Nuclear polarization by electron paramagnetic resonance in paramagnetic crystals

NUCLEAR POLARIZATION BY ELECTRON PARAMAGNETIC RESONANCE IN PARAMAGNETIC CRYSTALS

MILAN ODEHNAL Inst i tute oJ Nuclear Research, Czechosl. Acad. Sci., Prague

The possibility o] the polarization oJ nuclei in paramagnetic salts by saturation o] electron 2oaramagnetic resonance is theoretically analyzed. The proposed method assumes saturation o] the "]orbidden" transition o] the type A M = 4- 1, A m = 4- 1, ~= 2, ]or mutual ly perpendicular external magnetic and high-Jrequeney ]ields. The analysis is carried out for the case oJ a large quadrupole moment o] the nucleus. The degree o] orientation attained is comparable in order o] magnitude with O v e r h a u s e r ' s method. This method is particularly suitable ]or the polarization o] nuclei o] transuranium elements.

H O J I f [ P H 3 A I I H H H ~ E P H P H H O M O I ~ H H A P A M A F H H T H O F O P E 3 O H A H C A B I I A P A M A F H H T H L I X H P H C T A J I J I A X

B pa6ome cOe~an meopemuuec~u~ paa6op eo3aoo, cuoemu no~apu3a~uu :~Oep e napa~aenum- ~b~x co.~ax nyme.~ lcaeb~u4e~u:~ napct~ae~cumtcoeo aaenmpouuo~o pe3ouauca. [[peO~oo~ce~m,~(~ aemoO npeOnoaaeaem nacbru~euue ,,3anpeu4eu~oeo" nepexoOa muna A M ~ 4- 1, A m = = -4- 1 , :J:2 npu eaau.~no nepneuOu~y~:~pnb~x e~ezuuea nocmoa~r u eb~eo~couaemom~oa ~aeuurnubrX no.~ax. 06ey~mOe~cue cOe~a~o c9.~:~ c.~yuaa 60.~bzuOeo ~r aoaenma ~Opa. /~ocmu:>tcu~taa cmene~tb no~puaa~tuu no nopaO~y eez~uuu~tb~ cpae~tu~a c pe~y~bmama,~tu ~uemoOa O e e p x a y e e p a . tlpeO~o~ceuTt~ft ~emoO oco6eu~o yOo6e~ 0 ~ non~pu3a~4uu nOep mpa~cypattoe~x a~eme~moe.

I. I N T R O D U C T I O N

Physicists have recently shown great interest in oriented nuclei. Apart from a series of static methods of nuclear orientation [1] Several dynamic methods have been proposed, using electron-nuclear spin interaction. The latter methods use electron or nuclear paramagnetic resonance. Their advantage lies in the fact that they do not need demagnetizing temperatures. In some cases they give measurable orientation at room temperature. The degree of orientation obtained, however, is smaller than with static methods. Dynamic methods of orientation can be used in all materials in which there exists a coupling between the magnetic moment of the unpaired electron and the magnetic moment of the nucleus and in which this coupling is the strongest relaxation mechanism of the nucleus. O v e r h a u s e r was the first to propose the use of the saturation of electron paramagnetic resonance for the polarization of nuclei [2 to 3]. His method, which was originally proposed for metals, was later extended [4 to 6] to paramagnetic materials generally. A b r a g a m [6] showed tha t Overhauser polarization can be attained by saturating the allowed transitions in the resolved hyperfine structure of electron paramagnetic resonance (EPR). Polarization during saturation is excited by a relaxation process where the number of projections of the electron and nuclear spins remains constant. EPR is observed experimentally for mutually perpendicular magnetic and high-frequency fields. In this case allowed transitions i n s hyperfine structure of EPR are understood to be transitions where the projection of electron spin changes by one and the :projection of nuclear spin remains as it was (AM = 4- 1, Am = 0; M and m are quantum numbers characterizing the projection of the electron and nuclear spins). If the electron and nuclear spin-spin interaction is anisotropic or if the nucleus has a large quadrupole moment, a whole series of "forbidden" lines of the type AM = 4- 1, Am

Czechosl. Journ. Phys. 9 (1959) 42]:

