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One- and two-step processes in natural- and unnatural-parity 208Pb(p , t)206Pb reaction at Ep = 22 MeV
Nuclear Physics A470 (1987) 377-396 North-Holland, Amsterdam
ONE-
AND
TWO-STEP PROCESSES IN NATURALAND ‘08Ph(j, t)mPb REACTION AT Ep = 22 MeV UNNATURAGPARITY M. KUROKAWA,
Y. AOKI,
Y. TAGISHI
and
K. YAGI
Insrirure of Physics and Tandem Accelerator Center, Uniwrsiry of Tsukuba, Ibaraki 305, Japan
Received
10 April
1987
Abstract: Reaction mechanism and nuclear-structure dependence of the natural- and unnatural-parity zOsPb(p t)206Pb(2’, 3+, 4’, 5- and 7-) reaction are studied by using a 22 MeV polarized proton beam. Angular distribution of the analyzing power for the second 3+(3;) transition has an opposite sign to that for the 3: one and the difference is explained in terms of the j-dependence in the sequential two-step (p, d)(d, t) process. The observed analyzing powers and cross sections for the two 3’ transitions are reproduced by the finite-range first- and second-order DWBA calculation. Two-step analysis is necessary to explain the variety of angular distributions of the analyzing powers for the natural-parity transitions. Microscopic analysis in terms of the one- and two-step (p, t) calculation is made for the transitions to the 2;_,, 4:_,, 5,, and 7; states in ‘“Pb. The calculation reproduces the absolute values of the observed cross sections within a factor of 2.
E
NUCLEAR REACTIONS 208Pb(polarized, t), E = 22.0 MeV, measured cr(E,, @), A(&, 0); deduced reaction mechanism. “%J levels deduced normalization factor. Magnetic spectrograph. First- and second-order DWBA analysis.
1. Introduction 208Pb(p, t)206Pb reaction leading problem of an unnatural-parity state of ‘OaPb has been solved by Tsukuba group by measuring power A,( 0) of this transition ‘*2).The (p, t) reaction to the unnatural-
A long-standing to the first 3+(3:)
the analyzing parity states of even-even nuclei is completely forbidden in the framework of simple zero-range (ZR) distorted-wave Born approximation (DWBA). However, the reaction can proceed via a (p, d)(d, t) two-step process since spin transfers = 1 is realized owing to twice transfer of spin-i neutrons. Actually the differential cross section g( 8) data of the reaction 208Pb(p, t)206Pb(3T, 1.34 MeV) at E, = 35 MeV by Lanford et al. 3, have been well reproduced by the second-order (p, d)(d, t) DWBA calculation by de Takacsy 4, and Charlton ‘). On the other hand, the same cross section data have been analyzed by Nagarajan et al. 6, as a one-step rather than the sequential (p, d)(d, t) process because the process is not forbidden in the finite-range (FR) DWBA if a realistic triton wave function containing a mixture of S, S’ and D state ‘) is used. Nagarajan et al. reported that their calculation can predict the absolute 03759474/87/503.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
M. Kurokawa
378
magnitude.
Thus the cross-section
al. “) are not powerful
enough
er al. /
experiments
to distinguish
ism for the 2”8Pb(p, t)206Pb(3 :) excitation In order to solve this problem,
20RPb(@, r)'"Ph by Lanford
whether
et al. ‘) and Wienke
the dominant
is a one-step
the first measurement
reaction
or two-step
et
mechan-
process.
of a vector analyzing
power
A,(8) for the 208Pb(p, t)2”6Pb(3:) excitation has been carried out by the Tsukuba group ‘) at E, = 22 MeV. The experimental result has been analyzed by Igarashi and Kubo ‘) by making the precise first- and second-order FR-DWBA calculations. The result has clearly shown the predominance of the (p, d)(d, t) two-step mechanism because the analyzing power at forward angles shows positive (negative) sign when the two-step (one-step) process is dominant. Thus the analyzing power experiment excludes the conclusion of Nagarajan et al. “). In addition to the unnatural-parity 3: transition, the Tsukuba group has carried out measurement and analysis of the natural-parity 2”8Pb(p, t)‘“Pb(Og’, g.s. and O:, 1.165 MeV) reactions lo.“). The predominance of the (p, d)(d, t) two-step process is confirmed also in the natural-parity (p, t) transitions and the large difference observed in the analyzing power for the two 0’ transitions is explained in terms of the nuclear-structure dependence of the two-step processes to the two 0’ states. Recently the Tsukuba group has extended the (p, t) A,,( 0) measurement to higher incident energies such as E, = 35 and 50 MeV and confirmed the predominance of the two-step processes over the one-step (p, t) process over a wide proton energy state range “.I’). In the analysis of these data, the effect of a deuteron unbound (S =O, T= 1) in the intermediate channels of the (p, d)(d, t) processes has been taken into account 12-14). The purpose of the present following essential
paper
is to extend
the above-mentioned
two points. In doing so, analyzing powers role in investigating the reaction dynamics
work to the
for (p, t) reactions play an and the nuclear structures
involved because analyzing powers are much more sensitive to interferences between various competing reaction process than the corresponding cross sections I’). (i) We extend analyzing power measurements to the (p, t) transition to the second of 3+(1.34 MeV) and 3’(3Z, 3.12 MeV) state of 206Pb. The wave functions to be almost pure configurations of 3’(3.12 MeV) states of “‘Pb are predicted (P ,,2, f,,,) and (P’,~, f,,2), respectively 16). These wave functions are confirmed from the angular distribution of cross section and spectroscopic factors for 207Pb(p, d)‘06Pb reaction I’). It is expected that the analyzing power for the 3; transition has an opposite sign to that of the 3; transition; see subsect. 3.2. (ii) We measure the cross sections and the analyzing powers of natural-parity “‘Pb(p, t)206Pb(2’, 4’, 5- and 7-) reaction. Then we analyze these transitions in term of the first- and second-order DWBA method so as to study the degree of contributions of the one- and the two-step processes and the dependence of the cross section and the analyzing power on the nuclear shell structure. Recently Wienke et al. Ix) h ave measured only the cross sections for the naturalparity 208Pb(p, t)206Pb reactions but have not measured the analyzing powers. We have briefly reported the “‘Pb(p, t)20”Pb(31) result elsewhere ‘9*20).
M. Kurokawa et al. /
“‘Pb( ~3,t)zwPb
379
2. Experimental procedures and results A polarized
proton
beam
was produced
source 2’). The 22.0 MeV polarized of Tsukuba
12 UD Pelletron.
proton
with a Lamb-shift beam was accelerated
The beam was analyzed
type polarized
ion
with the University
by a 90” analyzer
magnet
and
focused onto a 2o8Pb target by a switching magnet and a pair of quadruple magnets. The beam intensity on target was about 30 nA with a spot size less than a diameter of 2 mm. We used a 0.48 mg/cm2 thick 208Pb target backed by a 3 pm aluminum foil. Enrichment of the “‘Pb target is 98.69% was determined by weighing and checked with proton elastic section is estimated
“‘Pb. The thickness of the 208Pb target by comparing optical model prediction
scattering data. The error in the absolute values of the cross to be about 10%. A polarization monitor target was placed
upstream of the 208Pb target (fig. 1). The monitor target was 60 pg/crn’ thick LiF evaporated onto a carbon foil. Stability of the beam polarization was monitored with two solid state detectors (SSD’s) which measured the analyzing power for the ‘Li(p, a)4He reaction at 6,, = 130”. The amplitude of the effective analyzing power is about 0.7 at 8,, = 130”. Typical beam polarization was 80%. The error in the beam polarization is about 4%. Spin direction of the incident protons was flipped every 50 nC of beam current integration by the fast-spin-state-interchange system (FASSICS) 22). This system controls a spin-flipping pulse of the polarized ion source and supplies gate signals to each spin state for the counting system. Tritons from the (p, t) reactions were analyzed by a magnetic spectrograph (ESP-90) 23) and detected by a single-wire position-sensitive-proportional counter system 24). This counter system has sensitive length of 30 cm and is composed of a Magnetic spectrograph \
Position counter Ep=22 spin
Fig. 1. Schematic view of the detecting system.
380
M. Kurokawa et al. / “‘Pb( ~7,t)‘06Pb
208Pb(p,t)206Pb
Ep=22.0
8 l&,=25’
SPIN
800
MeV
UP
1200
2000
1600
CHANNEL NUMBER Fig. 2. Triton momentum spectrum of *“Pb(p
position
counter
and a AE counter.
of 0.0 to 3.8 MeV in *“Pb
, t)‘06Pb reaction at E, = 22 MeV, elab= 25”, with up spin.
This system covered
and no unknown
an excitation
peaks were observed
energy range
in a momentum
spectrum as shown in fig. 2. Coincidence requirement between position and AE signals lowered the background level. The FWHM of momentum spectrum was about 30 keV which was almost determined by energy loss in the target. As the excitation energies of all levels are well known from the literature 25), no effort is made to determine them in the present experiment. We measured angular distributions of cross sections a(0) and analyzing powers A,,( 0) from 5” to 90” in 5” steps. The horizontal acceptance of the magnetic spectrograph was +1.5”, which corresponded to a solid angle of 2.0 msr. The observed cross sections and analyzing powers are shown in figs. 3 to 6.
3. Opposite sign character of analyzing powers for unnatural-parity transitions to two 3+ states 3.1. ANALYZING-POWER
AND CROSS-SECTION
ANGULAR
(p, t)
DISTRIBUTIONS
Fig. 3 shows measured and calculated angular distributions of the differential cross sections and the analyzing powers for the unnatural-parity transitions
M. Kurokawa
_._._...-...._._______..
20
40
t)*“Pb
381
‘\ ‘\..,_ -.___..__‘_-- r - a
One-step
0
et al. / ‘08Pb($,
***-..___I\-
60
80
0
20
40
60
80
e,,(dw.) Fig. 3. Experimental and calculated cross sections a(0) and analyzing powers A,(0) for the *‘*Pb(p t)2”Pb(3:, 1.341 MeV and 3:, 3.122 MeV) reaction. The curves 2BS show the contribution from the bo&d deuteron channel, in which only the L = 0 component is taken into account in both (p, d) and (d, t) transfer steps, while the curves 2BT show the one in which the L = 2 component is taken into account. The curves 2US0 show the one from the ‘S, unbound deuteron channel.
