Introduction

Infrared to visible upconversion luminescence (UCL) has been extensively studied for its fundamental value [1,2,3] and its various potential applications in upconversion lasers, bioimaging, infrared imaging, solar cells, etc. [4,5,6,7,8]. The co-doping of Er3+ and a high concentration of sensitizer Yb3+ forms the most attractive energy transfer (ET) upconversion system [1]. Under 980 nm infrared excitation of the sensitizer Yb3+, this system can generate green and red emission originating from the (2H11/2, 4S3/2) → 4I15/2 and 4F9/2 → 4I15/2 transitions of Er3+, respectively [9]. Selection of appropriate host material is essential in the synthesis of lanthanide-doped nanocrystals (NCs) with favorable optical properties such as high UC efficiency and controllable emission profile. The practical applications require the development of more efficient, high stability UC materials with low excitation density [10, 11]. Oxide materials are usually very stable chemically, mechanically, and thermally, and could therefore be promising hosts for UC applications [3, 12,13,14,15,16]. The cubic sesquioxide materials (such as Y2O3, Lu2O3, Sc2O3, etc.) display particular structural characteristics and physical properties. For example, Y2O3 shows up the outstanding UCL as the typical oxide host [3, 17]. The Sc2O3 has the smallest lattice parameter. The short Sc–Sc bond length in Sc2O3 can produce the short distance within an Yb3+-Er3+ pair, speeding up the Yb3+ → Er3+ energy transfer. In our previous work, Sc2O3: Er3+, Yb3+ nanostructures were obtained using a biphasic solvothermal (ST) method [17]. The red UCL in this samples are enhanced, compared with the bulk sample synthesized using a solid-state (SS) reaction. The average crystal size of nanostructures has reduced to about 200 nm, which favors the application in fluorescence imaging.

A variety of chemical techniques, including coprecipitation, solvothermal synthesis (ST), hydrothermal method (HT), sol–gel processing, thermal decomposition, etc., have been demonstrated to synthesize lanthanide-doped UC NCs [14, 18,19,20,21,22]. Optimization of synthesis procedure is critical to obtain NCs with tailored crystal size, morphology, surface functionalization, and optical properties. The HT approach is a good choice due to its convenience, exemption from pollution, and the possibility of achieving satisfying crystallinity at a relatively low temperature [23]. Zhao et al. utilized an oleic acid-mediated HT method for the synthesis of UC NaYF4 nanorods, nanotubes, and flower-patterned nano-disks [20]. Chen et al. prepared Fe3+ co-doped NaYF4: Er, Yb UC NCs by a HT method using oleic acid as a capping ligand and a surface modifier [24]. In this work, Sc2O3: Er3+, Yb3+ nanoparticles (NPs) of 19 nm in average diameters have been first synthesized through a simple oleic acid-mediated HT method. We found the stronger UCL in this Sc2O3: Er3+, Yb3+NPs samples, of which the red UCL are enhanced by a factor of 4, in comparison with that in the same optimized concentration Sc2O3 samples by ST method. The UCL enhancement can be attributed to the reduced surface groups and longer lifetimes. Additionally, the UCL property and mechanism of HT-Sc2O3: Er3+, Yb3+ NPs were investigated by the spectra distribution, power dependence, and lifetime measurement.

Experimental

Sample Preparation

The Sc2O3: Er3+, Yb3+ samples were prepared by the HT method via the hydrolysis of relevant mineral salts in an ethanol scheme. The high purity raw materials of Sc2O3, Er2O3, and Yb2O3 powers were dissolved in dilute HNO3 and deionized water to obtain cationic nitrates solutions, respectively. The Sc(NO3)3, Er(NO3)3, and Yb(NO3)3 solutions with corresponding mole ratios were dissolved in absolute ethanol (20 ml), stirring to form a homogeneous solution. Then an aqueous sodium hydroxide solution (2 ml) was added dropwise to the above mixture with stirring for 30 min, followed by adding oleic acid (1 ml), then vigorous stirring for 1 to 2 h. The resulting suspension was placed in a close Teflon-lined stainless steel autoclave with 50 ml capacity and heated at 180 °C for 24 h. After the autoclave was cooled to room temperature, naturally the precipitate was then centrifuged and washed several times with deionized water and absolute ethanol, respectively. The powder was obtained after being dried in a vacuum oven at 80 °C for 15 h and annealed 700 °C for 2 h. For comparison, we prepared Sc2O3 samples prepared by the ST method at the same sintering temperature 700 °C for 2 h [17].

