Appendix A Mathematical Operations
This appendix presents a number of mathematical techniques and equations for the convenience of the reader. Although we have attempted to sum• marize accurately some of the most useful relations, we make no attempt at rigor. A bibliography is included at the end of this appendix.
A-1 COMPLEX NUMBERS A complex quantity may be represented as follows: u = x + iy = re+i
391 392 ELECTRON SPIN RESONANCE
Imaginary axis
u
iy
~E2------t-- Real axis
u' Fig. A-l Representation of a point u and of its complex conjugate u* in complex space (Argand diagram).
(y). Note that the real component of u is equal to one-half the sum of u and u*; the product of u and its complex conjugate is the square of the absolute magnitude, that is, (A-2)
A-2 OPERATOR ALGEBRA A-2a. Properties of operators An operator A is a symbolic instruction to carry out a stipulated mathematical operation upon some function which is called an operand. Unless its form is explicitly indicated, an operator will be designated by a circumflex. One of the simplest operators is a constant multiplier; for example, ka = ka. An operator n is said to be linear if the result of operation upon a sum of functions is the same as that obtained by operating on each function separately; i.e., if na = {3, then (A-3) Also, if e is a constant,
n(ca) = ena = e{3 (A-4)
If ai = !(qi) , then a/aqi is a linear operator. An example of a nonlinear operator is "Y". The reader will be familiar with such operators as the summation operator ~ n 2: ai == a1 + a2 + a3 + . . . + an (A-5) i=1 Its use permits a concise representation of a series. Frequently one wishes to summarize a set of equations with constant coefficients such as MATHEMATICAL OPERATIONS 393
t/h = C11 c/>1 + c 12c/>2 + c 13c/>3 +
t/12 = C21 c/>1 + C22c/>2 + C23c/>3 +
t/13 = C31 c/>1 + C32c/>2 + C33 c/>3 + + C3nc/>n (A-6) The sum of the t/1j can be represented by a double summation, viz.,
L t/1j = L L Cjkc/>k (A-7) j j k Here one encounters a juxtaposition of two operators, which in the general case are represented as Aii. It is understood that AB implies operation first with B and then with A. Interchange of order of the operators may give a different result; for example,
but
If AB = BA, then A and B are said to be commuting operators. The dif• ference A B - BA is called the commutator of A and B. It is represented by the symbol [A ,B] == (AB - BA). The magnitude of the commutator of two operators is of profound significance in quantum-mechanical systems. The commutators of angular-momentum operators will be treated in Appendix B. An operator fl is said to be hermitian if it obeys the following relation: (A-8) A useful aspect of the hermitian property is that (with care!) one may operate "backwards," i.e., to the left, when the operator occurs between two operands, as in Eq. (A-8). An example of operation to the left is found in Sec. B-4. Hermitian operators have the important property that if the result
Table A-l Classical and quantum-mechanical dynamical variables
Dynamical variable Classical quantity Quantum-mechanical operator
Position q q Time t dq , ... d Linear momentum pq= m dt Pq=-/l, dq
Pd> = r x p:j: Pd> = r x P Angular momentum P",= (xPy - ypx) pd>z=-ili (x :y - y :J = -iii a~ § 2 2 2 1f Kinetic energy T= Po" :!f=-!L.=--, P -h a associated with 2m 2m 2m aq2 coordinate q Potential energy V(q) V(q)
:j: The evaluation of vector products is described in Sec. A-4. § The angle measures rotation about the z axis. 1f This form of the kinetic-energy hamiltonian is valid only for cartesian coordinates. 394 ELECTRON SPIN RESONANCE of operating upon a function is the function itself multiplied by a constant, one is assured that the constant is real. (See Sec. A-2b.) Some of the most important operators of quantum mechanics are those associated with observable properties of a physical system, i.e., the "dy• namical variables." A few important linear operators are listed in Table A-I. Some of these operators are identical with the variable itself, whereas others involve derivatives.
A-2b. Eigenvalues and eigenfunctions If the result of the application of an operator A to a function I/In is Al/ln = A.nl/ln (A-9) where A.n is a constant, then I/In is said to be an eigenfunction of A with eigenvalue A.n- (The set of functions I/In is often called a "basis set. ") The spin functions I/Ia and 1/1/3 introduced in Sec. 1-5 are examples of eigenfunc• tions, in this case of the operator Sz; that is, Szl/la = +tl/la (A-lOa) Szl/l/3 = -tl/l/3 (A-lOb) (Angular momentum operator expressions are considered in detail in Sec. B-4.) A given set of eigenfunctions I/In may simultaneously be eigenfunctions of several operators. Operators having the same set of eigenfunctions have the very useful property that the operators must commute. In the case of the particle in a ring considered in Sec. 1-3, the wave functions 1/1 are eigen• functions both of the angular-momentum operator PcP and of the hamiltonian operator:fr. The eigenvalue equations are (A-II) and Schrodinger equation (A-12)
Table A-I gives PcP = -ih d/d~, where ~ measures the angular position ofthe particle. The kinetic energy of a classical particle having an angular mo• mentum PcP and moment of inertia 1 is
2 W=PcP (A-13) 21 The quantum-mechanical operator for a system with V = 0 is
~ _ PcP2 _ (-ih)2 d 2 _ -h2 d2 flt - 2i - 2I d~2 - 2T d~2 (A-14) Substitution of:fr from Eq. (A-14) into (A-12) gives MATHEMATICAL OPERATIONS 395
(A-I5)
Rearranging, 2 d 1/1 = -21WI/1 = -M2./. (A-I6) d1J2 1i2 'I' Here the constant 21W/1i2 has been set equal to M2. Two solutions of (A-16) are (A-17a) and 1/12 = A e-iMe/> (A-I7b) as is evident by substitution. From the requirement that the functions 1/1 be normalized, i.e., that
J:7T 1/1*1/1 d1J = 1 (A-IS) one finds that A = (27T)-!. Hence 1/11 = (27T) -!eiMe/> (A-I9a) and 1/12 = (27T)-!e-iM e/> (A-I9b) Insertion of 1/11 into Eq. (A-I2) gives -1i2 d 2 (_1_ iMe/» _M21i2 (_1_ iMe/» 21 d1J2 Y2ii e - 21 Y2ii e (A-20) Hence, the eigenvalue W of the operator it, corresponding to the eigenfunc• tion 1/11, is M21i2/2/. Use of 1/12 gives an identical energy value. Operation by Pe/> on 1/11 and 1/12 leads to the following equations:
-iii ~ (_1_ eiMe/» = Mli (_1_ eiMe/» (A-2Ia) d1J Y2ii Y2ii
-iii ~ (vk e-iMe/» = - Mli ( vk e-iMe/» (A-2Ib)
Hence, the eigenvalues of Pe/>' corresponding to the eigenfunctions 1/1] and 1/12, are +MIi and -MIi, respectively. The wave functions can be eliminated from Eq. (A-20) or (A-2I) by multiplication on the left by the corresponding complex conjugate function 1/1* followed by integration. This yields expressions for the energy and for the angular momentum of a particle moving in a circle. [See Eq. (1-21) and Prob. A-3.] 396 ELECTRON SPIN RESONANCE
A-3 DETERMINANTS A determinant is a scalar quantity which represents a linear combination of products of terms. It may be represented by a square array, for example
(A-22a)
Determinants are denoted in this book by boldface type enclosed by vertical lines. More generally a determinant of order k is represented as
(A-22b)
A determinant may be expanded by the "method of minors." The minor of any element aij is the determinant remaining after the row and column con• taining the element au are removed. The expansion is carried out by mul• tiplying the elements of a specific row or column by their corresponding minors as follows:
IAkl = 2: (-l)(i+j) au IA"-llu (A-23) iorj
Here IA"-llu is the minor corresponding to the element aij' For a determinant of order 3 this expansion may be carried out as follows:
all a l2 a21 a22 a31 a32 = alla22a33 - alla23a32 - al2a2la33 + al2a23a31 + a l 3a 21 a32 - al3a22a31 (A-24a)
Here the elements of the first row have been used. One could equally well have used the elements of any other row or column. The method of minors is a valuable technique for stepwise reduction of the order of a determinant. For example, a determinant of order 4 can be reduced in one step to a linear combination of four determinants of order 3. The value of a determinant is not affected by the addition (or subtrac• tion) of the elements of one row to those of another. This is also true for columns. Thus if any row or column is a mUltiple of another, the value of the determinant is zero. A special procedure applicable only to determinants of order 3 in- MATHEMATICAL OPERATIONS 397 volves a diagonal multiplication in the following manner. The sign of a term is positive if one proceeds diagonally downward, and it is negative if one proceeds diagonally upward.
IA31 = a lla22a 33 + a 12a 23a3l + a 13a 32a 2l - a lla23a 32 - a21Q 12Q33 - Q31 Q 22Q 13 (A-24b) Determinants are most frequently used for obtaining solutions to sets of simultaneous equations. Consider the following set of simultaneous equations relating the dependent variables Yl' Y2, Y3 to the independent variables Xl, X2, X3:
Yl = CllXl + C12X 2 + C 13X 3
Y2 = C21X l + C22X 2 + C23X 3
Y3 = C31Xl + C32X 2 + C33X 3 (A-25) The solutions may be represented as follows: 1..111 1..121 1..131 Xl=W X2=W X3=W (A-26)
Here
Cll C12 C13 1..11 = C2l C22 C23 (A-27a) C3l C32 C33 and
Yl C12 C13 1..111 = Y2 C22 C23 (A-27b) Y3 C32 C33 1..121 or 1..131 is obtained in an analogous fashion by replacing the column 2 or column 3 of 1..11 by
If the simultaneous equations are not independent, the value of 1..11 will be zero. One of the important applications of determinants is the solution 398 ELECTRON SPIN RESONANCE of secular equations (see Sec. 5-2) for the energies of a quantum-mechanical system. Determinants are often used to represent anti symmetrized wave func• tions because interchange of two electrons corresponds to interchange of two rows of the determinant. This changes the sign of the wave function as required by the Pauli principle. For example, a two-electron wave function is written
'1'_ 1 /o/(I)a(1) 0/(1)13(1)/ - V2f 0/(2)a(2) 0/(2)13(2)
= 0/(1)0/(2) ~ [a(1)f3(2) - a(2)f3(1)] (A-28a)
Equation (A-28a) is usually written in an abbreviated form
'I' = .~ 110/(1)a(I)0/(2)f3(2)11 (A-28b) v2! or 1 - 'I' = V2! 110/ 0/11 (A-28c) where the bar indicates 13 spin.
A-4 VECTORS: SCALAR, VECTOR, AND OUTER PRODUCTS Vectors (as distinct from scalars) are quantities for which a direction is as• sociated with a magnitude. One may add similar vectors by drawing arrows, tail-to-head, proportional to their magnitudes; the resultant is a vector drawn from the origin to the headofthe last vector. It is usually more expedient to express vector quantities analytically in terms of their components. Vectors are indicated in this book by boldface type, e.g., H. If the coordinates of a point with respect to fixed axes x, y, and z are 7, - 3, and 4.5, the vector locating the point is written as r = 7i - 3j + 4.5k (A-29a) Here i, j, and k are unit vectors in the x, y, and z directions, respectively. Addition of such vectors implies summation, respectively, of their x, y, and z components. Suppose a second vector is given by
s = 3i + 4j - 6k (A-29b) then the sum and difference are r+s= (7+3)i+ (-3+4)j+ (4.5-6)k = lOi + j - 1.5k (A-29c) r-s=4i-7j+ lO.5k (A-29d) MATHEMATICAL OPERATIONS 399
Multiplication of two vector quantities may give a scalar (scalar product) or a vector (vector product). The scalar product A . B of vectors A and B is defined to beAB cos eAB , where eAB is the angle between the A and B vectors. If A = axi + ayj + azk and B = bxi + byj + bzk, then
(A-30) since i . i = j . j = k . k = 1 and i . j = i . k = j . k = O. The vectors i, j, and k are said to be orthogonal. If A and B are complex quantities, the scalar product is taken as A * . B. The vector product C = A X B of vectors A and B is a vector C per• pendicular to the plane containing A and B; it is drawn from the origin of A and B and is of length AB sin eAB' The sense of the vector is obtained from the right-hand rule; if the right forefinger is parallel to A and the middle finger parallel to B, then the thumb indicates the direction of the vector product C. Considering the unit vectors i, j, and k, one notes that i X i = j X j = k X k = 0; i X j = k, j X k = i, i X k = -j, etc. Expansion of Ax B in terms of its components yields
C = A X B = (axi + ayj + azk) X (bxi + byj + bzk) = axbyk - axbzj - aybxk + aybzi + azbxj - azbyi = i(aybz - azby) + j (azbx - axbz) + k(axby - aybx) (A-3Ia)
Note that the result in Eq. (A-3I a) could have been obtained directly by writing a mnemonic "determinant" with the unit coordinate vectors in the first row and the components of A and B in the second and third rows, respectively,
j k A X B = ax ay az (A-3Ib) bx by bz
One of the important uses of vector products is in the description of the components of angular momentum. (See Table A-I and Appendix B.) A third type of product of two three-dimensional vectors is called an outer product. The result is a second-rank tensor. (See Sec. A-6.) The results of such a mUltiplication are as follows:
(A-32a) or AB=C (A-32b) 400 ELECTRON SPIN RESONANCE
A-5 MATRICES A matrix is defined as any rectangular array of n x m numbers or symbols ("matrix elements") where n is the number of rows and m the number of columns. The symbol for a matrix will be a boldface capital letter, e.g., B. If n = m = 1, then the matrix is a representation of a scalar quantity. If n = 1 and m > 1, then the resulting row matrix R may be regarded as one representation of a vector (a row vector). If n > 1 and m = 1, the column matrix C may similarly be regarded as a representation of a vector (a column vector).
C= (A-33 )
Such representations of vectors are a common practice. Hence the notation used for a matrix will be the same as that for a vector. A square matrix is one in which n = m. This special type of matrix is said to be an nth-order matrix. The square matrix B may be written
(A-34)
If bij = bi;, the matrix is said to be symmetric.
A-5a. Addition and subtraction of matrices The operation D=A+B or E=A-B (A-35) is accomplished by adding or subtracting corresponding matrix elements of A and B; e.g., the element dij is equal to aij + bij, and the element eij is equal to aij - bu. The following numerical examples will illustrate the procedure: 3 -2 7] [6 4 -2] [9 2 5] [-2 5 -4 + 4 2 3 = 2 7 -1 (A-36) 7 -4 8 -2 3 -5 5 -1 3 and
[-2 3 -25 -47] -[6 4 4 2 -2] 3 = [-3-6 -63-7 9] (A-37) 7 -4 8 - 2 3 - 5 9 - 7 13 Note that only matrices of the same dimensions can be added or subtracted. MATHEMATICAL OPERATIONS 401
A-5b. Multiplication of matrices The multiplication of a matrix by a scalar is accomplished by mUltiplying each element by the scalar, for example,
3 -2 7] [18 -12 42] 6 [ -2 5 -4 = -12 30 -24 (A-38) 7 -4 8 ~ -~ ~ The rules of matrix multiplication can be summarized as follows:
1. Two matrices can be multiplied only if the first is a z x n matrix and the second an n X y matrix, i.e., the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting product matrix will have dimensions z x y. 2. Each element of the product matrix is obtained as follows:
(ab )ik = L aiZblk (A-39) z As a first example, consider the multiplication of a column matrix by a row matrix [3 5 2] -4][ -1 = [(6) + (-5) + (-4)] = [-3] =-3[1] =-3 1 (A-40) Note that in this case the answer is a scalar. The result of this type of multiplication is called a "scalar product" since it corresponds to the scalar product of vectors. (See Sec. A-4.) For example, the scalar product H . S of the two vectors with components H x' H y, Hz and Sx, Sy, Sz is
[Hx Hy Hz] [Sx] ~: = [HxSx + HySy + HzSz] (A-41 )
Next consider the product of a 1 x 3 matrix and a 3 x 3 matrix [3 5 -4] [ 3 -2 7] -2 5 -4 = [-29 35 -31] (A-42) 7 -4 8 The product of a 3 x 3 matrix and a 3 x 3 matrix results in a 3 x 3 product matrix; for example,
I I 3 -2 7 j1 6 4 (-2)1 (-2)-----(5)----(-4) -(4)----(2)---~ (!) ~ [:i -i~ -~~J r 44 -66 7 -4 8 -2 3 (-5) (A-43) 402 ELECTRON SPIN RESONANCE
Ie will perhaps be clearer if one calculates a few elements of the above product. For example, the element au in the product matrix is (3)(6) + (-2)(4) + (7)(-2) = -4. The element a23 of the product matrix is (-2) (-2) + (5)(3) + (-4)(-5) = 39, etc. The location of an element resulting from multiplication of a particular row and column corresponds to that obtained by mentally (or actually) drawing lines through the row and the column being multiplied. This is shown in (A-43) for the element a23. In general, AB =;f BA. If AB = BA, A and B are said to commute. For example, if the order of multiplication in (A-43) is reversed, the result is quite different
6 4 -2] [ 3 -2 7] [-4 16 10] [ 4 2 3 - 2 5 -4 = 29 - 10 44 (A-44) -2 3 -5 7 -4 8 -47 39 -66 Each row of the product matrix (A-43) is the same as a column of the product matrix (A-44). This is a result of the fact that each original matrix is sym• metric. Had the original matrices not been symmetric, the product matrices would, in general, be dissimilar. The multiplication of matrices is associative, that is,
ABC = (AB)C = A(BC) (A-45) The student may satisfy himself as to the validity of this statement. In this book there will be occasion to perform the following type of multiplication.