M. Odehnal

= -4- 1, -4- 2 are;observed in t h e spectrum of hyperfine structure, for which a change i n orientation of the nuclear spin also occurs. When the external magnetic field N and high-frequency field N~ are parallel, the field I-I, induces only forbidden transitions of the type A(3I 4- m) = 0. J e f f r i e s [7] proposed and experimentally verified the method of nuclear polarization, where polarization is attained by the saturation of such a "forbidden" transition in parallel fields, In the present paper the case of saturation of a forbidden transition is theoretically analyzed for perpendicular fields (N _l_ N~) and on the assumption of a large quadrupole moment of the nucleus (i. e. the case of transuranium elements for example), After a brief analysis of the spin Hamiltonian the wave functions of the levels of hyperfine structure are calculated and the Hamiltonian describing all possible relaxation mechanisms of a nucleus and electron is constructed. If the wave functions and "relaxation" Hamiltonian are known, a matrix can be constructed for all possible relaxation processes. By solving a system of differential equations, describing the time changes in the population of the different levels during saturation of the transition of the type AM = -4- 1, Am = -4- 2, we obtain the values of the population from which we can easily determine the degree of polarization attained.

II. SPIN HAMILTONIAN

In order to describe the spectrum of EPl% we use the spin Hamiltonian [8], whose general form for axial symmetry of the electric field of the crystals is

~ = fl~[g,,S~ 4- g.(HxSx + H~S~)] + D[S~ -- �89 4- ])] + ASzI~ 4- B(Sflx + 4- SyI~) 4- Q[I~ ' - ~aI(I 4- ])] - ~fl~HI, (1)

where S and I are opera tors of electron and nuclear spins, H~, H v and Hz the components of the magnetic field in the coordinate system x, y, z, connected with the electric field of the crystal, gll and g• components of the spectro- scopic splitting factor lying in the z axis and in the (x,y) plane, fie Bohr's magneton, D the constant of fine structure, A and B the constants of hyperfine structure, Q the constant of quadrupole coupling,y I the nuclear factor of spec- troscopic splitting, fii the nuclear magneton. Calculation of the eigenvalues of this spin Hamiltonian was carried out by B l e a n e y [9] for the case of a strong magnetic field in which gfl,H >~ A, B and on the assumption that A, B > > Q. In this paper he also calculated the quantum-mechanical probabilities of the allowed and forbidden transitions in hyperfine structure of the E P R spectrum. We refer the reader to his paper and shall confine ourselves only to the remarks and equations absolutely necessary for understanding the following text.

As a result of the anisotropy of the coupling Of the orbital momentum of the electron with lattice (gH ~: g A ) a n d the interaction of the magnetic moments of.the electron and nucleus (A # B) S and I do not quantize about the z axis but about the z' and z" axes. Thus the operator S in the z', x', y' system and I in the z", x", y" system are represented by diagonal matrices.

! ~ t? # The transformation equations of S~, S~, S~ (I , Ix, I~) to S', S~, S~ (I~, Ix, I~) are:

S~ eosO' S~ 4- s i n O ' . S : I ~ = eosx]~4- sinx;~ S ~ = s inO' S ' - - c o s O ' . g x Ix== s inx I~- - cosxI~

p S~ - - S~ I~ - - 1[" = = ~ ( 2 )

422 ~ e x o c ~ . d p ~ l a . ~ y p H . 9 ( 1 9 5 9 )

.Nuclear Polarization by Electron Paramagnetic Resonance in Paramagnetic Crystals

where c o s O ' g, co sO s inO ' g_, s i n O Ag, cosO s i n x Bg• - - , - - - - ~ COS X ~ ~ - -

g g Kg Kg and 0 the angle be tween the ex te rna l magnet ic field H and the axis z of the axial electric field. I f the spin Hami l ton ian (1) is t r ans formed to these new coordinate systems, we obta in the following expressions for its different t e rms (the ex te rna l field H lies in the (zx) plane and N, in the y axis)

A B , 1"~ " " ~ 1 In x (I ,I , 4- l~I~)] 4- - ~ - S~l~ + BS~I~, . . . . Q[~ = Q [cos 2 x ~ 4- sin 2 x l , ~- eos x s . . . . . .

g = pg , cos" e + g'_, sin e , K = eos: e + B'g' , sin e Y

a L = g~ Ig-~ cos 0 sin 0 Kg 2

Z ~

/ F i g . 1. C o o r d i n a t e s y s t e m z, x, y c o n n e c t e d w i t h a x i a l e l e c t r i c f i e l d of c r y s t a l . E l e c t r o n s p i n q u a n t i z e s a b o u t z" ax i s , n u c l e a r s p i n

a b o u t z" ax i s .

