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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS -1963
Orru-p-1792
cost- 651118-4
MASTER
NUCLEAR SPIN-PARITY ASSIGNMENTS
DEC 2 1 1965
RIMEASED FOR ANNOUNCEMENT
IN MUCLEAR SCIENCE ABSTRACTS
LEGAL NOTICE
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privatoly owned recta; or

COULOMB EXCITATION
R. L. Robinson
Oak Ridge National Laboratory
Oak Ridge, Tennessee
Coulomb excitation, which is effected by bombarding
nuclei with charged particles, results from the electra-
magnetic field of the projectile acting on the nucleus. Since
this long-range interaction is well understood, the theory has ,
· been highly developed. The excitation probability predicted
by the theory contains a nuclear matrix element which is i-
dentical to that appearing in the transition rate of the gamma ·
ray from the excited state to the ground state. It is this :
fact which has made Coulomb excitation a valuable method for
ascertaining information about spins and parities of states in
stable nuclei.
For most studies the probability for excitation of a
nucleus in a single encounter with a projectile has been :
small. The cross section is then adequately determined by the
first order perturbation theor. The cross sections for
electric multipole excitation are given by
orga = Cipta fl-2 [E - ( A2.) QE]-2 B(En)ex fin (5). (1)
The value of the constant ce depends on the mass A, and.
charge 2, of the projectile and Ag and Z, of the target
nucleus. E and AE are the energies of the projectile and ex-
cited state, respectively. To avoid the complication of
nuclear interactions, E needs to be kept well below the
Coulomb barrier. Values for the function in ) can be found
in several references. Its argument & is defined by
• 2,22€ (1 11
5=hltet!
where V, and ve are the velocities of the projectile before
and after excitation of the target nucleus. The reduced
E
'
-
*
NUCLEAR SPIN-PARITY ASSIGNMENTS
transition probabillity for excitation, B(ER) , is related to
the B(en), of the gamma ray from the excited state, I, to the
ground ståte, Ic, by
B(ER)ex = 2X+11 B(ER) a
(3)
The only two unknowiis in Eq. (1) are B(EN), and AE. The
latter can easily be determined by a study of the ensuing
gamma rays or conversion electrons. Thus, an investigation of
the cross section can be used to determine the multipolarity
and magnitude of B(ER). The multipole order à in turn pro-
vides information about the spins through the relation
..
11.-I/ 515 1.+1.
(4)
-
-
Furthermore, knowledge of the multipolarity uniquely relates
the relative parity of the ground state and excited state.
The cross sections calculated from Eq. (1) for i = 1 to 4
for a hypothetical case are given in Fig. 1. These are for
excitation with alpha particles of a state at 500 keV of ap-
propriate spin and parity in a nucleus with As = 112 and zo =
48. This figure also contains cross sections predicted for M
and M2 excitation. Single-particle values were taken for the
reduced transition probabilities.
Because of the radiative background encountered experi-
mentally, Coulomb excitation of a level is observed only if
the cross section is greater than about 10 ub. Then for the
example given in Fig. 1, Coulomb excitation would only be dis-
cernible for the El, E2, and E3 multipolarities. However, ex-
citation via other raultipolarities could also be detected if
B(MI), B(M2), and B(E4) were sufficiently enhanced over the
single-particle estimates. The enhancement for B(MI) would
need to be about 10%. Also ML excitation would compete with
El excitation except for the spin sequence of - it and
and would generally require a still greater enhance-
ment to cause a noticeable change from pure E2 excitation.
Thus, investigators involved in Coulomb excitation studies
have normally not considered explicitly the Mi mode. The same
argument has been applied to M2 and Et excitation. However,
neglect of these multipolarities is founded on a possibly fal-
lacious assumption that all B(MI)'s, B(M2)'s, and B(E+)'s are
not large compared to the single-particle estimate
sumption could cause the unusual situation, such as an
unheard-of strongly enhanced M transition, to be wrongly
terpreted.
.................
