Effect of large neutron excess on the dipole response in the region of the Giant Dipole Res

a r X i v :n u c l -t h /9707026v 1 17 J u l 1997

E?ect of large neutron excess on the dipole

response

in the region of the Giant Dipole Resonance

F.Catara a ,d ,E.

https://www.sodocs.net/doc/1d10199748.html,nza a ,M.A.Nagarajan b ,1and A.Vitturi c

a

Dipartimento di Fisica and INFN,Catania,Italy

b Laboratori

Nazionali di Legnaro,INFN,Italy

c

Dipartimento di Fisica and INFN,Padova,Italy

d Departamento

de Fisica Atomica,Molecular y Nuclear,Sevilla,Spain

Currently,there has been interest in the study of the e?ect of neutron skin on collective states in neutron-rich nuclei [1–4].One of the collective modes of interest is the Isovector Giant Dipole resonance (GDR)in such nuclei.In neutron-rich nuclei,one expects the neutron and proton densities to have di?erent shapes.This would allow the possibility of exciting the isovector GDR by isoscalar probes through hadronic interactions [5–7],in addition to the usual method through Coulomb excitation [8].The excitation of the isovector GDR by isoscalar probes through hadronic interactions is possibly the most sensitive measure of the “neutron skin”.This property can be assessed by the evaluation of the isoscalar and isovector dipole transition densities in neutron-rich nuclei.

0.0 2.0

4.0 6.0r (fm)

10

?5

10

?4

10

?3

10

?2

10?1

10

ρ (n u c l e o n s /f m 3

)

28

O

0.0 2.0

4.0 6.08.0

r (fm)

60

Ca

Fig.1.Hartree-Fock proton and neutron densities obtained for 28O

and

60Ca

with

SGII interaction.The arrows indicate the r.m.s.radii.

In order to investigate the e?ect of neutron skin on collective dipole states of neutron-rich nuclei,we have performed microscopic calculations for several isotopes of oxygen and calcium nuclei,based on spherical Hartree-Fock (HF)method with Skyrme SGII interaction.The proton and neutron densities in 28

O and 60Ca are shown in Fig.1.The HF calculations predict the last neutron to be bound by 3.25MeV in 28O and 5.1MeV in 60Ca.These relatively large values of the neutron separation energy hinder the occurrence of “unusual”concentration of dipole strength at the continuum threshold,an e?ect that is directly associated with very small binding energy [3,9].Note that the use of other Skyrme interactions may give rather smaller neutron separation energy,leading to di?erent features in the very low energy part of the response.This should not alter the medium and high energy regions,which are the object of the present investigation.

The collective dipole excitations of these nuclei were calculated in RPA,using the full residual interaction.After obtaining the dipole states by the RPA cal-culation,we calculated the response to the isovector dipole operator

i z i τ3i .

Similarly we can calculate the isoscalar response to the operator

i z i ,which should vanish if the RPA states were proper eigenstates of a translationally invariant hamiltonian,together with the occurrence at zero excitation energy of the spurious state.Since the last neutrons in these nuclei are not too weakly bound,the continuum states needed for the RPA were expanded in oscilla-tor functions of di?erent principal quantum number,a procedure which was found to be adequate in these systems.In actual RPA calculations,because

0.010.0

20.030.0

E (MeV)

10

?1

100

101

102

isovector

10

?1

100

10

1

102

103

60

Ca, 1?

states

isoscalar

B (E 1) (e 2

f m 2

)

Fig.2.Isoscalar and isovector dipole response obtained in HF+RPA for 60Ca.Most of the spurious isoscalar response is concentrated in one low-lying state which can be easily eliminated.

of the truncation involved and the additional e?ect of the isospin mixing in-troduced by the Coulomb interaction,the spurious isoscalar mode occurs at very low excitation energy (carrying a large fraction of its strength)and there are insigni?cant spurious center-of-mass components at higher energies.The spurious state can thus be eliminated.

An example of the calculated B (E 1)’s for isoscalar and isovector dipole strength distributions is shown for 60Ca in Fig.2.The spurious state occurring at low excitation energy is marked with an arrow.

