ceramics
ArticleIron (III) Oxide-Based Ceramic Material for
Radiation Shielding
Hiroyuki Mori 1, *, Yohei Oku 2 , Yudo Mannami 1 and Takahiro Kunisada 3, *
1
2
3

*

Mitsuishi Refractory Brick Co., Ltd., 868, Mitsuishi, Bizen-shi, Okayama 705-0132, Japan;
mtaika-mori@abelia.ocn.ne.jp
Ofuna Chuo Hospital, Kamakura 247-0056, Japan; okuoku159@msn.com
Graduate School of Medicine, Gifu University, 1-1 Yanagido, Gifu 501-1194, Japan
Correspondence: mori4x@gmail.com (H.M.); tkunisad@gifu-u.ac.jp (T.K.)

Received: 3 June 2020; Accepted: 23 June 2020; Published: 24 June 2020


Abstract: We developed a new ceramic from raw material mainly composed of iron (III) oxide.
The measured attenuation coefficient of the ceramic for high-energy gamma rays was in the range
0.268–0.355, which is approximately 40% of that of lead and twice that of concrete. The measured
penetrating dose of the ceramic is half of that of concrete. Thus, the novel ceramic material named
RASHIX may serve as a novel ceramic alternative for the wide variety of radiation shielding materials
used in construction.
Keywords: ceramic; brick; radiation shielding; iron (III) oxide; Fe2 O3

1. Introduction
Shielding from radiation is key for personnel working in the field of medical radiation, suchas clinical radiotherapy and radiophysics, as well as for those in industrial radiation [1]. Lead ishighly effective for this purpose [2] and has been commonly used for local shielding, but its use isrestricted by the RoHS (Restriction of Hazardous Substance) directive due to its toxicity [3]. Hence,concrete, being far less toxic, is the first choice in the construction of walls and flooring of radiation
facilities. Although other materials such as ceramic (brick) [4] and granite [5] have been tested fortheir usefulness as shielding materials, their linear attenuation coefficient µ is 0.08 to 0.13 cm−1 for the
60 Co gamma ray source, which is even lower than those of the various heavyweight concrete mixtures(0.15–0.19) [6]. Hence, radioprotective shields for medical radiation facilities such as linear acceleratorsand for industrial gamma ray sterilization systems are constructed of concrete or concrete with an
inserted iron plate.
In this study, we developed a new ceramic material with a high shielding capacity against thehigh-energy gamma radiation from 137 Cs and 60 Co sources. As we experienced difficulties during
the reconstruction of the concrete shielding wall of the linac room, brick type of shielding material
which can be fast assembled in situ, with a flexibility to be shaped any form was expected to replace
the conventional concrete shielding. Although the material containing 65% Fe (Fe3 O4 and Fe2 O3 ) has
been used for the production of bricks [7], we were not sure that nearly 99% Fe2 O3 (Iron (III) oxide)
purified from the waste material of steel sheet production could be sintered into brick. Unexpectedly,
we successfully burned raw material containing iron (III) oxide into a hard-ceramic brick that showeda shielding capacity of 0.278 µ (cm−1 ) for the 60 Co gamma ray source, which is twice that of the bricks
commonly used for construction. It was comparable to that of the heavy concrete specifically designed
for radiation shielding as well.

Ceramics 2020, 3, 258–264; doi:10.3390/ceramics3020023

www.mdpi.com/journal/ceramics

Ceramics 2020, 3259

2. Experimantal
2.1. Raw Material and Preparation of Ceramic Specimens
A powder containing 99% iron oxide used in this study was provided by TETSUGEN Co. Ltd.,
Japan. The mean diameter of the powder measured by laser diffraction was approximately 1 mm.
The chemical composition of the raw materials is listed in Table 1.
Table 1. Chemical composition of raw material used for ceramic fabrication.
H2 O/mass %Ig. Loss/mass %Fe2 O3 /mass %ClSO4 /mass %MnO/mass %SiO2 /mass %CaO/mass %

