Anisotropies of Raman scattering cross-sections of YBa2Cu3O6+δ compounds

PhysicaC 177 (1991) 213-222 North-Holland

Anisotropies of Raman scattering cross-sections of YBa2Cu306+ 6 compounds D. Braithwaite, SNCIKNRS,

P.J.M.

J.Q. Liu and G. Martinez

25 Avenue des Martyrs, 38042 Grenoble, France

Van Bentum

University of Nijmegen. Toernooiveld. 6525 ED Nijmegen. The Netherlands

P. Lejay CRTBTKNRS,

25 Avenue des Martyrs, 38042 Grenoble, France

Received 23 November 1990 Revised manuscript received 16 April 199 1

We present Raman scattering results on superconducting single crystals of YBaZCu306+dfor 820.5 at temperatures below and above the critical temperature. A complete polarization analysis of these crystals allows us to deduce the cross-sections of the Raman active phonon modes in the three crystallographic directions. The electronic Raman scattering of 4 symmetry and with the polarization vector in the a-b-plane reveals an energy gap 24 of 3.3 to 3.6kaT, for crystals with 620.8. No clear gap is found in B,, symmetry or for polarizations parallel to the c-axis.

1. Introduction Following the discovery [ 1 ] of new high-T, superconductors, an unprecedented amount of both experimental and theoretical research has been devoted to understanding the nature of the superconducting state. Among the experimental investigations, the Raman technique has played an important iole because it is directly sensitive to various possible excitations (phonons, spin fluctuations and free carrier excitations). A good review on these aspects has recently been published [ 2 1. However one aspect has not yet received much attention, that is the anisotropic nature of the Raman response. Since the superconducting properties of these compounds are often related to their anisotropic character, it is important to try to quantify this aspect. We have addressed ourselves to this problem by investigating the Raman spectra of YBa2Cu306+6 single crystals for different values of the oxygen content and at temperatures below and above the superconducting crit- . ical temperature T,.Care has been taken to char-

acterize these samples and define an experimental procedure allowing for a quantitative comparison between different Raman spectra. This is described in section 2. We will then analyse the anisotropic character of the phonon scattering in section 3 and the electronic scattering in section 4. Section 5 will be devoted to the discussion and interpretation of the electronic response of these compounds. We will show that, at least along certain crystallographic directions, it is possible to define a superconducting gap in these compounds.

2. Samples and experimental

procedure

The single crystals were grown from the flux as defined in ref. [ 3 1. Suitable crystals with an a-b-plane area up to 3 mm2 were removed from the crucible. As such they are superconducting with a T,around 62 K as determined by their magnetic (shielding) transition (fig. 1 (c) ). Two of such samples will. be referred to in the following as samples E and F. When

092 l-4534/9 l/$03.50 0 199 I - Elsevier Science Publishers B.V. (North-Holland)

D. Braithwaite et al. /Anisotropies ofRaman scattering on YBa2Cu30,+d

I.8

“.I~.~..~.*

Temperature

nl....l (K)

Fig. 1. Different traces of the susceptibilities of YBaZCupOo6 samples as a function of temperature. (a) 6~0.9 (sample A); (b) 6~0.8 (samples B, C and D); (c) 6~0.5 (samples E and F).

such samples are subjected to a 20 days annealing procedure in 02, their transition temperature increases to above 90 K. We used two batches of samples with good and reproducible superconducting properties. Sample A, is micro-twinned with a T, of 92.5 K and a magnetic transition width of 0.2 K (fig. 1 (a) ). This corresponds to 6~ 0.9 as was estimated from the position of the 500 cm-’ phonon. Samples B, C and D show a wider transition ( 2: 3 K) around 90 K (fig. 1 (b) ) but have large untwinned domains on which it is possible to perform a detailed polarization analysis of Raman spectra. They correspond to 6- 0.8. All the samples are thin platelets, 20 to 30 urn thick, with the c-axis perpendicular to the plane