(A-46)
The result is a scalar. Multiplication of the first two matrices yields a row matrix, and a row matrix times a column matrix is a scalar. [See Eq. (A-40).J It will be left to the reader to work out the result. A further example of matrix multiplication is the operation of rotation through an arbitrary angle cp in the xy plane. After counterclockwise rota• tion of the x and Y axes through the angle cp, the coordinates of a point in a rigid body will change from (Xl,yl) to (X2,y2). Reference to Fig. A-2 yields the relations between the new and old coordinates:
X2 = Xl cos cp + YI sin cp (A-47a) Y2 = -Xl sin cp + YI cos cp (A-47b) Alternatively, both initial and final coordinates may be expressed as column vectors:
[ X2] = [ co.s cp sin cp] [Xl] = [ Xl c~s cp + YI sin cp ] (A-48) Y2 -sm cp cos cp YI -Xl sm cp + YI cos cp MATHEMATICAL OPERATIONS 403
y y'
Fig. A-2 Coordinates of a point P before (XI,yI) and after (X2,y2) coun• terclockwise rotation of the co• ordinate axes through an angle It is clear that Eq. (A-48) is equivalent to Eqs. (A-47). The square matrix in Eq. (A-48) is called a coordinate-rotation matrix. A-5c. Special matrices and matrix properties A given matrix may be trans• formed into various related matrices, some of which have especially useful properties. Table A-2 defines several matrices derived from a matrix A; Table A-2 Matrix symbol Components Example A i1;j 3; IJ [;i Transpose A (A)u = aj; [3 ! i ~J Complex conjugate A * (A*)u=a~ [_24i 3~ iJ 4i Adjoint At (Atlu = aJ~ [3 ~ i -5 J Inverse A-I (A-')u=_I_~:j: 14 + 12i [ 5 -3 - iJ§ IAI aaj; 340 -4i 2 :j: IAI refers to the determinant with elements identical with those of the matrix. The inverse exists only for a square matrix, the determinant of which is non• zero. The symbolism "aIAI"/aaji implies that in taking the derivative with re• spect to aj;, all other elements will be kept fixed. For a 2 x 2 determinant IAI = lall a'21 = a"a22 - a21 al2 a21 a22 Hence alAI 12 oa Z1 = -a § The numerator and denominator have been multiplied by 14 + 12i to ratio• nalize the denominator. 404 ELECTRON SPIN RESONANCE Table A·3 Matrix Alternative definitions Unit 1 ali = 1, au = 0 if i # j Diagonal dA au=Oifi#j Symmetric A=A aij=aji Antisymmetric A=-A au=-aji Real A* =A ai} = aij Orthogonal A-I =A Hermitian At =A Unitary A-I = At various examples illustrating the relations among matrix components are given in the third column. The properties of some matrices of especial importance are defined in Table A-3. A-5d. Dirac notation for wave functions and matrix elements A shorthand no• tation, introduced by Diract for wave functions, is employed in this book; for example, the wave function t/ln is represented by In), where n is an identifying label, usually a quantum number. The function In) is called a "ket," For example, Eq. (A-9) can be written (A-49) The ket In) may be labeled with the quantum number n, since the eigenvalue An is a function only of n. Spin functions corresponding to Ms = +t and Ms = -t are conventionally represented by la) and 1,13), respectively. Cor• responding to each ket In) there will be a function called a "bra," written as (nl. A bra has meaning only when combined with another ket. For the bra (n I and the ket 1m), the notation (n 1m) implies integration over the full range of all variables, that is, (nlm) = i t/I;'t/ln dr (A-50) If m ¥- nand (nlm) = 0, the wave functions are said to be orthogonal. If m = nand (mlm) = 1, the wave functions are said to be normalized. It is always possible and usually convenient to choose the angular-momentum wave functions to be orthogonal and normalized, i.e., orthonormal. Fre• quently integrals of the form f t/I; Bt/lm dr are encountered as elements of a matrix [See Eq. (A-57).] In the Dirac notation such integrals are repre• sented by J7 t/I;Bt/lm dr = (niB 1m) (A-51 ) t P. A. M. Dirac, "The Principles of Quantum Mechanics," 3d ed., p. 18, Oxford University Press, London, 1947. MATHEMATICAL OPERATIONS 405 The expression (nI8Im) is then called a "matrix element." If n = m, the function is called a "diagonal matrix element," and if n =.F m then it is called an "off-diagonal matrix element." The average value (or expectation value) (b) of any observable b for the state described by the orthonormal wave function lfin is obtained by the following operation: (A-52) where 8 is the operator corresponding to the observable b. An important property of the bra and ket functions is given by the relation (nlm) = [(min)] * (A-53) In the matrix element (nI8Im) it is assumed that 8 is to operate in the forward direction of the ket 1m). To operate backwards on the bra (n I, one must take the adjoint of the matrix element, using Eq. (A-53). (n I 8 1m) = [( n I 8 1m) ] t = [( m I 8 In) ] * = x; [ (m In') ] * = x; (n' 1m) ~ ~ ~ (A-54) Here Xn can be a complex number. If 8 is a nonhermitian operator (e.g., the ladder operators which will follow) the effect of 8 is to alter the wave function (In) ~ In'», as well as to multiply it by a constant X n . If the operator 8 corresponds to an observable quantity, and is therefore hermitian, then Xn must be real and n = n'. Hence in this case it does not matter whether 8 operates forward or backward in the matrix element of Eq. (A-54). A-5e. Diagonalization of matrices Many matrices of interest (for example, the energy matrix) are encountered in structural problems; they are usually hermitian but not necessarily diagonal. It is possible to transform such matrices into a diagonal form. As an example, consider the spin operators Sx, Sy, and Sz for S = t. The eigenfunctions of Sz are usually taken as a basis set for the spin func• tions (see Sec. 1-5); that is, Szla) = +t la) (A-55a) Szlf3> = -t 1,8) (A-55b) Multiplication of Eq. (A-55a) from the left by (al gives (aISzla) = +t(ala) = t (A-56a) Similarly (,81 Szla) = +t (,8la) = 0 (A-56b) (aISzl,8) = -t(alf3> = 0 (A-56c) 406 ELECTRON SPIN RESONANCE (A-56d) It will prove to be very convenient to deal with a single matrix Sz instead of numerous operator equations such as Eqs. (A-56). These equations may be combined into the matrix equation (A-57) Sz is a matrix which includes all possible matrix elements of the operator Sz between the states a and f3. In general, if there are n states in the system, Sz will be a square matrix of order n. If such a matrix is diagonal, then the basis wave functions must be eigenfunctions of the operator. Thus la) and 1(3) are eigenfunctions of Sz. Examples of spin operators and spin matrices are given in Sec. B-6. Operation by Sxon la) and 1(3) gives the following results: Sxl a ) = tl(3) (A-58a) Sxl(3) = t la) (A-58b) The corresponding matrix is then (al~xlf3)J = [0 tJ = S (A-59) (f3I Sxlf3) t 0 x Sx is not a diagonal matrix because the basis functions are not eigenfunctions of Sx. Assume that the unknown eigenfunctions of Sx can be represented as linear combinations of Ia) and I(3), that is, IcPl) = cula ) + Cl2 1(3) (A-60a) IcP2) = c2l!a) + cd(3) (A-60b) or in matrix form (A-6I) Each row of the square matrix C in Eq. (A-61) may be considered as a row vector c. These vectors have the property that (A-62) Hence, the vectors c are called eigenvectors of the matrix Sx with eigen• values A. Insertion of the unit matrix 1 into Eq. (A-62) and rearrangement gives (Sx - Al)c = 0 (A-63) Equation (A-63) represents a series of simultaneous equations (called sec- MATHEMATICAL OPERATIONS 407 ular equations) which are not independent if A is an eigenvalue. If c # 0, then these equations may be solved by expansion of the following deter• minantal equation, often called the secular determinantt (A-64) Since the matrix Sx is of order 2, there will be two roots in Eq. (A-64); these are two values of A. On expansion, Eq. (A-64) becomes (A-65) '11. 2 -t= 0 A=±t (A-66) For '11.= t, the two simultaneous equations corresponding to Eq. (A-63) are (0 - t )cu + ic't2 = 0 tcu + (0-t)CI2 =0 or (A-67) For A = -t, C21 = -c22 • If IcpI) and ICP2) are to be normalized, CU 2 + Cl22 = 1 and C212 + C222 = 1. Hence, the final eigenfunctions of Sx are (A-68) The square matrix C in Eq. (A-68) is a unitary matrix, since C-I = Ct. It has the additional property that CSXC-I = CSxC t = A (A-69) where A is a diagonal matrix. For Sx CSP~r~ -~H; ~H~ -~1 ~ l-:t :t1r~ -~1 ~ r~ -;1 (A-70) t See, e.g., G. G. Hall, "Matrices and Tensors," chap. 4, Pergamon Press, Oxford, England, 1963. 408 ELECTRON SPIN RESONANCE This procedure for diagonalizing a hermitian (in this case, symmetric) matrix with the use of a unitary matrix and its reciprocal is general for any hermitian matrix. The most frequent examples encountered in this book are the diagonalization of hamiltonian matrices (see Chaps. 10, 1 I, and 12 and Appendix C) and the diagonalization of g and of hyperfine tensors. (See Chap. 7.) If C is taken to be the two-dimensional coordinate-rotation matrix [this is a unitary matrix, see Eq. (A-48)], a general method for the diag• onalization of any 2 x 2 matrix may be developed. The appropriate diag• onalization procedure is as follows: (A-71 ) [~~i:: ~~~: ] [~ ~ ] [~~:: ~~i;: ] = [~ ~ ] After matrix multiplication, the general solutions for zero off-diagonal ele- ments are 2c tan2w =--b• (A-72a) a- cos2w = ~ [1 + (I + (b ~2a)2rtJ (A-72b) sin2 w = 4[ I - (1 + (b ~2a) 2rt] (A-72c) . C Sill w cos w = [(b _ a)2 + 4c2]f (A-72d) x = a cos2 w + b sin2 w + 2c sin w cos w (A-72e) y = a sin2 w + b cos2 W - 2c sin w cos w (A-72j) The advantage of this method is that with w given by Eqs. (A-72), the co• ordinate rotation matrix on the left of (A-71) become the eigenvector matrix. That is, the two elements in each row are the coefficients of one of the two eigenvectors. A-6 TENSORS In Chap. 1 it was explicitly indicated that the simple resonance expression H r = hv/ gf3 is applicable only to systems which behave as if they were isotropic (i.e., which exhibit physical properties independent of orientation). The same restriction (isotropic behavior) applies to the expression H = hv _ aM = hv _ Aoh M (A-73) r gf3 I gf3 gf3 I for systems showing hyperfine splitting [see Eq. (3-12)]. Essentially, g and a (or Ao) have been assumed to be scalar quantities. In an anisotropic system, the response to an external stimulus cannot be described by a single MATHEMATICAL OPERATIONS 409 constant. Most physical properties will, in general, require six independent parameters to describe the response. The basic reason for requiring this large number is that for a general orientation, the response occurs in a direc• tion different from that of the applied stimulus. Except for systems of low symmetry, there are three special directions ("principal axes") along which the response will occur in the same direction as the stimulus. It is commonly found for anisotropic systems that the symmetry axis of highest order will be a principal axis (e.g., for a system with hexagonal symmetry, the sixfold rotation axis is a principal axis). t The magnetic susceptibility of a solid illustrates many ofthe properties encountered in both isotropic and anisotropic systems. The magnetization ".({ (response) of an isotropic solid is related to the field H (stimulus) by the expression ".({ = XH (A-74) Here X is the magnetic susceptibility. Consider H applied along the z direc• tion; Eq. (A-74) is then written as ~xl = [~ ~ ~l [ ~ 1= [ ~ 1 (A-75) [ JIt: 0 0 X Hz XHz The physical property X has been represented as a matrix. Such matrix representations of physical properties are called tensors. Tensors repre• sented by a square matrix are called tensors of the second rank.* In the following discussion, "tensor" will imply a second-rank tensor. The notation for second-rank tensors will be boldface italic type, e.g., A. Note that in the above case there is a component of magnetization only along the field direction. If the field is applied successively in the x or y directions or any other arbitrary direction, an identical numerical value of .I( is ob• tained in the same direction as the stimulus H. Next consider a crystal with axial, e.g., hexagonal, symmetry, exem• plified by graphite. Experiments indicate that when the field is along the hexagonal axis, the absolute magnitude of the diamagnetic susceptibility is much larger than the constant value of susceptibility for H in the layer plane. Taking the z direction as that of the hexagonal axis, [JltJIt xly = [xxx0 Xxx0 0 1[H Hy xl = [XxxHXXXHy xl (A-76) Jltz 0 0 Xzz Hz XzzHz Each element of the susceptibility tensor requires two subscripts. (Note that the number of sUbscripts equals the rank of the tensor.) Consider first the application of a field along one of the principal axes. The magnetization t This is not necessarily the case for the hyperfine tensor. *Vectors are tensors of the first rank, whereas scalars are tensors of zero rank. A fuller defini• tion of second-rank tensors is given later. 410 ELECTRON SPIN RESONANCE L,z Fig. A-3 An illustration of the diamagnetic susceptibility of a graphite crystal. The zaxis corresponds to the hex• agonal axis of the crystal. For H in the xz plane, H can be resolved into components H x and Hz. Upon multiplica• tion of Hx and Hz by Xxx and Xzz (negative quantities), re• spectively, the components Atx and .ftz are obtained. The resultant .It is obviously not parallel to H since Xxx"" Xzz' is then along the same direction. If the crystal is rotated in the magnetic field, there will be nonzero field components H x, H y, and Hz along the x, y, and z axes; hence, there will be components.4tx = XxxHx,.4ty = XXXHy, and .4tz = XzzHz such that.A will not be parallel to H. Figure A-3 illustrates this for H lying in the xz plane. In this case it is clear that .A and H are not parallel. For an axial system, one usually writes X.L for Xxx or Xyy and XII for Xzz to designate the susceptibility components perpendicular or parallel to the symmetry axis. This same tensor X is applicable to systems with three• or fourfold axes of symmetry. The extension to rhombic or lower symmetry requires only the condi• tion Xyy "" Xxx, that is, [.4txly = [xxx0 Xyy0 0 1[H Hy xl = [XxxHXyyHy xl (A-77) .4tz 0 0 Xzz Hz XzzHz Again, only in the case of H along one of the principal axes will the magne• tization be parallel to H; an arbitrarily oriented field gives magnetization components determined by the field components H x' H y, Hz and the sus• ceptibility tensor x. At this point it is desirable to consider a representation for X ifthe axes which are used are not the principal axes of an orthorhombic crystal. Spe• cifically, consider the use of the x', y', z' axis system, rotated through cos-1 Ixx with respect to the x axis, and having the x' axis further described by cos-1 [Xy and cos-1 lxz. These Ijj quantities, which measure the cosine of MATHEMATICAL OPERATIONS 411 \ y y'\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x z I Z' Fig. A-4 A set of new axes x', y', z' derived from the old axes x, y z by a rotation about an arbitrary direction through the origin. the angle between axes i and j, are aptly termed direction cosines. (See Fig. A-4.) The first subscript refers to the "new" axes and the second to the "old." The "new" and "old" axes may be related by use of the direction cosines in Table A-4. Figure A-5 illustrates the components of H along, say, x'. Since the old axes x, y and z are not perpendicular to x', H x' H y, and Hz will all con• tribute to the field component H x'. The contribution from H x will be H x cos (x' ,x), that of H y is H y cos (x' ,y) and that of Hz is Hz cos (x' ,z). Hence H x' = Hx1xx + Hy1xy + H)xz (A-78a) Similarly (A-78b) and (A-78c) Table A-4 Old New x y z x' Lxx lxy lxz y' lyx lvv lyz <.