0.06

0.17

99.08

0.054

0.52

0.019

0.012

The powder was fabricated into specimens by pressing gently into an iron mold. After most of
the air was removed, the pressure was increased up to 3 MPa/cm2 . The powder was then subjected
to a firing process up to 1320 ◦ C and cooled to room temperature, and allowed to rest for 10 days.
Four kinds of the samples with thicknesses of 20, 30, 40, and 50 mm were prepared for the accurate
measurement of the radiation shielding.
The physical properties of the samples were analyzed at the Okayama Ceramics Center, Japan.
The specific gravity and apparent porosity were measured according to Japanese Industrial Standards(JIS R2205). The cold compressive strength, modulus of rupture, and refractoriness were measured
based on the standards JIS 2206, JIS R2213, and JIS R2204, respectively. The thermal conductivity wasmeasured by the laser flash method. The coefficient of linear expansion was measured at a heating rateof 4 ◦ C/min from 990 ◦ C to 1000 ◦ C.
To determine the chemical properties of the sample, X-ray diffraction analysis was performed at
the Tokyo Metropolitan Industrial Technology Research Institute, Japan. Y.MG452 generator and Y.TU
450-D02 X-ray tube (Yxlon International GmbH, Hamburg, Germany) were used as X-ray source, anddiffracted X-rays were measured by the RAMTEC-1000D ionization chamber irradiation dosimeter(TOYO MEDIC Co. Ltd., Tokyo, Japan). The lead equivalent of each sample was determined on theattenuation rate curve created from four different standard lead plates.
The microstructure of the sample was observed with the help of field-emission scanning electron
microscopy (FE-SEM, JSM-6490, JEOL Co. Ltd., Tokyo, Japan) at the Okayama Ceramics Center, Japan.
Prior to observation, the samples were polished to a mirror-finish and etched by 5% hydrofluoric acidsolution for 30 s.
2.2. Gamma Radiation Spectroscopy of the Iron (III) Oxide Ceramic
A narrow beam of gamma rays from 3.66 × 107 Bq Cs-137 or 1.63 × 106 Bq Co-60 sealed radionuclide
sources was irradiated onto the ceramic samples and the high-resolution gamma spectra were measured
by Falcon 5000TM detector (Mirion Technologies Inc., San Ramon, CA, USA). The calibrated detector
was programmed to record the spectra for 300 s for each measurement. Detailed measurement
procedures and apparatus used have been described in the official report of Savannah River National
Laboratory (SRNL-TR-2015-00244).
The linear attenuation coefficients were determined by the following equation:
I = I0 e−xwhere µ is the linear attenuation coefficient (cm−1 ) and I is the number of uncollided photons detected
when a sample of thickness x cm was placed in front of the detector. I0 is the number of photons
without any sample.

Ceramics 2020, 3

260

2.3. Dosimetry of The Iron (III) Oxide Ceramic by a Low Scatter Irradiator
The penetrating dose through the sample was measured by a low-scatter irradiator (LSI: Model
N40, Hopewell Designs Inc., USA). Sealed 2.62 × 1012 Bq Cs-137 and 6.03 × 1010 Bq 60 Co were used as
photon sources, and 8.51 × 109 Bq Cf-252 was used as a spontaneous fission neutron source. A RO-20
ion chamber survey meter (Thermo Fisher Scientific Inc., Waltham, MA, USA) was used for measuring
the photon dose rate. The Rem500 neutron survey meter (Far West Technology Inc., Goleta, CA, USA)was used for neutron detection. Detectors were placed at a distance of 74 cm from the radiation sources.
Ceramics 2020, 3 FOR PEER REVIEW
3
Detailed dosimetry procedures and the apparatus used have been described in the official report ofSavannahRiver National
Laboratory
measuring
the photon
dose rate. (SRNL-TR-2015-00244).
The Rem500 neutron survey meter (Far West Technology Inc.,
Goleta, CA, USA) was used for neutron detection. Detectors were placed at a distance of 74 cm from

3. Resultand Discussion
the radiation
sources. Detailed dosimetry procedures and the apparatus used have been described in
the official report of Savannah River National Laboratory (SRNL-TR-2015-00244).