(A~ - B ~ ) . (3)

Ib~=/~ 2, O> /c~,=@~ I>

M:-~- /o~=/--~,o~ l.f ~=t--~,- l>

Fig . 2. W a v e f u n c t i o n s of l e v e l s of h y p e r f i n e s t r u c t u r e of i o n w i t h S = �89 I = 1 fo r n e g l i -

g i b l y s m a l l r a t i o B / H a n d fo r Q = 0.

Since t h roughou t this paper we sha l l confine ourselves to the ion having S = �89 and I = 1 (S and I are the m a x i m u m components of the e lect ron and nuclear spins), the eigenvalue of t h e opera tor D[S~ - - �89 + 1)] is zero and this t e rm can thus be disregarded. B y means of (3) we calculate the wave functions for the different levels of hyperf ine s t ruc ture and t h en the intensities of the magnet ic dipole transit ions. For example, for a fo rb idden t rans i t ion of the type (21/, m) (M -- 1, m • 2) the in tens i ty J ~ is p ropor t iona l to the expression J• ~ [ ( M -- 1, m 4- 2) IS~l M, m}] 2 where IM, m} denotes the wave funct ion Of the level character ized b y the q u a n t u m numbers M and m. B le a- h e y gives expressions for these intensit ies [9]. I f the in tens i ty of the allowed transi t ions of t ype (21/, m) ~ (M - - 1, m) is p u t equal to one, t h en the forb idden t ransi t ions of the t ype (M, m) ~-~ (M - - 1, m 4- 1) and (M, m) ~ (21//-- t , m 4- 2) are of the order (QJK) ~, The in tens i ty of these forb idden lines la rgely depends on the ang le O. Wi th the t ransi t ions (21/, m) ~ (M - - 1, m 4 - 1) it is greatest for O = 45 ~ and is zero for O = 0 ~ or O = 90 ~ The t rans i t ions (M, m ) ~ - ( M - - 1, m 4- 2) are the most intensive for O = 90 ~ and for O = = 0 ~ t hey vanish. Bo th types of t r ans i t i ons can be used for the proposed me- t hod of polarization. A theore t ica l analysis is carried out for the sa tu ra t ion of the t rans i t ion (AM = 4- 1, Am = 4- 2) and in the last section a generaliza- t ion is carried out to the t ransi t ions (AM = 4- 1, Am = ~ 1).

Czechosl. Journ . Phys . 9 (1959) 4 2 3

M. Odehnal

III. CALCULATION OF WAVE FUNCTIONS OF LEVELS OF HYPERFINE STRUCTURE

Let us first consider the case of a paramagnetic ion with S ---- �89 and I = I on the assumption that Q ---- 0 and A ~-~ B (if B were very small, the intensity of the forbidden transitions would be negligible [9]). The Hamiltonian describing the energy level of such an ion will have the form (/-/[I z):

~'8 = gflsHS~ § ASJ~ ~- B(S,I , + S~Ii). (4)

The eigenvalues of this Hamiltonian are obtained by solving the secular determinant constructed from the matrix representation (4) and the wave functions of the different levels are obtained by solving six homogeneous equations of this secular determinant. The result is six wave functions describing the six levels of hyperfine structure of our paramagnetic ion

1 1 } ~ _ B I 1 [ 1 1} B 1

i 0> q_ B I i I B 1 ]b>-- V5 u 2) le> - o> VSH , - 2 >

1 1 2 > . Ic> "-- 2 ' 1> I/> ~" 2 '

The term B(S,I , ~- SvI~) thus leads to a mixture of the states. The ratio B / H is equal to 10 .2 with most paramagnetic ions for H of the order of 104 Oe. Thus the degree of mixture is very small. The probability of forbidden transitions for N~ J_ N is in this ease of the order ( B / H ) 2 ~ 20 -4 and is thus much smaller than the intensity of forbidden transitions for a large quadrupole moment, which are of the order (Q/K)2~.~IO -2. This degree of mixture will be neglected below and the wave functions in the form given in Fig. 2 will be considered.

