This paper explains the rotation and displacement as well as the couple and force stress in material with unpaired electrons/nucleons subjected to a magnetic field. This phenomenon is described in terms of quantum mechanics for nanoparticle and quantum statistical mechanics for loose nanomaterial. Quantitative calculations are carried out based on experimental data collected under the magnetic field of an EPR spectrometer from a set of nanocrystallites of hydrated copper sulfate

Bidzińska, Ewa

Sikoń, Marek

Bulletin of the Polish Academy of Sciences Technical Sciences

English

Jagiellonian Univeristy Repository

BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 71(1), 2023, Article number: e144575 DOI: 10.24425/bpasts.2022.144575MATERIAL SCIENCE AND NANOTECHNOLOGYQuantum magnetostriction effect resulting from the asymmetric structure of matter measured with EPR spectrometerMarek SIKO Ń 1® * and Ewa B IDZINSKA21 Cracow University of Technology, Kraków, Poland 2 Jagiellonian University, Kraków, PolandAbstract. This paper explains the rotation and displacement as well as the couple and force stress in material with unpaired electrons/nucleons subjected to a magnetic field. This phenomenon is described in terms o f quantum mechanics for nanoparticle and quantum statistical mechanics for loose nanomaterial. Quantitative calculations are carried out based on experimental data collected under the magnetic field o f an EPR spectrometer from a set o f nanocrystallites o f hydrated copper sulfate.Key words: atomic arm o f action; non-paired electron/nucleon; polar rotation; Cosserat mechanics; EPR.1. INTRODUCTIONThe formation o f  asymmetric matter is favored by nature at the stage o f  filling electron/nucleon shells, when the most en­ergetically favorable number o f  unpaired electrons/nucleons is as large as possible (the Hund rule) [ 1] . However, the com­mon occurrence o f symmetrization in nature (electronic, for the formation o f chemical compounds, and nucleonic, for the for­mation o f  elements) removes the asymmetry. A  magnetostric­tion [2, 3] is created in the case o f  incomplete symmetrization. Mechanically, incomplete symmetrization results in the polar continuum [4] . The asymmetric matter is revealed by the angu­lar momentum o f  the atom. This state comes from both spinons, which are carriers o f  electron/nucleon spin, and orbitons, which are associated with the orbital motion o f  the electron/nucleon.2. ASYM M ETRIC ATOM IN MAGNETIC FIELD  2.1. C lassical descriptionThe interaction between an external magnetic field B  (Bx,By,Bz) and an asymmetric atom k (shown in Fig. 1) generates a mechanical moment:(1)*e-mail: sikon@autograf.plManuscript submitted 2022-05-08, revised 2022-09-16, initially accepted for publication 2022-12-30, published in February 2023.Fig. 1. Asymmetric atom in a magnetic fieldk containing one electron ek and one proton p k. Ck is the cen­ter o f  mass o f  the atom k.We define the arm o f  action pk o f  an asymmetrical atom k as:(2)where po is the reduced Bohr radius o f the atom k, and n is the principal quantum number. Based on equations ( 1) and (2) , we can write the action force P k(B ) o f the moment M k (B ) in the form, APPE N D IX  I:where j j  is the magneto-mechanical ratio, and J k is the angular momentum o f the atom k (Fig. 1) .The F ig .1 shows formation o f  the moment M k, force P k and polar rotation ę k in a magnetic field B  (Bx,By,Bz) for an atom(3)where Pk, Jk and B are modules o f the vectors P k, J k, B.The moment M k(B )  induces additional motion o f  the atom in the form o f precession, as shown in Fig. 1, and the force P k(B )  displaces the atom. W e define the polar rotation o f  an atom k under the action o f a magnetic field B  as the precession angledf / r n\ę  k =  <  (Jk, B ).Bull. Pol. Acad. Sci. Tech. Sci., vol. 71, no. 1, p. e144575, 2023©  2023 The Author(s). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)M. Sikon and E. BidzinskaThe energy o f an atom k in a magnetic field B  can be ex­pressed as:(4)The precession a n g l e c a n  be also shown to have compo­nents:(5)where ( i) =  (x ), (y ), (z ) is the selected component o f  the po­lar rotation vector and the angular momentum vectoro f  the atom k, in relation to the axes o f  the Cartesian coordinate system (x ,y,z), Fig. 1.2.2. Quantum  descriptionTaking into account the solutions o f  quantum mechanics, where the quantum angular “ vector”  does not have a direction in the sense o f classical mechanics, and only one o f the angular mo­mentum projections has values that are simultaneously de­fined with the square o f the angular momentum , we can rewrite the classical equation (5) as an operator in the form:(6)where is the operator o f . Similarly, we can write the operators:(7)(8)(9)wher is the so-called g-ratio, is the re­duced Bohr magneton, are unit vectorsparallel to the x , y and z directions, i, j , k change cyclically.W e can formulate operator equations for the parameters o f Cosserat magnetostriction on the atomic scale in the form:( 10 )( 11)(1 2 )(13)where formula s related to equation ( 17) .3. STATISTICAL DESCRIPTIONW e consider a probabilistic space in a Cosserat material in the form o f an elementary surface (analogous to the Saint- Venant infinitesimal interaction area), which is connected with a material exposed to a magnetic field , as shown in Fig. 2. In the analysis we take into account the asymmetric atoms filling the surface The surfaceis small enough that the action o f  the magnetic field is almost constant, but large enough for the set o f  asymmetric atoms to be subjected to statistical anal­ysis.