Preamble

Double shape quantum phase transitions in the SU3-IBM: new -soft phase and the shape phase transition from the new -soft phase to the prolate shape De-hao Zhao, Ying-xin Wu, Li Gong, Ze-yu Yin, Xiao-shen Kang, and Tao Wang 1 School of Physics, Liaoning University, Shenyang 110036, People’s Republic of China School of Innovation and Entrepreneurship, Liaoning University, Shenyang 110036, People’s Republic of China College of Physics, Tonghua Normal University, Tonghua 134000, People’s Republic of China Shape quantum phase transition is an important and hot topic in nuclear structure. In this paper, we begin to study the finite- shape quantum phase transition in the SU3-IBM. In this new proposed model, spherical- -soft spectra was found to resolve the spherical nucleus puzzle, which is a new -soft rotational mode.

In this paper, the shape phase transition along the new -soft line is first discussed, and then the neighbouring case at the prolate side is also studied.

Some key quantities are discussed. We find that double shape phase transitions occur along a single parameter path. The new -softness is really a shape phase and the shape phase transition from the new -soft phase to the prolate shape is found. The experimental supports are also found Pd may be the critical nucleus.

Keywords

SU3-IBM, shape quantum phase transition, new spherical-like -soft spectra,

INTRODUCTION

Phase transitions are widely found in nature [ ]. A com- mon example is that, under standard atmospheric pressure, when the temperature rises, the ice becomes water and then water vapor. If the atmospheric pressure is raised to a certain level, the water and water vapor cannot be distinguished. In the field of atomic nuclei, nuclear shape can change when the number of the protons or neutrons varies, and shape quantum phase transition can occur [ ]. Since this control parame-

ter is discrete and finite, it becomes even more interesting to 10

identify these phase transitions [ The interacting boson model (IBM) was proposed by Arima and Iachello [ ], which is an influential algebraic model for describing the collective behaviors of nucleons. In

the simplest case, only the s ( L = 0 ) and d ( L = 2 ) bosons 15

are considered, and the Hamiltonian has the U(6) symmetry. 16

There are four dynamical symmetry limits (see Fig. 1 FIGURE:1 left): (1) the U(5) symmetry limit can present the spherical shape and its vibration; (2) the SU(3) symmetry limit can describe the prolate shape and its rotation; (3) the O(6) symmetry limit can describe the -soft rotation; and (4) the SU(3) symmetry limit can present the oblate shape and its rotation [ This simple model can describe the shape phase transitions between the spherical shape to various quadrupole deforma- tions or among different deformed shapes (see Fig. 1(a) left) ]. In these studies, along a single parameter path, the shape of the nucleus changes only from one to another. Af- ter 2000, an important class of shape phase transition has attracted attentions and created controversies, which is the prolate-oblate shape phase transition [ ]. In previous IBM, the prolate-oblate shape phase transition is described via changing from the SU(3) symmetry limit to the SU(3) Supported by the National Natural Science Foundation of China (No.12575087) symmetry limit, and the O(6) symmetry limit is just the first- order phase transitional critical point [ ], which implies that the O(6) -softness is not a shape phase. In this description, the spectra of the prolate and oblate shapes are the same, and they are not found in realistic nuclei. In [ ], for realistic nu-

clei in the Hf-Hg region, the energy ratio E 4 / 2 = E 4 + 1 /E 2 + 1 38

of the states is 3.33 for the prolate shape while 2.55 for the oblate shape ( is not related to the boson number ). Thus this mirror symmetry appears not to exist.

Recently , an extension of the interacting boson model with SU(3) higher-order interactions (SU3-IBM) was proposed ], which incorporates the idea of previous IBM and the SU(3) correspondence of the rigid triaxial shape [ In this new model, the role of the SU(3) symmetry is raised to a new level, dominating all the quadrupole deformations of nuclei (see Fig. 1(a) right). It contains only the U(5) sym- metry limit and the SU(3) symmetry limit. In the SU(3) sym- metry limit, higher-order interactions are needed. The SU(3) second-order Casimir operator SU(3) can present the prolate shape while the SU(3) third-order Casimir operator SU(3) can describe the oblate shape, which is very dif- ferent from the SU(3) description in previous IBM. The two interactions, together with the square of the SU(3) second- order Casimir operator SU(3) , can describe any rigid tri- axial shapes.