-' lzx lzv lzz 412 ELECTRON SPIN RESONANCE y, ,, , x' / /. ./ \ \ \ \ \ \ " \ \ \ \ \ \ .J'!''----+--~H;-;-X-+,-----<'+, -'---- x I , I I I I I I I I I I I I i / ______1~/ Fig. A-5 Representation of the component H x' and the separate contributions to it from H x, H y, and Hz. The results are conveniently summarized in matrix form as follows: (A-79a) or H'=.2'H (A-79b) The corresponding transformation of.,l( to the .,1(' representation is given by (A-80a) or (A-80b) The task is to relate the vfti' components to the Hi' components. With the expectation that the appropriate susceptibility tensor is more complicated than that in (A-77), one writes XXY [Al}[XUvfty' - XYX XYY XYZX"Wrj H y' (A-8Ia) vftz' XZX XZy Xzz Hz, X' or .,1(' = X'H' (A-8Ib) MATHEMATICAL OPERATIONS 413 In general, the principal axes for a given system are not known in advance; hence one uses some convenient set of arbitrarily chosen axes x', y', z'. The tensor X' represents the form in which the data are initially obtained. The task is to transform X' to a diagonal form which is designated dX. This is equivalent to the rotation of the x', y', z' axes to the principal axes x, y, and z. Substitution of Eq. (A-79b) into Eq. (A-8Ib) relates .At' to H in the principal-axis system. A similar substitution of Eq. (A-80b) into Eq. (A-8I b) yields ,2'.At = X'2H (A-82a) Multiplication from the left by ,2'-1 yields (A-82b) since ,2'-1,2' = 1. The direction-cosine matrices are unitary; hence, ,2'-1 =,2't. The direction cosines are real and thus ,2't =2. Equation (A-82b) then becomes .At=2X'2H (A-82c) or lyx I"Wx,~ XX'y' lxy 1x.][Hx] [Al}[l>"Jlty lxy lyy lzy Xy'X' Xy'y' X~,,]Xy'Z' r"lyx lyy lyz Hy Jltz lxz lyz lzz Xz'x' XZ'y' Xz'z' lzx lzy lzz Hz dX (A-82d) It is useful at this point to carry out the matrix multiplication indicated in Eq. (A-82d) to obtain the transformation equations relating X' to dX. For example, the element Xxx of d X is = 2: 2: IjxlixXij (A-83) j=x~Y~Z i=x,Y~z There will be a similar transformation equation for each element of the tensor. A tensor of second rank is then defined as any physical property which transforms according to equations such as Eq. (A-83). If the direc• tion cosines are such that Xu = 0 for i "# j, then X will be diagonal, and the direction cosine matrices are eigenvector matrices. (See Sec. A-5e.) Diag• onalization of g tensors and of hyperfine tensors is accomplished in a fashion identical to that described above and is a very important aspect of the study of anisotropic systems. (See Chap. 7.) The tensors 9 and A must be ob• tained by taking the square roots of the tensors g2 and A2. The square of a 414 ELECTRON SPIN RESONANCE tensor 8 is defined as: 8 2 == 88t. This definition requires 8 2 to be sym• metric, even if 8 is not. One is thus unable to distinguish between such components as (8)yx and (8)xy without additional information.t A-7 PERTURBATION THEORY In numerous structural problems, solutions are available for the major terms in the hamiltonian operator. For example, the rotational motion of a di• atomic molecule is well approximated by the solutions to the quantum• mechanical rigid-rotor problem. If there is an additional term in the hamiltonian operator which is small, then perturbation theory may be ap• plied to ascertain its effect on the wave functions and energies ofthe system_ Centrifugal distortion in the rigid-rotor problem is an example of a small perturbation. The hamiltonian for such systems is usually written as (A-84 ) where *0 is the hamiltonian for which solutions are known and *' is the perturbation operator. A is a perturbation magnitude parameter, having values between °and 1. Suppose the eigenfunctions and eigenvalues of *0 are given by (A-85a) or (A-85b) Here the Dirac notation for wave functions has been utilized. i ranges over the full set of "zero-order" eigenfunctions 11 )0, 12)0 ... In)o. The Wio values are called "zero-order" energies. The unknown eigenfunctions and eigenvalues of * are given by (A-86) Since the solutions of Eq. (A-86) must go continuously into those of Eq. (A-85b) as A ....,. 0, it is assumed that Ii) and Wi can be expanded as a power series in A; that is, Ii) = li)O + Ali)' + A2Ii)" + (A-87) Wi = Wio + AWI + A2WI' + (A-88) Substitution of Eqs. (A-84), (A-87), and (A-88) into Eq. (A-86) yields t If the true spin of a system is greater than!. there is some imprecision in referring to "g" and "A" as tensors. (See A. Abragam and B. Bleaney, "Electron Paramagnetic Resonance of Transition Ions", pp. 650 to 653, Oxford University Press, London, 1970.) MATHEMATICAL OPERATIONS 415 C~o + A*")(li)O + Ali)' + A2Ii)" + ... ) = (WiO + AW; + A2 W;' + ... )(li)O + Ali)' + A2Ii)" + ... ) (A-89) or *'oli)O + A(*"li)O + *'oli)') + A2(*"Ii)' + *'oli)") + ... = WiOli)O + A(W;li)O + WiOli)') + A2(W;'li)0 + W;Ii)' + WiOIi)") + ... (A-90) Equation (A-90) must be valid for all possible values of A; this is possible only if the coefficients of a given power of A are equal on both sides of the equation. Thus one may write *'oli)O = WiOli)O (A-91a) *', li)O + *'oli)' = W; li)O + Wioli)' (A-91h) *"Ii)' + *'oli)" =W;'Ii)° + W;Ii)' + WiOli)" (A-91c) It was assumed that the solutions to Eq. (A-91 a) are known. In Eq. (A-91h) the functions Ii)' are unknown; similarly the effect of *', on Ii)O is unknown. Assume that these functions of the zero-order basis set may be expanded as follows: Ii)' = Alll)O + A212)0 + ... + Ajlj)° + ... (A-92) *"li)O = HliIO° + H2i12)0 + ... + Hjilj)° + . (A-93 ) Multiplication ofEq. (A-93) from the left by °UI gives H ji = °UI*"li)O (A-94) since the kets are assumed to be orthonormal. Rearrangement of Eq. (A-91h) gives (*'0- WiO) Ii)' = (W; -*")li)O (A-95) Substitution of Eqs. (A-92) and (A-93) into Eq. (A-95) gives 2: (WjO - WiO)Ajlj)° j = W;li)O - HliIO° - H 2i 12)0 -. . - Hjilj)° - . .. (A-96) Multiplication from the left by O(iI gives W; = Hi; = O(il*"li)O (A-97) The quantity W; is the first-order correction to the energy. Multiplication of Eq. (A-96) from the left by °U I gives A - -Hji i#j (A-98) j- W.o- W.O J • 416 ELECTRON SPIN RESONANCE The coefficient Ai remains to be determined [since for i = j, Eq. (A-98) is not valid]. By requiring Ii) to be normalized, it is readily shown that Ai = Ot Hence the first-order correction to the wave function is Ii)' = - '"" Hji 1")0 (A-99) W.o-W.o J 4-J J , Here the prime on the summation signifies that the term i = j is not included. A similar approach to the solution of Eq. (A-91c) yields the second• order corrections to the energies and wave functions (A-IOO) Examples of first- and second-order corrections to energies and wave functions are given in Secs. C-7 and 11-6. Owing to the occurrence of energy differences in the denominator of Eqs. (A-98) or (A-1 00), it is necessary to modify the above procedure when applying perturbation methods to systems with degenerate energy levels. t A-a EULER ANGLES Some experimenters prefer to specify the orientation of the set of axes X, Y, and Z relative to x, y, and z by the Euler angles shown in Fig. A-6. t H. Eyring, J. Walter, and G. Kimball, "Quantum Chemistry," pp. 94 to 101, John Wiley & Sons, Inc., New York, 1944. z Fig. A-6 Representation of the Euler angles 8, x N The projection of the z axis on the XYplane determines the line OM. ON represents the intersection of the xOy and XOYplanes. With the magnetic field along the z direction, the energies of an electron spin system depend only upon the angles e and f/J, since 1/1 measures the rotation about z and such rotation does not alter the energy of the system. BIBLIOGRAPHY Anderson, J. M.: "Mathematics for Quantum Chemistry," W. A. Benjamin, Inc., New York, 1966 (good for operator algebra and perturbation theory). Bak, T., andJ. Lichtenberg: "Vectors, Tensors and Groups," W. A. Benjamin, Inc., New York, 1966 (discussion of vectors is recommended). Hall, G. G.: "Matrices and Tensors," Pergamon Press, Oxford, England, 1963. Nye, J. F.: "Physical Properties of Crystals," chaps. 1 and 2, Oxford University Press, Oxford, England, 1957 (excellent discussion of second-rank tensors). PROBLEMS A-I. Write the sum S = A + B, difference D = A - B, and products P = AB, Q = BA of the matrices 2 0 -IJ A=[ ! -21 -2J3 and B = [-4 1 2 -2 1 -I 1 -I 0 A-2. If and B= [-I 2J 3 -2 show that (A + B)(A - B) 7" A2 - B2. A-3. Multiply the wave functions in Eqs. (A-20) and (A-2l) from the left by the complex• conjugate wave functions. Then integrate the resulting expressions over the appropriate range of", to obtain the energy and the angular momentum for a particle moving in a circle of fixed radius. A-4. Find x, y, and z such that i -; -!] [~] = [-~] [-I 3 -2 z 6 A-S. Obtain the eigenvalues of the following matrix by the determinant method: 5 0 -2] [ o -3 0 -2 0 2 A·6. Diagonalize the matrix 1 A= [ ~ 3 -I]-I -I -I 3 418 ELECTRON SPIN RESONANCE using the eigenvector matrix 1 1 0 -- v'2 v'2 1 1 1 c= - -- - v'3 v'3 v'3 2 1 1 -- -- Y6 Y6 Y6 A-7. Perform the following matrix multiplication: A-S. The coordinates of the corners of the rectangular base of a parallelepiped are (2.191, 0.448,0); (-1.673, 1.484,0) (~2.191 ,-0.448,0); (1.673,-1.484,0). (a) Determine the angle between the x axis and the long side of the rectangle. (b) Using the appropriate rotation matrix (A-48), transform the coordinates to an axis system in which the long side of the rectangle is parallel to the x axis. A-9. A radical R is produced in a single crystal of RH by irradiation. Axes x, y, and z of R and RH are chosen to coincide with simple directions on the faces of a single crystal. The direction cosines of the principal axes 1, 2, and 3 with respect to the x, y, and z axes are, re• spectively, (0.48,0.88,0.05), (0.07,-0.10,0.99), and (0.87,-0.47,-0.1 I). The elements of the g tensor are as follows: g"" = 2.0039, gyy = 2.0030, gzz = 2.0035, gXY = -o.0007,gxz = -0.0001, gyZ = O. Find the principal elements gIl, g22, and g3:> A-tO. Prove that the coordinate-rotation matrix COS cf> sin cf> O~J [ -s~n cf> co~ cf> is unitary. A-H. Verify by matrix mUltiplication of Eq. (A-7!) that Eqs. (A-72) represent the require• ment for diagonalization of a symmetric 2 x 2 matrix. Appendix B Quantum Mechanics of Angular Momentum 8-1 INTRODUCTION Those properties (such as the total energy or angular momentum) which are conserved in an isolated system are called "constants of motion." These constants and their corresponding operators play an important role in quan• tum mechanics. Consider the operator :ft for the total energy, i.e., the hamiltonian operator. When:ft operates on an eigenfunction t/ln(x,y,Z) of a particular system the result is (B-1 ) Here W n is the exact total energy of the nth state of the system. Equation (B-l) is the Schrodinger equation. Such exact values W n are called "eigen• values." Operators which do not correspond to constants of motion do not yield eigenvalues. Suppose now that An is another constant of motion with a corresponding operator A. I t is an important result of quantum mechanics that t/ln(x,y,Z), the eigenfunctions of :ft, can always be chosen in such a way that they are also eigenfunctions of A. Angular momentum is a constant of motion for an isolated system; this implies that the operator for the total angular momentum (or more exactly the square of the total angular mom en- 419 420 ELECTRON SPIN RESONANCE tum) should also produce an exact result when applied to an appropriately chosen !/In(x,Y,Z). If this operator is designated by j2, then (B-2) Here Aj is an eigenvalue of j2 and is expressed in units of (h/27T)2 = liZ. By an appropriate choice of an axis system, !/In(x,y,Z) can be made to be an eigenfunction of anyone of the components of j, i.e . .ix.iy, or jz, in addition to being an eigenfunction of j2 and of it. However, !/In(x,Y,Z) can never be a simultaneous eigenfunction of two or more of the components of J If !/In(x,Y,Z) is to be a simultaneous eigenfunction of any group of operators, each of which corresponds to a constant of motion, all of the operators must commute among themselves. For instance, if !/In(x,Y,Z) is to be simulta• neously an eigenfunction of it, jz, and jz, then the following relations must hold: (itj2 -j2it)!/ln(x,y,Z) = 0 (B-3a) (itjz -j$)!/ln(x,y,Z) = 0 (B-3b) and (B-3c) Aj, the eigenvalue of j2, has an important property. Its value is de• termined only by the symmetry properties of !/In. Thus, although the values of W n vary according to the wave function used, Aj will be the same for all wave functions which have the same symmetry characteristics. For ex• ample, all spherically symmetric wave functions have Aj = O. Hence, if one can obtain a set of solutions to Eq. (B-2), these eigenvalues will be generally applicable to all wave functions of the same symmetry. !/In(x,Y,Z) has been represented as a three-dimensional wave function. In this case it is found that Aj is a function of the quantum number j, which is an integer. For instance,j could be the electron orbital angular-momentum quantum number lor the molecular rotation quantum number J. However, electrons and nuclei possess an intrinsic angular momentum (called spin angular momentum) which cannot be described in terms of the spatial wave functions !