In view of the interest in the e?ect of neutron excess on the isovector dipole states,the RPA results of the di?erent oxygen and calcium isotopes are shown in Fig.3.One of the e?ects of the neutron excess is the spreading of the isovector dipole strength.It should be noted that in both the 28O and 60Ca cases,the most “collective”state only exhausts approximately 15%of the total EWSR.The increased spreading in the neutron rich isotopes is more clearly seen in Fig.4,where we have averaged the dipole response with a lorentzian with a width of 2MeV.In both cases the full width at half maximum

0.010.0

20.0E (MeV)

10

?1

100

10110260

Ca

10

?1100

10110210340

Ca

0.010.0

20.0E (MeV)

10

?1

100

101

102

16

O

0.010.0

20.030.0

E (MeV)

70

Ca

48

Ca

0.010.0

20.030.0

E (MeV)

28

O

B (E 1) (e 2

f m 2

)

B (E 1) (e 2

f m 2

)

Fig.3.Isovector dipole response obtained in HF+RPA for a sequence of Oxygen and Calcium isotopes.Spurious states have been eliminated (cf.Fig.2).

(FWHM)is seen to have increased by about 50%going from the N =Z to the most neutron rich isotope.In addition,one observes that the centroid of the strength function shifts to lower energy in the neutron rich isotopes.This shift is larger than can be accounted by the A ?1/6dependence given by the Goldhaber-Teller [10]model.For example the centroid changes from 19MeV in 40Ca to 16MeV in 60Ca,a shift which is a factor two larger than the prediction of the Goldhaber-Teller model.The shift is closer,although still slightly larger,to the prediction of the A ?1/3scaling associated with the hydrodynamical model [11].

Even though the isoscalar B (E 1)to all states must identically vanish,the corresponding isoscalar transition densities to the di?erent states need not to be identically zero.For example,within the collective Goldhaber-Teller model for the GDR,only in the particular case where the neutron and proton densi-ties have the same shape (and scale with N and Z )would the isoscalar dipole transition density vanish.If the neutron and proton transition densities have di?erent shapes,as is the case for very neutron rich nuclei,the corresponding isoscalar dipole transition density will be non zero.The isoscalar and isovector-dipole transition densities to a selected state in 28O and 60Ca are shown in Fig.5.The states selected have 18and 17.4MeV of excitation energy and carry 8and 16%of the EWSR,respectively.In these ?gures also are shown the separate neutron and proton transition densities.

0.0

10.0

20.030.0

E (MeV)

0.02.04.06.08.00.02.04.06.0

8.010.00.0

10.0

20.030.0

E (MeV)

0.01.02.03.0d B (E 1)/d E (e 2 f m 2

M e V ?1

)

0.0

10.0

20.030.0

40.0

E (MeV)

0.010.0

20.030.040.0

E (MeV)

d B (E 1)/d E (

e 2

f m 2 M e V ?1

)

16

O

28

O

40

Ca

48

Ca

60

Ca

70

Ca

Fig.4.Dipole response for 16,28O and 40?70Ca obtained from the RPA response shown in Fig.3,by averaging the discrete spectra with a lorentzian with Γ=2MeV.

We can compare the microscopic transition densities with those predicted by the macroscopic Goldhaber-Teller transition density

δρisoscalar GDR

(r )=δρp GDR +δρn

GDR =α1

2N

dr

?

2Z

dr

where α1,given by

α2

1

=

πˉh 2

NZE x

is the amplitude of the oscillation,derived from the dipole EWSR.E x is the excitation energy of the dipole state and it is assumed that the state exhausts the full EWSR.The feautures of this isoscalar transition density can be bet-ter evidenced by expanding it in the neutron excess parameter,according to Satchler [7],in the form

δρisoscalar GDR

(r )≈α1γ

N ?Z

dr

+

R 0

dr 2

where R 0=(R n +R p )/2is the average radius of the total nuclear density

ρ(r )and the parameter γis related by γ(N ?Z )/A =3/2(?R/R 0)to the measure of the neutron skin ?R =R n ?R p .The isoscalar transition density is therefore,to leading order,directly proportional to the neutron skin.The corresponding isovector transition density has a weaker dependence on

0.0

2.0 4.0

6.08.010.0

r (fm)

?0.20

?0.10

0.00

0.10

δρ ×€r 2

(n u c l e o n s /f m )

0.0

2.0

4.0 6.0

r (fm)

