l μ M Right hand Screw rule

Transcription

1 Magnetic materials

2 Magnetic property The response of the materials to external magnetic field All the materials are magnetic, only the degree of response varies, which is measured in terms of their magnetization (strong or weak) The parameters used to study the magnetic behaviors of the materials are as follows: 1.Magnetic dipoles & magnetic moment: Magnetic dipoles are analogous to electric dipoles ; consists of a north pole and a south pole of strength m each separated by a small distance l Magneic moment = mxl For a circular current loop equivalent to a magnetic dipole, magnetic moment μ M = NIxA ( amp. m 2 ) & torque on the dipole τ = μ M XB Where I is the current in the loop and A is the area of the loop,n is no. of turns N l μ M S I Right hand Screw rule

3 2. Magnetisation = dipole moment / volume M = μ / V ( amp. / met.) 3.Magnetic susceptibility = magnetization / mag. field strength χ = M / H (no unit) 4.Magnetic permeability = magnetic induction / mag. field strength μ =μ 0 μ r μ = B / H (Wb / amp. met. = H/met) 5.Relative permeability μ r = μ / μ 0, μ 0 = absolute permeability = 4πX 10-7 H/m 6.Relation between H,B & M is B = μ H = μ 0 μ r H so 7. μ r =(1 + χ) B = μ 0 (M+H) = μ 0 (χh + H) = μ 0 (1 + χ) H

4 Classification of magnetic materials Diamagnetism Paramagnetism Ferromagnetism 1.Normally referred as nonmagnetic as the response is very weak 2.In ext. magnetic field magnetic moment induced in a direction opposite to applied field repelled by the field H=0, M=0 H=H M= - M 3. Permeability μ<1 4. susceptibility χ <0 5. Susceptibility does not depend on temperature 5. Ex; Cu, Ag, Hg, Au, Zn,SC 1.Normally referred as nonmagnetic as the response is very weak 2. They posses permanent magnetic moments, which are randomly oriented in the absence of ext. magnetic field., hence net magnetization is zero. When a field is applied moments get aligned in the field direction, giving positive magnetization H=0, M=0 H=H M= M 3. Permeability μ>1 4. susceptibility χ is positive, small and temp. dependant χ = C/ T Curie law 1.Referred as magnetic as response is strong(exchange coupling) 2. Posses permanent dipoles 3. Show spontaneous magnetization even in the absence of ext. field, magnetization shown is high, when field is applied, M increases 4.Permeability μ>1 5.susceptibility χ is positive, large and temp. dependant χ = C/ T-θ Curie Weiss law Ex; Fe, Co, Ni ferromagnetic domains

5 Origin of magnetic moment The three sources of magnetic moment in an atom are 1. orbital motion of the electron 2.spin motion of the electron 3. nuclear spin If the vector sum of all the contribution is zero then net magnetization is zero 1. orbital motion of the electron Motion of the electron (charged particle)around the nucleus in a circular orbit (orbital motion) is equivalent to a circular current and behaves as a magnetic dipole. Associated magnetic moment is μ M = IxA ( amp. m 2 ) I= - q/t, where T is the time period for one rotation = 2 π r / v v is the velocity of the electron in the orbit μ M = (- q v / 2 π r ) (π r 2 ) = - qvr/2 = - q/ 2m ( mvr) = - (q/ 2m ) L Orbital magnetic moment μ orb = - (q/ 2m ) L

6 2.spin motion of the electron Similarly, the spin motion of the electron around their own axis give rise to spin magnetic moment μ spin = - (q/ 2m ) S Total magnetic moment due to electron motion inside the atom is μ M = - (q/ 2m ) (L+S) = - (q/ 2m ) J, J is the total angular momentum ( from L+S to L- S Quantum no. associated with L is * l(l+1) ħ+, l is the orbital quantum no. l=0---s shell, l=1---p shell, l=2-----d shell, l= f shell Quantum no. associated with S ± ħ/2 Total magnetic moment = μ M = - g(q/ 2m ) J If magnetic field is applied along z-direction, the component of the total magnetic moment in that direction is, μ M = -g(qħ/2m)m j