/In(x,Y,Z). In order to account for this spin angular momentum, a "spin coordinate" must be included in the wave function. It is found that the quantum number j can then take on half-integral values as well (that is, j = 0, t, I,!, ...). This spin angular momentum has no classical counter• part; however, it can be accommodated in the quantum mechanics if a gen• eralized angular momentum is defined. (See Sec. B-3.) B-2 ANGULAR-MOMENTUM OPERATORS In order to obtain the quantum-mechanical operators for angular momentum, one must first consider the classical expression for the angular momentum P PrI> = r X P (B-4) Here r represents the position vector of the particle, and P is its linear• momentum vector. The vector product indicated in Eq. (B-4) can be com• puted as in Eq. (A-31h). The components of PrI> are PrI>x = ypz - ZPy (B-5) PrI>y = ZPx - xpz (B-6) PrI>z = XPy - YPx (B-7) The operator r remains the same as r in classical mechanics but P must be replaced by -iliV, where V=ia/ax+ja/ay+ka/az. Thus in quantum mechanics, the components of PrI> become P• =-In'J:. (ay--Z- a) (B-8) P• =-In'J:. (aZ --x- a) (B-9) rl>y ax az and P• =-In'J:. (ax-- y-a) (B-10) J· =-1 . (ay--z- a) (B-ll ) x az ay (B-12) and iz = -i (x a~ - ya:) (B-13 ) As in the classical mechanics of angular momentum, the square of a vector operator is equivalent to the sum of the squares of its three component operators, that is, (B-14 ) In classical mechanics the magnitude and direction of an angular• momentum vector are well defined. In quantum mechanics only the mag• nitude of the total angular momentum vector and one of its components may simultaneously have well-defined values. It is possible to determine two observables simultaneously only if the operators corresponding to them commute. (See Sec. B-1.) It will be seen in Sec. B-3 that]2 commutes with 1z; however,1z does not commute with ix or i y. 422 ELECTRON SPIN RESONANCE B-3 THE COMMUTATION RELATIONS FOR THE ANGULAR-MOMENTUM OPERATORS By expanding i 2, it can be shown that i2 commutes with anyone of the components ix, i y, or i z, that is, (B-15) Here, [i2,Jx] = [i2ix - ixi2] etc. However, no two components com• mute among themselves. For example, A A (a a)( a a) J x J y =- y--z-az ay z--x-ax az a a2 a2 a2 a2 ] = - [y ax + yz az ax - xy az2 - Z2 ax ay + xz ay az (B-16) JA J A =- (az--x- a )(y--z- a a ) y x ax az az ay [ a2 a2 a2 a2 a ] = - yz ax az - Z2 ax ay - xy az2 + xz az ay + x ay (B-17) (B-18) Similar expressions hold for the commutators of the other components. Consequently, it is not possible simultaneously to determine three or even two components of the angular-momentum operator. However, it is possi• ble to determine the square of the magnitude of j and one of the components of j, which is taken as i z. These important commutation relations are sum• marized as follows: [ix,Jy] = ilz (B-19a) [iy,Jz] = iix (B-19b) [iz,Jx] = iiy (B-19c) and (B-15) A generalized angular momentum (i.e., one which includes spin) is defined as any function which obeys the commutation relations of (B-J9) and (B-J5). At this point it is convenient to introduce the so-called "ladder" oper• ators, which are linear combinations of i x and i y: i+=ix+Jy and (B-20) QUANTUM MECHANICS OF ANGULAR MOMENTUM 423 } + is called the "raising" operator and}_ the "lowering" operator. The significance of these operators will become apparent later. As can readily be verified by substitution of their definitions, they obey the commutation relations [}2,1+] = [12,1-] = 0 [1z,1-] = -1- [1z,1+] =}+ and (B-21 ) 8-4 THE EIGENVALUES OF }2 AND }z Let the eigenvalues of }2 and }z be Aj and Am, respectively. The angular• momentum eigenvalues depend only on the quantum numbers j and m. j is characteristic of the total angular momentum, and m is characteristic of the z component of the angular momentum. Consequently, as far as the angular• momentum properties are concerned, the eigenfunctions are solely a func• tion of j and m. Since the angular-momentum properties of the wave functions depend only onj and m, the functions can be represented by the kets Ij, m). (See Sec. A-5d.) The eigenvalue equations for }2 and }z can then be written (B-22) and (B-23) Here Aj and Am are the eigenvalues of}2 and }z, respectively. Note that the Ij, m) kets are orthonormal. This means that (j', m'lj, m) = 1 for j' = j and m' = m =0 for j' =F j or m' =F m (B-24) The operator }2 may be expanded as 2 }2 =}x 2 +} y +} z 2 (B-14 ) } 2 2 =J~2 2 x +} y _JA z (B-25) Thus the operator (}x2 + }y2) also has the discrete eigenvalues (}x2 +}y2)lj, m) = (}2 _}z2)1j, m) = (Aj - Am2 ) Ij, m) (B-26) Since the operators }x2 and }y2 correspond to experimental observables (and hence are hermitian), when they are applied to Ij, m), they must give real 424 ELECTRON SPIN RESONANCE numbers. Hence the eigenvalues of (Jx 2 +J/) must be real and positive, that is, (B-27) To establish the exact form of the eigenvalues Am, it is convenient to examine the matrix elements of the commutator [1z,1+] =J+ from Eq. (B-21), (B-28) Evaluation of the left-hand side of Eq. (B-28) demonstrates the effect of J+ on the wave functions Ij, m). The left-hand matrix element can be ex• panded into two matrix elements (j, m'IJzi+U, m) - (j, m'IJ+JzU, m) Use of Eq. (B-23) allows the second matrix element to be reduced to (j, m'IJ+Jzlj, m) = Am(j, m'IJ+lj, m) The first matrix element can be reduced with the help of Eq. (A-54), (j, m'IJzi+lj, m) = A-;;',(j, m'IJ+U, m) (B-29) ~ Since Jz is a hermitian operator, Am' must be real, and 1..-;;', = Am'. Thus Eq. (B-28) reduces to (Am' - Am)(j, m'li+U, m) = (j, m'IJ+lj, m) (B-30) This means that the only nonzero matrix elements of J+ are those for which (Am' - Am) = + 1. Hence J+lj, m) = xmU, m + 1) (B-31) This is easily seen from the fact that if Am' - Am '" + 1, then Eq. (B-30) is satisfied only if (j, m'IJ+U, m) = o. Similarly, an examination ofthe matrix element of J_ shows that the only nonzero matrix elements of J_ are those for which (Am' - Am) = -1, that is, LU, m) = YmU, m - 1) (B-32) Here Xm and Ym of (B-31) and (B-32) may be complex numbers. The factor ei, where cf> is a phase angle, may appear in Xm and Ym. (See Sec. A-I.) It is apparent from Eqs. (B-31) and (B-32) why J+ and L are called raising and lowering operators, respectively. This analysis shows that for a given value of Aj, one may obtain a whole series of states U, m) having the eigenvalues This series must terminate at both ends, since from Eq. (B-27), Am2 :% Aj. Since the Am values differ by integers and the m quantum number is assumed to increase in integral steps for a given value ofj, one may equate Am and m. QUANTUM MECHANICS OF ANGULAR MOMENTUM 425 Within the above series, the lowest eigenvalue of lz is designated by t!!. and the highest eigenvalue of iz bym. Therefore i+Jj, m) = ° (B-33) LJj,t!!:,) = 0 (B-34) Otherwise, there would be a value of mhigher than A} and a value of t!!.less than -A}; this is contrary to the limitation imposed by Eq. (B-27). Next apply L to Eq. (B-33). When Li+ is expanded, Li+ = Ox - ily) Ox + iiy) = i/ + ii + i[ix,iy] =i2 -iz 2-i z (B-35) Therefore Li+lj, m) = (Aj - m2-m) Ij, Yi1) = 0 (B-36) that is, Aj=m(m+ 1) (B-37) Similarly, by applying i + to Eq. (B-34) Aj = t!!. (t!!. - 1) (B-38) Equations (B-37) and (B-38) are compatible only if m= -t!!.. Since successive values of m differ by unity, (m- t!!.) is a positive integer which is denoted by 2j. Hence j can have the values j=O,i, l,i, ... Then from m - t!!. = 2j and m = - t!!., m=j and t!!.=-j (B-39) Hence m = j, j - 1, .. , -j + 1, -j. There are thus 2j + 1 permissible values of m for each value of j. From Eqs. (B-37) and (B-38) Aj =m (m+ I) = j(j + 1) (B-40) The eigenvalues of i2 and iz are then i2Jj, m) = j (j + 1) Ij, m) (B-4l) and izJj, m) = mJj, m)t (B-42) t Note that if the integer (m - mJ had been set equal toj, then the eigenvalue of}2 would be (/2) [(j12) + 1]. Hence half-integral quantum numbers appear quite naturally in this treatment. Half-integral values of j occur for electron and nuclear spin angular momenta. 426 ELECTRON SPIN RESONANCE 8-5 THE MATRIX ELEMENTS OF i+, L, i"" AND iy The quantities Xm and Ym of Eqs. (B-31) and (B-32) remain to be evaluated. Consider the matrix element U, mILi+U, m) = xmU, mlLlj, m + 1) = xmYm+1U, mU, m) = XmYm+1 (B-43) But from Eq. (B-36) (j, mILi+U, m) = [j(j + 1) - m2 - m] (B-44) Thus XmYm+1 = [j(j + 1) - m(m + 1)] (B-45) The matrix element of Eqs. (B-43) may be evaluated in a third way: (j, mILi+U, m) = xm[U, mILU, m + 1)]t ~ = xmx;,(j, m + Ilj, m + 1) (B-46) Equation (A-54) has again been used together with the fact that i":. = i +. Thus from Eqs. (B-43) and (B-44) Ym+1 = x;' and or Xm = [j (j + 1) - m (m + 1)] t (B-47) The value of Xm in Eq. (B-47) should have been multiplied by ei i+U, m) = [j(j+ 1) - m(m + 1)]tu, m + 1) (B-49) and LU, m) = [j(j + 1) - m(m -l)]tu, m - 1) (B-50) QUANTUM MECHANICS OF ANGULAR MOMENTUM 427 At this point one may write the nonzero matrix elements of i +, i _, ix, and i y as follows: (j, m + II i + Ij, m) = [j (j + I) - m (m + 1) Jt (B-51) (j, m-lILlj, m) = [j(j+ I) -m(m- I)J! (B-52) (j, m + llix)j, m) =Hj(j+ I) - m(m + l)J! (B-53 ) (j, m -llixlj, m) =Hj(j+ 1) - m(m - I)J! (B-54) (j, m + 1Iiy )j, m) = ~i [j(j+ 1) - m(m + l)J! (B-55) and (j, m -lliy)j, m) =~ [j(j+ I) - m(m - I)J! (B-56) Equations (B-53) to (B-56) follow from Eq. (B-20); that is, ix = t(i+ + i_) and i y= (l/2i)(i + - L). B-6 ANGULAR-MOMENTUM MATRICES For a given value of j, the matrix elements such as those in Eqs. (B-51) to (B-56) are conveniently arrayed as a square matrix. The order of the matrix will be (2j + 1), corresponding to the possible values of m. Consider the spin matrices for j = t. These will be directly applicable to the electron spin case, where S = t, and to the nuclear spin cases with I = t- It, t ) It, -t ) Jx=~!: ~tl [~ ~ J=t [~ bJ (B-57) The elements appearing in the Jx matrix (B-57) are obtained by inserting the ix operator between the corresponding bra to the left of a given matrix element and the ket above that element. For example, the a12 element of Eq. (B-57) is computed as (t, t lixlt, -t) = HH!) - (-t)(t)]t = t In a similar fashion Jy and Jz are written as Jy = f~ -~j = 1. [0 -ij (B-58) .!.. 0 2 i 0 2 - [t OJ - 1 [I OJ (B 59) Jz - ° -t - 2" ° -I The matrices on the right of Eqs. (B-57) to (B-59) are often called the Pauli spin matrices, symbolized by u x , U y , and U z • Hence i = x, y, or Z (B-60) 428 ELECTRON SPIN RESONANCE One can obtain the J+ and L matrices either from Eqs. (B-51) and (B-52) or from matrix addition, that is, (B-61) and (B-62) Since the eigenvalue of i2 for each spin function in the case of j = i must be Hi + 1) = to the matrix J2 is (B-63 ) This can be verified by computing J2 = Jx 2 + Jy2 + Jz2 (B-64) For instance, matrix multiplication of Jx by itself gives with identical results for J/ and Jz2. Addition of the matrices in (B-64) yields the desired result (B-63). 8-7 ADDITION OF ANGULAR MOMENTA One often encounters problems in which there are two angular momenta which mayor may not be coupled by an interaction. The necessity for con• sidering interaction of angular momenta arises in the following cases: I. Coupling of electron spin and orbital angular momenta. 2. Coupling of the angular momenta of two different particles. We shall begin by considering two angular momenta JI and J2 which initially are not coupled. The eigenfunctions for JI and J2 will be taken as Ijl' ml ) and Ij2' m2), respectively. Thus i j2 ljj, mj) =jjUj + I)Uj, mj) illj2' m2) =j2U2 + I)U2' m2) ijzUj, mj) = mlljl> mj) i 2Z U2' m2) = m2U2, m2) (B-65) The direct-product representation Ijl' m j ) Ij2' m2) == Ijl> j2, m j , m2) will be called the uncoupled representation. The total angular momentum J is defined by (B-66) QUANTUM MECHANICS OF ANGULAR MOMENTUM 429 If J is to be an angular momentum, its components must satisfy the com• mutation relations Eqs. (B-19). For example, [ix,iy] = [ilx + i 2X , i ly + i zy ] = [ilx,ilY] + [ilx,iZY] + [izx,ilY] + [izx,iZY] (B-67) Each of the middle two commutators is zero, since angular momenta in dif• ferent spaces commute. Thus (B-68) The representation jjbj2,j, m) of the eigenfunctions ofil2,ii,iz, and iz is called the coupled representation. Thus ilZjjl, j2, j, m) = jl (jl + 1) Ijl> j2, j, m) i Zjjl,j2,j, m) =j(j+ 1)ljl,j2,j, m) i 22Ijl,j2,j, m) =j2(j2 + I) Ijl, j2,j, m) izjjb j2, j, m) = miA, j2, j, m) (B-69) The coupled and uncoupled representations are connected by the transformation jjbj2,j, m) = L C(jljzj;mlm2m)ljl>j2' ml> m2)t (B-70) ml,m2 The factors CUljzj;mlm2m) are variously called vector-coupling, Clebsch• Gordan, or Wigner coefficients. lfthe operator iz = i lZ + i 2Z is applied to (B-70) one gets mljl,j2,j, m) = L (ml + mZ)C(jljzj;mlm2m )ljbj2, mb m2) ml,m2 or L (m - ml - m2)C(jJzj;mlm2m) Ijl, j2, ml, m2) = 0 (B-71) 1nl·m2 Since the Ijl, j2, ml, m2) functions are linearly independent, the above sum can vanish only if the coefficient of each term is identically zero; hence (m - ml - m2)C(jljzj;mlm2m) = 0 (B-72) Thus m, ml, and m2 are not independent, and the sum in Eq. (B-70) can be replaced by a sum over ml, since m2 = m - ml; that is, Ijl>j2,j, m) = L C(jljzj; ml> m - ml )Ul,j2, ml , m - m l) (B-73 ) m, t The function on the left-hand side of this equation is in the coupled representation, whereas the functions on the right hand side are in the uncoupled representation. The "equals" sign here should be taken to mean equivalence. 430 ELECTRON SPIN RESONANCE Further restrictions on the vector-coupling (VC) coefficients can be derived from the orthonormal properties of the Ijl,j2,j, m) eigenfunctions. That is, Ud2,j', m'Ul>j2,j, m) = 6j j'6mm, = 2: 2: CUJ0; ml> m - ml )C(jIj2j': m;, m' - m;) m{ mt The VC coefficients have been assumed to be real in Eq. (B-74). Note thatj and j' are obtained from the same values of ji and j2' Equation (B-74) thus restricts the sum in Eq. (B-73) to functions that have the same values of j and m. Since the values ofjl andj2 must also be the same, the notation may be simplified to Ij, m) = 2: CUIj0; m l , m - m l ) Iml , m2) (B-75) m, where m = m i + m2• Nothing has yet been said about the ranges of j and m. Since J is a generalized angular momentum, the restriction found in Eq. (B-39) will apply; that is, m=j and ,!!j =-j (B-39) Here m and,!!j are, respectively, the maximum and minimum values of m. Since m = mi + m2, the maximum value of m for all values of j is ji + j2' This must also be the maximum value of j; otherwise there would exist a larger value of m. Thus (B-76) where T is the maximum value of j. When j and m have their maximum values, the relation between the coupled and uncoupled representations is especially simple, since there is only one permissible value of m i (and hence of m2). Thus from Eq. (B-75) Ihm) = CUJ2,jl + j2;jJ2) I'm;, m;) (B-77a) =I'm;,m;) The standard convention takes CUJ2, ji + j2; jIj2) = 1. Similarly Ij,,!!j) = I~,~) (B-77b) (B-77a) and (B-77b) are two of the 2j + 1 functions in the set with j = T. The other members of this set can be obtained by applying 1- = 11- + 12- to (B-77 a) or by applying 1+ = 11+ + 12+ to (B-77 b); for example, or QUANTUM MECHANICS OF ANGULAR MOMENTUM 431 Y'm'Ij, m- 1) = Ym; I~- 1, 1n;') + Y1ii2I~, 1n;'-1) (B-78) The Y coefficients are obtained from Eq. (B-48). Hence .. . + .. c:-1 1'--') - Y'm'l C (J1.12,11 12, m 1 - ,m? -- (B-79a) - Y'm' and (B-79b) Sequential application of j _ to Eq. (B-78) will generate all of the 2j + 1 functions in the set with j = j. The set of functions corresponding to j = T- 1 will be fewer in number by two than the set with j = T and will be bounded by the functions IT- 1, m- 1) and IT- 1, ~+ 1). The use of Eq. (B-75) demonstrates that IT- 1, m- 1) must be related to the same uncoupled functions as IT. m - 1). Further, there cannot be any other functions with m = m - 1. If one writes (B-80a) and IT- 1, m - 1) = C~I~- 1, 1n;') + C~I~, 1n;' - 1) (B-80b) then the orthonormal properties of the functions require that C~ = C 2 and C~= -C. Now that the function IT- 1, m - 1) has been defined, all the other members of that set can be obtained by the application of j _. The function I~ 2, m - 2) is obtained from the condition that it must be orthogonal to IT- 1, m- 2) and to 1T.'fi? - 2). The above sequence of processes is con• tinued until all of the functions have been generated. The number of uncoupled states must be the same as the number of coupled states, that is, j L (2j + 1) = (2jl + 1) (2j2 + 1) (B-81) 1 This counting procedure will determine j, since T= jl + j2. The left -hand side of (B-81) can be evaluated using '-' (j 1 Lj=- [,8(,8+ 1) -a(a-1)] (B-82) ('( 2 where j, a, and ,8 must be integers or half-integers. Thus (jl + j2) (jl + j2 + 1) -1(1- 1) + jl + j2 -1 = (2jl + 1)(2j2 + 1) or 432 ELECTRON SPIN RESONANCE Sincej ~ 0, j = Iii - j21 (B-83 ) '-' Thus j is restricted to the values j = ji + j2, ji + j2 - 1, ... , Iji - j21 (B-84) A number of symmetry relations exist among the VC coefficients; also a general relation may be derived for these coefficientst; however, it is often easier to evaluate the VC coefficients from relations such as Eqs. (B-79). The following example will illustrate the method: Consider two angular momenta such thatji = j2 = 1. From Eq. (B-77 a) 12,2) = 11, 1) (B-85a) Application of J_ gives 1 12, 1) = V2 (11,0) + 10, 1) (B-85b) Equations (B-79) have been used to obtain the VC coefficients. A second application of L gives 12,0) = ~ (210,0) + 11, -1) + 1-1,1» (B-85c) Further application of L or the use of J+ with 12, -2) gives 1 12, -1) = V2 (1-1,0) + 10, -1» (B-85d) and 12, -2) = 1-1,-1) (B-85e) In general (B-86) m, Thus only the firstj + 1 members of aj set need be evaluated. The members of the set with j = 1 are evaluated by using the condition that 12, 1) and 11, 1) must be orthogonal. This requires that 1 11, I) = V2 CI1, 0) - 10, 1) (B-87a) t See Bibliography on page 434. QUANTUM MECHANICS OF ANGULAR MOMENTUM 433 Application of i_to (B-87 a) gives 1 11, 0) = V2 (11, -1) - 1-1, 1)) (B-87b) The third member of the set is 1 11, -1) = V2 (1-1,0) -10,-1)) (B-87c) The single function 10, 0) of the set with j = 0 is obtained from the orthog• onality with 12,0) and 11,0). The result is 1 10,0) = v'3 (10,0) -11, -1) -1-1,1») (B-88) The use of the coupled representation becomes especially convenient when the angular momenta J1 and J 2 are coupled by an interaction term in the hamiltonian; for example Land S are coupled through the hamiltonian term AL . S. The coupled representation functions Ii, m} are still eigen• functions of i 2 ,1., i 12 , and i22; however, the uncoupled representation func• tions Im1' m2) are no longer eigenfunctions of i 1• and i 2•• Examples of the application of the methods outlined in this section are found in Chap. 12 and Appendix C. B·8 SUMMARY For convenience in reference, the essential results of this chapter are sum• marized below. For orbital, spin, or nuclear angular momenta, the appro• priate expressions are obtained from those below by substituting L, S, or I, respectively, for j. The quantum number expressed here as m is then M L , M s , or Mf, respectively. 1. Operations giving nonzero values i 2 1i, m} =j(j+ 1) Ii, m) (B-41 ) i.li, m} = mlj, m} (B-42) i+lj, m} = [j(j+ 1) -m(m+ 1)]!Ii, m+ 1) (B-49) Llj, m) = [j(j+ 1) - m(m -1)]!1i, m-1) (B-50) 2. Matrix elements (j', m'lj, m) = 1 ifj' =j and m' = m =0 ifj' ¥-jorm' ¥- m (B-24) (j, mli.lj, m) = m from (B-42) (B-89) (j, mli21i, m) = j (j + 1) from (B-41) (B-90) (j, m + Ili+lj, m) = [j(j+ 1) - m(m + 1)]! (B-51 ) (j, m -lILlj, m) = [j(j+ 1) - m(m -1)J! (B-52) 434 ELECTRON SPIN RESONANCE Matrix elements of ix and iy, less often used, are given as Eqs. (B-53) to (B-56). 3. Angular momentum matrices j-1.. 1 [0 1 [1 - 2· Jx ="2 1 ~J Jy =! [~ -~J Jz="2 0 -~J J+ = [~ ~J J- = [~ ~J (B-91 ) 1 -i j= 1: 1[0 0 1[0 Jx= V2 ~ !] Jy= V2 ~ 0 -J] 0 J.~ [~ 0 0 J 1 0 J+ ~ VI [~ 0 J- ~ VI [! 0 0 !] ~] (B-92) BIBLIOGRAPHY Edmonds, A. R.: "Angular Momentum in Quantum Mechanics," Princeton University Press, Princeton, N.J., 1960. Rose, M. E.: "Elementary Theory of Angular Momentum," John Wiley & Sons, Inc., New York,1957. PROBLEMS B-1. Show that [iy,Jz] = jix' B-2. Derive the commutation relations in (B-2 I). B-3. Show that the matrix element (j, m' Ii-Ij, m) is nonzero only if m' = m - I. B-4. Establish the angular-momentum matrices Jx , Jy , Jz , J+, L, and J2 for j = !. B-5. For j =!. show that Jx 2 + J/ + J} = J2. B-6. By matrix addition and multiplication, find the commutators [J+,J-], [J+,J2], and [J_,J2] for j = I. 2 B-7. For j = t, Jx = Jy 2 = J}. By calculation of the appropriate matrices for j = 1 andj =!. show that these relations are not satisfied for j;;" I. On the other hand, show that the trace (sum of diagonal elements) is the same for these matrices for a given value of j. B-8. Verify Eq. (10-9b) by matrix mUltiplication. B-9. For jl = 2, j2 = I, show that 13, ±3) = 1±2. ±I) QUANTUM MECHANICS OF ANGULAR MOMENTUM 435 I 13, ±2) = - (21±1, ±I) + v'21±2, o)} Y6 I 13, ±I) =- (2v'310, ±I) + 41±1, 0) + v'21±2,:': I») v'3O I 13,0)= .~ (v'31I,-1)+310,0)+ v'31-1, I») v 15 I 12, ±2) = - (v'2I±I, ±I) - 21±2, 0») Y6 I - 12, ±I) = - l V310, ±I) -I±I, 0) - v'21±2, :':1») Y6 I 12, 0) = v'2 (I-I, + I) - 1+ I, -I)) I II, ± I) = • /TA (10, ± I) - v'31± I, 0) + Y61±2, :,: I) ) vlO I 11,0) = ./TA (v'3II, -I) -210, 0) + v'31-1, I») vlO Appendix C Calculation of the Hyperfine Interaction in the Hydrogen Atom and in an 'RH2 Radical The hyperfine interaction in the hydrogen atom was treated in an approxi• mate manner in Chap. 3. An exact calculation will be presented in this appendix. C-l THE HAMILTONIAN FOR THE HYDROGEN ATOM A more exact spin hamiltonian than that of Eq. (3-8) for an isotropic system of one electron (S = t) and one proton (l = !) in a magnetic field H is k = g{3H . S + hAoS . i - gN{3NH • i (C-l) For H along the z axis, Eq. (C-l) becomes k = g{3HSz + hAo(SJz + six + Sly) - gN{3NHiz (C-2) Using the operators S+, S_, i +, and L, defined by Eq. (B-20), one finds the quantity 5+L + 5_i+ to be 2 (Sxix + Sly). Hence Eq. (C-2) can be re• arranged to k = g{3HSz + hAo[SJz + HS+i - + s_i+)] - gN{3NHiz (C-3) 436 CALCULATION OF HYPERFINE INTERACTION IN HYDROGEN ATOM AND IN AN ·RH, RADICAL 437 C-2 THE SPIN EIGENFUNCTIONS AND THE ENERGY MATRIX FOR THE HYDROGEN ATOM The bra and ket notation (see Sec. A-Sd) will be used for the spin eigen• functions, that is, IMs, M[). There will be four independent spin eigen• functions as given in Sec. 3-3. The energy matrix consists of the matrix elements of the spin hamil• tonian between all the spin eigenfunctions (i.e., (Ms, M[I*IMs, M~»). It will thus be a 4 x 4 matrix. Use of the angular-momentum matrices for S = t (computed in Sec. B-6), and Sy = r~ -~1 S+ = [0 11 s_ = fO 01 ~ 0 0 0 1 oj permits computation of these matrix elements. The matrices Iz, 1+, and L are exactly the same as the corresponding electron spin matrices, since these apply to any system with total angular momentum (1) = V1 (J + 1), with 1=t. The matrix elements are divided into two classes: 1. Diagonal matrix elements. A diagonal matrix element is one in which the bra and the ket have the same labels. Inspection of the spin matrices shows that only Sz and Iz have nonzero diagonal elements; hence the only nonzero diagonal matrix elements will be (Ms, M[ISzizIMs. M[) (Ms, M[ISzIMs, M[) and (Ms, M[lizIMs, M[) A typical diagonal matrix element is (ae, anlg{3HS z + hAoSJz - gN{3NHizlae, an) = tg{3H + thAo - tgN{3NH (C-4) ae and an correspond, respectively, to Ms = t and M[ = t; {3e and (3n correspond, respectively, to Ms = -t and M[ = -to 2. Off-diagonal matrix elements. Inspection of the spin matrices shows that S+, S_, 1+ and L have only off-diagonal nonzero elements. Hence for the operators S)_ and s_i +, the nonzero off-diagonal matrix ele• ments of the hamiltonian in Eq. (C-3) will be of the type 438 ELECTRON SPIN RESONANCE and For example (ae, ~nIS+LI ~e, an) = I The energy matrix is then constructed as follows: lae,an) I (!g~H + tMo I - !gN~NH) I ------r------, o I (!g~H - tMo !hAo I o I + !gN~NH) I (C-S) I I I I o I !Mo (-!g~H - tMo I o IL ______- !gN~NH) + I ______o o 0 I (-!g~H + tMo I + !gN~NH) Note that this matrix is factorable into blocks along the principal diagonal, with all other elements zero. When this factorization is possible, one may deal in succession with each of the sub matrices in turn. This pro• cedure results in a considerable simplification of the calculations. Similar considerations apply to determinants. C-3 EXACT SOLUTION OF THE DETERMINANT OF THE ENERGY MATRIX (SECULAR DETERMINANT) To obtain the energies of the four states, one must diagonalize the energy matrix (C-S). Diagonalization may be accomplished by subtracting a vari• able (say W) from each diagonal element and setting the resulting determi• nant (secular determinant) equal to zero. The four roots of the quartic equation in W will be the state energies. By inspection one can see that two of the state energies will be (C-6a) and (C-6b) The other two state energies can be obtained by expanding the re• maining 2 X 2 determinant CALCULATION OF HYPERFINE INTERACTION IN HYDROGEN ATOM AND IN AN ·RH, RADICAL 439 3 11,1>=la"on> 11,0> --I 0e ,,/3n> 2 N I =I,/3e,,/3n> -3 10,0> -1,/3e ,an> o 0.5 1.0 1.5 2.0 Magnetic field, kilogauss Fig. C-l Energy levels of the hydrogen atom at low and moderate magnetic fields (Breit-Rabi diagram); allowed transitions at moderate magnetic fields are shown. !gf3H - thA10 + !gNf3NH - W l zhAo One obtains the energies W(a,/3.) = H (gf3 + gNf3N)2H2 + h2A02]t - thAo (C-8a) and (C-8b) The parentheses on cxef3n and f3ecxn are meant to give notice that the cor• responding states are mixtures of cxef3n and f3ecxn. The eigenvalues W(a,/3.) and W(/3,a.) are subscripted as shown, since cxef3n and f3ecxn are the respective correct eigenstates in the limit of very high magnetic field. Equations (C-6) and (C-8) are called the Breit-Rabi formulas.t The Breit-Rabi energies are plotted as a function of magnetic field in Fig. C-l. The unusual aspects of this diagram at low magnetic fields are considered in Sec. C-8. C-4 SELECTION RULES FOR HIGH-FIELD MAGNETIC DIPOLE TRANSITIONS IN THE HYDROGEN ATOM The interaction of electromagnetic radiation with the hydrogen-atom sys• tem can lead to transitions between certain energy levels. The transition t G. Breit and l. l. Rabi, Phys. Rev., 38:2082L (1931); J. E. Nafe and E. B. Nelson, Phys. Rev., 73:718 (1948). 440 ELECTRON SPIN RESONANCE probability betwee~ states IMs, M I) and IM~, M;) is proportional tot I %' = gf3HJ)z (C-l1) A general matrix element of %' will be (M~, M;I%'IMs, M 1 ) = gf3HI(M~, M;ISzIMs, M I ) = gf3HI flMs = 0 (C-13) Under these conditions, one cannot expect to see absorption, since no transitions are allowed. With HI II X %' = gf3HISx (C-14) The general matrix element of %' will be (M~, M;I%'IMs, M I ) = gf3HI(M~, M;ISxIMs, M I) = gf3HI(M~ISxIMs)(M;IMI) (C-15) From Eqs. (B-53) and (B-54) one notes that the matrix element of Sx is zero unless M~ = Ms ± I. Hence, the selection rules are flMs = ±1 (C-16) Identical results are obtained with HI II y. t H. Eyring, J. Walter, and G. Kimball, "Quantum Chemistry," chap. 8, John Wiley & Sons, Inc., New York, 1944. CALCULATION OF HYPER FINE INTERACTION IN HYDROGEN ATOM AND IN AN ·RH, RADICAL 441 These selection rules are valid only when the kets IMs, M I ) are eigen• functions of the spin hamiltonian, Eq. (C-3), that is, when the term hAo A A A A 2 (5+L + 5-1+) is neglected. This approximation is valid when g(3H »hAo. The case of very low magnetic fields will be treated in Sec. C-S. It is desirable to be able to calculate relative intensities of possible transitions between energy levels. Equation (C-15) demonstrates that H1 must be perpendicular to z for transitions to occur. The transition prob• ability is proportional to the square of the matrix element of itt, between the initial and final states. Hence, relative intensities of different transi• tions can be obtained by computing the square of the corresponding matrix element of Sx; that is, the intensities will be proportional to Since Sx = HS+ + S_], the matrix elements of S+ and S_ can also be used. In a specific instance, only one of these will be effective in causing a tran• sition; if LlM s = + 1, then governs the intensity. If LlMs = -1, then determines the intensity. C-S THE TRANSITION FREQUENCIES IN CONSTANT MAGNETIC FIELD WITH A VARYING MICROWAVE FREQUENCY When H is constant, the separation of the energy levels is fixed. Upon scanning the microwave frequency, one will observe resonance when LlW = hv. With the selection rules LlMs = ±1, LlMI = 0, resonance will be observed at the following two frequencies [see Eqs. (C-6) and (C-S)]: The difference (Vk - vm) is exactly Ao, the hyperfine coupling constant. 442 ELECTRON SPIN RESONANCE C-G THE RESONANT MAGNETIC FIELDS AT CONSTANT MICROWAVE FREQUENCY Use of a constant microwave frequency and a varying magnetic field is the typical experimental arrangement. This situation is more complicated, since the magnetic field will not be the same for the two transitions. H k will be taken as the resonant field for the transition l.Be, an> -? lae, an> and Hm the resonant field for the transition l.Be, .Bn> -? lae, .Bn>. If A02 « (g.B + gN.BN)2h-2H2, one can carry out a binomial expansion of the square root term, that is, t [(g.B + gN.BN)2h-2H 2 + A02]i = h-1 (g.B + gN.BN)H I A02h I A04h3 +"2 (g.B + gN.BN)H -"8 (g.B + gN.BN)3H3 + . . . (C-18) Only the first two terms will be retained. For Vk = Vm = Vo (the fixed micro• wave frequency), Eqs. (C-17), can be transformed by substitution of (C-18) and multiplication by Hk or H m , respectively, to h- 1 g QH 2 - ( v --A1) H +--=0I Ao2h (C-19a) fJ k 0 2 0 k 4 g.B and (C-19b) Since gN.BN ~ 1O-3g.B, gN.BN is neglected compared to g.B in Eqs. (C-19). Solution of Eqs. (C-19a) and (C-19b) gives (C-20a) and (C-20b) It is clear that a = H m - H k # hAol g.B. a has been called the hyperfine splitting constant (i.e., the separation in gauss between the experimentally observed lines on an ESR spectrum), as opposed to Ao, the hyperfine cou• pling constant (in megahertz). For the H atom, Ao = 1,420.40573 MHz and g = 2.002256.* With Vo = 9,500 MHz, substitution in Eqs. (C-20) gives and Hm = 3,625.67 G t (a+b)t=at+ia-tb-!a-~b2+ . ... :j: R. Beringer and M. A. Heald, Phys. Rev., 95:1474 (1954); P. Kusch, Phys. Rev., 100:1188 (1955). CALCULATION OF HYPERFINE INTERACTION IN HYDROGEN ATOM AND IN AN ·RH, RADICAL 443 Hence a = Hm - Hk = 509.74 G. Since Molgf3 = 506.86 G, one sees that a differs from Molgf3 by 2.88 G. The difference is not large, but it is sig• nificant. It is of interest to note the average value (Hm + Hk)/2 = 3,370.80 G. This is 19.18 G lower than the field HI = hVo/gf3. (See Fig. 3-5b.) There would thus be a considerable error in determining the g factor from the mean position. For 7T-electron radicals this effect is much smaller. C-7 CALCULATION OF THE ENERGY LEVELS OF THE HYDROGEN ATOM BY PERTURBATION THEORY The hamiltonian [Eq. (C-3)] may be separated into two parts it= ito + it, (C-21) where ito = gf3HSz + hAosiz - gNf3NHiz (C-22) and (C-23) If it, « ito, then one may use the eigenfunctions of ito as a basis set for determining the energy corrections due to it,. The zero-order energies W(O) are just the matrix elements of ito (see Sec. A-7), namely, W}?'h, = igf3H + thAo - igNf3NH (C-24a) W}?'b" = igf3H - thAo+ igNf3NH (C-24b) Wh%" = -igf3H + thAo + hNf3NH (C-24c) and (C-24d) Note that these are just the diagonal elements of ito in the energy matrix (C-5). The effects of each of the terms in Eqs. (C-24) are indicated sequen• tially in Fig. C-2. Note that the addition of the nuclear Zeeman interaction in the hydrogen atom does not affect the zero-order transition energies. Regardless of the form of it', one may use the general expression Eq. (A-88) with (A-97) and (A-l 00) for the energy due to the perturbation as W=W(O)+(·lit'I·)+~' (ilit'ln)(nlit'li) +... (C-25) I I L" W.