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Nuclear dynamics and the X-ray spectrum of muonic atoms
Nuclear Physics AI09 (1968) 539--560; (~) North-Holland Publishino Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
NUCLEAR DYNAMICS AND T H E X-RAY S P E C T R U M OF M U O N I C A T O M S WILHELM PIEPER and WALTER G R E I N E R
lnstitut fiir Theoretische Physik der Universitiit, Franl~furt/Main t Received 7 August 1967 Abstract: The effect of nuclear rotations, surface vibrations and giant multipole oscillations on the X-ray spectrum of a muonic atom is discussed. The polarization of nuclear states due to the muon is calculated for various nuclei. A detailed comparison of 3d-2p and 2 p - l s transitions with theoretical predictions is given. Compared to non-dynamical fits of muonic X-ray spectra we have 1 parameter less, namely the surface thickness. It is due to the zero point rotations and vibrations that the intrinsic homogeneous charge distributions gets smeared out. The excitation o f fl- and y-vibrations by the muon is by a factor 10 s less probable than the excitation of the first rotational states.
1. Introduction
In recent years X-rays from muonic atoms have been an interesting tool for the investigation of nuclear charge distributions 1-5). The parameters of the charge distribution, e.g. the nuclear radius, the surface thickness and the deformation have been obtained. Since the invention of germanium detectors the experimental precision increased by an order of magnitude and one can obtain more detailed and more accurate information about nuclear structure from muonic atoms. Nuclear polarization effects, e.g. the interaction of the muon with a dynamic nuclear charge distribution become appreciable. Especially the interaction with the nuclear collective degrees of freedom is - because of the long-range behaviour of the Coulomb forces - very important. Therefore, for deformed nuclei, it seems necessary to take rotations, 1% and 7-surface vibrations and giant multipole resonance oscillations into account. We develop in this paper a theory of muonic atoms and discuss the various dynamic effects on the X-ray spectrum. In sect. 2 the total Hamiltonian, consisting of the muon, the nucleus and the mutual interaction is given. Rotations, surface vibrations and giant multipole oscillations are described within the dynamic collective model 6). The Coulomb interaction of the oscillating deformed nucleus is especially carefully considered; also the vacuum polarization due to the monopole charge distribution is taken into account. In sect. 3 the diagonalization of the total Hamiltonian is discussed and in sect. 4 the cascade of the muon is calculated. The polarization of the nucleus due to the presence of the muon is investigated in sect. 5. Results are obtained for various nuclei. They are discussed in the last section. t This work has been supported by the Deutsche Forschungsgemeinschaft with a contract for studies in nuclear structures. 539
540
W.
P1EPER
2.
The
AND
W.
GREINER
Hamiltonian
2.1. THE NUCLEAR HAMILTONIAN The total Hamiltonian of the muon-nucleus system contains three parts: (i) the nuclear Hamiltonian, HN, which in our treatment describes the nuclear collective degrees of freedom, (ii) the Hamiltonian of the muon, H , , represented by the Dirac equation and corrected with respect to the vacuum polarization and (iii) the Coulomb interaction, Hint, between the deformed nuclear charge distribution and the muon. We neglect in the nuclear Hamiltonian the single-particle excitations 1. Because of the long-range muon-nucleus Coulomb interaction, the collective nuclear modes e.g. rotations, surface vibrations, giant multipole resonance oscillations and their mutual interaction terms are most important in the dynamics of the muon-nucleus system for even nuclei. The nuclear collective modes are described in the framework of the dynamic collective theory 6), e.g. the low-energy spectrum is described in the rotation vibration model 7) and the giant multipole excitations are added for the high-energy spectrum (10-35 MeV excitations). The surface-vibrational coordinates, ao and a2, are defined by the expansion of the nuclear surface R = Ro(1 + ao Y20 + a2(YE2 + Y2- 2)).
(1)
Here, because of volume conservation,
ago+2a
Ro = Ro 1+
,
Ro
__
R , A ~.
(2)
The Hamiltonian of the dynamic collective theory is then 6, 7) 2
HN = Hrot+Hvib-i-Hvibrnt-{- Z H~')(ao, a 2 , q(l)),
(3)
/=0
where //rot and Hvib ,or contain the hydrodynamical moments of inertia, which are expanded to second order in terms of the vibrational variables a~ and a~. The parameter R1 ( ~ 1.2 fm) will be fitted later on. The variables a~ and a~ characterize the deviation of the surface parameters ao and a2 from their equilibrium 130 and 0, respectively, ao = flo+a'o, a2 = O+a'2, la~/fio[ << 1. (4) One has Hro t --
Hvi b =
Hvib r o t
--
M2-M + (M -a oy 2~ o --
16Ba22
h ( ~2 lf~2,:~+½Coa'o2+C2a'22 ' __ 2B ~ + 2 aa'22] ( _ 2- <- , + 3ao __ 2 + 2a~2~ ME+ME( 2Jo
/30
f12
fig ]
2Jo
'
, oa_q ,
flo
/32 ]
t
+ 2e ---a o.
(5)
/30
t For odd nuclei the last nucleon and its excitation modes have also some importance. This case will he discussed elsewhere.
MUONIC X-RAYS
541
Here M is the total angular momentum and M3 its projection along the symmetry axis of the nucleus, d~(° are the components of the angular momentum of a giant /-pole phonon along the intrinsic axis. B is the inertia parameter which is related to the moment of inertia J o and the deformation flo by J o = 3Bfl~ ", Co and C2 are the stiffness parameters for the fl- and y-vibrations, respectively. The Hamiltonian for the giant/-pole resonance oscillations including the interaction with the surface vibrations is given in the adiabatic approximation by ~,, H ~ = hot(a o , a2) ~~~, a(l)+-(t)
(6)
l,/t
where ~o~(a0a2) are the frequencies of the giant multipole resonances and ~,a(°÷, q~O the creation and annihilation operators for a giant/-pole phonon with the projection /~ of angular momentum. The dependence of o)t(a o, a2) on the surface coordinates leads in general to a very important coupling between surface modes and the giant multipole resonances 6, 8). This coupling is, however, of minor importance in the present discussion of muonic X-rays and will be neglected here. The giant multipole frequencies or(0) are given by giant monopole he) o = 168.4 • A -~ [MeV], giant dipole h o 1
=
78.0. A -~- [MeV],
(7)
giant quadrupole hco2 = 125.0- A -~ [MeV]. For the muon-nucleus system are the low-energetic collective nuclear excitations of greatest importance. They really couple strongly to the muon and are consequently excited in the decay of the muonic atom. The giant multipole resonances, having excitation energies > 12-15 MeV, describe somehow the most important parts of the nuclear continuum. The influence of this on the muonic X-rays is mainly a small level shift, which we call the nuclear polarization due to the giant resonances. It will be calculated later. We discuss now the low-energy solutions of (5), e.g. those solutions where no giant phonon is excited. We have
H o l l M i K n 2 n o ) = (Hrot+H,,~b)llMiKn2no) = EiK,,2nollMxKn2no),
(8)
with the energy
E,Kn,,,o = h ( n o + ½ ) E p + ( 2 n z + ½ K + l ) E , + ( I ( I + l ) - K 2 ) ½ e ,
(9)
and the parameters (see table 1) Ea = h
E, = h
~ - 3Bfl~"
(10)
Here I, M and K are the total angular momentum and its projection on the z-axis and the symmetry axis of the nucleus, respectively, n o and n 2 are the quantum numbers of the fl- and ?-vibrations. The term H, ib rot is diagonalized later on together with the muon-nucleus interaction.
542
W.
PIEPER
AND
GREINER
W.
2.2. T H E H A M I L T O N I A N O F T H E M U O N A N D T H E I N T E R A C T I O N W I T H T H E N U C L E A R DEGREES OF FREEDOM
To get the muon wave functions Inxjmj> we solve numerically the Dirac equation
H~,lnxjmi> = (T~ + ~'oo)ln~cjrnj> = E,~jlnxjrnj>
(11)
including only the monopole term of the Coulomb potential of the deformed nucleus. It is assumed that the charge distribution inside the surface (1) is homogeneous. Therefore we have for the potential
Vc(r, f2) = - e (
p(r), dz'
Jlr-r
l
2-~-11
-
Y,.(o')/
dr'r '2 rt
+I
(12)
.
TABLE 1 T h e low-energy nuclear p a r a m e t e r s o f eq. (10) in M e V Nucleus
Z
A
E#
Er
e
R1
flo
~mSm 1roW 1saW la~W lSSOs 19°Os ea2Th ~3sU
62 74 74 74 76 76 90 92
152 182 184 186 188 190 232 238
0.6841 1.104 1.500 1.300 1.766 1.585 0.7175 0.994
1.040 1.2246 0.8632 0.6837 0.569 0.4731 0.77 1.0473
0.0316 0.0283 0.031 0.0325 0.040 0.04563 0.01458 0.01356
1.1928 1.2003 1.1968 1.1933 1.1933 1.1923 1.1898 1.1892
0.320 0.240 0.230 0.220 0.180 0.165 0.240 0.267
T h e s e values are fitted c o m p a r i n g the experimental energies with the eigenvalues o f eqs. (3) a n d (47). T h e errors, w h i c h arise f r o m fitting flo a n d R1, are Aflo = -4-0.006 a n d ARt = +0.0009.
If we now define R I = min R(f2) and R A = max R(f2), we can calculate the potential Vc for l = 0, 2, 4 and r -<_ R~ o r R A ~ r. In these regions the integration is easy to perform and yields the potentials
l/c(r, 12) = Voo(r)+ V20(r)Y20+ V22(r)(Y22 + Y2-2)+ V40(r)Y40 + v.2(r)(Y.2 + y._ 2)+ v . . ( r ) ( Y . + L - . ) ,
(13)
where the radial functions V~m are simple r-powers Vl~ for r < RI and VlA for R A ~ r:
= N R'o/r, ~00 V~o = N ~
ao-- ]~ ao+~
~(a2--2a22) , ;(ag-2a~)
,
MUONIC X-RAYS
a2+~1 V! a°a2)
vh = N~ 3 (~)3(
4
vA=N~ ~-7 (~)" 1
543
' (14)
~/! aoa2),
V2o=-N ~o 2"74~ (3a~+a~)'
(R_~]' 1 (3aoz+aza), V2° = N \ r / 7~/n
v L = - n t w o / ~ - 7 ~°a:'
v4A2 = N
7 - - n aoa 2 ,
V:4= N
1-~na~,
(,T,,/,
N = --ZeZ/R'o . It is not possible to give equivalent expressions for the Vim in the region R i < r < RA. Therefore we make use of the fact that a; and a; are small compared with/30 and calculate the potentials VoZoand VZo in the intermediate region R i < r < RA only for the equilibrium shape (a; = 0)
-
*e=
(l_
{
--
x, 3
+ 5.7 V~o
=
- - -
-
R; a 2
-2
?
--
R;
5
r2-
(~,i ' ~; R5o! + 4 7
-5
r 3]
(~,l~tl1 / ' \R-7o!
(15)
+ ~ r (x, - 1)
R~o {-}x~-2x,2+l + 2floo(~-x, +2/3ooX,-~ 27 6 - ~-x, z7 4 + 3x,2 - 1)} + 4 r 2 1 + 2/3°° 0 { l - x ' - l/1-;flflS
("
arctg 1-+ x,3/3oo/(1_/3oo ) /3oo = ao ~
~,
1/ rmRi
x, = ~/
RA--R I
,
RI = Ro(1-floo),
(16)
_
,
RA = Ro(1 +2floo). (17)
These functions are continuations of the inside and outside solutions, but it is not possible to expand them in power series of ao, because of the denominators floo and x/floo- Apart from this the boundaries R i and R A themselves are floo dependent. But one can find for the exact functions very good approximations VZo and VZo, since their av dependency is similar to that of the inside solutions (14). We approximate
544
W.
