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Magnetic absorption in the spin system of some paramagnetic salts at about 1325 MHz
Hadders, H. Locher, P.R. Gorter, C. J. 1958
Physica XXIV 839-847
MAGNETIC ABSORPTION IN THE SPIN SYSTEM OF SOME PARAMAGNETIC SALTS AT ABOUT 1325 MHz by H. HADDERS, P. R. LOCHER and C. J. GORTER Communication No. 311¢ from the Kamerlingh Onnes Laboratorium, Leiden, Nederland
Synopsis An apparatus (fig. 1) has been constructed for the measurement of pargmagnetic losses and paramagnetic susceptibilities at a frequency of 1325 MHz. The properties of this set up are discussed; particularly a method is given b y which the measured X" can be related to the static susceptibility S0. Results have been presented on the losses of powdered CrK(SO4)2.12H20, FeNH4(SO4)2.12H20, Mn(NH4)2(SO4)2.6H~O and Cu(NH4)2(SO4)2.6H~O at liquid hydrogen temperatures as a function of a parallel and of a perpendicular static magnetic field (figs 3, 4, 5 and 6). I n a provisional theoretical discussion it is suggested that in parallel fields the absorption connected with the nondiagonal elements of the magnetic moment plays a dominating r61e.
1. Introduction. In the framework of the investigations carried out in this laboratory on the magnetic behaviour of paramagnetic substances at high frequencies, the magnetic absorption of a series of salts has been investigated previously at liquid hydrogen temperatures by the caloric method 1) in the frequency interval between 10 and 100 MHz. At the low frequency side these investigations have been extended by a bridge method 2), in which the dispersion (%') as well as the absorption (%") may be measured. The present investigation is an extension of the same researches to the high frequency side. So far it concerns only a frequency of about 1325 MHz (23 cm). The method employed shows similarity to the usual microwave techniques employed for the study of magnetic resonances and the results in perpendicular external fields are rather similar indeed to the usual microwave paramagnetic resonance data, with the difference that in the present research also both %' and %" may be measured. One may say that the frequency used lies between the customary frequency regions for the study of paramagnetic relaxation and of paramagnetic resonance and that both phenomena and their connections may be studied here. In order to avoid confusion, it may be stressed that the ultra high frequency power used is so low that saturation and the connected spin-lattice relaxation time 8) play no r61e, while on the other side the frequencies are so high and the temperatures so low that the conventional spin-lattice relaxation phenomena 4)5) do not occur either. In the whole group of investigations mentioned, one has only to do with interactions between -
-
839
-
-
840
H. HADDERS, P. R. LOCHER AND C. J. GORTER
magnetic spins and between the spins and the constant term in the crystalline electric fields. Accordingly X0, Z' and Z" in the almost ideal paramagnetic salts studied are found to be inversely proportional to T in all fields. The method used to measure Z" and Z' will be described but of the results obtained only a few data on X" will be given and discussed in the present paper.
2. Experimental methods. The magnetic absorption and dispersion have been measured w i t h the aid of a single-loop coupled resonant cavity ½t in length; the signal, reflected by this cavity is separated from the incidant signal by means of a slotted diplexer, which acts as a magic T. The signal, reflected in resonance is a measure for the absorption. The condition of resonance is realised by adjusting on the minimum of the reflected signal upon variation of the frequency of the signal source. The signal source consists of a Rhode & Schwarz signal generator hr. BN 41026. With the help of fig. 1 the method and the set-up m a y be described in some detail. The signal of the generator is first divided by means of the magic TI in order to have a reference signal (from arm B). This reference signal passes through a variable attenuator and is detected with the crystal detector I. The other half (from arm A) passes a fixed 10 dB attenuator in order to avoid reaction on the generator and then reaches magic TII, which divides the signal into the arms A and B. The signal, reflected from the cavity divides into the arms E and H from which the first is absorbed mostly in the 10 dB attenuator, while the latter is detected with the crystal detector II. The variable small reflection on arm B is tuned in order to compensate the small leakage directly from E to H. Further there are, wherever necessary, adjustable stub-linestretcher combinations in order to match the several components. These stubs and stretchers have not been indicated in fig. 1 for the sake of simplicity. In order to minimize the dependance of the detector II output on the phase of the signal, reflected from the cavity, a magic T matching procedure is followed as described in M.I.T. Radiation Laboratory Series vol. 11 chapter 9. The transmission lines all are coaxial and are , as well as the attenuators, stubs, strechers, detectors and several small components, of the General Radio type 874. The magic T's, the cavity with accessories and the variable small reflection component, which consists of a double slug tuner with characteristic impedance Z0 termination, have been constructed in the laboratory workshops. The cavity consists of two parts, each ¼t in length, which can be taken apart in order to exchange the sample. After joining the two parts, t h e y are soldered hermetically. Before measuring at liquid hydrogen temperature, the cavity is evacuated and filled with helium gas at atmospheric pressure through a slot in the coaxial conductor. The coupling loop can be rotated,
PARAMAGNETIC ABSORPTION AT 1325 M H z
841
while one end of the loop slides over a small gold strip fixed on the inside of the outer conductor of the coaxial line. This slide contact proved to be good and sufficiently stable for our experiments. Thus the coupling is adjustable during a measurement. I B
E
VQ
I
TI H
A
Zo
att scr
=~
L\
=~ H ~
vsr
l
dct X
Fig. 1. B l o c k d i a g r a m of t h e e x p e r i m e n t a l set-up. C = cavity. G = generator. TI a n d TII = m a g i c T's. d e t I a n d d e t I I = c r y s t a l d e t e c t o r I a n d c r y s t a l d e t e c t o r II. V ---- electronic D.C. m i l l i v o l t m e t e r . M = m a g n e t - p o l e s for t h e s t a t i c field. S = sample. L = loop. s c r = s c r e w for r o t a t i n g t h e loop. a t t = 10 d B a t t e n u a t o r , w = variable attenuator. v s r = v a r i a b l e small r e f l e c t i o n e l e m e n t . D D " : see fig. 2.