7 Where mj is the magnetic quantum no. Varying from (J to -J) & g is called Lande s g-factor g = Calculation rules 1.if electrons are in the s-orbit, orbital magnetic moment is zero (l=0) 2. for completely filled shell, orbital magnetic moment is zero (l=0) As ml = l to l (s shell,l=0, p shell, l=1, d shell, l=2 and f shell, l=3..) 3. partially filled p, d and f shells contribute to orbital magnetic moment 4.If all electrons are paired, spin magnetic momentum is zero 3. Nuclear spin 1 J ( J 1) S( S 2J ( J 1) 1) Due to the spin of nucleus, a magnetic moment is associated which is very small as compared to the electronic contribution as heavy mass is involved (10-3 times) is masked by electronic mag. Mom. L( L 1)

8 Bohr magnetron If there is only a single electron it will have only spin magnetic moment μ s = -2 (qħ / 2m) ±1/2, as g=2 and m j = ±1/2 This is the fundamental magnetic moment called Bohr magnetron μ B All the magnetic moments are expressed in terms of μ B μ B = q = 9.27x Am 2 2m Ex;-1. hydrogen atom One electron in s shell. So l=0 and s= +1/2 or -1/2 implies that orbital magnetic mom. Is zero and spin mag. Mom. Is same as the Bohr magnetron ( μ B is the magnetic moment possesed by hydrogen electron) 2. Helium atom?...calculate

9 For other atoms apply the rules to calculate Hund s rule: 1. spins of electrons remain parallel to each other to the max. Extent 2. max. Value of L is consistent with the spin S 3. if shell is less than half filled J= L- S, if more than half filled J= L+ S & if exactly half filled then L=0 and J=S

10 Langevin s theory of diamagnetism motion of the electron in the orbit around the nucleus is equivalent to a current in the closed circuit. When magnetic field is applied electric field is induced in the circuit. The induced emf is opposite to the applied field and electrons are accelerated in the opposite direction. Acceleration a =- ee i /m, E i = V i /d, where V i is induced emf and d is the path covered by the electron in the orbit, d=2πρ ρ is the radius of projection of the orbit in the X-Y plane. V i = induced emf = - (rate of change of magnetic flux.) If MF is applied along Z-direction, flux= B(πρ 2 ) Putting all these, we get acceleration a= - e m(2 ) d dt ( B 2 Or, dv dt e m(2 2 ) db dt

11 dv e 2m Or, v 2 -v 1 = v= (eρ/ 2m) B B 0 db indicates that the velocity of the electron in the orbit changes, so angular momentum also changes the magnetic moment. μ M = IxA = -ea/t = -ea v / 2π ρ = - eρ/2 ( eρ/2m)b = - (e 2 ρ 2 /4m ) B As the plane of the orbit varies continuously due to applied MF we take average value of ρ av 2 = x av2 +y av 2 if r is the radius of the atom, then r av2 = x av2 +y av2 +z av 2 For spherical symmetric atom x av2 =y av2 = z av2 = r av2 / 3 So, ρ av2 = (2/3) r av 2

12 Hence, μ M = - (e 2 r av 2 /6m ) B If there are Z no. of electrons, then, μ M = - Z(e 2 r av 2 /6m ) B If N is no. of atoms per vol. Then magnetization M= - NZ(e 2 r av 2 /4m ) B Negative sign indicates that Magnetization is opposite to applied field So the diamagnetic susceptibility χ = μ 0 M/B or, χ = -NZ(μ 0 e 2 r av 2 /6m) It indicates that Susceptibility depends on average mean square radius and independent of temperature

13 Theories of Paramagnetism Classical theory (Langevin's): Let n = no. of dipoles in a system = n 0 exp ( - E/ k β T) (Boltzmann s distribution formula) B = magnetic field applied τ = torque experienced by the dipoles in the magnetic field = μ M X B = μ M B sinθ E = energy of the dipoles = τ dθ = μ M B sinθ dθ = - μ M B cosθ = - μ M.B Now we have n = n 0 exp (μ M B cosθ / k β T ) Total dipole moment for all these dipoles = M cos dn < μ M > = ; μ M cosθ is the component of μ M dn along B n 0 M dn μ M θ B