(17)(18)(19)(20)By solving equations ( 10)- ( 13) , we can write a whole spec­trum o f  solutions depending on the magnetic quantum number as:(16)where is the magnetic quantum number and takes the values:(15)where is the quantum number o f the resultant angularmomentum o f atom is the orbital quantum number, and is the spin quantum number. W e write the wave function in the form:(14)where is the wave function, and arethe eigenvalues o f the operators and re­spectively, s modules o f  the vector W e can write the operato A PPE N D IX  II, for the direction ( i) as:Quantum magnetostriction effect resulting from the asymmetric structure of matter measured with EPR spectrometerFig. 2 . Definition of the magneto-mechanical state of the Cosserat magnetostrictionW e can write the canonical distribution for the asymmetric atoms :(2 1 )where k is the Boltzmann constant and T  is the absolute tem­perature.Based on the above, we can write the average values o f  the random variables o f the atoms on the el­ementary surface as:(22)(23)(24)(25)Components o f the couple stress , force stress and po­lar rotation , related to a surface element W e introduce the concentration coefficients:(26)and finally, obtain:(27)o f  an EPR spectrometer as shown in Fig. 3, where is a vari­able magnetic field rotating in a vertical plane with angular ve­locity is a constant magnetic field operating in the hori­zontal direction, and t is time.Measurements were carried out for two sample positions: the first when the magnetic induction was parallel to the y axis ( i) =  (y ), and the second after rotating the sample around the z axis by , when was parallel to the x axes, ( i ) =  (x), (Fig. 3) .The magneto-mechanical parameters were determined for the magnetic field acting on the sample under resonanceconditions: .(37)4. EXPERIM ENTAL STUDYIn the first stage, our experimental analysis was limited to the electron level, ng. 1 . The sample tested wasa powder o f mass 0.005 g, which was obtained from a hydrated copper sulfate crystal, , by grinding in a mortar.This material contained asymmetric copper ions with an unpaired electron in the configuration . The sample was sub­jected to the magnetic field:(36)(35)(34)(33)we can write the hyperbolic cotangent in equation (31) in the form o f a power series. Using only the first two terms, we write:(32)is the Brillouin function. A t room temperature, when:(31)(30)where:(29)M. Sikoń and E. BidzinskaFig. 3. Nanoparticle in the magnetic field of the EPR spectrometerEPR measurements were carried out at the Chemistry D e­partment o f  the Jagiellonian University, using an X-band (9.2 G H z) Bruker ELEXSYS  500 (Karlsruhe) with 100 kHz field modulation. Spectra were recorded at 293 K  with a mod­ulation amplitude o f 5 mT, a microwave power o f 10 m W  and a receiver gain o f 30. The EPR parameters were determined by simulation, using the software program EPR Sim 32.5. RESULTSThe results are shown in Fig. 4 as the first derivative o f  the absorption curve. The spectra for (i) =  (x ), (y ) are almost the same. Tables 1 and 2 show the results o f quantitative mea­surements. The arm o f action o f  the Cu2+ ion was taken asm (Periodic Table o f  Elements). The amount o f measured in the volume o f the sample by the comparative method (EPR Sim 32 program) was converted into a unit area, .Fig. 4. EPR resonance spectra for loose nanomaterial composed of hydrated copper sulfate, for two directions (i) =  x, y o f magnetic field Bo defined by Fig. 3. (i) =  x -  solid line, (i) =  y -  dashed lineTable 1Results of EPR measurementsTable 2Results of quantum statistical computing6. DISCUSSION AND CONCLUSIONDisturbance o f  symmetry o f the atom, where the center o f  mass and center o f  action o f  the Coulomb forces do not coincide, cre­ates an arm o f  action, which together with the angular momen­tum o f the atom generates the torque, couple stress and rotation o f  the material in a magnetic field. This gives rise to displace­ment, force, and force stress. A lso in [5] it is noticed that the basic variable o f magnetostriction is not the elastic strain ten­sor. The authors o f [5] associate local rotation with position- dependent magnetization.Based on quantum statistical mechanics and experimental re­sults, it was found that magnetic induction Tusing an EPR spectrometer at a temperature o f 293 K  gen­erates a rotation in loose nanomaterial o f  hydrated copper sulfate o f with couple stress o fand force stress o f Themagnetostriction energy per unit volume o f material was calcu­lated as .The asymmetric structure o f  the atom was not taken into account in initial studies o f atomic