The sample consists of many crystals of 0.2 to 1 mm diameter contained between two small strips of perspex. As indicated in fig. 2 the sample has been made so thin that the variation in direction of the ultra high frequency magnetic field is only small so that the constant field can be set in a good approximation perpendicular or parallel to the u.h.f, field. Assuming the u.h.f, field lines to be concentric circles, t h e contribution of Z±" to the observed absorption in a parallel external field, can be calculated from 7~" = (cos~ 9) Zll" + (sin~9)Z±" where 9 is the angle between the u.h.f, and the static field. For our set-up this yields about 0.5% o5 Z±". Experimentally an amount of 0.4% of Ztl" has been found in a measurement on CuC12. •2 H 2 0 with the static field at the value of resonance absorption. Considering now the dependance of the deflection V of the millivoltmeter on the absorption coefficient Z" in the Case that the cavity is in resonance, we m a y remark that the crystal detector II is always used in the square-law Physica X X l V
849.
H. HADDERS, P. R. LOCHER AND C. J. GORTER
region so t h a t the D.C.-output of this crystal is proportional to F 2 where F is the reflection coefficient of the cavity. So we can write for the D.C.output ~ deflection of the mV-meter V = c F 2, if the reference signal is zero. For several substances the absorption is found not to vary any more in high fields of the order of 10000 Oe. We shall assume the magnetic absorption to be zero in that case, though it is clear that a field independent term of the absorption could thus be ignored. We shall indicate the resonance reflection coefficient in the supposedly zero absorption case by/'oo and in a field H
P
Fig. 2. Full scale cross section of the lower part of the cavity, at DD' as indicated in fig. 1. S = sample, i ----inner conductor. P = perspex. by I n . Near resonance the impedance of the cavity is given by Z -----j(coL -- - 1/o~C) + R at a suitable point of the transmission line as described in M.I.T. Radiation Laboratory Series vol. 8 chapter 7. With the general expression r = (z -
lo)/(z + Zo)
•in which Z0 = the characteristic impedance of the transmission line, it can be shown that F 2 ( = [F[ 2) has a minimum (resonance by definition) for Z = R if the quality factor is sufficiently high. Thus in resonance Z = RH = ---- Roo + RM in which Roo corresponds to the losses in the cavity without magnetic absorption and R• corresponds to the magnetic losses. RM = = 4~qoJrLpZ" and RM/Roo = 4~qQupZ" with Qu = o~rL/Roo = the unloaded quality factor, q = the filling factor, p - ~ the density of the sample and X" = minus the imaginary component of the u.h.f, mass susceptibility. For the resonance reflection coefficient with magnetic absorption we can write FH = (Roo + R ~ --Zo)/(Roo + RM + Zo)
while Foo = (Roo - - Zo)/(Roo + Zo). Eliminating Z0 we get 2 ( r n -- too) 4~qQupZ" -----RM/Roo = (1 + 1"oo)(1 -
rH)
This is the undercoupled case. In the overcoupled case F n a n d / ' ~ change sign. With the change of the deflection of the mV-meter A V -~ C(lr'n2 - - F~o2)
PARAMAGNETIC
ABSORPTION
AT 1325 MHz
e43
and the deflection w i t h o u t m a g n e t i c losses V00 = cFoo 2 we can write ~n the u n d e r c o u p l e d case
4~qQupz" = RM/R~o =
2/"°°{%/(1 + AV/V00) -- 1} (1 + F00){1--F00%/(I+,dV/Voo)}
and developing this in t e r m s of A V/V00
4~qQupz"= r00 =
(i - / ' o o ) ( 1 + too)
(,~ v/v00)
F00(1 - 3 r ~ ) -
4(1 - F00)2(1 + F00)
(A v / v o j 2 + . . .
t~
tC
"~ x. ~.
\
Hj.
k~H X~mc
i.
F
__ Hc
,.
500
1000
1500
2000
~.'
