On the Generalized Bloch Precession Equations

The Bloch equations, which describe spin precession and relaxation in external magnetic fields, can be generalized to include the evolution of polarization tensors of various ranks in arbitrary multipole fields. The derivation of these generalized Bloch equations can be considerably simplified by using a particular bra-ket notation for irreducible tensors.


Hans-Jürgen Stöckmann
Fachbereich Physik der Philipps-Universität Marburg, Renthof  The Bloch equations, which describe spin precession and relaxation in external magnetic fields, can be generalized to include the evolution of polarization tensors of various ranks in arbitrary multipole fields. The derivation of these generalized Bloch equations can be considerably simplified by using a particular bra-ket notation for irreducible tensors.
with the Larmor angular frequency vector L γ = B ω ω ω ω . This equation holds for arbitrary angular momentum quantum number 1 2 j ≥ , and is all one needs for a complete quantum description of this simple system. Equation (2) is called semiclassical only because the external field B is not quantized, i.e., there is no back action of the spins on the magnetic field.
While Eqs. (1) and (2) were originally derived in the context of magnetic resonance, it was soon recognized that they describe the evolution of any system with "effective" or "pseudospin" J, the best known examples being the maser [2], or the two atomic states involved in an optical transition [3], which obey the "optical Bloch equations", both systems having effective spin-½. Even multiple-quantum transitions are covered by these equations.
The Bloch equations can be generalized to include the evolution of the higher multipole moments of spin polarization in the presence of arbitrary multipole fields. Starting point is the Liouville equation for the density matrix ρ . For a given angular momentum state j, its rank is 2 1 The Liouville equation (3) may be rewritten in terms of a generalized spin precession equation with the statistical tensors L ρ , also called state multipoles, with 2 1 L + polarization coefficients Ω . These tensors will be defined below, as will be their products and the coefficients c j . For the special case that there are only interactions with uniform magnetic fields, Eqs. (4) reduce to Eq. (2). Eqs. (4) had been derived by Fano [4] in an important paper, which regrettably never found the recognition it deserves. A possible explanation for this may lay in the fact that it uses an unfamiliar notation, and that the derivation is technical and tends to obscure the essential underlying ideas.
The purpose of the present letter is to show that the derivation of Eq. (4) from Eq. (3) can be considerably simplified and becomes nearly trivial by using a special bra-ket notation for the irreducible tensors. With this notation Eq. (4) will be obtained by a number of simple steps involving essentially the Wigner-Eckart theorem and nothing else.
The same procedure then is applied to derive the equivalent of Eqs. (1), namely the generalized Bloch equations, which, under conditions to be discussed later on, read where | jmLM jm′ 〈 〉 is a Clebsch-Gordon coefficient.
The rank of the ρ L is limited to 2 L j ≤ , as follows from the triangular rule for the Clebsch-Gordon coefficients. The rank-one polarization vector 1 ρ is often called orientation or simply polarization, and the rank-two tensor 2 ρ alignment. In Eq. (4), the tensor products A main use of the statistical tensor elements LM ρ is in the angular distribution or correlation of particles emitted in nuclear, atomic, or molecular reactions, see for instance ch. 19 of [5]. These distributions can be written in terms of the elements of the rotation matrices, for instance, for the case of a rotationally symmetric configuration, as 0 ( ) (cos ) with Legendre polynomials P L and some coefficients r L . Equations (3) and (4) are mathematically equivalent, but the spin precession equation has a number of obvious advantages as compared to the original Liouville equation: (i) The LM ρ allow a direct interpretation in terms of polarization, alignment etc. Very often just one tensor component is prepared in the production of the ensemble, all other initial components being zero. The spin precession equation is ideally suited to cope with this situation.
(ii) The tensor products allow an easy visualization of what is going on. From the selection rules of the Clebsch-Gordon coefficients and the prefactors c j (see Eqs. (28) and (29) below) it is obvious, e.g., that a magnetic interaction can only change the M state of a tensor component, but never its rank L, whereas a quadrupole interaction always changes the rank by 1 ± . (iii) If there are symmetries, e. g., rotation symmetries about the axis of an external magnetic field, the generalized spin precession equation systems for 2 (2 1) j + variables will decompose into smaller uncoupled differential equations.
To derive the generalized precession equations we first recall the definition of an irreducible tensor operator via its commutation relations with the elements L z and L ± of the orbital angular momentum operator The close similarity between this definition of the tensor operators T LM and the well known relations for the eigenstates | lm〉 , can be made even more suggestive by introducing a special notation for the tensor operators, see also [6]. We define tensor bras and kets (using round brackets) At the same time, we assume that an operator A acts on the tensor bras and kets as With these definitions, Eqs. (9) and (10) become identical even with respect to notation. One only has to replace the bras and kets in Eqs. (10) by the corresponding tensor bras and kets, i.e., Eqs. (12) with ± = z A L ,L , to recover Eqs. (9) for the tensor operators.
We then define the tensor matrix element of an operator A as † The last identity holds because of the commutativity property of the trace. It is thus irrelevant whether operator A acts on the bra or on the ket. The matrix elements of the product AB of two operators A and B follow from the completeness relation for the irreducible tensor operators as as can be seen by writing down explicitly all matrix elements from Eq.
