Some experiments on paramagnetic relaxation at low temperatures

Physica XVI, no 3

Maart 1950

SOME E X P E R I M E N T S ON PARAMAGNETIC RELAXATION AT LOW T E M P E R A T U R E S b y D. B I J L Commun. No. 280b from the Kamerlingh Onnes Laboratory at Leiden

Summary Paramagnetic relaxation at audio-frequencies has been studied in manganese ammonium sulphate, manganese sulphate, gadolinium sulphate and copper potassium sulphate. There appears to be a continuous distribution of relaxation constants except in the case of the manganese salts. In all cases second order absorption and emission processes are still important in the liquid helium region.

1. Introduction. In this paper we present the results of some measurements on p a r a m a g n e t i c relaxation between 1 and 4°K and 14 and 20°K. A n u m b e r of substances have been examined and will discussed below. The interpretation of the results is carried out as in a previous paper 1) (hence forth quoted as loc. cit.). We therefore only briefly summarize the formulae required. According to C a s i m i r and D u P r 6 1 1 ) 1~): •'/•0 = ( l - - F ) +

F/(1 + O2v2) :

(1-

F)+

½ F ( 1 - - t g h y)

X"/Zo = Fov/(1 + 02.'2) = ½F sech y,

(1) (2)

where Z' and Z" are the real and i m a g i n a r y part of the differential susceptibility resp., S0 = the static susceptibility, v = the frequency of the alternating magnetic field, Q = the spin-lattice relaxation constant, y = In 0v. If the magnetic m o m e n t M is a function of H / T (M = / ( H / T ) , H is the magnetic fieldstrength and T is the absolute temperature, F is given b y F =/'H2/(b +/'H2), (3) w h e r e / ' = the first derivative of / of its argument, b = the constant in the expression of the specific heat of the spin system in zero field cM = b/T 2. Finally 0 is determined b y 0 = 2~CH/a,

- -

2 6 9

- -

(4)

270

i). BIJL

where cn is the specific heat of the spin system in a magnetic field H, given b y cn = (b + / ' H 2 ) / T 2, and a is the coefficient of t h e r m a l contact between the spin system and the lattice. According to (1) and (2) the Z" versus X' plot is a semicircle with centre on the z'-axis, which is passing t h r o u g h the points Z' = X0 and Z' ---- 1 - - F = Zoo on the same axis. In m a n y cases where no satisfactory agreement between the experimental results and the C a s i m i r - D u P r 6 formulae (1) and (2) exists, the Z"- versus Z'- plot still is a circular arc, b u t flatter t h a n previously, passing through the same points as before on the z'-axis, but now with its centre above the z'-axis. In this case (1) and (2) have to be replaced b y ~'/Z0 =: (1 - - F )

+ ½F(1--

sinh (1 - - a)~r ) cosh (1 - - a)y + sin ½a~r COS ½a.~

x " l x o = ½F

cosh (1 - - a)y + sin ½oar

,

(5)

(6)

which contain (1) a n d (2) as special cases for a =: 0 and where ½oar = the acute angle between the z'-axis and the radius of the arc drawn to the point Zoo. These formulae can be interpreted in terms of a continuous distribution G of relaxation constants, which is given b y x4) I sin aJr ds G(s)ds - - " 2~r cosh (1 - - a)s - - cos a~ where s = In (e/ear) and Oav is the m e a n value of the relaxation constants. G has a m a x i m u m for Q ---- eav. The width of G can be described by the ratio el/2/Qav, where 0~/2 is the highest value of Q for which the value of G is half the m a x i m u m value; 01/~/Q~vis determined b y the equation 2 - - cos av-r = cosh {(1 - - a) In (01yJ0av)}According to V a n V 1 e c k's 15) t h e o r y at sufficiently high t e m peratures Q = ~(0) (l + x2)/(1 + Qx2) (7) or a -- A(1 +

px 2) =

Q(0) (1 +

X2)/O.

where x 2 ---- H2(C/b), e(0) and A are functions of T only and p should be independent of T(0 < p < 1), while e(0) should be proportional to T -7. At very low temperatures Q should be proportional to H-2T-I.