14 but dn= n0 k M B T exp M B cos k T sin d So, < μ M > = If we put cosθ = x, dx= - sinθdθ & μ M B / k β T = a then the above expression becomes < μ M > = 1 1 M 0 0 M x 1 1 cos exp( exp( M exp( ax) dx = μ M [ coth a- 1/a ] M exp( ax) dx B cos / k T )sin d B cos / k T )sin d

15 < μ M > = μ M L (a) ; L(a) is called Langevin s function Now magnetisation M= N μ M L(a) Case-I :- a is large when T is small or B is large L(a) 1 hence M = N μ M = M s = saturation magnetisation Case-II :- a is small when T is large or B is small coth = 1/a + a/3 a2/ /a + a/3 L(a) = a/3 magnetisation M = Nμ m μ m B/ 3k β T = N μ m2 B/ 3k β T Susceptibility or, χ = M/ H = N μ m2 μ 0 / 3k β T χ = C/T Curie law

16 Quantum Theory : The energy of the system in the magnetic field applied in Z- direction is E = - μ z.b μ z = -gm j μ B so, E = 2m j μ B B For a system of one electron (l=0), m j =+ ½ or ½ E = - μ B B when m j =- ½ & E = μ B B when m j = + ½ One corresponding to parallel and other to antiparallel spin magnetic moment orientation w.r.t. Magnetic field Difference in energy between the two E= 2 μ B B Let out of N total atoms per volume In the system N1 are parallel and N2 are antiparallel E B=0 E2 E1 B 0 m j =- ½ Z- dir m j =+ ½

17 Distribution of atoms at thermal equilibrium in the two corresponding states are given by Maxwell Boltzmann distribution function N1/ N = e E 1 / k β T / N & N2/ N = e E 2 / k β T /N Net magnetization M = (N1-N2) μ z Or M = N B e e k k B B B T B T e e BB k T k B B T M = N μ B tanh ( μ B B/ k β T) In case μ B B < < k β T M= N μ B ( μ B B/ k β T) = N μ 2 BB/ k β T χ = M / H or χ = C/T Curie law χ = N μ 0 μ 2 B/ k β T

18 Ferromagnetic theory ( Weiss ) As per Weiss in ferromagnetic materials spontaneous magnetization is observed, which is due to a strong internal field arising from an exchange interaction between the magnetic moments in the neighborhood domains exchange interaction between two atoms I and j = U= -2 J S i S j J is called the exchange integral Internal field H is proportional to the magnetisation H int α M Or H int = λm H tot = H appl + H int H tot = H appl + λm As, χ = M / H = M / H appl + λm = C / T by Curie law

19 C/ T = M/ H appl. +λ M C H appl. = MT - C λ M or, M = C H appl. / T C λ or, M = C H / T-Tc Susceptibility Curie-Weiss law χ = M / H = C / T-Tc Ferromagnetic domains: B

20 Domain wall 10-2 μm Ms Ms(0) Tc T Ferromagnetic hysteresis M s = saturation magnetisation M r = remanent magnetisation H c = coercive field

21 Soft ferromagnetic 1. Can be easily magnetized or demagnetized 2. Thin and long hysteresis loop 3. High permeability and low coercive field 4. Large susceptibility & low remanent mag. 5. As area of the loop is small, magnetic energy loss per volume is less during magnetisation and demagnetisation 6. Application: electromagnet, in motors, generators, dynamos and switching circuits 7. Ex: Fe-Si alloy, Fe-Co-Mn alloy and Fe-Ni alloy Soft and hard FM materials Hard ferromagnetic 1. Can not be magnetised or demagnetised easily 2. Wide hysteresis loop 3. Low permeability and high coercive field 4. small susceptibility & high remanent mag. 5. Large area of the loop indicates, magnetic energy loss per volume is high during magnetisation and demagnetisation 6. For permanent magnet in speakers, clocks 7. Rare earth alloys with Mn, Fe, Co, Ni