Recently, this construction of the SU3-IBM is found to be strongly supported by the experimental results of the large-deformed nuclei U and ], which found that the large-deformed nuclei previously thought to be prolate are in fact with small rigid triaxiality. This conclusion was proposed by Otsuka et al. ]. Thus rigid triaxiality plays a more important role than previ- ously expected. Moreover in previous IBM, the simple model with up to second-order interactions can not de- scribe the rigid triaxial rotor [ ]. In the simplest IBM- 1 without distinguishing protons and neutrons, if con- sidering the rigid triaxiality, the higher-order interac- tions should be contained. However the 6- interaction proposed in [ ] can not describe the

while (a) right represents the phase diagram of the in the SU3- IBM [ ]. In (b) the real blue and real red lines are two evolutional paths discussed in this paper. Along the real red line, the double shape phase transitions can occur.

small rigid triaxiality. SU(3) symmetry can provide a uni- 72

fied description for any rigid triaxiality [ ], thus the SU3-IBM seems to be very reasonable. Recently the small rigid triaxialities in Sm and Er have been realized by the SU3-IBM successfully [ The SU3-IBM can be used to explain the B(E2) anomaly [ ] with higher-order interactions [ to resolve the Cd puzzle [ with the newly proposed new spherical-like -soft spectra describe the prolate-oblate asymmetric shape phase transi- tion [ to describe -softness in Pt at a better level [ to describe the E(5)-like spectra in ], and to verify the boson number odd-even effect in which was observed in ]. Together these results reveal that, the SU3-IBM can better describe the collective behav- iors in nuclei.

Thus investigating the shape phase transition in the SU3- IBM is also important. In the SU3-IBM, for resolving the Cd puzzle in Cd nuclei and other nuclei previously thought to be spherical [ ], the spherical-like -soft cleus was proposed [ ], which is a new collective excita- tion and has been verified in realistic nuclei recently [ Cd and the Pd all have the spherical-like spectra.

This new shape was not mentioned by previous nu- clear theories. Moreover the prolate-oblate asymmetric shape phase transition in the Hf-Hg region can be better described by the SU3-IBM [ and recently the large-deformed nuclei have been verified to have small rigid triaxiality These unexpected new studies imply that real- istic nuclei can show more complex shape phase transition behaviors, which may be described by the SU3-IBM. So it becomes even more important to study the characteristics of shape phase transitions in the SU3-IBM to help us understand the shape phase transition of actual nuclei.

This is the first paper on this topic, and the simplest for- malism is discussed [ ], whose large- limit has been clarified [ ], see Fig. 1(a) right.

In the SU3-IBM, the ex- istence of the spherical-like -soft spectra is the most important feature . We study the shape phase transitions from this new collectivity. This has been first discussed in [ ] for

small boson number N = 7 . In this paper, we discuss them 112

with N = 60 for the ground state and N = 35 for the excited 113

states. This evolutional path can be represented by the real blue line in Fig. 1(b) from the U(5) symmetry limit to the SU(3) degenerate point. In this paper, the nearby evolutional path (denoted by the real red line) is also discussed, and we find that, along the real red , double shape quantum phase transitions can occur. The first is from the spherical shape to the new -soft rotation, and the second is from the new soft mode to the prolate shape. Here the new -softness is really a shape phase, which is different from the O(6) critical -softness.

The experimental results supporting this new shape phase transition are also shown, and it is found that 108 Pd may be the critical nucleus.

These results look more

realistic and very meaningful. 126

HAMILTONIAN

The simplest Hamiltonian for describing the shape phase 128

transition related to the new spherical-like -softness in the SU3-IBM is as follows [ SU(3) SU(3) where are two controlling parameters and is the en-

ergy scale parameter. 0 ≤ η ≤ 1 and κ ≥ 0 . If η = 0 , it 133

presents the spherical shape. If η = 1 and κ = 0 , it describe 134

the prolate shape. The two cases are the same as the ones

in previous IBM [ 16 ]. If η = 1 and κ varies, this Hamilto- 136

nian describes the prolate-oblate shape phase transition [ 62 ], 137

which is a finite- N effect. 138

the large- limit [ ]. Above the real green line, the spheri- cal shape exists, and under the real green line, the deformed

shapes exist. The deformed region is divided by the real blue line which is part of the connected line between the U(5) symmetry limit to the O(6) symmetry limit. The left part of the blue line presents the prolate shape while the right part presents the oblate shape. The blue line is the critical line between the prolate and oblate shapes. The crossover point of the green line and the blue line is the triple point with the spherical, prolate and oblate shapes [ ]. Obviously, the -softness not a shape phase.