(O) _ W (0) n ' n The second term in Eq. (C-25) is the first-order correction W{l) to the en• ergy; it is given by the diagonal matrix elements of it', taken over the zero• order wave functions. However, since it, involves only the raising and 444 ELECTRON SPIN RESONANCE ,ir-----« : __ ---,,-+-....--~~) , ~----~ ,,I ,,I , \ \,, \ ",./'"--• -+--.l.._{3{3 \ ,.-----" Zero \, field Electron Zeeman interaction , First• '~'------~" order Nuclear Zeeman ''--'-----({3a) hyperfine interaction Second-order interaction hyperfine interaction la) (b) (c) (d) Fig. C-2 Energy levels and allowed transitions for the hydrogen atom (at constant magnetic field) showing effects of successive terms in the spin hamil• tonian [Eq. (C-3)]. (0) Electron Zeeman interaction gf3H5z• (b) Addition of the first-order proton hyperfine interaction hAof;Jz. (c) Addition of the nu• clear Zeeman interaction - gNf3NH i z. (d) Addition of the second-order proton hyperfine interaction derived from H5J_ + 5_1+). lowering operators, all diagonal matrix elements of ft' will be zero. Thus the state energies previously calculated with the use of fto are correct to first order. Hence, it is necessary to utilize the third term of Eq. (C-25) to obtain additional (second-order) corrections W(2). The prime on I im• plies that the summation extends over all the zero-order states In) except for the state Ii). W/O) and Wn(O) are the zero-order energies for the states Ii) and In), respectively. Writing the states in Eq. (C-25) in terms of the Ms and M/ values, one finds the second-order correction to be given by W(2) = '" (Ms, M/lft'IM~, M;)(M~, M;lft'IMs , M/) (C-26) M,M, M4-M , W~) M - W~i M' s / S I J S I As in the exact treatment (see Sec. C-3), the only nonzero off-diagonal matrix elements of ft' are ({3e, anl.Li+lae, {3n) = 1 and CALCULATION OF HYPERFINE INTERACTION IN HYDROGEN ATOM AND IN AN ·RH, RADICAL 445 Therefore only the energies of the states lae, {3n> and l{3e, an> will be af• fected; the second-order energy corrections to these states are W(2) h2A02 W(2) = _ h2A02 _1_ = _1_ and (C-27) ('(,/3. 4 g{3H Il.". 4 g{3H The energies to second order are shown in Fig. C-2. The transition fre• quencies at constant field will be - h-1 {3H IA + Mo2 (C-28a) Pk - g + 2 0 4g{3H and _ -1 _ I hAo2 Pm - h g{3H 2 Ao + 4g{3H (C-28b) The term gN{3NH has been neglected in comparison with g{3H. As before, Pk - Pm = Ao; when P is held constant at Po and the field is scanned, the expressions obtained on solving Eqs. (C-28) for Hk and Hm are exactly the same as Eqs. (C-20). C-S WAVE FUNCTIONS AND ALLOWED TRANSITIONS FOR THE HYDROGEN ATOM AT LOW MAGNETIC FIELDS The fact that off-diagonal elements appear in the energy matrix (C-S) means that the basis spin functions are not all eigenfunctions of the hamiltonian equation (C-3). It is desirable to find a set of spin functions which are eigenfunctions of :ft. One first notes that lae, an) and l{3e, {3n> are already eigenfunctions of :ft; thus one need only diagonalize a 2 x 2 matrix. The four eigenfunctions of :ft may be expressed in the coupled rep• resentation (see Sec. B-7) IF, M F ), where F= IS+Ii, IS+I-ll,· .. ,IS-II (C-29) For the hydrogen atom, F = 0, 1. If F = 1, MF = 0, ± 1; if F = 0, MF = 0. The eigenfunctions Ia e, an) and l{3e, {3n> then become 11, 1) and 11, -1), respectively. The remaining two eigenfunctions 11, 0> and 10, 0> are obtained by diagonalization of the 2 x 2 matrix in (C-S). This is best accomplished using the coordinate-rotation matrix method outlined in Sec. A-Se. The two eigenfunctions are expressed as linear combinations, that is, 11, 0) = cos wlae, {3n> + sin wl{3e, an) (C-30) 10, 0> = -sin wlae, {3n> + cos wl{3e, an> (C-31) The use of Eqs. (A-72) gives (C-32a) 446 ELECTRON SPIN RESONANCE (C-32b) Now as H ~ 00, 11, 0) ~ lae, fin), and 10, 0) ~ lfie, an). However, atH=O 1 11,0) = v'2 [Iae, fin) + lfie, an)] (C-33) 1 10, 0) = v'2 [Iae, fin) - lfie, an) J (C-34) Because of the mixing of states, four transitions are possible at low magnetic fields. The relative intensities can be computed by evaluating the matrix elements of Sx' For example, the Sx matrix element for the 11, -1) ~ 11,0) transition is (C-35) The relative intensities of the four transitions computed in the above manner are given as follows 10, 0) ~ 11, -1) intensity 0:: sin2 w 10, 0) ~ 11, 1) intensity 0:: cos2 W (C-36) 11, -1) ~ 11,0) intensity 0:: cos2 W 11, 0) ~ 11, 1) intensity 0:: sin2 w It is clear from (C-36) that one should be able to detect resonance at zero magnetic field. These measurements have been carried out with ex• tremely high precision in atomic beams of hydrogen atoms. t Zero field measurements have also been reported for Cr3+ in MgO.:j: C-9 THE ENERGY LEVELS OF AN ·RH. RADICAL When more than one magnetic nucleus is interacting with the electron, the calculation of the state energies must be preceded by a careful inspection of the spin wave functions. If the nuclei are equivalent, it is usually con• venient to use a "coupled" representation for the nuclear spin states. (See Sec. B-7.) The nuclear spins of the two protons of ·RH2 are added vec• torially to obtain one set of wave functions with total nuclear spin 1 = 1 and another set with 1 = O. The new nuclear spin functions will be repre• sented by 11, M J ); these are related to the wave functions IMI" Miz) by tN. F. Ramsey, Jr., "Molecular Beams," pp. 263ff., Oxford University Press, London, 1956. :j: T. Cole, T. Kushida, and H. C. Heller, J. Chem. Phys., 38:2915 (1963). CALCULATION OF HYPERFINE INTERACTION IN HYDROGEN ATOM AND IN AN ·RH, RADICAL 447 11,1) == If, f) I (C-37a) 11,0) == ~ [It, -f) + I-t, f)] J=1 (C-37b) 11, -1) == I-t, -f) (C-37c) and 0 0) == _1_ [1.1 _.1) _ 1_.1 .1)] 1 , V2 2, 2 2,2 J=O (C-37d) Because of the fact that the square of the total nuclear angular momentum [characterized by J ( J + 1)] is a constant of the motion, mixture of states with different values of J is not allowed; that is, there are no nonzero matrix elements between J = 0 and J = 1 states. Including the electron spin, the total spin wave functions will be designated by IMs, J, MJ)' The spin hamiltonian for the . RH2 fragment will again be separated into two parts so that perturbation theory may be used, i.e., (C-38a) and (C-38b) The nuclear Zeeman terms have been omitted since they do not affect the transition energies. These expressions are analogous to Eqs. (C-22) and (C-23), with the total nuclear spin operator j replacing the individual nuclear spin operators i. The zero-order energies are again the diagonal matrix elements of *'0, = fgf3H + fhAo = fgf3H (C-39) = fgf3H W~O)1 -1 = !gf3H - fhAo 2' , Since the selection rules are tlMs = ± 1, tlMJ = 0, the spectrum at constant microwave frequency will consist of three lines occurring at the resonant fields, H _hvo_hAo k - gf3 gf3 Hz = hvo (C-40) gf3 448 ELECTRON SPIN RESONANCE The line at HI is twice as intense as the lines at H k or H m because the states contributing to the line at HI are doubly degenerate. (See Fig. 4-1 a.) As before [see Eq. (C-27)], the second-order energy corrections in• volve only off-diagonal matrix elements; from Eqs. (B-5l) and (B-52), only the following four are nonzero: (+t. 1, -ti.s+LI-t. 1,0) = V2 (+t. 1, OIS+LI-t. 1, 1) = V2 (-t. 1, 0 IS _1+ It. 1, -1) = V2 (C-4l) and The energies to second order are given in Fig. (C-3). The transition frequencies at constant field will be - h-1 (3H A 1 hAo2 lJk - g + 0 + "2 g(3H (C-42a) 1 +.1.1> I t· 1•O> I t·o.o> 1 +.1.-1) 1_-.1.1-1)2 •• 1- t·o.o) l-t·1•o) 1-+. 1,1> Fig. C-3 Energy levels of the . RHz fragment to second order. (Mo has been neglected in comparison with gf3H in the second-order correction terms.) CALCULATION OF HYPERFINE INTERACTION IN HYDROGEN ATOM AND IN AN ·RH, RADICAL 449 v' = h-lgQH + hA02 I ,.., gf3H (C-42b) VI' = h-1gf3H (C-42c) - h-1 QH A + 1 hAo2 Vm - g,.., - 0 "2 gf3H (C-42d) If hAo is small compared to gf3H, the latter can be set equal to hvo in the correction terms, and hence the resonant fields at constant Vo are (C-43a) (C-43b) (C-43c) and 2 H = hvo + hAo _1. hAo (C-43d) m gf3 gf3 2 gf3vo Thus the spectrum will now consist of four lines, with all lines except that at HI' shifted downfield from the zero-order positions. These are illustrated in Fig. 4-29a. PROBLEMS C-l. Consider the spectrum of the hydrogen atom shown in Fig. 3-1. Use the expressions developed in this appendix to compute an accurate value of the hyperfine coupling constant Ao (megahertz) and the g factor. Comment on the differences from the corresponding values for the free hydrogen atom. C-2. Use the methods developed for . RH2 to calculate to second order the energies of the states for a radical containing three equivalent protons. For Ao = 100 MHz, g = 2.00232, and microwave frequency 9.500 GHz, calculate the field positions and relative intensities of all allowed transitions. C-3. The effects of the nuclear quadrupole moment upon the energy levels of a system with I = 1 in an inhomogeneous electric field may be seen by examining the shifts of levels of the deuterium atom. These levels are given by Eq. (3-13). The quadrupole moment operator for an axial electric field is , 3eQ 2 4'R =Q'[/'2_ I U+I)] av f'ft quad z 3 Q = 4/(2/- I) aZ2 where e is the nuclear charge, Q is the quadrupole moment, and a2 v/az 2 is the gradient of the electric field av/az in which the nuclear quadrupole is located. Here Vis the potential energy. (a) Plot the energy of the six levels of Eq. (3-13). (b) Show the shift of each of these levels by %quad, expressing the shift in terms of Q'. (c) Show the allowed transitions (still fl.Ms = ± 1, fl.Ml = 0). (d) Can one detect a quadrupole interaction from the ESR spectrum? [These shifts must be taken into account when interpreting electron-nuclear double resonance (ENDOR) spectra; see Chap. 13.] Appendix D Experimental Methods; Spectrometer Performance This appendix is designed to acquaint the reader with some experimental methods and procedures which may be used to obtain the optimum per• formance from an ESR spectrometer. The reader is referred to several bookst,:!:,§ which provide a wealth of detail on many aspects of ESR measurement. D·' SENSITIVITY The optimization of the signal-to-noise ratio of a spectrometer is frequently a prerequisite to the successful performance of an ESR experiment. Such optimization requires familiarity with the various factors which affect either the noise level or the signal level. The minimum detectable number of t C. P. Poole, Jr., "Electron Spin Resonance: A Comprehensive Treatise on Experimental TechniEjues," Interscience Publishers, a division of John Wiley & Sons, Inc., New York, 1967. *T. H. Wilms hurst, "Electron Spin Resonance Spectrometers," Hilger. London, 1967. § R. S. Alger, "Electron Paramagnetic Resonance Techniques and Applications," Interscience Publishers, a division of John Wiley & Sons, Inc., New York, 1968. 450 EXPERIMENTAL METHODS; SPECTROMETER PERFORMANCE 451 paramagnetic centers N min in an ESR cavity with a signal-to-noise ratio of unity is given by (0-1) Here Vc = the volume of the cavity (assumed to be operated in the TE102 mode) k = Boltzmann's constant Ts = sample temperature r = half-half-width (gauss) of the absorption line H r = magnetic field at the center of the absorption line Q~ = the effective unloaded Q factor of the cavity (see Sec. 0-2e) T d = detector temperature b = bandwidth in S-1 of the entire detecting and amplifying system Po = microwave power (erg S-I) incident on the cavity F = a noise figure (> 1) attributable to sources other than thermal detector noise. An ideal spectrometer would have F = 1. The derivationt of Eq. (0-1) assumes that the absorption shape is lorentz• ian, that the Curie law applies, and that microwave saturation does not occur. All quantities are in cgs units. An estimate of N min is obtained by inserting the following typical values: Q~ = 5,000 Ts = Td = 300 K Hr= 3,400 G r = 1 G g = 2.00 S =t Vc = 11 cm3 (for a TE102 cavity at X band) F = 100 b = 1 S-1 Po = 106 erg S-1 = 100 mW These factors give N min = 1011. For a typical sample, the minimum detect• able concentration of paramagnetic centers is = 10-9M. Figures such as these are typically quoted by manufacturers of ESR spectrometers. How• ever, unless the conditions of measurement are given, a quoted N min value may be misleading. In particular, many samples are so readily saturated t G. Feher, Bell System Tech. J., 36:449 (1957); C. P. Poole, Jr., "Electron Spin Resonance: A Comprehensive Treatise on Experimental Techniques," Interscience Publishers, a division of John Wiley & Sons, Inc., New York, 1967, pp. 554ff. 452 ELECTRON SPIN RESONANCE that power levels in excess of 1 mW are out of the question. Hence, N min may be effectively larger by a factor of 10. When an ESR spectrum contains hyperfine lines, the intensity of a given line is only a fraction of the total intensity. Hyperfine splitting in• creases N min by the factor :LDj f!/t =--L-• (0-2) Dk where Dk is the degeneracy of the most intense line and :L D j is the sum of j the degeneracies of all the lines in the spectrum. For the p-benzosemiqui• none anion (Fig. 4-6) f!/t = 2.67, and for the naphthalene anion (Fig. 4-16) f!/t=7.11. 0-2 FACTORS AFFECTING SENSITIVITY AND RESOLUTION The preceding section dealt primarily with sensitivity. However, many spectra may be so rich in hyperfine components that resolution becomes an additional factor to be optimized. Increased resolution often results in a decreased sensitivity. It may then be necessary to sacrifice some sensi• tivity in order to gain the requisite resolution. Six factors affecting sensi• tivity and resolution will be considered in this section. D-2a. Modulation amplitude It was noted in Chap. 2 that small-amplitude field modulation techniques may be employed to improve the sensitivity 1.0 0.8 ~.. .!:! ~ 0.6 o .5 o 2 4 6 8 10 Fig. 0-1 Normalized peak-to-peak amplitude (App) for first derivatives of lorentzian and gaussian lines as a func• tion of modulation amplitude (H m). t:.Hpp(Hm --> 0) is the peak-to-peak derivative amplitude as Hm goes to zero. EXPERIMENTAL METHODS; SPECTROMETER PERFORMANCE 453 8 _6 o t ":J:::'E ~4 ":J::: o 2 8 10 Fig. D-2 Relative first-derivative linewidths at increas• ing values of the relative modulation amplitude H",/b..H"p (Hm -> 0). of a spectrometer. However, an excessive modulation amplitude or an excessively high modulation frequency can lead to line distortion. The modulation amplitude H m t should be a small fraction of the peak-to-peak derivative linewidth AHpp' The portion of an absorption line scanned during a half-cycle of field modulation must be nearly linear in order to obtain an output which is essentially the first derivative of the absorption line. (See Fig. 2-8.) As Hm approaches and exceeds AHpp , the derivative line amplitude first increases linearly with H m, then reaches a maximum, and finally decreases slowly. (See Fig. 0-1.) However, long before the line amplitude reaches a maximum, the linewidth AHpp is signif• icantly increased. (See Fig. D-2.) This phenomenon has been analyzed for lorentzian:j: and for gaussian lines. § The results are given in Table 0-1 and displayed in Figs. 0-1 and 0-2. It is apparent that a maximum deriva• tive amplitude App is obtained when Hm == 3.5 AHpp for lorentzian lines and Hm == 1.8 AHpp for gaussian lines. At these settings the lines are con• siderably broadened (by a factor of 3 for lorentzian lines or of 1.6 for gauss• ian lines). The optimum setting of H m will depend on how much sensitivity can be sacrificed for faithfulness of line shape or vice versa. If resolution and true line shape are important, then the modulation amplitude should satisfy the t The modulation amplitude H", at the modulation frequency w'" is defined by H = H 0 + t H m sin Wm t t H. Walhurst, J. Chem. Phys., 35:1708 (1961). § G. W. Smith, J. Appl. Phys., 35:1217 (1964). 