?0.15

?0.05

0.05

δρ ×€r 2

(n u c l e o n s /f m )

60

Ca

Fig.5.Transition densities to selected RPA states.Both isoscalar and isovector densities are shown,together with the separate proton and neutron contributions.The states selected in 28O and 60Ca have 18and 17.4MeV of excitation energy and carry 8and 16%of the EWSR,respectively.All densities are multiplied by r 2.

the neutron excess,to ?rst order in ?R/R 0and (N ?Z )/A ,given by

δρisovector GDR (r )=δρn GDR ?δρp

GDR

=?α1

2N

dr

+

2Z

dr

≈?α1

4NZ

dr

In Fig.6,we show the comparison between the microscopic transition densities

and the macroscopic ones for the state in 28O,already illustrated in Fig.5.The amplitude α1of the macroscopic model has been rescaled according to the proper percentage of the EWSR exhausted by the state.Both the

0.0

2.0

4.0 6.0

r (fm)

?0.15

?0.10?0.050.000.05

?0.20?0.15?0.10?0.05

0.000.0528

O , Dipole

δρ ×€r 2

(n u c l e o n s /f m )

https://www.sodocs.net/doc/1d10199748.html,parison of the microscopic proton and neutron transition densities and their isoscalar and isovector combinations (lower part)with the correspond-ing macroscopic expressions obtained within the Goldhaber-Teller model (upper part).The ?gure refers to the state in 28O,shown in Fig.5.We also show in the upper ?gure as curves with triangles and circles the approximated expression of Satchler for the isoscalar and isovector transition densities,respectively.In this ap-proximate expressions,a value ?R/R =0.186was used,taken from the HF density calculation.

“exact”Goldhaber-Teller expressions and the approximated forms suggested by Satchler are used (note that these latter practically coincide with the exact ones).One can see from the comparison that the relevant features of the microscopic RPA transition density are well reproduced by the collective GT model.

One can observe from Figs.5and 6the occurrence in the case of very neutron-rich systems of a node in the isoscalar transition density at the nuclear surface.Furthermore,at large radii both the isoscalar and isovector densities have similar radial dependence and magnitude.This is a re?ection of the fact that in this region it is only the tail of the neutron density that contributes and thus both isoscalar and isovector components contribute equally in this region.

0.0

10.0

20.030.040.0E (MeV)

0.0500.01000.01500.02000.02500.00.0100.0200.0300.0400.0

0.010.0

20.030.040.050.0

E (MeV)

40

Ca

48

Ca

d B (E 1)/d E (

e 2

f m 6 M e V ?1

)

60

Ca

70

Ca

Fig.7.Isoscalar dipole response to the RPA states obtained using the operator i z i r 2

i ,for a sequence of Calcium isotopes,after averaging the discrete spectra with a lorentzian with Γ=3MeV.

The fact that the isoscalar dipole transition density has di?erent sign at small and large radii has the consequence that the Coulomb-nuclear interference will be destructive at small radii and constructive at larger radii beyond the node.This has been pointed out by several authors [5–7]and this feature has been exploited as a speci?c tool in order to obtain a measure of the neutron skin [12,13].

Finally,we consider the possible compressional Isoscalar Dipole Resonance

(IDR)which is generated by the operator i z i r 2

i

.This operator is the leading non-spurious term in the expansion of j 1(qr )Y 10(?r )in the electromagnetic ?eld.The excitation of this isoscalar giant dipole resonance via inelastic scattering of αparticles has been investigated in ref.[14],and the model has been developed and compared to microscopic RPA calculations in ref.[15],who considered systems with di?erent masses in the stability region.In Fig.7,we present the evolution of the response for this operator in RPA with increasing number of neutrons,considering the sequence of isotopes 40,48,60,70Ca.It can be observed that there is an increased strength in the response as one moves to neutron-rich nuclei,approximately according to the ”standard”A 7/3scaling predicted by the EWSR.The neutron skin has in fact only a mild e?ect on the total EWSR,which is given by [14–16]

EW SR =

ˉh 2A

3

r 2 2

0.0

2.0 4.0

6.08.010.0

r (fm)

?0.15

?0.10?0.050.000.05?0.15

?0.10?0.050.000.050.10

p n p+n

IDR (high peak)

δρ ×€r 2

(n u c l e o n s /f m )