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271

2. R e s u l t s . 1. M a n g e n e s e s a 1 t s. a. Mn(NH4)2(SO4) 2. 6H20. We s t u d i e d a sample p r e p a r e d from MnSO 4. 6H20 and (NH4)2SO 4, b o t h Analar Analytical Ragent. This sample has been e x a m i n e d b o t h at liquid h y d r o g e n and liquid helium t e m p e r a t u r e s . At all t e m p e r a tures the C a s i m i r - d u P r 6 formulae (loc. cir., (1), (2)) and the B r o n s-V a n V 1 e c k f o r m u l a (loc. cit., (8), c o m p a r e fig. 1) were satisfied satisfactorily. T h e value of p was the same for all t e m p e r a tures, as can be seen from t a b l e I.

/

/ l+X 2

~ 5

10

1S

20

F i g . 1. M n ( N H 4 ) 2 ( S O 4 ) 2 . 6 H 2 0 (l + x2)/103 X @ as a f u n c t i o n of x 2. (~) T = 14.3 ° K X T ~- 20.3 ° K TABLE

1

Mn(NH4)2 (SO~)~. 6HzO

T

OK

p

~(0) X l0 s

20.3 14.3 4.21

0.43 0.43 0.44

1.06 5.86 50.3

Q(oo) × I0'

2.47 13.6 117

b/C X 10 -e 0.64 0.64 0.64

Our value of b/C agreed v e r y well with B r o e r's results at higher t e m p e r a t u r e s 2), b u t our value of p is slightly lower. B r o e r

272

D. BIJL

f o u n d p =- 0.50. B e t w e e n 14 and 20°K e is n e a r l y p r o p o r t i o n a l to T - 5 ; n e a r 4°K Q is p r o p o r t i o n a l to a lower negative power ( - - 2 to - - 3). b. M n S O 4 . 4 H 2 0 . The sample studied was t a k e n from a puriss i m u m sample of K a h 1 b a u m. I t was found, t h a t b o t h dispersion and a b s o r p t i o n satisfied the C a s i m i r - d u P r 6 formulae well. We c a l c u l a t e d b/C == 6.2 × 106 Oersted 2, which agrees with the value obtained b y G o r t e r and T e u n i s s e n 8). The values of Q are collected in table II. C o n t r a r y to Mn(NH4)2(SO4) 2. 6H20 t h e y do not fit the B r o n s - V a n Vleck formula (compare fig. 2). In the t e m p e r a t u r e range used o~-~ T -4"7. T A B L E II MnSO4.4H20 H(Oersteds)

I I

Q X 10~

I 670

1120

1685

2250

3370

4030

20.5 18.4 14.4

1.37 2.56 9.3

1.52 2.82 10.5

1.82 3.30 12.5

2.13 3.90 14.5

2.64 4.90 18.2

2.78 5.12 19.0

F

0.070

T °K

0.165

0.315

0.445

0.640

0.705

c. Previous e x p e r i m e n t s on p a r a m a g n e t i c relaxation at v e r y low t e m p e r a t u r e s (1 to 4°K) and at t e m p e r a t u r e s above 77°K indicate, t h a t at higher t e m p e r a t u r e s the C a s i m i r - D u P r 6 formulae are satisfied m u c h b e t t e r t h a n at lower t e m p e r a t u r e s . The results at lowest t e m p e r a t u r e s often can be b e t t e r i n t e r p r e t e d in terms of a continuous distribution of r e l a x a t i o n constants t h a n with one relaxation constant. No exceptions have been found and the present results on manganese salts agree with this tendency. At all t e m p e r a t u r e s investigated Mn(NH4)2(SO4)2.6H20 satisfies the B r o n s-V a n V 1 e c k formula loc. cir. (8); a p a r t perhaps from a slight decrease of the value of p with decreasing t e m p e r a t u r e . We therefore m a y conclude t h a t in this substance even at t e m p e r a tures as low as 14 to 20°K second order processes are p r e d o m i n a n t in the relaxation process. It should he r e m a r k e d however, t h a t the t e m p e r a t u r e coefficient of Q(Q ~ T -5) is lower t h a n V a n Vleck and previously Waller and K r o n i g predicted Q ~ T - 7 ) . Generally V a n

