- Email: [email protected]
IV. Relaxation Times and Line Width
IV. Relaxation Times and Line Width In many cases, the resolution of the fine structure and hyperfine structure is limited by the line width of the absorption lines. This line width is dependent on the interaction between the paramagnetic ions and the surroundings, on the interaction among the paramagnetic ions themselves, and on inhomogeneities in the crystal lattice. The most important sources of broadening are: ( 1 ) spin-lattice relaxation; (2) dipolar interaction between spins of the same kind; (3) dipolar interaction between unlike spins, that is, between spins with different resonance frequencies; (4) exchange interactions; ( 5 ) saturation by the radiation field; (6) hyperfine structure interaction; (7) inhomogeneities in the crystal lattice; (8) inhomogeneities in the applied magnetic field. Some of these factors have been dealt with by Pake6 in his discussion on nuclear magnetic resonance. We shall emphasize here only the aspects of importance for paramagnetic resonance.
22. SPIN-LATTICE RELAXATION TIME The spin-lattice relaxation time measures the energy transfer from the paramagnetic ions to the surroundings, that is to say, to the crystal lattice. This relaxation time T I , often called the longitudinal relaxation time, is defined for two energy levels by d
- (AN - AN,)
=
dt
-
(AN
- ANo) Ti
(22.1)
where A N is the population difference between the two levels; AN0 is this difference a t equilibrium. The differential equation expresses the rate at which the system approaches the equilibrium after having been disturbed by the absorption of energy. Here AN0 is defined by the Boltzmann distribution and is given by
ANO= N tanh (hv/2kT)
(22.2)
where N is the total population of the two levels. For microwave frequencies and high temperatures such that (hu/2kT) << 1, AN0
-
Nhv/2kT.
149
(22.2a)
150
IV. RELAXATION TIMES AND LINE WIDTH
The temperature T is the so-called spin temperature. One assumes that the system of magnetic dipoles, often called the spin system is in internal equilibrium. In absence of resonance absorption or external fields, this spin temperature corresponds to the lattice temperature. Systems having more than two levels behave essentially the same although the treatment is slightly more complex. Assume now that the equilibrium distribution is being disturbed, for example, by resonance absorption. What mechanisms are operative to restore the equilibrium distribution? This problem has been considered originally by Casimir and Du PrB, 160 Waller, 161 Kronig,IE2 Van Vleck,’68 and Heitler and Teller.Is4 According to these authors, there are two types of relaxations. One type is predominant a t high temperatures. This is an indirect process in which there is an energy transfer between the spins and the lattice by means of inelastic scattering of phonons. The spins absorb one phonon and scatter another. This process is sometimes called the Raman process. It leads for the case of S = to an expression in which T1 is proportional to A0/X2T7for T smaller than the Debye temperature, and proportional to l/Tz for T larger than the Debye temperature. These formulas indicate that the relaxation will be a function of the orbital contribution through A. Large splittings, as found in “ spin-only ” magnetic substances, will show long relaxation times. This is indeed observed for V2+, Cr3+, Mn2+, Fe*+,and Gd3+.On the other hand, the rare earth and uranium group or Ti3+,Fe2+,Co2+in octahedral complexes have short relaxation times. A detailed comparison of the relaxation times of various substances with these theoretical predictions has not been undertaken. The other type of interaction is the ‘(direct interaction” of the spin system with the lattice. In this case, the spin system emits phonons “on speaking terms” with the spins, that is, phonons having an energy distribution coinciding with the energy distribution of the spin system. This mechanism leads to a relaxation time proportional to A4/X2T. It would be of importance only a t low temperatures in which case the Raman process is ineffectual. One of the problems with the direct process, already discussed in detail by Van Vleck,166is that this process is not sufficient to explain the relaxation times. The number of phonons which are on ‘(speaking
+
H. B. G. Casimir and F. K. Du PrB, Physica 6 , 507 (1938). I. Waller, Z . Physik 70, 370 (1932). IE4 R. de L. ICronig, Physica 6, 33 (1939). 163 J. H. Van Vleck, Phys. Rev. 67, 426 (1940). 164 W. H. Heitler and E Teller, Proc. Roy. Soc. A166, 629 (1936). 166 J. H. Van Vleck, Phys. Rev. 69, 724 (1941). 180
161
22.
SPIN-LATTICE RELAXATION TIME
151
terms” with the spins is relatively small. These phonons will be unable to transfer the energy fast enough from the spin system to the lowtemperature bath to account for the short relaxation time. The coupling between the phonons on speaking terms with the majority of the phonons which contain the total thermal energy of the lattice (the lattice reservoir) is very small and is, in the main, through anharmonic terms in the elastic forces. Therefore, the temperature of the phonons on speaking terms will be essentially that of the spin system. A severe bottleneck is created in the transfer of the energy from the phonons on speaking terms to the lattice reservoir. In recent years, many more difficulties have appeared which cannot be accounted for by a simple direct phonon-spin interaction. One of the best mays of measuring relaxation is t o measure the susceptibility as a function of the frequency, a method employed in particular by Gorter and his zmsociatcs.166-16* These measurements often show a relaxation time by a factor of 10 larger than those derived from paramagnetic resonance a t microwave frequencies. 16* Moreover, the measurements of audiofrequencies have shown that the relaxation times of low temperatures depend on the size of the ~ r y s t a l , ~ ~ ~ - ~ ~ ~ and that the dependence on temperature and constant magnetic field is very much different from that expected on the basis of the theory by Waller*61and Van Vleck.IB3A distribution of relaxation times is sometimes observed rather than a single relaxation time.’?’ Two ways out this impasse have been suggested by G ~ r t e r ’ ’and ~ T o w ~ i e s . ~Gorter ’~ suggests that there may be transport of energy by phonons to cryst(a1boundaries or defects where the transformation t o heat may be easier. Gorter also considers the possibility that measurements by microwave saturation, that is, a t high power levels, show a rapid conversion of energy from the phonons t o heat. In this case, the bottleneck would be mainly between the spins and the phonons on speaking terms. A t low-frequency measurements in which the spin temperature is not raised appreciably, the bottleneck would be the small number of low-frequency phonons, that is, the transfer of the C. J. Gorter, “Paramagnetic Relaxation.” Elsevier, Amsterdam, 1947. A . H. Cooke, Repls. Progr. in Phys. 13, 276 (1950). I E R C. J. Gorter, “Progress in Low Temperature Physics,” Vol. 2, Chapter 9. North Holland, Amsterdam, 1955. 169 L. C. Van der Marel, J. Van der Brock, and C. J. Gorter, Physica 23,361 (1957). F. W. de Vrijer and C. J. Gorter, Physica 18, 549 (1952). 171 It. J. Benzie and A. H. Cooke, Proc. Phys. SOC.A63, 201 (1950). C. J. Gorter, L. C. Van der Marel, and B. Bolger, Physica 21, 103 (1955). J. A. Giordmaine, L. E. Alsop, F. R. Nash, and C. H. Townes,Phys. Rev. 109, 166
16’
302 (1958).
152
nT.
RELAXATION TIMES AND LINE WIDTH
energy from the phonons on speaking terms to the majority of the lattice phonons. Townes et aL1Ia suggests that there are two kinds of relaxation mechanisms a t low temperature, the usual spin-lattice relaxation and the lattice-bath relaxation. These authors feel that in many crystals, the latter mechanism predominates and this relaxation time, usually larger, is being measured. I n order t o overcome Van Vleck's contention of the small number of phonons on speaking terms, these authors suggest that the lattice modes are broadened by the interaction with the spins. The spin-lattice relaxation time is considered to be short of the order of sec. Each phonon then is interrupted many times and correspondingly, the frequency band of phonons on speaking terms is much larger. Townes et al. suggest a width of a few hundred Mc/sec, much wider than the usual paramagnetic line width. The measured relaxation refers to the transfer of energy from these phonons to the bath and it is predicted that this will show a concentration and crystal size dependence. I n addition, these may be a distribution of relaxation times because the lattice-bath relaxation will be influenced by transport phenomena as by diffusion of phonons to the lattice boundaries. No one so far has considered the influence of covalent bonding on relaxation times. It is possible that the bonding electrons overlapping with the magnetic electrons may be modulated by the spin flip. The bonding electrons probably are on speaking terms with a much larger number of phonons. Gorter"J8 has remarked that "the theoretical analysis of the behavior of the oscillators 'on speaking terms' is in a shockingly primitive state." It is impossible to predict quantitatively what relaxation time a given ion will have or how it is influenced by crystal field strength or crystal symmetries. An example will emphasize this point. In an octahedral symmetry Cr3+has a large relaxation time (lo+ sec) and the paramagnetic resonance can be observed with ease a t room temperature. In fluorite symmetry Co2+ has the same energy level scheme as Cr3+. No absorption lines were found down to 70°K.173a In general T 1may range from second's to lo-" second for various paramagnetic ions. The paramagnetic resonance spectrum of most ions in the 4 j and 5f group and of many ions in the other transition groups can be observed only a t liquid helium temperature because of the short spin-lattice relaxation time. Note Added in Proof. Absorption has been detected at 20°K (M. Dvir and W. Low, to be published). The strength of the crystal field is one-fifth for Co*+inthe
*78a
23.
DIPOLAR BROADENING
153
23. DIPOLAR BROADENING
The theory of dipolar broadening has been developed by Van Vleck in a classic paper.174He has treated in detail the particular case of free spins. The mutual potential of such dipoles is given by (23.1)
Van Vleck has shown that if the external field Ho is larger than the dipole-dipole interaction, then Eq. (23.1) can be approximated by
where eij is the angle between rij and the z axis, along which H O is applied. Here the so-called “satellite lines” appearing a t 2gj3H, 3gj3H, etc., have been omitted. It is very difficult to evaluate the detailed line shapes. Van Vleck has given expressions for the second and fourth moment of the absorption lines. The nth moment of an absorption line is defined J(Au)”x”(v) du (23.3) Sx”(u) du * In most spin resonance experiments, the frequency is held constant and the magnetic field is varied. In this case the moment is given by (23.4)
The second moment for the particular case of free spins is given by
The first term in Eq. (23.5) denotes the interaction of magnetically similar ions and the second term summed over k is the interaction between dissimilar ions. The broadening is caused by two actions. First, the various ions set up magnetic fields at any other ion. These fluorite symmetry in comparison with Cr*+ in octahedral symmetry and, therefore, results in a shorter relaxation time. 174 J. H. Van Vleck, Phys. Rev. 74, 1168 (1948).
