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MAGNETIC RESONANCE IN DILUTE MAGNETIC ALLOYS : HYPERFINE SPLITTING OF Ag : Er167
R. Chui, R. Orbach, B. Gehman
To cite this version:
R. Chui, R. Orbach, B. Gehman. MAGNETIC RESONANCE IN DILUTE MAGNETIC ALLOYS : HYPERFINE SPLITTING OF Ag : Er167. Journal de Physique Colloques, 1971, 32 (C1), pp.C1- 909-C1-912. �10.1051/jphyscol:19711323�. �jpa-00214356�
JOURNAL DE PHYSIQUE Colloque C I, suppliment au no 2-3, Tome 32, Fivrier-Mars 1971, page C 1
-
909MAGNETIC RESONANCE IN DILUTE MAGNETIC ALLOYS
:HYPERFINE SPLITTING OF
Ag - :Eri6'
R. C H U I and R. ORBACH (*) Department of Physics University of California
Los Angeles, California, 90024, U. S. A.
and B. L. G E H M A N
Department of Physics University of California San Diego, California, 92037
R h r n b . - L a diffraction hyperfine de la raie de resonance d u moment localis6 de I'erbium dans l'argent metallique a ete observke k la fois en reflexion et en transmission pour des alliages de la solution solide & : Er de concentrations nominales 28, 65, 86 et 1 500 ppm. Une constante de couplage hyperfin A = 75 f 1,O cer- sted a etb trouvke pour rendre compte des positions individuelles des raies, dans la limite des erreurs experi- mentales.
Les rbsultats sont analyses en termes du couplage electron localid-electron de conduction. Trois regions sont theoriquement identifikes : 1 ) pas de resonance magnetique en forme de goulot 2) un goulot prbsent, mais l/Tse est petit vis-a-vis de la diffraction hyperfine ; et 3) un goulot present, et 1/Tse est grand vis-A-vis de la diffraction hyperfine. Les mesures sur & : Er semblent satisfaire A la categorie 1). On montre que la categorie 3) s'applique probablement a la plupart des experiences antkrieures sur les alliages de mktaux de transition, mais que la catCgorie 2) peut ttre obtenue a suffisamment basses temperatures. Des mesures de la largeur de raie dans la region de categorie 2) montreraient des elargissements d'khange et rendraient possible la determination de la valeur du couplage d'khange A partir des mesures de resonance magnetique, mtme dans le regime avec le goulot.
Abstract. - The hyperfine splitting of the erbium localized moment resonance line in silver metal has been observed both in reflection and transmission for nominal concentrations of 28, 65, 86 and 1.500 ppm solid solution A g : Er alloys.
A hyperfine coupling constant A = 75 f 1.0 ersteds was found to fit the individual line positions to within experimental accuracy.
The results are analyzed in terms of the coupled localized-conduction electron response function. Three regions are identified theoretically : 1 ) no magnetic resonance bottleneck ; 2) a bottleneck present, but 1 / T s e small compared to the hyperfine splitting ; and 3) a bottleneck present, and l/Tse large compared to the hyperfine splitting. The A x : Er mea- surements appear to fit category 1). It is shown that category 3) probably applies to most previous experiments on tran- sition metal alloys, but that category 2) might be obtainable at sufficiently low temperatures. Line width measurements in the region of category 2) would exhibit exchange broadenings, and enable the value of the exchange coupling to be extracted from magnetic resonance measurements even in the bottlenecked regime.
This paper reports the observation of the hyper- fine splitting of a localized moment resonance in a metal [12]. We have observed the resonance spectrum of Ag : Er16' for erbium concentrations (nominal) of 28,
65,
86 a n d 1,500 ppm. A representative observation is displayed in figure 1. I t will be seen that a strong central line (isotopes of erbium for which I = 0) is surrounded assymetrically by at least six hyperfine split sattelites (a seventh can be seen (?) on the right hand side of the I = 0 line). The central line corres- ponds t o a g value of 6.852+
0.05, now in agreement with recent measurements of Griffiths a n d Coles [I].The hyperfine line positions can be fitted (using second order calculations) by a coupling constant
A = 75 f 1 .O oersteds
.
Table I lists the observed a n d calculated hyperfine line positions. The cc missing n lines are hidden by the (I = 0) central line.