O0 ~1
Fig. 3. The absorption in CrK(SO4)2.12H~O relative to that in zero field as a function of the static field He for H'e parallel to the u.h.f, field (Hit) and He perpendicular to the u.h.f, field (Ha_). The dashed line gives 1 -- F = b/(b + CHe2). T = 20.4°K. P a r t i c u l a r l y if we adjust the coupling loop so t h a t F00 = I/3, the second terrn vanishes and the d e f l e c t i o n / 1 V is in a good a p p r o x i m a t i o n p r o p o r t i o n a l to )('. F o r e x a m p l e Qu was m e a s u r e d to be a b o u t 2500 and q a b o u t 0.01, t h e n with p = 2, and •" = 2 × 10 -4 e.m. units 4~qQup)(' = 1/8 and AV/V00 = = 1/3. In this case the deviation f r o m the linear a p p r o x i m a t i o n is calculated to be 2 % . Mostly )(' was still less. In one case the linearity has been t e s t e d b y r o t a t i n g the m a g n e t and p r o v e d to be good. The real p a r t Z' of the mass susceptibility has been m e a s u r e d as the shift of the resonance f r e q u e n c y when the e x t e r n a l static field was varied. J u s t as in the case of )(' a field i n d e p e n d e n t c o n t r i b u t i o n to ;~' c a n n o t be detected. I n m a n y cases Z' m a y be supposed to vanish in a high parallel field. The frequencies could be directly read on a specially m o u n t e d microscale on the generator.
844
H. HADDERS, P. R. LOCHER AND C. J. GORTER
Assuming the frequency shift Avo between large parallel and large perpendicular fields to correspond with the static susceptibility go, it is possible to express the measured g"'s in terms of S0 if Foo is known. It gives some difficulty to determine Foo with reasonable accuracy, b u t if Poo is about 1/3 or less, as in our case, the relative accuracy of if~go is a few times better than that of Foo. The possibility of this can be outlined as follows. With g" = 0 we measure the deflection A V' out of the resonance value Voo caused b y the frequency shift Av0 out of resonance. With
P ---- i(oL -- 1/coC) + Roo -- Zo i(coL -- 1/o~C) + Roo + Zo and L = L0(1 + 4~zqpg0) it can be calculated in a w a y analogous to that used in the calculation of A V/Voo that if 4~qQupg0 is sufficiently small
AV'
__
Voo
=
(1 -- T'oo)(1 +/'oo) s (4~qQupgo) ~. 4_F',~Z
In combination with the expression for A V/V~ we can eliminate qQu and write
g" ' go =
(1 + Foo ~'
AV/Voo
½ i -- to./
In practice A V is measured over the full scale of the mV-meter b y eliminating the greater part of Voo b y means of the reference signal detected b y crystal I. Thus also amplitude variations of the signal generator are to a high degree compensated. The first measurements were carried out with a 1000 Hz square wave modulated signal and narrow band amplifiers after the detectors, b u t the sensitivity reached in this w a y is probably not better and the measurement o f / ' ~ is more complicated. The accuracy of g" is as a rule a few percents of the. highest if-value. That Of g'. is not quite so good. With the cavity used the spread of the measuring points corresponds to g" = 10-B and g' ---- 10-5 e.m. units. In future other forms of cavity m a y be tried out perhaps leading to a larger possible filling factor. :3. Results. A number of early results on Z" as a function of an external parallel and perpendicular field, obtained on powdered CrK(SO4) z. 12H~O, FeNH4(S04) 3.12H20, Mn(NH4) 2(S04) z. 6H9.0 and Cu(NH4)9.(SO4)9..6H20, have been given in the figures 3, 4, 5 and 6. The values of g" have been devided b y the values in zero external field under the same conditions. The measurements were made at the boiling temperature of hydrogen. In a few cases also measurements have been carried out at a liquid nitrogen temperature. Within the limits of the accuracy the same values of g"~/g"n=O were found. The top of the resonance curve for Cu(NH4)z(S04)2.6H~O in figure 6 was aboutffH/ffH=O = 5 b u t this result is
PARAMA.GNETIC ABSORPTION AT 1325 MHz
845
n o t reliable since F ~ was not equal to l/3 in this case. A recent m e a s u r e m e n t
with Foo = 1/3 gave %n"/%"n=o = 5.7. ,o
04
t-F He L
t0OO
2000
--I 3000
4OOOIZl
Fig. 4. T h e a b s o r p t i o n in F e N H 4 ( S 0 4 ) 2 . 1 2 H ~ O r e l a t i v e t o t h a t i n zero field as a f u n c t i o n
of t h e s t a t i c field H e for H e p a r a l l e l t o t h e u.h.f, field (Eli) a n d H e p e r p e n d i c u l a r t o t h e u.h.f, field (H_L). T h e d a s h e d line gives 1 - - F = b/(b + CHe~). T = 20.4°K. 1.o
0.8
0.4
O
'
~
I
|O00
:)000
3000
40000
Fig. 5. The absorption in Mn(NH4)~(SO¢)2.6H~O relative to that in zero field as a function of the static field/-/c for He parallel to the u.h.f, field (HII) and/-/o perpendicular to the u.h.f, field (H_L). The dashed line gives ! -- F = b/(b + C/-/eg'). T = 20.4°K.