154
W. RELAXATION TIMES AND LINE WIDTH
ions then see a different magnetic field than the externally applied field. In the case of paramagnetic ions, this is of considerable magnitude. The field of a nearest neighbor with a magnetic moment of about 1 Bohr magneton a t a distance of 3 A is of the order 350 gauss and the sum of the nearest neighbors may be over five hundred gauss. The interaction of the nuclear moments of the surrounding nuclei is negligible in comparison. In the case of hydrated salts, the mean half-width due to this cause is about 6 gauss and in deuterated salts, about 2 gauss. The second mechanism responsible for broadening is operative in like ions only. Two ions precessing with the same Larmor frequency will interact and cause their spin orientation to change. This reduces the average lifetime of the ion in a given state, and therefore gives rise to broadening. The relaxation time T 2is defined by 7'2 = + b ( v ) l m = (23.6) where g ( v ) is the normalized line shape function. The spin-spin interaction time Tr is approximately given by the inverse of the half-width of the absorption line if this width is caused only by the dipolar broadening.% The problem of the line shape as a function of concentration of paramagnetic ions in a lattice has been considered by Kittel and Abraham~."~ Their solution is essentially an extension of Van Vleck's results. The mean line width for similar ions I - 3 C O S ~eij )2 (23.7) h2< (Av2)> = 3s'(~!!' 4- 1) $ $ j
2
(
j
wherefis the probability that the lattice site is occupied by a magnetic ion. The sum is over all the lattice sites, whether occupied by a magnetic ion or not. The second moment is, therefore, proportional to the concentration. The fourth moment has been solved rigorously for cubic lattice with H o along 100 directions. Their results are h4
=
3h4112
1
(0.098 - 0.021(S2-I- S)-l) .
(23.8)
Iff
= 1, that is, all sites occupied, Eq. (23.8) reduces to Van Vleck's result. In this case, the ratio of (I lf)/(l If) = 1.25 with S = If the distribution were Gaussian, the ratio would be 1.32. The departure from Gaussian distribution is therefore small and this holds true for fractional concentration off > 0.1.
+.
1'6
C. Kittel and E. Abrahams, Phye. Rev. 90, 238 (1953).
24. EXCHANGE
INTERACTION
155
For small concentrations, the value of the last bracket in Eq. (23.8) shows a sharp increase. The fourth moment increases relative to the second moment and results in peaking the lines. This is similar to exchange narrowing to be discussed in Section 24. The line shape resembles approximately a Lorentaian shape with a cutoff frequency. The half-width is proportional to the concentration, but the maximum intensity is independent of the concentration. In general, the line shape will be rather complicated with a Lorentaian shape in the center and approximately Gaussian in its tails. As the concentration is increased, the shape will approach more and more a Gaussian distribution. The general case of ions other than free spins is more complicated. The line width depends considerably on the direction along which it is measured. In magnetically dilute samples, one may resolve satellite lines symmetrically displaced with respect to the central line.176In more concentrated samples, the broadened line is made up of a large number of satellites. The detailed shape of the line will depend on the arrangement of the neighbors. Only if the arrangement is of high symmetry will the line shape approximate a Gaussian distribution. In general, the line shape will be anisotropic, since the dipolar arangement depends on (3 cos2 eij - l).1764
24. EXCHANGEINTERACTION Line width measurements in crystals containing large concentration of paramagnetic ions hardly ever agree with the theory of dipolar broadening. One of the main reasons for this is the exchange interactions between electrons. Diracl” proved that the exchange coupling is equivalent approximately to a potential of the form
-2
2 Jij(Si Sj). *
(24.1)
ij
The effect is electrostatic in origin and is connected with the overlap of the orbital wave functions on the type of symmetry in the represen*I6 B.
Bleaney, R. J. Elliott, and H. E. D. Scovil, PTOC. Phys. SOC.A84,933 (1951). Actually the true line shape is never a Gaussian. If the off-diagonal elements in the dipolar interaction are taken into account, the line shape shows small bumps near the wings. These deviations in the line shape result in a new relaxation mechanism called “cross-relaxation,” used by the spin-spin interaction (N. Bloembergen, s. Shapiro, P. s. Persham, J. 0. Artman, to be published). This cross-relaxation may in part explain some of the peculiarities encountered when measuring relaxation times in more concentrated crystal. P. A. M. Dirac, “The Principles of Quantum Mechanics,” Chapter 10. Oxford University Press, London and New York, 1935.
1760
156
N. RELAXATION TIMES AND LINE WIDTH
tation of the permutation group. The exchange integral J decreases very rapidly with distance and is negligible in dilute crystals. In concentrated crystals, one can, therefore, approximate Eq. (24.1) by an interaction between the nearest neighbors. I n this case, the interaction is the scalar product -2JZ(Si * Sj)summed over the nearest neighbors only. This form of an interaction is isotropic in contrast to the dipolar interaction which is a second-rank tensor and not spatially invariant. The scalar invariant of the exchange interaction is correct only when the spin-orbit interaction is neg1e~ted.l~~ The reason is that the orbital wave functions are not spherically symmetrical in crystals. The spins feel the departure from spherical symmetry by means of the spin-orbit coupling. The anisotropic exchange has been shown to give a second-rank tensor similar to that of the dipole interaction, that is,
jj>i >l
3(Si. rj)(Si. rii) r..2 V
(24.2)
This interaction is of importance in explaining the ferromagnetic anisotropy in cubic crystals. It has only seldom been employed in the discussion of line width measurements in concentrated paramagnetic ~ r y s t a 1 s . l The ~ ~ measurements of line width and shape and their interpretation are rather difficult, even when assuming dipolar and isotropic exchange interaction only. Van V l e ~ k , Pryce ' ~ ~ and StevenslBOand Anderson181~182 have calculated the combined effect of exchange and dipolar interaction on the line width. Their calculations which are long and laborious can be summarized as follows. 1. The spins are all identical and S' = +. Van Vleck's calculations show that the isotropic exchange interaction contributes to the fourth moment and not to the second moment. Since the total area of the line cannot change, the effect is that the center part of the line is narrowed and the tails broadened. This peaking of the line shape is called exchange narrowing. 2. The spins are not identical. This may occur when the ions do not precess about parallel axes, as for example in the crystals where there is a zero field splitting of the energy levels. In this case, the isotropic exchange interaction contributes to the second moment and the line J. H. Van Vleck, Phys. Rev. 62, 1178 (1937). J. F.Ollom and J. H. Van Vleck, Physica 17, 205 (1951). 180 M.H.L. Pryce and K. W. H. Stevens, Proc. Phys. SOC A68, 36 (1950). *a1 P. W.Anderson, J . Phys. SOC.Japan 9,316 (1954). 181 P.W.Anderson and M. T. Weiss, Revs. Modern Phys. 26, 269 (1953).
1'8 1'9
24.
EXCHANGE INTERACTION
157
width is broadened. In general, both exchange broadening and narrowing are present and the resultant line width depends on the relative contribution of each type of interaction. 3. Strong exchange interaction between dissimilar ions. The only case considered so far is that of S' = 6. For slow rate of exchange, the resonance spectrum of different magnetic types of ions in a unit cell consists of distinct resolved lines having different g values. The exchange will contribute to the broadening of the lines as outlined in Section 2. As the rate of jumping increases, the lines will draw together. For J >> (gl - g2)/3H,the lines will coalesce into one line a t a mean g value given by hv = +(gl g 2 ) P H ; g1 and gz are the g factors of two dissimilar ions. At very strong exchange, the line will show exchange narrowing. This effect has been observed on copper sulfate ~ r y s t a l s . ~Bagguley ~ ~ - ~ ~and ~ Griffiths found that in order to resolve the two lines of the two inequivalent copper ions they had to increase the frequency to 24,000 Mc. Theirresultsindicatethat (gl - gz)PH J which gives J 0.15 cm-' for this crystal. Similar effects have been observed in I
+
-
-
+
+
J S i * Sz = +J[S'(S'
+ 1) - 4
(24.3)
where S' = S1 Sz = 0 or 1. The energy levels are given by - 3 J / 4 and J / 4 respectively. Which level is lowest will depend on the sign of J , whether it is ferro- or antiferromagnetic. If J is positive, the lowest level will be S = 0 and diamagnetic. The triplet spectrum will only be observed a t high temperatures. From a measurement of the relative intensities of the absorption lines at room temperature and a t a temperature where kT << J , the sign and magnitude of J can be determined. D. 31. S. Bagguley and J. H. E. Griffiths, Proc. Roy. Soc. A201, 366 (1950). M. H. L. Pryce, Nature 162, 538 (1948). K. Yosida, Progr. Theoret. Phys. (Kyoto) 6 , 1047 (1950). 186 H. Abe, K. Ono, I. Hayashi, J. Shimada, and K. Iwanaga, J . Phys. SOC. Japan 9, 814 (1954). lS7 B. Bleaney and K. D. Bowers, Proc. Roy. SOC.A214, 451 (1952). I B 8 H. Kumagai, H. Abe, and J. Shimada, Phys. Rev. 87, 385 (1952). F. W. Lancaster and W. Gordy, J . Chem. Phys. 19, 1181 (1951). 19" H. Abe, Phys. Rea. 92, 1572 (1953). IB3 IS4
158
by
IV. RELAXATION TIMES AND LINE WIDTH
The spin Hamiltonian with isotropic exchange interaction is given
+ g&H S2 + AiSi 11 + A2Sz I2 + JSi . (24.4) If A1 = A2, SI = S1 + S’, 11+ I , this can be rewritten X, +(gi + g2)BH S’ + +J[S’(S’ + 1) - 4 + +(gi - g2)BH (Si + +AS‘ I + +A(Ii - (Si - S2). (24.5) X, = g9H * Si
*
S2,
=
*
S2.
*
S2 =
I2
=
*
*
*
S2)
12)
*
If (gi - g2)PH << J , gBH >> A, J >> A perturbation theory shows that the triplet level is given by +J and +J f +(gl g2)OH f +Am, where m = ml m2. With the selection rules AM = & 1 and Am = 0, we have one electronic transition a t hu = gBH with g = (gl g2)/2 and 2m 1 hyperfine lines with separation of +A (and not as usual with a separation of A). The relative intensity of the hyperfine lines is given by the number of ways each value of m is formed. The foregoing spectrum is similar to that of nickel in a cubic field. Anisotropic exchange as explained above make the spins feel distortions from spherical symmetry. The Hamiltonian given by Eq. (24.4) will now contain a quantity ZiJ,S1iS2i. This term reflects the crystal symmetry of the spin Hamiltonian. I n the case of axial symmetry, this is equivalent to adding a term #D(SS2- +S’(S’ 1)) to the Hamiltonian. The effect is then similar to the nickel ion in an axial field in which the spin triplet has been split into a singlet at -D and a doublet a t D/2; D is related to exchange int,egral J by
+
+
+
+
+
D = -+J[HS11
-
212 - (Sl - 2121.