The magnetic resonance of Er3+ in T h o , has been observed by Abraham et al. [2]. They reported a g factor of 6.752
+
0.005 a n d a hyperfine coupling constant A = 73.8 f 0.1 cersteds. The dominance of the orbital contribution t o the hyperfine field allows us t o simply scale A according t o the resonance gfactor. Such a procedure results in A (scaled to Ag : Er) = 74.9, in excellent agreement with our measurements. Though the sattelite lines are t o o
FIG. 1. - Absorption derivative of an 86 ppm Ag : Er alloy made in reflection at X-band. The central ( I = 0) line occurs at Ho = 916,3 oersteds. The full chart sweep is one kilo-cersted.
The separation of the small peaks fit well with the predictions of second order hyperfine calculations :
in oersteds. The coefficient 3.07 = A212 Ho, where A = 75 mrs- (*) Supported in part by the National Science Foundation teds for this system (see Table I). Higher order corrections and the U. S. Office of Naval Research, Contract No. N00014- are negligible. The missing ),peaks of the eight line pattern
69-0200-4006. are hidden by the I = 0 central line.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711323
C 1-910 R. CHUI A N D R. ORBACH AND B. L. GEHMAN weak to determine their precise line shape ( ~ r ' ~ ~ is
only 22.8 % abundant, and the I = 712 spectrum intcn- sity is divided amongst eight lines) the central line (I = 0) exhibits the characteristic Bloembergen [3]
shape for the derivative response of magnetic resonance of fixed centers in a thick metallic slab. The ratio of derivative peak heights (AIB in the notation of Feher and Kip [4]) equals 2.7 (+ 0, - 0.3). The widths of the individual hyperfine components is approximately equal to the width of the central line.
All of these characteristics lead us to assert that we have indeed observed the hyperfine splitting of a localized moment in a metal. The question which must be resolved, however, is why no previous measu- rements in other dilute magnetic alloys have exhibited similar hyperfine split spectra. Residual (strain induced) linewidths, and line narrowing Ruderman-Kittel spin-spin exchange couplings could certainly explain ((away )> a number of measurements for which no splitting was observed, though it was known that a substantial hyperfine coupling existed. The former is certainly the reason that Coles and Griffiths [I] failed to observe the hyperfine splitting of Ag : Er alloys (their residual width amounted to 42Trsteds ; our samples exhibited residual widths as low as 13 aer- steds). The latter must be important in most transition metal alloys at usual reflection resonance conccntra- tions (Hirschkoff el al. [S] report Weiss constants of approximately 0.2, 0.05 and 0.02OK for conccntra- tions of 106, 30 and 9.0 ppm of
Cu
: Mn, respecti- vely). Yet, neither of these mechanisms can be opcra- tive to any effective degree in the very low concentra- tion conduction electron transmission measurements on Cu : Mn conducted by Schultz et al. [6]. Concen- trations as low as 13 ppm exhibited no evidence of hyperfine splittings. A scaled Weiss constant using the resultsofreference 5would indicate a mean exchange field of 70 cersteds, whereas the hyperfine coupling constant for these alloys is of ordcr 40 cersteds [7].Surely, some evidence of hyperfine splittings (or at least anomalous broadening) should have been obser- ved in such measurements, unless another reason exists for the absence of the effects of hypcrfine inter- actions.
We shall argue below that the exchange coupling bet- ween the localized momentand theconduction electrons determines the character of the resonance spectrum.
Specifically, there exist three regions, corresponding to weak, intermediate and strong exchange, which exhibit quite different magnetic resonance spectra.
We suggest that the results reported in this paper are appropriate to a system in the (( weak )> region, and that most of the transition metal alloys investigated so far lie on the borderline between intermediate and strong. In such a situation, the determining factor for the observation of a hyperfine splitting is the ratio between that splitting and the localized-to-conduction electron relaxation rate, l/Tsc. When the ratio is large, the splitting is observed ; when it is small, the reso- nance narrows, approaching a width appropriate t o the resonance in the absence of a hyperfine interac- tion. Because l/Tsc is proportional to temperature, it may be possible to move smoothly from one limi- ting regime to another by simply changing the tempe-
rature. Estimates indicate that l/Tsc for Cu : Mn is substantially greater than the hyperfine splitting for temperatures in the He4 temperature regime. However, at He3 temperatures, there exists a possibility that 1/TSe is comparable to the hyperfine splitting, and a t least a broadening of the resonance line might obtain.
The uncertainty results because of an incomplete knowledge of the magnitude of the exchange coupling constant. If such a splitting (or even broadening) could be observed, it would enable the extraction of an accurate value for the exchange constant. Conside- rable interest exists in a comparison of its magnitude with that determined from conductivity experiments [81.