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273

V 1 e c k's theory does not give a satisfactory description of the dependence of @ on T. This has been pointed out previously by B r 0 e 12). M n S Q . 4 H 2 0 clearly deviates from V a n V 1 e c k's theory. This may be due to the magnetic interaction of the Mn ++ions which certainly is much stronger in this salt than in Mn(NH4) 2 (S04)2.6H20. V a n V 1 e c k neglected the possible influence of

/ 1.0

/

Y

0.5

~

0 ~ X

2

F i g . 2. M n S O 4 . 4 H 2 0 Q T = 20.3°K,

2

(1 + x2)/103 >( Q a s a f u n c t i o n o f x 2. ~ T = 18.4°K, × T = 14.4°K.

magnetic interaction on the relaxation process. This may be a too crude approximation in the case of a not very dilute salt. It is of some interest to compare the values of @of Mn(H4N)2 (S04) 2. 6H20 and FeNH4(S04) 2. 12H20. The difference in magnitude of @ probably can be qualitatively understood by remembering the difference in size of Mn ++ and Fe +++. These ions are iso-electronic with the same lowest state (3d 5 6S) when free. The radii are Physica XVI

[8

274

D. BIJL

0.80 and 0.67 A respectively. In both substances the paramagnetic ion is surrounded by an octahedron of water molecules. The splitting of the higher orbital levels of the free ion probably is larger for Fe + + + than for Mn ++ as a consequence of the difference in size, and therefore the distance between the lowest orbital levels in the crystal is smaller. According to V a n V 1 e c k's theory the relaxation constants of the Fe +++ should be smaller than those of Mn ++. This agrees with the experimental results. The remarkable difference in the dependence of ~ on H - - only the manganese salt agrees with B r o n s-V a n V 1 e c k's formulae loc. cit. (8) -- is more difficult to explain as according to V a n V 1 e c k's theory both substances should be expected to behave similarly.

2. Gadolinium sulphale, Gd2(SO4)3.8H20. a. This substance has been studied between 1 and 4°K by D e H a a s and D u P r 6 4). Magnetic fields up to 2000 Oersteds and frequencies between 25 and 60 cycles/sec, were used. In the present experiments we extended both the range of field strength and the range of frequency. The sample used was taken from the same batch of crystals as before. b. We found that Z' and Z" agreed well with the formulae (5) and (6) of loc. cit. Table III contains some values of F together with vaTABLE

III

Gd,ISO4)~. 8H,O T ~ 4.15°K

H Oersteds

~Oav x I 0 '

1120 1685 2250 3370 4030

25 29 37 45 55

[

[

T =

~hlJ0a v

0 a v x 10 ~

2.06 2.14 ] .9 "t 1.5 ~ 1.3 g

(1oo) (12o) (15o) (19o) (230)

3.00°K ]

F

~t/,/~a v

(I.8) (1.6) (1.8) (1.6)

0.170 0.442 0.578 0. 720 0.800

lues of Qav and &/JQa, for different values of field strength at two temperatures. The values at 3.00°K only could be estimated by extrapolation and therefore are uncertain. From the results at 4.15°K we may conclude, that ~l/2/~av decreases with increasing H. The dependence of Q~/2/Qavon T probably is small. The relaxation constant Qav does not seem to agree with formula (8) of loc. cil. We estimated that ~v ~ T --~ in the temperature region used so that probably second order processes still are important in the liquid helium range. It must be remarked that the previous values