Measurements and Characterization

Powder X-ray diffraction (XRD) datum was collected using Cu-Kα radiation (λ = 1.54056 Å) on an X-ray powder diffractometer (Rigaku D/Max IIA). Transmission electron microscopy (TEM) image was obtained by using a transmission electron microscope (JEM-2000EX) operating at an acceleration voltage of 200 kV. The UCL spectra were recorded with a spectrophotometer (Hitachi F-7000) and infrared spectra were performed by using a Triax 550 spectrometer (Jobin-Yvon) pumped with a power-controllable 980 nm diode laser at room temperature. Infrared spectra in transmission mode were measured on a Thermofisher Nicolet IS50 FT-IR spectrometer, using pressed KBr tablets. In fluorescence lifetime measurements, an optical parametric oscillator (OPO) was tuned to 980 nm as an excitation source, and the signals were detected by a Tektronix digital oscilloscope (TDS 3052).

Results and Discussion

The structures characterized by the XRD patterns are shown in Fig. 1a for samples by HT method with the nominal compositions of Sc2O3: 1%Er3+, y%Yb3+ (x = 0, 5, 10, 15). The pure phase Sc2O3 was synthesized in agreement with JCPDS card 84-1884. The host lattice exhibits the mineral bixbyite structure with the \( Ia\overline{3} \) (Th2) symmetry [25]. In this structure, Sc3+ is sixfold with the effective ionic radius (0.745 Å). The Yb3+ ions owned the large ionic radius (0.868 Å) occupy Sc3+ sites to expand the lattice cell volume, making XRD peaks shift to smaller angles as Yb3+ concentration increases as shown in the magnified patterns of Fig. 1b. To further reveal the morphology and size distribution, the as-prepared Sc2O3 samples were characterized by TEM. Figure 2a shows the TEM image of HT-Sc2O3: 1%Er3+, 5%Yb3+. We obtained the sphered NPs with relatively uniform size and good monodispersity. Figure 2b depicts the histogram of the size distribution; these data were obtained from the TEM image of more than 300 NPs. The average diameter of NPs was determined to be about 19 nm.

Fig. 1
figure 1

a XRD patterns for HT-Sc2O3: 1%Er3+, y%Yb3+ (x = 0, 5, 10, 15) NPs. b Magnified patterns in the diffraction angle ranged from 30° to 33°

Fig. 2
figure 2

a TEM image and b histogram of size distribution of HT-Sc2O3: 1%Er3+, 5%Yb3+ NPs

Figure 3 shows the UCL spectra of Sc2O3: 1%Er3+, 10%Yb3+ (a) and Sc2O3: 1%Er3+, 5%Yb3+ (b) samples prepared by HT and ST methods under 980 nm excitation with an output power density of 3 mW mm−2. The strong emission bands centered at ~ 550 and 660 nm are attributed to the 4f - 4f electronic transitions of Er3+: (2H11/2, 4S3/2) → 4I15/2 and 4F9/2 → 4I15/2 transitions, respectively. The insets present the digital photographs of corresponding samples. It reveals that UCL has been dramatically enhanced for the HT sample, compared with the ST one. For HT-Sc2O3 samples, the calculated enhancement factor of red UCL is around 4, compared with corresponding ST-Sc2O3 samples. It is known that the size of samples has an influence on UCL intensity, which decreased with the decreasing of the size. However, for HT-Sc2O3 sample, it owns smaller size and more intensive UCL. It indicates the HT-Sc2O3 sample is an excellent material owned intense UCL with small size for the biological fields.

