 Standard Model goes pear-shaped in CERN experiment
The Register
 Physicists get a good look at pear-shaped atomic nuclei
The Los Angeles Times
 Exotic atoms could explain why Big Bang created more matter
Newstrack India
The lopsided nuclei, described today (May 8) in the journal Nature,
could be good candidates for researchers looking for new types of
physics beyond the reigning explanation for the bits of matter that
make up the universe (called the Standard Model), said study author
Peter Butler, a physicist at the University of Liverpool in the United
Kingdom.
The findings could help scientists search for physics beyond the
Standard model, said Witold Nazarewicz. An electric dipole moment
would provide a way to test extension theories to the Standard
Model, such as supersymmetry, which could help explain why there
is more matter than antimatter in the universe.
Until now, only one pear-shaped nucleus had been found experimentally:
radium 226, whose shape was sketched out back in 1993.
H.J. Wollersheim, H. Emling, H. Grein, R. Kulessa, R.S. Simon, C. Fleischmann,
J. de Boer, E. Hauber, C. Lauterbach, C. Schandera, P.A. Butler, T. Czosnyka;
Nucl. Phys. A 556, 261 (1993).
The Standard Model describes four fundamental forces or interactions
„If equal amounts of matter and antimatter were created at the Big Bang, everything would have been annihilated,
and there would be no galaxies, stars, planets or people“ (Tim Chupp Univ. Michigan)
I
II
III
Three Generations of Matter
Force Carriers
Gauge bosons
Leptons
Quarks
Elementary Particles
Shape parameterization
  

R ,    R0  1       Y  ,  
  0    

axially symmetric quadrupole
=2
20≠0, 2±1= 2±2= 0
axially symmetric octupole
=3
30≠0, 3±1,2,3 = 0
20≠0, 2±1,2 = 0
Octupole collectivity
Y30 coupling
(W.u.)
Δℓ=3 Δj=3
226
88


B E 3; 0  3 
Ra 138
9  Z 2  R6 2
 3
16   2
some subshells interact via the r3Y30 operator
R.H. Spear At. Data and Nucl. Data Tables 42 (1989), 55
e.g. in light actinide nuclei one has an interaction between
j15/2 and g9/2 neutron orbitals
i13/2 and f7/2 proton orbitals
Octupole collectivity
Octupole correlations enhanced at the
magic numbers: 34, 56, 88, 134
Microscopically …
Intruder orbitals of opposite parity and
ΔJ, ΔL = 3 close to Fermi level
226Ra close to Z=88 N=134
  f 7 / 2 i13/ 2   g9 / 2  j15 / 2 
Octupole collectivity
226Ra
+
+
+ 226Ra +
+
In an octupole deformed nucleus the center of mass
and center of charge tend to separate,
creating a non-zero electric dipole moment.
The double oscillator
V
 2  2 V0
2


H   




a
  E 
3
2  B  32 a 2
V0
-a
a
2V
1
2 V0  0
1


Eeven     even      
e
2
2
  


Eodd
2V
1
2 V0  0
1


    odd      
e
2
2
  










β3
β3
octupole deformed
octupole vibrational
Merzbacher ‘Quantum Mechanics’
Coulomb excitation
impact parameter
scattering angle
d i  f
dcm
 Pi  f 
2
d E 2
d Ruth
dcm

A 
A
 4.819 1  1   12  EMeV  BE 2; I i  I f  dfE 2  ,   b
 A2  Z 2
Scattered α-spectrum of
226Ra (t = 1600 yr)
1/2
0+
counts
→ 226Ra
Eα = 16 MeV
θlab = 1450
2+
3-
4He
4+
226Ra-target
beam direction
Si-detector

channel number
λ
 M  E  0
eb 
/2
Pi  f 
βλ (exp)
βλ (theo)
2
2.27 (3)
0.165 (2)
0.164
3
1.05 (5)
0.104 (5)
0.112
4
1.04 (7)
0.123 (8)
0.096
d i  f
d el
226Ra:

d i  f
d Ruth
t1/2= 1600 yr
Scattered α-spectrum of
0+
counts
4He
→ 226Ra
Eα = 16 MeV
θlab = 1450
2+
3-
226Ra
4+
226Ra-target
beam direction
Si-detector

channel number
λ
2
 M  E  0
2.27 (3)
eb 
/2
βλ (exp)
βλ (theo)
0.165 (2)
0.164
3
1.05 (5)
0.104 (5)
0.112
4
1.04 (7)
0.123 (8)
0.096
4+
If
E2
2+
In
E2
Ii
E4
0+
Experimental set-up