SOME EXPERIMENTS

ON P A R A M A G N E T I C

RELAXATION

275

of Qav are 4) at least a factor 10 too small. T h e y had only a provisional character, however, as the frequencies used were comp a r a t i v e l y low. F r o m the values of F we obtained b/C = 3.5 × 106 Oersted 2 or H h = 1900 Oersteds, in reasonable agreement with the value found at higher t e m p e r a t u r e s (b/C = 3.9 × 106 Oersted 2, G o rt e r et al. 5)) and the values deduced from caloric measurements (b/C=3.8× 106 Oersted 2, V a n D i j k and A u e r e ) ; b/C= 3.9 × l 0 6 0 e r s t e d 2, G i a u q u e and M c D o u g a l l V ) ) . T h e v a l u e given b y D e H a a s and Du P r 6 i s b / C = 3 . 0 × 1060ersted 2 and is lower t h a n the present value. Finally it m a y be added t h a t a more systematic s t u d y of the relaxation constants would have been desirable, but was not possible as a consequence of the large values of 0av.

3. Copper potassium sulphal,,, CuK2(SO4) 2. 6H20. a. This substance has been studied in the liquid helium range previously s). A sample of the same stock has been reinvestigated in wider ranges of constant magnetic field and t e m p e r a t u r e t h a n before. b. B o t h absorption and dispersion agreed with the formula (5) and (6) from loc. cit. At the highest t e m p e r a t u r e ( T - - 4.015°K), e.~,, and ~l/~/eav could be estimated. The results are collected in table IV. The present values of Qav are m u c h larger t h a n the previous values, but t h e y are certainly much more reliable. It was not possible to s t u d y the dependence of Q,v on H and T carefully; 0~v clearly increases with increasing field strength and ~v seems to increase rapidly at lower temperatures. Again el/2/eav decreases with increasing field strength. F r o m the values of F we obtained b/C ---- 0.10 × l06 Oersted 2 in good agreement with B r o e r and K e m p e r m a n ' s value 9) b/C = 0.12 × 106 0 e r s t e d 2. The present value of b/C proved to be T A B L E IV CuK,(SO~)~.6H20 H Oersteds 113 225 340 450 657

I [ 0av x 103 (25) 30 40 45 50

T = 4.015°K 0'/J~av

F

1.6 1.5 1.4 1.3

0.125 0.340 0.540 0.670 0.810

276

D. BIJL

independent of temperature and we believe the dependence on T found previously to be spurious.

4. Concluding r*.:marks, a. According to the results of the present experiments between 1 and 4°K both absorption and dispersion can be better interpreted in terms of a continuous distribution of relaxation constants than with a single relaxation constant as C as i m i r and Du P r ~'s theory does. Moreover the details of this distribution are strongly structure sensitive and depend in a complicated way on H and T. (The manganese salts are not necessarily an exception since they were examined at liquid hydrogen temperatures). This probably means that not all the assumptions underlying C a s i m i r and D u P r ~ ' s theory (compareloc. cir.) are satisfied. Without doubt in our experiments thermodynamical equilibrium in the spin system was established all the time in agreement with C a s i m i r and D u P r 6's second assumption. Consequently we believe that only C a s i m i r and D u P r 6's first assumption - - the substance is homogeneous - - often is not satisfied. A detailed picture of such an inhomogeneity is lacking, but it m a y have the character of a domain structure. Recent experiments on the heat conductivity of chromium potassiumalum seem to point into that direction 10). b. According to V a n V 1 e c k's theory at liquid helium temperatures only first order absorption and emission should be important and consequently Q ~ T - l . The much higher negative power of T found experimentally indicates that second order processes still are important, even if we take into account that V a n V 1 e c k's theory does not very well account for the observed dependence of Q on T. V a n V l e c k ' s theory namely predicts a too high power of T even when the dependence of Q on H is predicted allright. (Compare for instance manganese ammonium sulphate at liquid hydrogen temperatures). It m a y be remarked that for a rigorous test of V a n V 1 e c k's theory of first order processes still lower temperatures would be required. In conclusion I want to thank Prof. C. J. G o r t e r for his kind interest and Mr. P. W i n k e 1, nat. phil. cand., for his valuable help ~vith the experiments and calculations. Received 7-1-50.

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