of the platelet. When cleaved, in a direction parallel to the c-axis, the cleavage plane appears to be always the b-c-plane that is parallel to the Cu-0 chains. This is observed for all samples with 6 ranging from 0.8 to 0.9 including the twinned samples. So when analysing the response of the samples with the q vector of the light perpendicular to the c-axis (on the cleaved side of the sample) the results can be considered as obtained on untwinned samples. All results reported here have been obtained on freshly annealed or cleaved samples for which the Rayleigh scattering can be minimized to a negligible amount beyond 80’to 100 cm-‘. The polarized Raman spectra were recorded in backscattering geometry. The resolution of the spectrometer was 4 cm-‘. The polarized 5 14.5 nm output of an Ar+ laser was focussed onto a spot of 25 x 300 urn2 on the sample, through a cylindrical lens. The total incident power was kept below 3 mW corresponding to a power density of 100 W/cm2. Though non-negligible, this power density prevents excess heating or strong non-equilibrium generation of quasi-particles by pairbreaking. The temperature of the sample was monitored by a calibrated Pt resistor. The maximum laser heating was estimated to be less than 15 K by monitoring the relative intensity of the RI and R2 luminescence lines of a ruby crystal mounted near the sample. We will report on measurements performed at 30 and 100 K, that is below and above T, for these compounds. In order to get reliable quantitative measurements of relative Raman cross-sections, the procedure adopted was to record alternatively at each temperature successive spectra with different analysing configurations. Four successive iterations are completed in this way. One can then compare for a given sample and at a given temperature the absolute Raman cross-sections with an error estimated to be of the order of If:20% provided by appropriate corrections for the different polarised responses of the spectrometer have been made.

3. Anisotropies of the phonon Raman cross-sections The intrinsic phonons related to the YBa2Cu306+6 system are now well documented [ 21. The system crystallises in the orthorombic structure. Recently a

D. Braithwaite et al. /Anisotropies ofRaman scattering on YBatCu30d+d

complete polarization analysis [4] has been formed room temperature with a quite high power density orders than we used). the 5 5 5 Raman active modes have been identified. our case only the are served. Our is to anisotropies vary with oxygen doping and with temperature and above T,.Though orthorombit, the structure is not far from tetragonal and in such a way there is one Ag phonon at 340 cm-’ which has Raman tensor components (Y,= -a;~ and o!..=O and which, in fact, is a B,, mode in the tetragonal system. Some of the electronic Raman scattering has also the same symmetry. In the following we will refer to this symmetry as Bi, symmetry. This B,, phonon which involves the out-of-phase motion of 011 and 0111 atoms in the Cu-0, planes, is known to soften [ 51 for temperatures lower than T,. When following the experimental procedure outlined previously one gets from untwinned parts of sample C the spectra displayed in fig. 2 for different polarization geometries ki (ei ess)ks. The unit propagation vectors ki and k, represent the incoming and outgoing light and ei and e, are the unit polarization

215

vectors of the incident and scattered light. X, Y, Z refer to the crystallographic directions a, b, c and X’, Y’ to unit vectors rotated by 45”, in the a-b plane, with respect to X and Y. Spectra at low temperature displayed in fig. 2 are in very good agreement with previous results [ 5,6]. The Z(X’ Y’ )Z configuration provides a direct measurement of the B,, contribution but also of the anisotropy of the A, components since the cross-section is proportional -a;,,)*. The B1, scattering is also visible t0 Ci(aLx in the Z( XX)Z and Z( Y Y)Z configurations in addition to the A, scattering and A, phonons at 115, 140,340,440 and 500-l. Some additional structures are also seen especially in the ( Y Y) polarization at 220,240, 580 and 640 cm-‘. These modes are probably induced by O-vacancies in the Cu-0 chains since they disappear when the oxygen doping is more complete (see below). Figure 3 shows the relative scattering cross-sections for (ZZ) and (Y Y) polarization geometries of sample A. In general all zz components of the Raman tensor are enhanced but this is spectacular for the 500 cm-’ mode which involves the vibration of the apex oxygen OIV against the Cu-0, plane. When such an analysis is performed for different temper-

c z(x',y')S

100

/

K 1

4

30 K

Fig. 2. Comparison of scattering cross sections at 30 and 100 K for sample C (6~0.8) in the Z(XX)Z, urations. The base lines for the upper spectra have been shifted to the level of the dashed lines.