454 ELECTRON SPIN RESONANCE Table D-l Parameters for lorentzian and gaussian first-derivative absorption lines as a function of the relative modulation amplitude Lorentzian line G allssian line Hm f:l.Hpp(obs.) App Hm f:l.Hpp(obs.) A1,p f:l.Hpp f:l.Hpp(H m -;> 0) (normalized) ilHpp ilHpp(Hm ~ 0) (normalized) 0.000 1.000 0.000 0.000 1.000 0.000 0.173 1.006 0.130 0.141 1.001 0.148 0.346 1.029 0.248 0.282 1.007 0.291 0.694 1.114 0.478 0.564 1.039 0.551 1.388 1.432 0.784 1.128 1.178 0.887 2.08 1.903 0.930 1.692 1.454 0.993 2.78 2.387 0.987 1.848 1.560 1.000 3.46 3.000 1.000 1.974 1.645 0.995 4.16 3.564 0.992 2.26 1.862 0.983 4.86 4.221 0.974 2.82 2.343 0.943 5.56 4.884 0.952 3.38 2.856 0.898 6.24 5.537 0.929 3.94 3.384 0.857 6.94 6.288 0.905 4.52 3.922 0.819 10.40 9.55 0.800 5.08 4.465 0.785 13.84 13.0 0.721 5.64 5.013 0.755 17.34 16.4 0.659 8.46 7.786 0.639 27.72 26.5 0.541 11.28 10.6 0.564 34.64 33.7 0.488 14.10 13.5 0.497 69.4 68.2 0.353 00 00 0.000 00 00 0.000 condition Hm ,;;; 0.2 I1Hpp. However, if sensitivity is the prime concern and some line distortion can be tolerated, then Hrn should be increased until a maximum derivative amplitude is obtained. A reasonable compromise between sensitivity and resolution is then achieved by reducing H m by a factor of 4 to 5 from the value which makes App a maximum. Thus for I1H pp < 0.1 G, line distortion will occur unless Hrn is kept very small. The effect of increasing H rn for a very narrow line is seen in Fig. 0-3. If Hm » D.Hpp (true width at low modulation amplitude), then Hrn can be determined directly from the peak separation in Fig. 0-3. D·2b. Modulation frequency The observed line will also be distorted if the modulation frequency approaches the magnitude of the linewidth in Hz, i.e., if Wrn = (g{31 h )D.H pp" Since the crystal detector is a nonlinear device, its out• put contains the sum and the difference of the microwave and the modulation frequencies. This results in the production of side-band resonance lines spaced wrnlYe apart and extending over a range of H m gauss. For a modula• tion frequency of 100 kHz, wmlYe = 36 mG. The development of these side bands as the amplitude of the modulation is increased is shown in Fig. 0-4 for a line having a width less than 20 mG (the F center in CaO-see Sec. 8-5c). The side bands have a phase difference of 1800 with respect to b) (a) ~ ~ (j) Fig. 0-3 Wide-line N MR signals of protons in an aqueous solution of Cr(N 0 3). (ilH pp = 0.19 G) as a function of modulation amplitude H m' The field scan is the same for each trace. Values of Hm/ilHpp(Hm -> 0) are as follows: (a) 0.150; (b) 0.398; (e) 0.552; (d) 0.552; (e) 1.052; (j) 2.28; (g) 4.94; (h) 10.14; (il 20.6; (j) 28.8. The gain for traces (a) to (e) is twice that for traces (d) to 01. [From G. W. Smith, f. Appl. Phys., 35:1217 (1964).] (a) (b) (d) f4-0.250G- I , I I I I 1 ~ ~I I I I Fig. 0-4 Modulation side bands on the ESR spectrum of the F center in CaO, for which the linewidth is less than 20 mG. (al Modulation amplitude 4 mG. (b) 20 mG. (e) 50 mG. Cd) The phase is adjusted so that the center line is not seen. The two lines are the first modulation side bands. The modulation side bands are opposite in phase to the central line; their positions are indicated by dotted lines in (a), (b), and (e). The separation of the side bands corresponds to the wm, which for 100 kHz is 36 mG. 'Ye 455 456 ELECTRON SPIN RESONANCE the central line; the latter can be made to vanish (as in Fig. D-4d) by ap• propriate phase adjustment of the phase detector. As expected, the side bands are separated by about 36 mG. D·2c. Microwave power level At power levels in excess of 10-4 watt, the signal output voltage from the crystal detector of a reflection-cavity ESR spectrometer will be proportional to Pot, viz., to the square root of the power incident upon the sample cavity. This assumes that the microwave power is low enough so that no saturation occurs. For homogeneously broadened lines, the line shape is usually lorentz• ian. If the saturation effect of the microwave magnetic field HI is includedt and if the half-half width r is expressed in terms of T 2 , the absorption line• shape function and its first derivative become (see Table 2-1) Y 1 HIT2 (D-3) =;. 1 + (H - Hr)2"lT22 + H I2"y2T1 T2 Y' _ 2 H 1 T 3'l(H - Hr) 2 (D-4) - -;. [1 + (H - HrVlT22 + H12y2TIT2F As long as H12y2TIT2 « 1, this "saturation" term can be neglected, and both Y and Y' will be proportional to HI (or to Pot). When the absorp• tion line is strongly saturated (H12y2T 1 T2 » 1), Y' will decrease with in• creasing microwave power (see Fig. D-5a). By computing the peak-to-peak derivative amplitude 2Y~ax and differentiating, one finds that for a value of Po which gives a maximum derivative amplitude the spin-lattice relaxation time is given by 1 (D-5) TI = 2Hl 2y2T2 Further, by computing the derivative peak-to-peak width, ( AH )2 __4_ 4H12TI (D-6) L.l PP - 3y2T22 + 3T2 The increase in linewidth as saturation sets in can be explained in terms of the uncertainty principle. A higher microwave power produces spin transi• tions at a faster rate and hence decreases the spin lifetime. This results in an increased uncertainty in the energy and hence to an increased linewidth. It is instructive to note that when the derivative amplitude is at its maximum (Fig. D-5a), the linewidth has risen to only ~ 1.2 times the width in the absence of saturation (see Fig. D-5b). Hence, at the maximum deriv- t H, is defined by the expression H(w) = 2H, cos wI where w is 21T times the microwave fre• quency. Eqs. (D-3) and (D-4) are derived from the Bloch equations; a particularly good treat• ment is given in J. A. Pople, W. G. Schneider, and H. J. Bernstein, "High Resolution Nuclear Magnetic Resonance," pp. 31ff., McGraw-Hill Book Company, New York, 1959. EXPERIMENTAL METHODS; SPECTROMETER PERFORMANCE 457 (0) ~------/ ,;'"..- a. a. '<{ 1 2 °4 D-2d. The concentration of paramagnetic centers. It was noted in Chap. 9 that the intermolecular electron spin-electron spin exchange interaction contributes to 1/T2• At moderate concentration levels this contribution to the linewidth is directly proportional to the concentration of the lJara• magnetic center. However, the peak-to-peak derivative amplitude is in• versely proportional to the square of the linewidth. (See Table 2-1.) Hence, as long as the modulation amplitude is kept constant and the line• width is mainly determined by spin exchange, the derivative amplitude will actually increase as the concentration is decreased. Of course, a point is eventually reached at which a further decrease in concentration will result in a decreased signal amplitude. Other concentration-independent line• broadening mechanisms are then important. Linewidths of - 50 mG are not uncommon in the ESR spectra of free radicals in solution; these line• widths require concentrations of less than 10-4M to avoid exchange broadening. For free radicals in solution it is often convenient to have a concentra• tion gradient within the sample tube. Then the concentration in the cavity may easily be adjusted by moving the sample tube up or down. If electron transfer with a diamagnetic molecule contributes to the linewidth, a decrease in the overall concentration of all species will have the same effect as if the broadening were due to electron spin exchange. In solids, a major source of line-broadening is the dipole-dipole inter• actions between neighboring electron spins. To minimize this problem, it is necessary to dilute the paramagnetic centers by incorporating them in a diamagnetic host. Since limiting linewidths in solids are usually 2 I G, concentrations of 10-2 to 10-3 M can be tolerated if there is no tendency toward pairing or aggregation. D-2e. Temperature Even if the sample temperature does not govern the linewidth, for maximum sensitivity, one should work at as Iowa tempera• ture as feasible, since the signal amplitUde is inversely proportional to the absolute temperature (Curie's law). However, in many cases, the tempera• ture also has an effect on the linewidth. For each system there will usually be an optimum temperature at which the linewidth is a minimum. (See Fig. 9-20.) This temperature is usually below room temperature; hence, a decrease in sample temperature often produces improved spectra. If the linewidth is determined by a short value of Tl as in transition• metal ions, a significant decrease in sample temperature can have a dramatic effect on the ESR spectrum. The reason is that Tl is a strong function of the sample temperature (e.g., in some cases Tl is proportional to the inverse seventh power of the sample temperature or even increases exponentially as the temperature is lowered). For some samples liquid helium tempera• tures (4 K or less) are necessary to obtain sufficiently narrow lines. This is especially true of many of the rare-earth and actinide ions. EXPERIMENTAL METHODS; SPECTROMETER PERFORMANCE 459 0-2f. Q Factor of the cavity The Q factor was defined in Chap. 2 as 27T (maximum microwave energy stored in the cavity) (0-7) Q = energy dissipated per cycle The value of Q under the condition that only losses within the cavity are considered (i.e., resistive losses in the walls) is called the unloaded Q fac• tor Qu' However, a cavity must be coupled by an iris to the waveguide sys• tem. Since this entails additional losses, there will be a further lowering of Q. This coupling loss is measured by 1 energy lost through coupling holes per cycle {1. 27T (stored energy) The ratio {3 = Qu (D-S) Qr is called the coupling parameter. For optimum coupling (i.e., maximum transmission of power into the cavity) {3 = 1. The overall or loaded quality factor QL is then defined by _1 =_1 +_1 (D-9) QL Qr Qu Hence, when {3 = 1, QL = tQu. If within the cavity there are materials which have a nonvanishing imaginary part of the dielectric constant, additional losses can occur. To account for these losses, the dielectric Q factor is defined as Q = 27T (maximum microwave energy stored in the cavity) • energy lost per cycle in dielectric loss (D-lO) A factor Q~ is defined such that _1 = l... +_1 (D-II) Q~ Q. Qu Q~ is the factor which should be used in sensitivity calculations. Most of the dielectric losses usually occur within the sample or the sample tube. Thus it is important to position the sample in a region of the cavity for which the microwave electric field is a minimum. Positioning is extremely critical for samples (such as aqueous solutions) which exhibit a high dielec• tric loss. For these, the most satisfactory container is a flat high-purity silica cell which can be accurately oriented along the nodal plane of the E field. It is found that optimum sensitivity occurs when Qu = Q .. i.e., for a reduction in QL to one-half of its value in the absence of dielectric loss. t t Charles P. Poole, Jr., "Electron Spin Resonance: A Comprehensive Treatise on Experi• mental Techniques," p. 587, Interscience Publishers, a division of John Wiley & Sons, Inc., New York, 1967. 460 ELECTRON SPIN RESONANCE For aqueous solutions this requires that the silica plates be separated by - 0.3 mm for X-band frequencies. At higher frequencies, the dielectric loss for water is even more serious. For organic solvents with E < 10, cylindri• cal sample tubes with an internal diameter of = 3 mm are permissible. Since most glass and silica materials give strong ESR signals, they are not suitable as sample-tube materials. Fused silica of high purity avoids these difficulties. This material is also advantageous since it has a very low dielectric loss and hence is a desirable material for the fabrication of dewar inserts. One should be forewarned that uv, 'Y, or x irradiation of many materials, including fused silica, generally gives rise to defects which show ESR signals. Cylindrical cavities operated in the TEol1 mode (Fig. 2-4) generally have a significantly larger Q factor than does a rectangular cavity in the TEIo2 mode. Thus if the sample has a low dielectric loss, use of such a cylindrical cavity may be advantageous. D-2g. Microwave frequency The microwave frequency is a parameter which is varied but little in most ESR work. The principal reason is that most spectrometers permit frequency variations of no more than ± 10 per• cent of the center frequency of the klystron. However, there are numerous situations for which the inconvenience (and expense!) of introducing a major change-usually an increase-in the microwave frequency can result in a very significant improvement in sensitivity. Several cases will be considered. 1. Constant factors: filling factor, microwave power, and dielectric loss. As the microwave frequency increases, the size of the cavity (for the same mode) must decrease. If the sample size is scaled in the same proportion as the cavity dimension, then the filling factor remains con• stant. If in addition, the sample has a low dielectric loss, then the sensi• tivity will increase as vo~, where Vo is the microwave frequency. If the dielectric loss is significant, then this factor will usually be smaller, and it may even have a negative exponent. Since the manipulation of a sample in a small cavity is difficult, there would be little advantage of an increase in frequency in this case. Indeed, if the ,sample is readily satu• rated, then the improvement is minimal; for constant HI at the sample, the sensitivity increases only as vo~. 