IDR (low peak)

Fig.8.Isoscalar transition densities to the selected states in 60Ca collecting a large fraction of the isoscalar dipole strength.The state in the upper part (E=31.3MeV)belongs to the high-energy peak of the strength distribution,the one in the lower part (E=9.4MeV)to the low-energy peak.The isoscalar transition densities are shown together with the separate proton and neutron components.

and which can be in leading order estimated to be

EW SR≈3ˉh2r40

A

?R

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随机过程作业

第三章 随机过程 A 简答题： 3-1 写出一维随机变量函数的均值、二维随机变量函数的联合概率密度（雅克比行列式）的定义式。 3-2 写出广义平稳（即宽平稳）随机过程的判断条件，写出各态历经随机过程的判断条件。 3-3 平稳随机过程的自相关函数有哪些性质功率谱密度有哪些性质自相关函数与功率谱密度之间有什么关系 3-4 高斯过程主要有哪些性质 3-5 随机过程通过线性系统时，输出与输入功率谱密度之间的关系如何 3-6 写出窄带随机过程的两种表达式。 3-7 窄带高斯过程的同相分量和正交分量的统计特性如何 3-8 窄带高斯过程的包络、正弦波加窄带高斯噪声的合成包络分别服从什么分布 3-9 写出高斯白噪声的功率谱密度和自相关函数的表达式，并分别解释“高斯”及“白”的含义。 3-10 写出带限高斯白噪声功率的计算式。 B 计算题： 一、补充习题 3-1 设()()cos(2)c y t x t f t πθ=?+，其中()x t 与θ统计独立，()x t 为0均值的平稳随机过程，自相关函数与功率谱密度分别为：(),()x x R P τω。 ①若θ在（0，2π）均匀分布，求y()t 的均值，自相关函数和功率谱密度。 ②若θ为常数，求y()t 的均值，自相关函数和功率谱密度。 3-2 已知()n t 是均值为0的白噪声，其双边功率谱密度为：0 ()2 N P ω= 双，通过下图()a 所示的相干解调器。图中窄带滤波器（中心频率为c ω）和低通滤波器的传递函数1()H ω及2()H ω示于图()b ，图()c 。

试求：①图中()i n t （窄带噪声）、()p n t 及0()n t 的噪声功率谱。 ②给出0()n t 的噪声自相关函数及其噪声功率值。 3-3 设()i n t 为窄带高斯平稳随机过程，其均值为0，方差为2 n σ，信号[cos ()]c i A t n t ω+经过下图所示电路后输出为()y t ，()()()y t u t v t =+，其中()u t 是与cos c A t ω对应的函数，()v t 是与()i n t 对应的输出。假设()c n t 及()s n t 的带宽等于低通滤波器的通频带。 求()u t 和()v t 的平均功率之比。

自相关函数与偏自相关函数

自相关函数与偏自相关函数 上一节介绍了随机过程的几种模型。实际中单凭对时间序列的观察很难确定其属于哪一种模型，而自相关函数和偏自相关函数是分析随机过程和识别模型的有力工具。 1、自相关函数定义 在给出自相关函数定义之前先介绍自协方差函数概念。由第一节知随机过程{t x }中的每一个元素t x ，t = 1, 2, … 都是随机变量。对于平稳的随机过程，其期望为常数，用μ表示，即 ()t E x μ=，1,2,t =L 随机过程的取值将以 μ 为中心上下变动。平稳随机过程的方差也是一个常量 2()t x Var x σ=，1,2,t =L 2x σ用来度量随机过程取值对其均值μ的离散程度。 相隔k 期的两个随机变量t x 与t k x -的协方差即滞后k 期的自协方差，定义为： (,)[()()]k t t k t t k Cov x x E x x γμμ--==-- 自协方差序列：k γ，0,1,2,k =L 称为随机过程{t x }的自协方差函数。当k = 0 时，2 0()t x Var x γσ==。 自相关系数定义：k ρ= 因为对于一个平稳过程有：2 ()()t t k x Var x Var x σ-== 所以2 20 (,) t t k k k k x x Cov x x γγρσσγ-= = =，当 k = 0 时，有01ρ=。 以滞后期k 为变量的自相关系数列k ρ（0,1,2,k =L ）称为自相关函数。因为k k ρρ-=，即(,)t k t Cov x x -= (,)t t k Cov x x +，自相关函数是零对称的，所以实际研究中只给出自相关函数的正半部分即可。