(24.6)
A spectrum corresponding to the above theory has been observed in copper a ~ e t a t e ~ *and ~ - ~copper ~~ propionate crystals.1g0 Similar spectra have been observed in Si containing As atoms as donors,191,192 to be discussed in Section 27. Griffiths et a1.Ig3have observed an exchange interaction in ammonium chloroiridate. In semidilute crystals, there is a reasonable chance that (IrC16)2- complexes occupy nearest-neighbor positions. There is then an exchange interaction of the two unpaired electrons via their transfer on the intervening chlorine ions. This gives rise to an exchange C. P. Slichter, Phys. Rev. 99, 479 (1955). G. Feher, R. C. Fletcher, and E. A. Gere, Phys. Rev. 100, 1784 (1955). 108 J. H. E. Griffiths, J. Owen, J. G. Park,and M. F. Partridge, Phys. Rev.106,1345 191 192
(1957).
25.
METHODS OF MEASURING RELAXATION TIMES
159
coupled pair of Ir ions. Assuming the Hamiltonian given by Eq. (24.5) and anisotropic exchange, they find
J,
=
T0.84,
J, = 10.64,
J, = k0.20 cm-I.
These results indicate that paramagnetic resonance techniques are a powerful technique in measurement exchange interactions. No doubt this technique will be applied to many other ions in the nd transition elements.
25. METHODS OF MEASURING RELAXATION TIMES The methods of measuring the various contributions to the line width do not differ essentially from those used in nuclear resonance. The measurements are on the whole, however, more difficult to perform since the spin-lattice relaxation time is very short. I n addition, there may be other factors contributing to the line widths. We shall discuss these factors briefly.
a. Other Sources of Broadening At very large dilution, one may expect the dipolar and exchange contribution to the line width to be negligible. In practice, one finds that in many crystals, there is a residual line width which cannot be attributed to any of the relaxation mechanisms so far considered. This line width is particularly great for paramagnetic crystals having a large zero field splitting. I n dilute nickel fluosilicate, for example, the dipolar width should not be more than 6 gauss and in a deuterated crystal, about 2 gauss. In practice, it is found that the line width may be anywhere between 20-50 gauss, depending on the care with which the crystal was grown. Moreover, the line width is anisotropic. It is thought that the line width is caused by small variations in zero field splittings. Small differencesin D may result from three causes: (a) defects and dislocations in the neighborhood of the magnetic ions; (b) thermal vibrations of the surroundings; (c) twinning and mosaic structure of the crystal. For large zero field splittings, these factors will contribute considerably to the line width. There seems to have been no experimental systematic study of these effects. A special case of this broadening is found in crystals of cubic symmetry. In some of these crystals, as in V2+or Cr3+ in MgO, the ground state of the paramagnetic ion is not split by the cubic field. The theoretical line width for very large dilution is less than the experimental width of about 6 gauss. It is possible that some of the
160
JS'. RELAXATION TIMES AND LINE WIDTH
reasons advanced above may be operative in splitting the ground state. Possibly, the Jahn-Teller effect may cause a small splitting as well. Other sources of broadening are unresolved hyperfine structure lines (see Section 26) and hyperfine structure interaction with the surrounding diamagnetic nuclei by means of covalent bonding. b. Measurements of Ta
In concentrated salts, there is always present the dipolar and exchange interaction. The method used so far is to measure carefully the line shape from the wings to the center and establish the second and fourth moments. In these absorption measurements (x"), one has to be careful to exclude dispersion effects (x'). One has to take precautions to prevent the possibility of frequency pulling of the oscillator by the strong absorption of the concentrated paramagnetic salt in the cavity. One calculates the dipolar width, appropriately summing over all the lattice points. This may not be easy in some cases, in particular when the spectrum is highly anisotropic and when the crystals contain many inequivalent ions with different g factors. One subtracts the measured dipolar width from the measured width and estimates the isotropic and anisotropic contributions from the exchange interaction. This method has been used by a number of authors, in particular on nickel salts.194-196 Another method, which has not been tried so far, is to measure the width as a function of the dilution. At not too high dilution, the contribution from the exchange interaction should already be relatively small and the line width a line or function of the concentration of the paramagnetic ions. A measure of the exchange interaction may be obtained occasionally from the spectra of isolated pairs of paramagnetic ions. Assuming that the exchange contributions will be the same in concentrated crystals, the dipolar width can be estimated from the measured line width. c. Measurement of T I
The methods of measuring T I are the same as in nuclear resonance, that is, either by saturations or by pulse methods. The saturation method has been extensively used.lQ7J98Bloch's equations yield the K. W. H. Stevens, Proc. Roy. SOC.A214, 237 (1952). E. Ishiguro, K. Kambe, and T. Usui, Physica 17, 310 (1951). 108 H. Kumagai, K. Ono, I. Hayashi, H. Abe, J. Shimada, H. ShBro, H. Ibamoto, and S. Tachimori, J . Phys. SOC.Japan 9, 369 (1954). 197 G. Feher and H. E. D. Scovil, Phys. Rev. 106, 760 (1957). I98 A. H. Eschenfelder and R. T. Weidner, Phys. Rev. 92, 869 (1953).
194
196
25.
METHODS OF MEASURING RELAXATION TIMES
161
imaginary part of the rf susceptibility as
At resonance frequency X'I
LJ
=
wo and
= ~xowo
1
1'2
p2Hvr2T T
+
7
1
(25.2) 2
Here p is the dipole moment for the transition, i.e., I( = gBI; Hr/ is the rotating component of the applied field. The saturation factor S is defined as 1 (25.3) Since the absorbed power is proportional to XI', one measures the absorption a t intense excitation and a t relatively low power. Here T Zis measured from the line shape a t very large dilution and a t low power. From the knowledge of S and T2,T I can be obtained. The procedure gives a unique relaxation time for a system with two levels. For systems with many levels, there may be more than one relaxation process operative and one may not be able to associate one single relaxation time with each pair of levels. I n this case, this method measures T I which is the rate a t which the energy is transferred to the lattice via all relaxation processes. 1c180 A variation of this method has been used by Gi~rdmaine."~The spin system was saturated by pulsed or cw energy. A second oscillator with too little rf power to produce saturation was used t o probe the microwave absorption as a function of the time after the saturating energy had been removed. The two oscillators were swept synchronously past the paramagnetic resonance. The time difference of the two oscillators could be varied by either changing the frequency difference or the rate of sweep. This way, the rate at which equilibrium was approached could be measured d i r e ~ t l y . 1 ~ ~ " Methods analogous to the spin echo method used by Hahnlg9in nuclear resonance could be used in principle and have, indeed, been proposed. This technique would probably give the most accurate and Another method used in the determination of 2'1 involves a measurement of the z component of the magnetization [R. W. Damon, Revs. Modern Phys. 26, 239 (1953)l. This method is suitable for crystals with a large concentration of paramagnetic substances. lee E. L. Hahn, Phys. Rev. 80, 580 (1950). l9*0
162
IV. RELAXATION TIMES AND LINE WIDTH
direct measurements of TI. It is, however, difficult to perform since the pulse would have to last for a very short time for substances having short relaxation times. Fast pulse techniques and circuitry would have to be used and these are not particularly easy in the microwave range. 26. SOLIDSTATEMASER
The name “maser” has been coined by Townes el aZ.200 for molecular systems which give “microwave amplification by stimulated emission of radiation.” Bloembergen201and Bassov and Prokhorov2021208 have proposed that paramagnetic systems may be suitably employed for low noise microwave amplifiers or oscillators. Shortly afterwards, a number of laboratories operated such a paramagnetic solid state maser s ~ ~ ~ e s ~ f ~The l l practical y . ~ ~ ~ interest - ~ ~ in~ the usefulness of this device has stimulated fundamental research in the basic phenomena of energy exchange between the spin system and the surroundings. General radiation theory shows that transition between two levels a and b of a free atomic system is determined by three Einstein coefficients Ad, Bd,Bb.. The A d is the probability of an atom to make a spontaneous transition to a lower state and is given by’ (26.1)
where pd is the matrix elements of the dipole transition and v d is the transition frequency between the two levels. In a typical experiment, p d is of the order of Bohr magneton, v d , about loLo cps which yield Ad sec-l. Spontaneous transitions are, therefore, very rare for paramagnetic systems and will hardly affect the population difference between the two levels. It will, however, because of its lack of coherency, be a source of noise. At given temperature, the population difference between the two levels is given by Eq. (22.2). If it were possible to invert the population of the spin system, we would have a system in which the stimulated emmission would be larger than the stimulated absorption. The transition probability of free atomic system between two *Oo J. P. Gordon, H. J. Zeiger, and C. H.Townes, Phys. Rev. 99, 1264 (1955).
-
*01
101
N. Bloembergen, Phya. Rev. 104, 324 (1956). N. G. Basaov and A. M. Prokhorov, J . Ezpll. Theoret. Phye. (USSR) 27, 4131 (1954).
N. G. Baesov and A. M. Prokhorov, Proc. Acad. Sci. (USSE) 101, 37 (1945). lo*A. L. McWhorter and J. W. Meyer, Phy8. Rev. 109, 312 (1958). 20) J. Combrisson, A. Honig, and C. H. Townes, Compt. rend. 242, 2451 (1956). so( H. E. D. Scovil, G. Feher, and H. Seidel, Phy8. Rev. 106, 762 (1957). 008
26.
163
SOLID STATE MASER
levels a and b caused by a stimulating rotating field H,, is given by
The instantaneous rate of transfer of energy is P
=
P(t - ti)
=
d huo - Wd.
(26.3)
at
The probability that the spin will have an interruption due to interaction with phonons is given by 1 e-(t-td/T: -
dt,
7'2
1-
The average power is given then by pAV =
t
~
~
(
1 -t ti> e-(t-f1)/T: 7'2
at.
(26.4)
For T I# Tn,this yields
A t resonance v = vo and for N ions
The average power omitted or absorbed is therefore a function of TI,T2, v , and of the density N of the paramagnetic ions. The ratio of the emitted power for excess AN ions in the upper state to the incident power at resonance is
where the incident power is given by
cAH,,' 8*
and A is the cross section. The gain of the system will be given by
G
= ea'.
