MONTE CARLO ALGORITHMS FOR MOMENTS OF TRANSITION ARRAYS

HAL Id: jpa-00227435

https://hal.archives-ouvertes.fr/jpa-00227435

Submitted on 1 Jan 1988

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

MONTE CARLO ALGORITHMS FOR MOMENTS OF TRANSITION ARRAYS

A. Goldberg, S. Bloom

To cite this version:

A. Goldberg, S. Bloom. MONTE CARLO ALGORITHMS FOR MOMENTS OF TRANSITION ARRAYS. Journal de Physique Colloques, 1988, 49 (C1), pp.C1-79-C1-81. �10.1051/jphyscol:1988115�.

�jpa-00227435�

JOURNAL D E PHYSIQUE

Colloque C1, Suppl6ment au n 0 3 , Tome 49, Mars 1988

MONTE CARL0 ALGORITHMS FOR MOMENTS OF TRANSITION ARRAYS(')

A. GOLD BERG'^) and S.D. BLOOM

Lawrence Livermore National Laboratory, Livermore, C A 94550, U.S.A.

ABSTRACT

Closed expressions for the first, second, and (in some cases) the third moment of atomic transition arrays now exist. Recently a method has been developed for getting to very high moments (up to the 12th and beyond) in cases where a "collective" state-vector (i.e. a state-vector containing the entire electric dipole strength) can be created from each eigenstate in the parent configuration. Both of these approaches give exact results. Herein we describe a statistical (or Monte Carlo) approach which requires only

one

representative state-vector J R D for the entire parent manifold to get estimates of transition moments of high order. The representation is achieved through the random amplitudes associated with each basis vector making up IRD. This also gives rise to the dispersion characterizing the method, which has been applied to a system (in the M shell) with2250,OOO lines where we have calculated up to the 5th moment. It turns out that the dispersion in the moments decreases with the size of the manifold, making its application to very big systems statistically advantageous. A discussion of the method and these dispersion characteristics will be presented.

The use of transition moments for characterizing strength distributions is a constantly recurring theme in various fields,

including e.g. atomic physics and nuclear physics, amongst others. Here we describe a Monte Carlo method which is particularly apt for getting moments of that special class of strength distributions called transition arrays in atomic physics. In this paper we offer no proofs; these can be found in Ref.1. The best application of the Monte Carlo method is in highly complex cases consisting of many lines, say in the hundreds of thousands or more. Indeed the accuracy of the method improves markedly with the size of the problem, as we shall see.

Analytic expressions for first, second, and (in some cases) third moments of transition arrays were derived a few years ago.2 Most recently we have described a method for calculating exactly very high moments

(12th and higher)3 which is based on what we call the collective state- vector for the particular multipole transition of interest. We shall concentrate our attention here on El (electric dipole) transitions and so we shall denote this particular collective state-vector as

ICE^>.

(We emphasize however that our methods work equally well for all multipoles.) The Monte Carlo method is also based on the formation of a collective state-vector, though in a different context, as we shall see later. We now define

ICE^>,

CE1>

-

^(El)

¹

^p>

p> -= the "parent" state vector, (El) = the El operator.

"'work performed under t h e a u s p i c e s o f t h e US Dept. o f Energy by LLNL under c o n t r a c t number W-7405-ENG-48

( 2 ' ~ o w deceased

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988115

C1-80 JOURNAL

DE

PHYSIQUE

The (El) operator as defined here is a sum over all projection states. In the method of Ref. 3 Ip> was a particular eigenvector, as, e. g. , the ground state-vector, so that

I

CE1> would contain the entire

(El) strength pertaining to that particular Ip>. More explicitly this means the total strength St,, is given by,

S,,, = <CE1

I I

^CE1>. ( 4 )

In the Monte Carlo method, which is the principal interest here, Ip>

is replaced by a random sum over the set of all basis vectors Clbj>3 in the parent vector space. It can then be shown1 that this random vector, for which we use the symbol IRV>, is representative of the entire model space, in the statistical sense, for the purpose of calculating many properties, including, of course, transition moments. Thus one can calculate to a certain accuracy (see below) the properties of the whole transition array by studying the properties of a single state-vector, i.e. the collective state-vector. To make this more concise let us note that if the Ip> state-vector were represented by the expansion,

then the coefficients a, would, of course, be determined by the Hamiltonian such that Ip> would be the ground state, £of example. The expansion for the random vector would formally look much the same,

except now the set of coefficients

Cp,3

are random, i.e. totally

uncorrelated, except for the normalization condition <RV)

)

RV-1. We now form from IRV> the collective vector

I

CRE1> exactly as we formed

I

CE1>

from Ip>,

except now the quantity <CRElI ICREl> is a statistical average of the total strength summed over the eigen-states in the parent

configuration. It is thus subject to a statistical error as are all the quantities, derived from (RV> and ICREl>, which we discuss below.

Herein our goal is the evaluation of the q'th transition moment

<TY,

defined as follows,

-

N = the total El strength =

klp(p,,)

^(2Ja+1),

where (W,,E,) denotes the eigenenergies of the (parent,daughter) manifolds, denotes the unpolarized dipole strength for the a+i

(absorptive) transition, J, is the total spin of the parent state (in units of h), and (2J,+1) is the multip1icit)i of the parent state

Ips>

Using the binomial expansion one can reexpress Eq.7a as follows,

where (91 is the binomial coefficient. Note the double sums in E ~ S . 7(ayb) and Eq. 8c, which require, in general, knowledge of all the eigen-vectors in both the parent and daughter spaces. To get the corresponding equation to Eq.8c in the Monte Carlo approach we will require only two state-vectors, IRV> and

I

CRE~>. The statistical approximation to P(p,n) we call M(p,n), for which the equation is,

and the statistical approximation to <T% is given by,

where we see from Eq. 9a that I(o,o)=<cRE~~ ~ c R E ~ > , the statistical approximation to the total strength.

The actual evaluation of transition moments such as Eq.9a is much facilitated by the use of the methods of second quantisation, which indeed was the approach of Refs. 1 and 3. We now present some of these results in the following table wherein we compare the exact answers

(89>, q-1-5, from Ref.3) for one case (case MA, where M stands for M shell) with the Monte Carlo answers, W > . The dimensions were (630;6,211) for the (parent,daughter) model spaces and the number of lines was 5523. The corresponding numbers for the second case, MB, were (3780;44,048) and -250,000, much too large for the method of Ref.3.

(Neither case is descriptive of any particular M-shell atom.) All moments are centroidal except for the first moment, q=l. The dispersions (for

<W>) are based on 10 IRV>'s and show that the accuracy of the Monte

Carlo method varies inversely with the square root of the dimension of the parent space, roughly, as was noted earlier.

Moment order, q 1 2 3 4 5

MA, -Sq> 195.45 36.99 266.3 8.27x103 1.37x105

MA, UIP> 195.47 36.95 258.3 8.19x103 1.29x105

MA, dispersion .02% 1.5% 3.4% 2.7% 4.2%

MB, W> 198.89 54.53 213.86 1.14x104 2.39x105

MB, dispersion .01% .45% 1.2% 1.3% 2.3%

IS. D. Bloom and A. Goldberg, Phys. Rev. A, to be published.

2 ~ . Bauche-Arnoult, J . Bauche, and M. Klapisch, Phys. Rev. 30A, 3026(1984).

3 ~ . D. Bloom and A. Goldberg, Phys. Rev. 34A, 2865(1986).