“‘Pb(p, t)206Pb(3:, 1.34 MeV and 3:, 3.122 MeV). Note that the sign of the measured analyzing powers is opposite for the two 3+ transitions. The data for the 3: transition were obtained by Toba et al. ‘). Our new data for the 3: transition are consistent with them. The errors are almost determined by counting statistics, since the triton counts for the 3: transition were about 100 per 1 mC of beam current integration.
3.2. QUALITATIVE
ANALYSIS
BY USING
9-j COEFFICIENT
The opposite sign character is simply and qualitatively explained parison of 9-j coefficient for these two 3+ transitions. The scattering an analyzing power Ay are written as
A =Rek*h) ’
kt2+bi2
’
from a commatrix T and
(1)
382
M. Kurokawa
et al. /
20sPb(&r)'06Pb
1
0.803
MeV
0.803 MeV
;* I 0
*
1.304
! 40
I
MeV ' 80
1
120
@,,(deg.)
0
40
80
12'0
Q,,(deg.) Fig.
4. Experimental
and calculated cross sections a(O) and analyzing powers “‘Pb(d, t)*06Pb(2;, 2; and 3:) reaction at Ed = 17 MeV.
where g(h) denotes the spin-independent amplitude. We assume that the nuclear
A,(O)
for
the
(spin-dependent) part of the scattering structures of the 3: and 3: states to be
respectively; see sect. 1 and table 1. In this (P l/2, f5,2) and (pr12, f7,2) configuration, case the relative sign of the g and h for the (p, d)(d, t) two-step process is determined by the following 9-j symbol;
9-j=
( 1 11
12
1
s,
s2
s
iI
j2
,
.i
where 1, s and j (=3) show the transferred orbital, spin and total angular momenta, respectively. The 11(12) and j,(j,) are the orbital and total angular momenta of the shell-model states of the transferred neutrons. Of course we have S, = s2 = 4. The s = 0 part of the 9-j corresponds to the spin-independent amplitude g while the spin-dependent amplitude h is only due to the s = 1 part of the 9-j. The values of
M. Kurokaw~ et al.
-
/ 2*8Pb(@, t)*“l%
p1/2'p3/2
,
0
40
80
Two-step
120
0
40
80
120
B,,(dw.) Fig. 5. Calculated cross sections a(e) and analyzing powers A,,(B) of sa*~(p, t)*“Pb(2’) reaction leading to four con~garations. Experimental data show the a(@) and A,,(e) of 208Pb(p, tfZo6Pb(2~) transition. This state is predicted as almost pure configuration of (f,,*, p3,*).
the 9-j of our interest are given as follows; (P i/2, f5,X (P*,2, f7/2K 1=3 s=o 1=3 0.048 -0.036 1=2 1=2 -0.008 0.048 s=l 1=3 0.016 s=l 1=3 0.030 0.048 0.006. i 1=4 i 1=4 We can see that the signs of the s = 0 9-j coefficients are different for the two configurations while the signs for the main part of the s = 19-j coefficient are the same for the two configurations. Then from eq. (l), signs of the analyzing powers are opposite for the two 3+ transitions. Therefore the opposite sign character comes s=o
384
hf. Kurokawa
et al. / “*Pb( @, t)20dpb
0.803
MeV
, >_,--,
‘\ ‘\ 1
2.424
1 0
I 40
I 80
ecm(deg.
MeV
I
7 120
1
Fig. 6. Calculated and experimental cross sections and analyzing powers for the ‘OBPb(p, t)206Pb(2:_5, 4:-,, 5& 7;) reaction at E, = 22 MeV. The dot-dashed, broken and solid lines show one-step, two-step and combined zero-range DWBA calculations, respectively.
M. Karrokawa et al. / ***Pb(~, t)206Fb 0.803
0.2
MeV
,
,
MeV
,
0
-0.2 0.2
0
cd,
7.704
0.2
;r;
0
=T -0.2
r
1
, ,
2.149
0.2
0
-0.2
0
-0.2
Fig. 6-continued
MeV
385
386
M. Kurokawa
et al. / 20*Pb( $, t)‘“Pb 1 .684
MeV
.'.
(b’) I
43+ /,+. .+---..,r\\ ‘I\
o_ &'
I./’
h A.
3.014
I-1
+%y)J-
ZiT -
I
,.‘,2.928 MeV
,!,’ \:\
0.1-
I’
--
+v’ \(t
2.703
MeV
3.014
MeV
MeV
_--. ___* ____---71 -_*.______
10-2,
O 2.220 0
40
MeV 80
e,_,,(deg.
120
1
-0 . 2
2.220 0
40
80 Q,,(deg.)
Fig. bcontinued
MeV 120
M. Kurokawa
er al. / z”*l%( i, f)2wF6
387
TABLE 1 Eigenvalues
and two-neutron-hole
amplitude
of unnatural-parity
3’ states 16) and natural-parity
states ‘s)
in 2061%
J”
P!/29fv2
E,( MW
9
0.998
3+ 1st
1.433
“) 2nd
2.303
-0.036
3.015
-0.010
Pi/Z, f,,, 2’
b/2 P3/2
0.995 1.279
3rd
1.839
4th
2.270
-0.112
5th ‘)
2.542
-0.071
1.859
-0.111
2.121
-0.158
3rd
3.024
5-
-0.275 -0.153
f&2
0.832 0.549
Pi/Z. ‘13/2 2.457
0.124 -0.015
fs,2r P3/2
-0.924
-0.339
0.369
P~/~. ii3l2
3.286
0.926
0.241
f 5,2r i 13/2 3.047
f&2
0.817
-0.065
1st
-0.990
0.429
0.946
2nd
7- 1st
0.018 0.07 1
-0.009
P1/2. fV2
2nd
-0.013 -0.025
0.299
0.192
4+ 1st
0.035
P1/2. P3/2
-0.515
-0.908 -0.220
0.419
f x,2. i I),?.
-0.124
fs,2, P3/2 -0.114 0.032 0.000 -0.982 0.135
f5/2
9f7,2
0.075 0.040 -0.116
i is/21 f 7,2 -0.085
-0.701
0.066
p3/2. k/2
0.137
-0.980
fs,*. f-7/2
0.995 -0.018
0.810
1st 2nd
Pi/Z. f-7/2
0.083
“) Not observed. ‘)
From ref. 16).
from, through two-step
the 9-j coefficient,
(p, d)(d, t) process;
hole while 3:(~,,~,
3.3. FIRST-
AND
f,,J
the j-dependence
the 3:(p 1,2, fs,J
of the analyzing
state consists
power for the
of a j< = f,,, neutron
of a j, = f,,*.
SECOND-ORDER
DWBA
CALCULATIONS
Next we carry out a quantitative analysis of the two 3’ transitions. The finite-range DWBA calculation was made by Igarashi and Kubo 14) by using a computer program TWOFNRz6). Deuteron ground state (including D state) and unbound deuteron states (‘So and 3S, +3D,) are taken into account for the intermediate channels in the (p, d)(d, t) two-step process. The deuteron unbound state channels are evaluated by the discretized-continuum-channels method 27). The triton wave function, the
M. Kurokawa er al. / “*Pb( p, I)~“P~
388
deuteron ground state and unbound the Reid soft-core potential 2”). The distorting deuteron
potential
parameters
22 MeV by Satchler
state wave functions
parameters
were obtained
were calculated
by using
are given in table 2 (set A). The proton from ‘O’Pb(p, d)“‘Pb
29). The triton parameters
reaction
analysis
and
at E, =
were taken from optimal-model
fit to
elastic scattering data at E, = 17 MeV by Hardekopf et al. 3’). In fig. 3 the curve 2BS shows the contribution from the bound deuteron channel in which only the L = 0 component is taken into account in both (p, d) and (d, t) transition. The curve 2BT shows the one from the L = 2 component with the deuteron in its ground state. 2US0 and 2USl show the ones from the ‘So unbound deuteron and the ‘S, + ‘D, unbound deuteron channels, respectively. The calculation reproduces the opposite sign character of the analyzing powers for the two 3’ transitions well. Note that the pattern of the analyzing power is strongly dependent on the nuclear shell structure involved and the agreement between the experimental and calculated analyzing powers supports that the present nuclear structure wave functions are reasonable. The contribution of 2BS is the main part for the 3: transition. For the 3,’ transition the contribution of 2BS is the main part of the reaction at forward angles but the absolute value of cross section for the 2BT is as large as 2BS at backward angles. The absolute value of the cross sections of one-step and 2US0 are almost one to two orders of magnitude less than the total cross sections. The contributions of TABLE 2 Distorting Set
Particle
V
A, B A B C, E D A
P d d d d t
B, C E
t t
51.8 112.0 110.4 102.0 96.8 168.9 168.9 160.9
“) We define a distorting
potential
parameters
“), well depths
in MeV and lengths
in fm
W,
W,
r,
r,
r,,
0,
a,,
V, (,
r, 0
a, u
Ref.
0.0 0.0 0.0 0.0 -0.3 9.9 9.9 17.3
10.0 19.4 10.0 13.5 12.9 0.0 0.0 0.0
1.25 1.25 1.30 1.25 1.30 1.30 1.30 1.30
1.25 1.25 1.17 1.25 1.17 1.20 1.20 1.20
1.25 1.25 1.29 1.25 1.33 1.60 1.40 1.40
0.65 0.68 0.79 0.68 0.73 0.65 0.65 0.72
0.76 0.78 0.62 0.78 0.94 0.97 0.84 0.84
6.0 6.0 6.0 6.0 6.9 3.0 3.0 2.5
1.12 1.12 1.12 1.12 1.07 1.15 1.15 1.20
0.47 0.47 0.47 0.47 0.66 0.92 0.92 0.72
29) 29)
potential
as
U(r)=-[Vf,(r)+i(WJ,,(r)-4W,dJ,(r)/dr)] +2( h/m,&)? +
V, 0 /r)(dj,.,
(3 - r2/ Rf)zZeZ/2R,, I zZe2 / r ,
j;(r)=l/[l+exp[(r-R,)/a,]], Set A: used for unnatural-parity transition. Set B: used for natural-parity transition. Set C, D, E: see sect. 5.2.