164
JY. RELAXATION TIMES AND LINE WIDTH
In order to achieve a large gain, one has to use large length I in which the spins interact with the radiation field. This is achieved conveniently by means of a resonant cavity in which the sample is placed. In addition, Eq. (26.7) shows that the sample should have a large magnetic moment and long relaxation time. The gain of the induced emission has to be large enough to overcome the losses in the cavity. The noise figure of such an amplifier can be very small. The increment in noise power is given by207
dPnoi.. = ANI'P, dx
-
dx
+ ( ~ dx2 +, ~ hvrN2 d x .
(26.8)
The first term is the induced emission where AN = N2 - N 1 is the difference in the unpaired numbers of the spins in the two levels, and r is the matrix elements of the transition; P , is the noise power. The second term defines the losses in the cavity where a, is the attenuation constant. The third term is the blackbody radiation of the walls of the cavity at temperature T Oof the cavity, and the last term the spontaneous emission from level Nz to level N1. A detailed discussion of the nature and size of the noise has been given by Pound208and by Strandberg.:!Og Their analysis indicate that the limiting noise figure is given approximately as '
P,,,,, is the noise power a t the input and PnoiW, a t the output of the amplifying system. Equation (26.9) holds true only for the limiting case of large gain (G >> 1) and when the gain coefficient a exceeds the losses caused by the attenuation in the waveguide (a>> a,). If N 2 Nz - N I , the limiting noise temperature is given by hulk, which for 1-cm radiation is about 1.5"K. This is, of course, an amplifier having a very low noise. If such amplifiers could be constructed, one of the important uses would be for astronomical investigations. The source noise of interstellar space is about 1.3'K and a suitable radio telescope followed by such an amplifier could detect, for example, the 21-cm hydrogen lines far out in interstellar space. Negative temperatures, that is reversal of population, can be achieved by a number of processes. Three possibilities have been actively persued. These are (a) pulse methods, (b) rapid passage, and (c) transitions in a multilevel scheme.
-
J. P. Wittke, PTOC. IRE 46, 291 (1957). R. V. Pound, Ann. Phys. [13] 1, 24 (1957). 109 M. W. P. Strandberg, Phye. Rev. 106, 617 (1957).
I07
*o*
26.
SOLID STATE MASER
165
a. Pulsed System If a pulse of duration
is applied a t the resonance frequency to the spin system, then the spins of a free system will be reversed. This will hold true as well in a solid state system, provided t << TI or Tz. This method of inversion has some practical disadvantages. The pulse must be of the exact resonance frequency and must be of rather short duration. Moreover, for such short duration, the applied driving field must be of relatively large intensity. The system will amplify only a fraction of the time and will decay as equilibrium is reestablished.
b. Adiabatic Fast Passage Bloch210has shown that if the frequency (or the magnetic field) is swept quickly past the resonance frequency (or resonance field), an inversion of the energy levels may be obtained under certain conditions. The conditions are: (a) the passage must be rapid compared with T I or T z ;(b) the passage must be adiabatic, that is, slow compared with the driving frequency so that the magnetic moment can follow the changes in the frequency; (c) the rf field has to be relatively large; the number of ions in the crystal for maser action must be larger than
Here Q L is the loaded Q of the cavity, V , is the volume of the cavity, AH is the full width a t half-maximum of the spin resonance line, and and are the square of the microwave fields averaged over the cavity and the sample. This method suffers from many of the disadvantages of the pulsed system. It has, however, the advantage that one can sweep across the resonant frequency; that is, one does not have to work at the exact resonance frequency. Adiabatic fast passage effects have been observed in a number of paramagnetic systems. A maser built on this principle used a doped silicon sample having a long relaxation time.211 *lo 211
F. Bloch, Phys. Rev. 70, 460 (1946). G. Feher, J. P. Gordon, E. Buehler, E. A. Gere, and C. D. Thurmond, Phy.3. Rev. 109, 221 (1958).
166
W. RELAXATION
TIMES AND LINE WIDTH
c. Multilevel Scheme If a paramagnetic substance has more than two levels, that is, S' 2 1, it is sometimes possible to populate selectively one of the higher levels and thus obtain stimulated transitions to the ground state.201This is best illustrated for the three level case in Fig. 22. One saturates the transition v13 between levels one and three so that the population of these two levels is about equal, that is, N1- N3. A low
31'
v32
-J-L
FIG.22. Energy scheme for a three level ma8er.
power signal vzg will now cause transitions between levels three and two. Since, at low temperatures, N s> N 2 , this may give rise to stimulated emission a t v 2 3 . The conditions for the successful operation of such an inversion process are the following: (a) There must be at least three levels among which transitions are permitted. (b) The spin-lattice relaxation T l s must be long to permit saturation. (c) The relaxation time T s 2should preferably be larger than T21. If not, the population Na and N s will be equal after some time. The last condition is equivalent to
~21/T21f vaz/Tti. Bloembergen has shown that, under these conditions, the power emitted by the magnetic substance is
p = -Nh%2 (~21/T21- vsz/Tar)Waz 3kT 1/T2a 1/Tiz Wa2
+
+
(26.11)
26.
167
SOLID STATE MASER
where Wanis the rate a t which transitions caused by the stimulating signal occur between levels two and three, and N is the number of spins occupying the three levels. It is to be noted that Javan212and Yatziv213have shown that double quantum groups may modify condition (c) somewhat. A close inspection of the three conditions indicates that only a few inorganic crystals containing paramagnetic ions will be suitable for maser operation. Condition (a) rules out all substances which have S' = and also those substances for which the ground-state splitting is small. Transition probabilities corresponding to AM = + 2 are only large if the zero field splitting is of the order of the Zeeman energy. The requirement of large relaxation time is met in particular by substances which have an orbital singlet, that is, Cr3+, V2+, Fe3+, Mn2+, Cu2+, Ni2+, Eu2+, Gd3+ in octahedral complexes. It is conceivable that other ions in tetrahedral or fluorite symmetry might also have long relaxation times. To get reasonable amplification, one needs a large number of spins ( N ) in order to overcome the losses in the resonant structure. Preferably, one should use crystals (a) with one type of magnetic unit cell, (b) with S' not larger than one, (c) with ions which do not have a nuclear spin, (d) or those ions with a nuclear spin but a hyperfine structure splitting smaller than the width caused by the relaxation processes. Otherwise the levels will be split into 21 1 sublevels and the intensity of the stimulated emission would be reduced correspondingly. Therefore, the ions W+(Z = s), Mn2+(Z = $), EulS3(I= +), EuIKK(Z = 0 , Cu'18(I = +), Cusa(I= $) are not very suitable. Of the remaining ions Cr3+, Fe3+ and Gdx+ in nearly octahedral symmetry seem t o be the most promising substances. Nickel salts are not as promising. Although many of the nickel salts show the necessary initial splittings, the line width of all the transitions are relatively large, probably caused by inhomogeneous broadening. Only a fraction of these ions will give stimulated emission. It is therefore necessary to look for ions with small line width. I n order that transitions corresponding to AM = 2, 3, . . should be observed with fairly large intensities, one needs crystal field symmetries which cause initial splittings of the same order of magnitude as the microwave frequency to be amplified. The transition probability Wa2 is given by
+
+
.
A. Javan, Phys. Rev. 107, 1579 (1957). S. Yatziv, Phys. Rev. in press.
168
IV. RELAXATION TIMES AND LINE WIDTH
where 1 <21J,13> l2 is a measure of the intensity of the transition from levels 2 to 3 and H s ( V 3 Z ) is the signal frequency field. I n order that the power emitted should be larger than that dissipated in the microwave circuit, W32 should be as large as possible. This can be achieved if the matrix element <21J(3> is large, and if the line width caused by spin-spin interaction is small. Another advantage of the multilevel maser is now discernible. If the zero field separation of the levels is sufficiently large, i.e., of the order of the frequency to be amplified, only a small magnetic field would be necessary to obtain the proper energy level spacings corresponding to v13 and V Z S . Moreover, a t these low fields the energy levels are usually not equally spaced (since we are not working in the Paschen-Back region), and transitions corresponding to AM = 2, 3, . . . are of fairly large intensity. An octahedral cubic field splits up the energy levels of S > Q, that is for iron and gadolinium. These splittings are usually small. The largest splitting reported is for Gd3+ in ThOz for which 8c - 2d is about 0.18 Forbidden transitions corresponding to AM = 3 , 4 , 5 are about & to $a of the main AM = 1 absorption line at 3 cm wavelength. This crystal should be suitable for maser operation in the 3-10 cm range. Axial crystal field symmetries may result in fairly large splittings. These are in some cases large enough for millimeter microwave generation or amplification. Among the more promising materials are corundum (A1203),spinel (MgOA1203),beryl (3BeO.AL03.6Si02) and rutile (TiOz). A few tentative initial aplittings are reported here. Trivalent chromium: corundum (ruby) :0.385 cm-'; spinel: 0.989 cm-l; beryl splitting of about 1.78 cm-'; and in rutile 1.45. Trivalent iron: corundum: 1.1 cm-1; rutile about 4.13 cm-'. Beryl and spinel also show fairly large splittings. Gadolinium salts show splittings of the order of 0.1-2 crn-' in axial fields. Indeed in the first successful three level Maser a crystal of ethyl sulfate containing gadolinium was used. The relaxation of one of the transitions of gadolinium ( - Q + -4) was shortened by having its resonance frequency coincide with that of cerium in the same c r y ~ t a l . ~ ~ ~ ~ ~ ~ ~ This short review of the published data reveals only few salts which seem to be suitable for maser operation. Possibly, some of the organic substances may yield radical spectra suitable for maser operation. Three-level masers have been operated successfully by a number
26. SOLID STATE
MASER
169
of laboratories. The crystal used by most workers is K3Co(CN)6conThe group a t Lincoln Laboratory reports taining some Cr3+.204,214-216 noise temperatures less than 20°K, and the product of the square root of the gain times the band width about 2 X los sec-I. The maximum power emitted a t 2800 Mc/sec was about 3 pw for input of more than 10 mw of 9400 Mc power.216aIt should be mentioned here that noise measurements are rather difficult to make since the input and output connections, as well as some of the microwave components, reflect noise into the maser cavity and into the receiver. Another maser employed Cr3+ in ruby.217 The field of maser amplifiers is expanding and full of surprises. It has and will shed considerable light on the fundamentals of the relaxation processes in crystals. J. 0. Artman, N. Bloembergen, and S. Shapiro, Phys. Rev. 109, 1392 (1958). M. W. P. Strandberg, C. F. Davis, B. W. Faughman, R. L. Kyhl, and G. J. Wolga, Phys. Rev. 109, 1988 (1958). * l 6 A. L. McWhorter and J. W. Meyer, Phys. Rev. 109, 312 (1958). 214
*16
A. L.McWhorter, J. W. Meyer, and P. D. Strum, Phys. Rev.108,1642 (1957). G. Makhov, C. Kikuchi, J. Lambe, and R. W. Terhune, Phys. Rev. 109, 1399
2160 217
(1958).