interactions with the use o f  quantum mechanics (Schrödinger, Feynman, etc.) [2] , and has not been considered in contemporary attempts to represent mechanical stress through the application o f quantum mechan­ics (Kugler, Nielsen, Martin, Gosfrey, Folland, Rogers, Rappe, etc.) [6] .The quantum-classical model o f  the polar continuum is pro­posed, as one o f the first, in work [7] . The enclose numerical example demonstrates the possible applicability o f  the theory.Quantum magnetostriction effect resulting from the asymmetric structure o f matter measured with EPR spectrometerAPPEN DIX I. REFERENCE TO LORENTZ FORCEW e introduce the reduced mass o f the asymmetrical atom k and the velocity o f  this mass into equation (3) , and we write:(A I.1 )(A I.2 )which corresponds to the Lorentz force.(A I.3 )APPEN DIX II. REFERENCE TO PAULI MATRIXW e can describe the angular momentum o f  an atom in a mag­netic field with a magnetic quantum number , equation ( 16) . Positive values o f :(A II.1 )(A II.2 )(A II.3 )that defines the operator o f quantum magnetostriction. When L =  0, / =  S =  1/2, we get:In general, we present the force acting on the atom k in the magnetic field B  in the form:where expression:(A II.8 )(A II.9 )where q is an elementary charge. Based on the above we w ill write for an electron e  when L =  0, J =  S =  1/2:refers to a Pauli matrix for the “ up-down”  direction.(A III.1 )(A III.2 )which corresponds to the current state o f  knowledge, where: E  is Young’s modulus, l is the linear dimension.APPEN DIX IV. REFERENCE TO THE PRINCIPLE  OF CO RRESPONDENCEUsing classical mechanics, the direction o f  angular momen­tum for an atom k is unknown, and hence the rotation o f an atom k in a magnetic field B  cannot be known. However, we know the range o f the changes in the di­rection o f J k with respect to the direction o f the magnetic field and we can determine the average value o f the sine for this range as:(A II.4 )According to quantum formalism, the measured values o f correspond to the operators We expect operator equa­tions for and in the form:(A II.5 )(A II.6 )(A II.7 )corresponding to the average value o f the moment =  , equation (1 ). By transforming equation (A IV .1 ) we get: The angle associated withtakes the values:Using quantum statistical mechanics, at very low tempera­tures, when the follow ing conditions are met:Based on equations (A II.3 ) , (A II.4 ) , (A II.5 ) , and (A II.6 ) , we can write the formula:and counting the limits o f the hyperbolic cotangents:APPEN DIX III. REFERENCE TO JO U LE’S MAGNETOSTRICTIONFrom equation ( 36) , we can write the linear magnetostriction coefficient as:and Joule’ s magnetostriction can be written in the form:refer to atoms with precession momentum in the direction o f the magnetic field B, whereas negative values o f :refer to the opposite direction W e introduce to the states , the wave functions:(A IV .1 )(A IV .2 )(A IV .3 )M. Sikod and E. Bidzinskawe can write equation (27) as:(A IV .4 )(A IV .5 )which corresponds to the average quantum value o f the moment . For orbital quantum number L =  1 and spin S =  1/2, we can write which fulfils the Bohr correspon­dence principle with an error o f  0.79%.ACKNO W LEDGEM ENTSMany thanks to my two daughters A lic ja  Sikoń and Agata Wierzbicka for the inspiration to write this work.REFERENCES[1] W. Kołos and J. Sadlej, Atom and Molecule. Warsaw: WNT, 2007, pp. 99-135.[2] R.P. Feynman, R.B. Leighton, and M. Sands, Feynman Lecture on Physics. vol. 2, chapter 34-37, CA: Addison-Wesley, 2005.[3] J.P. Joule, “On a new class o f magnetic forces,”  Ann. Electr. Magn. Chem., vol. 8, pp. 219-224, 1842.[4] M. Sikon, “Physical interpretation o f the Cosserat mechanics for a collection o f atoms,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 64, no. 2, pp. 333-338, 2016, doi: 10.1515/bpasts-2016-0038.[5] M. Fähnle and R. Schäfer, “Magnetostriction in systems with in- homogeneous magnetization,”  J. Magn. Magn. Mater., vol. 523, p. 167551, 2021, doi: 10.1016/J.JMMM.2020.167551.[6] O.H. Nielsen and R.M. Martin, “Quantum-mechanical theory o f stress and force,”  Phys. Rev. B, vol. 32, no. 6, pp. 3780-3791, 1985, doi: 10.1103/PhysRevB.32.3780.[7] M. Epstein, “Hilbert boundless as quantum-classical continua,” Math. Mech. Solids, vol. 25, no. 6, pp. 1312-1317, 2020, doi: 10.1177/1081286519888964.

Quantum magnetostriction effect resulting from the asymmetric structure of matter measured with EPR spectrometer

Bidzinska, Ewa

Biblioteka Nauki - repozytorium artykuÅÃ³w

https://ruj.uj.edu.pl/xmlui/bitstream/handle/item/308581/sikon_bidzinska_quantum_magnetostriction_effect_resulting_2023.pdf?sequence=1&isAllowed=y

Quantum magnetostriction effect resulting from the asymmetric structure of matter measured with EPR spectrometer

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Quantum magnetostriction effect resulting from the asymmetric structure of matter measured with EPR spectrometer

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Jagiellonian Univeristy Repository

Biblioteka Nauki - repozytorium artykuÅÃ³w

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