(r)/dr)(l. rb R, r> R, R,=r,A”‘.
s)
30) 29) 34)
31) 21) “)
M. Kurokawa et al. / 20RPb($, t)‘06P6
2USl
and
the nonorthogonality
term
in the two-step
389
process
can be neglected
because both cross sections are about 10m4 pb/sr. The calculation angular distributions of the cross sections for the two 3’ transitions; distribution
has a peak at 20 degrees
and falls off gradually
reproduces the the 3: angular
while the 3: one is
almost constant. The absolute values of the calculated cross sections are lower than the experimental ones by a factor of 2. This discrepancy can be removed by modifying the distorting potential parameters for the tritons; subsect. 4.2. More details are given in refs. 13*14).
4. One- and two-step analysis of natural-parity 4.1. NUCLEAR-STRUCTURE
WAVE FUNCTIONS
(p, t) transitions
AND SPECTROSCOPIC
AMPLITUDES
We employ one-broken-pair-model wave functions of Hengeveld and Allaart ‘*) for 2’, 4’, 5- and 7- states in 206Pb (table 1). We refer to the paper by Wienke et al. lx) for these wave functions. We use the wave function of True for 2: state because the Hengeveld wave function is not available for this level. The phases of the main part of the two wave functions are consistent apart from the time reversal phase of i’. The spectroscopic amplitude of one-step (p, t) process for the pure configuration
is given by
s,, = Jpi while that of (p, d)(d, t) process
(2)
)
used for the pure configuration
s,,, = -_(
+ @Ti -) jl+j2-jajej2
(j,,jz)j is given by
ijji, ,
(3)
where j’ show the total angular momentum of the intermediate channel. In actual calculations we construct the transition amplitudes by using the pure-configuration spectroscopic amplitudes (2) and (3) and the configuration mixing coefficients given by table
1.
Investigation of the (p, t) transitions to the 0: and 0,’ states is not done present paper because the result has been already published in ref. ‘I). 4.2. DISTORTING
The distorting
in the
POTENTIALS
potential
parameters
used are given in table 2 (set B). We use the
proton optical potential parameters of Satchler 29). The deuteron parameters were obtained from ‘ORPb(p, d)207Pb reaction analysis at E, = 22 MeV by Dickey ef al. 3”). We use triton parameters of Hardekopf et al. 3’), but the imaginary-potential parameters ri( = 1.6 fm) and ai( =0.97 fm) are reduced to 1.4 fm and 0.84 fm, respectively. The absolute value of the cross section for the case of (ri = 1.4 fm, ai = 0.84 fm) is 1.85 times larger than that for the case of (ri = 1.6 fm, ai = 0.97 fm), but angular distributions of the cross sections and the analyzing powers are almost the same for the two ri’s.
M. Kurokawa et al. / zwPb( ~5,I)~~I%
3w
The absolute value of the (d, t) process also becomes about two times larger by this change. This fact can explain the discrepancy observed in the absolute values of the cross section
4.3. 0;
for the (p, t) (3: and 3;) transitions.
PARAMETERS
In the zero-range and the calculated
approximation the relation between one-step cross section is given by
the measured
cross section
. The trial value of D:(p, t) is determined by comparing the experimental cross section for 2: transition with that of the one-step calculation. The value of II;= 44x lo4 MeV’ . fm3 is used as a starting value of the calculation. The value of Dz(p, d) = 15 300 MeV’ * fm3 is used in the present calculation for (p, d) process. The calculation reproduces the experimental cross sections and analyzing powers for *08Pb(p, d)207Pb(p,,2, ~312, fS,2, f7,* and i13,*) reaction at E, = 22 MeV [ref. 32)]. But the analyzing power for the i,3,2 transition is not reproduced. For (d, t) process a value of Di(d, t) = 33 400 MeV’ * fm3 is used. The pattern of calculated cross sections reproduces experimental ones of 207Pb(d, t)206Pb(2:, 2: and 3:) reaction at Ed = 22 MeV [ref. 33)], but the absolute values of the calculated ones are about two times larger than that of the experimental ones (fig. 4). The calculated analyzing powers of the reaction roughly reproduce experimental ones. We use spectroscopic factors of Lanford “) which were determined from 207Pb(p, d)206Pb reaction. Finally the value of the D:(p, t) is determined by comparing the experimental cross section with that of combined one- and two-step calculations. Then the value of D:(p, t) = 61.6 x lo4 MeV* * fm3 is used throughout the calculation. The present calculation for the 3: and 3; transitions reproduce absolute value of the experimental cross sections. This fact is sufficiently enough to indicate that our calculation procedure for the reaction process is right and reasonable.
4.4. RESULTS
OF ANALYSIS
Numerical DWBA calculations of one-and two-step processes are carried out by using a computer program TWOFNR 2”). The zero-range approximation is used in the calculation for one- and two-step (p, d)(d, t) processes. First of all, calculations should explain the fact that angular distributions of the experimental analyzing powers are strongly dependent on each level of the same j cross sections are almost states of 206Pb, while those patterns of the experimental the same for the same j states. The patterns of calculated one-step analyzing powers and cross sections are almost independent of the nuclear shell structure for the 2+
M. Kurokawa
states as shown momentum two-step
in fig. 5 because
have unique analyses
and 1 transfers
ef al. /
transferred
value in one-step
predict
are allowed.
‘08Pb(t, ~)‘~pb
different
patterns
Therefore,
intrinsic process.
spin and transferred On the other hand,
as shown
contrary
391
in fig. 5 because
to the one-step
calculation,
angular (p, d)(d, t) both s = 0 two-step
calculations predict different patterns of a( 6) and A,( 0) depending on the configurations involved. Thus the nuclear structure dependence of the two-step analyzing powers can explain the variety of experimental analyzing powers. As shown in fig. 5, the calculated one-step cross section to a state having the configuration of (p ,,Z, pJIZ) is almost eight times larger than that of (fSJ2, and the calculated two-step cross section of (p ,,2, P~,~) is more than 20 times larger than that of (f5,2)2. Nuclear wave functions of natural-parity states in 206Pb consist of two to four substantial configurations as shown in table 1. The difference of the calculated absolute values of the cross sections due to the interference between each configuration term explains the observed difference of the experimental absolute values of the cross sections. Results of one-step (dot-dashed line), two-step (broken line) and combined (solid line) calculations are shown in fig. 6. General behavior of the cross sections and the analyzing powers for the transition to the 2: state at 0.803 MeV and the 2: state at 1.46 MeV is explained by the one-step analysis. But the calculation for the 2: state fails to reproduce the analyzing power around 20”. The 2: state at 1.784 MeV has a main configuration of (f&. The 2; state at 2.149 MeV is predicted to be almost pure configuration of (f 5,2, P~,~). Agreement between the calculated and experimental analyzing powers is improved around 20” by introducing two-step contributions for the 2; and 2: states. The absolute value of the two-step cross section for the 2: state is larger than that of one-step cross section at forward angles. The experimental absolute values of the cross sections reproduced by the calculation within a factor of 1.5.
for the five 2’ states
are
Calculation reproduces the absolute value of cross section for the 4: state at 1.684 MeV, but fails to reproduce the ones for the 4: state at 1.993 MeV and the 4f state at 2.928 MeV. The calculation fails to reproduce the cross-section angular distribution for the three 4’ states around 70”. No remarkable improvement can be seen by including
two-step
For two 5- states
contribution
the calculation
for the three 4+ states. reproduces
absolute
values
of cross sections
well, but oscillation pattern of the calculated cross sections is different from the experimental data around 20”. Agreement between the calculated and experimental analyzing powers is improved by introducing two-step contributions for the 5; state at 3.014 MeV. Two-step contributions to the analyzing powers for the two 5- states are larger than that for the 2+ and 4+ transition. Calculation reproduces a pattern of the cross section for the 7; state at 2.20 MeV but the absolute value is larger than the experimental one by a factor of 1.7. The angular distribution of the analyzing power is sensitive to the parameters of the distorting potential for the 7- state; this problem is discussed in subsect. 5.2.
392
M. Kurokawa et al. / “‘Pb( ~7,t)‘06Pb
The experimental are reproduced determined
absolute
values of the cross sections
by the calculation
by the one-step
within
contribution
a factor
except
for 2+, 4+, 5 and 7- states
of 2.0 and
they
are almost
for 2: state.
5. Discussion 5.1. NUCLEAR-STRUCTURE
WAVE FUNCTIONS
Absolute values of the cross sections for the “*Pb(p, t)206Pb(2f, 4+, 5- and 7-) reaction are strongly dependent on the nuclear wave functions; see subsect. 4.4. If analysis fails to reproduce experimental absolute value of the cross section, the wave function used in the calculation may be doubtful. The calculated one- and two-step absolute values of the cross sections are almost two times larger than the experimental ones for the 4:, 4: and 7, transitions. Also the analysis fails to reproduce the analyzing powers for these states. The calculated analyzing powers are dependent on the nuclear wave functions and the ratio of the one- to two-step calculated values of the cross sections. Agreement between the calculated and experimental analyzing powers becomes worse by introducing two-step contribution for the 4: state. From the above-mentioned reasons for the analyzing power and cross section, the wave function for 4: state may be wrong. If absolute value of the two-step cross section increases to 1.5 times for the 2: state, the agreement between the calculated and experimental analyzing power is improved. The present analysis by using the wave function of Hengeveld and Allaart ‘*) reproduces the absolute values of the cross sections better than the analysis by using the wave function of True for the most cases. The wave function of True 16) predicts the one-sixth of the experimental Agreement between the calculated
intensity of the cross section for the 4: state. and experimental analyzing power becomes worse
for the 2: state by introducing the two-step contribution. For the 2: state the wave function of True is used in the calculation. It is likely that the difference between the nuclear-structure model of Hengeveld et al. and that of True causes the abovementioned results.