:
:
C
NUCLEAR SPIN-PARITY ASSIGNMENTS
The vast majority of Coulomb excited states have been
populated by the E2 mode. Although the cross section for El
excitation is appreciably larger when single-particle values
are taken for the BCE 's, only one case of an El excitation
has been positively identified. 3 Perhaps this is because El
rates are so strongly hindered. This is certainly true for
the El rates which have been measured. Another reason may be
that few low-lying states, and therefore those which are ac
cessible to Coulomb excitation, have a spin which meets the
necessary conditions of I <l and parity opposite that ol.
the ground-state parity
Since the multipolarity for excitation is important in
both spin and parity assignments, let us review the methods
· which have been employed to determine it. This discussion
will be restricted to the El, E2, and E3 multipolarities which
are the only ones at present that have played a role in
Jomb excitation.
One method is to measure the cross sections for exci-
tation with different projectiles. If the energies of the
projectiles are selected to give the same value for 5, the de-
pendence of the cross section on the type of bombarding parti-
cle is approximately given by;
1
:
21/3
(5)
*
: ---
-
-
------
-
This states that the ratio A, /2, for the two projectiles must
be different for 1 to be ascertained. Therefore, the method
is most sensitive when a cross section pbtained with protons
is compared with that obtained with d, "He, or 160 ic.s. But
according to this approximation the multipolarity could not be
distinguished through a comparison of cross sections for exci-
tation with d, 4He, and 100 ions. There is, however, some
change when one uses the exact expression as given by Eq. (1).
For example, for the state considered in Fig. I the ratio of
cross sections resulting from bombardment with 45-MeV 100 ions
and 10.7-MeV 4He ions ( = 0.232 for both energies), is 15,3,
14.3, 13.4 for El, E2, and E3 excitation, respectively.
Study of the variation of the cross section with pro-
jectile energy is another technique of determining the multi-
polarity for excitation. This variation is shown in Fig. 2
for the same case cited in Fig. 1. The B(E 's are adjusted
so that the cross sections are identical when the energy of
the bombarding alpha particles is 5 MeV. The difference in .
the E2 and E3 curves, which is a little greater than a factor
of two for' E = 11 MeV, would be easily detectable. It would
---
--
--
---
.
NUCLEAR SPIN.PARITY ASSIGNMENTS
be more challenging to decide between El and E2 excitation
since their two curves differ by only 14% when E = 11 MeV.
The difference in these cuves becomes more pronounced
with decreasing excitation energy. This can be seen by com-
paring the crose sections for exciting a state at 250 keV as
shown in Fig. 3 with Fig. 2. The cross sections are again ad-
justed to the same value for E = 5 MeV. Now for ll-MeV alpha
particles, the E2 and E3 curves differ by over a factor of 3
and the E2 and El curves differ by 34%.
. The multipolarity for excitation can be established by
measuring the differential cross section of the inelastically.
scattered particles. An example of the theoretical angular
distributions, as taken from ref. 1, are given in Fig. 4.
They are normalized at 0 = 180°. The difference in differ-
ential cross section was employed by Elbek et al.*35 to es-
tablish spin and parity of 3” for the 660-keV state in 238U
and the 1060-keV state in 252Th. However, the use of this ap-
proach has been extremely limited.
The transition probability for gamma radiation with
multipolarity En is related to the reduced transition proba-
bility by
-
.
..
T, = C, (AE) 21+1 B(ER),
(6)
:
It follows from this that if the lifetime of a level 43 knowo,
a measurement of the absoluto cross section for Coulomo exci-
tation of this level at just one energy of the projectile can
be important in ascertaining the multipolarity for excitation.
By way of illustration, consider the 961-keV state of 6%Cu.
If the state is assumed to be populated by El, E2, and E3 ex-
the half-life of an unmixed transition to the ground
state obtained from the observed Coulomb excitation cross
sections by means of Eqs. (1), (3), and (6) is, re-
spectively, 1.1 X 10-15, 3.0 x 10^12, and 1.2 x 10-8 sec. The
lifetimes would be reduced by admixtures of magnetic radia.
tion. A direct measure of the level's half-life by nuclear
resonance fluorescence has yielded a value of 4 X 10-10 sec.?
This lifetime is about a factor of 400 slower than the life ,
time deduced for El Coulomb excitation and therefore elini. ''
nates the El multipolarity as a candidate for excitation. It
also makes E3 excitation unlikely since it would require a
mixture of M2 radiation that is enhanced by 20% over the
single-particle estimate. Thus, i = 2 is the preferred
assignment; further, comparison between the direct measure- .
ment of the lifetime and that obtained through Coulomb exci-
tation indicates that the gamma ray has an E2/MI admixture of
about 0.2. This is consistent with the admixture of 0.17
...
