Phys. kondens. Materie 4, 397--403 (1966)

O p t i c a l A b s o r p t i o n S p e c t r u m a n d C r y s t a l F i e l d

o f E r b i u m A l u m i n u m G a r n e t ( E r A 1 G )

K. I-I. I-IELL'WEG:E, S. t~fftcNER, ~[. SCIIINKIVIAN:N and H. SCHMIDT

Institut fiir technische Physik der Technischen Itochschule Darmstadt

Received August 17, 1965

The optical absorption spectrum of Erbium Aluminum Garnet (ErA1G) has been measured between 12000 cm -1 and 26000 cm -t. The crystal field splitting of the terms 4115/u, 419/2 , 4~9/2 , 4S3/2 , 2I-Ill/2, 4~712, 4~5/2, 4F3/2, 2H912 and 4Gl1/2 was determined from the spectra and has been analysed in terms of the usual parametrization scheme. Nine crystal field parameters A~<r ~> =- B~ which are necessary to describe the crystal field with the symmetry D2 at the site of the rare earth ion, were fitted to the experimental data. The values are (in cm -1) :

B~ ~ +160 B 0 = - 160 B ~ 30 B~= ~-80 B~ = - 100 B~ = + 140 B~ = -- 40

B 4 = - 1 8 0 0 B~= --700

This result is compared with those obtained from similar analysis of other garnets.

Le spectre d'absorption optique du grenat d'erbium-aluminum (ErA1G) a 6t6 mesur6 entre 12000cm -1 et 26000 cm -1. Le champ cristallin qui sgpare les termes 4115/2 , 419/2, 4~'9/2, 4S312, 21-111/2, 4F712, 'tF5/2, 4F3/2, 21-1912 et 4Gll/2 a 6t~ d6terming s partir du spectre et analys6 s l'aide du schema ~suel de param6trisation. On a ajust~ aux valeurs exp~rimentales les neuf param~tres du champ cristallin A2 (r'> --- B~ qui sent n6cessaires s la description du champ, lequel poss~de la sym6trie D2 s l'emplacement des ions de terre rare. Leurs valeurs sent (en cm-1):

B ~ B ~ 160 B~ ~ 30 B~= +80 B~ = - l O O B~ = + 14o B~ = - 40

B~ = - 1see B~ = -- 7o0

Ce r6sultat est compar6 s ceux obtenus M'aide d'anMyses similaires d'autres grenats.

Das Absorptior~sspektrum des Erbiumaluminiumgranats (ErA1G) wurde im Gebiet zwischen 12000 cm-1 und 26000 cm -1 aufgenommen. Die Kristallfeldaufspaltung der Terme 4115/2, 41912, 4F9/2,4Ss/2, 21-Ill/2, 4F7/2, aFs/~, 4Fa/2, 2H9/~ und 4Gll/2 wurde aus den Spektren ermittelt und im Rahmen des fiblichen Parametrisierungsverfahrcns analysiert. Neun Kristall- feldparameter A7 <r ~> = B7 ', die zur Beschreibung eines Kristallfeldes der Symmetrie Du erforderlieh sind, warden dureh Anpassung an die experimentell gefundenen Werte der Kristallfeldaufspaltung bestimmt. Die Werte sind (alle in cm-Z):

B~= @160 B ~ = - - 160 B ~ 30 B~= ~-80 B~=--10O B~=- t - 140 B ~ = - - 40

2~ = -- 1800 B~ = -- 700 Die Werte werden verglichen mit Ergebnissen aus der Kristallfeldanalyse yon anderen Granaten.

Introduction

The rare earth ions in the garnet lat t ice show a n u m b e r of in teres t ing magnet ic properties [1, 2], which arise from the combined influence of the crystal field a nd the exchange interact ion. An in te rpre ta t ion of the magnet ic properties in the ordered state ]mpJ[ies a detailed knowledge of the crystal field interact ion. I t is therefore convenient to s tudy this in te rac t ion in paramagnet ic garne t systems.