The T LM are orthogonal with respect to the trace operation, † 2 Tr( ) Tr |T | where 2 Tr |T | L is independent of M as is indicated by the notation. The proof follows exactly the same line as the corresponding proof for the orthogonality of the eigenfunctions | lm〉 which can be found in any textbook. Throughout this paper we assume that the T LM are normalized, i. e., 2 Tr |T | 1 L = . For 0 L = to 2 the normalized tensors are given by The matrix elements of the T LM in the | jm〉 basis are easily calculated by means of the Wigner-Eckart theorem, where the reduced matrix element shows up to be T 2 1 L j || || j L 〈 〉 = + , as an immediate consequence of the orthogonality relation for the Clebsch-Gordon coefficients.
We are now prepared to derive the generalized spinprecession equation (4) We then expand also the Hamiltonian H in terms of normalized irreducible tensor operators, The interaction frequencies therein are, for the mainly occurring magnetic dipole and electric quadrupole interactions, 10   Next we have to evaluate the tensor matrix element brackets. Because of the one-to-one correspondence of (9) and (10), each proof for the tensor functions is automatically true for the tensor operators, too, and vice versa. In particular, we can directly apply the Wigner-Eckart theorem to calculate the matrix elements with tensor bra-kets By a consequent use of the normalized irreducible tensors and the round-bracket tensor matrix elements, the derivation of the generalized spin precession equation (4) has been reduced to a small number of elementary steps. Only the Wigner-Eckardt theorem has been used in the derivation. Since this is a universal theorem holding in all symmetry groups, the generalized spin precession equation, too, is applicable in all groups. The individual group structures only enter in the calculation of the reduced matrix element in Eq. (28). It is only here where some computational effort is needed [4], see also Appendix A.3 of [7]. If this is done, one obtains (2 1)(2 1)(2 1) In the literature, statistical tensors are generally employed without using Fano's compact and suggestive tensor product expression (4). For the purely magnetic case, magnetic resonance line shapes of the LM ρ were treated in [8], including the line shapes of atomic double-resonance signals as known from [9]. The Hanle effect as based on atomic alignment was studied in [10]. For further experiments on the reorientation of atomic state multipoles, see [11,12], and references therein.
Let us now turn to the discussion of the relaxation term in Eq. (5). The relaxation of the LM ρ was first treated in [13], and independently in [14]. For more recent uses of the statistical tensors in magnetic resonance and relaxation, see [15][16][17], and the review [18]. In relaxation studies, the tensor bra-ket notation exhibits its full power. It is the usual situation in every experiment that there is a system Hamiltonian H S , which is under control of the experimentalist, the environment described by a bath Hamiltonian H B , which usually cannot be controlled, and a Hamiltonian H SB coupling the system to the environment. Removing It is beyond the scope of this letter to go into the details of relaxation theory. Instead we just follow the standard approach and treat relaxation in second order perturbation theory. To this end we integrate both sides of Eq. (31) over t, where in addition we relabelled the indices, and substitute this expression for ˆ( ) L M t ρ ′ ′ on the right hand side of Eq. (31), where in the second step the completeness of the irreducible tensor operators has been used. In addition we changed the integration variable from τ to t − τ.
To avoid a possible misinterpretation of the equation it is reminded of the operator multiplication convention (15). Equation (34) is still exact, but to proceed further we have to apply approximations: (i) First we assume that 1 ( ) H t varies rapidly with time as compared to ˆ( ) L M t ρ ′ ′ . Then we may average the right hand side over these rapid fluctuations by replacing 1 ( ) H t and ρ on the right hand side by t, and extend the upper limit of the integration to ∞. (iv) Finally, for the sake of simplicity let us restrict the sum over (L′M′) to just one term (LM). We then obtain Going back to the laboratory system by inverting transformation (30) we obtain the generalized spin precession equation (33) including the relaxation term. We see that on the level of the applied approximations, each tensor component decays exponentially with its own decay constant, but this is no longer true for more evolved relaxation theories, in particular if fluctuating quadrupole interactions are involved [14]. For tensor rank L = 1 we recover Bloch's equation, where we can identify τ 10 and τ 1±1 with T 1 and T 2 , respectively. We do not recover, however, a decay towards a thermal equilibrium in this way, since all tensor components at the very end decay to zero. A better approach repairing this deficiency would take into account the Boltzmann polarization of the bath, which is not difficult to do, but not of relevance in the present context. To get explicit expressions for the relaxation rates from Eq. (36) for various situations found in experiments still some effort is needed [19].
In conclusion, using a special bra-ket notation, the derivation of the generalized Bloch precession equations from the Liouville equation has been reduced to three to four nearly trivial steps, both for the precession and the relaxation term. The key ingredients were the introduction of the operator tensor matrix elements, Eq. (14), and the reinterpretation of the statistical tensors ρ LM in terms of expectation values of the normalized irreducible tensor, Eq. (21). Since nothing but the Wigner-Eckardt theorem (27) was used in the derivation, Eq. (4) is valid for arbitrary groups. Only in the calculation of reduced matrix elements, Eq. (29), or the explicit evaluation of the relaxation rates from Eq. (36), the individual group structures enter. Though it is only a special notation we used in this letter, we consider the advantages of the present approach as so convincing that the technique still should be of use for all people working in the field, in spite of the fact that nearly half a century has passed since Fano's original work.