4. Discussion. It is well known that the absorption in zero field can be described b y X" = Xop'v at frequencies where #'v ~ 1. If we take the results for#' thus found in earlier researches 4) and m u l t i p l y b y the present frequency
846
H. HADDERS, P. R. LOCHER AND C. ]. GORTER
of about 1325 MHz we obtain the folio'wing values of p'v for the four salts investigated: 2.1, 0.9, 1.05 and 7.5. For this reason already one should expect different results, the frequency used being relatively much larger in the
i
X~=o
x
0
~:~tr;--_. ~ ~ ~
0
H~;
~
m t
Fig. 6. The absorption in Cu (NH4) ~(SO4)3.6H~O relative to that in zero field as a function of the static field He for He parallel to the u.h.f, field (HII) and Hc perpendicular to the u.h.f, field (H±). The dashed line gives 1 -- F= b/(b+CHe2). T=20.4°K.
1000
copper salt than in the iron and manganese salts. As a m a t t e r of fact this difference appears very clearly when a perpendicular field is applied. In the copper salt, and to a lesser degree also in the chromium salt, a clear paramagnetic resonance is observed with a width of the order of 250 and 600 oersted respectively while in the manganese and iron salts only a broad excess in absorption over the parallel field value is observed. It m a y be mentioned that this agrees qualitatively with' the investigations on paramagnetic resonance at considerably higher frequencies e). The main results of the present research, however, are the data on Z" in parallel fields. It is evident from a comparison with the dotted 1-F curves
PARAMAGNETIC ABSORPTION AT 1325 M H z
847
in the figures 3, 4, 5 and 6 that there is no simple relation between tl~e field dependence of Z" and 1 - - F or (1 --F)Z, as was suggested by Russian investigators ~). It is striking that, with exception of the copper salt for which the accuracy is not sufficient, Z" decreases in strong fields faster than proportional with He -2. The rapid decrease of Z" in CrK(SO4)2.12H20 was already observed in the preceding research at lower frequencies in small fields. The increased absorption observed in the Cr-alums in fields of the order of 500 oersted 1), which has later been attributed to flip-flop processesS) involving the spins of two neighbouring ions, has disappeared at o u r m u c h higher frequency. The same applies to the spin-spin relaxation h u m p which in accordance with K r o n i g and B o u w k a m p ' s views 9) moves to lower frequencies when the field is increased. It seems therefore reasonable to ascribe the main part of Z" in paraUel fields to broad absorption bands connected with the non diagonal elements of the magnetic m o m e n t 10). As a m a t t e r of fact the absorption due to this cause should, with rising parallel field, not only decrease because the non diagonal elements decrease in magnitude but also because the remaining absorption bands are being displaced to higher frequencies. In a forthcoming paper we shall publish more results, in particular on Z' too, and we intend to present then a more complete discussion of the data. We are indebted to Mr. H. E. D e r k s e n who made the first draft of the experimental method used and to Mr. J. C. V e r s t e l l e for his stimulating interest. Received 20-6-58
REFERENCES 1) S m i t s , L. J., D e r k s e n , H. E., V e r s t e l l e , J. C. and G o r t e r , C. J., Commun. Kamerliagh Onnes Lab., Leiden No. 304d; Physica 22 (1956) 773. 2) V e r s t e l l e , J. C., D r e w e s , G. W. J. and G o r t e r , C. J., in preparation. 3)- E s e h e n f e l d e r , A. H. and W e i d n e r , R. T., Phys. Rev. 92 (1955) 869. G i o r d m a i n e , J. A., Alsop, L. E., Nash, F. R. and T o w n e s , C. H., Phys. Rev. 109 (1958) 302. G o r t e r , C. J., Van d e r Marel, L. C. and B61ger, B., Commun. Suppl. No. 109¢; Physiea 21 (1955) 103. 4) G o r t e r , C. J., Paramagnetic Ralaxation (Elsevier, Amsterdam 1947). 5) Van d e r Mare1, L. C., Thesis Leiden (1958). 6) B l e a n e y , B., P e n r o s e , R.P. and P l u m p t o n , B e t t y , I., Proc. roy. Soc. (London) A198(1949) 406. B a g g u l e y , D. M. S. and G r i f f i t h s , J. H. E., Proc. roy. Soc. (London) 204 (1950) 188. U b b i n k , J., P o u l i s , J. A. and G o r t e r , C. J., Commun. No. 283b; Physica 17 {1951) 215. 7) S h a p o s h n i k o v , I. G., Zhur. ~ksp. i teor. fizika 18 (1948) 533. G a r i f i a n o v , N.S., Zh. ~ksp. teor Fiz. 25 (1953) 359. S h a p o s h n i k o v , I. G., Izv. Akad. Nauk 20 (1956) 1255. 8) V e r s t e l l e , J. C., D r e w e s , G. W. J. and G o r t e r , C. J., Commun. No. 311b; Physica 24 (1958). 9) K r o n i g , R. and B o u w k a m p , C. J., Physiea 5 (1938) 521. 10) G o r t e r , C. J., Progress in low temperature Physics II (North Holland Publ. Comp., Amsterdam, 1957) Ch. IX.
