Nuclear Electric Quadrupole Interactions In Aluminum

NUCLEAR

ELECTRIC

QUADRUPOLE

INTERACTIONS

IN ALUMINUM*

T. J. ROWLANDt An experimental study of the nuclear magnetic resonance absorption of aluminum in solid solutions containing zinc or magnesium as solute is presented. The magnitude of the electric gradients caused at the Al lattice sites by the solute atoms is estimated and compared with the relative sizes of various solute atoms in a common solvent. Effects of precipitation and annealing have also been detected, and some possible applications are discussed. LES

INTERACTIONS

DES

QUADRUP6LES NUCLfiAIRES, DANS L’ALUMINIUM

BLECTRIQUES

On presente une etude experimentale de l’absorption magnetique, nucleaire de resonance de l’aluminium en solution solide contenant du zinc ou du magnesium comme solute. On a estime la grandeur des gradients electriques causes par la presence d’atomes de solute aux noeuds du reseau d’aluminium et on l’a compare aux dimensions relatives des differents atomes dans un solvant commun. On a constate aussi des effets de recuit et de precipitation; certaines applications possibles sont discutees. ELEKTRISCHE

KERNQUADRUPOLWECHSELWIRKUNGEN

IN ALUMINIUM

Es wird iiber eine Experimentaluntersuchung der kernmagnetischen Resonanzabsorption in festen Losungen von Zink oder Magnesium in Aluminium berichtet. Die Griissenordnung der Gradienten der elektrischen Felder, die an den Al-Gitterplatzen durch die gel&ten Atome hervorgerufen werden, wird abgeschatzt und mit den relativen Abmessungen der verschiedenen gel&ten Atome in einem gegebenen Lhungsmittel verglichen. Es wurden ausserdem Einfliisse von Ausscheidungen und Gliihungen aufgefunden. Einige miigliche Anwendungen wurden diskutiert.

INTRODUCTION

seen to be unchanged according to (1). Since all angles 0 are present in the polycrystalline samples, the satellite lines (transitions 5/2-+3/2, 3/2-+1/2, - l/2-+-3/2, and -3/2-+-S/2) take on a characteristic shape3 and cause a broadening, though no shift, of the resonance.’ The m= 1,/2--t- l/2 transition will be broadened in second-order perturbation. The shift of the individual components, using the notation above, is

It has recently been shown’ that nuclear magnetic resonance techniques are capable of detecting deviations from cubic symmetry on an atomic scale in metallic crystals of cubic lattice structure. The electric gradients at the nuclear sites having noncubic environment interact with the nuclear electric quadrupole moment to produce a change in energy of the nuclear spin levels from the value in the absence of the gradients. A12’, which has a nuclear quadrupole moment Q of +0.156 X 1O-24 cm2,2 and a spin I of 5/2, has a resonance line width in the pure metal which can be explained almost entirely by local field (dipole-dipole) broadening. Foreign atoms or lattice imperfections will cause additional broadening of the line in a powdered sample which, if sufficiently drastic, will cause an apparent decrease in intensity. The difference in energy between adjacent spacequantized levels of the aluminum nucleus, when in an external magnetic field Ho and an axially symmetric electric field is to a first approximation,

AW,,p-,-l,t=hvo+A

81(21-

1)’

A=9

(1)

g(v)= SW P(q){ (9/3%,(v)+ --m

10, 1954. t Division of Applied Science, Harvard University, Cambridge, Massachusetts; now at Metals Research Laboratories, Electra Metallurgical Co., Niagara Falls, New York. VOL.

3, JANUARY

1955

(2)

e4qzQ2 1)

hv,,

The shape of the absorption line in a polycrystalline material can be determined by the straightforward process used by Pake.4 Figure 1 shows that the line broadens and its center of gravity moves to lower frequencies. The dashed curve describes the theoretical central line shape and the full curve the absorption as it might be observed when broadened by local fields and other sources of symmetrical broadening. Nierenberg and Ramsey5 have discussed the same problem in molecular beam spectroscopy. The shape of the absorptions to be discussed can be written to a good approximation as a function

* Received August

METALLURGICA,

21-l-3 64 412(21-

where hvo=pH,/I, m and m-l are the magnetic quantum numbers of the adjacent levels, 0 is the angle between Ho and the axis of the electric field, and q is the electric gradient at the nucleus in the direction of that axis. The frequency of the m= l/2-+ l/2 transition is

