1.1. thehoppingfrequencyofthechargecarriers andthefrequencyofappliedacelectricfieldbecomesthe same, the energyshiftedtowardthefluctuatingionsandthuspresenceofdielectriclosspeakismaximum[80]. 1.2. ACConductivity

1.1.            
DielectricProperty

1.1.1.       
RealandImaginaryPartofDielectric

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Fig.6describes the real and imaginary
part of dielectric constant (and) forCoFe2-xGdxO4 nanoparticlesdepends on frequency.This isnotedfromFig.6thatthedielectricconstant inverse relate to the frequency.Onlow frequencyarea,thedielectricconstantfallsabruptly,but when frequency becomeshigh,then dielectric constant does not depend on frequency.This situationcanbedescribedbytheunderstandingthestructureofspinelferriteandKoop’smodel73.AsinterpretationofKoop’smodel, spinelferriteispossessedofunusuallyconductinglayers certified asgrains,associatedinpoorlyconductinglayersidentifiedsuch asgrainboundaries74.Whenfrequency is low,thegrainboundariesarefurtheractivethangrain,therefore,
thedielectricconstantbecomeshighon lowfrequency
since these are inverse relate to each otheranditreduceswhen
frequency is enlarged75.Bytheeffectofappliedacelectricfield, at the time that,electronsspreadonthepoorlyconductinggrainboundaries,theseelectronsaggregatethereanditformsspacechargepolarization.As a resultofthis,thedielectricconstantbecomesgreatatsmallfrequency,andamoreriseoffrequency,fallsthedielectricconstant.Thedielectricconstantis decreasedwhenthechargemovescannot obeytheappliedacelectricfieldoutsideaspecificcriticalfrequency. Further,now theopinionofRabinkinandNovikova76,thepolarizationinspinelferriteexiststhroughelectroninterchangeamongFe2+andFe3+.Theconfinedmovementoftheseelectronsinthe
samedirection as thatofappliedacelectricfielddefinesthepolarization. Byriseoffrequency, thispolarizationdropsandgetsaconstantvaluesince,outsideaspecificfrequencyofappliedacelectricfield, theelectroninterchangeamongFe2+andFe3+cannotfollowtheappliedacelectricfield77.Such
astheparticlemassfallstoNano-
regime,thevolumeelementoftheparticlerises,andthereforespacepolarizationshowsamost importantpartonthedielectricconstantofthematerial.It canbenotedfromFig.6that thedielectricconstantrisesthroughriseofGd3+innanoparticles ofcobalt ferrite.ByriseofGd3+incobaltferrite, orina
differentobservation,by thereductionofgrainsizeofthisferritenanoparticles,thesurfaceinfluencerisesand sinceinslightergrainsizespinelferritenanoparticlesthegrainboundariesaremorein effectonthelowerfrequency,due to this
resultthevalueofdielectricconstant is rises.Inanothertechnique,
by thereplacementofGd3+incobaltferritenanoparticleseffectsintherelocationtheCo2+ionstowardtetrahedralsites,producesthetensioninthelatticeandtoreleasethistension,the samequantityofFe3+ionsbe able tomigratefromthetetrahedralsitetowardoctahedralsite. This
onerisesthejumpingofelectronamongFe2+ andFe3+onoctahedralsitesinspinelferriteandthereforethevalueofdielectricconstantrisesby rise ofGd3+replacementincobaltferritenanoparticles(Table7).

1.1.2.       
Dielectricloss

Theexpression of frequencydependence ofdielectriclossofCoFe2-xGdxO4nanoparticlesareexposed in figure 7.It couldbeseenthatthedielectriclossfallsbyriseoffrequencyonshortfrequency,whileitdoes not depend onfrequencyongreaterfrequency. When the frequency
is small,thegreater thevalueofthedielectriclossisrelatedthroughlargeresistivity(i.e.,poorconductivity)ofgrainboundariesthataregreatsignificantonlowfrequency78.As a
resultoflargeresistivityofgrainboundariesgreater the energyisneededforelectroninterchangeamongFe2+andFe3+andas a resulthigherenergyloss79.
As well as,byriseoffrequency,a lesser amount ofenergyisneededtoelectronsubstitutionandthusa smaller amount ofdielectricloss occur.Moreover,
when thehoppingfrequencyofthechargecarriers andthefrequencyofappliedacelectricfieldbecomesthe same, the energyshiftedtowardthefluctuatingionsandthuspresenceofdielectriclosspeakismaximum80.