Physica XXIV 839-847
MAGNETIC ABSORPTION IN THE SPIN SYSTEM OF SOME PARAMAGNETIC SALTS AT ABOUT 1325 MHz by H. HADDERS, P. R. LOCHER and C. J. GORTER Communication No. 311¢ from the Kamerlingh Onnes Laboratorium, Leiden, Nederland
Synopsis An apparatus (fig. 1) has been constructed for the measurement of pargmagnetic losses and paramagnetic susceptibilities at a frequency of 1325 MHz. The properties of this set up are discussed; particularly a method is given b y which the measured X" can be related to the static susceptibility S0. Results have been presented on the losses of powdered CrK(SO4)2.12H20, FeNH4(SO4)2.12H20, Mn(NH4)2(SO4)2.6H~O and Cu(NH4)2(SO4)2.6H~O at liquid hydrogen temperatures as a function of a parallel and of a perpendicular static magnetic field (figs 3, 4, 5 and 6). I n a provisional theoretical discussion it is suggested that in parallel fields the absorption connected with the nondiagonal elements of the magnetic moment plays a dominating r61e.
1. Introduction. In the framework of the investigations carried out in this laboratory on the magnetic behaviour of paramagnetic substances at high frequencies, the magnetic absorption of a series of salts has been investigated previously at liquid hydrogen temperatures by the caloric method 1) in the frequency interval between 10 and 100 MHz. At the low frequency side these investigations have been extended by a bridge method 2), in which the dispersion (%') as well as the absorption (%") may be measured. The present investigation is an extension of the same researches to the high frequency side. So far it concerns only a frequency of about 1325 MHz (23 cm). The method employed shows similarity to the usual microwave techniques employed for the study of magnetic resonances and the results in perpendicular external fields are rather similar indeed to the usual microwave paramagnetic resonance data, with the difference that in the present research also both %' and %" may be measured. One may say that the frequency used lies between the customary frequency regions for the study of paramagnetic relaxation and of paramagnetic resonance and that both phenomena and their connections may be studied here. In order to avoid confusion, it may be stressed that the ultra high frequency power used is so low that saturation and the connected spin-lattice relaxation time 8) play no r61e, while on the other side the frequencies are so high and the temperatures so low that the conventional spin-lattice relaxation phenomena 4)5) do not occur either. In the whole group of investigations mentioned, one has only to do with interactions between -
-
839
-
-
840
H. HADDERS, P. R. LOCHER AND C. J. GORTER
magnetic spins and between the spins and the constant term in the crystalline electric fields. Accordingly X0, Z' and Z" in the almost ideal paramagnetic salts studied are found to be inversely proportional to T in all fields. The method used to measure Z" and Z' will be described but of the results obtained only a few data on X" will be given and discussed in the present paper.
2. Experimental methods. The magnetic absorption and dispersion have been measured w i t h the aid of a single-loop coupled resonant cavity ½t in length; the signal, reflected by this cavity is separated from the incidant signal by means of a slotted diplexer, which acts as a magic T. The signal, reflected in resonance is a measure for the absorption. The condition of resonance is realised by adjusting on the minimum of the reflected signal upon variation of the frequency of the signal source. The signal source consists of a Rhode & Schwarz signal generator hr. BN 41026. With the help of fig. 1 the method and the set-up m a y be described in some detail. The signal of the generator is first divided by means of the magic TI in order to have a reference signal (from arm B). This reference signal passes through a variable attenuator and is detected with the crystal detector I. The other half (from arm A) passes a fixed 10 dB attenuator in order to avoid reaction on the generator and then reaches magic TII, which divides the signal into the arms A and B. The signal, reflected from the cavity divides into the arms E and H from which the first is absorbed mostly in the 10 dB attenuator, while the latter is detected with the crystal detector II. The variable small reflection on arm B is tuned in order to compensate the small leakage directly from E to H. Further there are, wherever necessary, adjustable stub-linestretcher combinations in order to match the several components. These stubs and stretchers have not been indicated in fig. 1 for the sake of simplicity. In order to minimize the dependance of the detector II output on the phase of the signal, reflected from the cavity, a magic T matching procedure is followed as described in M.I.T. Radiation Laboratory Series vol. 11 chapter 9. The transmission lines all are coaxial and are , as well as the attenuators, stubs, strechers, detectors and several small components, of the General Radio type 874. The magic T's, the cavity with accessories and the variable small reflection component, which consists of a double slug tuner with characteristic impedance Z0 termination, have been constructed in the laboratory workshops. The cavity consists of two parts, each ¼t in length, which can be taken apart in order to exchange the sample. After joining the two parts, t h e y are soldered hermetically. Before measuring at liquid hydrogen temperature, the cavity is evacuated and filled with helium gas at atmospheric pressure through a slot in the coaxial conductor. The coupling loop can be rotated,
PARAMAGNETIC ABSORPTION AT 1325 M H z
841
while one end of the loop slides over a small gold strip fixed on the inside of the outer conductor of the coaxial line. This slide contact proved to be good and sufficiently stable for our experiments. Thus the coupling is adjustable during a measurement. I B
E
VQ
I
TI H
A
Zo
att scr
=~
L\
=~ H ~
vsr
l
dct X
Fig. 1. B l o c k d i a g r a m of t h e e x p e r i m e n t a l set-up. C = cavity. G = generator. TI a n d TII = m a g i c T's. d e t I a n d d e t I I = c r y s t a l d e t e c t o r I a n d c r y s t a l d e t e c t o r II. V ---- electronic D.C. m i l l i v o l t m e t e r . M = m a g n e t - p o l e s for t h e s t a t i c field. S = sample. L = loop. s c r = s c r e w for r o t a t i n g t h e loop. a t t = 10 d B a t t e n u a t o r , w = variable attenuator. v s r = v a r i a b l e small r e f l e c t i o n e l e m e n t . D D " : see fig. 2.