We now introduce the operator QI~ as the operator of the perturbation disturbing the level of hyperfine structure. The wave functions of the levels disturbed in this way are given by the following relation according to the theory of perturbations for non-degenerated states

= ~ + o Ii> ( 6 ) ~.k El~ - - E~ '

where [1~}, }k} ~ are wave functions of the disturbed and undisturbed levels, E ~ and E~ the energies of the undisturbed levels 1~ and ], W~-k the matrix element of the perturbation W between the states j and k. In our ease the operator of the perturbations W is given by Eq. (3):

W = QI~ = Q[cos 2 x I; ~ -? sin 2 x I'; ~ @ cos x sin x (l;l; ~- I;I;) (7)

and (6) will thus have the form

I k ) = M , m } + (M,m• m} M,m~4-2> EM m- - EM,~nzk l

[ ~ " 2 l - f f 2 I [ _~ < i , m ~ 2 I ~:~ s in Xl x, i M, m>

o - - - ~ - ~ - - - - ] M , m -~ 2 } . (8) EM, ra - - EM, r a ~

424 "-lexoca. dp~u. ~ypIL 9 (1959)

Nuclear Polarization by Electron Paraznaynetic Resonance in Paramagnetic Crystals

By solving (8) we obtain the desired wave functions of the levels of hyperfine structure disturbed by the quadrupole term (7) in the form

[a> ---- ]�89 --I> -6 ~ [�89 O> -- fl [�89 I>, [d> ---- [-- �89 I> - -

- o,l - �89 o > - t l- �89 - 1 > ,

Ib> = 1�89 o> - - ~ [�89 - 1> - ~ [~, 1>, [e> - - I - �89 o> + -6 a l - � 8 9 + o~ I �89 -6 l> ,

Ic>-- 1�89 1> + ~ [�89 + / ~ [ � 8 9 1>, I/> = ] - � 8 9 - o ~ ] - �89 o> -6 ~ I - �89 1> , (9)

2 Q sin s x where ~ = -~ Q cos x sin x and /~ = 2K

IV. I - I A M I L T O N I A N O F R E L A X A T I O N P R O C E S S E S O F N U C L E A R S P I N

As we stated in the introduction, relaxation processes of nuclear and electron spin play the most important role in dynamic orientation. The degree of polarization attained depends on their type and intensity, In our case, we must take into consideration the following relaxation mechanisms:

a) relaxation of electron spin, b) relaxation of nuclear spin through scalar interaction of electron and nucleus, c) relaxation of nucleus through quadrupole interaction.

Relaxation of electron spin, which is very rapid, can be formally described by a Hamiltonian of the type 2/~H'S, where H' is the fluctuating, time variable magnetic field of the crystal lattice. The nature of this spin-lattice relaxation of electron spin does not interest us and we shall therefore only introduce a field H ' equivalent to the interaction between the electron spin and the lattice. In paramagnetic materials the nuclear spin is coupled by a strong magnetic interaction to the electron spin, which is again coupled by a spin-orbital coupling to the crystal lattice. Through this spin-orbital coupling the thermal motion of the crystal lattice modulates the scalar interaction of the nuclear spin with the electron spin, which is described by the operator A1SI, where A1 is a time variable function expressing the modulation of the nucleus and electron coupling by the thermal motion of the lattice. This is the quickest relaxation process of a nucleus of a paramagnctic ion. Similarly, a weaker relaxation of the nucleus through the coupling of the quadrupole moment with the electric field of the electron envelope can be formally written by the operator Q' l~, where Q' is also a time variable function expressing the modulation of this coupling by the thermal motion of the lattice.

The entire Hamiltonian of the relaxation processes, which must be taken into consideration, is thus [6]

§ - ~ (S+I- § S-I+) -6 Q'I~, (10)

where S+ --= S~ % i S~, S_ : Sz -- i S~ and similarly for I+ and L .

CzechosL Journ. Phys. 9 (1959) 42'5

M . Odehnal

V. IV~ATRIX: OF R E L A X A T I O ~ P R O C E S S E S

The mat r ix of re axat ion processes has its elements given by (/c[$fffl} where ] k} and ] l} are wave functions of the levels between which the relaxation processes

excite t ransi t ion and ~ is the Hamit tonian of these processes (10). In our case we have a system of six levels , described b y the wave funct ion (9) and the mat r ix will therefore have the form shown on the opposi te page.