398 K . H . HELLWEGE, S. H~FNER, 13~. SCt{IlgI4h{ANN and W. SOH1KIDT:

The analysis of optical and magnetic data in terms of the crystal field interaction is rather difficult because of the low symmetry at the rare earth sites in the garnets. According to the X-ray investigations of various groups [3, 4, 5] the point symmetry at the rare earth sites is D2 and therefore nine independent crystal field parameters are necessary to describe the interaction of the rare earth ion with the surrounding lattice. In recent years there has been a number of analysis of magnetic and spectroscopic data [6--17] in order to obtain a good set of para- meters for various garnet systems, but in most cases some of the nine crystal field parameters were a priori [6--14, 16] neglected. In the only case [15] (Yb) where all nine parameters were taken into account the number of independent experi- mental data is too small for a complete analysis. Therefore HUTCmNGS and WOLF [15] deduced the ratios B~/B ~ from a point charge calculation. Recently KoNI~c- sT~I~ [17] made a crystal field analysis of (EuY) GaG neglecting only the B~ term.

In the present investigation it was tried to fit all nine parameters to a large number of energy levels determined by optical spectroscopy. This implies the assumption, tha t the crystal field acts on each J term in the same way, which means tha t the crystal field parameters are no function of the total angular momentum J . The system investigated was erbiumaluminumgarnet (ErA1G). I t was chosen because in the crystal spectra of erbium salts all terms of the Er 3+ ion have been determined up to 40.000 cm -1 [18, 19] and the wave functions of the free Er a+ ion, which are necessary in the crystal field calculations have been determined by intermediate coupling calculations of WYBou~N~ [20] and KAI4LE [21]. Erbiumgarnets have previously been investigated by optical spectroscopy and the results have been analyzed in terms of the crystal field parametrization scheme. I~AI'I~ALAI~DO [11] investigated the absorption spectrum of (ErY) Ga garnet and made a crystal field analysis in the cubic approximation ; DR~YFVS et al. [14] refined both the experimental and theoretical results for this erbiumgarnet. These authors also used the cubic approximation in their analysis. KONINOST~I~ and GEUSIC [12] took absorption spectra of (ErY)A1 garnet; in their crystal field analysis of the spectrum they assumed a tetragonal symmetry of the crystal field, thus omitting some of the second order terms and one sixth order te rm*.

Experimental The absorption spectra were measured with a 3.5 m Eber t spectrograph.

A 1.200 lines per m m grating was used yielding a dispersion of approximately 2.5 ~ per mm. The spectra were photographed on Kodak and Perutz Peromnia S plates. The light source was an Osram high pressure xenon discharge tube XBO 450. The crystals were placed in conventional glas or quartz dewars directly into liquid helium or nitrogen. Some spectra were measured at room temperature to determine the excited crystal field levels of the 4Ii5/2 ground term. The crystals were grown in this institute using a method first described by NIELSEN [22].

Experimental Results In a crystal field of Du symmetry every free ion term of a Kramers ion (odd

electron number) is split into ( 2 J - ~ 1)/2 crystal field levels, where J is the quantum number of the total angular momentum. For an arbitraryly oriented

* A detailed comparison of the results of the various investigations is compiled in Table 3.

Absorp t ion S p e c t r u m a n d Crys ta l Fie ld of E r b i u m A l u m i n i u m Garne t 399

Tab le 1. Experimental and theoretical energy levels in ErAlG

4115/2 125

574,3 ~- 308,4 ~ 274 530,1 -~ 264,2 -~ 247 436,1 ~- 170,2 ~ 193 422,8 ~- 156,2 -~ 183

78,8 - - 187,1 - - 158 58,4 - - 207,5 - - 213 26,6 - - 239,3 - - 252

0,0 - - 265,9 - - 273

419/2 12363

12766,3 ~- 187,5 -~ 198 721,9 ~- 143,1 ~- 111 575,8 - - 3,0 ~- 15 528,7 - - 50,1 - - 14 301,3 - - 277,5 - - 309

