General inequalities for the relaxation constants of a spin density matrix

Volume 4 1A, number 5

23 October 1972

PHYSICS LETTERS

GENERAL

INEQUALITIES

FOR THE RELAXATION

OF A SPIN DENSITY

CONSTANTS

MATRIX

MI. DYAKONOV and V.I. PEREL A.F. Ioffe Physico-Technical

Institute of the Academy of Sciences of the USSR, Leningrad,

USSR

Received 22 August 1972 General inequalities are found for the relaxation constants of a spin density matrix in the irreducible tensor representation. These inequalities must be satisfied independently of the nature of the relaxation mechanism.

The spin state of a particle (for example an atom or a nucleus) is often described in terms of polarisation moments, which are related to the spin density matrix in the following way ft=TrfiJ),

(1)

where ?” is the irreducible tensor operator of rank K. polarisation moments have a more direct physical meaning than the elements of the density matrix itself. Thus$ gives the total number of particles, the quantitiesfj andft describe orientation and alignment, respectively. For any isotropic relaxation process different polarisation moments change independently. This offers another important advantage of this representation. The relaxation of polarisation moments is described by simple equations

that this is not the case. Let the angular momentum j = 1 and the initial state be such, that only the level m = 1 is populated, so that fil = 1, while other elements of the density matrix are zero at t = 0. Then the following components of orientation and alignment exist

The

fb”=fil +f-l,_l-

2f()()

(4)

(3)

and ft = 1, f 2 = 1 at t = 0. It is easy to see that the 20 alignment f, can not be destroyed arbitrarily quickly compared to the orientation fi. Indeed, if we assume y2 B y1 then after an interval t such that ~2~ < t <-yi’, the quantity fi.will practically disappear, while f: will remain very close to unity. This will contradict, however, the normalisation condition fil + foe t f_l,_l = 1 and the requirement that the occupation numbers are positive. Thus it is clear that the constants TI and 72 must satisfy some inequality. This inequality may be easily derived from the expression for df_,,_,/dt at t = 0. Eq. (4) gives f_l,_l = &f;-3f;+2). Thus

where in general all elements of the matrix I’ are nonzero. The irreducible tensor representation was first applied to a description of spin relaxation in gases by the present authors [ 1,2] and by Omont [3] . The description of relaxation in terms of constants 7, is now generally accepted. For different K these constants characterize different physical processes and are measured in independent experiments [e.g. 4,511. The question arises can the relaxation constants TK be arbitrary? The following simpie example shows

Since f_1,_1 = 0 at t T 0, the right-hand side of eq. (5) should be non-negative. In such a way we obtain an inequality 3~~ > 72. Similarly the condition df,,/dt > 0 at t = 0 leads to the inequality r2 > 0. We may now derive inequalities for the constants nfKfor arbitrary values of the angular momentum j. Let us choose a representation in which the density matrix is diagonal at t = 0. It will remain diagonal at t > 0 if the relaxation process is isotropic. Then we

qf,ldt = - ,Y& ,

(2)

while the equations for the elements of the density matrix are much more complicated

df ’ 2$-L_

C mlmi

plm\ mm'

fmlmi

’

451

Volume

41A, number

PHYSICS

5

dfmm 1

r~~mm,fmlml

’

(61

It is physically obvious that in the process of relaxation all the occupation numbers fmm should remain non-negative. The necessary and sufficient condition for this is the following mm

form,fm.

Indeed, let us choose such an initial state that only one level m. is occupied. Then form #m. we have (df,mldt)t=o = - T’meme from where the necessity of inequalities (7) fClEws: They are also sufficient, since if at a certain moment one of the occupation numbers fmm e q uals zero, then its time derivative at this moment is, according to eqs. (6,7) non-negative. From the physical point of view, inequalities (7) are obvious. They require the terms in the balance equation (6) describing transitions to a given state nz from other states, to be non-negative. Using the relation between rmrmr and y [2] we arrive finally at a system of ine$%ities forKyK, which must be satisfied for any relaxation mechanism. c (2K+1)(-1)m-m’ K

For j = 1 fromula (8) reduces to the conditions

452

1972

derived above. For J’= 9 we obtain

--r~~fmm-m~m

Plml
23 October

3yr > y2 > 0

may rewrite eq. (3) in the form

at

LETTERS

IY1 - Y3I G6tYltY3-Y2).

(10)

In this particular case the inequalities for yK were obtained recently by Papp and Franz [6]. It was assumed above that in the process of relaxation the total population of the state considered remains unchanged (y. = 0). If this is not the case then in formulas (8- 10) y, should be substituted by YK--Yu. The general inequalities derived here should be taken into account when spi_n relaxation is describea phenomenologically in the framework of irreducible tensor representation.

References

111M.I.

Dyakonov and V.I. Perel, Zh. Eksp. i Teor. Fiz. 47 (1964) 1483;48 (1965) 345;Soviet Phys. JETP 20 (1965) 997; 21 (1965) 227. 121M.I. Dyakonov, Zh. Eksp. i Teor. t:iz. 47 (1964) 2213: Soviet Phys. JETP 20 (1965) 1484. [31 A. Omont, J. Phys. (Paris) 26 (1965) 26. I41 A. Omont, C.R. Acad. Sci. 260 (1965) 3331. I51 E.B. Saloman and W. Happer, Phys. Rev. 144 (1966) 7. 161 J.F. Papp and F.A. Franz, Phys. Rev. A5 (1972) 1763; there is a misprint in eq. (Bla) of this work.