22 SOFT & HARD FERROMAGNETIC HYSTERESIS LOOP

23 Ferrimagnetic & Antiferromagnetic materials Ferrimagnetic material are special class of ferromagnetic material called ferrites with high permeability, saturation magnetisation and show hysteresis (square loop) Suitable for high frequency application and magnetic devices Some spin magnetic moments are in opposite direction in the magnetised state, but are of different magnitudes, giving rise to net finite magnetic moment. Molecular formula: Me 2+ Fe 2 3+ O 3 ; Me is a divalent atom like, Fe, Mn, Zn,Cd,Cu,Ni,Co,Mg Crystal structure: Inverse spinel In the cubic cell 8 atoms per unit cell. In the unit cell, 32 O -2 ions, 16 Fe 3+ ions, and 8 Me 2+ ions 8- Fe 3+ ions, 8- Me 2+ ions are surrounded by 6 oxygen ion octahedral site and spins are parallel

24 Rest 8- Fe 3+ ions are surrounded by 4 oxygen ions tetrahedral site and spins antiparallel Hence net spin moment of Fe 3+ ions cancel ( 8 up spin and 8 down spin) Only, 8 Me 2+ ions contribute to magnetic moment. As spin magnetic moment is μ = g μ B S since g=2, magnetic moment of one divalent atom μ di = 2μ B S in a unit cell 8 such atoms are present ( for Fe2+, s= 2, for Co =3/2,Ni=1, Cu=1/2, Mn=5/2 ) total moment in unit cell= 8 x μ di Magnetisation M = total moment per volume = 8 x μ di / a 3 Where a is lattice parameter Applications: resistivity of ferrites are very high so applied for high frequency application (eddy current energy loss less) Mixed ferrites are produced by combining two suitable divalent atoms

25 Ferrites have square hysteresis loop. So used for digital storage device ( two values of magnetisation +Ms & - Ms; so 1 or 0 ) Soft ferrites are used for high freq. Transformer core, computer memory, hard disc, floppy disk audio video cassette, recorder head Hard ferrites are used for permanent magnets in generator, motor, loud speaker, telephone Non-volatile memory called magnetic bubbles Antiferromagnetism: when exchange interaction between adjacent or neighboring domains give rise to ordered antiparallel spin arrangement, below a temp. called Neel temp. ex.- MnO,MnS,FeCl2, CoO. Net moment or magnetisation is Zero χ= C / (T +θ ) χ T N T

26 Fe 3+ octahedral Fe 3+ tetrahedral Me 2+ S= 2 S= 5/2 S= 5/2 Mn 2+ = 3d 5 μ = g μ B S = 2 x 5/2 x μ B = 5μ B Fe 2+ = 3d 6, μ = 4 μ B Co 2+ = 3d 7, μ = 3 μ B Ni 2+ = 3d 8, μ = 2μ B Cu 2+ = 3d 9 μ = 1μ B

27 Ferrites = Me 2+ O Fe 3+ 2O 3 Me= Fe, Mn, Co, Ni, Cu, Mg, Zn, Cd Magnetite has the empirical formula Fe 3 O 4, or Fe 2+ (Fe 3+ O 2 ) 2, ferrous ferrite. Its formula as a spinel would be Fe 3+ tetfe 2+ octfe 3+ octo 4, where "tet" and "oct" stand for tetrahedral and octahedral coordinations by the oxide anions. In the above model, the blue spheres represent the tetrahedral iron(iii) cations, and the red spheres are the octahedrally coordinated iron(ii) and (III) cations. The oxide anions are shown as the green spheres. Because of the fortuitous inverse nature of the magnetite structure, ferrous and ferric cations are both in the similar octahedral coordination by oxides. In "normal" spinels, such as the mineral spinel itself (magnesium aluminate), the A cation is tetrahedral and the M cations are both octahedral