ACTA

cos20) (1-cos2t?),

where

WQ

AWm+m--l= hv,,-(2m-1)(1--3~os~f?)-~

(l-9

(26,‘35)g,,(v))dq,

(3)

where P(q) denotes the probability that the gradient q exists at any particular lattice site of the nuclei under 74

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NUCLEAR

ELECTRIC

study and in an alloy is a function of the composition. P(q) is normalized so that Lmrn P(q)+= 1, thus NP(q), where N is the total number of nuclei of gyromagnetic ratio y and quadrupole moment Q, is the number of nuclei having quadrupole coupling / eQ/h 1. The function g,,(v) describes the central line shape for a gradient q and gsq(v), which is made up of functions representing all the satellites, describes the total resultant satellite shape for gradient q; the functions g(v) are also normalized so that Kmrn g(v)&= 1. By integrating the resonance curve the total amount of power absorbed by the nuclear spin system can be determined. Experimentally the integration is usually extended over only a rather narrow region surrounding vO, thus only those transitions will contribute to the observed absorption which occur within this region, say from vO--Av to vo+Av. The shape of the satellites suggests anot.her simplification which will be used to interpret the data. The major satellite contribution to the observed absorption derives from their peaks, or infinities, which occur at 8= n/2_ The limited integration automatically excludes all those peaks contributed by nuclei at which q>20kAv/3e2Q (calculated for the m= S/2-+3/2 transition). Let us consider a primary solid solution of c atomic fraction of solute. If we are observing the solvent resonance and a solute atom in the $z’th-position causes a qn, > 2ohAv/3ezQ while that in an IZ’+ 1 neighbor position causes a q,,c+l< 2OhAv~3e~~ then the plot of the integrated intensity (from VO-Av to vo+Av) should follow the function (1 -c)“, where n is the number of neighboring lattice sites contained in a sphere passing between the n’ and n’f 1 neighbors. Equivalently n is the number of solvent atoms, in a sphere around a solute atom, which have an electric gradient q greater than 2OhAv/3&). (l-c)” is the probability that a solvent nucleus has no solute atom inside this sphere of critical radius, provided the solution is ideally random. In the present experiment aluminum alloys of various zinc and magnesium contents were used to determine the extent of the influence of the solute atoms and the magnitude of the gradients caused by them in the aluminum lattice. Pure aluminum (99.99) was alloyed with spectroscopically pure zinc to provide the Al-Zn alloys and with “commercial pure” (99.9) magnesium to provide the Al-Mg alloys used. They were made in graphite crucibles and heat treated prior to the resonance experiment in such a way as to bring about single phase equilibrium. The papers of Fink and Van Horn6 and Fink and Smith’ were used as a guide in establishing the desired state. Inasmuch as all of the specimens were used in the form of filings the final solution heat treatment was carried out on the cold-worked (as filed) material, immediately before use. The detection equipment consisted of a PoundKnight-Watkins type of radiofrequency spectrometer.8,9 The external magnetic field of about 5560 gauss was

QUADRUPOLE

INTERACTIONS

75

FIG. 1. The theoretical shape of the *ST=)-+- 4 absorption in a powdered material when QQ is nonvanishing. Second-order quadrupole interaction broadens the line and shifts its center of gravity.

supplied by a permanent magnet. Over the volume of the sample the field inhomogeneity was less than 0.3 gauss, very small compared to the width of the Al resonance in the metal. The magnetic field was sinusoidally modulated at 280 cps with a peak amplitude of 1.1 gauss. The frequency of the r-f oscillator was slowly swept through. the resonance and the oscillator output fed into a phase sensitive detector with a band width of l/S set-I. The particle size of the samples was in every case smaller than the skin depth so that the experimentally observed quantities do indeed correspond to nuclear absorption only.‘O RESULTS