To make our considerations more quantitative, we solve the molecular field equations for a dilute collec- tion of localized moments, each acted upon by a hyperfine ficld originating from their respective nuclei, dissolved in a metal. We specify the magnitude of this field in terms of the equavalent electronic frequency o,, for a nuclear spin of I =
4.
It is easy to generalize our results to arbitrary nuclear spin, and the results will be quoted a t the end of this paper. We define two electronic systems, appropriate the two orientations of nuclear spin<
I,>
= $-i.
We assume equal populations of the nuclear levels, so that two local moment magnetizations M I and M, (appropriate to I, =+
and -4
respectively) obey the following equations of motion :+
(11Tcs) [me -Ile(H +
i(M1+
M2))]- [(l/Ts,)+As] [MI,,)-$ X, (H+iLme)]
.
(1) Similarly, the conduction electron magnetization obeys the following equation of motion :
In these equations, identical to those of Langreth et al. [9], g, and g, are the localized and conduction elec- tron g factors, respectively ; me the conduction clec- tron magnetization ; H the total (static plus micro- wave) external magnetic iield ; H,,, the hyperfine field, equal to o,,,/g, (we take units where h = /*B = 1) ;
L is the exchange coupling appropriate to the magne- tizations ; X, and 1, are the static susceptibilities of the localized and conduction electrons ; A , and A , are the lattice relaxation rates for the localized and conduc- tion electrons ; l/Tse is the so-called Korringa exchange relaxation rate for the ct direction >) localized-to- conduction electrons, and finally l/Tc, is the so-called Overhauser exchange relaxation rate for the c< direc- tion )) conduction-to-localized electrons [lo]. At tem- peratures high compared to the Zeeman splittings, detailed balance requires that
x,/x,
= TcS/Tqc. Hence, at usual concentrations in the helium temperature range, l/Tes % ]ITs,.The solution of these equations in the limit of small exchange (no bottleneck) is trivial. Ignoring for the moment the (small amplitude) conduction electron root, one finds two resonance frequencies, each broa-
MAGNETIC RESONANCE IN DILUTE MAGNETIC ALLOYS : HYPERFINE SPLIlTING C 1 - 91 1
dened by the relaxatiori rates 1/TSe and A,, and split apart from one another in frequency by 2 o H F :
a + = as - i[(l/Ts,)
+
A,]+
UHFW - = o, - i[(l/Tsc)
+
A,] - o H F . (3) Here, cc), is the Zeeman frequency appropriate to the localized spins. It is this limit we refer to as region l), and to which we believe our results for - Ag : Er are appropriate.For a larger value of the exchange coupling, there occurs a new region where the magnetic resonance bottleneck begins to become important. In simple terms, this occurs when l/T,,
+
l/Tse becomes com- parable to, or larger than,I
o, - o, I and A, (where o, is the Zeeman frequency of the conduction electrons, and we assume A, is small). In the process of working out the solution to (1) and (2), for this region, one finds he must make a decision concerning the magni- tude of the ratio :This parameter can be thought of as a measure of the degree to which the hyperfine frequency shift is allowed to take place before the local magnetization is trans- mitted to the conduction magnetization through the agency of the exchange coupling (i. e. through l/T,,).
After this transfer of magnetization to the conduction electrons, a succeeding transfer occurs back to the local rnagnctization (through l/Tcs) but with esserrtially eq~ralprobability of returning to the same sublattice or to the other sublattice (with opposite hyperfine fre- quency shift). The fact that l/Te, is much larger than l/T,, means that the latter (slower) rate is the impor- tant one to compare with the hyperfine frequency, hence (4). The requirement that one be in the bottle- neck regime for this ratio to be important is related to the necessity of a phased relation between the conduc- tion and localized electron spin systems. If such a coherence exists, then the result of a small value for (4) implies a narrowing of the hyperfine spectrum (not a broadening as would occur in region 1) ; see eq. 3, where no coherence esists because the localized magnetization is rapidly transferred between sites with opposite nuclear spin directions (opposite hyper- fine frequencies). In the limit of vanishing (4), the bottlenecked resonance lineshape and width approa- ches that to be expected for an alloy with the same parameters, only in the absence of a nuclear spin interaction. We refer to this limit as region 3). In the regime where a bottleneck exists, but (4) is large, the hyperfine coupling is in fact fully resolved, but pos- sesses a width equal to HALF of the width exhibited in (3) for I =
3.