Z(X’ Y’ )Z and A’( Y Y)Xconfig-

D. Braithwaite et al. /Anisotropies of Raman scattering on YBazCuJOs,a

216

A

30K

500 -

200

400

600

Raman

Shift

(cm-‘)

II

Fig. 3. Comparison of scattering cross sections with k, and k, parallel to the a-axis for sample A (6~0.9).

atures and oxygen content, one can by standard deconvolution procedures extract the integrated intensity of each phonon and compare the relative polarisabilities with respect to a single component of the Raman tensor. The results are shown in table 1 where the reference is the zz component of the 500 cm- ’ phonon mode. These results will be discussed in section 5.

4. Anisotropies sections

of the electronic Raman cross-

The origin of the electronic Raman scattering in these materials is still a matter of debate. One point on which everybody agrees is that this scattering is surprisingly strong. However until recently the related anisotropy [6] between (XX) and (Y Y) polarization (fig. 2 ) was not established. Before we go further, it is important to know to what extent this

scattering continuum is intrinsic to the electronic structure of the material. We find that the intensity of the electronic background in (XX) polarization spectra is fairly reproducible for sample from different batches. However this is not the case for the ( Y Y) polarization spectra. In fig. 4 we show the ( Y Y) spectra for four different samples with S ranging between 0.8 and 0.9. One clearly sees that in addition to the phonon structures that we have already assigned to 0 vacancies, the (Y Y) electronic background is strongly sample-dependent. For samples with 6=0.8 we find roughly the same spectral response as for 6=0.9, except that it is superimposed on an additional nearly flat background. The fact that this additional scattering is absent in (XX) polarization indicates that it is probably related to electronic excitations along the Cu-0 chains. In order to search for the opening of an energy gap, related to the condensation into the superconducting state, it is especially interesting to compare the Raman response at temperatures below and above T,. This has been done in fig. 2 for all configurations with k]]Z. A similar comparison can also be made for the X(ZZ)X configuration (fig. 5). The (XX) and ( Y Y) polarized spectra show a clear depletion of the electronic scattering at low energies when lowering temperatures below T,, whereas the (X’ Y’ ) and (Z Z) polarized spectra do not really exhibit such a depletion. Indeed this later polarization shows after deconvolution of the 440 and 500 cm-’ phonon modes, an increase of the electronic scattering independent of temperature. Figures 6 and 7 show the investigated polarization geometries for samples with 6~0.5. The situation is now different for the (Y Y) polarisation where no clear temperature dependent depletion of the Raman signal is observed below T,. Notice that in this case the Raman response is not flat at low frequencies in the normal state like it is found for samples with 62 0.8, showing directly that this particular response is not specific of high-T, materials. We will discuss in the next section the different peculiarities of these electronic backgrounds and try to define, when possible, a superconducting gap. Before starting this discussion it is important to note that, within our experimental resolution and sensitivity, the Z(XY)Z and X( YZ)X configurations do not give a significant signal as compared to

211

D. Braithwaite et al. / Anisotropies of Raman scattering on YBa,Cu306+d

1 Relative polarizabilities of Raman active phonon modes of YBazCu306++ The reference is cu&(500 cm- ’ ) Typical uncertainty is f 20% Mode

wj

“500” cm-’

xx YY zz xx YY zz

“440” cm-’

“340” cm- ’

“130”cm-’

“llS’cm-’

xx YY zz xx YY zz xx YY zz

Temperature (K)

those reported

&&z 6=0.5

&& 6=0.8-0.9 0.015 0.06 1 0.02 0.02 0.20 0.135 0.10 0 0.006 0.008 0.085 0.011 . 0.055 0.065

0.02 0.07 1 0.015 0.015 0.235 0.13 0.11 0 0.01 0.01 0.10 0.01 0.08 0.075

30

100

in the other polarizations.

5. Discussion of the results 5.1. Phonon scattering In table 1 the results for samples with 62 0.8 have been gathered in a single column because, within the experimental errors ( + 2OW), we did not find a significant difference between samples. Many aspects are interesting in these results. We first notice that in each case the cross-sections of the B,, phonon, though a little bit bigger in the (XX) polarization than in the ( Y Y), do not differ significantly. This confirms that this phonon has an approximate B,,-like symmetry. For each phonon, the temperature and oxygen doping dependences of the anisotropy are different. The xx and yy components of the “440” cm-’ phonon (in-phase motion of 011 and 0111 in the CuO2 planes) are more pronounced in the superconducting state for samples with 6> 0.8. For low values of 6 the intensity decreases slightly at low temperature. The B,, phonon, though having specific anomalies for 620.8, has components which do not vary much either with temperature or 6. The anisotropy