5.2. DISTORTING
POTENTIALS
Calculated (p, d)(d, t) two-step analyzing powers are dependent on the deuteron distorting potential parameters used. In order to investigate the effect of the difference of several deuteron potentials, we compare experimental cross sections and analyzing Powers for *‘*Pb(p, d)207Pb(p,,, , fs,z, p3/2, &3/z and fsi2) reaction at Ep = 22 MeV with those of the zero-range DWBA calculation. We employ the deuteron potential parameters of Dickey et al. 30) (set B), Satchler *‘) (set A) and Daehnick et al. 34) (set D); see table 2. An analysis by using these three potentials almost reproduces the cross sections and the analyzing powers for the P,,~, pX12, and f7,* transitions
M. Kurokawa
but it fails to reproduce is large at forward A. The calculated
the analyzing
angles
et al. / zOsl%( @, I)‘~P~
power for the i13,* transition.
in the case of the calculation
analyzing
393
The discrepancy
by use of the potential
power with set D fails to reproduce
set
the experimental
one for the f5,2 transition. Therefore we have employed the deuteron potential of set B. Since 5- and 7- states in 2MPb contain an i ,3,2 hole in their main configuration, it is likely that the disagreement between the measurement and the analysis of the (p, t) analyzing powers for these states is due to the deuteron potential for the 208Pb(p, t)206Pb(5 and 7-) reaction. But note that the (p, d)(d, t) process in the analysis reproduces the experimental analyzing power for the 208Pb(p, t)20”Pb(5;) reaction. Calculated (p, d)(d, t) two-step analyzing power for the 7; state in ‘“Pb is positive at forward angles by use of the deuteron potential of Daehnick et al. (table 2, set D) and it becomes negative when we use the deuteron potentials of Satchler (table 2, set E) and Dickey er al. (table 2, set C). One- and two-step calculation for the 7- transition reproduces the analyzing power by using the deuteron potential of Dickey et al. better in comparison with the use of Satchler and Daehnick potentials. In consequence the distorting potential parameters of Dickey et al. are suitable for the present calculation. Calculated (p, t) one-step analyzing powers are dependent on the triton distorting potential parameters. We use the triton potential parameters of Hardekopf et al.* (ri = 1.4 fm and ai = 0.84 fm); see subsect. 4.2 and of Becchetti and Greenlees a modification of the imaginary(BG) “); see table 2. We add a star * indicating potential parameters ri and ai from the original values of Hardekopf et al. The one-step calculation using the potential of BG roughly reproduces the analyzing power for the *‘*Pb(p, t)““Pb(7;) reaction but it fails to reproduce the analyzing powers for the 2: and 2: states. The calculation using the potential of Hardekopf et al.* almost reproduces the analyzing powers for the 2:, 2: states. The main difference between these potentials is the potential depth of the imaginary part. The potential of BG.
of Hardekopf
Normalization
et al.* reproduces
factor
(disagreement
the experimental
TABLE
3
factor)
N for
results better than that
“‘F%(p, I)?‘~(J”)
transitions
J,”
2+I
2:
2;
2:
2;
3:
3;
E,(MeV)
0.803 1.0
1.467 0.8
1.784 0.5
2.149 1.2
2.424 1.2
1.304 1.0
3.122 1.3
N
J,”
4;
4;
4:
5,
5;
7;
E,(MeV)
1.684 1.0
1.998 0.6
2.928 0.5
2.783 1.0
3.014 1.0
2.200 0.5
N
394 5.3. FINITE-RANGE
M. ~uro~wa
et al. / 208Pb(A t)206Fb
EFFECT
To evaluate the finite-range effect in the (p, d)(d, t) two-step process in the natural-parity transitions, a numerical calculation is made in the case of the reaction “‘Pb(p, t)206Pb(2+). Procedure of the calculation is the same as in subsect. 3.3 but only the S state of deuteron and triton is taken into account. We calculated only for the configuration (P,,~, f&. Angular distributions of the cross sections in the two-step analysis are almost the same for the zero- and finite-range calculations and the magnitudes obtained from the analysis agree with each other within a factor of 1.04. The analyzing power of the finite-range calculation is smaler by 24% than the zero-range calculation at forward angles (OO-1S’) and the difference between the two calculations is limited at forward angles (Y-40”). It would be worthwhile to perform a finite-range calculation for the 2:, 5; and 4c3 transitions because the two-step process yields large reaction amplitudes for these states. 5.4. NORMALIZATION
FACTORS
We calculate normalization factors N which are obtained by taking the ratio of the measured cross section to the results of the combined direct- and sequentialtransfer calculations: (5),,,=~($X:“* The obtained values of N for the *‘*Pb(p, t)‘06Pb( J”) transitions are given in table 3. These values indicate the degree of disagreement between the experimental cross sections and the calculated ones presented by the solid lines in fig. 6. 6. Conclusion
We measured differential cross sections and analyzing powers for 208Pb(p, t)206Pb(0;___,,2;_,, 3;,2, 4:--3, 5L2 and 7;) reaction at E,=22 MeV. U~~~turu~-~~~~t~ fra~sition~: (i) The measured analyzing power for 3: transition has an opposite sign to that of 3: transition. (ii) This character is explained in terms of j-dependence of the analyzing power in the sequential two-step (p, d)(d, t) process. The measured analyzing powers and cross sections for the two 3’ states are reproduced with the finite-range DWBA calculation. Natural-parity transitions: (iii) Angular-distribution patterns of the measured cross sections for naturalparity transition are almost the same for the common transferred J transition but patterns of the analyzing powers of those are different. Two-step analysis is necessary
M. Kurokawa et al. / *‘*Pb(@,t)Z06Pb
395
to study the above-mentioned character because the one-step calculation predicts almost similar angular distributions of the analyzing powers. (iv) The one- and two-step calculation by using the nuclear wave functions of Hengeveld and Allaart reproduces the absolute values of the cross sections within a factor of 2. The absolute values of two-step cross section are 0.1 to 1.0 times smaller than those of a direct one-step depending on the nuclear structure. The present analysis by using nuclear wave functions of Hengeveld reproduces our measurement better than the analysis by using the wave function of True for the most cases. (v) The present calculation almost reproduces the analyzing powers and the cross sections for the 2:, 2:, 4: and 5: transitions. For 2: and 5: transitions agreement between calculated and experimental analyzing powers is improved by including the two-step contribution. But improvement is not clear in other transitions. (vi) For the 5- and 7- transitions two-step contributions of the analyzing powers are large over the measured angular range. On the other hand the effect of the two-step processes for 2+ transitions is limited around 20”. (vii) More work should be done for the deuteron and triton distorting potentials in the one- and two-step processes. We would like to thank Professors K.-I. Kubo and M. Igarashi for their analysis of the unnatural-parity (p, t) transitions. References 1) Y. Toba, Y. Aoki, S. Kunori, K. Nagano and K. Yagi, Phys. Rev. C20 (1979) 1204 2) K. Yagi, in hoc. of the Tsukuba, Int. Workshop on Deutron involving reactions and polarization phenomena, Tsukuba, 1985, ed. Y. Aoki and K. Yagi (Word Scientific, Singapore, 1986) p. 3 3) W.A. Lanford and J.B. Mcgrory, Phys. Lett. 45B (1973) 238 4) N.B. de Takacsy, Phys. Rev. Lett. 31 (1972) 1007; Nucl. Phys. A231 (1974) 243 5) L.A. Charlton, Phys. Rev. Cl4 (1976) 506 6) N.A. Nagarajan, M.R. Strayer and M.F. Werby, Phys. Lett. 68B (1977) 421; M.F. Werby, M.R. Strayer and M.A. Nagarajan, Phys. Rev. C21 (1980) 2235 7) M.R. Strayer and P.U. Sauer, Nucl. Phys. A231 (1974) 1 8) H. Wienke, H.P. Blok, J.F.A. van Hienen, J. Blok, Y. Iwasaki, A.G. Drentje and W.A. Sterrenburg, Phys. Lett. 135B (1984) 13 9) M. Igarashi and K.-I. Kubo, Phys. Rev. C25 (1982) 2144 10) K. Yagi, in Proc. of 5th Int. Symp. on Polarization phenomena in nuclear physics, Santa Fe, 1980, AIP Conference Proceedings 69 (1981) p. 254 11) Y. Toba, Y. Aoki, H. Iida, S. Kunori, K. Nagano and K. Yagi, Phys. Lett. 1OOB (1981) 232 12) K. Yagi, Y. Aoki, K. Hashimoto, H. Iida, H. Sakamoto, Y. Tagishi, M. Matoba, H. Igarashi and K.-I. Kubo, Phys. Rev. C31 (1985) 676 13) K.-I. Kubo, M. Igarashi and K. Yagi, ibid. ref.2), p. 273 14) M. Igarashi and K.-I. Kubo, in Proc. of the 6th Int. Symp. on Polarization phenomena in nuclear physics, Osaka, 1985, J. Phys. Sot. Jpn. 55 (1986) Suppl. p. 134 15) K. Yagi, H. Iida, Y. Aoki, K. Hashimoto and Y. Tagishi, Phys. Rev. C31 (1985) 120, and references therein especially refs. 1-21 16) W.W. True, Phys. Rev. 168 (1968) 1388
396 17) W.A. Lanford
M. Kurokawa
et al. 1 ““Pb(@.