The sample consists of many crystals of 0.2 to 1 mm diameter contained between two small strips of perspex. As indicated in fig. 2 the sample has been made so thin that the variation in direction of the ultra high frequency magnetic field is only small so that the constant field can be set in a good approximation perpendicular or parallel to the u.h.f, field. Assuming the u.h.f, field lines to be concentric circles, t h e contribution of Z±" to the observed absorption in a parallel external field, can be calculated from 7~" = (cos~ 9) Zll" + (sin~9)Z±" where 9 is the angle between the u.h.f, and the static field. For our set-up this yields about 0.5% o5 Z±". Experimentally an amount of 0.4% of Ztl" has been found in a measurement on CuC12. •2 H 2 0 with the static field at the value of resonance absorption. Considering now the dependance of the deflection V of the millivoltmeter on the absorption coefficient Z" in the Case that the cavity is in resonance, we m a y remark that the crystal detector II is always used in the square-law Physica X X l V
849.
H. HADDERS, P. R. LOCHER AND C. J. GORTER
region so t h a t the D.C.-output of this crystal is proportional to F 2 where F is the reflection coefficient of the cavity. So we can write for the D.C.output ~ deflection of the mV-meter V = c F 2, if the reference signal is zero. For several substances the absorption is found not to vary any more in high fields of the order of 10000 Oe. We shall assume the magnetic absorption to be zero in that case, though it is clear that a field independent term of the absorption could thus be ignored. We shall indicate the resonance reflection coefficient in the supposedly zero absorption case by/'oo and in a field H
P
Fig. 2. Full scale cross section of the lower part of the cavity, at DD' as indicated in fig. 1. S = sample, i ----inner conductor. P = perspex. by I n . Near resonance the impedance of the cavity is given by Z -----j(coL -- - 1/o~C) + R at a suitable point of the transmission line as described in M.I.T. Radiation Laboratory Series vol. 8 chapter 7. With the general expression r = (z -
lo)/(z + Zo)
•in which Z0 = the characteristic impedance of the transmission line, it can be shown that F 2 ( = [F[ 2) has a minimum (resonance by definition) for Z = R if the quality factor is sufficiently high. Thus in resonance Z = RH = ---- Roo + RM in which Roo corresponds to the losses in the cavity without magnetic absorption and R• corresponds to the magnetic losses. RM = = 4~qoJrLpZ" and RM/Roo = 4~qQupZ" with Qu = o~rL/Roo = the unloaded quality factor, q = the filling factor, p - ~ the density of the sample and X" = minus the imaginary component of the u.h.f, mass susceptibility. For the resonance reflection coefficient with magnetic absorption we can write FH = (Roo + R ~ --Zo)/(Roo + RM + Zo)
while Foo = (Roo - - Zo)/(Roo + Zo). Eliminating Z0 we get 2 ( r n -- too) 4~qQupZ" -----RM/Roo = (1 + 1"oo)(1 -
rH)
This is the undercoupled case. In the overcoupled case F n a n d / ' ~ change sign. With the change of the deflection of the mV-meter A V -~ C(lr'n2 - - F~o2)
PARAMAGNETIC
ABSORPTION
AT 1325 MHz
e43
and the deflection w i t h o u t m a g n e t i c losses V00 = cFoo 2 we can write ~n the u n d e r c o u p l e d case
4~qQupz" = RM/R~o =
2/"°°{%/(1 + AV/V00) -- 1} (1 + F00){1--F00%/(I+,dV/Voo)}
and developing this in t e r m s of A V/V00
4~qQupz"= r00 =
(i - / ' o o ) ( 1 + too)
(,~ v/v00)
F00(1 - 3 r ~ ) -
4(1 - F00)2(1 + F00)
(A v / v o j 2 + . . .
t~
tC
"~ x. ~.
\
Hj.
k~H X~mc
i.
F
__ Hc
,.
500
1000
1500
2000
~.'