.
..
NUCLEAR SPIN.PARITY ASSIGNMENTS
which was determined from the angular distribution of the
961-keV gamma ray.°
The fact that the Coulomb excitation process and the
Lifetime of a state are so closely related has made Coulomb
excitation studies an important source of ML and E2 tran- .
sition probabilities; they have also recently begun supple-
menting our knowleage of lifetimes for E3 transitions. The
relevance or this to the present conference was discussed in
an earlier paper by Gove.
Spiris for a large nunber of low-lying states in even-odd
nuclei nave been established through a measurement of the
angular distribution of gamma rays resulting from Coulomb ex-
citation; this tool was also used a few years ago to corrobo-
rate the spin 2 assignment for the first excited state in
several even nuclei. The angular distribution of the gamma
ray from a Coulomb excited state, as shown in Fig. 5a, rel.,-
tive to the direction of the incident beam can be treated in
the same manner as the angular correlation of gamma-ray 2 in
Fig. 55 relative to gamma-ray 1. The two would give the same
distribution except in the former, one needs to include
particle parameters, & 's, in the equation
W(0) = 1 + 2nA Ps (cose) + QLALP2 (cose),
...
-.
-
where P (cose)'s are Legendre polynomials and A's are theo .
retical coefficients which depend on the spins end gamma-ray
admixture. The particle parameters have been evaluated for
El, Ml, and E2 excitation. Although it has never been done,
it is possible that the dependence of these parameters on
multipolarity could be used to establish it.
For angular distributions which are preceded by E2 exci-
tation, ay, is unfortunately usually < 10.1 and Ay, can there.
fore only be poorly determined. The parameter a, is generally
between 0.5-1.0.
The information acquired from the angular distribution of
gamma rays following Coulomb excitation has complemented in .
several ways that obtained from the angular correlations be-
tween gamma rays resulting from the decay of radioactive
nuclei. There is the obvious situation in which the angular :
distribution is of gamma rays from a state that is populated
only by Coulomb excitation. There are other gamma rays from
radioactive nuclei whose distribution cannot be measured be-
cause coincident transitions either do not exist or experi-
mentally are inaccessible. Often in gamma-gamma angular .
correlation studies, both transitions are admixtures of two..
types of radiation with the consequence that many combinations
of spins and admixtures give the observed correlation. In ...
Coulomb excitation this problem is reduced because the
NUCLEAR SPIN.PARITY ASSIGNMENTS
.
.
-
excitation process is unmixed.
There are, of course, many instances where angular dis-
tribution studies cannot uniquely determine spins and radi-
ative admixtures. McGowan and Stelson have been able to re-
move some of these ambiguities by measuring the linear po.lari-
zation of the gamma rays following Coulomb excitation as well
as their angular distribution. Frequently, other arguments
ruch as reasonable transition rates have been employed to help
establish these properties. This is illustrated by the case
Oſ 77se. The angular distributions of the 440- and 201-keV
ganma rays which de-excite the Coulomb excited state at 440
keV in this nucleus are compatible with six different combi-
nations of spins and admixtures. 10 However, five of these
appear unlikely because the 201-keV transition would have a
B(E2) which was between 400 and 8000 times the single-particle:
estimate.
Other spins and parities have been determined through the
study of conversion electrons resulting from Coulomb exci.
tation. Although a great deal of effort has been given to
this type of investigation, nothing will be said here as the
internal conversion process has been thoroughly covered by
other papers at this conference
When the probability of the nucleus being Coulomb excited
in a single encounter with a projectile is large, excitation
by a double prouess or more generally a multiple process be-
comes important. Then the states excited are no longer re-
stricted to those for which I-II <3. In order to obtain an
energy for the projectile which is sufficiently high to effect
multiple coulomb excitation and still be below the Coulomb
barrier, one needs to use heavy ions. The most popular pro-
jectile has been tºo. Multiple Coulomb excitation studies
have so far been concentrated in two areas: 1) investigation
oſ the "two-phonon" states in even, medium-weight nuclei, and
2) investigation of the rotational and vibrational states in
permanently deformed nuclei.