Fig. 3
figure 3

UCL spectra of Sc2O3: 1%Er3+, 10%Yb3+ (a) and Sc2O3: 1%Er3+, 5%Yb3+ (b) samples prepared by HT and ST methods, respectively, pumped under 980 nm excitation. The insets present the digital photographs of corresponding samples

The FTIR spectra of HT-Sc2O3: 1%Er3+, 5%/10%Yb3+ and ST-Sc2O3: 1%Er3+, 5%Yb3+/10%Yb3+ samples are shown in Fig. 4. The broad band around 3429 cm− 1 is attributed to the stretching vibration of –OH in the oleic acid (OA) and water [26, 27]. The 2925 and 2850 cm− 1 absorption bands are assigned to the asymmetric and symmetric stretching vibrations of the methylene (CH2) in the long alkyl chain of the OA molecules. The sharpness of the bands indicates that the hydrocarbon chains are well ordered. The anti-symmetric methyl stretch (CH3) is seen as a shoulder on the peak at 2975 cm−1. The bands at 1200–1750 cm−1 can be assigned to the vibrations of C=O in the oleic acid molecule and CO2 in the air [28]. The transformation to carbonate might have occurred on the surface of crystallites during the heat treatment. These results evidence the existence of capping ligands on the surfaces of samples. Figure 4 shows the absorption intensities of –OH vibration for ST-Sc2O3 samples are stronger. The intensities of surface groups for HT/ST-Sc2O3: 1%Er3+, 10%Yb3+ samples are both stronger than that in co-doped 5%Yb3+ samples. The abundant surface groups with available large vibrational quanta may efficiently enhance the MPR processes, inducing the decline of luminescence.

Fig. 4
figure 4

FTIR spectra of HT-Sc2O3: 1%Er3+, 5%/10%Yb3+ (a) and ST-Sc2O3: 1%Er3+, 5%Yb3+/10%Yb3+ (b) samples

In order to exactly describe the population mechanism in Er3+/Yb3+ co-doped HT-Sc2O3 sample, the dependence of spectral distributions on the Er3+/Yb3+ concentrations has been studied in detail.

The UCL spectra of HT-Sc2O3: x%Er3+, 10%Yb3+ (x = 0, 0.5, 1, 2) under 980 nm excitation are presented in Fig. 5a. For the fixed Yb3+ concentration at 10%, the strongest UCL is observed for Er3+ concentration around 1%. When Er3+ concentration exceeds 1%, the intensity begins to diminish because of the cross relaxation (CR) of Er3+ ions [17]. The UCL spectra of HT-Sc2O3: 1%Er3+, y%Yb3+, (y = 0, 5, 10, 15) are presented in Fig. 5b. For the Er3+ singly doped Sc2O3, its UC emission is very faint, which has been magnified 100 times. The ET process of Yb3+ → Er3+ plays a dominant role for UCL enhancement. The strongest UCL is observed for Yb3+ concentration 5% when fixed the optimal Er3+ concentration 1%.

Fig. 5
figure 5

UCL spectra of HT-Sc2O3: x%Er3+, 10%Yb3+ (x = 0, 0.5, 1, 2) (a) and HT-Sc2O3: 1%Er3+, y%Yb3+, (y = 0, 5, 10, 15) (b) under 980 nm excitation

The near infrared emission spectra in the range of 1000–1700 nm for the same variety samples are shown in Fig. 6. In the Er3+/Yb3+ co-doped samples, 980-nm photon excites Yb3+: 2F7/2 → 2F5/2 which exhibits fluorescence at 1000–1200 nm exciting Er3+ ions into 4I11/2 level through a nonresonant phonon-assisted ET process [9]. The Er3+ ions in 4I11/2 level decay nonradiatively to 4I13/2 level, then radiatively to the ground state emitting the photon around 1550 nm [9]. In Fig. 6a, as Er3+concentration increases, the Yb3+ emission has a steady decline which evidences the efficient Yb3+ → Er3+ ET. The Er3+ emission gradually increases when Er3+ concentration increases from 0 to 1%, then declines slightly as a result of the self-absorption of Er3+ ions. In Fig. 6b, Er3+: 4I13/2 emission gradually enhances when Yb3+ concentration increases from 0 to 5% but subsequently begins to decrease. As Yb3+ concentration increases, Yb3+ capacity of 980 nm photon absorption is enhanced. The Yb3+ emission intensity is shown to increase. Meantime, as the distance of Yb-Yb and Yb-Er pairs decreases, the enhanced energy migration among Yb3+ ions speeds up ET from Yb3+ to Er3+. It leads to the increased population of Er3+: 4I13/2 level but the decreased one of Yb3+: 2F5/2level. Due to the quenching of Er3+ by Yb3+ ions, the emission of Er3+: 4I13/2 → 4I15/2 reaches a maximum then drops down.