O0 ~1
Fig. 3. The absorption in CrK(SO4)2.12H~O relative to that in zero field as a function of the static field He for H'e parallel to the u.h.f, field (Hit) and He perpendicular to the u.h.f, field (Ha_). The dashed line gives 1 -- F = b/(b + CHe2). T = 20.4°K. P a r t i c u l a r l y if we adjust the coupling loop so t h a t F00 = I/3, the second terrn vanishes and the d e f l e c t i o n / 1 V is in a good a p p r o x i m a t i o n p r o p o r t i o n a l to )('. F o r e x a m p l e Qu was m e a s u r e d to be a b o u t 2500 and q a b o u t 0.01, t h e n with p = 2, and •" = 2 × 10 -4 e.m. units 4~qQup)(' = 1/8 and AV/V00 = = 1/3. In this case the deviation f r o m the linear a p p r o x i m a t i o n is calculated to be 2 % . Mostly )(' was still less. In one case the linearity has been t e s t e d b y r o t a t i n g the m a g n e t and p r o v e d to be good. The real p a r t Z' of the mass susceptibility has been m e a s u r e d as the shift of the resonance f r e q u e n c y when the e x t e r n a l static field was varied. J u s t as in the case of )(' a field i n d e p e n d e n t c o n t r i b u t i o n to ;~' c a n n o t be detected. I n m a n y cases Z' m a y be supposed to vanish in a high parallel field. The frequencies could be directly read on a specially m o u n t e d microscale on the generator.
844
H. HADDERS, P. R. LOCHER AND C. J. GORTER
Assuming the frequency shift Avo between large parallel and large perpendicular fields to correspond with the static susceptibility go, it is possible to express the measured g"'s in terms of S0 if Foo is known. It gives some difficulty to determine Foo with reasonable accuracy, b u t if Poo is about 1/3 or less, as in our case, the relative accuracy of if~go is a few times better than that of Foo. The possibility of this can be outlined as follows. With g" = 0 we measure the deflection A V' out of the resonance value Voo caused b y the frequency shift Av0 out of resonance. With
P ---- i(oL -- 1/coC) + Roo -- Zo i(coL -- 1/o~C) + Roo + Zo and L = L0(1 + 4~zqpg0) it can be calculated in a w a y analogous to that used in the calculation of A V/Voo that if 4~qQupg0 is sufficiently small
AV'
__
Voo
=
(1 -- T'oo)(1 +/'oo) s (4~qQupgo) ~. 4_F',~Z
In combination with the expression for A V/V~ we can eliminate qQu and write
g" ' go =
(1 + Foo ~'
AV/Voo
½ i -- to./
In practice A V is measured over the full scale of the mV-meter b y eliminating the greater part of Voo b y means of the reference signal detected b y crystal I. Thus also amplitude variations of the signal generator are to a high degree compensated. The first measurements were carried out with a 1000 Hz square wave modulated signal and narrow band amplifiers after the detectors, b u t the sensitivity reached in this w a y is probably not better and the measurement o f / ' ~ is more complicated. The accuracy of g" is as a rule a few percents of the. highest if-value. That Of g'. is not quite so good. With the cavity used the spread of the measuring points corresponds to g" = 10-B and g' ---- 10-5 e.m. units. In future other forms of cavity m a y be tried out perhaps leading to a larger possible filling factor. :3. Results. A number of early results on Z" as a function of an external parallel and perpendicular field, obtained on powdered CrK(SO4) z. 12H~O, FeNH4(S04) 3.12H20, Mn(NH4) 2(S04) z. 6H9.0 and Cu(NH4)9.(SO4)9..6H20, have been given in the figures 3, 4, 5 and 6. The values of g" have been devided b y the values in zero external field under the same conditions. The measurements were made at the boiling temperature of hydrogen. In a few cases also measurements have been carried out at a liquid nitrogen temperature. Within the limits of the accuracy the same values of g"~/g"n=O were found. The top of the resonance curve for Cu(NH4)z(S04)2.6H~O in figure 6 was aboutffH/ffH=O = 5 b u t this result is
PARAMA.GNETIC ABSORPTION AT 1325 MHz
845
n o t reliable since F ~ was not equal to l/3 in this case. A recent m e a s u r e m e n t
with Foo = 1/3 gave %n"/%"n=o = 5.7. ,o
04
t-F He L
t0OO
2000
--I 3000
4OOOIZl
Fig. 4. T h e a b s o r p t i o n in F e N H 4 ( S 0 4 ) 2 . 1 2 H ~ O r e l a t i v e t o t h a t i n zero field as a f u n c t i o n
of t h e s t a t i c field H e for H e p a r a l l e l t o t h e u.h.f, field (Eli) a n d H e p e r p e n d i c u l a r t o t h e u.h.f, field (H_L). T h e d a s h e d line gives 1 - - F = b/(b + CHe~). T = 20.4°K. 1.o
0.8
0.4
O
'
~
I
|O00
:)000
3000
40000
Fig. 5. The absorption in Mn(NH4)~(SO¢)2.6H~O relative to that in zero field as a function of the static field/-/c for He parallel to the u.h.f, field (HII) and/-/o perpendicular to the u.h.f, field (H_L). The dashed line gives ! -- F = b/(b + C/-/eg'). T = 20.4°K.