PIEPER AND W.
--
GREINER
7.~ % :
a3)
~-~V~o (% O.18) =
/
QlO
,~
V2o(ao: 0 3 0 )
/ 7 - - V2o(ao=f]o: 0.24) = ~o (/3o) /---
V2o(ao: O.Ta)
0.05
0'5
RilR~,
f.
Re~R;
i.5
.
uS_
R~
Fig. 1. The exact quadrupole potential V~o (r, ao, a2 = 0) is s h o w n for a0 = 0.18, 0.24 and 0.30. In the region between the small and large axis R ! ~ r ~ R A the approximations VZ are given for flo = 0.24 and a'o = ±0.06.
Voo(¢ %) _ vs (,r)
%:
03 ~x
V~ (r, ao) % : 024
ao= 018
0
~z
"~'~T- 1o{<%)- vs (r)
o.ol
0'5
/vdz (r. ao
-~ ' n"
Fig. 2. In this figure the difference between the potential of a fl0 deformed nucleus and the spherical nucleus o f the same volume is plotted. In the region between the small and large axis the dashed lines are the a p p r o x i m a t i o n Vz for flo = 0.24 and a'o = ~0.06.
M U O N I C X-RAYS
545
e.g. VZo by the function
_ r i o + a ; - i4 V2Zo(r, ao) =- V2Zo(r,flo +a'o) = V2Zo(r,rio)
(ri°+a;)2 ,
(18)
and neglect the dynamical a; dependency of R~ and RA. The exact solution V2Zo(r, ao) and the approximation vZo(r, ao) are shown in fig. 1. To get a similar function vZo we first subtract the potential of a homogeneous charged sphere Vs(r) with a radius R ; , because in the inside region this difference is proportional to ag (see V~o in eq. (14)) V~(r) = N({-½(r/R'o) 2) for 0 < r -< R ; ,
= NR'o/r
for
#
(19)
Ro < r.
The function VoZo(r, ao) --- l/oZ(r, rio+a'o) = V~(r)+(VZo( r, rio)- V~(r))(rio +a'o)2/ri 2
(20)
approximates the exact solution VZo (r, no). In fig. 2 the differences VZo- Vs and VZo- V~ are compared. By this procedure we have split the monopole potential Voo into a static and dynamic part, Voo and Foo, respectively:
Voo = Voo(r, rio)+ Foo(r, a'o).
(21)
We put the static part
Voo(r, rio) = N[{(1-ri~/4TO-½(r/R'o) 2] = VZo(r, rio) = NR'o/r
for 0 _< r _< R, for R, < r < RA for n A ~ r
(22)
into the Dirac equation and the dynamical rest Foo(r, a'v) = -3N(2rioa'o+a'o2+2a'22)/(8~r)
=
(V£(r, rio)-
, ,2 +2a2 )/rio2 V~(r))(2rioao+ao
= 0
for
0 < r < R~,
for for
RI -< r -< R A <
r,
R A,
(23)
is diagonalized. The relative errors of the matrix elements of V2o and ~0o are less than 0.2 ~o. As a part of the Hamiltonian the Coulomb potential is invariant with respect to rotations of the coordinate system
Vc(r, f2) = -N[lYoo(r)+ l?o2(r)[a x a] [°] - -~[o3 ~ 11o3 + l?21(r)[a × y2]t°]+ ~22(r)[aar2_l + ~'42(r)[aar4.1 . (24)
546
W.
PIEPER AND W.
GREINER
Here use is made of the tensor character of the collective coordinates av. The symbol [ x ]to~ denotes a scalar contraction and the radial functions in the inside (0 _< r _< R~) and outside (RA <_--r) region are
( ½-½(r/R'°)2]
~oo(r) = t
i'
R'olr
t?o2(r) =
- ~
3 i(R'o/r)2
x/~
o
'
~z,(r) = ~5 \(R'o/r)3] ' $
I722(r) = [
(fiR!) 2 ~3V5
[?42(r) = (--(r(R~))4~ 3 V 5 \2(Ro/r )s ] 2 14~"
\-4(Rolr)~i ?l ~"
(25)
The expansion is stopped as in eq. (13) after the second order of the av coordinates. This kind of notation shows that the different av powers have apart from ClebschGordan coefficients the same radial functions. We therefore use the potential VZo also as a continuation of V212
l/2Z(r, ao, a2) =
VZo(r, flo)
, a2+a2a°~lI V ~r
(26)
and for the same reason we put the term 2a~z into Foo for the intermediate region R~ - r <_ RA. All the other Vu, we continue from inside and outside r-region to r = R~. The interaction potentials (13) are given in the intrinsic system. It is therefore necessary to rotate the spherical harmonics into the laboratory system before calculating matrix elements. Since the energy of the muon (ca. 12 MeV in the ls level for Z = 92) is not very small compared with the muon rest mass (105.65 MeV), the vacuum polarization is not negligible. We use here the well-known formula of Ford and Wills 14) and describe the corrections by the potential Vp
Vp(r) = 3- (VL(r)-~Voo(r, flo)), VL(r) =
21ref°~dr'r'p(r' ) r
do
X {[r--r'[ (ln ( ~ ] r - - r ' ] ) - - 1 ) - - ( r + r ' ) ( l n ( ~ ( r + r ' ) ) - - 1 ) ) , --¢ = 4.61237" lO-3[fm
-1
].
(27)
The charge distribution p in VL is computed by smearing out the dynamical
"
2 " " "
.45o
O S 90
j
Fig. 3. Here the isodensity lines o f l~Sm and 19°Os calculated with eq. (28) are shown for ~0 = 0 °, 22.5 ° and 45 °. Since (Y22+Y~-~) = 0 for 99 = 45 °, the surface thickness in this direction is caused only by the/~-vibrations (eq. (1) and eq. (28)). Therefore the hard E-vibrator 180Os has nearly no surface thickness for ~o = 45 °. The three curves are the lines o f 30, 50 and 10 ~ density c o m p a r e d with the density at the origin.
........
548
W. PIEPER AND W. GREINER
l
frr)
50
~rn~Z~ " ~ 190 05.
•
,
I
085
0 90
I
I
095
10
i
~05
rlR~
110
Fig. 4. The dynamically "smeared out" monopole charge distribution p(r) (eq. (28)) of 19°Os and 15zSm are compared with the static ones p(r) = Po for r ~ Ri and p(r) = p0(l - - ~ / ( r - - R I ) / ( R A - - R I ) ) for RI <_%r _--..R A (dashed curves).
TABLE 2 Vacuum polarization energies 1 ls
2P~r
2p{
3d~
3d½
4f{
4rex
xs~Sm
46.5
17.0
15.6
4.1
3.9
0.9
0.9
ls~W
58.7
25.9
23.7
7.2
6.7
1.9
1.9
ls~W
58.5
25.7
23.6
7.1
6.7
1.9
1.9
lssW
58.3
25.6
23.5
7.1
6.6
1.9
1.9
9°Os
60.0
27.0
24.8
7.6
7.1
2.1
2.0
~a2Th
71.8
37.9
35.0
12.5
11.5
3.8
3.6
~ssU
73.6
38.9
36.0
13.3
12.2
4.1
3.9
The vacuum polarization energies for several nuclei are calculated using eqs. (27) and (28). The values are given in keV.
MUONIC X-RAYS
549
homogeneous charge distribution
p(r)=
~ f dOp(r,O)= ~ f dO.
(2s)
The matrix element contains the wave function of the nuclear ground state and the step function 0. Only at this point we have changed the order of integration, while in the interaction energy Hi,t we first calculated the potentials and then we folded them with the nuclear wave functions. To give an impression of the vibration amplitudes fig. 3 shows p(r, f2) for different directions 12 and fig. 4 shows p(r) together with the static monopole charge distribution which one gets out of Voo using the Poisson equation. The matrix elements of Vp are given in table 2.
2.3. THE INTERACTION
WITH THE GIANT
RESONANCES
Another correction term of the Hamiltonian is the Coulomb interaction of the muon with the giant resonance oscillations. In the dynamic collective theory 6) the total charge distribution,
p(r) = po(r) + q(r, t),
(29)
contains a fluctuating giant resonance part q(r, t)
,(r. z) = Z a2.(t),2.(r). 2/l
(30) The F~ are normalizing constants so that I q j 2 are normalized to 1 over the nuclear volume and the Ax. can be expressed by the giant resonance phonon creation and annihilation operators
A2u = ~ . - - ~ -
(q+u+qau).
(31)
The energies hcoz(2 = 0, 1, 2) are given in eq. (7), tc is the asymmetry energy constant of the mass formula 9). The smallness of this perturbation (1 keV compared with several MeV of the muon energies) permits to neglect the nuclear deformation (ka~ - k~) and vibration. The corresponding Coulomb potential is 2
v.. = E v"~ • gr l=O
(32)
550
W . PIEPER AND W . GREINER
with 1 . {~o(p~)+0.21724
V~°) = -4ne2F°A°°Y°° k~
for for
r < Ro, Ro < r,
1 . t3j,(p'l)-p'~0.41908 V~g)) = -- 4zce2Fa( ~ Alm(-)mYm) k~ [1.8900/pl 2 V~f) = - 4ne2F2( ~ Az,,,(-)"Y2,,)
~ . (5jz(p'2)-p'220.082409 ~22.9015/p~3
r
for
O~
Ro < r, (33)
TABLE 3 Giant resonance polarization angles Nucleus
ls~
2p~.
2 p ?r
3d~r
3 d ,z
4f{_
4f~_
152Sm
--0.072 --0.356 --0.002
<10 -a --0.187 --0,011
< 1 0 -4 --0.183 --0.011
< 1 0 -a --0.008 < 1 0 -3
<10 -a --0.007 < 1 0 -3
<10 -s <10 -a <710 -5
<10 -s <10 -a <710 -5
182w
--0.136 --0.674 --0.006
--0.001 --0,413 --0.036
--0.001 --0.413 --0.034
< 1 0 -6 --0.025 --0.002
< 1 0 -8 --0,021 --0.002
< 1 0 -6 --0,001 <7 10 -4
< 1 0 -6 --0.001 < 10 -4
la4 W
--0.137 --0.675 --0.006
--0.001 --0.416 --0.036
--0.001 --0.416 --0.033
< 1 0 -6 --0.025 --0.002
< 1 0 -e --0.022 --0.002
< 1 0 -6 --0.001 < 10 -4
< 1 0 -6 --0.001 < 10 -4
ls6 w
--0.137 --0,678 --0.006
--0.001 --0.418 --0.036
--0.001 --0.418 --0.034
< 1 0 -6 --0.025 --0.002
< 1 0 -6 --0.022 --0.002
< 1 0 -6 --0.001 < 10 -4
< 1 0 -6 --0.001 <7 10 -4
--0.148 --0.647 --0.007
--0.001 --0.461 --0.037
--0.001
<10 -6
<10 -6
< 1 0 -6
< 1 0 -6
lSSOs
--0.462 --0.035
--0.031 --0.003
--0.025 --0.002
--0.002 < 1 0 -4
--0,002 <710 -4
19OOs
--0.149 --0.651 --0.007
--0.001 --0.465 --0.037
--0.001 --0.467 --0.035
< 1 0 -6 --0.031 --0.003
< 1 0 -6 --0.025 --0.002
<10 -° --0.002 < 10 -4
< 1 0 -5 --0.002 < 10 -4
2a2Th
--0.259 --1.179 --0.021
--0.006 --0.951 --0.096
--0,003 --0.985 --0.096
< 10 -5 --0.084 --0.010
< 10 -5 --0.070 --0.009
< 10 -5 --0,005 < 10 -3
< 10 -5 --0.005 <7 1 0 - a
23s U
--0.277 --1.204 --0.023
--0.007 --1.039 --0.102
--0.003 --1.081 --0.104
< 10 -5 --0.096 --0.011
< 10 -5 --0.080 --0.011
< 10 -5 --0.006 < 1 0 -3
< 10-5 --0.005 < 1 0 -~
T h e g i a n t r e s o n a n c e p o l a r i z a t i o n e n e r g i e s AEnrj ~ Enrj--Enr.j a r e g i v e n f o r d i f f e r e n t m u l t i p o l a r i t i e s 2 = 0, I, 2 i n k e V . T h e m a t r i x e l e m e n t s o f e q . (34) a r e c a l c u l a t e d w i t h t h e r e l a t i v i s t i c w a v e f u n c t i o n s o f e q . (1 I). T h e c o n t r i b u t i o n s o f a l l levels u p t o n" = 4 a n d o f a l l p o s s i b l e j " a r e t a k e n i n t o a c c o u n t .