Fig. 6
figure 6

Near-infrared emission spectra in the range of 1000–1700 nm for HT-Sc2O3: x%Er3+, 10%Yb3+ (x = 0, 0.5, 1, 2) (a) and HT-Sc2O3: 1%Er3+, y%Yb3+, (y = 0, 5, 10, 15) (b) under 980 nm excitation

The pumping power dependences of Er3+: (2H11/2, 4S3/2) → 4I15/2 and Er3+: 4F9/2 → 4I15/2 intensities in HT-Sc2O3: 1%Er3+, 10%Yb3+ are measured under 980 nm excitation and plotted in a double logarithmic scales in Fig. 7. For the UCL processes, the UCL intensity (IUCL) depends on the pumping laser power (P) as the equation: IUCLPn where n is the number of pumping photons absorbed per upconverted photon emitted [29]. The n value can be obtained from the slope of the linear plots between log (I) and log (P). For the two-step ET process, the n value is theoretically less than 2 due to the competition between linear decay and UC processes. Figure 7 shows the slope n values for red and green emissions are 2.5 and 2.1 in the low pump power density, respectively. It indicates, except for two-step process, that there are also the three-photon processes in HT-Sc2O3: 1%Er3+, 10%Yb3+ NPs [30, 31].

Fig. 7
figure 7

Power dependence curves for Er3+: (2H11/2, 4S3/2) → 4I15/2 and 4F9/2 → 4I15/2 transitions in HT-Sc2O3: 1%Er3+, 10%Yb3+ NPs

The upconversion mechanism is drawn in Fig. 8. The ET process is as follows:

  • The ET①: Yb3+: 2F5/2 + Er3+: 4I15/2 → Yb3+: 2F7/2 + Er3+: 4I11/2

    • Er3+: 4I11/2 → Er3+: 4I13/2 (MPR)

  • The ET②: Yb3+: 2F5/2 + Er3+: 4I13/2 → Yb3+: 2F7/2 + Er3+: 4F9/2

  • The ET③: Yb3+: 2F5/2 + Er3+: 4I11/2 → Yb3+: 2F7/2 + Er3+: 4F7/2

    • Er3+: 4F7/2 → Er3+: (2H11/2, 4S3/2) (MPR)

  • The ET④: Yb3+: 2F5/2 + Er3+: 4F9/2 → Yb3+: 2F7/2 + Er3+: 2H9/2

    • Er3+: 2H9/2 → Er3+: (2H11/2, 4S3/2) (MPR)

    • Er3+: (2H11/2, 4S3/2) → Er3+: 4F9/2 (MPR)

  • The ET⑤: Yb3+: 2F5/2 + Er3+: (2H11/2, 4S3/2) → Yb3+: 2F7/2 + Er3+: 2G7/2

Fig. 8
figure 8

The energy level diagrams and dominant upconversion mechanism in Sc2O3: Er3+, Yb3+ NPs under 980 nm pump

To verify and make a theoretical interpretation of the UCL results mentioned above, we utilize the simplified steady-state equations.

$$ \frac{dn_0}{dt}=0 $$
(1)
$$ \frac{dn_1}{dt}={n}_2{W}_{21}-{C}_2{N}_1{n}_1-\frac{n_1}{\tau_1} $$
(2)
$$ \frac{dn_2}{dt}={C}_1{N}_1{n}_0-{C}_3{N}_1{n}_2-{n}_2{W}_{21}-\frac{n_2}{\tau_2} $$
(3)
$$ \frac{dn_3}{dt}={C}_2{N}_1{n}_1-{C}_4{N}_1{n}_3-\frac{n_3}{\tau_3} $$
(4)
$$ \frac{dn_4}{dt}={C}_3{N}_1{n}_2-{C}_5{N}_1{n}_4-\frac{n_4}{\tau_4} $$
(5)
$$ \frac{dN_1}{dt}=\sigma {IN}_0-{C}_1{N}_1{n}_0-{C}_2{N}_1{n}_1-{C}_3{N}_1{n}_2-{C}_4{N}_1{n}_3-{C}_5{N}_1{n}_4-\frac{N_1}{\tau_{Yb}}=0 $$
(6)

Where σ is the absorption cross section of Yb3+ ions, I is the incident pumping power, Ni is the population density of the ith level of Yb3+, ni is the population density of ith level of Er3+ involved in the upconversion process, τi is the lifetime of ith level of Er3+and τYb is the lifetime of 2F5/2 level of Yb3+, Ci represents the ET coefficient of Yb3+ → Er3+ for steps i = 1, 2, 3, 4, 5, and W21 represents the nonradiative rate between 1 and 2 levels of the Er3+ ions.

