Induced Fields in Magnetized Mateials: Calculations and the Uses S.ARAVAMUDHAN Depatment of Chemisty Noth Easten Hill Univesity PO NEHU Campus: N.E.H.Univesity Shillong 79 Abstact The induced field distibution in mateials, which inheently possess lage intenal magnetic fields, o in mateials which get magnetized when placed in lage extenal Magnetic Fields, is of impotance to mateial scientists to adequately categoize the mateial fo its possible uses. It addesses to the questions petaining to the stuctue of the mateial in the given state of matte by inquiing into the details of the mechanisms by which the mateials acquie the popety of magnetism. To aive at the equied stuctual infomation ultimately, the beginning is made by studying the distibution of the magnetic field distibutions within the mateial (essentially magnetization chaacteistics) so that the field distibution in the neighbohood of the magnetized (magnetic) mateial becomes tactable. The consequences extenal to the mateial due the intenal magnetization is the pime concen in finding the utilization pioities fo that mateial. In the mateials known conventionally as the magnetic mateials, the intenal fields ae of lage magnitude. To know the magnetic field inducing mechanisms to a geate detail it may be advantageous to study the tends and pattens with a moe sensitive situation of the smalle vaiations in the aleady small values of induced fields can be studied and the Nuclea Magnetic Resonance Technique tuns out to be a technique, which seems suitable fo such studies. When the magnetization is homogeneous though out the specimen, it is a simple matte to associate a demagnetization facto fo that specimen with a given shape-detemining facto. When the magnetized (magnetic) mateial is in-homogeneously magnetized, then a single demagnetization facto fo the entie specimen would not be attibutable but only point wise values. Then can an aveage demagnetization facto be of any avail and how can such aveage demagnetization facto be defined and calculated.it is a tedious task to evaluate the demagnetization facto fo homogeneously magnetized, spheical (ellipsoidal) shapes. An altenative simple mathematical pocedue could be evolved which epoduces the aleady available tables of values with good accuacy. With this method the questions petaining to induced field calculations and the infeences become moe elevant because of the feasibility of appoaches to find answes. KEY Wods: Magnetic mateials, Bulk susceptibility, Induced fields, Nuclea shielding, Demagnetization factos. Intoduction: The Nuclea Magnetic Resonance Technique can measue the Nuclea Shielding paametes. This Nuclea Shielding, paticulaly in solid single cystalline samples, aises due to the induced fields at the nuclea site in the molecules when these molecules themselves ae placed in the cystal lattice points as detemined by the symmety elements of the Cystallogaphic Space Goups. As depicted in Fig., even though it is the pime concen in NMR expeiments to detect the changes in the inta molecula electon-chage ciculations by measuing the Nuclea Shielding, in ode to obtain this specific molecula contibutions, it becomes necessay to estimate the intemolecula [fom the nea neighbos and the fa away bulk] contibutions to induced fields at the nuclea site and appopiately take into account quantitatively to etieve eliably, the inta molecula contibutions only(). As illustated in Fig., the NMR specta typically consist of one spectal line fom one equivalent set of nuclei. The PROTON SHIELDING Molecula + Region I + Region II Bulk Susceptibility Effects I II Loentz Sphee Contibution Fig. Induced fields elevant in NMR Measuements line positions change when the inta molecula contibutions vay fom one molecule to anothe molecule.