Second order perturbation theory has predicted the cross
sections for Coulomb excitation of the two-phonon states by a
double E2 process via the first 27 state. 2 Although it has
not yet been done, it appears feasible for the spin of the
two-phonon state to be ascertained by comparing its experi-
mental excitation function with theory. The excitation
functions oredicted for the double excitation of a hypo-
thetical state at 1000 keV via a 27 state at 500 keV in a
nucleus with Zo = 48 ara illustrited in Fig. 6. With the
curves normalized at an oxygen crergy of 35 MeV, the functions
for excitation of the ope and the second 27 states differ by
16% at 55 MeV and for excitation of the 47 and 27 states
differ by 23%. However, the second 27 state is also populated
-
-- --
-
St
.
..
NUCLEAR SPIN.PARITY ASSIGNMENTS
-
-
-
-
.
-
-.-
--
'
by direct E2 Coulomb excitation and by interference between
the direct and double E2 processes. The latter is small and
goes to zero when the energy of the second 2 state is just
twice the energy of the first 27 state.
Many of these second 2" states were first observed by
Stelson and McGowanii in direct Coulomb excitation studies
where alpha particles were the projectiles. The spin and
parity were inferred from level and gamma-ray properties.
These are, the energy of the state is about twice the energy
of the first 2* state, ühe B(E2), for the transition to the
ground state is ~ B(E2), and thë B(E2), of the transition to.
the first 2" state is 10-60 B(E2) coi these are properties
found for well established 2 states in other medium-weight
nuclei. A much stronger basis for the 27 assignment has been
provided by investigations carried out by Eccleshall
et al.12 1% and by McGowa: et al.15 They measured the gamma-
ray yields from these same states in Se, Ru, Pa, and ca nuclei
following bombardment with oxygen ions. The yields were found
to be in agreement with the theoretical predictions for direct
and double Coulomb excitation if the levels had IT = 2*, and
the B(E2)'s reported by Stelson and McGowanii were used.
Agreement would not be actained for I" = 1" or 3%, which are
the only other assignments consistent with alpha-particle ex-
citation, except for a fortuitous combination of nuclear
properties.
The other states with energies of about twice that of the
firsi 2+ state populated by double Coulomb excitation have
been treated as if IV = OT or 47. For some states there is
good evidence for such an assignment. But for others the only
evidence is that there are no states in this energy region
with different spins and parities except the 27 states, and it
has been argued that these would have been seen in direct Cou-
lomb excitation studies made with alpha particles. In actu-
ality, direct excitation of a 27 state would have gone unde-
tected if the B(E2) were reduced by as little as 10 below
estimate. Coulomb excitation studies do
favor even parity for these states. In order to have oda
parity, the excitation would be by the E2-El or 2-E3 mode.
This would require the unusually large B(El) and B(E3) for the
cascade gamma ray of ~ B(El), and ~ 108 B(E3) ani
Eccleshall et al.12114 have shown that a spin 4 assign-
ment for a double E2 Coulomb excited state in each of the
nuclei 76se, 785e, 108Pa, and 11 ºpa is consistent with the
angular distribution of the gamma rays in coincidence with
the back scattered oxygen ions. For ions scattered exactly
through 180°, the resulting distribution is independent of
the nuclear reaction mechanism. 16 The predicted distributions
d
..
aindwa kuwaosat
in
inoráz,
........
....
.
..
.
.
.
.
.
.
.
.
NUCLEAR SPIN-PARITY ASSIGNMENTS
---
-
-----
-
agna a ta
ana mt
tentang terhadater marking handa na
for transitions irom spin 0, 2, and 4 states can be found in
ref. 12. Eccleshall et al. found the angular distribution of
the 895-keV gamma ray from the 1515-keV state in Tose, which
was typical of the other distributions, was fitted by taking
A = 0.47 $ 0.12 and Aj = - (0.40 + 0.22). These are in good
ent with the theoretical values for I = 4 of A, = 0.51
and Aj = - 0.37. However, they are also obtained for I = 2 if
the admixture of the transition is 5. The spin cannot
be zero since the predicted distribution is isotropic.