The squares of the mat r ix elements (11) a r e direct ly proport ional to the intensities of the various relaxat ion transitions. L e t us now est imate the order- of-magnitude probabil i ty W~=~ of the different transit ions. ~2 and/5~ are proport ional to (Q/K) 2 and this is of the order of 10 -~ for most paramagnet ie salts. The probabilities of the quickest t ransi t ions are

Wb ~ - e ~-~/5~ (H'~ z § He 3) (1 - - o~ 2 --/52)~

W~ ~ d ~ fl~ (H,~'~ + H~'~) (1 c~ /53)2 + ~A~ (a~fl~ _ 2~f l 3 + o~) "

(12)

Le t us now neglect a 2,/52 compared wi th 1 and the te rm with A~ compared with the te rm /52(H'~3 + H~ 3) a n d denote the probabil i ty of these very strong t ransi t ions , corresponding in essence to the relaxation of electron spin, by W. Let us now est imate i n order of magni tude the P~robabili~ies of other transit ions wi th respect to these very strong transit ions.

, , A ~ . W a ~ r ~ W b ~ d "~ Wb.~_ f ~ Wc~_e e~-,).lW-~- ~ 1 W, where )'1 ~ 4 ~ 3 a n d )'~ ~ 2 W '

W~_a ~-~ W ~ e - ~ W , where ).3 ~ 4 f l 2 ; , ~2/52C'3

W ~ = b ~ W b = , ~ - . W e ~ _ , ~ - . ~ W , ~ _ ~ . ~ ) , 3 W § where ) ' 3 ~ - 4W '

, ~3A~ ,, ~3Q,3 ) . 3 = 4 W ' ) ' 3 = W '

, C%x 4 , A~3 2 W a ~ r 2 4 7 Where ) ' 4 " ~ and ) ' 4 ~ W

Transit ions having their probabil i ty proport ional to ()'1 § ).1) W correspond in the zero-th approximat ion to the t ransi t ions AM = § 1, Am = =~ 1; ),3W

n

corresponds to the transit ions AM = ~ 1, Am = • 2; ()'3 § § ~3) W transi t ions to AM = 0, Am = ~ 1 and ()'4 § )'~) W transi t ions to AM ---- = 0, Am ---- • 2. The elements of the ma t r ix (11) sat isfy the physical assump-

t ion t h a t Wz-~ = Wk~_~, i. e, [ (k l~r [ / ) ] 2 [ f l [ ~ [ k ) ] . 21 and ~3 are smaller C'2 . 10 .4

by an order of 10 -3 t han W, 23 and ),~ of an order of W ' )'s of the order

A l l 0 -2 of W times smaller. See Fig. 3.

426 "-!exoc~. qb~a. ~KypH, 9 (1959)

Nuclear Polarization by Electron ParamagnetiC Resonance in Paramagnetic Crystals

~ a _ �84 ~ ~ ~ .

§ §

~ ~ ~ 1 ~ ; ~ + ~ +

A I §

A

A m

A

I

%

I

4- *1 v

+ ~

A

I § I I % "

~+ ~ +

I +

e;

L

4- I

I I

~ ~ I

e + I

~ 1 ~ ~ + + ~

I

�9 ~ ~

I

I

I §

%

§ I §

I §

f §

i ~ ~

L

3"

I §

+

LI

~q

II b

A A A A A A

Czeehosl . Journ, Phys . 9 (1959) 427

11I. Odehnal

VI, D I F F E R E N T I A L EQUATIONS FOR TIME CHANGES IN P O P U L A T I O N OF L E V E L S

In the case of the rmodynamica l equilibrium of the spin sys tem and latt ice and in the absence of a high frequency field, the populat ion of the different levels mus t sat isfy the following equations

dp~ _ ~ dt (p~g~ - - p~g~) W ~ ~ O, (13)

where Pi is the populat ion of the i level, gi = exp (EdicT), Wi~ = W ~ is the probabil i ty of t ransi t ion (relaxation) between the levels i and ], p~ the popula- t ion of the ] level and g~ = exp (E~/kT). If a certain t ransi t ion is sa tura ted

+ s ~ I - 2 ~,' (7

§ I, *, '~

-.4' + ~

/ ~ -A'-8

Fig. 3. All possible relaxation processes of paramagnetic ion with S = �89 I = 1.