概率统计讲课稿第十二章第五节平稳过程的相关函数与谱密度

第十二章 第五节 平稳过程的相关函数与 谱密度 一、 相关函数的性质 平稳过程)(t X 的自相关函数 )(τX R 是仅依赖于参数间距τ的函数。它有如下性质： 性质1 )(τX R 是偶函数， 即)(τ-X R )(τX R =； （事实上)]()([)(ττ+=t X t X E R X ， )]()([)(ττ-=-t X t X E R X )()]()([ττττX R t X t X E =+--= ） 性质2 2 )0(|)(|X X X R R ψ=≤τ ， 2 )0(|)(|X X X C C στ=≤， 就是说，自相关函数)(τX R 和自协方差函数 )(τX C 都在 0=τ 处达到最大值。事实上 （利用不等式|)(|XY E 2 1 2 2 1 2 ] [][EY EX ?≤） | )]()([||)(|ττ+=t X t X E R X )0()]([)]([21 2 2 1 2 X R t EX t EX =+≤τ，

| ))]()(())()([(||)(|τττ+-+?-=t EX t X t EX t X E C X 2 1 2 2 1 2 ] ))()(([]))()(([ττ+-+?-≤t EX t X E t EX t X E 2 )0(X X C σ== 。 性质3 )(τX R 非负定。即对任意实数n τττ,,,21Λ和任意函数)(τg 有 0)()()(1 ,≥-∑=j i j i n j i X g g R ττττ 。 事实上 )()()(1,j i j i n j i X g g R ττττ-∑= )()()]()([1 ,j i j i n j i g g X X E ττττ∑== 0)]()([21 ≥=∑=i i n i g X E ττ。 性质4 如果)(t X 是以T 为周期的周期平稳过程，即满足 )()(t X T t X =+，那么，)(τX R 也是以 T 为周期的函数。 事实上 )]()([)(T t X t X E T R X ++=+ττ

第十二章 平稳随机过程

第十二章平稳随机过程 平稳随机过程是一类应用相当广泛的随机过程．本章在介绍平稳过程概念之后，着重在二阶矩过程的范围内讨论平稳过程的各态历经性、相关函数的性质以及功率谱密度函数和它的性质． §1 平稳随机过程的概念 在实际中，有相当多的随机过程，不仅它现在的状态，而且它过去的状态，都对未来状态的发生有着很强的影响．有这样重要的一类随机过程，即所谓平稳随机过程，它的特点是：过程的统计特性不随时间的推移而变化．严格地说，如果对于任意的 和任意实数A，当时，n维随机变量 具有相同的分布函数，则称随机过程具有平稳性，并同时称此过程为平稳随机过程，或简称平稳过程． 平稳过程的参数集T，一般为 ．当定义在离散参数集上时，也称过程为平稳随机序列或平稳时间序列．以下若无特殊声明，均认为参数集． 在实际问题中，确定过程的分布函敷，并用它来判定其平稳性，一般是很难办到的．但是，对于一个被研究的随机过程，如果前后的环境和主要条件都不随时间的推移而变化，则一般就可以认为是平稳的． ．376． 恒温条件下的热噪声电压过程以及第十章§1例2、例3都是平稳过程的例子．强震阶段的地震波幅、船舶的颠簸过程、照明电网中电压的波动过程以及各种噪声和干扰等等在工程上都被认为是平稳的． 与平稳过程相反的是非平稳过程．一般，随机过程处于过渡阶段时总是非平稳的．例如，飞机控制在高度为丸的水平面上飞行，由于受到大气湍流的影响，实际飞行高度H（他）应在A水平面上下随机波动，H（他）可看作是平稳过程，但论及的时间范围必须排除飞机的升降阶段(过渡阶段)，因为在升降阶段内由于飞行的主要条件随时间而发生变化，因而H(t)的主要特征也随时间而变化着，也就是说在升降阶段内过程II(t)是非平稳的．不过在实际问题中，当仅仅考虑过程的平稳阶段时，为了数学处理的方便，我们通常把平稳阶段的时间范围取为一oo<他<+oo．