f)‘06Pb
and G.M. Crawley, Phys. Rev. CXJ (1974) 646 18) H. Wienke, H.P. Blok, J.F.A. Van Hienen and J. Blok, Nucl. Phys. A442 (1985) 397 19) M. Kurokawa, K. Aoki, Y. Aoki, K. Hashimoto, A. Manabe, Y. Sakai, M. Takei, Y. Tagishi, M. Tomizawa and K. Yagi, ibid., ref. 2), p. 313 20) M. Kurokawa, K. Aoki, Y. Aoki, K. Hashimoto, A. Manabe, T. Sakai, M. Takei, Y. Tagishi, M. Tomizawa and K. Yagi, ibid., ref. 14), p. 636 21) Y. Tagishi and J. Sanada, Nucl Instr. Meth. 164 (1979) 411 22) T. Murayama, H. Sakamoto, M. Oyaizu, Y. Tagishi and S. Seki. UTTAC University of Tsukuba Annual Report (1984) 4 23) J.E. Spencer and H.A. Enge, Nucl. lnstr. Meth. 49 (1967) 181 24) H. Iida, Y. Aoki, K. Yagi and M. Matoba, Nucl. lnstr. Meth. 224 (1984) 432 25) C. Lederer er al., Table of Isotopes (Wiley, 1967) p. 1325 26) M. Toyama and M. Igarashi, computer code TWOFNR, private communication 27) M. Yahiro, M. Nakano, Y. lseri and M. Kamimura. Prog. Theor. Phys. 67 (1982) 1467 28) R.V. Reid, Jr., Ann. of Phys. SO (1968) 411 29) G.R. Satchler, Phys. Rev. C4 (1971) 1485 30) S.A. Dickey, J.J. Kraushaar and M.A. Rumore, Nucl. Phys. A391 (1982) 413 31) R.A. Hardekopf, R.F. Haglund, Jr., G.G. Ohlsen, W.J. Thompson and L.R. Veeser, Phys. Rev. C21 ( 1980) 906 32) Y. Toba, K. Nagano, Y. Aoki, S. Kunori and K. Yagi, Nucl. Phys. A359 (1981) 76 33) M. Kurokawa, unpublished; H. Iida, M. Kurokawa, Y. Aoki, K. Hashimoto, K. Nagano, Y. Tagishi, M. Takei and K. Yagi, UTTAC University of Tsukuba Annual Report (1983) 29 34) W.W. Daehnick, J.D. Childs and Z. Vrcelj, Phys. Rev. C21 (1980) 2253 35) F.D. Becchetti, Jr. and G.W. Greenlees, in Polarization phenomena in nuclear reactions, ed. H.H. Barschall and W. Haeberli (Univ. of Wisconsin Press, Madison, 1971) p. 682
ONE-
AND
TWO-STEP PROCESSES IN NATURALAND ‘08Ph(j, t)mPb REACTION AT Ep = 22 MeV UNNATURAGPARITY M. KUROKAWA,
Y. AOKI,
Y. TAGISHI
and
K. YAGI
Insrirure of Physics and Tandem Accelerator Center, Uniwrsiry of Tsukuba, Ibaraki 305, Japan
Received
10 April
1987
Abstract: Reaction mechanism and nuclear-structure dependence of the natural- and unnatural-parity zOsPb(p t)206Pb(2’, 3+, 4’, 5- and 7-) reaction are studied by using a 22 MeV polarized proton beam. Angular distribution of the analyzing power for the second 3+(3;) transition has an opposite sign to that for the 3: one and the difference is explained in terms of the j-dependence in the sequential two-step (p, d)(d, t) process. The observed analyzing powers and cross sections for the two 3’ transitions are reproduced by the finite-range first- and second-order DWBA calculation. Two-step analysis is necessary to explain the variety of angular distributions of the analyzing powers for the natural-parity transitions. Microscopic analysis in terms of the one- and two-step (p, t) calculation is made for the transitions to the 2;_,, 4:_,, 5,, and 7; states in ‘“Pb. The calculation reproduces the absolute values of the observed cross sections within a factor of 2.
E
NUCLEAR REACTIONS 208Pb(polarized, t), E = 22.0 MeV, measured cr(E,, @), A(&, 0); deduced reaction mechanism. “%J levels deduced normalization factor. Magnetic spectrograph. First- and second-order DWBA analysis.
1. Introduction 208Pb(p, t)206Pb reaction leading problem of an unnatural-parity state of ‘OaPb has been solved by Tsukuba group by measuring power A,( 0) of this transition ‘*2).The (p, t) reaction to the unnatural-
A long-standing to the first 3+(3:)
the analyzing parity states of even-even nuclei is completely forbidden in the framework of simple zero-range (ZR) distorted-wave Born approximation (DWBA). However, the reaction can proceed via a (p, d)(d, t) two-step process since spin transfers = 1 is realized owing to twice transfer of spin-i neutrons. Actually the differential cross section g( 8) data of the reaction 208Pb(p, t)206Pb(3T, 1.34 MeV) at E, = 35 MeV by Lanford et al. 3, have been well reproduced by the second-order (p, d)(d, t) DWBA calculation by de Takacsy 4, and Charlton ‘). On the other hand, the same cross section data have been analyzed by Nagarajan et al. 6, as a one-step rather than the sequential (p, d)(d, t) process because the process is not forbidden in the finite-range (FR) DWBA if a realistic triton wave function containing a mixture of S, S’ and D state ‘) is used. Nagarajan et al. reported that their calculation can predict the absolute 03759474/87/503.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
M. Kurokawa
378
magnitude.
Thus the cross-section
al. “) are not powerful
enough
er al. /
experiments
to distinguish
ism for the 2”8Pb(p, t)206Pb(3 :) excitation In order to solve this problem,
20RPb(@, r)'"Ph by Lanford
whether
et al. ‘) and Wienke
the dominant
is a one-step
the first measurement
reaction
or two-step
et
mechan-
process.
of a vector analyzing
power
A,(8) for the 208Pb(p, t)2”6Pb(3:) excitation has been carried out by the Tsukuba group ‘) at E, = 22 MeV. The experimental result has been analyzed by Igarashi and Kubo ‘) by making the precise first- and second-order FR-DWBA calculations. The result has clearly shown the predominance of the (p, d)(d, t) two-step mechanism because the analyzing power at forward angles shows positive (negative) sign when the two-step (one-step) process is dominant. Thus the analyzing power experiment excludes the conclusion of Nagarajan et al. “). In addition to the unnatural-parity 3: transition, the Tsukuba group has carried out measurement and analysis of the natural-parity 2”8Pb(p, t)‘“Pb(Og’, g.s. and O:, 1.165 MeV) reactions lo.“). The predominance of the (p, d)(d, t) two-step process is confirmed also in the natural-parity (p, t) transitions and the large difference observed in the analyzing power for the two 0’ transitions is explained in terms of the nuclear-structure dependence of the two-step processes to the two 0’ states. Recently the Tsukuba group has extended the (p, t) A,,( 0) measurement to higher incident energies such as E, = 35 and 50 MeV and confirmed the predominance of the two-step processes over the one-step (p, t) process over a wide proton energy state range “.I’). In the analysis of these data, the effect of a deuteron unbound (S =O, T= 1) in the intermediate channels of the (p, d)(d, t) processes has been taken into account 12-14). The purpose of the present following essential
paper
is to extend
the above-mentioned
two points. In doing so, analyzing powers role in investigating the reaction dynamics
work to the
for (p, t) reactions play an and the nuclear structures
involved because analyzing powers are much more sensitive to interferences between various competing reaction process than the corresponding cross sections I’). (i) We extend analyzing power measurements to the (p, t) transition to the second of 3+(1.34 MeV) and 3’(3Z, 3.12 MeV) state of 206Pb. The wave functions to be almost pure configurations of 3’(3.12 MeV) states of “‘Pb are predicted (P ,,2, f,,,) and (P’,~, f,,2), respectively 16). These wave functions are confirmed from the angular distribution of cross section and spectroscopic factors for 207Pb(p, d)‘06Pb reaction I’). It is expected that the analyzing power for the 3; transition has an opposite sign to that of the 3; transition; see subsect. 3.2. (ii) We measure the cross sections and the analyzing powers of natural-parity “‘Pb(p, t)206Pb(2’, 4’, 5- and 7-) reaction. Then we analyze these transitions in term of the first- and second-order DWBA method so as to study the degree of contributions of the one- and the two-step processes and the dependence of the cross section and the analyzing power on the nuclear shell structure. Recently Wienke et al. Ix) h ave measured only the cross sections for the naturalparity 208Pb(p, t)206Pb reactions but have not measured the analyzing powers. We have briefly reported the “‘Pb(p, t)20”Pb(31) result elsewhere ‘9*20).
M. Kurokawa et al. /
“‘Pb( ~3,t)zwPb
379
2. Experimental procedures and results A polarized
proton
beam
was produced
source 2’). The 22.0 MeV polarized of Tsukuba
12 UD Pelletron.
proton
with a Lamb-shift beam was accelerated
The beam was analyzed
type polarized
ion
with the University
by a 90” analyzer
magnet
and
focused onto a 2o8Pb target by a switching magnet and a pair of quadruple magnets. The beam intensity on target was about 30 nA with a spot size less than a diameter of 2 mm. We used a 0.48 mg/cm2 thick 208Pb target backed by a 3 pm aluminum foil. Enrichment of the “‘Pb target is 98.69% was determined by weighing and checked with proton elastic section is estimated
“‘Pb. The thickness of the 208Pb target by comparing optical model prediction
scattering data. The error in the absolute values of the cross to be about 10%. A polarization monitor target was placed
upstream of the 208Pb target (fig. 1). The monitor target was 60 pg/crn’ thick LiF evaporated onto a carbon foil. Stability of the beam polarization was monitored with two solid state detectors (SSD’s) which measured the analyzing power for the ‘Li(p, a)4He reaction at 6,, = 130”. The amplitude of the effective analyzing power is about 0.7 at 8,, = 130”. Typical beam polarization was 80%. The error in the beam polarization is about 4%. Spin direction of the incident protons was flipped every 50 nC of beam current integration by the fast-spin-state-interchange system (FASSICS) 22). This system controls a spin-flipping pulse of the polarized ion source and supplies gate signals to each spin state for the counting system. Tritons from the (p, t) reactions were analyzed by a magnetic spectrograph (ESP-90) 23) and detected by a single-wire position-sensitive-proportional counter system 24). This counter system has sensitive length of 30 cm and is composed of a Magnetic spectrograph \
Position counter Ep=22 spin
Fig. 1. Schematic view of the detecting system.
380
M. Kurokawa et al. / “‘Pb( ~7,t)‘06Pb
208Pb(p,t)206Pb
Ep=22.0
8 l&,=25’
SPIN
800
MeV
UP
1200
2000
1600
CHANNEL NUMBER Fig. 2. Triton momentum spectrum of *“Pb(p
position
counter
and a AE counter.
of 0.0 to 3.8 MeV in *“Pb
, t)‘06Pb reaction at E, = 22 MeV, elab= 25”, with up spin.
This system covered
and no unknown
an excitation
peaks were observed
energy range
in a momentum
spectrum as shown in fig. 2. Coincidence requirement between position and AE signals lowered the background level. The FWHM of momentum spectrum was about 30 keV which was almost determined by energy loss in the target. As the excitation energies of all levels are well known from the literature 25), no effort is made to determine them in the present experiment. We measured angular distributions of cross sections a(0) and analyzing powers A,,( 0) from 5” to 90” in 5” steps. The horizontal acceptance of the magnetic spectrograph was +1.5”, which corresponded to a solid angle of 2.0 msr. The observed cross sections and analyzing powers are shown in figs. 3 to 6.