A detailed study on the shape phase transitions in previous IBM can be seen Up to two-body interactions, the IBM can not de- scribe the rigid triaxial shapes [ The phase diagram of in the large- limit was first dis- cussed in [ ], see Fig. 1(a) right (or see Fig. 14 [FIGURE:14] in [ The key difference between the SU3-IBM and previous IBM is to use the SU(3) third-order Casimir operator SU(3) instead of the SU(3) symmetry limit to describe the oblate nuclei. Above the real green line, the spherical shape exists.

Under the real green line, the deformed shapes exist. For the deformed region, the phase diagram becomes more compli- cated than the ones in previous IBM [ ]. From the SU(3) prolate to the SU(3) oblate, there exists a SU(3) degenerate point (the blue point). At the SU(3) prolate side of the degen- erate point, the shape of the ground state is always the prolate shape, and at the SU(3) oblate side of the degenerate point, it is always oblate. Thus across the SU(3) degenerate point, the shape changes abruptly.

For small the prolate shape can exist near the new spherical-like -soft phase. In pre- vious IBM, this is impossible for small (this will be dis-

cussed in next paper [ 70 ]). This finite- N first-order shape 171

phase phase transition was studied in [ ]. Connected with the SU(3) degenerate point, in the middle of the deformed re- gion, there exists a narrow region with rigid triaxial shapes, see the region between the two blue lines in Fig. 1(a) right.

At the left side of the narrow region, the prolate shape exists, while at the right side, the oblate shape exists. Thus along the dashed blue line in Fig. 1(a) right, the shape changes from the prolate to the oblate via a narrow region with rigid triaxiality ]. Although the rigid triaxial region is narrow, the rigid triaxial shape really exists. The crossover point between the green line and the two blue lines is a fourfold point.

Ref. [ 23 ] found an important result. For finite- N , the 183

rigid triaxial shape becomes the new spherical-like -soft ro- tational mode.

The rigid triaxiality can only exist in the large- limit.

This new -softness is a shape phase and not a critical phenomenon. Along the dashed blue line in Fig. 1(a) right, the shape changes from the prolate to the new soft, and then to the oblate, which was used to describe the prolate-oblate asymmetric shape phase transition in the Hf- Hg region [ ], the real blue line in Fig. 1(b) is a critical line be- tween the prolate shape and the new -soft rotation in the de- formed region, along which the states are degenerate states are also degenerate. However for large- , this critical line is actually a curve and curves to the left. Thus for large- , the critical line can not be de- scribed by the real blue line [ ]. At the right side of the degenerate line, there exists another line, along which the states are also degenerate [ ]. Between the two de- generate lines of the states, it was supposed that the spherical-like -softness exists, however in this paper it is shown that this new spherical-like -soft region may be larger, which is unexpected. And importantly we find that double shape quantum phase transition can be observed along a single parameter path.

In this paper, the shape phase transition along the real blue line is studied, and not stress the degenerate line, which is dif-

ficult to discuss. The SU(3) degenerate point is at κ 0 = 3 N 2 N +3 209

and the red point is at . Through previous analysis, the red point presents the prolate shape. This is very interest- ing. For the real red line from the U(5) symmetry limit to the red point, intuitively, the shape changes from the spherical shape to the prolate. Through later numerical calculations, we find that this shape transition is not direct but through the spherical-like -soft region, and the double shape quan- tum phase transition can occur. The right part of the real blue line is not discussed in this paper, and will be studied in next paper for investigating the scope of the new spherical-like -soft region.

The differences between the O(6) -softness and the new spherical-like -softness should be clarified here. For the two -soft rotation, the B(E2) values are very similar ], so in [ ], this new spherical-like -softness was mis- understood as the previous O(6) -softness. (A detailed comparison on the B(E2) values will be discussed in fu-

ture.) The significant differences can be found in the en- 227

ergy spectra of the two -softness [ ]. For the O(6) -softness, the two states are nearly degenerate while for the new -softness, the three states are nearly degenerate. This feature is similar to the ones in the spherical spectra, so this new -softness is called spherical-like. There are also different degeneracies in higher energy levels. For the O(6) -softness, the four states are nearly degenerate while for the new -softness, the four states are nearly de- generate. In the spherical spectra, the five states are nearly degenerate, so in the new -soft spectra, there is no state near the four degenerate states, which is the most important feature.