3.1. Fabrication of Novel Iron (III) Oxide-Based Brick and Analysis of its Physical Properties
3. Result and Discussion

Although we first intended to manufacture bricks capable of radiation shielding, it is equallyimportantthat their ofphysicalproperties
are suitable
forAnalysis
use asofregularbuilding
materials, to avoid
3.1. Fabrication
Novel Iron
(III) Oxide-Based
Brick and
its Physical
Properties
the cumbersome preparation and brittleness of concrete or concrete with an inserted iron plate.
Although we first intended to manufacture bricks capable of radiation shielding, it is equally
As previously mentioned, common ceramic bricks have a poor radiation shielding capacity asimportant that their physical properties are suitable for use as regular building materials, to avoid
compared with that of concrete [4]. Hence, we aimed to produce ceramic material from raw materials
the cumbersome preparation and brittleness of concrete or concrete with an inserted iron plate. As
containing
a considerable
iron, which
is cheapandis expected
to be
an effective
radiation
previously
mentioned,amountcommonof
ceramic
bricks havea poorradiation
shieldingcapacity
as comparedshieldingmaterial.
with that of concrete [4]. Hence, we aimed to produce ceramic material from raw materials containingAsmagnetite (FehematiteO3 ) have
been
used inpast
for theradiation
productionof bricks
a considerable
amountof iron,which(Feis 2cheap
and isexpected
to thebe aneffective
shielding
3 O4 ) andmaterial.
suitablefor heat storage with sufficient bending strength [7], we used nearly 99% Fe2 O3 (Iron (III)As magnetite(Fewaste
3O4) andhematite
2O3)sheet
have beenused in the
pastprecise
for the productionof bricks
oxide) purified
from thematerial
of(Fe
steel
production.
Thechemical composition
suitableforheatstoragewithsufficientbending
strength
[7],
we
used
nearly99%
Fe
2O3 (Iron (III)of the raw materials is shown in Table 1. X-ray diffraction analysis (Figure 1) revealed that the peakoxide)
purifiedfrommaterialof thesteelmain
sheetingredientproduction.Iron
The (III)
precisechemicalcomposition
positions(sensor
angle
ofthe
thewastedetector)
from
oxide
did notchange afterof the raw materials is shown in Table 1. X-ray diffraction analysis (Figure 1) revealed that the peakfiring, indicating that the micro crystal structure of Iron (III) oxide was maintained even after firing.
positions (sensor angle of the detector) from the main ingredient Iron (III) oxide did not change after
The field-emission
scanning electron microscopy shows that the fibrous structures composed of small
firing, indicating that the micro crystal structure of Iron (III) oxide was maintained even after firing.
particles
in
raw
Iron
(III)
oxide electronmaterials(Figure 2A)
developlarge,
smoothpebblesofsimilar
The field-emission
scanning
microscopy
showsthat theinto
fibrousstructures
composedsmall toa stoneparticles
wall (Figure2B). (III)Theoxideuniform
size (Figureof these2A)
pebbles
inintothelarge,
ceramic
indicates
normalin raw Iron
materials
developsmoothpebblesasimilar
to grainagrowthduringtreatment
Tablesize
2 presents
physicalof the aresultant
ceramic
stonewall heat(Figure2B). The[8].
uniformof these thepebblesin theproperties
ceramic indicatesnormal grain
growthheatthetreatment
2 presents
the physicalproperties
the resultant
material.
It is during
seen that
ceramic[8].
hasTablea physicalstrengthexceedingthat ofofconcrete
and aceramichigh bulk
material.
It
is
seen
that
the
ceramichasaphysicalstrengthexceedingthatofconcrete
and
a
high
bulk
density close to the true specific gravity of Iron (III) oxide (5.24). These physical properties attest
to thedensity
close
to
the
true
specific
gravity
of
Iron
(III)
oxide
(5.24).
These
physicalproperties
attesttosuitability of the ceramic to be used as a basic building material.
the suitability of the ceramic to be used as a basic building material.

Figure
1. X-raydiffraction
profilesofof
Ironmaterialandandfiredfiredceramic.
Figure
1. X-raydiffraction
profilesIron(III)(III)oxide
oxide raw
raw material
ceramic.