If the diamagnetic sample is homogeneously magnetized, then the intemolecula contibutions to the induced fields at the nucleus would cause an additional shift of the line. But, if the sample is inhomogeneous magnetized, then the induced field contibutions at the nuclea sites ove the extent of the sample would vey and this can cause a line width and line shape vaiations. This aspect of the NMR technique is bone out by the consideations in Ref.() and the efeences cited thee in. The Oute Continuum in the Magnetized Mateial Specified Poton Site Loentz Sphee The Oute Continuum in the Magnetized Mateial σ i =Σ i i /R i [-(.RR i /R5 i )] Oute suface D out Sphee of Loentz Loentz Cavity D out = - D in Hence D out + D in = Added intemolecula Contibultions causes a shift downfield o upfield homogeneous NMR Line fo only Inta molecula Shielding To slide # Inhomogeneous Magnetization can Cause Line shape alteations Fig. Defining the tems elevant fo induced field calculations in magnetized mateials and the possible way line shape and width changes can aise in NMR specta due to vaiations in induced field distibutions. An effot to intepet the esults of High Resolution Poton Magnetic Resonance studies in oganic molecula single cystals made evident the vaious contibutions to the induced field as descibed. Futhe it led to the devising of a simple summation pocedue fo calculating demagnetization factos fo specimens with shapes descibable, as the ellipsoids of evolution and this seem to be simplifying the mattes with egad to estimating the tends of the induced field contibutions in such diamagnetic samples. It is the taget set in this pape to bing out this point of view to the mateial scientists and to make it obvious as to how this can be useful fo the undestanding of field distibution pattens inside magnetic mateials. At this stand point, it would be wothwhile to note also that tying to pin point the oigin of the field at a point within a magnetized mateial aises the issues of whethe it has to be a micoscopic aveage o a macoscopic aveage which is elevant ().. Magnetic Dipole Model fo the Calculation of Induced Fields: Fig. illustates the vaious demacations, in a Single Cystalline Spheical specimen, fo the calculation of the induced field at the specific site. By definition, the Loentz Sphee is a semi mico volume element suounding the specified site. The intemolecula contibutions fom within this Loentz Sphee at the cental point has to be calculated as a discete sum ove the molecula contibutions fom evey one of the molecules occuing within this spheical volume element. The contibution fom the oute egion is the macoscopic bulk demagnetization, which depends on the specimen shape factos. The equied discete summation of the contibutions fom each of the molecule 'i' is accomplished using the Equation. The Fig. depicts how at each individual lattice point; a spheical element can be placed with the consequence that due the susceptibility of the mateial in this spheical volume a magnetic moment is induced which is a dipole oiginating at the cente of the sphee. This magnetic moment esults in a Shielding at a distant point as pe the Equation and this shielding is popotinal to the induced filed at the distant point. When the adius of this spheical element is vey small compaed to the distance whee its induced field is being calculated, then the point dipole appoximation may be a valid appoximation fo the eliability of the induced filed values thus obtained. Fig.4 gives the equation with all the x matices ae witten out in the expanded fom so that the equied matix multiplications ae obvious. Thus the discete summation within the sphee appeas to be obviously simple to pogam on the compute. Even though, in all the effots to calculate induced fields, ascetaining the magnitude of the induced magnetic dipole moment, which causes the induced field distibution aound it, makes the beginning. Since the induced field
distibution aound a point magnetic dipole is simple to envisage and amenable by simple equations, if one can ensue that the point dipole appoximation would be applicable then the coesponding calculation of the induced field would become simple. The discete summation within the Loentz Sphee has this possibility inheently because it is possible to assess the magnetic susceptibility values fo the molecules and let this give ise to the magnetic dipole moment at the appopiate cental point in the molecule and ty to calculate the induced field by Equation at the site of the nucleus. It is usually possible that the distances ae much lage than the coesponding molecula dimensions (whee the magnetic moment oiginates) and hence point dipole appoximation would be valid and the summation ove all the molecula point dipole souces can be computed. σ i =Σ i i /R i [-(.RR i /R5 i )] Induced field Calculations using these equations and the magnetic dipole model have been simple enough when the summation pocedues wee applied as descibed in the pevious pesentations and expositions. Isotopic Susceptibility Tenso ~ = R = ( θ ) cos σ zz = 5 = σ xx xy xz xx xy xz xx xy xz yx yy yz yx yy yz yx yy yz xx xy xz zx zy zz σ σ σ zx zy zz zx zy zz σ σ σ = yx yy yz 5 σ σ σ zx zy zz Isotopic Susceptibility Tenso ~ = R = ( θ ) cos σ zz = = σ 5 Fig.. Equation fo calculation of Shielding (induced field) by discete summation of the contibutions fom within the Loentz sphee. Explicit expession in tems of the matix indicating the equied matix multiplication steps fo such calculation. While handling a continuum situation, it is necessay to execise discetion and hypothetically cave out small volume elements and associate the magnetic moment due to that volume susceptibility with that element and place it at the cente. This hypothetical division should extend ove the entie sample extent and fom all these subdivided elements the contibution to induced field at a paticula site can be calculated by an appopiate