It is important to emphasize here why the O(6) -softness in previous IBM and actual -soft nuclei do not match. In actual nuclei, there exists many nuclei in the -soft region, such as Os, Pt, Xe, Ba nuclei [ ], so it is hard to believe that this is just the O(6)-softness in previous IBM, a critical point of shape phase transition. In the SU3-IBM, such a concep- tual conflict does not exist. We believe that the shape phase transition given by the SU3-IBM is an accurate description of the realistic shape phase transitions in nuclei. This point has been preliminarily confirmed by the prolate-oblate asymmet- ric shape phase transition in the Hf-Hg region [ For understanding the B(E2) anomaly, the values are necessary. The operator is defined as

ˆ T ( E 2) = q ˆ Q, (2) 254

where is the boson effective charge. The evolutions of

values are discussed. DOUBLE SHAPE QUANTUM PHASE TRANSITION Ground state

Shape quantum phase transition is first manifested by the 260

energy evolution of the ground state. This is not an observable quantity, but very useful for understanding the shape quan- tum phase transition. Fig. 2 FIGURE:2 shows the energy evolution of the ground state of along the real blue line in Fig. 1(b)

for N = 10 (dashed blue line) and for N = 60 (real blue 265

line). Clearly, around η = 0 . 2 (denoted by the left dashed 266

line), the shape phase transition from the spherical to the new

spherical-like γ -soft occurs. Between η = 0 . 2 and η = 1 , 268

the new spherical-like γ -softness exists. η = 0 . 5 is the typ- 269

ical position to study the Cd and Pd nuclei. It should be noted that the SU(3) degenerate point is not -soft.

state of ˆ H along the real red line in Fig. 1(b) for N = 10 273

(dashed red line) and for N = 60 (real red line). For N = 60 , 274

it is shown that, around η = 0 . 5 (denoted by the middle 275

dashed line), a new shape phase transition appears. The part of the real red line deviating from the real blue line is promi-

nent, which has a steeper descent. Between η = 0 and 278

η = 0 . 2 , there is the spherical shape, and between η = 0 . 5 279

and η = 1 , there is the prolate shape. Obviously, between the 280

two shapes, the new spherical-like -softness exists. When changes from , the new spherical-like -soft region reduces, but it does exist, see Fig. 1(b) In next paper ], the shape phase transitions in the whole parameter regions will be discussed, it can show that, when reduces to 0, the new spherical-like -soft region will re- duce to zero too, see Fig. 1(b). At the left side of the blue

line, for finite- N , there also exists a new spherical-like γ - 288

soft region.

A key point is that, between η = 0 . 2 and η = 0 . 5 , the red 290

and blue lines are nearly degenerate, so when changes from , the energies of the new spherical-like -softness are nearly the same and do not reduce. This implies that, new spherical-like -softness is really a shape phase Thus along the left real red line, double shape quantum phase transition can occur. This can not occur for the shape phase transition along the real blue line and in previous IBM. should be noted that despite the parameter variation from is small, the shape phase transitions along the

blue line and the red line exhibit significant changes when 300

η > 0 . 5 . The reason is that, even at finite- N , the prolate 301

shape is near the new spherical-like -soft region.

In this paper, when increases to 35, the critical value

of η is kept at 0.5 when κ = 0 . 9 κ 0 . In the large- N limit, 304

κ 0 = 1 . 5 , the critical line can curve to the left, and κ ≈ 305

1 . 32 when η = 0 . 5 , which is nearly 0.88 κ 0 [ 61 ]. Thus I 306

can guess that, this critical point η = 0 . 5 when κ = 0 . 9 κ 0 307

should remain unchanged when changes from small to

infinity, which will be also discussed in next paper [ 70 ]. 309

along the real blue (red) line in Fig. 1(b) for (dashed blue (red) line) and for (real blue (red) line). ( ) The evolution of the ground state of along the real blue (red) line in Fig. 1(b) for (dashed blue (red) line) and for (real blue (red) line).

Some details need further elaboration. For N = 10 , the 310

real blue line in Fig. 1(b) is nearly the critical line with the

4 + 1 , 2 + 2 degeneracy . For N = 35 , the critical line curves 312

to the left, so the new shape phase transition becomes more prominent.