PPAC ring counter
150≤ΘL≤450, 00≤φL≤3600
4 PPAC counter
226Ra
226RaBr
2
530≤ΘL≤900, 00≤φL≤840
(400 μg/cm2) on C-backing (50 μg/cm2) and covered by Be (40 μg/cm2)
Coulomb excitation of 226Ra
γ-ray spectrum of
226Ra
208Pb
→ 226Ra
Elab = 4.7 AMeV
counts
150 ≤ θlab ≤ 450
00 ≤ φlab ≤ 3600
energy [keV]
γ-ray spectrum of
224Ra
224Ra
→ 60Ni / 120Sn
Elab = 2.83 AMeV
Signature of an octupole deformed nucleus
R   R0  1   2  Y20    3  Y30     4  Y40  
π: +
π: -
226
88
Ra
β2= 0.16
β3=-0.11
β4= 0.10
E2
Single rotational band with spin sequence:
I = 0 +, 1 - , 2 +, 3 - , …
excitation energy E ~ I·(I+1)
competition between intraband E2 and
interband E1 transitions
E1 transition strength 10-2 W.u.
E1
Signature of an octupole deformed nucleus
π: +
π: -
E2
224Ra
Observation of unexpectedly small E1 moments in 224Ra, R.J. Poynter, P.A. Butler et al., Phys. Lett. B232 (1989), 447
E1
Signature of an octupole deformed nucleus
Energy displacement δE between the
positive- and negative-parity states
if they form a single rotational band
E I  1  E I  1
 E I   E I  
2
2


for rigid rotor
2

W. Nazarewicz et al.; Nucl. Phys. A441 (1985) 420


Electric transition quadrupole moments in 226Ra
negative parity states
positive parity states
rigid rotor model:
I  2 M E 2  I 
15
I  I  1

 Q2  e
32 
2I  1
liquid drop:
Q2 
 
3  Z  R02
  2  0.36022  0.33632  0.32842  0.9672 4  fm2
5 
Q2(exp) = 750 fm2
β2 = 0.21
Q2(theo) = 680 fm2
H.J. Wollersheim et al.; Nucl. Phys. A556 (1993) 261
W. Nazarewicz et al.; Nucl. Phys. A467 (1987) 437
Static quadrupole moments in 226Ra
negative parity states
positive parity states
rigid rotor model:
QS I 

Q0
I M E 2  I
I  2 I  1

I  1 2 I  1 2 I  3 21 M E 2 01
rigid triaxial rotor model:
Qs 21 
6  cos3 

Q0
7  9  8  sin 2 3 
H.J. Wollersheim et al.; Nucl. Phys. A556 (1993) 261
Davydov and Filippov, Nucl. Phys. 8, 237 (1958)
Electric transition octupole moments in 226Ra
liquid drop:
 
β3
3  Z  R03
Q3 
 3  0.841 2 3  0.7693  4  fm3
7 
0.18
0.12
   0.10 MeV
0.06
I 7
0.1
β3 0
35
I  I  1 I  2
I  3 M E3 I  

Q e
2I  3 2I  3 3
32
21
I  1 M E3 I 

32
I 1 I  I  1  Q  e
2I  3 2I  3 3
- 0.1
   0.15 MeV
0.1
I  13 
I  15 
β3 0
I  13 
- 0.1
0.16
β2
H.J. Wollersheim et al.; Nucl. Phys. A556 (1993) 261
W. Nazarewicz et al.; Nucl. Phys. A467 (1987) 437
Electric transition octupole moments in 224Ra
Intrinsic electric dipole moments in 226Ra
liquid-drop contribution:
Q1LD  CLD  A  Z  2 3 1.458 34 
with CLD  5.2 104
 fm
rigid rotor model:
I  1 M E1 I  
H.J. Wollersheim et al.; Nucl. Phys. A556 (1993) 261
3
 I  Q1  e
4
G. Leander et al.; Nucl. Phys. A453 (1986) 58
Intrinsic electric dipole moments in 226Ra
liquid-drop contribution:
Q1LD  CLD  A  Z  2 3 1.458 34 
with CLD  5.2 104
 fm
224Ra:
Q1 (e fm)
Iπ
1989
2013
3-
0.027(3)
0.031(6)
5-
0.040(10)
0.028(9)
7-
< 0.05
< 0.08
G. Leander et al.; Nucl. Phys. A453 (1986) 58
Collective parameters



In general, R ,    R0  1       Y*  ,  
  


forbidden – density change!
λ=0
forbidden – CM moves!
λ=1
time
OK...
λ=2
λ=3
Center of mass conservation
  0  x  d 
 


0     o  y  d     0  r  d


   0  z  d 
0
 1 3



z
m

0


 2 

r  Y1m  ,    

3

 x  iy  m  1


2


The coordinates (x, y, z)
can be expressed by
4
  0  r  Y10  ,   d   r 3  dr  Y10  d
3