MUONIC X-RAYS
551
where Pl = k~r, k~ = wt/u and u is the velocity of sound in nuclear matter u = x/8~~/A. The potential (32) gives a correction to the Dirac energies E.~j of the muon due to the virtual excitation of giant multipole resonances
z = E. j+ Z
i[2
= o .'~'j, E,~j - (E.,~, i, + h~o~)
(34)
The matrix elements occurring in (34) can be easily calculated after inserting (33) into (32) and (34). We need in the calculation the numerical values of kxR'o = 4.4934, 2.0816 and 3.3420 for 2 = 0, 1 and 2, respectively. The explicit values of the shifts E , ~ j - E , ~ i are given in table 3.
3. The diagonalization The total Hamiltonian,
H = Ho+Hu+Hint,
(35)
consists of three parts, the diagonal nuclear part H o [eq. (8)], the diagonal muon Harniltonian Hu (eq. (11)), in which the energies are corrected corresponding to eq. (34) and the off-diagonal part
Hi.t = Hvlbrot+Vp+Foo+
Z Vt.,(r)Ylm.
(36)
1=2,4 m
For the basis system we choose the nuclear wave functions [IMiKn2no) and the muon wave functions [nKjmj) coupled to total angular momentum F
IIKn2no, nxj; F M ) = ~ (IjFIMIM-MIM)[IMIKn2no) " IntcjM-M~).
(37)
MI
We take into account the first three rotational states of the first vibrational bands: 11000) (I = 0, 2, 4; ground band), 11200) (I = 2, 3, 4; 7-band) and 11001) (I = 0, 2, 4; fl-band). In heavy nuclei we assume that the muon is captured in a high orbit and cascades down by Auger and El transitions to the 4f level and reaches by E1 transitions the 3d, 2p and 1s levels. The 4f levels shall be statistically occupied initially. The possible mixing of the 1s with the 3d and of the 2p with the 4f levels via the higher multipole terms of the Coulomb potential is neglected, because the connecting matrix elements are of the order of 10 -2 MeV and the energy differences are 9 MeV and 4 MeV respectively. Therefore we have to diagonalize the 1s state, the 2p states and the 3d states separately, Vp becomes diagonal. In the 4f states the muon is far away from the nucleus. Thus it is a good approximation to assume that the nucleus is in its ground state and the muon levels are not mixed, when the muon reaches this orbit. The matrix elements ofHvi b rot are given in full detail in the work ofA. Faessler et aL7).
552
W.
PIEPER
AND
W.
GREINER
For the Coulomb part of Hin t the Wigner-Eckart theorem yields
(I'K'n'2n~, n~c~'; FM[Voo+ ~ ~,,,Yt,,llKn2no, nxj, FM) /=2,4 nl t
t
t
p
!
p.t
= (1 K n2n o , n xj ; FMIFoolIKn2no, nxj, FM)fin,,,f,:,,,6jy + ½((2I + 1)(2I' + 1)(2j + 1)(2j' + 1))~(47t(1 + fifo)(1 + 6r,o))- ~(-)½- v
+(Q4°w4°+2Q42W42w2Q44w4')3( j '
40 --J½){J'
I'j F}I ,
(38)
where Q"=
(/
ml -K'I' ) +(_)i+r( I_K ml K'I')
ml / ' K ) + ( - ) I ( / K
m
K'
(39)
and
W~m = fo odr(K'n'2n'ol V~,.(r, a 0 , a2)lKn2no)(f.,~,f.~+9.,~,9.~ ).
(40)
The radial functions in Wtm contain the small (f) and the large (0) radial components ao) of the Dirac wave function normalized to 1:
fo~(f;,
2 +9n~2 ) dr = 1.
(41)
The given total angular momentum F and the angular momentum j of the muon select the different possible nuclear spins I and determine the size of the matrices. 4. The cascade
We start the cascade calculation with statistically filled unperturbed 4f levels, and the nucleus is assumed to be in its groundstate I = K = n 2 = no = 0. Therefore the two possible total angular momenta are F = ~ and F = ~. Then the muon drops down into the 3d levels by an E1 transition. The transition probability is different from zero only if the 3d eigenvector has a component with I = 0. Thus only those eigenstates, which belong to the F = ½ and F = ~ matrices can be reached. Since in these eigenvectors also components of higher rotational and vibrational levels are admixed, all F-values between ½ and z2 can be reached in the 2p states after the next transition and between F = ½ and F = ~ in the ls states. We therefore have to diagonalize 11 matrices: n ---- 3, F = ~, gk; n = 2, F = ½, ~, ~, ~; n = 1, F = ½, ~, -~, ~, ~2. The experiments do not measure transition probabilities per unit time but only the number of transition between two levels. If the upper level can decay only into one lower level, the intensity is proportional to the population of the upper level.
MUONIC X-RAYS
553
If, however, the upper level decays into two or more different lower levels the branching ratios are given by the transition probabilities. In this case the sum of the intensities of the different branches is proportional to the population of the upper level. The intensity m a x i m u m of the total transitions n, l ~ n - 1, 1 - 1 is fitted to the experimental spectrum. The transition probability per unit time of an E1 transition between an initial state
[omF) = ~. a~.v,plntcaj11,111K11n2,11no,11;F)
(42)
I £ n - i F ' ) = ~ a.,.-lr, r[n- 1 x~j~, IvK~n2, rno, 7; F')
(43)
11
and a final state 7
is given by
(anFIBEI ]~'n -- 1F') = c2(E,nF-- E,,,,,_ 1F,)3(ZF' + 1)1 Z • acmF, # a,,,,_ lr',~ r 11 •
x
1
fo odr r(g.- 1~.g.~a+f.-~ ~,f.~,)l 2.
qtY (44)
The population of a lower state is proportional to the sum of intensities of all transitions to this level. 5. Nuclear polarization When the muon has reached the ls orbit after the last 2 p - l s transition, there is a probability to find the nucleus in an excited state• In about 50 ~ of all cases it is in a 2 + state and in about 5 ~ or less it is in a 4 + state. I f one measures this nuclear Xrays one finds that they are shifted to higher energies compared with the transition in the absence of the m u o n 11). These shifts can easily be understood in our model. Since the muon is in an s-orbit only the monopole potential, that is Vo~o and vz0, can contribute to this effect. I f we introduce the rms radius of our charge distribution
r2=fpr2dz/fPdz=3R'°2[l+
5~a(~+2aZ42'rl)c
(45)
the potential can be rewritten
],110 =
Ze2 I~ R~
r2+r21 ~-R-~ ] .
(46)
The depth of the monopole potential of the muon depends on the dynamical deformation of the nucleus. But in spite of r 2 acting only on the vibrations, the several rotational states are shifted by the rotation vibration interaction (centrifugal stretching) by different amounts. The rotational states of the nucleus in absence of the m u o n have admixtures of the different vibrational bands (ground-state, 7- and fl-
554
W.
PIEPER
AND
W.
GREINER
band), which one gets as eigenvectors of the matrix HN
1(1+ 1)(1 + 3y 2 + x2)½e -- (I(I + 1)(1+ 2)(1-- 1))½x½ex/z -- (1(1+ 1)(1+ 2)(I-- 1))½x½ex/z [Er + [1(1+ 1)-- 4]½5(1 + 3y 2 + 2x2)](1 -- 6,0) ( - 1(1 + l)ey + 2ey) ((1 - 1)1(1+ 1)(I + 2)6)½½~xy
\ ((1-1)1(l+l)(I+2)6)~½exy 1. [E~ + 1(1+ 1)(1 + 9y 2 + x2)½e] ]
(47)
The presence of the muon adds constants/-independent elements to this matrix B = do
"
2x 2 + y2
\
2y
0
x 2 = 38/E~,
,
(48)
x2+3y2,1
y2 = 3e/(2Ep).
(49)
These are the matrix elements of F o o with the 1s radial matrix element of the muon ~Ze 2
do
=
..~_
flo2 ~ . /
,,Rx
S~R~Jo
_
dr (g~s+f~s)+ 2 z |
~RA
dr(g~s+fzs)[VZo(r, flo)-V~(r)].
(50)
d Rt
Here the density g~s 2 +fl~2 is • not a constant over the nuclear volume as it is assumed in electron atoms. If we now compare the eigenvalues of H i with those of HN+B for different values of I, we get the shift of the low-energy nuclear levels due to the presence of the muon, which is called the nuclear monopole polarization. The results are given in table 4 for different spins I of several nuclei. One of these values, the 2 + - 0 + shift in 152Sm has been measured recently ~2). The measured value ( 1.03 __+0.15 keV) and the theoretical one (3.87+0.2 keV) differ by a factor 3.7___0.6. This difference is not yet understood. The theoretical value may change by a few percents, if one enlarges the matrices by taking more vibrational bands into account or if one expands Voo to higher order in the vibrational parameters av. It seems, however, that this discrepancy between theory and experiment is similar to the one observed in electron atoms with the help of the Mt~ssbauer effect. This might indicate that the protons do not follow completely the vibrational motion of the nuclear mass. In other words: The protons and neutrons might have different vibrational amplitudes even they are vibrating in phase. This is similar to different proton-neutron deformations observed in gR factor measurements 13). Instead of a diagonalization of Voo there is the possibility to treat it in first and second order perturbation theory. The corresponding terms are called isomer shift and polarization, respectively. This procedure is perhaps not quite as accurate as the diagonalization, but it gives explicit expressions for the level shifts and therefore makes the physics better understandable. First we give good approximations for energies of the 0 + level E0 ÷ ,N in absence and Eo ÷, ~ in presence of the muon Eo+,N = - - 6 e (2~, ]- "
\E~/
(51)
100.1
110,9
121.0
154.4
187.1
50,1
44.7
ls2W
ls4W
la6W
188Os
la°Os
~32Th
~asU
147,6
163.7
569.8
483.1
386.4
359.1
324.4
380.4
4+-0 +
45.2
50.9
187.8
154,9
121.9
111.5
101.1
125.6
2+-0 +
149.3
166.6
571.8
484.6
388.8
361.0
327.5
391.4
4+-0 +
With muon
0.5
0.9
0,7
0.5
0.8
0,6
1.0
3,9
1.7
2.8
2.0
1.5
2.4
1.9
3,1
11.0
3(2+-0 +) A(4+-O +)
Shifts
3.90
4.32
6.71
5.91
6,00
5.07
3.98
6.02
EO+~-Eo+N
4.88 (4.89)
5.79 (5.80)
7.42 (7.38)
6.43 (6.42)
6.92 (6.93)
5.77 (5.79)
5.21 (5.22)
10.93 (11.09)
z112+
--0.47 (--0.48)
--0.55 (--0.60)
--0.05 (--0.05)
--0.05 (--0.05)
--0.14 (--0.14)
--0.12 (--0.12)
--0.23 (--0.25)
--1.06 (-- 1.07)
,d~2+
4.40 (4,40)
5.24 (5.19)
7.37 (7.33)
6.38 (6.37)
6.77 (6,79)
5.65 (5.67)
4.97 (4.98)
9.87 (t0.02)
d22+q-zl~2 +
First and second order perturbation
76.61
59.63
20.93
24.96
35.52
38.86
42.28
52.41
do
The 4+-2+-0 + differences of the nuclear low-energy spectrum are given due to the absence and presence of the muon. The energies are given in keV. The first six columns contain the energies calculated by diagonalization of H N and HN+B (eq. (47) and eq. (48), respectively). In column seven we noticed the 0 ÷ shift calculated with eq. (51) and (52). The energy shifts of the 2 + level computed in first (All), and second (A2) order perturbation theory and their sum are given in the columns 8 to 10. The upper numbers are calculated with the exact eigenvectors of HN, while the values in brackets correspond to eq. (53) and eq. (54), The last column shows the matrix element do (eq. (50)) in keV.