In previous IBM, similar result of the ground energy evo- lution along the real blue line can be also obtained along the evolution path from the U(5) symmetry limit to the O(6) sym- metry limit. If the O(6) symmetry limit is the prolate-oblate critical point, the connected line in the deformed region be- tween the U(5) symmetry limit and the O(6) symmetry limit is a prolate-oblate critical line. When deviating from the crit- ical line, the energies of the deformed region reduce, large

change is impossible for finite- N , and the double shape 323

phase transitions can not be observed [ This is a fun-

damental difference between the SU3-IBM and previous IBM, which can not be clearly observed in the large- limit.

The mean value of the boson number in the ground state is also important. Fig. 2(b) shows the evolution along

the real blue line in Fig. 1(b) for N = 10 (dashed blue line) 330

and N = 60 (real blue line). The phase transition behaviors 331

from the spherical to the new spherical-like γ -soft across η = 332

is clear. Fig. 2(b) also shows the evolution along the

real red line in Fig. 1(b) for N = 10 (dashed red line) and 334

N = 60 (real red line). The double shape quantum phase 335

transitions are also clear. When η > 0 . 5 , the two red lines are 336

deviated from the two blue lines obviously. And importantly

between η = 0 . 2 and η = 0 . 5 the red and blue lines are nearly 338

degenerate for N = 10 and N = 60 . The new spherical-like 339

-soft phase really exists.

The ¯ γ evolution of the ground state of ˆ H along the real blue (red) line in Fig. 1(b) for N = 10 (dashed blue (red) line) and for N = 60 (real blue (red) line).

The mean value of the deformation parameter in the ground state is also studied, which can be in fact ob- served by experiments now [ ]. For any SU(3) irre- ducible representation , the corresponding value

is γ = tan − 1 √

. The value is the average of the value of the whole possible irreducible representations ]. Fig. 3 [FIGURE:3] depicts the evolution along the real blue line

in Fig. 1(b) for N = 10 (dashed blue line) and N = 60 348

(real blue line). The phase transition behaviors from the

spherical to the new γ -soft across η = 0 . 2 is clear. Fig. 350 [FIGURE:350]

3 also depicts the evolution along the real red line in

N = 10 (dashed red line) and N = 60 352

(real red line). The double shape quantum phase tran- sitions are also clear.

An important discovery is that, the two evolutional trends are opposite for small

η > 0 . 2 . If κ = κ 0 , the ¯ γ value increases slightly while if 356

κ = 0 . 9 κ 0 , the ¯ γ value decreases prominently. We expect 357

these unique trends can be verified in future experiments 358

] with Cd and Excited states Now we discuss some other observable quantities, such as the excited energies, the B(E2) values, and the electric quadrupole moment of the state. Previous discussions can help us confirm that the double shape quantum phase transi- tions do exist. These observable quantities can further us find them experimentally. along the real blue line in Fig. 1(b); (b) The evolutional behaviors of the (black line), (blue line), (red line), (green line) along the real blue line in Fig. 1(b).

We first study the shape phase transition along the real blue line in Fig. 1(b) from the U(5) symmetry limit to the SU(3) degenerate point. This study has been performed in [ ] for

N = 7 . Here the evolutional behaviors of the partial low- 370

lying states for N = 10 are shown in Fig. 4 FIGURE:4. The 4 + 1 and 371

2 + 2 states are nearly degenerate. η = 0 . 5 presents the typi- 372

spherical-like -soft spectra, in which the energy of the state is nearly twice the one of the state. The

spherical-like -soft spectra was confirmed in Pd recently ]. Thus the shape phase transition discussed in this pa- per can be found in Pd nuclei. Fig. 4(b) shows the evolu- tional behaviors of the B(E2) values of the along the real blue line in Fig. 1(b). The results are similar to the evolutions from the U(5) symmetry limit to the O(6) symmetry limit [ along the real blue line in Fig. 1(b); (b) The evolutional behaviors of the (black line), (blue line), (red line), (green line) along the real blue line in Fig. 1(b).

tial low-lying states for N = 35 along the real blue line in 384

tion from the spherical shape to the new -soft rotation be-

comes more prominent. When η > 0 . 2 , the 4 + 1 and 2 + 2 states 387

begin to separate because the degenerate line curves to the left. Besides, the level-anticrossing of the states becomes more clear [ ]. Fig. 5 FIGURE:5 shows the evolu- tional behaviors of the B(E2) values of the along the real blue line in Fig. 1(b). The shape phase tran- sition becomes more clear. When increases, the values of becomes smaller. along the real red line in Fig. 1(b); (b) The evolutional behaviors of the (black line), (blue line), (red line), (green line) along the real red line in Fig. 1(b).