R04 
0
  1  4   *1m1Y1m1  6   *1m1 *2 m2 Y1m1Y 2 m2    Y10  d
4 
 1m1
 1m1 2 m2

0  4   6
*
10

 1m1 2 m2
*
 1m1

*
 2 m2
 2  1  2 2  1  3 
 1

4


3
 2  1  2 2  1  3 
     *1m1 *2m2   1

2 1m1 2 m2
4


1/ 2
*
10
1/ 2

  1
0

  1
0
2
0
2
0
1  1

0   m1
1  1

0   m1
2
m2
2
m2
1

0 
1

0 
The dipole coordinate is not an independent quantity. It is non-zero for nuclear shapes with both quadrupole
and octupole degrees of freedom.
3
9
3 12

 3   4
1  

  2  3 
4

4
63
35
3 5  7  3
1    
2  4 
1/ 2
2
 2 3 1
   2   3   3   2 
 
 0 0 0
 2 3 1
3

  
0
0
0
35


Intrinsic electric dipole moment
Q1LD  e   z   proton  d 
4
 e    p  r 3  dr  Y10  ,   d
3
The local volume polarization of electric charge can be derived from the requirement of a
minimum in the energy functional. (Myers Ann. of Phys. (1971))
 proton  neutron
1

 e VC r 
 proton  neutron
4CLD
2




3  r 
3 1  r 
 Ze
      Y 0  
VC r         
2 2  R0   1 2  1  R0 
R



 0
where ρp and ρn are the proton and neutron densities, CLD is the volume symmetry energy coefficient
of the liquid drop model and VC is the Coulomb potential generated by ρp inside the nucleus (r < R0)
p  
0
4  CLD
 e VC r    n
0   p  n
p 
0 

1
 1 
 e VC r 
2  4  CLD

2
3
 3 1  r  2  r 
 Ze
3  r 
3  r 








VC r            1  Y10       2  Y20       3  Y30  
5  R0 
7  R0 
 2 2  R0   R0 
 R0
Keeping the center of gravity fixed, the integral
0    0  r 3  dr  Y10  d
1  
3
9

  2  3
4
35
Intrinsic electric dipole moment
Q1LD 

 
4
1
 e   0  1 
 e VC r   r 3  dr  Y10  ,    d
3
2  4CLD

Q
2
2
3


4 3 3  A
1 Z
3  r 
3  r 
3 1  r   r 
 3









e








Y





Y





Y

1
10
2
20
3
30   r  dr  Y10  d
3









3
8    R0 4CLD R0
2 2  R0   R0 
5  R0 
7  R0 




Q1LD
 3 R04

2
1  4  1Y10   2Y20   3Y30   6  1Y10   2Y20   3Y30     Y10  d 


 2 44

 1 R0 1  6   Y   Y   Y   15  Y   Y   Y 2    Y  d
1 10
2 20
3 30
1 10
2 20
3 30
10
 2 6 


4 3 3  A
1
Z 
R04

e

 


1Y10  5  1Y10   2Y20   3Y30   1Y10   Y10  d

3

3
8    R0 4CLD R0 
5

3 R04


 2Y20  6  1Y10   2Y20  3Y30   2Y20   Y10  d




5 6
4


3 R0








Y

7


Y


Y


Y


Y



Y

d

3 30
1 10
2 20
3 30
3 30
10
7 7 


Q1LD
 3 R04

41  6   21 2Y10Y20  2 2  3Y20Y30   Y10  d 

 2 44

 1 R0 6  15 2  Y Y  2  Y Y   Y  d 
1
 1 2 10 20 2 3 20 30 10 
 2 6

4 3 3  A
1
Z 
R4

e

 
 0 1  5   1 2Y10Y20  Y10  d

3
3
8    R0 4C LD R0 
5

3 R04



6   1 2Y10Y20   2  3Y20Y30   Y10  d


5 6
4


3 R0



7    2  3Y20Y30  Y10  d
7 7


LD
1














Intrinsic electric dipole moment
2
57 3  2 3 1

 
Y
Y
Y

d


20
30
10

4  0 0 0 
Q1LD
3
4
3
35
 3

9 3 3
 2 3 
 1 
2 4 35
 2

5 3 3
 1

 2 1  2 4 35  2  3 


4 3 3  A  Z 
1


e 

 1

3
32    C LD 
5

3 3 3


  5 4 35  2  3 


 3 3 3  

2 3


7 4 35
Q1LD  e3 
3 A Z
60

  2  3
32   CLD 35 35
Q1LD  0.01245
e  A Z
  2  3
CLD
 fm
CLD ≈ 20 MeV
Summary
 single rotational band for I 10
 no backbending observed
 β2, β3, β4 deformation parameters are in excellent agreement
with calculated values
 octupole deformation is three times larger than in octupolevibrational nuclei
 equal transition quadrupole moments for positive- and negativeparity states
 static quadrupole moments are in excellent agreement with an
axially symmetric shape
 electric dipole moments are close to liquid-drop value ( I 10 )
 octupole deformation seems to be stabilized with increasing
rotational frequency
Coulomb excitation of 226Ra
226Ra
target broken after 8 hours
Christoph Fleischmann