121.8
2+-0+
Without muon
1~2Sm
Nucleus
TABLE 4
556
w.
PIEPER AND W.
GREINER
Eo +, u = do(x2 + y2) + [ _ 4y2(e + do)2 + do(y4_ x,*)]/(Ea+ 2do(x 2 + y2)),
(52)
which we get out of the diagonalization of the 2 x 2 matrix of HN and HN + B. The error of Eo+, is less than 0.01 keV and that of EO+N is still an order of magnitude smaller compared with the exact roots. The term proportional to y 4 x4 is negligible. Then the first term of Eo+ ~ e.g. is 6.02 keV and the second one is - 1 . 9 0 keV for aS2Sm. As we see, both terms are comparable in magnitude. The easiest way to get similar formulas for the 2 + level is to calculate the matrix elements (2g+lVool2+g) and (2g+lFoo12+13) with the physical 2 + state of the groundband 12+g) and the betaband 12+fl). If we take this physical states out of the diagonalization of HN, we get the numbers A ~2+ = (2g + tFool2+g) and A22+ = 1(2g + IFoo12+13)2/(E2.g-E2+a) given in the columns 8, 9 and 10 of table 4. If we, however, do not diagonalize HN, but treat H~i b rot in first order perturbation theory, we get slightly different physical eigenstates of HN. These vectors yield up to the second order in e the expressions A12 + _- (2+glFool2+g) = 3d0e
A22+ -1(2+gl~°°lE+fl)12/(E2÷g-E2÷p)=
1 + 2E~
6dZe E-7 (1 \ +3 (2e12) \Ep/ /
/ (E2+g--E2+#). (54)
The corresponding values are given in brackets in columns 8, 9 and 10 of table 4. As we see the approximation is good. On the other hand, if we do not use the formulas (51) and (52), but calculate dl 0+ and d20 + similarly to eqs. (53) and (54), it turns out that Zi20+ = A22+ with an error of a few eV. Thus it is shown, that the polarization is not measurable.
6. Comparison with the experiments We compare our spectra with the experiments of the CERN group. The theoretical lines are folded with the energy dependent experimental line shapes of CERN. To get the best fit we are only allowed to vary Rt and 13o, because all the other parameters (eq. (10)) are given by the low-energy spectrum. Both parameters, R1 and 13o, are fitted using the most sensitive 2p-ls transition. R 1 is determined by the total energy position of this transition and 13o is fixed by the 2p~-2p~ fine-structure splitting. Then the number of free parameters is exhausted and the 3d-2p and 4f-3d spectra should agree with the experimental ones. Proceeding in this way we find that the hyperfine structure of the 3d-2p transitions are in good agreement with the experiments (see fig. 5). The only discrepancy between theory and experiment is an energy shift of 3-8 keV of the theoretical 3d-2p "mountain" to higher energies. The shift depends on the nucleus. The question is, how significant this shift is. The experimental error of the 2p-ls calibration lines are 4 keV. Therefore the uncertainty of the Rt fit yields an uncertainty of the theoretical 3d-2p energies of 0.7 keV. Even if one adds the 3d-2p
URANIUM Z = 92 3D- 2P TRANSITIONS
tl
2.9
Fig.5.
3.2
URANIUM Z = 9 2 2p ~TS TRANSITIONS
E[MeV)
I
6.I
Fig.6.
6.5
THORIUM Z = 90 3D - 2 P TRANSI TION5
2.90
Fig.7.
3.20
THORI UM 2 P - tS
Z = 90 TRANSITIONS
t 5.90
6.50
Fig. 8. TUNGSTEN Z = 74 3D -2P TRANSITIONS
60~ 500 400 3O0 200 I00
~!',~,'
I' p'I'"!ill~,i'I~i I I
' 19
2.3
1.95
'
1
225
~,v
'1 ~
Fig. 9. TUNGSTEN Z= ?4
I
2P- IS TRANSIT/ONS 60d 50~ 400 3OO 20O I00 II qrTr,l,r~L ', II '
50
5!
~,,1~1~.... r,,* rrn,rl r
'1]"
'l "
52
53
Fig. 10.
5.4
55
M UONIC X-RAYS
559
TUNGSTEN Z= 74 l
2 P - IS [RANSIT/ONS
300 i
200 tO0
,,,. l.-r[[,., 50
, 5.!
'1 5.2
53
' "it" 54
" I'rl'[I~
MeV
5 5
Fig. 11. Figs. 5-11. These figures s h o w the 3 d - 2 p a n d 2 p - l s spectra o f 2~8U, ~3~Th a n d 184W. T h e dotted line in the 2 p - l s s p e c t r u m o f 23sU c o r r e s p o n d s to a static t r e a t m e n t (a'0 = a'~ = 0) o f the C o u l o m b potential. In fig. 5 a n d fig. 7 the total theoretical curves have to be shifted to higher energies as indicated by t h e length o f the arrows A (3.6 keV for 238U a n d 9 keV for ~3~Th). Figs. 9 a n d 10 give the theoretical s p e c t r u m o f 184W c o m p a r e d with the s p e c t r u m o f n a t u r a l tungsten. In the last s p e c t r u m (fig. 11 ) the spectra o f ~s2W, 184W a n d ls6W are calculated with the s a m e R, = 1.1968 f m in R'0 = R I A a; a n d a d d e d with respect to their a b u n d a n c e s . T h e A~- low is obviously n o t fulfilled.
experimental error of 1-2 keV, there remains a significant difference between the theoretical and experimental 3d-2p energies of about 0-5 keV. This difference comes from the fact that the intrinsic nucleus itself has a small surface thickness, which is not included in our a n s a t z . We have assumed a homogeneous charge distribution for the intrinsic nucleus. This is certainly an approximation. The present results indicate, however, that the zeropoint-fluctuations and -rotations smear out the sharp edged intrinsic distribution in the right way and to nearly quantitative agreement. The small additional surface thickness which is needed in order to explain the 0-5 keV discrepancy in the 3d-2p transition might come from higher multipole vibrations (octupole, hexadecupole) not taken into account here. For convenience we have plotted in fig. 3 the smeared out [see eq. (28)] charge distribution for a S 2 S m and 19°Os. It can be noticed how the soft fl-vibrations in 152Sm and the soft y-vibrations in ag°Os determine the various surface thickness at the pole and at the equator of the ellipsoid, respectively. After averaging over the orientations in space (zeropoint rotations) one obtains the charge distribution shown in fig. 4. It is noticed that a broad surface thickness is obtained. An additional surface thickness for the intrinsic nucleus would soften the shape of fig. 4 somewhat more. One of the aims of this work was to look for vibrational lines in the spectra of muonic atoms. It results that the excitation of these lines is very
560
W.
PIEPER
AND
W.
GREINER.
weak. Their intensities are smaller than the maximum intensity of the 2p-1 s transition by a factor of 10- 3. But after all the mixing of the eigenvectors is changed due to the presence of vibrations. To test this effect we calculated the spectra with a static Coulomb interaction (a~ = a~ = 0, flo ~ 0). The dotted line in fig. 6 shows the result for the 2 p - l s transition in 238U. The change of the higher transitions is so small, that one cannot see a difference in the theoretical curves. The difference in the 2 p - l s transition, however, seems to be indicated in new high precision experiments of the Ottawa group 15) which are not yet published. In figs. 5-10 the 2 p - l s and 3d-2p spectra of 23su, 232Th and la4W are shown. For W and Os we can now only fit the spectra of the natural admixtures of isotopes. In these cases it is not useful to fit the parameters R~ and flo of every isotope separately, since one has too many free parameters. But if we choose the R1 parameter of 184W for 182W and 186W too and add the spectra of these three isotopes in such a way, that the maxima of the intensities of the different isotopes are proportional to the abundance, then we find that the A ~ low is not fulfilled for the A-dependence of the Ro parameter of the Wisotopes (fig. 11). The A-dependence is weaker. We cannot conclude the same for 19°Os and la8Os, because the abundances of these isotopes are only 26.4 70 and 13.3 70, respectively. It seems to be obvious that high precision experiments should be done for pure isotopes only, if useful theoretical information shall be obtained. We are very grateful to Dr. G. BackenstoB and Dr. H. Daniel (CERN) for many fruitful discussions and for giving us their experimental spectra. One of the authors (W.P.) wishes also to thank Prof. V. L. Telegdi (Chicago) and the muonic group of the Columbia University, especially Prof. C. S. Wu for clearing discussions in connection with sect. 4. The numerical work was done on the computer of the Deutsches Rechenzentrum Darmstadt.
References 1) 2) 3) 4) 5)
L. Wilets, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 1 (1954) B. A. Jacobson, Phys. Rev. 96 (1954) 1637 S. A. De Witt et al., Nuclear Physics 87 (1967) 657 S. Raboy et aL, Nuclear Physics 73 (1965) 353 H. L. Anderson, contribution to the International Conference on Electromagnetic sizes of nuclei, Ottawa, Canada (May 1967) 6) M. Danos and W. Greiner, Phys. Rev. 134 (1964) B284 7) A. Faessler, W. Greiner and R. K. Sheline, Nuclear Physics70 (1965) 33 8) H. J. Weber, M. G. Huber and W. Greiner, Z. f. Phys. 190 (1966) 25, 192 (1966) 182, 192 (1966) 223 9) D. Drechsel, Z. f. Phys. 192 (1966) 81 10) M. E. Rose, Relativistic electron theory (John Wiley & Sons, New York, 1961) 11) J. Htifner, Nuclear Physics 60 (1964) 427 12) S. Bernow et al., Phys. Rev. Lett. 18 (1967) 787 13) K. Afewu, Diplomarbeit, Institut far Theor. Physik, Frankfurt/M, Germany, (1967) 14) K. W. Ford and J. G. Wills, Report Lams 2387 (1960) 15) C. K. Hargrove and R. J. McKee, private communication
NUCLEAR DYNAMICS AND T H E X-RAY S P E C T R U M OF M U O N I C A T O M S WILHELM PIEPER and WALTER G R E I N E R
lnstitut fiir Theoretische Physik der Universitiit, Franl~furt/Main t Received 7 August 1967 Abstract: The effect of nuclear rotations, surface vibrations and giant multipole oscillations on the X-ray spectrum of a muonic atom is discussed. The polarization of nuclear states due to the muon is calculated for various nuclei. A detailed comparison of 3d-2p and 2 p - l s transitions with theoretical predictions is given. Compared to non-dynamical fits of muonic X-ray spectra we have 1 parameter less, namely the surface thickness. It is due to the zero point rotations and vibrations that the intrinsic homogeneous charge distributions gets smeared out. The excitation o f fl- and y-vibrations by the muon is by a factor 10 s less probable than the excitation of the first rotational states.