This reduction in width is caused by the character of the eigenstates in the bottlenecked regime, and will be discussed in detail in a forthcoming publication [ll]. We refer to this limit as region 2).It permits an experiment to exhibit the effects of exchange even in a bottlenecked region. The value
of the temperature dependent part of the linewidth of each hyperfine component can be used to extract an explicit value for the exchange coupling. When the generalization is made to arbitrary nuclear spin I, the same hyperfine splitting occurs as would be expected in an ionic material with the same parameters, but explicit considerations regarding the character of the eigenvectors lead to a conclusion [I I ] that each hyperfine component possesses a width equal to 2 I/(21
+
1) of that given in (3). In summation, the appropriate resonance frequencies and widths for each of the three regions are :region (1)
(weak exchange, no bottleneck.)
i
a+ Z (0, - 1'[(1/T,,) f A,]+
OH^ ;w - =(us - i[(liq,)
+
A,] - a,,,.region (2)
(stronger exchange, bottleneck present, but (4) $- 1.) o + z o, - i[2 1/(2 1
+
1)] [(l/Tsc)+
A,]+
o,, ;o- 2 0, - i[2 1/(2 1
+
l)] [(l/TSc)+
A,] - o,, .region (3)
(very strong exchange, bottleneck present, and (4)
<
1 .)The Cu : Mn alloy system has a hyperfine splitting of 40 G r s t e d s between adjacent hyperfine lines.
The value of 1/TSe is unknown because of the strong bottleneck appropriate to this alloy. However, (4) may increase to the vicinity of unity in the He3 temperature regime for sensible values of the exchange broadening (estimates indicate rates as low as l/Tse = 500 erstedsldegreej. Other systems which have promise are Mn in Ag, Au, Mg and Zn, as well as Eu in Ag and Mg metals.
Resonance line positions (in arsteds) for Er167 ( I = 712) : Ag. The calculated values are appropriate to a lzyperjin~oupling constant A = 75 arsteds. The line appropriate to M I = - 312 may just be seen in jigure 1 ; that appropriate to M , = - 112 is obscured completely by the I = 0 line. The asymmetry and shift of the line positions relative to the central line are caused by second order hyperfine eflects. The central line ( I = 0) occurs at 916.3 arsteds.
References
Experimental Calculated
- -
641.3
+
5 643.1696.3
+
5 699.6761.3
+
5 '762.4831.3 _+ 5 831.2
- - - 906.2
- - -
987.41,071.3
+
5 1,074.61,161.3 +_ 5 1.168.1
[I] GRIFFITHS (D.) and COLB (B. R.), Phys. Rev. Letters, J. Appl. Phys, 1968, 39, 844. A more recent g 1966, 16, 1093 ; and HIRST (L. L.), WILLIAMS value reported for &g : Er is 6.85 f 0.04 (pri- (Gwyn), GRIFFITHS (D.) and COLES (B. R.), vate communication, B. R. Coles).
C 1
-
912 R. CHUI AND R. ORBACH AND B. L. GEHMAN ABRAHAMS (M.), WEEKS (A.), CLARK (G. W.) andFINCH (C. B.), Phys. Rev., 1965, 137, A 138.
BLOEMBERGEN (N.), J. Appl. Phys., 1952, 23, 1379.
DYSON (F. J.), Phys. Rev., 1955 ; 98,349 ; FEHER (G.) and KIP (A. F.), ibid., 1955, 98, 349.
HIRSCHKOFF (E. C.), SYMKO ( 0 . G.) and WHEATLEY (J. C.), Phys. Letters, 1970, 33A, 19.
SCHULTZ (S.), SHANABARGER (M. R.) and PLATZ-
MAN (P. M.), Phys. Rev. Letters, 1967, 19, 749 ; more details are available from the thesis of M. R. Shanabarger, University of California, San Diego, California, 1970, unpublished.
CAMPBELL (I. A.), COMPTON (J. P.), WILLIAMS (I. R.) and WILSON (G. V. A.), Phys. Rev. Letters, 1967, 19, 1319.
BLANDAN (A.), J. Appl. Phys., 1968, 39, 1285.
LANGRETH (D. C.), COWAN (D. L.) and WILKINS (J. W.), Solid State Communications, 1968, 6 , 131.
For a review of the general theory and microscopic calculations for 1/Tse and 1/Tes, see DUPRAZ (J.), GIOVANNINI (B.), ORBACH (R.), RILEY (J. D.) and ZITKOVA (J.), Magnetic Resonance (Plenum Press, New York, 1970), p. 197.
BARNES (S, E.), DUPRAZ (J.) and ORBACH (R.), Sub- mitted for publication to J. Appl. Phys., 1970.
A preliminary presentation of these results has been published see CHUI (R.), ORBACH (R.) and GEH-
MAN (B. L.), Phys Rev., B, 1970,2, 2298.