0.08 1

0.10 1

0.02 0.15

0.025 0.17

0.11 0

0.10

”

0

0.015 0.065

0.03 0.07

0.07 0.07

0.08 0.10

30

100

-

between the yy and zz components of the “ 130” cm-’ mode (motion of the Cu atoms in the CuOZ planes) does not depend on temperature but increases significantly when 6 increases. This is unexpected because the structural anisotropy of the system is known to decrease with increasing 6. So the yy component of this mode is increasingly screened by the doping of the layers. But the more spectacular temperature dependence is the one related to the yy component of the “115” cm-’ phonon (motion of Ba atoms) which is directly apparent on the spectra (fig. 2 ) . This dependence is accompanied by a strong line-shape distortion already reported [ 2,6,7] and analysed as due to a Fano effect between the phonon and the lowenergy scattering continuum (see below). As expected and already known [ 2 1, these effects are much less pronounced in the low doping case (fig. 6). Result in table 1 have not been corrected for the anisotropic character of the absorption coefficients for electric field E parallel or perpendicular to the caxis. This anisotropic absorption is known for semiconducting compounds [ 81 but not to our knowledge for superconducting ones. So the overall anisotropy between (Y,, and (Y, or olyv is certainly due in part to this effect. However the strong anisotropy of the zz component with respect to the yy or the xx component, for the “500” cm-’ mode which in-

218

D. Braithwaite et al. IAnisotropies ofRaman scattering on YBa2Cu306+d

9c )-

-=-T--l

8C I_

7c l-

v) .e c II

6C l-

4 _0

50

x .e F al c -

---_.

40 30

I 20

L A

10

_

Raman

Shift

__--

I

1

200

Raman

X(YlY>~

I

I

I

I

I

I

400

600

Shift

(cm-‘)

I

L

Fig. 5. Comparison of scattering cross sections at 30 and 100 K in the X(ZZ)Xconfiguration. Sample B: 6=0.8. The base line for the upper spectrum has been shifted to the level of the dashed line.

(cm-‘)

Fig. 4. Comparison of scattering cross sections in the ( Y Y) polarization configuration for four different samples with 6 ranging from 0.9 (sample 4) to 0.8 (samples B, C, D). The base line for each spectruxq has heen shifted to the level of the dashed line.

volves the apex oxygen in these compounds is still significant. One can compare for instance this anisotropy with that of the “115” cm-’ mode, a method which allows to compensate for respective absorption anisotropies. This anisotropy is not much affected by temperature or doping. The same behaviour is observed, at least qualitatively, in all high-T, materials including Pb2Sr2Yo.,5Cao.zsCu308+6 compounds [ 93 or Thallium based compounds [ 10,111. It turns out that the anisotropy of the cation-oxygen bond is commonly found in perovskite structures [ 121. It is worth questioning the eventual influence of such a high polarizability on the superconductivity. It is possible that this strong possible that this strong polarizability, could give the necessary inter-

action between CuO, planes to stabilize a quasi twodimensional superconductivity. It is clear, at least for these modes [ 13 1, that the electron-phonon interaction is nat providing the pairing mechanism for high-T, compounds. However, some models for instance require transfer excitations [ 141 between planes and chains with a large oscillator strength. These excitations are supposed to provide the phase coherence between superconducting planes, therefore stabilizing the superconductivity. 5.2. Electronic scattering

5.2.1. The B,, electronic scattering As already mentioned, the Z(X’ Y’ )Z configuration shows essentially the B1, part of the Raman scattering. When looking at fig. 7, the first question is to know whether the electronic scattering is of B,, symmetry or due to a strong anisotropy between (XX) and (Y Y) polarizations. Since the three different polarized spectra are obtained in the same run they

D. Braithwaite et al. /Anisoiropies of Raman scattering on YBatCu30,+d

II 200

I.