AND

DISGUSSION

The width and shape of the absorption line in Al-Zn alloys remain approximately constant over the range of composition to be considered. Consequently the integrated intensity in the region 2Av centered at vo is proportional to the maximum absorption g(vmtrx). In Fig. 2 these maxima are plotted as a function of the zinc concentration. They represent the absorption per Al atom in the alloy under like conditions of r-f field strength and modulation of the external magnetic field. The absorption is seen to drop rapidly from the value characteristic of pure Al to about. one-quarter of the pure Al value. Since the central line intensity is, according to theory, precisely 9/35 of the total of all five component lines this remaining absorption is identified with it. The lower part of Fig. 2 is a double logarithmic plot of the satellite intensity only (i.e., g(v,,,)/go(v,,,) -9/35) versus the zinc concentration. The slope of the line drawn through the exper~ental points determines the exponent n to be about 98; this number is to be associated with a sphere around a Zn atom outside of which q is less than approximately 6X 102l crne3, corresponding to a Av of about 5X lo3 set-l. More precisely, two values of n corresponding to gradients necessary to

ACTA

METALLURGICA,

FIG. 2. (a) Intensity of absorption per Al atom as a function of the Zn content in annealed alloys. (b) The satellite intensity alone closely obeys the function (1- c)~*, the m = )-3 transition of Al is unaffected by a near neighbor zinc atom in a 5560 gauss external field.

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cause each satellite pair to fall outside of vof AV could be defined, but this is unwarranted by the data. From the unaffected central line we can estimate an upper limit for ~1, the gradient at an Al atom with a zinc nearest neighbor. Let us say that if the central line were broadened to 2 kc/s it would show up as a broadening of the observed absorption. Then, using Eq. (2) we find that eZqrQ/h<6.3X106 cps or q1<1.2X1P3 cmw3, where v0=6.18XlO~ cps and Q=0.156X10-24 cm2 have been used. From the data given above for the satellites one can estimate whether or not it is possible to satisfy both conditions simultaneously. It was stated that outside of a sphere (around an impurity) containing about 98 atoms, q is less than 6X10z1 cme3. This sphere will include 6th neighbors but not 7th. Let us therefore take q7= 6X 102l cme3. We find ql= (r7/r1)3q7= 18.5X6X 10zL = l.lX1w3 cmP3 indicating that indeed the nearest neighbor’s central line can escape broadening while 6th neighbors satellites are significantly broadened. In two ways the data resulting from the work on Al-Mg alloys differ markedly from that on the Al-Zn discussed above. The intensity drops more abruptly with small solute additions and the m= 1/2--t-l/2 transition is affected. The latter is observed as an asymmetrically broadened line. In Fig. 3 the ratio ~(v~~~)/R~(v~~~) has been plotted for the Al-Mg alloys as a function of the atomic fraction Mg. The exponent n, describing the number of atoms for which q> 6X 10zl, is 130 for this case; thus magnesium atoms must cause greater strain at neighboring sites than do zinc in the aluminum lattice. Using the simplest elasticity theory for homogeneous isotropic media” the relation between the radial strain surrounding a foreign atom and the size of this atom is sI= ja”/r”, where j is the fractional difference of the radii of the impurity and the solvent atoms in the alloy, a is the solvent bulk atomic radius and r is the distance from the impurity center. The uncertainty in the meaning of j makes this of doubtful quantitative use, but perhaps as a proportionality it is valid. Let us assume that the electric gradient at a lattice point is proportional to the strain at that point. Then at points of equal q in the vicinity of two different impurity species the relation jla3/r13= jid/r23will hold. Of course rr= (3%1/47&)1’3, nl being the experimentally determined exponent for the first impurity and d, the atomic density of the alloy in atoms/cm3. Similarly r2 is related to n2 and thus ji/ji=nl/n2. The number of nuclei affected by gradients caused by an impurity are directly proportional to the fractional difference in radius of the impurity and solvent atoms in any one solvent. If nl pertains to Mg, of radius al and n2 to Zn of radius az, each in Al, we find ji= 1.3ji, i.e., al-a= 1.3(an-a). As mentioned above, further evidence that a Mg atom causes larger gradients than does a Zn atom is found in the shape of the central component for high Mg content. It no longer remains symmetrical, as was the case in

ROWLAND:

NUCLEAR

ELECTRIC

Al-Zn, but becomes somewhat asymmetrical and broadened and the recorded peak-to-peak amplitude drops. A higher magnetic field HO would decrease the secondorder interaction, and according to the estimate of the gradients obtained from the satellite behavior, an increase in Ho by a factor of about 1.7 should make the central line behave as it does in Al-Zn at 5560 gauss. Because of the rather slight drop in g(v,& and the asymmetry of the central line it is probable that its perturbation is due to nearest neighbors only. Using notation introduced previously it can be described by a shape function gc(v)= (~-c)‘~g~o(~)+c1-(~--c)‘~lg,,(v),