Compared with two-step process, the UC efficiency of three-photon processes from NIR to visible is decreased [32]. Additionally, the high-photon process is prominent when pumping power is high enough. The excitations of Er3+: 4F9/2 by ET to Er3+: 2H9/2 can be neglected due to the weak pump in our experiment. By Eq. (4), the red emission intensity (IRed) can be obtained by

$$ {I}_{Red}={\gamma}_3{n}_3={\gamma}_3{C}_2{\tau}_3{I}_{Yb}{I}_{n_1} $$

Due to the CR of the Er3+– Er3+ interaction is not considered, the lifetime, τ3, is a constant. That is to say, \( {\mathrm{I}}_{\mathrm{Red}}\propto {\mathrm{I}}_{\mathrm{Yb}}{\mathrm{I}}_{{\mathrm{n}}_1} \), where IYb and In1 represent the emission intensity of Yb3+: 2F5/2 and Er3+: 4I13/2, respectively. The γ3 is radiative rate of red emission. The calculated IRedvalues at various Er3+/Yb3+concentrations are presented in Fig. 9, scaled to the maximum. For comparison, the IRed values obtained directly from the UCL emission spectra are also depicted. The calculated and experimental IRed trends are consistent with each other and obtain the best value at the same Er3+/Yb3+ concentrations, demonstrating the validity of experimental data.

Fig. 9
figure 9

Calculated and experimental red emission intensities (IRed) values at various Er3+/Yb3+concentrations. The intensities are scaled to the maximum

The three-photon green and red UC processes occurred simultaneously result in the increase of the corresponding n values. Meanwhile, the n value of red UC process increases more effectively than that of green UC process. In Fig. 8, the green and red UCL can be populated by CR, as Er3+: 4G11/2 + Er3+: 4I15/2 → Er3+: (2H11/2, 4S3/2) + Er3+: 4I13/2 and Er3+: 4G11/2 + Yb3+: 2F7/2 → Er3+: 4F9/2 + Yb3+: 2F5/2, respectively [31]. The three-photon green UCL is via a cross-relaxation process between two Er3+ ions; however, the cross-relaxation in the three-photon red UCL is between Yb3+ and Er3+ ions. Since the Yb3+ concentration is much higher than Er3+ in our experiment, the three-photon red UC process is more effective than the three-photon green UC process, resulting in a rapid increase of n value for red UCL. In addition, it should be noted that all the three-photon processes are few, so the n values deviate obviously from 3. At the high pump power density, two slopes gradually drop to 1 because UC process becomes dominant [33].

The decay curves of the Er3+: (2H11/2, 4S3/2) → 4I15/2 and 4F9/2 → 4I15/2 transitions in HT-Sc2O3 and ST-Sc2O3 samples under the 980 nm excitation wavelength have been measured and shown in Fig. 10. The decay times for red and green emissions are calculated by integrating the area under the corresponding decay curves with the normalized initial intensity. Figure 10a, b shows the green and red emission lifetimes in HT-Sc2O3: 1%Er3+, 5%Yb3+ are longer than those in ST-Sc2O3: 1%Er3+, 5%Yb3+. The lifetime is proportional to population of level. The longer values indicate the stronger red and green UCL in HT-Sc2O3 sample. In our previous report, we found our samples own the shorter decay lifetime values than that in the literature. Actually, the decay times of Er3+: (2H11/2, 4S3/2) → 4I15/2 and 4F9/2 → 4I15/2 emissions for HT/ST-Sc2O3: 1%Er3+, 5%Yb3+ samples are both close to each other. If Er3+: 4F9/2 level is populated by the MPR process from Er3+: (2H11/2, 4S3/2) levels, the decay time of Er3+: 4F9/2 level approaches to that of Er3+: 4S3/2 level. However, this MPR process is inefficient for population of Er3+: 4F9/2 level [17]. There is another non-MPR mechanism for populating the Er3+: 4F9/2level from Er3+: 4S3/2 level. The mechanism involves CR ET: Er3+: (2H11/2, 4S3/2) + Yb3+: 2F7/2 → Er3+: 4I13/2 + Yb3+: 2F5/2; then, in the same Er3+–Yb3+ pair, an energy back transfer (CRB) Yb3+: 2F5/2 + Er3+: 4I13/2 → Yb3+: 2F5/2 + Er3+: 4F9/2occurs [1]. If the CRB process dominates the main way for the population of Er3+: 4F9/2 level, the decay time of Er3+: 4F9/2 level should be almost equal to the decay time of Er3+: 4S3/2 level. The CRB process is fast and efficient at low excitation density.