22. SPIN-LATTICE RELAXATION TIME The spin-lattice relaxation time measures the energy transfer from the paramagnetic ions to the surroundings, that is to say, to the crystal lattice. This relaxation time T I , often called the longitudinal relaxation time, is defined for two energy levels by d
- (AN - AN,)
=
dt
-
(AN
- ANo) Ti
(22.1)
where A N is the population difference between the two levels; AN0 is this difference a t equilibrium. The differential equation expresses the rate at which the system approaches the equilibrium after having been disturbed by the absorption of energy. Here AN0 is defined by the Boltzmann distribution and is given by
ANO= N tanh (hv/2kT)
(22.2)
where N is the total population of the two levels. For microwave frequencies and high temperatures such that (hu/2kT) << 1, AN0
-
Nhv/2kT.
149
(22.2a)
150
IV. RELAXATION TIMES AND LINE WIDTH
The temperature T is the so-called spin temperature. One assumes that the system of magnetic dipoles, often called the spin system is in internal equilibrium. In absence of resonance absorption or external fields, this spin temperature corresponds to the lattice temperature. Systems having more than two levels behave essentially the same although the treatment is slightly more complex. Assume now that the equilibrium distribution is being disturbed, for example, by resonance absorption. What mechanisms are operative to restore the equilibrium distribution? This problem has been considered originally by Casimir and Du PrB, 160 Waller, 161 Kronig,IE2 Van Vleck,’68 and Heitler and Teller.Is4 According to these authors, there are two types of relaxations. One type is predominant a t high temperatures. This is an indirect process in which there is an energy transfer between the spins and the lattice by means of inelastic scattering of phonons. The spins absorb one phonon and scatter another. This process is sometimes called the Raman process. It leads for the case of S = to an expression in which T1 is proportional to A0/X2T7for T smaller than the Debye temperature, and proportional to l/Tz for T larger than the Debye temperature. These formulas indicate that the relaxation will be a function of the orbital contribution through A. Large splittings, as found in “ spin-only ” magnetic substances, will show long relaxation times. This is indeed observed for V2+, Cr3+, Mn2+, Fe*+,and Gd3+.On the other hand, the rare earth and uranium group or Ti3+,Fe2+,Co2+in octahedral complexes have short relaxation times. A detailed comparison of the relaxation times of various substances with these theoretical predictions has not been undertaken. The other type of interaction is the ‘(direct interaction” of the spin system with the lattice. In this case, the spin system emits phonons “on speaking terms” with the spins, that is, phonons having an energy distribution coinciding with the energy distribution of the spin system. This mechanism leads to a relaxation time proportional to A4/X2T. It would be of importance only a t low temperatures in which case the Raman process is ineffectual. One of the problems with the direct process, already discussed in detail by Van Vleck,166is that this process is not sufficient to explain the relaxation times. The number of phonons which are on ‘(speaking
+
H. B. G. Casimir and F. K. Du PrB, Physica 6 , 507 (1938). I. Waller, Z . Physik 70, 370 (1932). IE4 R. de L. ICronig, Physica 6, 33 (1939). 163 J. H. Van Vleck, Phys. Rev. 67, 426 (1940). 164 W. H. Heitler and E Teller, Proc. Roy. Soc. A166, 629 (1936). 166 J. H. Van Vleck, Phys. Rev. 69, 724 (1941). 180
161
22.
SPIN-LATTICE RELAXATION TIME
151
terms” with the spins is relatively small. These phonons will be unable to transfer the energy fast enough from the spin system to the lowtemperature bath to account for the short relaxation time. The coupling between the phonons on speaking terms with the majority of the phonons which contain the total thermal energy of the lattice (the lattice reservoir) is very small and is, in the main, through anharmonic terms in the elastic forces. Therefore, the temperature of the phonons on speaking terms will be essentially that of the spin system. A severe bottleneck is created in the transfer of the energy from the phonons on speaking terms to the lattice reservoir. In recent years, many more difficulties have appeared which cannot be accounted for by a simple direct phonon-spin interaction. One of the best mays of measuring relaxation is t o measure the susceptibility as a function of the frequency, a method employed in particular by Gorter and his zmsociatcs.166-16* These measurements often show a relaxation time by a factor of 10 larger than those derived from paramagnetic resonance a t microwave frequencies. 16* Moreover, the measurements of audiofrequencies have shown that the relaxation times of low temperatures depend on the size of the ~ r y s t a l , ~ ~ ~ - ~ ~ ~ and that the dependence on temperature and constant magnetic field is very much different from that expected on the basis of the theory by Waller*61and Van Vleck.IB3A distribution of relaxation times is sometimes observed rather than a single relaxation time.’?’ Two ways out this impasse have been suggested by G ~ r t e r ’ ’and ~ T o w ~ i e s . ~Gorter ’~ suggests that there may be transport of energy by phonons to cryst(a1boundaries or defects where the transformation t o heat may be easier. Gorter also considers the possibility that measurements by microwave saturation, that is, a t high power levels, show a rapid conversion of energy from the phonons t o heat. In this case, the bottleneck would be mainly between the spins and the phonons on speaking terms. A t low-frequency measurements in which the spin temperature is not raised appreciably, the bottleneck would be the small number of low-frequency phonons, that is, the transfer of the C. J. Gorter, “Paramagnetic Relaxation.” Elsevier, Amsterdam, 1947. A . H. Cooke, Repls. Progr. in Phys. 13, 276 (1950). I E R C. J. Gorter, “Progress in Low Temperature Physics,” Vol. 2, Chapter 9. North Holland, Amsterdam, 1955. 169 L. C. Van der Marel, J. Van der Brock, and C. J. Gorter, Physica 23,361 (1957). F. W. de Vrijer and C. J. Gorter, Physica 18, 549 (1952). 171 It. J. Benzie and A. H. Cooke, Proc. Phys. SOC.A63, 201 (1950). C. J. Gorter, L. C. Van der Marel, and B. Bolger, Physica 21, 103 (1955). J. A. Giordmaine, L. E. Alsop, F. R. Nash, and C. H. Townes,Phys. Rev. 109, 166
16’
302 (1958).
152
nT.
RELAXATION TIMES AND LINE WIDTH
energy from the phonons on speaking terms to the majority of the lattice phonons. Townes et aL1Ia suggests that there are two kinds of relaxation mechanisms a t low temperature, the usual spin-lattice relaxation and the lattice-bath relaxation. These authors feel that in many crystals, the latter mechanism predominates and this relaxation time, usually larger, is being measured. I n order t o overcome Van Vleck's contention of the small number of phonons on speaking terms, these authors suggest that the lattice modes are broadened by the interaction with the spins. The spin-lattice relaxation time is considered to be short of the order of sec. Each phonon then is interrupted many times and correspondingly, the frequency band of phonons on speaking terms is much larger. Townes et al. suggest a width of a few hundred Mc/sec, much wider than the usual paramagnetic line width. The measured relaxation refers to the transfer of energy from these phonons to the bath and it is predicted that this will show a concentration and crystal size dependence. I n addition, these may be a distribution of relaxation times because the lattice-bath relaxation will be influenced by transport phenomena as by diffusion of phonons to the lattice boundaries. No one so far has considered the influence of covalent bonding on relaxation times. It is possible that the bonding electrons overlapping with the magnetic electrons may be modulated by the spin flip. The bonding electrons probably are on speaking terms with a much larger number of phonons. Gorter"J8 has remarked that "the theoretical analysis of the behavior of the oscillators 'on speaking terms' is in a shockingly primitive state." It is impossible to predict quantitatively what relaxation time a given ion will have or how it is influenced by crystal field strength or crystal symmetries. An example will emphasize this point. In an octahedral symmetry Cr3+has a large relaxation time (lo+ sec) and the paramagnetic resonance can be observed with ease a t room temperature. In fluorite symmetry Co2+ has the same energy level scheme as Cr3+. No absorption lines were found down to 70°K.173a In general T 1may range from second's to lo-" second for various paramagnetic ions. The paramagnetic resonance spectrum of most ions in the 4 j and 5f group and of many ions in the other transition groups can be observed only a t liquid helium temperature because of the short spin-lattice relaxation time. Note Added in Proof. Absorption has been detected at 20°K (M. Dvir and W. Low, to be published). The strength of the crystal field is one-fifth for Co*+inthe
*78a
23.
DIPOLAR BROADENING
153
23. DIPOLAR BROADENING
The theory of dipolar broadening has been developed by Van Vleck in a classic paper.174He has treated in detail the particular case of free spins. The mutual potential of such dipoles is given by (23.1)
Van Vleck has shown that if the external field Ho is larger than the dipole-dipole interaction, then Eq. (23.1) can be approximated by
where eij is the angle between rij and the z axis, along which H O is applied. Here the so-called “satellite lines” appearing a t 2gj3H, 3gj3H, etc., have been omitted. It is very difficult to evaluate the detailed line shapes. Van Vleck has given expressions for the second and fourth moment of the absorption lines. The nth moment of an absorption line is defined J(Au)”x”(v) du (23.3) Sx”(u) du * In most spin resonance experiments, the frequency is held constant and the magnetic field is varied. In this case the moment is given by (23.4)
The second moment for the particular case of free spins is given by
The first term in Eq. (23.5) denotes the interaction of magnetically similar ions and the second term summed over k is the interaction between dissimilar ions. The broadening is caused by two actions. First, the various ions set up magnetic fields at any other ion. These fluorite symmetry in comparison with Cr*+ in octahedral symmetry and, therefore, results in a shorter relaxation time. 174 J. H. Van Vleck, Phys. Rev. 74, 1168 (1948).