4F0/2 15306

15518,4 ~ 127,7 -~ 126 474,0 ~- 83,3 -~ 88 355,3 - - 35,4 - - 8 314,7 - - 76,0 - - 86 291,2 - - 99,5 : - 115

4S~:/~ 18411 18455,5 ~- 30,1 ~- 21 395,2 - - 30,2 - - 21

2Hl112 19300

19370,6 ~ 131,2 ~- 134 365,6 ~- 126,2 ~ 70 347,6 ~- 108,2 -~ 55 149,2 - - 90,4 - - 59 112,6 - - 127,0 - - 89

91,2 - - 148,4 - - 112

4F7/2 20580

4F5/2 22 246

20697,5 ~- 89,7 ~- 93 649,3 ~- 41,3 -~ 48 568,2 - - 39,6 - - 45 516,4 - - 91,4 - - 96

22284,2 ~- 37,3 ~- 25 238,8 - - 8,1 ~ 17 217,8 - - 29,2 - - 42

4F~/2 22 576 22660,9 -~ 36,5 ~- 26

587,9 - - 36,5 - - 26

2H9/2 24731

24784,8 -~ 159,2 -~ 167 763,7 -~ 138,1 ~- 93 589,3 - - 36,3 ~- 22 572,8 - - 52,8 - - 15 417,3 - - 208,3 - - 266

2G1~:/~ 26 758

26500,7 -~ 130,1 ~- 172 469,7 -[- 99,1 ~- 129 461,4 -~ 90,8 ~- 79 314,9 - - 55,6 - - 80 269,6 - - 101,0 - - 141 206,8 - - 163,1 - - 178

F i r s t c o l u m n : T e r m s accord ing to reL [21J. - - Second co lumn : E n e r g y of t he t e r m s accord ing to reL [21]. - - T h i rd co l umn : E x p e r i m e n t a l ene rgy levels in ErA1G. - - F o u r t h co lumn : E n e r g y levels in ErAIG of t h e s epa ra t e t e rms . - - F i f t h co lumn : Calcula ted ene rgy levels in ErA1G.

400 K. t t . HELLWEGE, S. H~FNEI%, M. SCHINKMANN and H. SCtB~IDT:

crystal there are no selection rules for electric dipole radiation [23]. Hence the crystal field levels of each term are directly obtained from the absorption spectra at 4.2 ~ as only the lowest crystal field level of the aI15/2 ground term is populated at this temperature. All higher crystal field levels could be determined from measurements at 77 ~ and 293 ~ at these temperatures the excited levels of the ground term get thermally populated, thus making possible transitions from those levels to excited terms.

All experimental results are contained in Table 1. In the first two columns the terms and their energies are labelled according to KAItLE [21]. The third column shows the crystal field components relative to the lowest component of the aI15/2 ground term. In the fourth column the energies of the crystal field levels of each term are listed relative to their center of gravity; the fifth column shows the calculated levels.

I t is evident from Table 1 tha t for each te rm (2J + 1)/2 crystal field com- ponents have been observed. This, as well as the large (60 cm -1) crystal field splitting of the 4S3/2 term shows at once, tha t the deviations of the point symmetry from the cubic one are remarkable.

The positions of the crystal field levels given in Table 1 differ slightly from those determined by KOlqNGSTEIN and GEusIc [12] in (ErY)A1G*, thus indica- ting tha t the crystal field in ErA1G and (ErY)A1G is not very different. The lowest crystal field levels of the ground term at 26.6 cm -1, 58.2 cm -1 and 79 cm -1 are also in good agreement with those from the far infrared investigations of SI~v]~s and TINKKAM [24].