1. Introduction
In recent years X-rays from muonic atoms have been an interesting tool for the investigation of nuclear charge distributions 1-5). The parameters of the charge distribution, e.g. the nuclear radius, the surface thickness and the deformation have been obtained. Since the invention of germanium detectors the experimental precision increased by an order of magnitude and one can obtain more detailed and more accurate information about nuclear structure from muonic atoms. Nuclear polarization effects, e.g. the interaction of the muon with a dynamic nuclear charge distribution become appreciable. Especially the interaction with the nuclear collective degrees of freedom is - because of the long-range behaviour of the Coulomb forces - very important. Therefore, for deformed nuclei, it seems necessary to take rotations, 1% and 7-surface vibrations and giant multipole resonance oscillations into account. We develop in this paper a theory of muonic atoms and discuss the various dynamic effects on the X-ray spectrum. In sect. 2 the total Hamiltonian, consisting of the muon, the nucleus and the mutual interaction is given. Rotations, surface vibrations and giant multipole oscillations are described within the dynamic collective model 6). The Coulomb interaction of the oscillating deformed nucleus is especially carefully considered; also the vacuum polarization due to the monopole charge distribution is taken into account. In sect. 3 the diagonalization of the total Hamiltonian is discussed and in sect. 4 the cascade of the muon is calculated. The polarization of the nucleus due to the presence of the muon is investigated in sect. 5. Results are obtained for various nuclei. They are discussed in the last section. t This work has been supported by the Deutsche Forschungsgemeinschaft with a contract for studies in nuclear structures. 539
540
W.
P1EPER
2.
The
AND
W.
GREINER
Hamiltonian
2.1. THE NUCLEAR HAMILTONIAN The total Hamiltonian of the muon-nucleus system contains three parts: (i) the nuclear Hamiltonian, HN, which in our treatment describes the nuclear collective degrees of freedom, (ii) the Hamiltonian of the muon, H , , represented by the Dirac equation and corrected with respect to the vacuum polarization and (iii) the Coulomb interaction, Hint, between the deformed nuclear charge distribution and the muon. We neglect in the nuclear Hamiltonian the single-particle excitations 1. Because of the long-range muon-nucleus Coulomb interaction, the collective nuclear modes e.g. rotations, surface vibrations, giant multipole resonance oscillations and their mutual interaction terms are most important in the dynamics of the muon-nucleus system for even nuclei. The nuclear collective modes are described in the framework of the dynamic collective theory 6), e.g. the low-energy spectrum is described in the rotation vibration model 7) and the giant multipole excitations are added for the high-energy spectrum (10-35 MeV excitations). The surface-vibrational coordinates, ao and a2, are defined by the expansion of the nuclear surface R = Ro(1 + ao Y20 + a2(YE2 + Y2- 2)).
(1)
Here, because of volume conservation,
ago+2a
Ro = Ro 1+
,
Ro
__
R , A ~.
(2)
The Hamiltonian of the dynamic collective theory is then 6, 7) 2
HN = Hrot+Hvib-i-Hvibrnt-{- Z H~')(ao, a 2 , q(l)),
(3)
/=0
where //rot and Hvib ,or contain the hydrodynamical moments of inertia, which are expanded to second order in terms of the vibrational variables a~ and a~. The parameter R1 ( ~ 1.2 fm) will be fitted later on. The variables a~ and a~ characterize the deviation of the surface parameters ao and a2 from their equilibrium 130 and 0, respectively, ao = flo+a'o, a2 = O+a'2, la~/fio[ << 1. (4) One has Hro t --
Hvi b =
Hvib r o t
--
M2-M + (M -a oy 2~ o --
16Ba22
h ( ~2 lf~2,:~+½Coa'o2+C2a'22 ' __ 2B ~ + 2 aa'22] ( _ 2- <- , + 3ao __ 2 + 2a~2~ ME+ME( 2Jo
/30
f12
fig ]
2Jo
'
, oa_q ,
flo
/32 ]
t
+ 2e ---a o.
(5)
/30
t For odd nuclei the last nucleon and its excitation modes have also some importance. This case will he discussed elsewhere.
MUONIC X-RAYS
541
Here M is the total angular momentum and M3 its projection along the symmetry axis of the nucleus, d~(° are the components of the angular momentum of a giant /-pole phonon along the intrinsic axis. B is the inertia parameter which is related to the moment of inertia J o and the deformation flo by J o = 3Bfl~ ", Co and C2 are the stiffness parameters for the fl- and y-vibrations, respectively. The Hamiltonian for the giant/-pole resonance oscillations including the interaction with the surface vibrations is given in the adiabatic approximation by ~,, H ~ = hot(a o , a2) ~~~, a(l)+-(t)
(6)
l,/t
where ~o~(a0a2) are the frequencies of the giant multipole resonances and ~,a(°÷, q~O the creation and annihilation operators for a giant/-pole phonon with the projection /~ of angular momentum. The dependence of o)t(a o, a2) on the surface coordinates leads in general to a very important coupling between surface modes and the giant multipole resonances 6, 8). This coupling is, however, of minor importance in the present discussion of muonic X-rays and will be neglected here. The giant multipole frequencies or(0) are given by giant monopole he) o = 168.4 • A -~ [MeV], giant dipole h o 1
=
78.0. A -~- [MeV],
(7)
giant quadrupole hco2 = 125.0- A -~ [MeV]. For the muon-nucleus system are the low-energetic collective nuclear excitations of greatest importance. They really couple strongly to the muon and are consequently excited in the decay of the muonic atom. The giant multipole resonances, having excitation energies > 12-15 MeV, describe somehow the most important parts of the nuclear continuum. The influence of this on the muonic X-rays is mainly a small level shift, which we call the nuclear polarization due to the giant resonances. It will be calculated later. We discuss now the low-energy solutions of (5), e.g. those solutions where no giant phonon is excited. We have
H o l l M i K n 2 n o ) = (Hrot+H,,~b)llMiKn2no) = EiK,,2nollMxKn2no),
(8)
with the energy
E,Kn,,,o = h ( n o + ½ ) E p + ( 2 n z + ½ K + l ) E , + ( I ( I + l ) - K 2 ) ½ e ,
(9)
and the parameters (see table 1) Ea = h
E, = h
~ - 3Bfl~"
(10)
Here I, M and K are the total angular momentum and its projection on the z-axis and the symmetry axis of the nucleus, respectively, n o and n 2 are the quantum numbers of the fl- and ?-vibrations. The term H, ib rot is diagonalized later on together with the muon-nucleus interaction.
542
W.
PIEPER
AND
GREINER
W.
2.2. T H E H A M I L T O N I A N O F T H E M U O N A N D T H E I N T E R A C T I O N W I T H T H E N U C L E A R DEGREES OF FREEDOM
To get the muon wave functions Inxjmj> we solve numerically the Dirac equation
H~,lnxjmi> = (T~ + ~'oo)ln~cjrnj> = E,~jlnxjrnj>
(11)
including only the monopole term of the Coulomb potential of the deformed nucleus. It is assumed that the charge distribution inside the surface (1) is homogeneous. Therefore we have for the potential
Vc(r, f2) = - e (
p(r), dz'
Jlr-r
l
2-~-11
-
Y,.(o')/
dr'r '2 rt
+I
(12)
.
TABLE 1 T h e low-energy nuclear p a r a m e t e r s o f eq. (10) in M e V Nucleus
Z
A
E#
Er
e
R1
flo
~mSm 1roW 1saW la~W lSSOs 19°Os ea2Th ~3sU
62 74 74 74 76 76 90 92
152 182 184 186 188 190 232 238
0.6841 1.104 1.500 1.300 1.766 1.585 0.7175 0.994
1.040 1.2246 0.8632 0.6837 0.569 0.4731 0.77 1.0473
0.0316 0.0283 0.031 0.0325 0.040 0.04563 0.01458 0.01356
1.1928 1.2003 1.1968 1.1933 1.1933 1.1923 1.1898 1.1892
0.320 0.240 0.230 0.220 0.180 0.165 0.240 0.267
T h e s e values are fitted c o m p a r i n g the experimental energies with the eigenvalues o f eqs. (3) a n d (47). T h e errors, w h i c h arise f r o m fitting flo a n d R1, are Aflo = -4-0.006 a n d ARt = +0.0009.
If we now define R I = min R(f2) and R A = max R(f2), we can calculate the potential Vc for l = 0, 2, 4 and r -<_ R~ o r R A ~ r. In these regions the integration is easy to perform and yields the potentials
l/c(r, 12) = Voo(r)+ V20(r)Y20+ V22(r)(Y22 + Y2-2)+ V40(r)Y40 + v.2(r)(Y.2 + y._ 2)+ v . . ( r ) ( Y . + L - . ) ,
(13)
where the radial functions V~m are simple r-powers Vl~ for r < RI and VlA for R A ~ r:
= N R'o/r, ~00 V~o = N ~
ao-- ]~ ao+~
~(a2--2a22) , ;(ag-2a~)
,
MUONIC X-RAYS
a2+~1 V! a°a2)
vh = N~ 3 (~)3(
4
vA=N~ ~-7 (~)" 1
543
' (14)
~/! aoa2),
V2o=-N ~o 2"74~ (3a~+a~)'
(R_~]' 1 (3aoz+aza), V2° = N \ r / 7~/n
v L = - n t w o / ~ - 7 ~°a:'
v4A2 = N
7 - - n aoa 2 ,
V:4= N
1-~na~,
(,T,,/,
N = --ZeZ/R'o . It is not possible to give equivalent expressions for the Vim in the region R i < r < RA. Therefore we make use of the fact that a; and a; are small compared with/30 and calculate the potentials VoZoand VZo in the intermediate region R i < r < RA only for the equilibrium shape (a; = 0)
-
*e=
(l_
{
--
x, 3
+ 5.7 V~o
=
- - -
-
R; a 2
-2
?
--
R;
5
r2-
(~,i ' ~; R5o! + 4 7
-5
r 3]
(~,l~tl1 / ' \R-7o!
(15)
+ ~ r (x, - 1)
R~o {-}x~-2x,2+l + 2floo(~-x, +2/3ooX,-~ 27 6 - ~-x, z7 4 + 3x,2 - 1)} + 4 r 2 1 + 2/3°° 0 { l - x ' - l/1-;flflS
("
arctg 1-+ x,3/3oo/(1_/3oo ) /3oo = ao ~
~,
1/ rmRi
x, = ~/
RA--R I
,
RI = Ro(1-floo),
(16)
_
,
RA = Ro(1 +2floo). (17)
These functions are continuations of the inside and outside solutions, but it is not possible to expand them in power series of ao, because of the denominators floo and x/floo- Apart from this the boundaries R i and R A themselves are floo dependent. But one can find for the exact functions very good approximations VZo and VZo, since their av dependency is similar to that of the inside solutions (14). We approximate
544
W.
