in contrast to higher-order ("Whispering Gallery") modes of surface wave which exist at energies above a certain cutoff value E c that is different for each mode, and for which c(E) descends from an infinite value at E c and approaches (for E -* oo) the bulk wave speed m nuclear matter. The limiting speed c(~o) o f the Rayleigh wave shown in fig. 1, however, represents the speed o f a Raylmgh wave on the flat boundary o f semi-infinite nuclear matter which in general hes below the bulk-speed value. We may state, therefore, that our analysis o f the heav3~-ion scattering in question indicates the existence of Rayleigh waves on the curved surface o f nuclear matter, while higher order or Whispering-Gallery modes that may exist for such a system will be discerned in one o f the following cases. The plot Of0e(E ) in fig. 1 indicates that a t E =
T, t
T,2c6o
14 July 1980
,io I , ~ ,| I ee(E)' rod 5 O"
I '
.
.
.
L2C+12C c(O3)
~ ~
k
20
>
.
~
~
]
_
R
30
~E E
L
i
h
I0
L
J
20
t
i
30
/
i
40
50
Ecm ,MeV
Fig 2 Dispersion curve of Raylelgh-type wave on the surface of 24Mg, excited m 12C + 1 2 C scattering, as well as graph of 0e(E). Error bars show the experimental [2] width.
19.7 MeV, the surface wave amplitude decreases to 1/e of its original value after moving through an angle 0 e = 790 °, while at E = 25.2 MeV the same decay takes place after 0 e = 126 °. Thus especially in the first case, a substantial clrcumnagivation o f the system by the resonant creep wave is seen to take place. Figs. 2 and 3 present corresponding curves for the systems [2] of 24Mg, and [4] of 44Ti. While c(E) for 12C + 12C (fig. 2) again indicates the presence of a Raylelgh wave on the surface o f the 24Mg system, with the Rayleigh speed limit on a flat nuclear surface agreeing with that from 28Si (fig. 1) to with,
r-
,
,
,
£8Sl
R >
>
2
O"
2O
2
3O
E
w u
v
L
c(~)
R
o,
'
Ib
'
' 20
'
3' 0
'
40'
'
50
Ecm, MeV
Fig. 1. Dispersion curve of Raylelgh-type wave on the surface of 2BSI, excited m 12C + 1 6 0 scattering, as well as graph of Oe(E).Error bars show the experimental [3] width. 26
i
0
l
I0
'
z'o
'
30'
'
4b
'
50
Ecm,MeV
Fig 3. Dispersion curve of Whispering-Gallery-type wave on the surface of 44T1, excited in 160 + 28Si scattering. Error bars show the experimental [4] width.
Volume 94B, number 1
PHYSICS LFTTERS
m 5%, the character of the dispersion curve for 44T! (fig. 3) indicates the excitation of (presumably the lowest-order branch of) nuclear Whispering Gallery waves on this system in the corresponding heavy-ion reaction. The fact that Raylelgh wave resonances are observed at EcM ~ 1 5 - 3 5 MeV for the lighter nuclei under consideration, and Whispering Gallery wave resonances at EcM ~ 20--35 MeV for a heavier nucleus, may be simply due to the fact that heavier nuclei are larger and thus have a flatter surface, so that at a given energy we are here closer to the flat-surface limit, and thus the pattern of the dispersion curves shifts to lower energies as the nucleus becomes larger. It therefore remains to search for the WhisperingGallery wave resonance in the 12C + 12C and 12C + 160 reactions at higher energies, and for the Raylelghwave resonance in the 160 + 28S1 system at lower enerDes, as well as for the higher-order WhisperingGallery resonances in all of these reactions at higher energies yet In conclusmn, it may be said that our analysis of the resonances observed in heavy-ion scattering in terms of nuclear surface waves on the intermediate (nuclear-molecular) systems seems to provide the first indication for the existence, as well as their classification, of nuclear Rayleigh and Whispering Gallery waves on the curved surface of nuclear matter. These waves agree with their classical counterparts insofar as they constitute a solution of the wave equation (in the nuclear case, the SchriSdinger equation), propagate over a boundary of matter and show an equivalent dispersion behavior. They differ from the classical waves as these are actual density waves, while the nuclear creep waves are only the probability amphtudes described by an optical model wave function of two colliding heavy ions. Therefore, rather higher angular momenta, or short wavelengths, are completely acceptable here even for a relatively light intermediate nuclear system. We wish to acknowledge the support of the U.S. National Science Foundation, International Programs
14 July 1980
(Latin America) Section permitting the attendance of one of us (H.U.) at the Third Latin American Condensed Matter workshop in Buenos Aires, Argentina, July 2 - 1 3 , 1979, as well as our support at his workshop by CONICET and CNEA of the Repubhc of Argentina.
References [1] K.A Erb and D A. Bromley, Physics Today, January 1979, p. 34 [2] T.M Cormler et al., Phys. Rev. Lett 40 (1978) 924 [3] D. Shaplra et al., Phys. Rev. Lett. 40 (1978) 371 [4] J Barrette et al, Phys. Rev Lett. 40 (1978) 445. [5] KW McVoy, Phys Rev. C3(1971) 1104, R.C Fuller and O. Dragfin, Phys Rev Lett 32 (1974) 617 [6] A.R. Farhan, J. George and H i)berall, Nucl Phys A305 (1978) 189 [7] J.D Murphy, J George, A Nagl and H. i~lberall,J. Acoust. Soc. Am. 65 (1979) 368, H Uberall et al., J. Acoust. Soc. Am., to be pubhshed [8] H i~lberall,Surface waves m acoustics, m Physical acoustics, eds W P Mason and R.N Thurston (Academic Press, New York, 1971) Vol 10, p 1, H. i~/berall,m Proc IUTAM Symp on Modern problems m elasnc wave propagation, eds. J Achenbach and J. Mlklowltz (Wiley, New York, 1978) p. 239 [9] G.V. Frisk, J W. Dickey and H i~lberall,J. Acoust. Soc. Am. 58 (1975) 996, G.V Frisk and H. Uberall, J Acoust. Soc. Am. 59 (1976) 46, J W Dickey, G.V. Frisk and H. Uberall, J. Acoust. Soc Am 59 (1976) 1339. [10] W Ewmg, W Jardetzky and F. Press, Elastic waves m layered media (McGraw-Hill,New York, 1957). [11] D. Brink, J, Grabowskl and E. Vogt, Nucl. Phys A309 (1978) 359. [12] K.W McVoy, m: Classical and quantum mechamcal aspects of heavy Ion colhslons, Lecture Notes l~ Physics No. 33, eds. 1 Ehlers and H A. WeldenmfiUer (Springer Verlag, Berhn, 1975) p 127. [13 ] A Sommerfeld, Partlelle Dlfferentlalglmchungen der Physlk (Akademlsche Verlagsgesellschaft, Leipzig, 1948) p 216 [14] J.W. Dickey and H. Uberall, J Acoust Soc. Am 63 (1978) 319.
27