154
W. RELAXATION TIMES AND LINE WIDTH
ions then see a different magnetic field than the externally applied field. In the case of paramagnetic ions, this is of considerable magnitude. The field of a nearest neighbor with a magnetic moment of about 1 Bohr magneton a t a distance of 3 A is of the order 350 gauss and the sum of the nearest neighbors may be over five hundred gauss. The interaction of the nuclear moments of the surrounding nuclei is negligible in comparison. In the case of hydrated salts, the mean half-width due to this cause is about 6 gauss and in deuterated salts, about 2 gauss. The second mechanism responsible for broadening is operative in like ions only. Two ions precessing with the same Larmor frequency will interact and cause their spin orientation to change. This reduces the average lifetime of the ion in a given state, and therefore gives rise to broadening. The relaxation time T 2is defined by 7'2 = + b ( v ) l m = (23.6) where g ( v ) is the normalized line shape function. The spin-spin interaction time Tr is approximately given by the inverse of the half-width of the absorption line if this width is caused only by the dipolar broadening.% The problem of the line shape as a function of concentration of paramagnetic ions in a lattice has been considered by Kittel and Abraham~."~ Their solution is essentially an extension of Van Vleck's results. The mean line width for similar ions I - 3 C O S ~eij )2 (23.7) h2< (Av2)> = 3s'(~!!' 4- 1) $ $ j
2
(
j
wherefis the probability that the lattice site is occupied by a magnetic ion. The sum is over all the lattice sites, whether occupied by a magnetic ion or not. The second moment is, therefore, proportional to the concentration. The fourth moment has been solved rigorously for cubic lattice with H o along 100 directions. Their results are h4
=
3h41
1
(0.098 - 0.021(S2-I- S)-l) .
(23.8)
Iff
= 1, that is, all sites occupied, Eq. (23.8) reduces to Van Vleck's result. In this case, the ratio of (I
+.
1'6
C. Kittel and E. Abrahams, Phye. Rev. 90, 238 (1953).
24. EXCHANGE
INTERACTION
155
For small concentrations, the value of the last bracket in Eq. (23.8) shows a sharp increase. The fourth moment increases relative to the second moment and results in peaking the lines. This is similar to exchange narrowing to be discussed in Section 24. The line shape resembles approximately a Lorentaian shape with a cutoff frequency. The half-width is proportional to the concentration, but the maximum intensity is independent of the concentration. In general, the line shape will be rather complicated with a Lorentaian shape in the center and approximately Gaussian in its tails. As the concentration is increased, the shape will approach more and more a Gaussian distribution. The general case of ions other than free spins is more complicated. The line width depends considerably on the direction along which it is measured. In magnetically dilute samples, one may resolve satellite lines symmetrically displaced with respect to the central line.176In more concentrated samples, the broadened line is made up of a large number of satellites. The detailed shape of the line will depend on the arrangement of the neighbors. Only if the arrangement is of high symmetry will the line shape approximate a Gaussian distribution. In general, the line shape will be anisotropic, since the dipolar arangement depends on (3 cos2 eij - l).1764
24. EXCHANGEINTERACTION Line width measurements in crystals containing large concentration of paramagnetic ions hardly ever agree with the theory of dipolar broadening. One of the main reasons for this is the exchange interactions between electrons. Diracl” proved that the exchange coupling is equivalent approximately to a potential of the form
-2
2 Jij(Si Sj). *
(24.1)
ij
The effect is electrostatic in origin and is connected with the overlap of the orbital wave functions on the type of symmetry in the represen*I6 B.
Bleaney, R. J. Elliott, and H. E. D. Scovil, PTOC. Phys. SOC.A84,933 (1951). Actually the true line shape is never a Gaussian. If the off-diagonal elements in the dipolar interaction are taken into account, the line shape shows small bumps near the wings. These deviations in the line shape result in a new relaxation mechanism called “cross-relaxation,” used by the spin-spin interaction (N. Bloembergen, s. Shapiro, P. s. Persham, J. 0. Artman, to be published). This cross-relaxation may in part explain some of the peculiarities encountered when measuring relaxation times in more concentrated crystal. P. A. M. Dirac, “The Principles of Quantum Mechanics,” Chapter 10. Oxford University Press, London and New York, 1935.
1760
156
N. RELAXATION TIMES AND LINE WIDTH
tation of the permutation group. The exchange integral J decreases very rapidly with distance and is negligible in dilute crystals. In concentrated crystals, one can, therefore, approximate Eq. (24.1) by an interaction between the nearest neighbors. I n this case, the interaction is the scalar product -2JZ(Si * Sj)summed over the nearest neighbors only. This form of an interaction is isotropic in contrast to the dipolar interaction which is a second-rank tensor and not spatially invariant. The scalar invariant of the exchange interaction is correct only when the spin-orbit interaction is neg1e~ted.l~~ The reason is that the orbital wave functions are not spherically symmetrical in crystals. The spins feel the departure from spherical symmetry by means of the spin-orbit coupling. The anisotropic exchange has been shown to give a second-rank tensor similar to that of the dipole interaction, that is,
jj>i >l
3(Si. rj)(Si. rii) r..2 V
(24.2)
This interaction is of importance in explaining the ferromagnetic anisotropy in cubic crystals. It has only seldom been employed in the discussion of line width measurements in concentrated paramagnetic ~ r y s t a 1 s . l The ~ ~ measurements of line width and shape and their interpretation are rather difficult, even when assuming dipolar and isotropic exchange interaction only. Van V l e ~ k , Pryce ' ~ ~ and StevenslBOand Anderson181~182 have calculated the combined effect of exchange and dipolar interaction on the line width. Their calculations which are long and laborious can be summarized as follows. 1. The spins are all identical and S' = +. Van Vleck's calculations show that the isotropic exchange interaction contributes to the fourth moment and not to the second moment. Since the total area of the line cannot change, the effect is that the center part of the line is narrowed and the tails broadened. This peaking of the line shape is called exchange narrowing. 2. The spins are not identical. This may occur when the ions do not precess about parallel axes, as for example in the crystals where there is a zero field splitting of the energy levels. In this case, the isotropic exchange interaction contributes to the second moment and the line J. H. Van Vleck, Phys. Rev. 62, 1178 (1937). J. F.Ollom and J. H. Van Vleck, Physica 17, 205 (1951). 180 M.H.L. Pryce and K. W. H. Stevens, Proc. Phys. SOC A68, 36 (1950). *a1 P. W.Anderson, J . Phys. SOC.Japan 9,316 (1954). 181 P.W.Anderson and M. T. Weiss, Revs. Modern Phys. 26, 269 (1953).
1'8 1'9
24.
EXCHANGE INTERACTION
157
width is broadened. In general, both exchange broadening and narrowing are present and the resultant line width depends on the relative contribution of each type of interaction. 3. Strong exchange interaction between dissimilar ions. The only case considered so far is that of S' = 6. For slow rate of exchange, the resonance spectrum of different magnetic types of ions in a unit cell consists of distinct resolved lines having different g values. The exchange will contribute to the broadening of the lines as outlined in Section 2. As the rate of jumping increases, the lines will draw together. For J >> (gl - g2)/3H,the lines will coalesce into one line a t a mean g value given by hv = +(gl g 2 ) P H ; g1 and gz are the g factors of two dissimilar ions. At very strong exchange, the line will show exchange narrowing. This effect has been observed on copper sulfate ~ r y s t a l s . ~Bagguley ~ ~ - ~ ~and ~ Griffiths found that in order to resolve the two lines of the two inequivalent copper ions they had to increase the frequency to 24,000 Mc. Theirresultsindicatethat (gl - gz)PH J which gives J 0.15 cm-' for this crystal. Similar effects have been observed in I
+
-
-
+
+
J S i * Sz = +J[S'(S'
+ 1) - 4
(24.3)
where S' = S1 Sz = 0 or 1. The energy levels are given by - 3 J / 4 and J / 4 respectively. Which level is lowest will depend on the sign of J , whether it is ferro- or antiferromagnetic. If J is positive, the lowest level will be S = 0 and diamagnetic. The triplet spectrum will only be observed a t high temperatures. From a measurement of the relative intensities of the absorption lines at room temperature and a t a temperature where kT << J , the sign and magnitude of J can be determined. D. 31. S. Bagguley and J. H. E. Griffiths, Proc. Roy. Soc. A201, 366 (1950). M. H. L. Pryce, Nature 162, 538 (1948). K. Yosida, Progr. Theoret. Phys. (Kyoto) 6 , 1047 (1950). 186 H. Abe, K. Ono, I. Hayashi, J. Shimada, and K. Iwanaga, J . Phys. SOC. Japan 9, 814 (1954). lS7 B. Bleaney and K. D. Bowers, Proc. Roy. SOC.A214, 451 (1952). I B 8 H. Kumagai, H. Abe, and J. Shimada, Phys. Rev. 87, 385 (1952). F. W. Lancaster and W. Gordy, J . Chem. Phys. 19, 1181 (1951). 19" H. Abe, Phys. Rea. 92, 1572 (1953). IB3 IS4
158
by
IV. RELAXATION TIMES AND LINE WIDTH
The spin Hamiltonian with isotropic exchange interaction is given
+ g&H S2 + AiSi 11 + A2Sz I2 + JSi . (24.4) If A1 = A2, SI = S1 + S’, 11+ I , this can be rewritten X, +(gi + g2)BH S’ + +J[S’(S’ + 1) - 4 + +(gi - g2)BH (Si + +AS‘ I + +A(Ii - (Si - S2). (24.5) X, = g9H * Si
*
S2,
=
*
S2.
*
S2 =
I2
=
*
*
*
S2)
12)
*
If (gi - g2)PH << J , gBH >> A, J >> A perturbation theory shows that the triplet level is given by +J and +J f +(gl g2)OH f +Am, where m = ml m2. With the selection rules AM = & 1 and Am = 0, we have one electronic transition a t hu = gBH with g = (gl g2)/2 and 2m 1 hyperfine lines with separation of +A (and not as usual with a separation of A). The relative intensity of the hyperfine lines is given by the number of ways each value of m is formed. The foregoing spectrum is similar to that of nickel in a cubic field. Anisotropic exchange as explained above make the spins feel distortions from spherical symmetry. The Hamiltonian given by Eq. (24.4) will now contain a quantity ZiJ,S1iS2i. This term reflects the crystal symmetry of the spin Hamiltonian. I n the case of axial symmetry, this is equivalent to adding a term #D(SS2- +S’(S’ 1)) to the Hamiltonian. The effect is then similar to the nickel ion in an axial field in which the spin triplet has been split into a singlet at -D and a doublet a t D/2; D is related to exchange int,egral J by
+
+
+
+
+
D = -+J[HS11
-
212 - (Sl - 2121.