PIEPER AND W.
--
GREINER
7.~ % :
a3)
~-~V~o (% O.18) =
/
QlO
,~
V2o(ao: 0 3 0 )
/ 7 - - V2o(ao=f]o: 0.24) = ~o (/3o) /---
V2o(ao: O.Ta)
0.05
0'5
RilR~,
f.
Re~R;
i.5
.
uS_
R~
Fig. 1. The exact quadrupole potential V~o (r, ao, a2 = 0) is s h o w n for a0 = 0.18, 0.24 and 0.30. In the region between the small and large axis R ! ~ r ~ R A the approximations VZ are given for flo = 0.24 and a'o = ±0.06.
Voo(¢ %) _ vs (,r)
%:
03 ~x
V~ (r, ao) % : 024
ao= 018
0
~z
"~'~T- 1o{<%)- vs (r)
o.ol
0'5
/vdz (r. ao
-~ ' n"
Fig. 2. In this figure the difference between the potential of a fl0 deformed nucleus and the spherical nucleus o f the same volume is plotted. In the region between the small and large axis the dashed lines are the a p p r o x i m a t i o n Vz for flo = 0.24 and a'o = ~0.06.
M U O N I C X-RAYS
545
e.g. VZo by the function
_ r i o + a ; - i4 V2Zo(r, ao) =- V2Zo(r,flo +a'o) = V2Zo(r,rio)
(ri°+a;)2 ,
(18)
and neglect the dynamical a; dependency of R~ and RA. The exact solution V2Zo(r, ao) and the approximation vZo(r, ao) are shown in fig. 1. To get a similar function vZo we first subtract the potential of a homogeneous charged sphere Vs(r) with a radius R ; , because in the inside region this difference is proportional to ag (see V~o in eq. (14)) V~(r) = N({-½(r/R'o) 2) for 0 < r -< R ; ,
= NR'o/r
for
#
(19)
Ro < r.
The function VoZo(r, ao) --- l/oZ(r, rio+a'o) = V~(r)+(VZo( r, rio)- V~(r))(rio +a'o)2/ri 2
(20)
approximates the exact solution VZo (r, no). In fig. 2 the differences VZo- Vs and VZo- V~ are compared. By this procedure we have split the monopole potential Voo into a static and dynamic part, Voo and Foo, respectively:
Voo = Voo(r, rio)+ Foo(r, a'o).
(21)
We put the static part
Voo(r, rio) = N[{(1-ri~/4TO-½(r/R'o) 2] = VZo(r, rio) = NR'o/r
for 0 _< r _< R, for R, < r < RA for n A ~ r
(22)
into the Dirac equation and the dynamical rest Foo(r, a'v) = -3N(2rioa'o+a'o2+2a'22)/(8~r)
=
(V£(r, rio)-
, ,2 +2a2 )/rio2 V~(r))(2rioao+ao
= 0
for
0 < r < R~,
for for
RI -< r -< R A <
r,
R A,
(23)
is diagonalized. The relative errors of the matrix elements of V2o and ~0o are less than 0.2 ~o. As a part of the Hamiltonian the Coulomb potential is invariant with respect to rotations of the coordinate system
Vc(r, f2) = -N[lYoo(r)+ l?o2(r)[a x a] [°] - -~[o3 ~ 11o3 + l?21(r)[a × y2]t°]+ ~22(r)[aar2_l + ~'42(r)[aar4.1 . (24)
546
W.
PIEPER AND W.
GREINER
Here use is made of the tensor character of the collective coordinates av. The symbol [ x ]to~ denotes a scalar contraction and the radial functions in the inside (0 _< r _< R~) and outside (RA <_--r) region are
( ½-½(r/R'°)2]
~oo(r) = t
i'
R'olr
t?o2(r) =
- ~
3 i(R'o/r)2
x/~
o
'
~z,(r) = ~5 \(R'o/r)3] ' $
I722(r) = [
(fiR!) 2 ~3V5
[?42(r) = (--(r(R~))4~ 3 V 5 \2(Ro/r )s ] 2 14~"
\-4(Rolr)~i ?l ~"
(25)
The expansion is stopped as in eq. (13) after the second order of the av coordinates. This kind of notation shows that the different av powers have apart from ClebschGordan coefficients the same radial functions. We therefore use the potential VZo also as a continuation of V212
l/2Z(r, ao, a2) =
VZo(r, flo)
, a2+a2a°~lI V ~r
(26)
and for the same reason we put the term 2a~z into Foo for the intermediate region R~ - r <_ RA. All the other Vu, we continue from inside and outside r-region to r = R~. The interaction potentials (13) are given in the intrinsic system. It is therefore necessary to rotate the spherical harmonics into the laboratory system before calculating matrix elements. Since the energy of the muon (ca. 12 MeV in the ls level for Z = 92) is not very small compared with the muon rest mass (105.65 MeV), the vacuum polarization is not negligible. We use here the well-known formula of Ford and Wills 14) and describe the corrections by the potential Vp
Vp(r) = 3- (VL(r)-~Voo(r, flo)), VL(r) =
21ref°~dr'r'p(r' ) r
do
X {[r--r'[ (ln ( ~ ] r - - r ' ] ) - - 1 ) - - ( r + r ' ) ( l n ( ~ ( r + r ' ) ) - - 1 ) ) , --¢ = 4.61237" lO-3[fm
-1
].
(27)
The charge distribution p in VL is computed by smearing out the dynamical
"
2 " " "
.45o
O S 90
j
Fig. 3. Here the isodensity lines o f l~Sm and 19°Os calculated with eq. (28) are shown for ~0 = 0 °, 22.5 ° and 45 °. Since (Y22+Y~-~) = 0 for 99 = 45 °, the surface thickness in this direction is caused only by the/~-vibrations (eq. (1) and eq. (28)). Therefore the hard E-vibrator 180Os has nearly no surface thickness for ~o = 45 °. The three curves are the lines o f 30, 50 and 10 ~ density c o m p a r e d with the density at the origin.
........
548
W. PIEPER AND W. GREINER
l
frr)
50
~rn~Z~ " ~ 190 05.
•
,
I
085
0 90
I
I
095
10
i
~05
rlR~
110
Fig. 4. The dynamically "smeared out" monopole charge distribution p(r) (eq. (28)) of 19°Os and 15zSm are compared with the static ones p(r) = Po for r ~ Ri and p(r) = p0(l - - ~ / ( r - - R I ) / ( R A - - R I ) ) for RI <_%r _--..R A (dashed curves).
TABLE 2 Vacuum polarization energies 1 ls
2P~r
2p{
3d~
3d½
4f{
4rex
xs~Sm
46.5
17.0
15.6
4.1
3.9
0.9
0.9
ls~W
58.7
25.9
23.7
7.2
6.7
1.9
1.9
ls~W
58.5
25.7
23.6
7.1
6.7
1.9
1.9
lssW
58.3
25.6
23.5
7.1
6.6
1.9
1.9
9°Os
60.0
27.0
24.8
7.6
7.1
2.1
2.0
~a2Th
71.8
37.9
35.0
12.5
11.5
3.8
3.6
~ssU
73.6
38.9
36.0
13.3
12.2
4.1
3.9
The vacuum polarization energies for several nuclei are calculated using eqs. (27) and (28). The values are given in keV.
MUONIC X-RAYS
549
homogeneous charge distribution
p(r)=
~ f dOp(r,O)= ~ f dO
(2s)
The matrix element contains the wave function of the nuclear ground state and the step function 0. Only at this point we have changed the order of integration, while in the interaction energy Hi,t we first calculated the potentials and then we folded them with the nuclear wave functions. To give an impression of the vibration amplitudes fig. 3 shows p(r, f2) for different directions 12 and fig. 4 shows p(r) together with the static monopole charge distribution which one gets out of Voo using the Poisson equation. The matrix elements of Vp are given in table 2.
2.3. THE INTERACTION
WITH THE GIANT
RESONANCES
Another correction term of the Hamiltonian is the Coulomb interaction of the muon with the giant resonance oscillations. In the dynamic collective theory 6) the total charge distribution,
p(r) = po(r) + q(r, t),
(29)
contains a fluctuating giant resonance part q(r, t)
,(r. z) = Z a2.(t),2.(r). 2/l
(30) The F~ are normalizing constants so that I q j 2 are normalized to 1 over the nuclear volume and the Ax. can be expressed by the giant resonance phonon creation and annihilation operators
A2u = ~ . - - ~ -
(q+u+qau).
(31)
The energies hcoz(2 = 0, 1, 2) are given in eq. (7), tc is the asymmetry energy constant of the mass formula 9). The smallness of this perturbation (1 keV compared with several MeV of the muon energies) permits to neglect the nuclear deformation (ka~ - k~) and vibration. The corresponding Coulomb potential is 2
v.. = E v"~ • gr l=O
(32)
550
W . PIEPER AND W . GREINER
with 1 . {~o(p~)+0.21724
V~°) = -4ne2F°A°°Y°° k~
for for
r < Ro, Ro < r,
1 . t3j,(p'l)-p'~0.41908 V~g)) = -- 4zce2Fa( ~ Alm(-)mYm) k~ [1.8900/pl 2 V~f) = - 4ne2F2( ~ Az,,,(-)"Y2,,)
~ . (5jz(p'2)-p'220.082409 ~22.9015/p~3
r
for
O~
Ro < r, (33)
TABLE 3 Giant resonance polarization angles Nucleus
ls~
2p~.
2 p ?r
3d~r
3 d ,z
4f{_
4f~_
152Sm
--0.072 --0.356 --0.002
<10 -a --0.187 --0,011
< 1 0 -4 --0.183 --0.011
< 1 0 -a --0.008 < 1 0 -3
<10 -a --0.007 < 1 0 -3
<10 -s <10 -a <710 -5
<10 -s <10 -a <710 -5
182w
--0.136 --0.674 --0.006
--0.001 --0,413 --0.036
--0.001 --0.413 --0.034
< 1 0 -6 --0.025 --0.002
< 1 0 -8 --0,021 --0.002
< 1 0 -6 --0,001 <7 10 -4
< 1 0 -6 --0.001 < 10 -4
la4 W
--0.137 --0.675 --0.006
--0.001 --0.416 --0.036
--0.001 --0.416 --0.033
< 1 0 -6 --0.025 --0.002
< 1 0 -e --0.022 --0.002
< 1 0 -6 --0.001 < 10 -4
< 1 0 -6 --0.001 < 10 -4
ls6 w
--0.137 --0,678 --0.006
--0.001 --0.418 --0.036
--0.001 --0.418 --0.034
< 1 0 -6 --0.025 --0.002
< 1 0 -6 --0.022 --0.002
< 1 0 -6 --0.001 < 10 -4
< 1 0 -6 --0.001 <7 10 -4
--0.148 --0.647 --0.007
--0.001 --0.461 --0.037
--0.001
<10 -6
<10 -6
< 1 0 -6
< 1 0 -6
lSSOs
--0.462 --0.035
--0.031 --0.003
--0.025 --0.002
--0.002 < 1 0 -4
--0,002 <710 -4
19OOs
--0.149 --0.651 --0.007
--0.001 --0.465 --0.037
--0.001 --0.467 --0.035
< 1 0 -6 --0.031 --0.003
< 1 0 -6 --0.025 --0.002
<10 -° --0.002 < 10 -4
< 1 0 -5 --0.002 < 10 -4
2a2Th
--0.259 --1.179 --0.021
--0.006 --0.951 --0.096
--0,003 --0.985 --0.096
< 10 -5 --0.084 --0.010
< 10 -5 --0.070 --0.009
< 10 -5 --0,005 < 10 -3
< 10 -5 --0.005 <7 1 0 - a
23s U
--0.277 --1.204 --0.023
--0.007 --1.039 --0.102
--0.003 --1.081 --0.104
< 10 -5 --0.096 --0.011
< 10 -5 --0.080 --0.011
< 10 -5 --0.006 < 1 0 -3
< 10-5 --0.005 < 1 0 -~
T h e g i a n t r e s o n a n c e p o l a r i z a t i o n e n e r g i e s AEnrj ~ Enrj--Enr.j a r e g i v e n f o r d i f f e r e n t m u l t i p o l a r i t i e s 2 = 0, I, 2 i n k e V . T h e m a t r i x e l e m e n t s o f e q . (34) a r e c a l c u l a t e d w i t h t h e r e l a t i v i s t i c w a v e f u n c t i o n s o f e q . (1 I). T h e c o n t r i b u t i o n s o f a l l levels u p t o n" = 4 a n d o f a l l p o s s i b l e j " a r e t a k e n i n t o a c c o u n t .