Ceramics
Ceramics2020,
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4261

Figure 2. Scanning electron micrographs of Iron (III) oxide raw material (A) and fired ceramic (B).
Figure 2. Scanning electron micrographs of Iron (III) oxide raw material (A) and fired ceramic (B).
Table 2. General physical properties of the Iron (III) oxide-based ceramic.
Table 2. General physical properties of the Iron (III) oxide-based ceramic.
Physical Property
Typical Value
Physical Property
Typical Value
Specific gravity
4.9
Specific gravity
4.9
Apparent porosity
1.0%
Apparent
porosity
1.0%
Cold compressive strength
200 MPa (2000 kg f/cm2 )2
Cold compressive
strength200(200
MPakg(2000
Modulusof rupture
20 MPaf/cm2kg) f/cm )2
Modulus Thermalof rupture
20 W/(mMPa (200
conductivity
4.26K) kg f/cm )◦ C)
Thermalconductivity
4.26 W/(mK)
Coefficientof linear expansion
1.14% (1000
Coefficient of linear
expansion
1.14% (1000 °C)
Refractoriness
SK26
RefractorinessSK26

3.2. Radiation Shielding Performance of the Iron (III) Oxide-Based Brick3.2. Radiation Shielding Performance of the Iron (III) Oxide-Based Brick
Peak energy counts representing the total energy of the uncollided photons through the various
Peak energy
representing
thetotal energy
uncollided
photons
through
thecalculated
variousthicknessesof thecounts
ceramic
samples areshown
in Tableof3.theTheattenuation
coefficientswerethicknesses
theintegral
ceramic counts
samples(bottom
are shownin TableThe
coefficientswere
calculated
60 Cofrom theseofraw
of Table3). 3.
Theµ attenuation
values of thesamples forphotons
60Co photons are
from
these
raw
integralcounts(bottom
of
Table
3).
The
μ
valuesofthesamplesforare approximately 40% of those of lead [2], 60% of that of stainless steel [2], twice that of concrete [6]approximately
40%of thoseof lead[2],heavy
60% ofconcretethat of stainlesssteelmade[2], twicethat of concrete[6] andand almost thesameas thoseof thespecificallyfor radiation
shielding[6,9].
almost
the
same
as
those
of
the
heavy
concrete
specificallymadeforradiation
shielding[6,9].
Commonly used bricks mostly show a µ of approximately 0.1 [4], indicating the superior radiation
Commonlyused bricks
mostly show
a μironof approximately
0.1brick.
[4], indicating the superior radiation
shielding capacity
performanceof this(III) oxide-based
shielding capacity performance of this iron (III) oxide-based brick.
Table 3. Peak energy counts and the calculated linear attenuation coefficients of the Iron (III) oxideTable
Peak energy counts and the calculated linear attenuation coefficients of the Iron (III) oxidebased3.ceramic.
based ceramic.
Thickness of
Raw Counts and the Calculated Linear Attenuation Coefficientsand the Calculated Linear Attenuation Coefficients
Sample(cm) (cm) Raw Counts
137 Cs)Thicknessof Sample
1332
keVkeV(60 Co)1173
keV(60(Co)
662
60Co)
1332
(60Co)
1173
keV
662keVkeV((137Cs)
3 300
4.114.11
× 10×3103
4.63
× 10
3.02
4.63
× 10
3.02×× 105533
3 (0.282)*
3 (0.285)
2.0
2.342.34
× 10× 10(0.282)
*
2.62
× 10
1.44
2.0
2.62
× 10(0.285)
1.44×× 10
1055 (0.370)
33
4
3 (0.272)
3 (0.285)
3.0
1.821.82
× 10× 10(0.272)
1.97
× 10
9.86
1.97
× 10(0.285)
9.86×× 10
104 (0.373)
3.0
334 (0.349)
4.0
3 (0.254)
3 (0.272)
1.491.49
× 10× 10(0.254)
1.56
× 10
7.30
4.0
1.56
× 10(0.272)
7.30×× 10
104 (0.349)
3
3
4 (0.329)
5.0
1.111.11
× 10× 10(0.262)
1.18
× 10
4.91
3 (0.262)
3 (0.273)
1.18
× 10(0.273)
4.91×× 10
104 (0.329)
5.0Linearattenuation
attenuation coefficient(µ)(μ)
calculatedby thein section
2.2 is indicatedin parentheses.
Mean values* *linearcoefficient
calculated
byequation
the equationin section
2.2 is indicatedin parentheses.
of µ for 1332 keV, 1173 keV, and 662 keV are 0.268, 0.279, and 0.355, respectively. The calculated HVL values forMeanof μ for11731.95,
keV,
and 662 keV are 0.268, 0.279, and 0.355, respectively. The
these values
mean µ values
are1332
2.58,keV,
2.48, and
respectively.
calculated HVL values for these mean μ values are 2.58, 2.48, and 1.95, respectively.