3. Opposite sign character of analyzing powers for unnatural-parity transitions to two 3+ states 3.1. ANALYZING-POWER
AND CROSS-SECTION
ANGULAR
(p, t)
DISTRIBUTIONS
Fig. 3 shows measured and calculated angular distributions of the differential cross sections and the analyzing powers for the unnatural-parity transitions
M. Kurokawa
_._._...-...._._______..
20
40
t)*“Pb
381
‘\ ‘\..,_ -.___..__‘_-- r - a
One-step
0
et al. / ‘08Pb($,
***-..___I\-
60
80
0
20
40
60
80
e,,(dw.) Fig. 3. Experimental and calculated cross sections a(0) and analyzing powers A,(0) for the *‘*Pb(p t)2”Pb(3:, 1.341 MeV and 3:, 3.122 MeV) reaction. The curves 2BS show the contribution from the bo&d deuteron channel, in which only the L = 0 component is taken into account in both (p, d) and (d, t) transfer steps, while the curves 2BT show the one in which the L = 2 component is taken into account. The curves 2US0 show the one from the ‘S, unbound deuteron channel.
“‘Pb(p, t)206Pb(3:, 1.34 MeV and 3:, 3.122 MeV). Note that the sign of the measured analyzing powers is opposite for the two 3+ transitions. The data for the 3: transition were obtained by Toba et al. ‘). Our new data for the 3: transition are consistent with them. The errors are almost determined by counting statistics, since the triton counts for the 3: transition were about 100 per 1 mC of beam current integration.
3.2. QUALITATIVE
ANALYSIS
BY USING
9-j COEFFICIENT
The opposite sign character is simply and qualitatively explained parison of 9-j coefficient for these two 3+ transitions. The scattering an analyzing power Ay are written as
A =Rek*h) ’
kt2+bi2
’
from a commatrix T and
(1)
382
M. Kurokawa
et al. /
20sPb(&r)'06Pb
1
0.803
MeV
0.803 MeV
;* I 0
*
1.304
! 40
I
MeV ' 80
1
120
@,,(deg.)
0
40
80
12'0
Q,,(deg.) Fig.
4. Experimental
and calculated cross sections a(O) and analyzing powers “‘Pb(d, t)*06Pb(2;, 2; and 3:) reaction at Ed = 17 MeV.
where g(h) denotes the spin-independent amplitude. We assume that the nuclear
A,(O)
for
the
(spin-dependent) part of the scattering structures of the 3: and 3: states to be
respectively; see sect. 1 and table 1. In this (P l/2, f5,2) and (pr12, f7,2) configuration, case the relative sign of the g and h for the (p, d)(d, t) two-step process is determined by the following 9-j symbol;
9-j=
( 1 11
12
1
s,
s2
s
iI
j2
,
.i
where 1, s and j (=3) show the transferred orbital, spin and total angular momenta, respectively. The 11(12) and j,(j,) are the orbital and total angular momenta of the shell-model states of the transferred neutrons. Of course we have S, = s2 = 4. The s = 0 part of the 9-j corresponds to the spin-independent amplitude g while the spin-dependent amplitude h is only due to the s = 1 part of the 9-j. The values of
M. Kurokaw~ et al.
-
/ 2*8Pb(@, t)*“l%
p1/2'p3/2
,
0
40
80
Two-step
120
0
40
80
120
B,,(dw.) Fig. 5. Calculated cross sections a(e) and analyzing powers A,,(B) of sa*~(p, t)*“Pb(2’) reaction leading to four con~garations. Experimental data show the a(@) and A,,(e) of 208Pb(p, tfZo6Pb(2~) transition. This state is predicted as almost pure configuration of (f,,*, p3,*).
the 9-j of our interest are given as follows; (P i/2, f5,X (P*,2, f7/2K 1=3 s=o 1=3 0.048 -0.036 1=2 1=2 -0.008 0.048 s=l 1=3 0.016 s=l 1=3 0.030 0.048 0.006. i 1=4 i 1=4 We can see that the signs of the s = 0 9-j coefficients are different for the two configurations while the signs for the main part of the s = 19-j coefficient are the same for the two configurations. Then from eq. (l), signs of the analyzing powers are opposite for the two 3+ transitions. Therefore the opposite sign character comes s=o
384
hf. Kurokawa
et al. / “*Pb( @, t)20dpb
0.803
MeV
, >_,--,
‘\ ‘\ 1
2.424
1 0
I 40
I 80
ecm(deg.
MeV
I
7 120
1
Fig. 6. Calculated and experimental cross sections and analyzing powers for the ‘OBPb(p, t)206Pb(2:_5, 4:-,, 5& 7;) reaction at E, = 22 MeV. The dot-dashed, broken and solid lines show one-step, two-step and combined zero-range DWBA calculations, respectively.
M. Karrokawa et al. / ***Pb(~, t)206Fb 0.803
0.2
MeV
,
,
MeV
,
0
-0.2 0.2
0
cd,
7.704
0.2
;r;
0
=T -0.2
r
1
, ,
2.149
0.2
0
-0.2
0
-0.2
Fig. 6-continued
MeV
385
386
M. Kurokawa
et al. / 20*Pb( $, t)‘“Pb 1 .684
MeV
.'.
(b’) I
43+ /,+. .+---..,r\\ ‘I\
o_ &'
I./’
h A.
3.014
I-1
+%y)J-
ZiT -
I
,.‘,2.928 MeV
,!,’ \:\
0.1-
I’
--
+v’ \(t
2.703
MeV
3.014
MeV
MeV
_--. ___* ____---71 -_*.______
10-2,
O 2.220 0
40
MeV 80
e,_,,(deg.
120
1
-0 . 2
2.220 0
40
80 Q,,(deg.)
Fig. bcontinued
MeV 120
M. Kurokawa
er al. / z”*l%( i, f)2wF6
387
TABLE 1 Eigenvalues
and two-neutron-hole
amplitude
of unnatural-parity
3’ states 16) and natural-parity
states ‘s)
in 2061%
J”
P!/29fv2
E,( MW
9
0.998
3+ 1st
1.433
“) 2nd
2.303
-0.036
3.015
-0.010
Pi/Z, f,,, 2’
b/2 P3/2
0.995 1.279
3rd
1.839
4th
2.270
-0.112
5th ‘)
2.542
-0.071
1.859
-0.111
2.121
-0.158
3rd
3.024
5-
-0.275 -0.153
f&2
0.832 0.549
Pi/Z. ‘13/2 2.457
0.124 -0.015
fs,2r P3/2
-0.924
-0.339
0.369
P~/~. ii3l2
3.286
0.926
0.241
f 5,2r i 13/2 3.047
f&2
0.817
-0.065
1st
-0.990
0.429
0.946
2nd
7- 1st
0.018 0.07 1
-0.009
P1/2. fV2
2nd
-0.013 -0.025
0.299
0.192
4+ 1st
0.035
P1/2. P3/2
-0.515
-0.908 -0.220
0.419
f x,2. i I),?.
-0.124
fs,2, P3/2 -0.114 0.032 0.000 -0.982 0.135
f5/2
9f7,2
0.075 0.040 -0.116
i is/21 f 7,2 -0.085
-0.701
0.066
p3/2. k/2
0.137
-0.980
fs,*. f-7/2
0.995 -0.018
0.810
1st 2nd
Pi/Z. f-7/2
0.083
“) Not observed. ‘)
From ref. 16).
from, through two-step
the 9-j coefficient,
(p, d)(d, t) process;
hole while 3:(~,,~,
3.3. FIRST-
AND
f,,J
the j-dependence
the 3:(p 1,2, fs,J
of the analyzing
state consists
power for the
of a j< = f,,, neutron
of a j, = f,,*.
SECOND-ORDER
DWBA
CALCULATIONS
Next we carry out a quantitative analysis of the two 3’ transitions. The finite-range DWBA calculation was made by Igarashi and Kubo 14) by using a computer program TWOFNRz6). Deuteron ground state (including D state) and unbound deuteron states (‘So and 3S, +3D,) are taken into account for the intermediate channels in the (p, d)(d, t) two-step process. The deuteron unbound state channels are evaluated by the discretized-continuum-channels method 27). The triton wave function, the
M. Kurokawa er al. / “*Pb( p, I)~“P~
388
deuteron ground state and unbound the Reid soft-core potential 2”). The distorting deuteron
potential
parameters
22 MeV by Satchler
state wave functions
parameters
were obtained
were calculated
by using
are given in table 2 (set A). The proton from ‘O’Pb(p, d)“‘Pb
29). The triton parameters
reaction
analysis
and
at E, =
were taken from optimal-model
fit to
elastic scattering data at E, = 17 MeV by Hardekopf et al. 3’). In fig. 3 the curve 2BS shows the contribution from the bound deuteron channel in which only the L = 0 component is taken into account in both (p, d) and (d, t) transition. The curve 2BT shows the one from the L = 2 component with the deuteron in its ground state. 2US0 and 2USl show the ones from the ‘So unbound deuteron and the ‘S, + ‘D, unbound deuteron channels, respectively. The calculation reproduces the opposite sign character of the analyzing powers for the two 3’ transitions well. Note that the pattern of the analyzing power is strongly dependent on the nuclear shell structure involved and the agreement between the experimental and calculated analyzing powers supports that the present nuclear structure wave functions are reasonable. The contribution of 2BS is the main part for the 3: transition. For the 3,’ transition the contribution of 2BS is the main part of the reaction at forward angles but the absolute value of cross section for the 2BT is as large as 2BS at backward angles. The absolute value of the cross sections of one-step and 2US0 are almost one to two orders of magnitude less than the total cross sections. The contributions of TABLE 2 Distorting Set
Particle
V
A, B A B C, E D A
P d d d d t
B, C E
t t
51.8 112.0 110.4 102.0 96.8 168.9 168.9 160.9
“) We define a distorting
potential
parameters
“), well depths
in MeV and lengths
in fm
W,
W,
r,
r,
r,,
0,
a,,
V, (,
r, 0
a, u
Ref.