Crystal Field Analysis ot the ErA1G Spectrum a) The Point Charge Model

The parametrizatio~ scheme of the crystal field interaction which was first introduced by STnV~NS [25] and is now generally used to describe the effect of the crystal field on rare earth ions is based on the strictly electrostatic model. The electrostatic interaction of the N 4f electrons of a rare earth ion (radii rj from the center) with surrounding point charges qi with distances R~ from the rare earth ion is :

N

j = l ~ IRi ~ r j i " (1)

Here, the denominator can be developped into spherical harmonics yielding

= ~ ~ ~ A? r ~ r?(~, ~), (2) i 1 m

where the coefficients A~ ~ contain the effect of the surrounding lattice on the rare earth ion. The matr ix elements of the t tamil tonian (2) are now evaluated according to STnV]~S [25] yielding

A*~ /r Z" 0 0 "~ H = A , z \ / ~ ~, (3) l, m

H = ~ B? O~ 07, (3')

* The m a x i m u m devia%ion is 62 cm-1; %he excited ground t e rm level a t 468 cm -1 in (ErY) A1G is at 530 cm -1 in ErA1G.

Absorption Spectrum and Crystal Field of Erbium Aluminium Garnet 401

where 02 , Ca, 06, are the Stevens ' mul t ipl icat ive factors ~, fi, 7 and the O~ r~ are mat r ix elements of angular m o m e n t u m operators defined for example by BAKER

[26] et al. I n the case of the garnets the nonvan i sh ing terms in the development (3') are :

B ~ B ~ B ~ B 0,

We shall drop the B ~ te rm as it only shifts the levels of each term by an equal a m o u n t and causes no spli t t ing.

b) Determination o] the Crystal Field Parameters B~ ~

For the de te rmina t ion of the crystal field parameters from the observed energy levels (Table l) by equat ion (3) the O~ for the terms listed in Table 2 were used. They were calculated from the tables of JUDD [27] using the in termedia te coupling wave funct ions of KA~LE [21] by the formula

Ol : ~ a~LSja2,L, S j ( ~ , i , S, JI Oil ~', L', S, J } . (4) 2"SLL"

Table 2. Operator equivalent factors for intermediate coupling

c~- 10 ~ f l - 1 0 4 y - l O 5

4Ij5/2 + 2.690 +0.452 +0.199 419/2 + 0.351 +1.824 +3.827 4F9/2 -- 5.864 +1.922 --0.915 4S~/2 --41.245 -- -- 2Hll/2 + 0.612 +0.720 +0.367 4F7/2 --12.278 --1.974 +8.715 4F5/2 -- 7 . 8 5 7 - - 6 . 3 3 7 --

4F3/2 - - 5 3 . 2 3 1 -- --

2H91~ -- 2.089 -~1.673 +3.170 4Gll/2 -- 0.773 --1.157 +0.290

Table 3. Comparison between crystal field parameters of various garnet systems

KONING- ]i~ONING- nUTCHINGS DREYFUS STEINand KONING- ]~ONING- ~J:O:h'ING- ]~ONING- STEINand this

A u t h o r s a n d WOLF et al. [ lg] GEUSIC STEIn[12] STEIN [17] STEIN[16] STEIN[12] GEUSIC work [15] [/s] [lZ]

Y b Y G a E r G a ~dYA1 EuYAI E u Y G a Eu I TbYA1 ErYAl ]~rAl Subst. Garnet Garnet Garne t Garnet Garnet Garnet Garnet Garne t Garne t

cln-1 el~l-1 cm-1 cm-1 cm-1 e m - i eln-1 era-1 eln-1

B~ -- 86 (100) + 270 + 268 +105 +105 + 266 +260 + 160 BI + 297 -- -- + 51 • 95 • 95 + 50 -- -- 100 B~ -- 193 -- 260 -- 250 -- 190 --170 --140 -- 184 --160 -- 160 B~ + 159 -- -- -- ~100 -- -- -- + 140 B~ + 535 --2180 +1250 + 950 +850 +700 + 920 +800 --1800 B~ + 72 + 40 + 92 + 94 + 50 + 30 + 93 + 45 + 30 B] -- 229 -- -- -- • . . . . 40 B] +1315 -- 850 -- 965 --1150 +945 +63O --1115 - -710 - 700 B~ -- 233 . . . . . . . + 80