1.2.            
ACConductivity

Thematerialconductivitytellsthemacroscopicdeterminationtothemicroscopicassociationofthechargecarriers.ThefrequencydependenceacconductivityofCoFe2-xGdxO4nanoparticlesisdisplayedinfigure
8.Itcouldbenotedthatatlowerfrequencytheelectricalconductivityofmanufacturedspinelferritenanoparticlesrisesgradually,while,itrisesquicklyonahigherfrequency.Theelectricalconductivityinspinelferritematerialisduetohoppingofelectronsbetweenthesameelementsexistinginhigherthansinglevalencestate.ThepresenceofFeandCoionsat octahedralsitesinto themanufacturedspinelferritenanoparticlesmaycauseconductionaccordingtofollowingrelation81:

Co2++Fe3+Co3++  Fe2+

TheelectroncantransferamongFe2+andFe3+existingattheoctahedralsiteinspinelferritenanoparticlesundertheinfluenceofappliedacelectricfieldanditscontributetotheelectricalresponseofsynthesizednanoparticles.Moreover, riseinFe3+ions at
theoctahedral sitesbytheexchangeofGd3+incobaltferritenanoparticles,showsanimportantroleinincreasingtheacconductivityofCoFe2-xGdxO4nanoparticles(Table7)38.

1.3.            
ImpedanceandModulusSpectroscopy

Fig.7shows the deviation the actual
quantity of resistance (Z) and modulus (M) of CoFe2-xGdxO4nanoparticles which made via sol-gel method.ItisnotedfromFig.7thattheZ’fallsbyriseoffrequency.Itassociatesthat theelectricalconductivityofpreparedspinelferritenanoparticles is riseofbyriseoffrequency.Forferritenanoparticles,thevalueof impedance (Z’)resembled on
high frequency,whichdemonstratethepossiblereliefofspacecharge. Since Fig.7 also
showsthatthe
impedance(Z’)fallsbyriseofexchangeofGd3+incobaltferritenanoparticles,whichhelptodetectedtheriseinconductivitythrougha
riseofGd3+inferritenanoparticles.Moreover,onshortfrequency,thevalueofactualpartofthemodulus(M’)becometoaverysmallvalue,i.e.,compare tozero, the donationofelectrodeeffect
demonstrate asnegligible82. Byriseoffrequency,aregulardissipationof Mbecause ofconductionmechanismassociatedto themobilityofchargecarriers83.

Fig.8illustratesthedeviationofunrealpartofresistance(Z”)andmodulus(M”)forall
manufacturednanoparticles.ItisnotedfromFig.8that at low concentrationspectrumhastwo M”peaks i.e.,sampleatx= 0.00,whichisrelatedbytheinfluenceofgrainboundaryandgrain,butwhen concentrationis high,
the spectrumhasasingleM”peak.ThegreaterfrequencysideoftheM”peak, demonstratetheseriesoffrequencieswhereintheionsarespatiallylimitedtotheirpotentialwells, while,thesmall
frequencysideoftheM”peak,showstheseriesoffrequencywherechargecarriersaremobileoverlongdistances84.Fig.9indicatestheCole-ColeplotofCoFe2-xGdxO4nanoparticles.Thecobaltferritenanoparticles(atx=0.00)showtwoconsecutivesemicircles, on smallfrequencyfirst semicircles, signifiestheroleofgrainboundary,and on largefrequency thesecondsemicircle,tells theroleofgraintoconductionmechanism.
By thereplacementofGd3+incobaltferritenanoparticles,the influenceofgrain inthesecondsemicircleswasreduced.ThepresenceofonesemicirclebygreaterabsorptionofGd3+incobaltferritenanoparticles,showthegrainboundaryeffectisdominantoneintheconductionmechanism.Thefactorsgb,Cgb,andRgbrelatetotherelaxationtime,capacitanceandresistanceforthegrainboundary,weredeterminedbythefollowingrelations85-87:

Thecalculatedfactorsgb,CgbandRgbwerepresentedinTable7.Thecalculatedvaluesofgrainboundaryrelaxationtime(gb)formanufacturedspinelferritenanoparticles weresurrounded bytherangeof0.9s-36.0s.Thecalculatedvalues ofgrainboundarycapacitance(Cgb) wereinsidetherangeof63.3pF–82.0pF.Moreover,thegrainboundaryresistance(Rgb)wereinsidetherangeof14.5K-513K.BythereplacementofGd3+incobaltferritenanoparticles thesefactorswere differentbythedifferenceofmicrostructureandgrainsize.