(24.6)
A spectrum corresponding to the above theory has been observed in copper a ~ e t a t e ~ *and ~ - ~copper ~~ propionate crystals.1g0 Similar spectra have been observed in Si containing As atoms as donors,191,192 to be discussed in Section 27. Griffiths et a1.Ig3have observed an exchange interaction in ammonium chloroiridate. In semidilute crystals, there is a reasonable chance that (IrC16)2- complexes occupy nearest-neighbor positions. There is then an exchange interaction of the two unpaired electrons via their transfer on the intervening chlorine ions. This gives rise to an exchange C. P. Slichter, Phys. Rev. 99, 479 (1955). G. Feher, R. C. Fletcher, and E. A. Gere, Phys. Rev. 100, 1784 (1955). 108 J. H. E. Griffiths, J. Owen, J. G. Park,and M. F. Partridge, Phys. Rev.106,1345 191 192
(1957).
25.
METHODS OF MEASURING RELAXATION TIMES
159
coupled pair of Ir ions. Assuming the Hamiltonian given by Eq. (24.5) and anisotropic exchange, they find
J,
=
T0.84,
J, = 10.64,
J, = k0.20 cm-I.
These results indicate that paramagnetic resonance techniques are a powerful technique in measurement exchange interactions. No doubt this technique will be applied to many other ions in the nd transition elements.
25. METHODS OF MEASURING RELAXATION TIMES The methods of measuring the various contributions to the line width do not differ essentially from those used in nuclear resonance. The measurements are on the whole, however, more difficult to perform since the spin-lattice relaxation time is very short. I n addition, there may be other factors contributing to the line widths. We shall discuss these factors briefly.
a. Other Sources of Broadening At very large dilution, one may expect the dipolar and exchange contribution to the line width to be negligible. In practice, one finds that in many crystals, there is a residual line width which cannot be attributed to any of the relaxation mechanisms so far considered. This line width is particularly great for paramagnetic crystals having a large zero field splitting. I n dilute nickel fluosilicate, for example, the dipolar width should not be more than 6 gauss and in a deuterated crystal, about 2 gauss. In practice, it is found that the line width may be anywhere between 20-50 gauss, depending on the care with which the crystal was grown. Moreover, the line width is anisotropic. It is thought that the line width is caused by small variations in zero field splittings. Small differencesin D may result from three causes: (a) defects and dislocations in the neighborhood of the magnetic ions; (b) thermal vibrations of the surroundings; (c) twinning and mosaic structure of the crystal. For large zero field splittings, these factors will contribute considerably to the line width. There seems to have been no experimental systematic study of these effects. A special case of this broadening is found in crystals of cubic symmetry. In some of these crystals, as in V2+or Cr3+ in MgO, the ground state of the paramagnetic ion is not split by the cubic field. The theoretical line width for very large dilution is less than the experimental width of about 6 gauss. It is possible that some of the
160
JS'. RELAXATION TIMES AND LINE WIDTH
reasons advanced above may be operative in splitting the ground state. Possibly, the Jahn-Teller effect may cause a small splitting as well. Other sources of broadening are unresolved hyperfine structure lines (see Section 26) and hyperfine structure interaction with the surrounding diamagnetic nuclei by means of covalent bonding. b. Measurements of Ta
In concentrated salts, there is always present the dipolar and exchange interaction. The method used so far is to measure carefully the line shape from the wings to the center and establish the second and fourth moments. In these absorption measurements (x"), one has to be careful to exclude dispersion effects (x'). One has to take precautions to prevent the possibility of frequency pulling of the oscillator by the strong absorption of the concentrated paramagnetic salt in the cavity. One calculates the dipolar width, appropriately summing over all the lattice points. This may not be easy in some cases, in particular when the spectrum is highly anisotropic and when the crystals contain many inequivalent ions with different g factors. One subtracts the measured dipolar width from the measured width and estimates the isotropic and anisotropic contributions from the exchange interaction. This method has been used by a number of authors, in particular on nickel salts.194-196 Another method, which has not been tried so far, is to measure the width as a function of the dilution. At not too high dilution, the contribution from the exchange interaction should already be relatively small and the line width a line or function of the concentration of the paramagnetic ions. A measure of the exchange interaction may be obtained occasionally from the spectra of isolated pairs of paramagnetic ions. Assuming that the exchange contributions will be the same in concentrated crystals, the dipolar width can be estimated from the measured line width. c. Measurement of T I
The methods of measuring T I are the same as in nuclear resonance, that is, either by saturations or by pulse methods. The saturation method has been extensively used.lQ7J98Bloch's equations yield the K. W. H. Stevens, Proc. Roy. SOC.A214, 237 (1952). E. Ishiguro, K. Kambe, and T. Usui, Physica 17, 310 (1951). 108 H. Kumagai, K. Ono, I. Hayashi, H. Abe, J. Shimada, H. ShBro, H. Ibamoto, and S. Tachimori, J . Phys. SOC.Japan 9, 369 (1954). 197 G. Feher and H. E. D. Scovil, Phys. Rev. 106, 760 (1957). I98 A. H. Eschenfelder and R. T. Weidner, Phys. Rev. 92, 869 (1953).
194
196
25.
METHODS OF MEASURING RELAXATION TIMES
161
imaginary part of the rf susceptibility as
At resonance frequency X'I
LJ
=
wo and
= ~xowo
1
1'2
p2Hvr2T T
+
7
1
(25.2) 2
Here p is the dipole moment for the transition, i.e., I( = gBI; Hr/ is the rotating component of the applied field. The saturation factor S is defined as 1 (25.3) Since the absorbed power is proportional to XI', one measures the absorption a t intense excitation and a t relatively low power. Here T Zis measured from the line shape a t very large dilution and a t low power. From the knowledge of S and T2,T I can be obtained. The procedure gives a unique relaxation time for a system with two levels. For systems with many levels, there may be more than one relaxation process operative and one may not be able to associate one single relaxation time with each pair of levels. I n this case, this method measures T I which is the rate a t which the energy is transferred to the lattice via all relaxation processes. 1c180 A variation of this method has been used by Gi~rdmaine."~The spin system was saturated by pulsed or cw energy. A second oscillator with too little rf power to produce saturation was used t o probe the microwave absorption as a function of the time after the saturating energy had been removed. The two oscillators were swept synchronously past the paramagnetic resonance. The time difference of the two oscillators could be varied by either changing the frequency difference or the rate of sweep. This way, the rate at which equilibrium was approached could be measured d i r e ~ t l y . 1 ~ ~ " Methods analogous to the spin echo method used by Hahnlg9in nuclear resonance could be used in principle and have, indeed, been proposed. This technique would probably give the most accurate and Another method used in the determination of 2'1 involves a measurement of the z component of the magnetization [R. W. Damon, Revs. Modern Phys. 26, 239 (1953)l. This method is suitable for crystals with a large concentration of paramagnetic substances. lee E. L. Hahn, Phys. Rev. 80, 580 (1950). l9*0
162
IV. RELAXATION TIMES AND LINE WIDTH
direct measurements of TI. It is, however, difficult to perform since the pulse would have to last for a very short time for substances having short relaxation times. Fast pulse techniques and circuitry would have to be used and these are not particularly easy in the microwave range. 26. SOLIDSTATEMASER
The name “maser” has been coined by Townes el aZ.200 for molecular systems which give “microwave amplification by stimulated emission of radiation.” Bloembergen201and Bassov and Prokhorov2021208 have proposed that paramagnetic systems may be suitably employed for low noise microwave amplifiers or oscillators. Shortly afterwards, a number of laboratories operated such a paramagnetic solid state maser s ~ ~ ~ e s ~ f ~The l l practical y . ~ ~ ~ interest - ~ ~ in~ the usefulness of this device has stimulated fundamental research in the basic phenomena of energy exchange between the spin system and the surroundings. General radiation theory shows that transition between two levels a and b of a free atomic system is determined by three Einstein coefficients Ad, Bd,Bb.. The A d is the probability of an atom to make a spontaneous transition to a lower state and is given by’ (26.1)
where pd is the matrix elements of the dipole transition and v d is the transition frequency between the two levels. In a typical experiment, p d is of the order of Bohr magneton, v d , about loLo cps which yield Ad sec-l. Spontaneous transitions are, therefore, very rare for paramagnetic systems and will hardly affect the population difference between the two levels. It will, however, because of its lack of coherency, be a source of noise. At given temperature, the population difference between the two levels is given by Eq. (22.2). If it were possible to invert the population of the spin system, we would have a system in which the stimulated emmission would be larger than the stimulated absorption. The transition probability of free atomic system between two *Oo J. P. Gordon, H. J. Zeiger, and C. H.Townes, Phys. Rev. 99, 1264 (1955).
-
*01
101
N. Bloembergen, Phya. Rev. 104, 324 (1956). N. G. Basaov and A. M. Prokhorov, J . Ezpll. Theoret. Phye. (USSR) 27, 4131 (1954).
N. G. Baesov and A. M. Prokhorov, Proc. Acad. Sci. (USSE) 101, 37 (1945). lo*A. L. McWhorter and J. W. Meyer, Phy8. Rev. 109, 312 (1958). 20) J. Combrisson, A. Honig, and C. H. Townes, Compt. rend. 242, 2451 (1956). so( H. E. D. Scovil, G. Feher, and H. Seidel, Phy8. Rev. 106, 762 (1957). 008
26.
163
SOLID STATE MASER
levels a and b caused by a stimulating rotating field H,, is given by
The instantaneous rate of transfer of energy is P
=
P(t - ti)
=
d huo - Wd.
(26.3)
at
The probability that the spin will have an interruption due to interaction with phonons is given by 1 e-(t-td/T: -
dt,
7'2
1-
The average power is given then by pAV =
t
~
~
(
1 -t ti> e-(t-f1)/T: 7'2
at.
(26.4)
For T I# Tn,this yields
A t resonance v = vo and for N ions
The average power omitted or absorbed is therefore a function of TI,T2, v , and of the density N of the paramagnetic ions. The ratio of the emitted power for excess AN ions in the upper state to the incident power at resonance is
where the incident power is given by
cAH,,' 8*
and A is the cross section. The gain of the system will be given by
G
= ea'.