Now we discuss the double shape phase transitions along the real red line in Fig. 1(b). Fig. 6 FIGURE:6 presents the evolutional

behaviors of the partial low-lying states for N = 10 . Between 398

the η = 0 . 2 and η = 0 . 5 , the 4 + 1 and 2 + 2 are nearly degener- 399

ate, and the spectra are also similar to the spherical-like -soft spectra, so this is the new spherical-like -soft phase.

When η > 0 . 5 , obviously it is the prolate shape. In Fig. 402 [FIGURE:402]

7(a) the case of N = 35 is shown and the shape phase transi- 403

tion from the new spherical-like -soft rotation to the prolate

shape becomes very prominent. Thus κ = 0 . 9 κ 0 and η = 0 . 5 405

is the phase transition critical point.

along the real red line in Fig. 1(b); (b) The evolutional behaviors of the (black line), (blue line), (red line), (green line) along the real red line in Fig. 1(b).

In previous IBM, the O(6) -softness is the shape phase transition critical point from the prolate shape to the oblate shape, so it is not a shape phase. There is no shape phase transition from the O(6) symmetry limit ( -soft rotation) to the SU(3) symmetry limit (prolate shape). In the SU3-IBM, the new -softness is a shape phase and the shape phase tran- sition from the new spherical-like -soft phase to the prolate shape really exists. the B(E2) values of the . The double shape phase transitions are also clear. Across

η = 0 . 5 , the value of B ( E 2; 0 + 2 → 2 + 1 ) increases first, and 419

then decreases, which can not be found in Fig. 4(b) and Fig. 5(b).

We look for some experimental evidences for the existence (a) The evolution along the real blue (red) line in (dashed blue (red) line) and for (real blue (red) line); (b) Different evolutional trends of the values of Cd and of the shape phase transition from the new spherical-like -soft phase to the prolate shape.

In previous IBM, the

E 4 / 2 = E 4 + 1 /E 2 + 1 value is often used to discuss various 425

shape phase transitions [ ]. In the U(5) symmetry limit, this value is 2. In the SU(3) symmetry limit, it is 10/3 while in the O(6) symmetry limit, it is 2.5. Fig. 8 FIGURE:8 displays the evolutional behaviors along the real blue (red) line

in Fig. 1(b) for N = 10 and N = 35 . For small N , when 430

κ = κ 0 , this value can increase first and then decrease, 431

which is very interesting. When κ = 0 . 9 κ 0 , this value 432

only increases. Thus the SU3-IBM can show two differ- ent evolutional trends within a parameter interval, see the two black circles. Fig. 8(b) shows the different evolutional trends of the experimental values of Cd and Pd, which can be verified by the theory. Due to the deviation of the critical line with the degen- eracy, this signature seems to be sensitive to . For the

(a) The evolution along the real blue (red) line in (dashed blue (red) line) and for (real blue (red) line); (b) Different evolutional trends of the values of Cd and Pd nuclei, across Pd, the degree of increasing becomes larger, so Pd may be the critical nucleus.

The staggering parameter in γ band energies J (4) = 442

is also a useful signature. are the energies in band while is the energy of the first state in the ground band. The energy evolutions of the state are shown in Fig. 4(a)- 7(a), which is nearly degenerate with the state and the signature of the shape phase transition is not clear. Fig. 9 FIGURE:9 displays the evolutional behaviors along the real

blue (red) line in Fig. 1(b) for N = 10 and N = 35 . 450

We find prominent feature. If κ = κ 0 , the J (4) value de- 451

creases first and then increases. At this turning point, if 452

κ = 0 . 9 κ 0 , the J (4) value still decreases, see the two black 453

circles. Fig. 9(b) presents the experimental evolu- tions of Cd and Pd. It is prominent that

the turning point is at 112 Cd, and at this point, it corre- 456

sponds to Pd. Due to the deviation of the critical line,

this turning point seems to be sensitive to N . The SU3- 458

IBM can reproduce the turning feature, which can prove 459

the validity of our idea. (a) The evolution along the real blue (red) line in (dashed blue (red) line) and for (real blue (red) line); (b) Evolutional trends of the experimental values of Cd and is also a shape phase transition signature usually used. Fig. 10 FIGURE:10 shows the evolutional behaviors along the real blue

(red) line in Fig. 1(b) for N = 10 and N = 35 . In the de- 464

formed region η > 0 . 2 , the phase transition is not so clear, 465

which is around 1.5. Fig. 10(b) presents the experimen- evolutions of Cd and Pd, which is also around 0.5. Due to large errors, this signature is also unclear.