MUONIC X-RAYS
551
where Pl = k~r, k~ = wt/u and u is the velocity of sound in nuclear matter u = x/8~~/A. The potential (32) gives a correction to the Dirac energies E.~j of the muon due to the virtual excitation of giant multipole resonances
z = E. j+ Z
i
= o .'~'j, E,~j - (E.,~, i, + h~o~)
(34)
The matrix elements occurring in (34) can be easily calculated after inserting (33) into (32) and (34). We need in the calculation the numerical values of kxR'o = 4.4934, 2.0816 and 3.3420 for 2 = 0, 1 and 2, respectively. The explicit values of the shifts E , ~ j - E , ~ i are given in table 3.
3. The diagonalization The total Hamiltonian,
H = Ho+Hu+Hint,
(35)
consists of three parts, the diagonal nuclear part H o [eq. (8)], the diagonal muon Harniltonian Hu (eq. (11)), in which the energies are corrected corresponding to eq. (34) and the off-diagonal part
Hi.t = Hvlbrot+Vp+Foo+
Z Vt.,(r)Ylm.
(36)
1=2,4 m
For the basis system we choose the nuclear wave functions [IMiKn2no) and the muon wave functions [nKjmj) coupled to total angular momentum F
IIKn2no, nxj; F M ) = ~ (IjFIMIM-MIM)[IMIKn2no) " IntcjM-M~).
(37)
MI
We take into account the first three rotational states of the first vibrational bands: 11000) (I = 0, 2, 4; ground band), 11200) (I = 2, 3, 4; 7-band) and 11001) (I = 0, 2, 4; fl-band). In heavy nuclei we assume that the muon is captured in a high orbit and cascades down by Auger and El transitions to the 4f level and reaches by E1 transitions the 3d, 2p and 1s levels. The 4f levels shall be statistically occupied initially. The possible mixing of the 1s with the 3d and of the 2p with the 4f levels via the higher multipole terms of the Coulomb potential is neglected, because the connecting matrix elements are of the order of 10 -2 MeV and the energy differences are 9 MeV and 4 MeV respectively. Therefore we have to diagonalize the 1s state, the 2p states and the 3d states separately, Vp becomes diagonal. In the 4f states the muon is far away from the nucleus. Thus it is a good approximation to assume that the nucleus is in its ground state and the muon levels are not mixed, when the muon reaches this orbit. The matrix elements ofHvi b rot are given in full detail in the work ofA. Faessler et aL7).
552
W.
PIEPER
AND
W.
GREINER
For the Coulomb part of Hin t the Wigner-Eckart theorem yields
(I'K'n'2n~, n~c~'; FM[Voo+ ~ ~,,,Yt,,llKn2no, nxj, FM) /=2,4 nl t
t
t
p
!
p.t
= (1 K n2n o , n xj ; FMIFoolIKn2no, nxj, FM)fin,,,f,:,,,6jy + ½((2I + 1)(2I' + 1)(2j + 1)(2j' + 1))~(47t(1 + fifo)(1 + 6r,o))- ~(-)½- v
+(Q4°w4°+2Q42W42w2Q44w4')3( j '
40 --J½){J'
I'j F}I ,
(38)
where Q"=
(/
ml -K'I' ) +(_)i+r( I_K ml K'I')
ml / ' K ) + ( - ) I ( / K
m
K'
(39)
and
W~m = fo odr(K'n'2n'ol V~,.(r, a 0 , a2)lKn2no)(f.,~,f.~+9.,~,9.~ ).
(40)
The radial functions in Wtm contain the small (f) and the large (0) radial components ao) of the Dirac wave function normalized to 1:
fo~(f;,
2 +9n~2 ) dr = 1.
(41)
The given total angular momentum F and the angular momentum j of the muon select the different possible nuclear spins I and determine the size of the matrices. 4. The cascade
We start the cascade calculation with statistically filled unperturbed 4f levels, and the nucleus is assumed to be in its groundstate I = K = n 2 = no = 0. Therefore the two possible total angular momenta are F = ~ and F = ~. Then the muon drops down into the 3d levels by an E1 transition. The transition probability is different from zero only if the 3d eigenvector has a component with I = 0. Thus only those eigenstates, which belong to the F = ½ and F = ~ matrices can be reached. Since in these eigenvectors also components of higher rotational and vibrational levels are admixed, all F-values between ½ and z2 can be reached in the 2p states after the next transition and between F = ½ and F = ~ in the ls states. We therefore have to diagonalize 11 matrices: n ---- 3, F = ~, gk; n = 2, F = ½, ~, ~, ~; n = 1, F = ½, ~, -~, ~, ~2. The experiments do not measure transition probabilities per unit time but only the number of transition between two levels. If the upper level can decay only into one lower level, the intensity is proportional to the population of the upper level.
MUONIC X-RAYS
553
If, however, the upper level decays into two or more different lower levels the branching ratios are given by the transition probabilities. In this case the sum of the intensities of the different branches is proportional to the population of the upper level. The intensity m a x i m u m of the total transitions n, l ~ n - 1, 1 - 1 is fitted to the experimental spectrum. The transition probability per unit time of an E1 transition between an initial state
[omF) = ~. a~.v,plntcaj11,111K11n2,11no,11;F)
(42)
I £ n - i F ' ) = ~ a.,.-lr, r[n- 1 x~j~, IvK~n2, rno, 7; F')
(43)
11
and a final state 7
is given by
(anFIBEI ]~'n -- 1F') = c2(E,nF-- E,,,,,_ 1F,)3(ZF' + 1)1 Z • acmF, # a,,,,_ lr',~ r 11 •
x
1
fo odr r(g.- 1~.g.~a+f.-~ ~,f.~,)l 2.
qtY (44)
The population of a lower state is proportional to the sum of intensities of all transitions to this level. 5. Nuclear polarization When the muon has reached the ls orbit after the last 2 p - l s transition, there is a probability to find the nucleus in an excited state• In about 50 ~ of all cases it is in a 2 + state and in about 5 ~ or less it is in a 4 + state. I f one measures this nuclear Xrays one finds that they are shifted to higher energies compared with the transition in the absence of the m u o n 11). These shifts can easily be understood in our model. Since the muon is in an s-orbit only the monopole potential, that is Vo~o and vz0, can contribute to this effect. I f we introduce the rms radius of our charge distribution
r2=fpr2dz/fPdz=3R'°2[l+
5~a(~+2aZ42'rl)c
(45)
the potential can be rewritten
],110 =
Ze2 I~ R~
r2+r21 ~-R-~ ] .
(46)
The depth of the monopole potential of the muon depends on the dynamical deformation of the nucleus. But in spite of r 2 acting only on the vibrations, the several rotational states are shifted by the rotation vibration interaction (centrifugal stretching) by different amounts. The rotational states of the nucleus in absence of the m u o n have admixtures of the different vibrational bands (ground-state, 7- and fl-
554
W.
PIEPER
AND
W.
GREINER
band), which one gets as eigenvectors of the matrix HN
1(1+ 1)(1 + 3y 2 + x2)½e -- (I(I + 1)(1+ 2)(1-- 1))½x½ex/z -- (1(1+ 1)(1+ 2)(I-- 1))½x½ex/z [Er + [1(1+ 1)-- 4]½5(1 + 3y 2 + 2x2)](1 -- 6,0) ( - 1(1 + l)ey + 2ey) ((1 - 1)1(1+ 1)(I + 2)6)½½~xy
\ ((1-1)1(l+l)(I+2)6)~½exy 1. [E~ + 1(1+ 1)(1 + 9y 2 + x2)½e] ]
(47)
The presence of the muon adds constants/-independent elements to this matrix B = do
"
2x 2 + y2
\
2y
0
x 2 = 38/E~,
,
(48)
x2+3y2,1
y2 = 3e/(2Ep).
(49)
These are the matrix elements of F o o with the 1s radial matrix element of the muon ~Ze 2
do
=
..~_
flo2 ~ . /
,,Rx
S~R~Jo
_
dr (g~s+f~s)+ 2 z |
~RA
dr(g~s+fzs)[VZo(r, flo)-V~(r)].
(50)
d Rt
Here the density g~s 2 +fl~2 is • not a constant over the nuclear volume as it is assumed in electron atoms. If we now compare the eigenvalues of H i with those of HN+B for different values of I, we get the shift of the low-energy nuclear levels due to the presence of the muon, which is called the nuclear monopole polarization. The results are given in table 4 for different spins I of several nuclei. One of these values, the 2 + - 0 + shift in 152Sm has been measured recently ~2). The measured value ( 1.03 __+0.15 keV) and the theoretical one (3.87+0.2 keV) differ by a factor 3.7___0.6. This difference is not yet understood. The theoretical value may change by a few percents, if one enlarges the matrices by taking more vibrational bands into account or if one expands Voo to higher order in the vibrational parameters av. It seems, however, that this discrepancy between theory and experiment is similar to the one observed in electron atoms with the help of the Mt~ssbauer effect. This might indicate that the protons do not follow completely the vibrational motion of the nuclear mass. In other words: The protons and neutrons might have different vibrational amplitudes even they are vibrating in phase. This is similar to different proton-neutron deformations observed in gR factor measurements 13). Instead of a diagonalization of Voo there is the possibility to treat it in first and second order perturbation theory. The corresponding terms are called isomer shift and polarization, respectively. This procedure is perhaps not quite as accurate as the diagonalization, but it gives explicit expressions for the level shifts and therefore makes the physics better understandable. First we give good approximations for energies of the 0 + level E0 ÷ ,N in absence and Eo ÷, ~ in presence of the muon Eo+,N = - - 6 e (2~, ]- "
\E~/
(51)
100.1
110,9
121.0
154.4
187.1
50,1
44.7
ls2W
ls4W
la6W
188Os
la°Os
~32Th
~asU
147,6
163.7
569.8
483.1
386.4
359.1
324.4
380.4
4+-0 +
45.2
50.9
187.8
154,9
121.9
111.5
101.1
125.6
2+-0 +
149.3
166.6
571.8
484.6
388.8
361.0
327.5
391.4
4+-0 +
With muon
0.5
0.9
0,7
0.5
0.8
0,6
1.0
3,9
1.7
2.8
2.0
1.5
2.4
1.9
3,1
11.0
3(2+-0 +) A(4+-O +)
Shifts
3.90
4.32
6.71
5.91
6,00
5.07
3.98
6.02
EO+~-Eo+N
4.88 (4.89)
5.79 (5.80)
7.42 (7.38)
6.43 (6.42)
6.92 (6.93)
5.77 (5.79)
5.21 (5.22)
10.93 (11.09)
z112+
--0.47 (--0.48)
--0.55 (--0.60)
--0.05 (--0.05)
--0.05 (--0.05)
--0.14 (--0.14)
--0.12 (--0.12)
--0.23 (--0.25)
--1.06 (-- 1.07)
,d~2+
4.40 (4,40)
5.24 (5.19)
7.37 (7.33)
6.38 (6.37)
6.77 (6,79)
5.65 (5.67)
4.97 (4.98)
9.87 (t0.02)
d22+q-zl~2 +
First and second order perturbation
76.61
59.63
20.93
24.96
35.52
38.86
42.28
52.41
do
The 4+-2+-0 + differences of the nuclear low-energy spectrum are given due to the absence and presence of the muon. The energies are given in keV. The first six columns contain the energies calculated by diagonalization of H N and HN+B (eq. (47) and eq. (48), respectively). In column seven we noticed the 0 ÷ shift calculated with eq. (51) and (52). The energy shifts of the 2 + level computed in first (All), and second (A2) order perturbation theory and their sum are given in the columns 8 to 10. The upper numbers are calculated with the exact eigenvectors of HN, while the values in brackets correspond to eq. (53) and eq. (54), The last column shows the matrix element do (eq. (50)) in keV.