0.0 0.0 0.0 0.0 -0.3 9.9 9.9 17.3
10.0 19.4 10.0 13.5 12.9 0.0 0.0 0.0
1.25 1.25 1.30 1.25 1.30 1.30 1.30 1.30
1.25 1.25 1.17 1.25 1.17 1.20 1.20 1.20
1.25 1.25 1.29 1.25 1.33 1.60 1.40 1.40
0.65 0.68 0.79 0.68 0.73 0.65 0.65 0.72
0.76 0.78 0.62 0.78 0.94 0.97 0.84 0.84
6.0 6.0 6.0 6.0 6.9 3.0 3.0 2.5
1.12 1.12 1.12 1.12 1.07 1.15 1.15 1.20
0.47 0.47 0.47 0.47 0.66 0.92 0.92 0.72
29) 29)
potential
as
U(r)=-[Vf,(r)+i(WJ,,(r)-4W,dJ,(r)/dr)] +2( h/m,&)? +
V, 0 /r)(dj,.,
(3 - r2/ Rf)zZeZ/2R,, I zZe2 / r ,
j;(r)=l/[l+exp[(r-R,)/a,]], Set A: used for unnatural-parity transition. Set B: used for natural-parity transition. Set C, D, E: see sect. 5.2.
(r)/dr)(l. rb R, r> R, R,=r,A”‘.
s)
30) 29) 34)
31) 21) “)
M. Kurokawa et al. / 20RPb($, t)‘06P6
2USl
and
the nonorthogonality
term
in the two-step
389
process
can be neglected
because both cross sections are about 10m4 pb/sr. The calculation angular distributions of the cross sections for the two 3’ transitions; distribution
has a peak at 20 degrees
and falls off gradually
reproduces the the 3: angular
while the 3: one is
almost constant. The absolute values of the calculated cross sections are lower than the experimental ones by a factor of 2. This discrepancy can be removed by modifying the distorting potential parameters for the tritons; subsect. 4.2. More details are given in refs. 13*14).
4. One- and two-step analysis of natural-parity 4.1. NUCLEAR-STRUCTURE
WAVE FUNCTIONS
(p, t) transitions
AND SPECTROSCOPIC
AMPLITUDES
We employ one-broken-pair-model wave functions of Hengeveld and Allaart ‘*) for 2’, 4’, 5- and 7- states in 206Pb (table 1). We refer to the paper by Wienke et al. lx) for these wave functions. We use the wave function of True for 2: state because the Hengeveld wave function is not available for this level. The phases of the main part of the two wave functions are consistent apart from the time reversal phase of i’. The spectroscopic amplitude of one-step (p, t) process for the pure configuration
is given by
s,, = Jpi while that of (p, d)(d, t) process
(2)
)
used for the pure configuration
s,,, = -_(
+ @Ti -) jl+j2-jajej2
(j,,jz)j is given by
ijji, ,
(3)
where j’ show the total angular momentum of the intermediate channel. In actual calculations we construct the transition amplitudes by using the pure-configuration spectroscopic amplitudes (2) and (3) and the configuration mixing coefficients given by table
1.
Investigation of the (p, t) transitions to the 0: and 0,’ states is not done present paper because the result has been already published in ref. ‘I). 4.2. DISTORTING
The distorting
in the
POTENTIALS
potential
parameters
used are given in table 2 (set B). We use the
proton optical potential parameters of Satchler 29). The deuteron parameters were obtained from ‘ORPb(p, d)207Pb reaction analysis at E, = 22 MeV by Dickey ef al. 3”). We use triton parameters of Hardekopf et al. 3’), but the imaginary-potential parameters ri( = 1.6 fm) and ai( =0.97 fm) are reduced to 1.4 fm and 0.84 fm, respectively. The absolute value of the cross section for the case of (ri = 1.4 fm, ai = 0.84 fm) is 1.85 times larger than that for the case of (ri = 1.6 fm, ai = 0.97 fm), but angular distributions of the cross sections and the analyzing powers are almost the same for the two ri’s.
M. Kurokawa et al. / zwPb( ~5,I)~~I%
3w
The absolute value of the (d, t) process also becomes about two times larger by this change. This fact can explain the discrepancy observed in the absolute values of the cross section
4.3. 0;
for the (p, t) (3: and 3;) transitions.
PARAMETERS
In the zero-range and the calculated
approximation the relation between one-step cross section is given by
the measured
cross section
. The trial value of D:(p, t) is determined by comparing the experimental cross section for 2: transition with that of the one-step calculation. The value of II;= 44x lo4 MeV’ . fm3 is used as a starting value of the calculation. The value of Dz(p, d) = 15 300 MeV’ * fm3 is used in the present calculation for (p, d) process. The calculation reproduces the experimental cross sections and analyzing powers for *08Pb(p, d)207Pb(p,,2, ~312, fS,2, f7,* and i13,*) reaction at E, = 22 MeV [ref. 32)]. But the analyzing power for the i,3,2 transition is not reproduced. For (d, t) process a value of Di(d, t) = 33 400 MeV’ * fm3 is used. The pattern of calculated cross sections reproduces experimental ones of 207Pb(d, t)206Pb(2:, 2: and 3:) reaction at Ed = 22 MeV [ref. 33)], but the absolute values of the calculated ones are about two times larger than that of the experimental ones (fig. 4). The calculated analyzing powers of the reaction roughly reproduce experimental ones. We use spectroscopic factors of Lanford “) which were determined from 207Pb(p, d)206Pb reaction. Finally the value of the D:(p, t) is determined by comparing the experimental cross section with that of combined one- and two-step calculations. Then the value of D:(p, t) = 61.6 x lo4 MeV* * fm3 is used throughout the calculation. The present calculation for the 3: and 3; transitions reproduce absolute value of the experimental cross sections. This fact is sufficiently enough to indicate that our calculation procedure for the reaction process is right and reasonable.
4.4. RESULTS
OF ANALYSIS
Numerical DWBA calculations of one-and two-step processes are carried out by using a computer program TWOFNR 2”). The zero-range approximation is used in the calculation for one- and two-step (p, d)(d, t) processes. First of all, calculations should explain the fact that angular distributions of the experimental analyzing powers are strongly dependent on each level of the same j cross sections are almost states of 206Pb, while those patterns of the experimental the same for the same j states. The patterns of calculated one-step analyzing powers and cross sections are almost independent of the nuclear shell structure for the 2+
M. Kurokawa
states as shown momentum two-step
in fig. 5 because
have unique analyses
and 1 transfers
ef al. /
transferred
value in one-step
predict
are allowed.
‘08Pb(t, ~)‘~pb
different
patterns
Therefore,
intrinsic process.
spin and transferred On the other hand,
as shown
contrary
391
in fig. 5 because
to the one-step
calculation,
angular (p, d)(d, t) both s = 0 two-step
calculations predict different patterns of a( 6) and A,( 0) depending on the configurations involved. Thus the nuclear structure dependence of the two-step analyzing powers can explain the variety of experimental analyzing powers. As shown in fig. 5, the calculated one-step cross section to a state having the configuration of (p ,,Z, pJIZ) is almost eight times larger than that of (fSJ2, and the calculated two-step cross section of (p ,,2, P~,~) is more than 20 times larger than that of (f5,2)2. Nuclear wave functions of natural-parity states in 206Pb consist of two to four substantial configurations as shown in table 1. The difference of the calculated absolute values of the cross sections due to the interference between each configuration term explains the observed difference of the experimental absolute values of the cross sections. Results of one-step (dot-dashed line), two-step (broken line) and combined (solid line) calculations are shown in fig. 6. General behavior of the cross sections and the analyzing powers for the transition to the 2: state at 0.803 MeV and the 2: state at 1.46 MeV is explained by the one-step analysis. But the calculation for the 2: state fails to reproduce the analyzing power around 20”. The 2: state at 1.784 MeV has a main configuration of (f&. The 2; state at 2.149 MeV is predicted to be almost pure configuration of (f 5,2, P~,~). Agreement between the calculated and experimental analyzing powers is improved around 20” by introducing two-step contributions for the 2; and 2: states. The absolute value of the two-step cross section for the 2: state is larger than that of one-step cross section at forward angles. The experimental absolute values of the cross sections reproduced by the calculation within a factor of 1.5.
for the five 2’ states
are
Calculation reproduces the absolute value of cross section for the 4: state at 1.684 MeV, but fails to reproduce the ones for the 4: state at 1.993 MeV and the 4f state at 2.928 MeV. The calculation fails to reproduce the cross-section angular distribution for the three 4’ states around 70”. No remarkable improvement can be seen by including
two-step
For two 5- states
contribution
the calculation
for the three 4+ states. reproduces
absolute
values
of cross sections
well, but oscillation pattern of the calculated cross sections is different from the experimental data around 20”. Agreement between the calculated and experimental analyzing powers is improved by introducing two-step contributions for the 5; state at 3.014 MeV. Two-step contributions to the analyzing powers for the two 5- states are larger than that for the 2+ and 4+ transition. Calculation reproduces a pattern of the cross section for the 7; state at 2.20 MeV but the absolute value is larger than the experimental one by a factor of 1.7. The angular distribution of the analyzing power is sensitive to the parameters of the distorting potential for the 7- state; this problem is discussed in subsect. 5.2.
392
M. Kurokawa et al. / “‘Pb( ~7,t)‘06Pb
The experimental are reproduced determined
absolute
values of the cross sections
by the calculation
by the one-step
within
contribution
a factor
except
for 2+, 4+, 5 and 7- states
of 2.0 and
they
are almost
for 2: state.