Penetratingdoses
doseswere
weremeasured
measuredininananirradiation
irradiationchamberchamberdesigned
designedtotominimize
minimizethe
thescatteredscattered
Penetratingradiationfrom
fromanyany
othermaterial
placedbetweenthe radiation
sourceandsurvey
the survey
radiationothermaterial
placedbetweenthe radiation
sourceand themeter.meter.
The
The measured
gamma
are indicatedin Table
4 alongwith
thosecomparator
materials.
measured
gamma
dosedose
ratesrates
are indicatedin Table
4 along
with
those
of of
thethecomparator
materials.
137 Cs, the ceramic showed expected
Forthethegamma
gammarays
raysfrom
from0.6
0.6to
to1.3
1.3MeV
MeVemitted
emitted fromfrom 6060Co
Coand
and137For
Cs, the ceramic showed expected
andsufficientsufficientreduction
reductionofofthe
thepenetrating
penetratingdose
doseinincomparison
comparisonwith
withthatthatofofstainlessstainlesssteel
steelofofsame
same
andthickness.It Itwas
wastwice
twiceasaseffective
effectiveasasconcrete.
concrete.However,
However,thethedosedoserate
rateforforneutrons
neutronsthroughthroughthe
thethickness.
ceramic was comparable with that of concrete (Table 4). High-energy electron linac greater than 10

Ceramics 2020, 3 FOR PEER REVIEW

5

MeV is known to emit photoneutrons and this may occasionally affect the radioprotection of the linac
facility [10]. Therefore, the shielding capacity of the iron (III) oxide-based ceramic against neutrons is
Ceramics 2020, 3262
an additional merit.
Tablewas
4. Doserate measurementof concrete
the Iron (Table(III) oxide-based
ceramicelectron
for gamma
and neutron
ceramic
comparablewith that of4). High-energy
linacraygreaterthan 10 MeV
from thetoindicatedradiation sources.
is known
emit photoneutrons
and this may occasionally affect the radioprotection of the linac
facility [10]. Therefore, the shielding capacityThicknessof the iron (III) oxide-based
ceramicis
60Co
137Csagainst neutrons
232Cf
Material
an additional merit.
(cm)(mSv/h)(mSv/h)(mSv/h)
Bare
0
3.5
43.5
1.55
Table 4. Dose rate measurement of the Iron (III) oxide-based ceramic for gamma ray and neutron fromFe2O3 based ceramic
2.0
2.5
29
2.4
the indicatedradiation
sources.
ceramic
3.0
2.1
22.5
2.2
Fe2O3 based
Fe2O3 based ceramic
4.060
1.7
17.5 232
2.0
137 Cs (mSv/h)Material
Thickness (cm)Co (mSv/h)
Cf (mSv/h)
5.0
1.4
13.9
1.7
Fe2O3 based ceramic
Bare steel
0
3.5
43.5
1.55
Stainless
5.08
0.7
5.0
1.05
Fe2 O3 based
ceramic
2.0
2.5
29
2.41.35
Lead
5.08
0.2
0.7
Fe2 O3 based ceramic
3.0
2.1
22.5
2.2
Concrete
4.75
2.4
27.0
1.55
Fe2 O3 based ceramic
4.0
1.7
17.5
2.0
Fe2 O3 based ceramic
5.0
1.4
13.9
1.7
3.3. Use of The
Iron (III)Oxide-Based Brick5.08
as A Construction0.7
Material for ClinicalFacility
Stainlesssteel
5.0 Radiotherapy 1.05
Lead
5.08
0.2
0.7
1.35
The use ofthis iron (III) oxide-based
brick as a building
material forthe shielding wallof theConcrete
4.75
2.4
27.0
1.55