121.8
2+-0+
Without muon
1~2Sm
Nucleus
TABLE 4
556
w.
PIEPER AND W.
GREINER
Eo +, u = do(x2 + y2) + [ _ 4y2(e + do)2 + do(y4_ x,*)]/(Ea+ 2do(x 2 + y2)),
(52)
which we get out of the diagonalization of the 2 x 2 matrix of HN and HN + B. The error of Eo+, is less than 0.01 keV and that of EO+N is still an order of magnitude smaller compared with the exact roots. The term proportional to y 4 x4 is negligible. Then the first term of Eo+ ~ e.g. is 6.02 keV and the second one is - 1 . 9 0 keV for aS2Sm. As we see, both terms are comparable in magnitude. The easiest way to get similar formulas for the 2 + level is to calculate the matrix elements (2g+lVool2+g) and (2g+lFoo12+13) with the physical 2 + state of the groundband 12+g) and the betaband 12+fl). If we take this physical states out of the diagonalization of HN, we get the numbers A ~2+ = (2g + tFool2+g) and A22+ = 1(2g + IFoo12+13)2/(E2.g-E2+a) given in the columns 8, 9 and 10 of table 4. If we, however, do not diagonalize HN, but treat H~i b rot in first order perturbation theory, we get slightly different physical eigenstates of HN. These vectors yield up to the second order in e the expressions A12 + _- (2+glFool2+g) = 3d0e
A22+ -1(2+gl~°°lE+fl)12/(E2÷g-E2÷p)=
1 + 2E~
6dZe E-7 (1 \ +3 (2e12) \Ep/ /
/ (E2+g--E2+#). (54)
The corresponding values are given in brackets in columns 8, 9 and 10 of table 4. As we see the approximation is good. On the other hand, if we do not use the formulas (51) and (52), but calculate dl 0+ and d20 + similarly to eqs. (53) and (54), it turns out that Zi20+ = A22+ with an error of a few eV. Thus it is shown, that the polarization is not measurable.
6. Comparison with the experiments We compare our spectra with the experiments of the CERN group. The theoretical lines are folded with the energy dependent experimental line shapes of CERN. To get the best fit we are only allowed to vary Rt and 13o, because all the other parameters (eq. (10)) are given by the low-energy spectrum. Both parameters, R1 and 13o, are fitted using the most sensitive 2p-ls transition. R 1 is determined by the total energy position of this transition and 13o is fixed by the 2p~-2p~ fine-structure splitting. Then the number of free parameters is exhausted and the 3d-2p and 4f-3d spectra should agree with the experimental ones. Proceeding in this way we find that the hyperfine structure of the 3d-2p transitions are in good agreement with the experiments (see fig. 5). The only discrepancy between theory and experiment is an energy shift of 3-8 keV of the theoretical 3d-2p "mountain" to higher energies. The shift depends on the nucleus. The question is, how significant this shift is. The experimental error of the 2p-ls calibration lines are 4 keV. Therefore the uncertainty of the Rt fit yields an uncertainty of the theoretical 3d-2p energies of 0.7 keV. Even if one adds the 3d-2p
URANIUM Z = 92 3D- 2P TRANSITIONS
tl
2.9
Fig.5.
3.2
URANIUM Z = 9 2 2p ~TS TRANSITIONS
E[MeV)
I
6.I
Fig.6.
6.5
THORIUM Z = 90 3D - 2 P TRANSI TION5
2.90
Fig.7.
3.20
THORI UM 2 P - tS
Z = 90 TRANSITIONS
t 5.90
6.50
Fig. 8. TUNGSTEN Z = 74 3D -2P TRANSITIONS
60~ 500 400 3O0 200 I00
~!',~,'
I' p'I'"!ill~,i'I~i I I
' 19
2.3
1.95
'
1
225
~,v
'1 ~
Fig. 9. TUNGSTEN Z= ?4
I
2P- IS TRANSIT/ONS 60d 50~ 400 3OO 20O I00 II qrTr,l,r~L ', II '
50
5!
~,,1~1~.... r,,* rrn,rl r
'1]"
'l "
52
53
Fig. 10.
5.4
55
M UONIC X-RAYS
559
TUNGSTEN Z= 74 l
2 P - IS [RANSIT/ONS
300 i
200 tO0
,,,. l.-r[[,., 50
, 5.!
'1 5.2
53
' "it" 54
" I'rl'[I~
MeV
5 5
Fig. 11. Figs. 5-11. These figures s h o w the 3 d - 2 p a n d 2 p - l s spectra o f 2~8U, ~3~Th a n d 184W. T h e dotted line in the 2 p - l s s p e c t r u m o f 23sU c o r r e s p o n d s to a static t r e a t m e n t (a'0 = a'~ = 0) o f the C o u l o m b potential. In fig. 5 a n d fig. 7 the total theoretical curves have to be shifted to higher energies as indicated by t h e length o f the arrows A (3.6 keV for 238U a n d 9 keV for ~3~Th). Figs. 9 a n d 10 give the theoretical s p e c t r u m o f 184W c o m p a r e d with the s p e c t r u m o f n a t u r a l tungsten. In the last s p e c t r u m (fig. 11 ) the spectra o f ~s2W, 184W a n d ls6W are calculated with the s a m e R, = 1.1968 f m in R'0 = R I A a; a n d a d d e d with respect to their a b u n d a n c e s . T h e A~- low is obviously n o t fulfilled.
experimental error of 1-2 keV, there remains a significant difference between the theoretical and experimental 3d-2p energies of about 0-5 keV. This difference comes from the fact that the intrinsic nucleus itself has a small surface thickness, which is not included in our a n s a t z . We have assumed a homogeneous charge distribution for the intrinsic nucleus. This is certainly an approximation. The present results indicate, however, that the zeropoint-fluctuations and -rotations smear out the sharp edged intrinsic distribution in the right way and to nearly quantitative agreement. The small additional surface thickness which is needed in order to explain the 0-5 keV discrepancy in the 3d-2p transition might come from higher multipole vibrations (octupole, hexadecupole) not taken into account here. For convenience we have plotted in fig. 3 the smeared out [see eq. (28)] charge distribution for a S 2 S m and 19°Os. It can be noticed how the soft fl-vibrations in 152Sm and the soft y-vibrations in ag°Os determine the various surface thickness at the pole and at the equator of the ellipsoid, respectively. After averaging over the orientations in space (zeropoint rotations) one obtains the charge distribution shown in fig. 4. It is noticed that a broad surface thickness is obtained. An additional surface thickness for the intrinsic nucleus would soften the shape of fig. 4 somewhat more. One of the aims of this work was to look for vibrational lines in the spectra of muonic atoms. It results that the excitation of these lines is very
560
W.
PIEPER
AND
W.
GREINER.
weak. Their intensities are smaller than the maximum intensity of the 2p-1 s transition by a factor of 10- 3. But after all the mixing of the eigenvectors is changed due to the presence of vibrations. To test this effect we calculated the spectra with a static Coulomb interaction (a~ = a~ = 0, flo ~ 0). The dotted line in fig. 6 shows the result for the 2 p - l s transition in 238U. The change of the higher transitions is so small, that one cannot see a difference in the theoretical curves. The difference in the 2 p - l s transition, however, seems to be indicated in new high precision experiments of the Ottawa group 15) which are not yet published. In figs. 5-10 the 2 p - l s and 3d-2p spectra of 23su, 232Th and la4W are shown. For W and Os we can now only fit the spectra of the natural admixtures of isotopes. In these cases it is not useful to fit the parameters R~ and flo of every isotope separately, since one has too many free parameters. But if we choose the R1 parameter of 184W for 182W and 186W too and add the spectra of these three isotopes in such a way, that the maxima of the intensities of the different isotopes are proportional to the abundance, then we find that the A ~ low is not fulfilled for the A-dependence of the Ro parameter of the Wisotopes (fig. 11). The A-dependence is weaker. We cannot conclude the same for 19°Os and la8Os, because the abundances of these isotopes are only 26.4 70 and 13.3 70, respectively. It seems to be obvious that high precision experiments should be done for pure isotopes only, if useful theoretical information shall be obtained. We are very grateful to Dr. G. BackenstoB and Dr. H. Daniel (CERN) for many fruitful discussions and for giving us their experimental spectra. One of the authors (W.P.) wishes also to thank Prof. V. L. Telegdi (Chicago) and the muonic group of the Columbia University, especially Prof. C. S. Wu for clearing discussions in connection with sect. 4. The numerical work was done on the computer of the Deutsches Rechenzentrum Darmstadt.
References 1) 2) 3) 4) 5)
L. Wilets, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 1 (1954) B. A. Jacobson, Phys. Rev. 96 (1954) 1637 S. A. De Witt et al., Nuclear Physics 87 (1967) 657 S. Raboy et aL, Nuclear Physics 73 (1965) 353 H. L. Anderson, contribution to the International Conference on Electromagnetic sizes of nuclei, Ottawa, Canada (May 1967) 6) M. Danos and W. Greiner, Phys. Rev. 134 (1964) B284 7) A. Faessler, W. Greiner and R. K. Sheline, Nuclear Physics70 (1965) 33 8) H. J. Weber, M. G. Huber and W. Greiner, Z. f. Phys. 190 (1966) 25, 192 (1966) 182, 192 (1966) 223 9) D. Drechsel, Z. f. Phys. 192 (1966) 81 10) M. E. Rose, Relativistic electron theory (John Wiley & Sons, New York, 1961) 11) J. Htifner, Nuclear Physics 60 (1964) 427 12) S. Bernow et al., Phys. Rev. Lett. 18 (1967) 787 13) K. Afewu, Diplomarbeit, Institut far Theor. Physik, Frankfurt/M, Germany, (1967) 14) K. W. Ford and J. G. Wills, Report Lams 2387 (1960) 15) C. K. Hargrove and R. J. McKee, private communication