5. Discussion 5.1. NUCLEAR-STRUCTURE
WAVE FUNCTIONS
Absolute values of the cross sections for the “*Pb(p, t)206Pb(2f, 4+, 5- and 7-) reaction are strongly dependent on the nuclear wave functions; see subsect. 4.4. If analysis fails to reproduce experimental absolute value of the cross section, the wave function used in the calculation may be doubtful. The calculated one- and two-step absolute values of the cross sections are almost two times larger than the experimental ones for the 4:, 4: and 7, transitions. Also the analysis fails to reproduce the analyzing powers for these states. The calculated analyzing powers are dependent on the nuclear wave functions and the ratio of the one- to two-step calculated values of the cross sections. Agreement between the calculated and experimental analyzing powers becomes worse by introducing two-step contribution for the 4: state. From the above-mentioned reasons for the analyzing power and cross section, the wave function for 4: state may be wrong. If absolute value of the two-step cross section increases to 1.5 times for the 2: state, the agreement between the calculated and experimental analyzing power is improved. The present analysis by using the wave function of Hengeveld and Allaart ‘*) reproduces the absolute values of the cross sections better than the analysis by using the wave function of True for the most cases. The wave function of True 16) predicts the one-sixth of the experimental Agreement between the calculated
intensity of the cross section for the 4: state. and experimental analyzing power becomes worse
for the 2: state by introducing the two-step contribution. For the 2: state the wave function of True is used in the calculation. It is likely that the difference between the nuclear-structure model of Hengeveld et al. and that of True causes the abovementioned results.
5.2. DISTORTING
POTENTIALS
Calculated (p, d)(d, t) two-step analyzing powers are dependent on the deuteron distorting potential parameters used. In order to investigate the effect of the difference of several deuteron potentials, we compare experimental cross sections and analyzing Powers for *‘*Pb(p, d)207Pb(p,,, , fs,z, p3/2, &3/z and fsi2) reaction at Ep = 22 MeV with those of the zero-range DWBA calculation. We employ the deuteron potential parameters of Dickey et al. 30) (set B), Satchler *‘) (set A) and Daehnick et al. 34) (set D); see table 2. An analysis by using these three potentials almost reproduces the cross sections and the analyzing powers for the P,,~, pX12, and f7,* transitions
M. Kurokawa
but it fails to reproduce is large at forward A. The calculated
the analyzing
angles
et al. / zOsl%( @, I)‘~P~
power for the i13,* transition.
in the case of the calculation
analyzing
393
The discrepancy
by use of the potential
power with set D fails to reproduce
set
the experimental
one for the f5,2 transition. Therefore we have employed the deuteron potential of set B. Since 5- and 7- states in 2MPb contain an i ,3,2 hole in their main configuration, it is likely that the disagreement between the measurement and the analysis of the (p, t) analyzing powers for these states is due to the deuteron potential for the 208Pb(p, t)206Pb(5 and 7-) reaction. But note that the (p, d)(d, t) process in the analysis reproduces the experimental analyzing power for the 208Pb(p, t)20”Pb(5;) reaction. Calculated (p, d)(d, t) two-step analyzing power for the 7; state in ‘“Pb is positive at forward angles by use of the deuteron potential of Daehnick et al. (table 2, set D) and it becomes negative when we use the deuteron potentials of Satchler (table 2, set E) and Dickey er al. (table 2, set C). One- and two-step calculation for the 7- transition reproduces the analyzing power by using the deuteron potential of Dickey et al. better in comparison with the use of Satchler and Daehnick potentials. In consequence the distorting potential parameters of Dickey et al. are suitable for the present calculation. Calculated (p, t) one-step analyzing powers are dependent on the triton distorting potential parameters. We use the triton potential parameters of Hardekopf et al.* (ri = 1.4 fm and ai = 0.84 fm); see subsect. 4.2 and of Becchetti and Greenlees a modification of the imaginary(BG) “); see table 2. We add a star * indicating potential parameters ri and ai from the original values of Hardekopf et al. The one-step calculation using the potential of BG roughly reproduces the analyzing power for the *‘*Pb(p, t)““Pb(7;) reaction but it fails to reproduce the analyzing powers for the 2: and 2: states. The calculation using the potential of Hardekopf et al.* almost reproduces the analyzing powers for the 2:, 2: states. The main difference between these potentials is the potential depth of the imaginary part. The potential of BG.
of Hardekopf
Normalization
et al.* reproduces
factor
(disagreement
the experimental
TABLE
3
factor)
N for
results better than that
“‘F%(p, I)?‘~(J”)
transitions
J,”
2+I
2:
2;
2:
2;
3:
3;
E,(MeV)
0.803 1.0
1.467 0.8
1.784 0.5
2.149 1.2
2.424 1.2
1.304 1.0
3.122 1.3
N
J,”
4;
4;
4:
5,
5;
7;
E,(MeV)
1.684 1.0
1.998 0.6
2.928 0.5
2.783 1.0
3.014 1.0
2.200 0.5
N
394 5.3. FINITE-RANGE
M. ~uro~wa
et al. / 208Pb(A t)206Fb
EFFECT
To evaluate the finite-range effect in the (p, d)(d, t) two-step process in the natural-parity transitions, a numerical calculation is made in the case of the reaction “‘Pb(p, t)206Pb(2+). Procedure of the calculation is the same as in subsect. 3.3 but only the S state of deuteron and triton is taken into account. We calculated only for the configuration (P,,~, f&. Angular distributions of the cross sections in the two-step analysis are almost the same for the zero- and finite-range calculations and the magnitudes obtained from the analysis agree with each other within a factor of 1.04. The analyzing power of the finite-range calculation is smaler by 24% than the zero-range calculation at forward angles (OO-1S’) and the difference between the two calculations is limited at forward angles (Y-40”). It would be worthwhile to perform a finite-range calculation for the 2:, 5; and 4c3 transitions because the two-step process yields large reaction amplitudes for these states. 5.4. NORMALIZATION
FACTORS
We calculate normalization factors N which are obtained by taking the ratio of the measured cross section to the results of the combined direct- and sequentialtransfer calculations: (5),,,=~($X:“* The obtained values of N for the *‘*Pb(p, t)‘06Pb( J”) transitions are given in table 3. These values indicate the degree of disagreement between the experimental cross sections and the calculated ones presented by the solid lines in fig. 6. 6. Conclusion
We measured differential cross sections and analyzing powers for 208Pb(p, t)206Pb(0;___,,2;_,, 3;,2, 4:--3, 5L2 and 7;) reaction at E,=22 MeV. U~~~turu~-~~~~t~ fra~sition~: (i) The measured analyzing power for 3: transition has an opposite sign to that of 3: transition. (ii) This character is explained in terms of j-dependence of the analyzing power in the sequential two-step (p, d)(d, t) process. The measured analyzing powers and cross sections for the two 3’ states are reproduced with the finite-range DWBA calculation. Natural-parity transitions: (iii) Angular-distribution patterns of the measured cross sections for naturalparity transition are almost the same for the common transferred J transition but patterns of the analyzing powers of those are different. Two-step analysis is necessary
M. Kurokawa et al. / *‘*Pb(@,t)Z06Pb
395
to study the above-mentioned character because the one-step calculation predicts almost similar angular distributions of the analyzing powers. (iv) The one- and two-step calculation by using the nuclear wave functions of Hengeveld and Allaart reproduces the absolute values of the cross sections within a factor of 2. The absolute values of two-step cross section are 0.1 to 1.0 times smaller than those of a direct one-step depending on the nuclear structure. The present analysis by using nuclear wave functions of Hengeveld reproduces our measurement better than the analysis by using the wave function of True for the most cases. (v) The present calculation almost reproduces the analyzing powers and the cross sections for the 2:, 2:, 4: and 5: transitions. For 2: and 5: transitions agreement between calculated and experimental analyzing powers is improved by including the two-step contribution. But improvement is not clear in other transitions. (vi) For the 5- and 7- transitions two-step contributions of the analyzing powers are large over the measured angular range. On the other hand the effect of the two-step processes for 2+ transitions is limited around 20”. (vii) More work should be done for the deuteron and triton distorting potentials in the one- and two-step processes. We would like to thank Professors K.-I. Kubo and M. Igarashi for their analysis of the unnatural-parity (p, t) transitions. References 1) Y. Toba, Y. Aoki, S. Kunori, K. Nagano and K. Yagi, Phys. Rev. C20 (1979) 1204 2) K. Yagi, in hoc. of the Tsukuba, Int. Workshop on Deutron involving reactions and polarization phenomena, Tsukuba, 1985, ed. Y. Aoki and K. Yagi (Word Scientific, Singapore, 1986) p. 3 3) W.A. Lanford and J.B. Mcgrory, Phys. Lett. 45B (1973) 238 4) N.B. de Takacsy, Phys. Rev. Lett. 31 (1972) 1007; Nucl. Phys. A231 (1974) 243 5) L.A. Charlton, Phys. Rev. Cl4 (1976) 506 6) N.A. Nagarajan, M.R. Strayer and M.F. Werby, Phys. Lett. 68B (1977) 421; M.F. Werby, M.R. Strayer and M.A. Nagarajan, Phys. Rev. C21 (1980) 2235 7) M.R. Strayer and P.U. Sauer, Nucl. Phys. A231 (1974) 1 8) H. Wienke, H.P. Blok, J.F.A. van Hienen, J. Blok, Y. Iwasaki, A.G. Drentje and W.A. Sterrenburg, Phys. Lett. 135B (1984) 13 9) M. Igarashi and K.-I. Kubo, Phys. Rev. C25 (1982) 2144 10) K. Yagi, in Proc. of 5th Int. Symp. on Polarization phenomena in nuclear physics, Santa Fe, 1980, AIP Conference Proceedings 69 (1981) p. 254 11) Y. Toba, Y. Aoki, H. Iida, S. Kunori, K. Nagano and K. Yagi, Phys. Lett. 1OOB (1981) 232 12) K. Yagi, Y. Aoki, K. Hashimoto, H. Iida, H. Sakamoto, Y. Tagishi, M. Matoba, H. Igarashi and K.-I. Kubo, Phys. Rev. C31 (1985) 676 13) K.-I. Kubo, M. Igarashi and K. Yagi, ibid. ref.2), p. 273 14) M. Igarashi and K.-I. Kubo, in Proc. of the 6th Int. Symp. on Polarization phenomena in nuclear physics, Osaka, 1985, J. Phys. Sot. Jpn. 55 (1986) Suppl. p. 134 15) K. Yagi, H. Iida, Y. Aoki, K. Hashimoto and Y. Tagishi, Phys. Rev. C31 (1985) 120, and references therein especially refs. 1-21 16) W.W. True, Phys. Rev. 168 (1968) 1388
396 17) W.A. Lanford
M. Kurokawa
et al. 1 ““Pb(@.
f)‘06Pb
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