summation. But this concept in the case of a continuum essentially leads to an evaluation of an Integal fo the effective summing. And it is well known in this case that the esulting integal thus set up becomes complicated even fo the simple case and evaluating it also is tedious. Howeve fo the case of the sample shapes which ae egula ellipsoids of evolution evaluations of such integals have been possible and fo the vaious shape detemining factos (the atio of the pola / axial lengths fo the shape) the Demagnetizing factos have been tabulated (4). In spite of this success, at the stages of the evolution the coesponding equations do not any longe etain the conceptualized simple physical pictue of the point dipoles and hence at the end when one has a convenient table in the hand to use, this does not povide a convenient physical insight fo intepeting the shape dependences fo the othe vaiety of shapes of specimen which occu and within which it is equied to envisage the induced field distibution. This is moe so because, fo othe shapes than the specific shape efeed to above (that of ellipsoids) the mateials have inhomogeneous magnetization ove the extent of the specimen and it may not be possible to associate a single numbe as the shape dependent demagnetization facto applicable at any point within the specimen.. The Simple Summation Pocedue in Place of the Integation Ove the Bulk of Sample: Fig.4 to Fig.6 illustate in steps the altenative simple summation pocedue, which evolved while being concened with the intepeting the esults of HR PMR
studies on oganic molecula single cystals fo the detemination of the shielding tenso of potons in molecules. This pocedue seems to be capable of epoducing with good accuacy the tables of Demagnetization factos efeed to ealie, which wee calculated evaluating the complicated integals which wee set up fo the case of shapes attibutable with a single demagnetization facto fo any point within the specimen. Fig.4 explains the essential pinciple used fo woking out such a summation pocedue. At the outset what was sought fo, was a possibility to divide the entie continuum bulk pat of the specimen (excluding the Loentz Cavity Fig.) into small, closely packed elemental spheical volume elements each of which can be assigned a susceptibility value popotional to the volume of that spheical element (to be multiplied with the unifom Volume Susceptibility of the specimen). By the close packing citeia it is to be ensued that the entie volume of the mateial is consideed fo the calculation. When placed in a magnetic field, these elemental spheical elements would be consideed as giving ise to a magnetic moment due to its inheent susceptibility assigned as above and thus induced Magnetic Dipole Moment is placed at the cente of that sphee fo its oigin. distance atio is small enough fo all spheical elements when a paticula point is specified whee the induced field is to be calculated and fo this paticula case the point dipole appoximation becomes automatically the valid appoximation. As can be seen in Fig.4, this fist step fo the basic citeion is to ensue not only that the point dipole appoximation is inheently valid but also that fom evey one of the elemental sphees along the length of a vecto the subdivision ensues the contibution is the same. Which means if one knows the numbe of subdivided elements along the line, it is only equied to know the numbe of such sphees to multiply with the contibution fom any one spheical element. Fig.5 explains the equation used fo evaluating the numbe 'n' of such point dipoles along the length of the vecto. And the equation fo 'n' was the esult of a simple deivation. The Fig.6 illustates the compehensive situation fo a macoscopic spheical sample, which will have to be consideed with the subdivision citeia as above fo aiving at the Fo the Point Dipole Appoximation to be valid pactical citeia had been that the atio : S = : ( θ ) cos σ zz = 5 = σ ( ( θ )) v v σ N = cos v = Volume Susceptibility V = Volume = (4/) π s 4 v π s σ = N [ cos ( θ )] s st n th R i : i = : o even bette and the atio R i / i = C can be kept constant fo all the n sphees along the line (adial vecto) 4 v π i σ N [ ( )] i = cos θ = σ i R i σ i will be the same fo all i, i=,n and the value of n can be obtained fom the equation below n = + log log C C R R n + Link to Details v = -.855 x -7 s / = 45.86= C σ = σ =.4 x - fo θ= Fig.4. The basic pinciple of the summation pocedue illustated. Thus this subdivision ensues that the total magnetic susceptibility of the entie specimen has been localized into closely packed elements with popotionate susceptibility values and the sum of all the elemental susceptibility values would yield the total susceptibility of the entie specimen. The main pupose of this subdivision is to ensue that, when, at any given point the induced field contibution is to be calculated, the distance fom the cente of the espective small volume element to that point would be much lage than the adius of the spheical element and hence the point dipole appoximation would be valid. Setting up such an inheent citeion fo the subdivision was the key to open up the possibility of this simple summation pocedue, so that the esulting adius to Fig.5. An illustation of the equation equied fo the calculation of the numbe of close-packed sphees and the simultaneous citeion fo the validity of point dipole appoximation Demagnetization facto and induced fields at a cental point in the sphee. This appoach fo evaluating the induced fields etained the simple physical basis of a dipole field distibution all though and at evey stage and made it possible to think out such unconventional combination of oute specimen shapes with the inne Loentz cavity shapes and deduce also the induced field vaiation tends by simple aguments. as depicted in Fig.7. In Fig.7 thee is also a pat, which explains the tends of discete summations within the Loentz Sphee.