. Multiple Coulomb excitation has made it possible to ob-
serve higher-lying levels of the ground-state rotational band
in permanently deformed nuc).ei than was previnusly feasible.
For exemple, Stephens et al.17 have reported Coulomb exci-
tation of all members of this band in 2080 with spins through
12; Diamond et al.18 observed all members in 165HO with spins
througi 17/2. Theory for multiple excitation of these states
and also vibrational states has been developed by Alder and
Winther.13320 These calculations in contrast to the pertur-
bation treatment depend on a nuclear model. De Boer, Goldring,
and Winkler21 have compared timeir experimental results for ex-
citation of the rotational states in twenty-two nuclei
(150 <A 192) with this theory. This is illustrated in Fig.
7. The clustering of data points in this figure into grours
which agree with the theoretical curves strengthens the 2
4+, 67, and 87 assignments which are suggested by the energies
of the levels.
According to the multiple Coulomb excitation theory the
excitation of a gamma-Vibrational band in even nuclei will be
primarily to its lowest member, whereas for the beta-
vibrational band there is comparable population of the lower
members for typical incident energy of the oxyger ions.
Yoshizawa et al.22 have used this information as supporting
evidence for assignments of I = 2, K = 2, and I = 0, 2, and 4,
K = 0 to states which they observed Coulomb excited in rare
earth nuclei.
Because of the fast decrease in cross section with in-
creasing AE, Coulomb excited states for the most part have
been below 1.5 MeV. However, in several medium-weight nuclei
there is excitation of a level at about 2 Mev. 15723324 The
observed yield of gamma rays between this level and the first
2+ level could be attributed to direct excitation of a lº, 2
or 3- state iſ the ensuing transition to the ground state had
B(El) = 0.3 B(El) so, B(E2) = 2 B(E2) sp, or B(E3) = 20 B(E3) s.
The B(EI) would be appreciably larger than those B(El)'s which
have been reported, but both the B(E2) and B(EB) have accepta.
ble values. Measurements made by Hansen and Nathanes of the
angular distribution of the cascade gamma rays in coincidence
--..
-
.
..
Voi'
LAY
. -
V
.
-
-
-
-
;
-
7
-
i
i
V
NUCLEAR SPIN.PARITY ASSIGNMENTS
with back scattered ions favor I" = 3. Also the enhance-
ments of about 20 found for the B(EZ)'s are compatible with
EZ enhancements deduced Irom the inelastic nucleon scaütering
studies. Thus, the evidence supports a 3º assignment and be-
cause of the enhanced transition probabilities, the levels
have been interpreted as resulting the collective octu-
pole oscillation of the nucleus.
In summary, Coulomb excitation studies have made signifi-
cant contributions to our knowledge of spins and parities for
low-lying levels on stable nuclei by 1) relating the spin and'
parity of the excited state to the spin and perity of the
ground state, and by 2) providing information on the charac.
teristics of the ensuing ganice, radiation and internal con-
version electrons. The weakness of Coulomb excitation
studies has been their reliance on the systematics of radia-
tive lifetimes. At the outset of this paper, it was pointed
out how these systematics had been used to argue that the sig-
nificant modes for Coulomb excitation were El, E2, and E3.
Examples of other arguments were given throughout the paper.
If the future should reveal that there are transitions with
B(MI) > 102B(MI).., B(El) > B(El) .ne etc., there are assign-
ments based on cotlomb excitation Work which should be re-
examined.
The author wishes co express his appreciation to Drs.
F. K. McGowan and P. H. Stelson for their critical reading of
the rianuscript,
i ci si
References
1. K. Alder, A. Bohr, T. Huus, B. Mottelson, and A. Winther,
Revs. Modern Phys. 28, 432 (1956).
2. These are listed in Table I of the reference: P. H.
Stelson and F. K. McGowan, fnn. Rev. Nuclear Sci. 13, 163
(1963).
R. Sherr, c. W. Li, and R. F. Christy, Phys. Rev. 96,
1258 (26954).
4. B. Elbek, G. Igo, F. S. Sternens, and R. M. Diamond, in
Proc. Second Conf. on Reactions between Complex Nuclei,
ed. by A. Zucker, F. T. Howard, and E. C. Halbert (John
Wiley & Sons, Inc., 1960), p. 102.