I 400

I

I

I1

I

I

600

Raman

yf+yyj,& 200

Shift

219

400

600

(cm-l)

Fig. 6. Comparison of scattering cross sections at 30 and 100 K for samples E and F (6% 0.5) in the X( Y Y)X and Z(X’ Y’ )Z configurations. The base lines for the upper spectra have been shifted to the level of the dashed lines.

can be compared quantitatively for a given sample. If we recalculate the (x’ Y’ ) electronic scattering expected from the anisotropy of the A, contributions in the (XX) and (Y Y) contributions, we obtain at most 15W of the actual scattering observed in the (x’ Y’ ) spectrum, mainly at frequencies above 500 cm-‘. So the observed (X’Y ? ) electronic scattering is essentially of B,, symmetry. Once again this procedure assumes no essential difference in absorption with E parallel to the a- or the b-axis. However, in this case the results obtained for the evaluation of the “ 115” cm- I mode support this assumption within the experimental errors. The origin of this scattering is still a matter of debate. It has been first assigned to superconducting gap excitations [ 15 1. However since it does not vary much below and above T,, this explanation has to be ruled out. An interesting model [ 161 has been proposed which assigns this scattering to interband transitions between two bands crossing at the Fermi level of the system. However, since this model depends strongly on subtle aspects of the band structure (and more specifically on the existence of the Cu-0 chains) it is surprising that a similar scat-

tering is also present in the B&lass of high-T, superconductors. On the other hand it is well-known that in the semiconducting phase (6~ 0.4) the Cu spins in the plane are anitferromagnetically ordered. In the Raman spectrum this gives rise to strong collective spin excitations appearing at high frequencies. This twomagnon scattering appears to be active in both B,, and Ag symmetries. Since only the long-range antiferromagnetic order, but not the effective moment on the Cu, is destroyed by doping, it is interesting to discuss the possibility that the B,, scattering at low frequencies is related to fluctuations of this spin degree of freedom of the system. Such an assignment could explain both the relative, temperature independence of the scattering and.ths absence of a gap in the low temperature B,, spectra. The selection rules for collective excitations in a tetragonal system with long-range antiferromagnetic order predict [ 17 ] that it should ba absent in (XY), (YZ) and (ZZ) polarization configurations, but should be allowed in (XX), (Y Y) and (X’ Y’ ). It is not clear whether these selection rules survive in a system with no longrange order, but we indeed find no scattering in (X Y)

220

D. Braithwaite et al. / Anisotropies of Raman scattering on YBa,Cu,O,+,

200

400

Raman

Shift

600

(cm-’

)

Fig. 7. Comparison of scattering cross sections at 30 and 100 K in the X( ZZ)X configuration. Sample F: 8% 0.5. The base line for the upper spectrum has been shifted to the level of the dashed line.

and ( Y Z). In (Z Z) polarization however, the scattering is not zero. It is interesting to note that for samples with a reduced oxygen content (S=O.5) the B,, continuum seems to weaken with respect to the intensity of the B ,g phonon. 5.2.2. The A, electronic scattering Since the Bi, scattering is also present in the (XX) polarization geometry, one has to subtract it from the spectrum in order to get the & contribution. This is what,has been done in fig. 8 for the two untwinned samples C and D at 30 and 100 K. This subtraction is performed after normalization to the Bi, phonon at the corresponding temperature and subtraction of the anisot-ropic contribution of the “115” and “140” cm-’ phonons. This normalization takes automatically into account any difference in the absorption coefficients along the a- and b-axis. In this figure one can recognize the structures due to phonons which can be used as a measure of the internal consistency and the reproducibility of the procedure. Within the

experimental errors, the scattering is close to zero below around 200 cm-‘. Two broad structures appear with threshold around 2 10 and 580 cm-‘. When the same procedure is applied at 100 K the overall picture is quite different: the high-energy structure remains unaffected whereas the low-energy one transforms into a scattering decreasing continuously with increasing frequency and which can be assigned to free carrier electronic scattering near the Fermi level. In a standard model, where the mean free path is long, the collision frequency w,, is small compared to q& and the cut-off is determined by the latter [ 2 1. However, as demonstrated by Jha [ 18 1, this condition is not fulfilled in high-T, compounds. When one evaluates this collision frequency from infra-red reflectivity measurements, as those reported for instance by Kamaras et al. [ 19 ] we end up with a value for w,, around 100 cm-‘. This corresponds to a ratio a,,/ (qt+) = 1 and within the model developed by Jha we expect an effective cut-off frequency around 400 cm- ’ as observed experimentally. The high-energy structure which is visible below and above T, is located an an energy close to that observed in infra-red reflectivity measurements [ 19 ] where such a structure has been assigned to electronic interband transitions which are not very temperature dependent. As already pointed out, this structure is also visible in the X( Z Z)X configuration. The low-energy structure is more interesting because it corresponds to what is expected from pairbreaking scattering in superconductors. The same procedure applied to the ( Y Y) spectra (sample A) provides the same kind of information (fig. 9). In this case the scattering does not go to zero at low temperatures below 200 cm-’ but is superimposed on a background which is sample dependent. However, the two thresholds and the relative intensities of these broad structures with respect to that of phonons are reproducible. Within the experimental errors, the low-energy thresholds do not depend on the polarization or the q vector of the light. In a BCS pairing scheme the transition probabilities are determined by well known coherence factors. The interaction with light of potential vector A is determined by the A-A term (A, symmetry) of the Hamiltonian (though the p-A term may not be negligible for multi-band or multi-valley transitions). In