In this paper, is also discussed. Fig. 11 FIGURE:11 shows the evolu- tional behaviors along the real blue (red) line in Fig. 1(b)

for N = 10 and N = 35 . In the deformed region around 473

(a) The evolution along the real blue (red) line in (dashed blue (red) line) and for (real blue (red) line); (b) Evolutional trends of the experimental values of Cd and

η = 0 . 5 , they show decreasing behaviors. Fig. 11(b) 474

presents the experimental evolutions of Pd, which increases first and then deceases at 108 Pd. Thus the evolutions of in realistic nuclei can not be reproduced by the simple theory. This may be due to the lack of the SU(3) higher-order interactions, which can affect the new spherical-like -soft rotation [ In the previous analysis, we see that, from the new spherical-like -soft phase to the prolate shape, the value of increases first, and then decreases. Here we define evolutional behaviors along the

real blue (red) line in Fig. 1(b) for N = 10 and N = 35 . 486

When κ = 0 . 9 κ 0 , it clearly shows a phase transition sig- 487

nature near η = 0 . 5 that increases first and then decreases. 488

, this feature can not appear. presents the experimental evolution of Pd and (a) The evolution along the real blue (red) line in (dashed blue (red) line) and for (real blue (red) line) ; (b) Evolutional trends of the experimental values of obviously it is clearly in line with the theoretical prediction.

Pd are two typical new -soft nuclei [ ] and 108 Pd may be a critical nucleus from the new spherical-like

γ -soft phase to the prolate shape. However for finite- N , the 494

spherical-like -soft spectra and the critical spectra may be not distinguished [ ]. A detailed discussions on the prop- erties of Pd will be given in future. ], the evidence confirming the existence of a spherical-like -soft spectra was found, which is the anoma- lous evolutional trend of the electric quadrupole moments of the first states in Cd nuclei. Now we further study this interesting phenomenon. Fig. 13 FIGURE:13 shows the evolutional be- haviors of the values along the real blue (red) line in Fig.

1(b) for N = 7 , N = 10 and N = 35 . Along the real red line 504

in Fig. 1(b), the double shape phase transitions can be clearly observed. The key result is the different evolutional trends of the two parameter paths. The blue one is anomalous. When

evolution along the real blue (red) line in (dotted blue (red) line), (dashed blue (red) line) and (real blue (red) line); (b) Different evolutional trends of the experimental values of increases, the value evolves to the oblate side. The red

line is just opposite. When N increases, its magnitude in- 509

creases too if η ≥ 0 . 5 . These different evolutional trends 510

are unique [ 60 ]. These trends are similar to the ones of the 511

value. experimental values of Cd and For Cd nuclei, the magnitude of the value decreases

slightly while for Pd nuclei the magnitude of the value 515

increases prominently, which are nearly the same as the

ones in theoretical results within 0 . 2 < η < 0 . 6 .

This 517

feature was first observed in [ ], and can be regarded as the strong support for the existence of the new spherical-like -soft spectra. The discussions in this paper further support this conclusion and is an indirect experimental support for the existence of the shape phase transition.

It should be noted that this double shape quantum phase transitions cannot be verified directly. In the last two decades, the experimental discovery that nuclei previously thought to be spherical cannot be confirmed to be spherical is a break- through in the field of nuclear structure [ ]. If the spherical nucleus is absent, it is difficult to confirm the shape phase transition from the spherical shape to the new -soft phase.

In our discussion, we also found that the spherical nucleus

and the critical nucleus at η = 0 . 2 are also difficult to be dis- 531

tinguished, which can help us further discuss those nuclei that look like spherical.