R. M. Diamond, B. Elbek, G. Igo, and F. S. Stephens, in
Proc. Int. Conf. on Nuclear Structure, ed. by D. A.
Bromley and E. W. Vogt (University of Toronto Press,
1960), p. 563.
R. L. Robinson, F. K. McGowan, and P. H. Stelson, Phys.
Rev. 134, 567 (1964).
Nuclear Data Sheets, Nuclear Data Group, Oak Ridge
National Laboratory.
nic i
unveri
c
ilmente
9
NUCLEAR SPIN-PARITY ASSIGNMENTS
8. F. K. McGowan and P. H. Stelson, Phys. Rev. 106, 522
(1957..
9. F. K. McGowan and P. H. Stelson, Phys. Rev. 109, 901
(1958).
R. L. Robinson, F. K. McGowan, and P. H. Stelson, Phys.
Rev. 125, 1373 (1962).
P. H. Steilson and F. K. McGowan, Phys. Rev. 12.1, 209
(1961); Phys. Rev. 126, 257 (1962).
D. Eccleshall, B. M. Hinds, and M. J. L. Yates, Nucl.
Phys. 32, 190' (1962).
D. Eccleshall, B. M. Hinds, M. J. I. Yates, and N.
MacDonald, Nucl. Phys. 37, 377 (1962).
14. W. Bygrave, D. Eccleshall, and M. J. L. Yates, Nucl.
Phys. 53, 385 (1964).
F. K. McGowan, R. L. Robinson, P. H. Stelson, and J. L.
C. Ford, Jr., Nuci. Phys. 66, 97 (1965), ard private
communication.
A. E. Litherland and A. J. Ferguson, Can. J. Phys. 39,
788 (1961). .
17. F. 3. Stephens, Jr., R. M. Diamond., and I. Perlman, Phys.
Rev. Letters 3, 435 (1959).
R. M. Diamond, B. Elbek, and F. S. Stephens, Nucl. Phys. .
43, 560 (1963).
K. Alder and A. Winther, Kgl. Danske Videnskab Selskab,
Mat.-fys. Meda. 32, No. 8 (1960).
K. Alder, Proc. Third Cení. on Reactions between Complex
Nuclei, ed. by A. Ghiorsi, R. M. Diamond, and H. E.
Conzett (University of California Press, 1963), p. 253.
J. de Boer, G. Goldring, and H. Winkier, Phys. Rev. 134,
B1032 (1964).
Y. Yoshizawa, B. Elbek, B. Herskind, and M. C. Olesen, .
Proc. Third Conf. on Reactions between Complex Nuclei,
ed. by A. Ghiorsi, R. M. Diamond, and H. E. Conzett
(University of California Press, 1963) p. 289.
23. 0. Hansen and 0. Nathan, Nucl. Phys. 42, 197 (1963).
24. D. G. Alkhazov, Yu. P. Gangrskii, I. Kh. Lemberg, and
Yu. I. Udralov, Izv. Akad. Nauk SSSR, Ser. Fiz. 28, 232
(1964).
Figure Captions
Fig. 1. Cross sections predicted by the first order pertur-
bation theory with reduced transition probabilities equal
to single-particle estimates.
Fig. 2. Relative probability for excitation of a 500-keV
state with alpha particles. !
NUCLEAR SPIN-PARITY ASSIGNMENTS
Fig. 3. Relative probability for excitation of a 250-keV
state with alpha particles.
Fig. lt. Relative differer:ial cross sections for the ine-
lastic scattering of 9.-MeV alpha particles to a 500-keV
state in a nucleus with Zo = 48.
Fig. 5. Two schemes which apart from particle parameters
will give the same angular distribution.
Fig. 6. Relative yields of gamma rays from a thick target
following double E2 excitation of a 1000-keV state of spin
I.
Fig. 7. Comparison of the excitation of rotational states in
deformed nuclei with theory (taken from ref. 21). R. is
the total de-excitation of the level with spin I as-
measured in coincidence with oxygen ions scattered through
an average angle of 160°. X depends on the B(E2) for the
0 - 2 excitation and projectile energy. The predictions
of the multiple excitation theory are given by dashed
lines for 5 = 0,0 = 160°, and continuous lines for
0 = 180°.
END

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