I -PZ~ ~ Q 0 b I - 2 A ' b ~ 1-2"1'

1 - 2 I t ' I I r r I

I i I 11 i I m ~

I+2,1'

e l e I 1 y t f V I

Fig. 4a. Population of different levels of paramagnetic ion with S = �89 I = 1 in thermodynamically steady statefor H, = 0. Fig. 4b. Population of levels for saturation of transition ] ~ c. Dashed line denotes

quick relaxation of electron spin.

(i. e. the populat ion of both levels, between which t ransi t ion occurs, are equalized) by a high-frequency field, then (13) will have the form

dp~ dt " ~ [(P~g~ - - Pigi) Wi~ -~ (pj - - Pi) Vi j ] , (14)

where V~j is the probabi l i ty of t ransi t ion between i and j induced by a high- f requency field, which apar t f rom other quanti t ies is proport ional to v2 and for sufficiently large in tens i ty /-/, approaches infinity. I f we consider, during the saturat ion of a certain transit ion, a thermodynamica l ly s t eady state, we have d p J d t ~ 0 for all i. In order to calculate the populat ion of the different levels after applying H,, we m u s t solve the system of homogeneous equations

[p~g~ - - p~g~) W~j + (pj - - p~) V~j] ----- O . (15) J :V i

For our ion the sys tem of different equations (14) will be as follows

1 da W dt - - / " exp (-- A' - - ~) ~- ~(1)e. exp (-- zJ') ~- ~2d.

exp (-- A' -~ ~) ~- ~(a)/. exp (A') -~ ~(4)c. exp (A ' - - ~) - -

-- a [1 + ~(1) + ~2 + ~(a) + ~(4)] . exp (A' + ($) ,

1 db -- e . exp (-- A') + ~(1)d. exp (-- A' + ~) + ~(1)].

W dt

4 2 8 q e x o c n , dpi43. ~ y p H . 9 (1959)

Nuclear Polarization by Electron Paramagnetic Resonance in Paramagnetic Crystalz

exp (-- A' -- 6) -4- 2(8)c �9 exp (A' -- 6) %. 2(a)a. exp (A' %. 6) -- b . exp (A') . [1 + 21 (1) + 21 (a)] ,

1 dc - - d , exp (-- A' %. ~) %. t(1)e, exp (-- A') %. I(2)] .

W dt exp (-- A' -- 6) %. 2 (s) b. exp (A') %. 1(4)a. exp (A! %. 6) Jr V ' �9 ] - - V ' . c - - c .

�9 [1%. 2 (a) + 22 + i(a)%. 1(4)]. e x p (A' - - 6 ) ,

1 de/ -W d--{ = c . exp (A' -- 6) %. 1(1)b . exp (A') %. 2~a.

exp (A' %. 6) + 2(a)e. exp (-- A) + )(4)/. exp (-- A' -- 6) -- d . exp (-- A' %. 6) . �9 [1%. 2 (1) %- 1~ %- 2 (a) %- 1 (4) ] ,

1 de -- b . exp (A') %- 1(1)a. exp (A' %- 6) %- 1(1)c.

W dt exp (A' -- 6) %- 2(a)d. exp (-- A' %- 6) %- 2(a)/. exp (-- A' -- 6) -- e.

�9 exp (-- A ' ) . [1%- 22 (1) %- 21(a)], 1 d/ W dt - - a ' e x p ( A ' %- 6) %-2(1)b exp (A ' ) %-22c.

exp (A' -- 6) + 2(a)e. exp (-- A') %. 1(4)d. exp (-- A' %- 6) %- % - V ' . c - - V ' . / - - / . e x p ( - - A ' - - 6 ) . [ 1 % - 1(1) %-2,%-4 (3) %-i(4)],

where V' = V/W, 4 (1) = 41%- 2~;, 4 (8) = 4 3 %- 2 ; %- 2 tt3, 2 (4) = 44 %- 2~, A' = = gfl~H/2kT, 6 = K / 2 k T and a, b, c, d, e, / denote the populat ion of the levels according to the nota t ion in Fig. 2. In Eqs. (16) we considered the case when the t ransi t ion / ~- c is sa tura ted (i. e. in the zero-th approximat ion of the wave functions the t ransi t ion AM = 4. 1, Am = 4- 2), see Fig. 4.