(5)

since the first term is less than 1O-3 for c greater than 0.05. The solid line in Fig. 3 is a plot of the function 0.743(1-~c)~~O+ (the dotted line in Fig. 3). The dotted and full lines coincide beyond about 4 atomic per cent Mg. Figure 1 shows g,, in a powder assuming q the same at each lattice site. The present situation demands that the quadrupole width be a fraction of the dipolar broadening instead of vice versa as is represented in Fig. 1. The peak at YO- 16A/9h contains about twice the intensity of the one at vof A/h ; consequently, when this line is superimposed on the essentially unperturbed

O.l-

1

0 0

.Ol

1 .oz

I 0.05 C, ATOMIC

I 0.10 FRACTION

77

INTERACTIONS

0.5 -

(4)

where gco is the unperturbed central line shape. Equation (4) merely represents the second term of g(v)=0.743(1-c)‘30g,,(v)+0.257g,(v),

QUADRUPOLE

1 I

0.15

Mq

FIG. 3. Intensity per Al atom in Al-rich Al-Mg solid solutions. The satellite intensity is proportional to (l-c)‘3a showing that a Mg atom causes higher gradients at neighboring Al sites than does a Zn atom.

0 6.160

6’70

MCh

6.RO

FIG. 4. Experimental absorption of Al in an alloy of AI-14 atomic per cent Mg: (a) as filed from the quenched material, (b) from the same sample after a 25O”C, 5 day anneal. A Mg-rich phase precipitates.

central line, an increase in the absorption on the lower side of the line takes place. Curve (a) of Fig. 4 illustrates this asymmetry in the central line of Al in an Al-14 atomic per cent Mg alloy. Curve (b) was obtained from the same alloy as curve (a) after it had been held at a temperature of 250°C for 5 days. During the precipitation process the Mg content of the matrix decreases so much that second order perturbation effects become unnoticeable. This is a relatively minute effect compared to that which could be expected if the impurity concentration in the solid solution should drop below that necessary for satellite disappearance. Drastic changes could then occur, and, using appropriate normalization conditions for nuclear absorption per atom in one phase of a two-phase system the precipitation process might be followed quite closely. Exact quantitative interpretation of the shape of the Al resonance in Al alloys of high Mg content is difficult, because it is probable that many Al atoms have more than one Mg neighbor. Throughout the discussion of quadrupole effects complete randomness of the solute atoms on the lattice sites has been assumed. The extension of the theory to ordered structures is fairly straightforward and carefully chosen alloys could give much information on the ordering process and degree of ordering as a function of temperature and composition. In the experiments discussed here the effects of short range order would be slight causing a very small deviation from the straight line plot of log intensity versus log composition. Even for maximum ordering the intensity will not deviate significantly from that expected of a random solution if it is dilute. This has been calculated by Bloembergenn for a face-centered cubic lattice. Greater deviations in intensity naturally accompany higher solute concentrations, and, according to the computation referred to above, a measurable effect might be observed in alphabrass of 10 to 30 per cent zinc.

ACTA

FREQUENCY

METALLURGICA,

MC/seC

FIG. 5. Experimental result of annealing a cold-worked specimen of Al-O.64 atomic per cent Mg. The absorption curves were obtained after the following treatments: (a) as filed; (b) after a 250°C 87-hour anneal; (c) after a 480°C 2-hour anneal. The pure Al curve (d) is included for comparison.