linac room of a hospital is illustrated in Figure 3A. Instead of the ordinally wall made by driving
concrete into formwork lined with iron plate, the bricks (Figure 3B) were piled up along with iron3.3. Use of The Iron (III) Oxide-Based Brick as A Construction Material for Clinical Radiotherapy Facility
plate. The leakage dose rate measured at point A in Figure 3A (distance from the electron target is 1.9
The usetheof thisbrick
as aμSv/3
building
material
thetheshieldingwall limit（1.3of the linac
m, including
60.1ironcm (III)
walloxide-basedof the brick)is 314months,lessforthanprescribedroom of a hospital is illustrated in Figure 3A. Instead of the ordinally wall made by driving concrete
mSv/3 months）of the controlled area boundary. The highest leakage dose rate recorded was 999
into formwork lined with iron plate, the bricks (Figure 3B) were piled up along with iron plate. The
μSv/3 months at point B in Figure 3A. In this case, the bricks were piled up as in Figure 3C to form aleakage dose rate measured at point A in Figure 3A (distance from the electron target is 1.9 m, including
rigid shielding wall. The construction was carried out just by manual stacking of the bricks, which
the 60.1 cm wall of the brick) is 314 µSv/3 months, less than the prescribed limit (1.3 mSv/3 months) of
was ideal for the hospital with limited space and a short construction period as shown in Figure 3D.
the controlled area boundary. The highest leakage dose rate recorded was 999 µSv/3 months at point B
In particular, the seismic strength of the iron (III) oxide-based brick (200 MPa) measured by the JIS
in Figure 3A. In this case, the bricks were piled up as in Figure 3C to form a rigid shielding wall. The
R2206-1method exceed that of concrete (20–30 MPa).
construction was carried out just by manual stacking of the bricks, which was ideal for the hospitalTh iron (III) oxide-based ceramic could be formed into thinner tiles, which were used for thewith limited space and a short construction period as shown in Figure 3D. In particular, the seismic
radioprotection of the disaster restoration office of the town hall. Tiles set into the iron frame opened
strength of the iron (III) oxide-based brick (200 MPa) measured by the JIS R2206-1method exceed thatwhen used. These ceramic shadings are expected to reduce the 137Cs derived gamma rays in the eventof concrete (20–30 MPa).
of a radiation accident of the nuclear power plant.

Figure3.3.Visual
Visualimages
imagesofofFeFe
ceramic
bricksusedprotectionshieldshieldininlinac
linacroom.
room.Shields
Shields
2O
3 basedFigure2O
3 basedceramic
bricksused
forforprotection
constructedbybythetheIronIron(III)(III)oxide-basedoxide-basedceramic
ceramicbricks
brickswere
wereshownshownininred.
red.
constructedCeramics 2020, 3

263

Th iron (III) oxide-based ceramic could be formed into thinner tiles, which were used for theradioprotection of the disaster restoration office of the town hall. Tiles set into the iron frame opened
when used. These ceramic shadings are expected to reduce the 137 Cs derived gamma rays in the eventof a radiation accident of the nuclear power plant.
4. Conclusions
We developed a ceramic product made mainly from iron (III) oxide (Fe2 O3 ). The µ values of theceramic for 60 Co photons are approximately 40% of that of the µ of lead and twice that of concrete. The
penetrating dose of the ceramic is approximately twice of that of stainless steel and half of concrete.
These data suggest that this ceramic material, with a thickness half of that of concrete, will be as
efficient. This was confirmed by using this brick as a building material for the shielding wall of
the linac room. Although the shielding effect of the iron (III) oxide-based ceramic brick is less thanthat of lead, the material is non-toxic, very stable and non-rusting, sufficiently hard even at a hightemperature, and easy to fabricate into various forms. We named this novel ceramic material RASHIX(RAdiation SHIelding ceramiX) and all these features indicate the possibility of a wide use of RASHIX
as construction or shading materials for any protection purposes from radiation.
Author Contributions: H.M. and Y.M. provided the experimental platform and prepared all the materials andequipment. Y.O. set up plans to measure radioactivity and analyzed the related data. H.M. and T.K. analyzed
all the experimental data and wrote the paper. All authors have read and agreed to the published version of
the manuscript.
Funding: This research received no external funding.
Acknowledgments: We would like to express our gratitude to the National Federation of Small
Business Associations and Okayama Federation of Small Business Associations for their support with
Manufacturing/Commerce/Service/Productivity Improvement Promotion Subsidy. We are also grateful to the
Ofuna Chuo Hospital and Ikata Town Office for kindly using RASHIX products. We thank the advice andencouragement from Kouichi Toyota and Michihiko Mannami, as well as the Japan Radioisotope Association.
We would like to thank Editage (www.editage.com) for English language editing.
Conflicts of Interest: The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the present work.

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