4. Discussion. It is well known that the absorption in zero field can be described b y X" = Xop'v at frequencies where #'v ~ 1. If we take the results for#' thus found in earlier researches 4) and m u l t i p l y b y the present frequency
846
H. HADDERS, P. R. LOCHER AND C. ]. GORTER
of about 1325 MHz we obtain the folio'wing values of p'v for the four salts investigated: 2.1, 0.9, 1.05 and 7.5. For this reason already one should expect different results, the frequency used being relatively much larger in the
i
X~=o
x
0
~:~tr;--_. ~ ~ ~
0
H~;
~
m t
Fig. 6. The absorption in Cu (NH4) ~(SO4)3.6H~O relative to that in zero field as a function of the static field He for He parallel to the u.h.f, field (HII) and Hc perpendicular to the u.h.f, field (H±). The dashed line gives 1 -- F= b/(b+CHe2). T=20.4°K.
1000
copper salt than in the iron and manganese salts. As a m a t t e r of fact this difference appears very clearly when a perpendicular field is applied. In the copper salt, and to a lesser degree also in the chromium salt, a clear paramagnetic resonance is observed with a width of the order of 250 and 600 oersted respectively while in the manganese and iron salts only a broad excess in absorption over the parallel field value is observed. It m a y be mentioned that this agrees qualitatively with' the investigations on paramagnetic resonance at considerably higher frequencies e). The main results of the present research, however, are the data on Z" in parallel fields. It is evident from a comparison with the dotted 1-F curves
PARAMAGNETIC ABSORPTION AT 1325 M H z
847
in the figures 3, 4, 5 and 6 that there is no simple relation between tl~e field dependence of Z" and 1 - - F or (1 --F)Z, as was suggested by Russian investigators ~). It is striking that, with exception of the copper salt for which the accuracy is not sufficient, Z" decreases in strong fields faster than proportional with He -2. The rapid decrease of Z" in CrK(SO4)2.12H20 was already observed in the preceding research at lower frequencies in small fields. The increased absorption observed in the Cr-alums in fields of the order of 500 oersted 1), which has later been attributed to flip-flop processesS) involving the spins of two neighbouring ions, has disappeared at o u r m u c h higher frequency. The same applies to the spin-spin relaxation h u m p which in accordance with K r o n i g and B o u w k a m p ' s views 9) moves to lower frequencies when the field is increased. It seems therefore reasonable to ascribe the main part of Z" in paraUel fields to broad absorption bands connected with the non diagonal elements of the magnetic m o m e n t 10). As a m a t t e r of fact the absorption due to this cause should, with rising parallel field, not only decrease because the non diagonal elements decrease in magnitude but also because the remaining absorption bands are being displaced to higher frequencies. In a forthcoming paper we shall publish more results, in particular on Z' too, and we intend to present then a more complete discussion of the data. We are indebted to Mr. H. E. D e r k s e n who made the first draft of the experimental method used and to Mr. J. C. V e r s t e l l e for his stimulating interest. Received 20-6-58
REFERENCES 1) S m i t s , L. J., D e r k s e n , H. E., V e r s t e l l e , J. C. and G o r t e r , C. J., Commun. Kamerliagh Onnes Lab., Leiden No. 304d; Physica 22 (1956) 773. 2) V e r s t e l l e , J. C., D r e w e s , G. W. J. and G o r t e r , C. J., in preparation. 3)- E s e h e n f e l d e r , A. H. and W e i d n e r , R. T., Phys. Rev. 92 (1955) 869. G i o r d m a i n e , J. A., Alsop, L. E., Nash, F. R. and T o w n e s , C. H., Phys. Rev. 109 (1958) 302. G o r t e r , C. J., Van d e r Marel, L. C. and B61ger, B., Commun. Suppl. No. 109¢; Physiea 21 (1955) 103. 4) G o r t e r , C. J., Paramagnetic Ralaxation (Elsevier, Amsterdam 1947). 5) Van d e r Mare1, L. C., Thesis Leiden (1958). 6) B l e a n e y , B., P e n r o s e , R.P. and P l u m p t o n , B e t t y , I., Proc. roy. Soc. (London) A198(1949) 406. B a g g u l e y , D. M. S. and G r i f f i t h s , J. H. E., Proc. roy. Soc. (London) 204 (1950) 188. U b b i n k , J., P o u l i s , J. A. and G o r t e r , C. J., Commun. No. 283b; Physica 17 {1951) 215. 7) S h a p o s h n i k o v , I. G., Zhur. ~ksp. i teor. fizika 18 (1948) 533. G a r i f i a n o v , N.S., Zh. ~ksp. teor Fiz. 25 (1953) 359. S h a p o s h n i k o v , I. G., Izv. Akad. Nauk 20 (1956) 1255. 8) V e r s t e l l e , J. C., D r e w e s , G. W. J. and G o r t e r , C. J., Commun. No. 311b; Physica 24 (1958). 9) K r o n i g , R. and B o u w k a m p , C. J., Physiea 5 (1938) 521. 10) G o r t e r , C. J., Progress in low temperature Physics II (North Holland Publ. Comp., Amsterdam, 1957) Ch. IX.