Nuclear resonance provides a sensitive method of detecting lattice imperfections.i2 The usefulness of the technique, when applied to metals, is demonstrated by an experiment on the annealing of a low-~purity sample of Al. The result should be of interest to physical metallurgy because of the correlation between hardness and “locked-in” dislocations.13*14J6 Here we have a method, superior in some ways to a conductivity study, of determining the amount of material subjected to strain due to lattice imperfections. A powdered sample of Al-O.64 atomic per cent Mg was prepared as described before; the absorption in Fig. 5a was observed soon after filing. This sample was heated to 250°C and held there for 87 hours; it was then cooled to room temperature, and curve Sb determined. A second specimen was filed (same result) and held at 480°C for two hours. After cooling, curve SC was obtained and further heat treatment caused no further change. Clearly the experiment involves no precipitation processes and therefore the interpretation must lie predominantly in the migration and recombination of dislocations which have been obstructed by the impurity atoms. The fact that the resonance in pure Al, Fig. 5d, shows no effect of working proves the necessity of having the impurity atoms present to tangle the dislocations. Enough of these imperfections are trapped in the 0.64 atomic per cent Mg sample to cause the complete disappearance of the satellites and to make the observed intensity fafl on the dotted line in Fig. 3. The point describing the effect of the impurity atoms alone, devoid of locked dislocations, is plotted at c=O.O064 near the solid curve in Fig. 3. The present data on pure Al, which are different from nucfear resonance results in pure, cold-worked copper,’ are consistent with the findings of workers in other fields. Dehlingerr6 first showed that the X-ray line broadening was “negligible” for pure Al and WoodI found the same result, attributable to the self-recovery during cold

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work of Al. In this respect Wood found Al to behave quite differently from Cu, Ag, Ni and MO at room tem~rature, and he reproduces X-ray photographs in support of his statement. A recent article on X-ray linebroadening from filed aluminum and tungsten by Williamson and HalP8 shows that the lines from highpurity Al have a very low value of broadening compared to that from a commercially pure (99.7% Al) sample, Both of these breadths were much less than were produced by the cold-worked tungsten. Seeger’$ has recently shown that the difference between purealuminum and copper at room temperature can be ascribed to the fact that the activation energy needed to create a jog when two dislocations lines cross each other is ten times lower in aiuminum than in copper. At room temperature the dislocations can readily move in the aluminum lattice, which consequently will have a very low random density of dislocations. On the contrary in cold-worked copper at room temperature the dislocations form a locked network and their density is high. Nuclear resonance data confirm this viewpoint. ACKNOWLEDGMENTS

The author wishes to express his gratitude to Professor Bloembergen for many inspiring discussions and to Drs. Willenbrock and Redfield for reading and commenting on this paper. A stipend from the General Electric Company is also gratefully acknowledged. This work was supported jointly by the Office of Naval Research, the Signal Corps of the U. S. Army, and the U. S. Air Force. REFERENCES and T. j. Rowland, Acta Met. 1, 731 (1953). 1. N. Bloembergen An error in equation (6) of this paper will be corrected by multiplying the first two terms on the right hand side by a farctmo;,“. Subsequent equations and conclusions are correct as H. Lew, Phys. Rev. 76, 1086 (1949). 3: R. V. Pound, Phys. Rev. 79,685 (1950). 4. G. E. Pake, J. Chem. Phys. 16,327 (1948). 5. ~~4~i

Nterenberg

and

N. F. Ramsey,

Phys.

Rev.

72, 1075

99, 132 6. W. L: Fink and K. R. Van Horn, Trans. A.I.M.E. (1932). 7. W. L. Fink and D. W. Smith, Trans. A.I.M.E. 124,162 (1937). 8. R. V. Pound and W. D. Knight, Rev. Sci. Inst. 21,219 (1950). G. Watkins, Thesis, Harvard University (19.52). 1:: N. Bloembergen, J. Appl. Phys. 23, 1383 (1952). Theory of Elasticity (McGraw-Hill, New 11. S. Timoshenko, York, 1934), p. 323. Report on the Bristol Conference (1954) 12. N. Bloembergen, Proc. Phys. Sot. (London). 13. A. H. Cottrell and M. A. Jaswon, Proc. Roy. Sot. A199, 104 (1949). Dislocations and Plastic Flow in Crystals. 14. A. H. Cottrell, Oxford, The Clarendon Press, 19.53. 15. A. H. CottreII, Progress in Metal Physics I (Butterworths, London, 1949). Chapter II. 16. U. Dehlinger, Z. Kristallogr. 44, 241 (1930). 17. W. A. Wood, Proc. Roy. Sot. A172, 231 (1939). 18. G. K. Williamson and W. H. Half, Acta Met. 1, 22 (1953). (1954), Proc. 19. A. Seeger, Report on the Bristol Conference Phys. Sot. (London).