The crystal field parameters B~ n were obta ined by a least square fit of the cal- culated energy levels to the observed energy levels (Table 1). The calculated energy levels were obta ined by diagonalizing the crystal field in te rac t ion mat r ix for each term. The opt imal set of crystal field parameters is given in Table 3. This table also contains all other crystal field parameters hi therto de te rmined for rare

402 K.H. ttELLWE(~]~, S. HiiFN]~R, M. SCHINKMANN and H. SCttMIDT:

earth garnets. The crystal field splitting calculated from the parameter set in Table 3 is contained in column 5 of Table 1; in Fig. 1 the caculated levels are compared with the experimental ones.

-1- l

- - -= , 1 Fig. 1. Comparison between calculated and experimental crystal field components of different terms in ErA1G.

Left: observed energy levels; Right: calculated energy levels

Discussion

The inspection of Table 3 reveals at once the relative magnitude of the para- meters B~ and B~ as compared to all other parameters. From this it can be con- cluded that, though the actual symmetry at the rare earth sites is Du, the cubic component is still predominant. This explains also the relative success of the calculations in the cubic approximation [8, 14]. The fitting procedure in the present case reveals that the quality of the fit is not very sensitive to the magnitude and the sign of the parameters B~, B~ and B~. Even the omission of these parameters changes the calculated crystal field splitting only by 5~o. This indicated that probably the symmetry of the crystal field is approximately tetragonal as was also pointed out by KO~IZCGST~ and G]~vsIc [12]. The parameters B~, B~ and B 6 are believed to be the most inaccurate ones and can easily be wrong by 100~/o . The other parameters should be accurate by 50 ~o.

The inspection of Table 3 shows that generally the parameters are of the same magnitude for all garnets. Especially the values for the YA1G system i~om KOZ~I~GSTEIN and G]~usic [12] are in reasonable agreement with the present results, as was to be expected. The most disquieting fact about the present parameter set seems to be the negative sign of B~; yet using a positive B~ we were unable to achieve a good agreement with the experimental results.

Fig. 1 shows, that though nine independent parameters were used the agreement between the experimental and the calculated energy levels is far worse as in the case of other ionic salts (ethylsulfates and anhydrous chlorides). This may be due

Absorpti[on Spectrum and Crystal Field of Erbium Ahiminium Garnet 403

to two reasons. Obviously covalent bonding which is neglec ted in the present t ype of analys is is ve ry s t rong in ga rne t crys ta ls as eompa rd with o ther rare ea r th ionic salts. This follows f rom the re la t ive large exchange effects in these compounds which are due to superexehange and hence to an over lap of the 4 f orb i ta l s with those of the sur rounding oxygen a toms. The second reason m a y be due to the fact t h a t the present ca lcula t ions were per formed only in first order neglect ing mix ing of var ious J / s t a t e s b y the c rys ta l field in terac t ion . Since the c rys ta l field in garne ts is re la t ive large the errors in t roduced b y omi t t ing second order effects get more p ronounced t h a n in eases wi th a small c rys ta l field in terac t ion . Inc lus ion of second order effects are beyond our present compu ta t i ona l abil i t ies.

The c rys ta l field analys is of the k ind per formed in the present inves t iga t ion neglects effects l ike nonl inear shielding [28] and configurat ion in te rac t ion [29]. I t is difficult to es t imate the magn i tude of these in te rac t ions and hence no conclusion can be d rawn wether the not sa t i s fac tory fit could be caused b y these two effects.

Acknowledgements. We wish to thank Prof. H. G. KXgLE (Teclmische Hochschule Karls- ruhe) for the permission to use his operator equivalent factors for intermediate coupling. We are very much indepted to Dipl.-Math. H. SCI~ELLm~S for his assistance in programming our computer calculations. We wish to thank Mr. EWALD for his technical assistance. This work was supported by the Deutsche Forschungsgemeinschaft.

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