Fig. 10
figure 10

Decay curves of the Er3+: (2H11/2, 4S3/2) → 4I15/2 and 4F9/2 → 4I15/2 transitions in HT-Sc2O3 and ST-Sc2O3 samples under the 980 nm excitation wavelength

Figure 11 shows the UCL spectra of three typical sesquioxides under 980 nm excitation. The Sc2O3: 1%Er3+, 5%Yb3+ sample exhibits the strongest UCL in the series of spectra. Furthermore, the emission line of Er3+: 4F9/2 level at the lowest energy side in Sc2O3 shifts to the longer wavelength side by 8 nm relative to that in Y2O3. The nearest Sc-Sc distance is 3.27 Å in Sc2O3 shorter than the Y-Y distance (3.752 Å) in Y2O3 [3, 17]. The mean Sc–O bond length (2.121 Å) in Sc2O3 is shorter than the mean Y–O bond length (2.263 Å) in Y2O3. The Er3+/Yb3+ on Sc3+ site in Sc2O3 experiences a stronger crystal field than on Y3+ site in Y2O3. The red shift of spectrum can be attributed to the large Stark splitting of Er3+ ions in Sc2O3 host. The morphologies of Y2O3 and Lu2O3 samples were also characterized by TEM as shown in the inset of Fig. 11a, b, respectively, for comparison. The obtained spherical particles are both agglomerated to bulk. The better dispersion and uniformity of Sc2O3 NPs synthesized by HT method favor its application in biological assays and medical image.

Fig. 11
figure 11

UCL spectra of Er3+/Yb3+ co-doped typical sesquioxide materials under 980 nm excitation

Conclusions

In summary, Sc2O3: Er3+, Yb3+ NPs about 19 nm were synthesized by a simple oleic acid-mediated HT process. The Sc2O3: Er3+, Yb3+ NPs by HT method shows the stronger UCL, of which the red UCL are enhanced by a factor of 4, in comparison with that in the same optimized concentration Sc2O3 samples by ST method. The UCL enhancement can be attributed to the reduced surface groups and longer lifetimes. The surface groups enhanced the MPR, inducing the decline of luminescence. Under the 980 nm excitation, the decay curves of Er3+: (2H11/2, 4S3/2) → 4I15/2 and 4F9/2 → 4I15/2 emissions for HT-Sc2O3: 1%Er3+, 5%Yb3+ samples are close to each other, resulting from the non-MPR mechanism for populating the Er3+: 4F9/2 level from Er3+: 4S3/2 level. The mechanism involves CR ET: Er3+: (2H11/2, 4S3/2) + Yb3+: 2F7/2 → Er3+: 4I13/2 + Yb3+: 2F5/2; then, in the same Er3+–Yb3+ pair, an energy back transfer (CRB) Yb3+: 2F5/2 + Er3+: 4I13/2 → Yb3+: 2F5/2 + Er3+: 4F9/2 occurs. Under the relatively low-power density, the slopes of the linear plots of log(I) vs log(P) for red and green emissions are 2.5 and 2.1, respectively, which are larger than 2 because of the existence of three-photon processes. Compared with the typical sesquioxides (Y2O3 and Lu2O3), the Sc2O3: 1%Er3+, 5%Yb3+ NPs exhibits the stronger UCL. Furthermore, in Sc2O3 the emission line of Er3+: 4F9/2 level at the lowest energy side shifts to the longer wavelength side by 8 nm relative to that in Y2O3 owing to the large Stark splitting of Er3+ ions in Sc2O3 host. Results show the Sc2O3: Er3+, Yb3+ nanoparticles (NPs) is an excellent material for achieving intense UCL with small size in the biological fields.