I f the left side of (16) is pu t equal to zero, we obtain a sys tem of six homogeneous linear equations which mus t be solved. The de te rminan t of this sys tem D = 0 a n d the system thus has a non-tr ivial solution. The ratio of the sub- d e t e r m i n a n t s of the f if th degree, which are denoted by Sa, S b . . . Sf , gives the ratio of the desired values of the new population:

S~ = -- V ' . exp (A' %- 6). 4(2 (~) %- 2 (~)) (1 -- 3A' ) , Sb = - - V' . exp (A' %- 6). 4(i (1) %- 1 (a)) . (1%- 2A') , S o = - - V ' exp ( A ' %-6) .4(1 <1) %-1 (a)) ( 1 - - A ' ) , Sa - - V ' . exp (A' %- 6). 4(2 (~) %- 2 (a)) (1 -- A ' ) , S~ -- V ' . exp (A' %- (~). 4(1(! ) %- i(a)) , S I - - - - - V ' . exp (A' %- 6). 4 ( i (1) %- i (3) ) . (1 F A ' ) . (17)

In the very laborious calculation of these sub-determinants all the products of the Coefficients 2 were neglected. A fu r the r simplification was the approxim- a t ion A' >~ 6: During sa tura t ion V' is very large and all the terms no t containing V' were neglected compared with those containing V'.

VII. DISCUSSION OF DEGREE OF ORIENTATION ATTAINED

The ratios of t h e populations of the different levels fo r sa tura t ion of the t ransi t ion / ~ c can easi ly be calculated from Eqs. (17)�9 The popula t ion of these levels will be the s a m e and thus c / / = 1. If we p u t c ---- / = 1 we obta in the relative populat ion of the different levels during sa tura t ion

Czechosl, Journ. Phys. 9 (1959) 429

M , O d e h n a l

a - - " 1 - - 2A, d~__ I + 2A' b ~ l - - A ' e ~ l - t - A ' (18)

c - -" 1 1 - - 1 .

The degree of or ien ta t ion is def ined as V = (P+ - - P _ ) / P , where P+ i s the number of nuclei in the s ta tes wi th ( + m), P - the n u m b e r of nuclei in the s ta tes wi th ( - - m), P0 the n u m b e r of nuclei in the s t a te wi th (m -=- 0) and P = P+ +

2 ? + P - + P0. I n our ease we get ~ ---- xA . The degree of or ienta t ion is ~A' and not A' because p a r t o f the nuclei (one third) are in the s t a te in which the p roper va lue of the opera to r I~ is zero (because I has an in teger value). I f the nucleus has spin �89 we have ~ ---- A', The r a t io P + / P _ is also somet imes regarded

M I - 2 A '

l - 2 A '

+ I - 2 A ~

1 - 2 z ~ ~

I--2A ~

l

' l ! --5 1

l

m 5 M l - 2 A ~ m 5

~ 1- 2,~-' _1_ _ i_ ~ + I - 2 A ' _1.2. 2

l - 2A' _

_ 5 l - 2 A ~ _ 5 2 2 2

l 3 2

- - 1 2

a) b)

Fig. 5a. Equilibrium population of levels of U ~'3a ion (S = �89 I = 0) for N~= 0. Fig. 5b. Equilibrium population of levels of U ~aa ion for saturation of transition AM =

= =t= 1, A m = =t= 1.

as a q u a n t i t y charac ter iz ing the degree of polar izat ion. I n our case P + / P - -= ---- 1 -F 2A', which is the same va lue as for Overhauser ' s effect in meta ls [1 to 3, 6]. Fo r ~ =- 10 k M H z and T ---- I~ 23% of the nuclei could theore t ica l ly be polarized. The prac t ica l value of ~ will of course be smal ler bu t even this va lue could enable m e a s u r e m e n t of the r a d i a t i o n an iso t ropy , for example , if the polar ized nuclei were radioact ive , and de t e rmina t ion of the nuclear m o m e n t l r om t h i s anisot ropy.