164
JY. RELAXATION TIMES AND LINE WIDTH
In order to achieve a large gain, one has to use large length I in which the spins interact with the radiation field. This is achieved conveniently by means of a resonant cavity in which the sample is placed. In addition, Eq. (26.7) shows that the sample should have a large magnetic moment and long relaxation time. The gain of the induced emission has to be large enough to overcome the losses in the cavity. The noise figure of such an amplifier can be very small. The increment in noise power is given by207
dPnoi.. = ANI'P, dx
-
dx
+ ( ~ dx2 +, ~ hvrN2 d x .
(26.8)
The first term is the induced emission where AN = N2 - N 1 is the difference in the unpaired numbers of the spins in the two levels, and r is the matrix elements of the transition; P , is the noise power. The second term defines the losses in the cavity where a, is the attenuation constant. The third term is the blackbody radiation of the walls of the cavity at temperature T Oof the cavity, and the last term the spontaneous emission from level Nz to level N1. A detailed discussion of the nature and size of the noise has been given by Pound208and by Strandberg.:!Og Their analysis indicate that the limiting noise figure is given approximately as '
P,,,,, is the noise power a t the input and PnoiW, a t the output of the amplifying system. Equation (26.9) holds true only for the limiting case of large gain (G >> 1) and when the gain coefficient a exceeds the losses caused by the attenuation in the waveguide (a>> a,). If N 2 Nz - N I , the limiting noise temperature is given by hulk, which for 1-cm radiation is about 1.5"K. This is, of course, an amplifier having a very low noise. If such amplifiers could be constructed, one of the important uses would be for astronomical investigations. The source noise of interstellar space is about 1.3'K and a suitable radio telescope followed by such an amplifier could detect, for example, the 21-cm hydrogen lines far out in interstellar space. Negative temperatures, that is reversal of population, can be achieved by a number of processes. Three possibilities have been actively persued. These are (a) pulse methods, (b) rapid passage, and (c) transitions in a multilevel scheme.
-
J. P. Wittke, PTOC. IRE 46, 291 (1957). R. V. Pound, Ann. Phys. [13] 1, 24 (1957). 109 M. W. P. Strandberg, Phye. Rev. 106, 617 (1957).
I07
*o*
26.
SOLID STATE MASER
165
a. Pulsed System If a pulse of duration
is applied a t the resonance frequency to the spin system, then the spins of a free system will be reversed. This will hold true as well in a solid state system, provided t << TI or Tz. This method of inversion has some practical disadvantages. The pulse must be of the exact resonance frequency and must be of rather short duration. Moreover, for such short duration, the applied driving field must be of relatively large intensity. The system will amplify only a fraction of the time and will decay as equilibrium is reestablished.
b. Adiabatic Fast Passage Bloch210has shown that if the frequency (or the magnetic field) is swept quickly past the resonance frequency (or resonance field), an inversion of the energy levels may be obtained under certain conditions. The conditions are: (a) the passage must be rapid compared with T I or T z ;(b) the passage must be adiabatic, that is, slow compared with the driving frequency so that the magnetic moment can follow the changes in the frequency; (c) the rf field has to be relatively large; the number of ions in the crystal for maser action must be larger than
Here Q L is the loaded Q of the cavity, V , is the volume of the cavity, AH is the full width a t half-maximum of the spin resonance line, and
F. Bloch, Phys. Rev. 70, 460 (1946). G. Feher, J. P. Gordon, E. Buehler, E. A. Gere, and C. D. Thurmond, Phy.3. Rev. 109, 221 (1958).
166
W. RELAXATION
TIMES AND LINE WIDTH
c. Multilevel Scheme If a paramagnetic substance has more than two levels, that is, S' 2 1, it is sometimes possible to populate selectively one of the higher levels and thus obtain stimulated transitions to the ground state.201This is best illustrated for the three level case in Fig. 22. One saturates the transition v13 between levels one and three so that the population of these two levels is about equal, that is, N1- N3. A low
31'
v32
-J-L
FIG.22. Energy scheme for a three level ma8er.
power signal vzg will now cause transitions between levels three and two. Since, at low temperatures, N s> N 2 , this may give rise to stimulated emission a t v 2 3 . The conditions for the successful operation of such an inversion process are the following: (a) There must be at least three levels among which transitions are permitted. (b) The spin-lattice relaxation T l s must be long to permit saturation. (c) The relaxation time T s 2should preferably be larger than T21. If not, the population Na and N s will be equal after some time. The last condition is equivalent to
~21/T21f vaz/Tti. Bloembergen has shown that, under these conditions, the power emitted by the magnetic substance is
p = -Nh%2 (~21/T21- vsz/Tar)Waz 3kT 1/T2a 1/Tiz Wa2
+
+
(26.11)
26.
167
SOLID STATE MASER
where Wanis the rate a t which transitions caused by the stimulating signal occur between levels two and three, and N is the number of spins occupying the three levels. It is to be noted that Javan212and Yatziv213have shown that double quantum groups may modify condition (c) somewhat. A close inspection of the three conditions indicates that only a few inorganic crystals containing paramagnetic ions will be suitable for maser operation. Condition (a) rules out all substances which have S' = and also those substances for which the ground-state splitting is small. Transition probabilities corresponding to AM = + 2 are only large if the zero field splitting is of the order of the Zeeman energy. The requirement of large relaxation time is met in particular by substances which have an orbital singlet, that is, Cr3+, V2+, Fe3+, Mn2+, Cu2+, Ni2+, Eu2+, Gd3+ in octahedral complexes. It is conceivable that other ions in tetrahedral or fluorite symmetry might also have long relaxation times. To get reasonable amplification, one needs a large number of spins ( N ) in order to overcome the losses in the resonant structure. Preferably, one should use crystals (a) with one type of magnetic unit cell, (b) with S' not larger than one, (c) with ions which do not have a nuclear spin, (d) or those ions with a nuclear spin but a hyperfine structure splitting smaller than the width caused by the relaxation processes. Otherwise the levels will be split into 21 1 sublevels and the intensity of the stimulated emission would be reduced correspondingly. Therefore, the ions W+(Z = s), Mn2+(Z = $), EulS3(I= +), EuIKK(Z = 0 , Cu'18(I = +), Cusa(I= $) are not very suitable. Of the remaining ions Cr3+, Fe3+ and Gdx+ in nearly octahedral symmetry seem t o be the most promising substances. Nickel salts are not as promising. Although many of the nickel salts show the necessary initial splittings, the line width of all the transitions are relatively large, probably caused by inhomogeneous broadening. Only a fraction of these ions will give stimulated emission. It is therefore necessary to look for ions with small line width. I n order that transitions corresponding to AM = 2, 3, . . should be observed with fairly large intensities, one needs crystal field symmetries which cause initial splittings of the same order of magnitude as the microwave frequency to be amplified. The transition probability Wa2 is given by
+
+
.
A. Javan, Phys. Rev. 107, 1579 (1957). S. Yatziv, Phys. Rev. in press.
168
IV. RELAXATION TIMES AND LINE WIDTH
where 1 <21J,13> l2 is a measure of the intensity of the transition from levels 2 to 3 and H s ( V 3 Z ) is the signal frequency field. I n order that the power emitted should be larger than that dissipated in the microwave circuit, W32 should be as large as possible. This can be achieved if the matrix element <21J(3> is large, and if the line width caused by spin-spin interaction is small. Another advantage of the multilevel maser is now discernible. If the zero field separation of the levels is sufficiently large, i.e., of the order of the frequency to be amplified, only a small magnetic field would be necessary to obtain the proper energy level spacings corresponding to v13 and V Z S . Moreover, a t these low fields the energy levels are usually not equally spaced (since we are not working in the Paschen-Back region), and transitions corresponding to AM = 2, 3, . . . are of fairly large intensity. An octahedral cubic field splits up the energy levels of S > Q, that is for iron and gadolinium. These splittings are usually small. The largest splitting reported is for Gd3+ in ThOz for which 8c - 2d is about 0.18 Forbidden transitions corresponding to AM = 3 , 4 , 5 are about & to $a of the main AM = 1 absorption line at 3 cm wavelength. This crystal should be suitable for maser operation in the 3-10 cm range. Axial crystal field symmetries may result in fairly large splittings. These are in some cases large enough for millimeter microwave generation or amplification. Among the more promising materials are corundum (A1203),spinel (MgOA1203),beryl (3BeO.AL03.6Si02) and rutile (TiOz). A few tentative initial aplittings are reported here. Trivalent chromium: corundum (ruby) :0.385 cm-'; spinel: 0.989 cm-l; beryl splitting of about 1.78 cm-'; and in rutile 1.45. Trivalent iron: corundum: 1.1 cm-1; rutile about 4.13 cm-'. Beryl and spinel also show fairly large splittings. Gadolinium salts show splittings of the order of 0.1-2 crn-' in axial fields. Indeed in the first successful three level Maser a crystal of ethyl sulfate containing gadolinium was used. The relaxation of one of the transitions of gadolinium ( - Q + -4) was shortened by having its resonance frequency coincide with that of cerium in the same c r y ~ t a l . ~ ~ ~ ~ ~ ~ ~ This short review of the published data reveals only few salts which seem to be suitable for maser operation. Possibly, some of the organic substances may yield radical spectra suitable for maser operation. Three-level masers have been operated successfully by a number
26. SOLID STATE
MASER
169
of laboratories. The crystal used by most workers is K3Co(CN)6conThe group a t Lincoln Laboratory reports taining some Cr3+.204,214-216 noise temperatures less than 20°K, and the product of the square root of the gain times the band width about 2 X los sec-I. The maximum power emitted a t 2800 Mc/sec was about 3 pw for input of more than 10 mw of 9400 Mc power.216aIt should be mentioned here that noise measurements are rather difficult to make since the input and output connections, as well as some of the microwave components, reflect noise into the maser cavity and into the receiver. Another maser employed Cr3+ in ruby.217 The field of maser amplifiers is expanding and full of surprises. It has and will shed considerable light on the fundamentals of the relaxation processes in crystals. J. 0. Artman, N. Bloembergen, and S. Shapiro, Phys. Rev. 109, 1392 (1958). M. W. P. Strandberg, C. F. Davis, B. W. Faughman, R. L. Kyhl, and G. J. Wolga, Phys. Rev. 109, 1988 (1958). * l 6 A. L. McWhorter and J. W. Meyer, Phys. Rev. 109, 312 (1958). 214
*16
A. L.McWhorter, J. W. Meyer, and P. D. Strum, Phys. Rev.108,1642 (1957). G. Makhov, C. Kikuchi, J. Lambe, and R. W. Terhune, Phys. Rev. 109, 1399
2160 217
(1958).