Quasispin

The seniority quantum number can also have a group-theoretical interpretation. If we define 

Usually the Wigner-Eckart theorem (5.15) is utilized to find the dependence of the matrix elements on the projections of angular and spin momenta. Its use in the quasispin space 

It would be more logical to denote the right side of Eq. (9.25) in the form ocQLSMq) because Mq defines unambiguously the shell lN, and, therefore, there is no need to indicate lN. However, we retain this symbol as a traditional definition of the configuration. 

Using the notation of a term as 2S+qL instead of 2S+ L we can learn that 

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Clebsch-Gordan coefficients have already occurred several times in our considerations in the Introduction (formula (2)) while generalizing the quasispin concept for complex electronic configurations, while defining a relativistic wave function (formulas (2.15) and (2.16)), in the addition theorem of spherical functions (5.5) and in the definition of tensorial product of two tensors (5.12). Let us discuss briefly their definition and properties. There are a number of algebraic expressions for the Clebsch-Gordan coefficients [9, 11], but here we shall present only one ... 

As was mentioned in the previous paragraph, the Wigner-Eckart theorem (5.15) is fairly general, it is equally applicable for both approaches considered, for tensorial operators, acting in various spaces (see, for example, Chapters 15,17 and 18, concerning quasispin and isospin in the theory of an atom). 

Second-quantization in the Theory of an Atom. Quasispin and Isospin... 

Moreover, the second-quantization technique is closely linked with the quasispin method, which turned out to be fruitful in atomic theory, since it enables physicists to improve significantly the theory of the spectra of many-electron atoms and ions, and to make it far more universal, and more simple and convenient for practical use. 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

The methods of theoretical description of many-electron atoms on the basis of tensorial properties of the orbital and spin angular momenta are well established [14, 18] and enable the spectral characteristics of these systems to be effectively found. The relation between the seniority quantum number and quasispin makes it possible to extend the mathematical tools to include the quasispin space and to work out new modifications of the mathematical techniques in the theory of spectra of many-electron atoms that take due account of the tensorial properties of the quasispin operator. 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

Tables of numerical values of the CFP with two detached electrons are available, for example, in [14]. But, as we shall see later, the quasispin method allows us to do without tabulation of these quantities, since they are simply related to other standard quantities of the theory - the submatrix elements of tensors W kK i.e.

The relationships between various CFP and submatrix elements of unit tensors have been derived in a number of works [103, 105]. Their derivations drew specifically on commutation relations (14.45)-(14.50). An explanation of these results and their generalization, can be obtained within the framework of the quasispin method. 

The concept of quasispin quantum number was discussed in the Introduction and Chapter 9 (see formulas (9.22) and (9.23)). Now let us consider it in the framework of the second-quantization technique. We can introduce the following bilinear combinations of creation and annihilation operators obeying commutation relations (14.2) - the quasispin operator ... 

Here t may represent Is or j, whereas z-projection of the quasispin operator is determined by the difference between the particle number operator and the hole number operator in the pairing state (a, / )... 

Operator (15.36), thus, has eigenvalue +1/2 depending on whether or not the pairing state is occupied. If only one particle is in that state, then the eigenvalue of operator Qz is zero. It is worthwhile mentioning the fact that vacuum states, both for holes and for particles, are the components of a tensor in the quasispin space... 

By computing the commutators of the components of the quasispin operator with electron creation and annihilation operators, we can directly see that the latter behave as the components of a tensor of rank q = 1/2 in quasispin space and obey the relationship of the type (14.2)... 

Quasispin formalism can be used in describing the properties of the occupation number space for an arbitrary pairing state. For one shell of equivalent electrons, these pairing states can be chosen to be two one-particle states with the opposite values of angular momentum projections. 

The operator of total quasispin angular momentum of the shell can be obtained by the vectorial coupling of quasispin momenta of all the pairing states. For a shell of equivalent electrons, instead of (15.35) we have... 

According to (14.17), the z-projection of the quasispin operator is related to the operator of the particle number in the shell... 

Let us recall that the seniority quantum number, by definition, is the number of unpaired particles in a given state, and two electrons are called paired when their orbital and spin momenta are zero. Since it follows from definitions (15.39) and (15.40) that operators and Q operating on wave function lNaLS) respectively, create and annihilate two paired electrons, the seniority and quasispin quantum numbers v and Q must be somehow related. Let a certain iV-particle state have absent the paired electrons ... 

Consequently, the quantum numbers of quasispin Q and seniority v are related by expression (9.22) which is also valid for the wave function in the general case at N v. Operator Qj1, acting on wave function (15.44), increases the number of particles by two paired electrons, leaving, by definition, number v unchanged. 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an AT-electron wave function the operator a s) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

Tensors (15.52) in the quasispin method are the main operators. Their irreducible tensorial products are used to expand operators corresponding to physical quantities. That is why we shall take a closer look at their properties. Examining the internal structure of tensor Wyields... 

For triple tensors a we shall express the irreducible components of quasispin operator (15.39)—(15.41) as follows ... 

Connection of quasispin method with other group-theoretical... 

We noted in Chapter 6 that the seniority quantum number v in reduction chain (14.38) unambiguously classifies the irreducible representations of S P4i+2 group. Then one may well ask how can the earlier group-theoretical schemes include a rotation group defined by the operators of quasispin angular momentum ... 

Operator 1 ), by (15.56), is proportional to the quasispin operator, and operator W<0kK by (15.59), to double tensor with an odd sum of ranks k + k. These tensors, as shown in Chapter 14, are generators of the Sp4i+2 group, and so the above selection of generator subsets corresponds to the reduction of the Rsi+4 group on the direct product of two subgroups ... 

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