DISCUSSIONS state in the new spherical-like -soft spectra is similar to the one in the spherical spectra, so the crit- ical feature can be also found in the shape phase transi- tion from the spherical shape to the prolate shape [ However this feature can be only found in realistic nuclei with large deformation, which is unexpected to be found in Pd nuclei. For Pd nuclei, in previous IBM, if the shape phase transition exists, it should be described by the shape phase transition from the spherical shape to the O(6) soft rotation, such as the descriptions in Xe, Ba nuclei.

In previous nuclear structure researches, no one believed that Pd nuclei would be related with the prolate shape.

Thus, in previous opinions, this feature is impossible in 547

Pd nuclei. In [ ], an opposite trend can be found. This feature found in realistic Pd nuclei is very important for the verification of the new shape phase transition and the SU3-IBM.

In this paper, the shape phase transition of the simplest SU3-IBM is discussed, thus only the most robust signa- tures can be observed. In [ ], it revealed that the other SU(3) higher-order interactions are also needed for a better fitting effect. These interactions are small for Cd, Pd nuclei, but they can also affect some quantities greatly.

Although there are some deficiencies in details, the dis- cussions in this paper are still very important, and they give the main evolutional ways. In previous studies on the transition from the prolate shape to the oblate shape [

the same simple hamiltonian was employed. Despite its 562

simplicity, it still clearly revealed the shape asymmetry phenomenon that could not be explained by earlier IBM models, including the IBM-2 that distinguishes protons and neutrons.

The phase diagram of the IBM-2 has been discussed in ]. In this extended model, the -softness is still the O(6) style, the spectra of the prolate shape and the oblate shape are still the same, and small rigid triaxiality can not be realized. In this model, different from the IBM-1,

the rigid triaxiality with γ = 30 ◦ can be obtained, and 572

the phase transition from the prolate shape to this rigid

triaxial shape with γ = 30 ◦ can be studied. 574

The main reason for the nonexistence of the phonon ex- citations in Cd nuclei results from the impossibility of ex-

plaining the new spectra with the IBM-2 [ 72 ]. In [ 60 ], 577

it was found that the spectra in Pd can be better ex- plained by the SU3-IBM, rather than the IBM-2. The key feature that there is no the state near the four states can not be described by the IBM-2, which is the fundamental reason. Thus the shape phase transi- tion from the new spherical-like -soft phase to the pro- late shape can not be realized by the IBM-2 with up to two-body interactions except for the introduction of the higher-order interactions.

In previous IBM, the Pd nuclei are usually discussed with the shape phase transition from the U(5) symme- try limit to the O(6) symmetry limit, and Pd is found to be a critical nucleus [ ]. In previous IBM, -soft nucleus can not be related to the prolate shape.

Thus it is very interesting to compare the results in this paper withe the critical symmetry in Pd. (This will be done in future) Moreover, Ref. [ ] found that Pd may be also the critical nucleus, thus the may be all the nuclei [ ]. This conclusion seems unrealistic. However this seems to fit the double quantum phase transition discussed in this pa- Pd are the new spherical-like -soft nuclei, and 108 Pd is the critical nucleus from the new spherical-like -soft phase to the prolate shape. And interestingly the 102 Pd may be the critical nucleus from the spherical shape to the new spherical-like -soft phase, which will be inves- tigated in future. If so, our theory will be further sup- ported.

In this paper, some critical quantities are discussed, such as . When fitting other SU(3) higher-order interactions are needed, and these quantities can fit well quantitatively. Especially the quadrupole moments of the states H. E. Stanley, Introduction to Phase Transitions and Critical

Phenomena (Oxford University Press, Oxford, 1971) 642

S. Sachdev, Quantum Phase Transitions, second edition (Cam-

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the quadrupole moment of the first state in Pd is smaller than the ones in adjacent nuclei Pd and which can not be explained by the simple SU3-IBM dis- cussed here. We expect that it can be resolved in future.

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CONCLUSION

Based on the existence of the new spherical -like -soft spectra [ ], we further discuss the related shape quantum phase transition. In this paper, we have drawn some new conclusions. First, the new spherical-like -softness is a shape phase, which is very different from previous O(6)- softness as a critical point. Then, we find the double quan- tum phase transitions along a single parameter path. We con- firm that there is indeed a shape phase transition from the new spherical-like -soft phase to the prolate shape, and we find experimental evidences that Pd may be a critical nu- cleus, which will be studied in detail later. In next paper [

the scope of the new γ -soft region in the SU3-IBM for finite- 639

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