D. Braithwaite et al. / Anisotropies of Raman scattering on YBa,Cu,O,+,

1

200

I

I.

1111111,

400

221

,,,,,,*,,,,,,,,

600

Raman

200

Shift

400

600

(cm-‘)

Fig. 8. A,(X) contribution of the electronic scattering for the untwinned samples C and D at 30 and 100 K. The base lines for the upper spectra have been shifted to the level of the dashed lines.

this case a discontinuous jump is expected [20] in the scattering cross-section just above the superconducting optical gap 24. In our case the jump near 210 cm-’ is not discontinuous, but extends over a limited frequency range of about 30 cm- ‘. Abrikosov et al. [ 2 1 ] predict that for high-T, materials the strong electron-electron interaction gives rise to higher-order scattering events which cancel the first-order term just at and slightly above the gap, leading to a more gradual increase of the scattering. On the other hand the electronic scattering is not limited to a singular point on the Fermi surface. In the case of an anisotropic energy gap this would also lead to a more gradual increase of the scattering. As already pointed out the measured spectrum does not seem to depend on the q vector of incident photons. This could be related to the non-conservation of momentum during the excitation process as discussed above. In that case the measured cross-section should correspond to an average value in the a-b-plane. Because of all these problems it is difficult to extract a precise value for the measured gap 24. It should lie between the threshold ( N 210 cm-’ ) and the inflexion point ( u 260 cm- ’ ). This corresponds

to a superconducting gap in the a-b-plane of the order of 3.3-3.6kBTc. This is in accordance with other results obtained in BizSr2CaCuzOs compounds by Yamanaka et al. [ 22 1. This finding does not agree with recent results [23] reported for RBaZCu307_6 where, using the properties of enhanced self energies of phonons below T,, the authors report a value of the gap of 3 16 cm-‘. This frequency corresponds to the maximum of the singularity we observed. However, we did not find in the (Z Z) polarization geometry any reliable indication of a gap structure. We have then to assume that the screening for polarizations along the c-axis reduces the fiossible excitations below the detection threshold or that these excitations are hidden in the strong phonon structures around 500 cm-‘. Neither did we find any structure in samples with 6-0.5. This may be due to an excessive heating of the sample or to a weakening of the superconducting excitations. Notice that in this case the free carrier scattering at 100 K is also significantly reduced.

222

D. Braithwaite et al. /Anisotropies ofRaman scattering on YBa2Cu,0,+s

to detect the presence of gap-like structures larizations along the c-axis.

for po-

References

200

400

Raman

Shift

600 (cm-‘)

Fig. 9. A,( Y) contribution of the electronic scattering for sample Aat30and lOOK.

6. Conclusions I

The investigation of the different polarized Raman scattering response of YBaZCu306+6 single crystals as a function of oxygen doping and of the temperature show that the phonon polarizabilities are very anisotropic. In particular, the polarizability along the c-axis of. the mode involving the motion of the apex oxygen is by far the strongest one in the system. The electron-phonon interaction associated with this mode could then play a role in the stabilization of the quasi two-dimensional superconductivity. The analysis of the A, electronic scattering in the a-b-plane of untwinned samples reveals structures which can be assigned to s*uperconducting transitions below T, and to free carrier scattering above. We find for,samples with 6- 0.8-0.9 a value of 26 ranging between 3.3 and 3.6kBTc. Additional investigations are necessary to determine the nature of the B,, scattering continuum and

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