I t follows f rom an analysis of the q u a n t u m mechan ica l probabi l i t ies of re lax- a t i o n t rans i t ions t h a t t he s t ronges t (i. e. the quickest) are those for which A M = -b 1 and A m ---- 0 ( re laxat ion of e lect ron spin itself). I f we p u t the i n t e n s i t y of these t rans i t ions equal to one, t hen fu r the r re laxa t ion t rans i t ions of the t y p e A M = ~ 1, Am = -b 1, :t: 2 are an order ,of 10 .2 t imes weaker . E v e n slower t rans i t ions are " p u r e " nuclear re laxa t ion mechan i sms of the t y p e A M = 0, A m = • 1, ~ 2. This means t h a t for s a tu r a t i on of cer ta in levels Of h y p e r f i n e s t ruc tu re the re laxa t ion t rans i t ions of the t y p e A M = • 1, Am = 0 v e r y rap id ly set u p t h e r m o d y n a m i c equ i l i b r i um be tween the levels, differ ing b y A M = -4- 1, Am = 0, a n d m a i n t a i n it. L e t u s d e n o t e b y T 0 the

430 ~ e x o c ~ (:[:)i, I3. ,~:ypH. 9 (1959)

Nuclear Polarization by Electron Paramagnetic Resonance in Paramagnetic Crystals

relaxation time of these transitions ands by T~ and T 2 the times of t he relaxation transitions of the type A M = ~= 1 , A m = 4-1 and A M = • 1, A m ~ 4- 2. I t is clear tha t T 0 ~ T l, T 2. Fig. 4a shows the relative populations of the levels of hyperfine structure of a paramagnetic ion with S = � 8 9 and I = 1. Let us consider our theoretically analyzed case of saturation of the transition ] ~ c. For total saturation the populations of these levels will be the same and we denote it / = c = 1. Between the levels c and d, however, there exists a fast relaxation time To, which maintains both levels in thermodynamic equilibrium so that the ratio of the populations Of the two levels is c/d ~ 1 -- 2A'. And this is also the c~se for the levels ] and a. The new populations of the levels will thus appear as plotted in Fig. 4b. The ratio P + / P - = 1 ~-

2A'. We therefore get the same result as in the case of the detailed solution of the system of six homogeneous equations for the simultaneous eXistence of all possible relaxation mechanisms.

Our consideration is thus justified and can be applied to the saturation of transitions of the type A M = • 1 , A m = 4- I. Let us analyse the case of U2a3, which has S = �89 and I = } and in the spectrum of which bo th the lines AM ----- 4- 1, Am = 4- 2 and A M = 4- I , A m = • 1, [10] are observed, and let us consider tha t one of the lines Of type A M = 4- 1, Am = 4- 1 are saturated. The result is given in Figs. 5a and 5b.

VIII. CONCLUSION

It was theoretically shown tha t a measurable degree of orientation of nuclei can be attained by the saturation of forbidden lines in the spectrum of E P R on levels of hyperfine structure of paramagnetic ions. The orientation is in no way disturbed by the large quadrupole moment of the nucleus and its relaxat ion mechanism as followed from solving the system of equations (16).

The advantage of this method lies in the fact tha t small high-frequency powers are sufficient to saturate a forbidden transition since the relaxation time between these levels is 10 2 times longer than the time of the spin-lattice relaxation of electron spin. The degree of orientation attained is the same as with the O v e r h a u s e r and J e f f r i e s methods. The disadvantage is that the method is not universal and can only be used with those paramagne- tie materials where forbidden lines are observed. The distance between them must be sufficiently large for several lines not to be saturated simultaneously.

In conclusion the author would like to thank J. B u r g e t , J. ~ a j k o , M. Ko1s and M. ~ o t t for helping in the laborious solution of system (16).

Received 20. 11. 1958.

Re]erences

[1] ~ a f r a t a S.: Cs. 5as. fys. 8 (1958), 112; [2] O v e r h a u s e r A.: Phys. Rev. 89 (1953), 689. [3] O v e r h a u s e r A . : P h y s . R e v . 94 (1954) , 411. [4] O v e r h a u s e r A.:' Phys. l~ev. 9d (1954), 768. [5] Blotch F.: Phys. Rev. 93 (1954), 944. [6] A b r a g a m A.- Phys. Rev. 98 (1955), 1729. [7] J e f f r i e s C. D.." Phys. Rev. 106 (1957), 164. [8] B l e a n e y B., S t e v e n s K. W. H.: Rep. Progr. Phys. 16 (1953), 108. [9] B l e a n e y B.: Phil . Mag. g2 (1951), 441.

[10] D o r a i n P. B., I - I u t c h i s o n C. A., W o n g E.: Phys. Rev. 105 (1957), 1307:

CzechosL J'ourn. Phys. 9 (1959) 431

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Nuclear polarization by electron paramagnetic resonance in paramagnetic crystals