Fig.6. A figue indicating the summation equied ove all the adial vectos with diffeent pola and azimuthal angula coodinates. Oute a/b= oute a/b=.5 Demagf=. Demagf=.78 inne a/b= inne a/b= Demagf=-. Demagf=-..-.=.78-.=.78 conventional combinations of shapes Fig.5[a] Conventional cases a b Cuent popositions of combinations Oute a/b= oute a/b=.5 Demagf=. Demagf=.78 inne a/b=.5 inne a/b=.5 Demagf=-.78 Demagf=-.78.-.78=-.78.78-.78= Fig.5[b] Fig.7. The summation pocedue in the case of homogeneously magnetized sample;leading to the consideation of the unconventional combination of oute and inne shapes and the simplicity of handling such combinations to infe on the induced fields within the sample 4. The Case of Homogeneous Magnetization and Inhomogeneous Magnetization: Fig.8 illustates the advantages of this summation pocedue, which is the diect consequence of the possibility to etain the physical pictue in view at evey stage duing the calculation. Besides this qualitative advantage which is to be emphasized as the most impotant achievement now, quantitative aspects also stand to gain fo the possibility of handling a spindle like shape o a cylindical shape and use these basic citeion fo subdividing the magnetized sample to aive at the induced filed values at any point and at evey point even if the induced filed values would not be the same at all points as ensued fo the special shapes of the egula ellipsoids of evolution. It is no consolation fo such shapes that the sample is made up of the same mateial with the same susceptibility value unifomly though out the spindleshaped specimen. The magnetization attibutable to the susceptibility of a small volume element would be popotional to its volume. But, due to these induced magnetic moments the induced fields at vaious points within the specimen would sum up to the same value because of the esulting geometical consideations fo that shape. Thus if one can account fo the fact that the ellipsoids esult in the same value fo the demagnetizing facto at evey point within the specimen, then an account can be made also fo the fact that fo a spindle shaped specimen of the same mateial, the point by point values fo the demagnetization facto within the specimen ae not the same. In fact, by tabulating the values of the induced fields as a function of the spatial coodinate within the specimen fo a given shape it is possible to compae the tends of the contibutions as specific sums fo compaable pola coodinates fo diffeent shapes and gain much geate insights than what was possible with the ealie method. These details would not be pat of this pape since it would add to the length of this publication to such an extent as to become a distaction fom the undestanding of essential pinciples. 5. Futhe Advantages and the NMR as a Sensitive Tool:. Fist and Foemost, it was a vey simple effot to epoduce the demagnetization facto values, which wee obtained and tabulated in vey ealy woks on magnetic mateials. Those Calculations which could yield such Tables of demagnetization facto values wee athe complicated and equied setting up elliptic integals which had to be evaluated.. Secondly, the pinciple involved is simply the convenient point dipole appoximation of the magnetic dipole. And, the method equies hypothetically dividing the sample to be consisting of closely spaced sphees and the adii of these magnetized sphees ae made to hold a
convenient fixed atio with thei espective distances fom the specified site at which point the induced fields ae calculated. This fixed atio is chosen such that fo all the sphees the point dipole appoximation would be valid while calculating the magnetic dipole field distibution.. The demagnetization factos have been tabulated only fo such shapes and shape factos fo which the magnetization of the sample in the extenal magnetic field is unifom when the magnetic susceptibility of the mateial is the same homogeneously though out the sample. This esticts the tabulation to only to the shapes, which ae ellipsoids of otation. Whee as, if the magnetization is not homogeneous though out the sample, then, thee wee no such methods possible fo getting the induced field values at a point o the field distibution patten ove the entie specimen. The pesent method povides a geatly simplified appoach to obtain such distibutions. 4. It seems it is also a simple matte, because of the pesent method, to calculate the contibutions at a given site only fom a pat of the sample and account fo this potion as an independent pat fom the emaining pat without having to physically cause any such demacations. This also makes it possible to calculate the field contibution fom one pat of the sample, which is within itself a pat with homogeneously, magnetized pat and the emaining pat being anothe homogeneously magnetized pat with diffeent magnetization values. Hence a single specimen which is inheently in two distinguishable pat can each be consideed independently and thei independent contibution can be added. of Physics, 5(8), pages 74 to 78, 98 4. As can be found fom the efeences cited in Refeence above. Fo the point mentioned above view the web page URL: http://saavamudhan.tipod.com/ Refeences:. "Pyomellitic Acid Dianhydide ; Cystal Stuctue and Anisotopic Poton Magnetic Shielding ", S.Aavamudhan, U.Haebelen, H.Ingatinge and C.Kiege, Molecula Physics, 8, 4 (979). "Effect of Demagnetization on Magnetic Resonance Line Shapes in Bulk Samples: Application to Tungston", geoge Mozukewich, H.I.Ringemache and D.I.Bolef, Physical Review B, Volume, Page, 979.. "Local-field Effects and Effective-medium theoy: A Micoscopic pespective", D.E.Aspnes, Ameican Jounal