2. Constant factors: sample size and microwave power, with negligible microwave loss. If the sample size is limited, as is often true for single crystals, then an increase in the microwave frequency may result in a dramatic improvement in sensitivity. In this case, the sensitivity in• creases as vol. The important factor here is that for constant sample volume, the filling factor increases as vo3. All other factors being equal, a change from X band to Q band would result in a SOO-fold increase in sensitivity! EXPERIMENTAL METHODS; SPECTROMETER PERFORMANCE 461 3. Aqueous samples. Unfortunately, the dielectric loss of water increases with frequency from X band to Q band. The resultant reduction in the Q factor largely cancels any gain in sensitivity which would otherwise be achieved. It is for this reason that almost all aqueous solution studies are carried out at X band or lower. D-2h. Signal averaging When the signal-to-noise ratio is very low, there are two techniques for improving this ratio by signal averaging. The first involves filtering the output of the phase-sensitive detector by the use of a resistive-capacitive filter (see Fig. D-6). The product RC has units of seconds if R is expressed in megohms and C in microfarads. The filter attenuates noise components with a fre• quency greater than - (RC)-l. However, the use of this filter restricts the rate at which a line may be scanned. A good working rule is to adjust the scan rate so that the time T required to go from peak to peak on a derivative line is such that T > lORe. Scan times less than this will dis• tort the line. However, if sensitivity is the prime factor, then T - RC gives the best sensitivity. It is desirable to have a large range of RC combina• tions (e.g., 10-3 to 100 s) depending on the signal-to-noise ratio required and the rate at which the spectrum is to be scanned. The use of an RC filter improves the signal-to-noise ratio in propor• tion to V RC, since the effective bandwidth of the spectrometer is usually governed by the time constant of the output filter, i.e., b = (RC)-l. How• ever, there are practical considerations which limit the magnitude of the RC product which can be used. These usually include: (1) a limited life• time of the paramagnetic species and (2) instrumental instability (baseline drift, drift of klystron frequency or power output). A second technique for signal averaging involves addition of spectra obtained by repetitive scans. This is accomplished by dividing the spec• trum into equal intervals (typically 512 or 1,024). Each portion is then stored in a separate channel of a time-averaging computer. The coherent signal in each channel will rise in proportion to n, the number of scans. However, due to its randomness, the noise signal will tend to cancel; in fact, it will increase in proportion to Yn. Thus there will be an improve• ment in the signal-to-noise ratio in proportion to Yn. If both the sample and the spectrometer are sufficiently stable, the time-averaging computer technique has little advantage over RC filtering. However, for short-lived paramagnetic species or for rapid kinetic studies, the improved bandwidth of the time-averaging computer technique makes possible studies which might otherwise be hopeless. From phase - senSit~ To recorder detector 0 T 0 Fig. D-6 An RC filter. 462 ELECTRON SPIN RESONANCE 0-3 ABSOLUTE INTENSITY MEASUREMENTS In any analytical applications, one desires a knowledge of the number of paramagnetic centers giving rise to the observed signal. The following factors determine the absolute intensity of the signal: 1. The area under the absorption curve. For derivative spectra this re• quires a double integration of the first-derivative curve. 2. The modulation amplitude at the sample. 3. The amplitude of the microwave magnetic field HI at the sample. This requires a knowledge of Po, Q~, and the distribution of HI within the cavity. 4. The overall spectrometer gain. S. The sample temperature. 6. The spin of the paramagnetic species. 7. The g factor. 8. The microwave frequency. 9. The filling factor "fl. For a small sample in a TE102 cavity, "fI ~ 2Vs/Vc, where Vs is the sample volume and V c the cavity volume. In practice, absolute determinations on a single sample are rarely carried out since so many errors can enter in. Standards are employed to minimize some of these errors. Such standards are of two types: (1) concen• tration standards and (2) absolute spin number standards. If one requires only the concentration of paramagnetic species in a liquid solution or solid, then a concentration standard can be employed. The following conditions should apply to both standard and unknown: 1. The same solvent or host and the same sample geometry should be em• ployed in order to ensure that the microwave magnetic field HI is the same for the unknown and the standard samples. 2. The amplitude of the ESR signals should be proportional to (Po)!; i.e., there should be no saturation for either sample or standard. Ideally, Po should be the same for sample and standard. 3. The modulation amplitude can be large [that is, Hm ~ (2 - 4) IlHpp] pro• viding that the area under the absorption curve is determined. This is especially important for weak signals where overmodulation must be employed to achieve adequate sensitivity. 4. The unknown and standard samples should be at the same temperature. Under these conditions the concentration of the unknown paramag• netic species will be given by [X] = [std]AxYl'xCscanx)2GstdMstd(gstd)2[S(S + 1) ]std ( D-12) Astd9?std (scanstd)2G xMxCgx)2[S (S + 1) Jx EXPERIMENTAL METHODS; SPECTROMETER PERFORMANCE 463 Here A is the measured area under the absorption curve. This may be in arbitrary units as long as they are the same for unknown and standard. Scan is the horizontal scale in gauss per unit length on the chart paper. G is the relative gain of the signal amplifier. M is the modulation ampli• tude in gauss, and Yt is defined by Eq. (D-2). The area under the absorption curve can be obtained by computing the first momentt,:!: by electronic double integration,§ digital integration, or by weighing a cut-out absorption curve. This procedure requires a constant base line or a base-line correction after each integration. One must use extreme care in evaluating the area under an absorption curve. Serious errors may result from failure to extend measurements sufficiently far from the center of the line. ~ The percentage error resulting from finite truncation of the first-derivative curve is shown in Fig. D-7, which may be used to apply corrections. The errors are especially large for lorentzian lines. The following concentration standards have proved useful: 1. a,a' -diphenyl-{3-picrylhydrazyl (D P PH). This substance can be weighed out and it dissolves readily in benzene; however its solutions are not stable over a long period of time. 2. Potassium peroxylamine disulfonate [K2NO(S03)2]. This is a good standard for aqueous solutions, since concentrations can be determined optically.tt Aqueous solutions should be prepared in 10% N~C03 but these are stable for only about one day. 3. MnS04 • H20 and CuS04 ·5H20. These are good intensity standards since they are readily available in pure form; however, their lines are rather broad. (Note that for Mn++, S = %.) 4. The following nitroxide compounds t The first moment is computed by evaluating the integral J:~ Y'(H - H') dH, where Y' is the amplitude of the first-derivative curve, H is an arbitrary value of the magnetic field, and H' is the field at the center of the derivative curve. t C. P. Poole, Jr., "Electron Spin Resonance: A Comprehensive Treatise on Experimental Techniques," p. 784, Interscience Publishers, a division of John Wiley & Sons, Inc., New York, 1967. § M. L. Randolph, Rev. Sci. lnstr., 31:949 (J 960). ~ M. L. Randolph, in H. M. Swartz, J. R. Bolton. and D. C. Borg (eds.), "Biological Applica• tions of Electron Spin Resonance," John Wiley & Sons, Inc., New York, 1972. tt M. T. Jones. 1. Chern. Phys., 38:2592 (1963). 464 ELECTRON SPIN RESONANCE 50 40 ~ '" +-c: 30 ~'" Q.'" 20 10 10 20 30 40 S Fig. 0-7 Percent error in determining the area under an absorption line when the first-derivative lorentzian or gaus• sian curve is truncated at the finite limits ±S, where S is measured in units of IlHpp. (Data taken from M. L. Ran• dolph, "Quantitative Considerations in the ESR Studies of Biological Materials," in H. M. Swartz, J. R. Bolton, and D. C. Borg, eds., "Biological Applications of Electron Spin Resonance," John Wiley & Sons, Inc., New York, 1972.) have proved to be very versatile standards. They can be obtained in a pure form and dissolve readily in a variety of solvents, including water. In addition, the solutions, stored in a refrigerator, are stable over periods of months. The determination of the absolute number of spins is more difficult, since the spatial variations of both the modulation field and the microwave magnetic field can cause serious errors. The unknown and the standard must be placed in equivalent positions and must have a small sample vol• ume. Weighed samples of DPPH, CuS04 ·5H2 0, or the nitroxide com• pounds mentioned above serve as satisfactory standards. The use of a dual-sample cavity (TE104) is advantageous when making absolute-intensity measurements since both standard and unknown can be run simultaneously. The standard and unknown samples should be inter• changed to ensure that differences between the two sample positions are accounted for. 0-4 MEASUREMENT OF g FACTORS ANO HYPERFINE SPLITTINGS The absolute determination of g factors and hyperfine splittings requires accurate measurements of the external magnetic field at the sample. Such measurements are usually made by determining the proton magnetic reso- EXPERIMENTAL METHODS; SPECTROMETER PERFORMANCE 465 Table 0-2 Radicals for which the g factors are accurately known Radical Solvelltt g factor'!; Reference Naphthalene- DME/Na -58°C 2.002743 ± 0.000006 ~ Perylene- DME/Na 2.002657 ± 0.000003 ~ Perylene+ Cone. H 2SO4 2.002569 ± 0.000006 ~ Tetracene+ Cone. H2 SO4 2.002590 ± 0.000007 ~ p-benzosemiquinone- Butanol with KOH at 23°C 2.004665 ± 0.000006§ ~ Wiirster's blue cation Absolute ethanol 2.003037 ± 0.000012 tt DPPH (Sec. D-3) None (powder) 2.0037 ± 0.0002 t DME is dimethoxyethane. *Not corrected for second-order shifts. § Temperature dependent. ~ B. G. Segal, M. Kaplan, and G. K. Fraenkel, J. Chern. Phys., 43:4191 (1965) as corrected by R. D. Allendoerfer, J. Chern. Phys. 55:3615 (1971). tt W. R. Knolle. Ph.D. thesis, University of Minnesota, 1970. nance frequency of a water sample doped with a paramagnetic salt such as FeCla. However, precautions must be taken to correct for field variations between the positions of the ESR and NMR samples.t The use of a dual-sample cavity considerably simplifies the measure• ment of g factors if a suitable secondary standard is employed. Table 0-2 lists some secondary standards useful for this purpose. Matched standard samples are inserted into the dual-sample cavity, and their spectra are run on a dual-channel recorder. The separation of the centers of the two spectra is a measure of the field difference at the two sample positions. One of the standard samples is then replaced by the unknown sample. The field difference AH (corrected for the standard-sample field difference) between the spectral centers is then measured. As long as AH is small (less than 1 percent) compared with the magnetic field Hs at the center of the standard ESR spectrum, the g factor gx for the unknown may be determined from -AH gx-gs=~gs (0-13) Here gs is the g factor of the standard. The magnetic-field sweep may also be calibrated conveniently by the use of a dual-sample cavity. Here it is convenient to use as a standard a substance giving a many-line spectrum for which an accurate determination of the hyperfine splittings has been made. One useful standard is Wurster's blue perchlorate.:j: Table 0-3 lists the line positions and relative intensities for some of the strong lines in this spectrum (some of the weak outermost lines are also included). t B. G. Segal, M. Kaplan, and G. K. Fraenkel,J. Chern. Phys .• 43:4191 (1965). :j: W. R. Knolle, Ph.D. thesis, University of Minnesota, 1970. 466 ELECTRON SPIN RESONANCE Table D-3 Field positions of the strong lines in half the ESR spectrum of Wurster's blue perchlorate in (degassed) absolute ethanol at 23"Ct a~H = 1.989 ± 0.009 G HaC" -@- /CHa a~H' = 6.773 ± 0.005 G /N 8 N" N 7.051 ± 0.007 G HaC CHa a = g = 2.003015 ± 0.000012 (corrected for second-order shifts) = 2.003051 ± 0.000012 (uncorrected) Line position, G, Relative - H - H relative to the center intensity MCH M CH, MN 0.000 16,632 0 0 0 0.278 9,504 0 -I 1 1.711 6,336 I -I 1.989 11,088 0 0 2.267 6,336 I -I 1 2.796 2,376 -2 1 0 3.978 2,772 2 0 0 4.785 9,504 -I I 0 5.062 7,392 -I 0 6.496 5,940 0 2 -I 6.773 14,256 0 0 7.051 11,088 0 0 7.329 4,752 0 -I 2 8.762 9,504 0 9.040 7,392 0 I 11.558 5,940 -I 2 0 11.836 6,336 -I I I 13.547 8,910 0 2 0 13.824 9,504 0 I 14.102 5,544 0 0 2 15.536 5,940 2 0 15.813 6,336 I 18.609 3,960 -I 2 I 18.887 3,168 -I I 2 20.598 5,940 0 2 20.875 4,752 0 1 2 22.587 3,960 2 22.864 3,168 2 25.382 1,760 -I 3 25.660 1,980 -I 2 2 27.371 2,640 0 3 1 27.649 2,790 0 2 2 29.360 1,760 3 29.638 1,980 2 2 32.433 880 -I 3 2 34.422 1,320 0 3 2 36.411 880 I 3 2 41.195 396 0 4 2 43.184 264 4 2 t W. R. Knolle, Ph.D. thesis, University of Minnesota, 1970. EXPERIMENTAL METHODS; SPECTROMETER PERFORMANCE 467 PROBLEMS 0.1. Suppose that kinetic studies are to be conducted on a radical for which the half-life is = 10-3 s at O°C. With an active sample volume of = 0.5 cm3 , Q~ of the cavity is - 3,000. The radical gives rise to a single lorentzian line with a peak-to-peak linewidth of 5 G. The derivative signal amplitude reaches a maximum when the microwave power is = 10 mW. The measurements are made at X band and a TEIO• cavity of volume 11 cm3 is used. The gfactor is 2.01. Assume F = 200, Td = 300 K, and b = 10-4 s. With the conditions as given above, determine the appropriate minimum radical con• centration detectable in this system for a signal-to-noise ratio of 10. 0.2. Determine the value of fJIl in Eq. (0-2) for the following radicals: I. Anthracene anion 2. Pyrazine anion 3. 13CD.H 0.3. For a certain radical TI = 10-5 s. Determine the value of HI which would give a maxi• mum derivative amplitude. The peak-to-peak linewidth is 1 G. D-4. Calculate the relative intensities of the two lines of Fig. 2-11, using the method of moments. Table of Symbols a designation of an orbitally nondegenerate level. a, b, c crystallographic axes; also unit-cell dimensions. a' zero field splitting parameter for an S-state ion in an octahedral field. a; isotropic hyperfine splitting due to the ith nucleus, G. A designation of an orbitally nondegenerate state. A area: Aeff effective hyperfine coupling. [See Eq. (7-37).] Au i, j component of the hyperfine tensor. Ao isotropic hyperfine coupling, MHz. App peak-to-peak amplitude of the first-derivative curve. At parameter of the tetragonal component of the crystal field. Au hyperfine coupling parallel to a symmetry axis. Al. hyperfine coupling perpendicular to a symmetry axis. (A) antisymmetric benzene orbital. (See Sec. 5-4) A hyperfine tensor. dA diagonal form of the hyperfine tensor. .sf surface area on a sphere. b bandwidth. B purely anisotropic part of the hyperfine coupling. [See Eq. (7-28).] B designation of an orbitally nondegenerate state. B reference energy in Tanabe-Sugano diagram. (See Sec. 11-7.) 468 TABLE OF SYMBOLS 469 Be parameter of the octahedral component of the crystal field potential. C speed of light. CiJ coefficient of the ith atom in the jth molecular orbital. C capacitance. d states of electron orbital angular momentum with I = 2. D states of total orbital angular momentum with L = 2. D zero field parameter defined in Eq. (lO-24a). D' zero field splitting parameter in magnetic field units, that is, D/ gf3. Dq crystal field splitting parameter. (See footnote p. 272.) D xx, D yy , D zz diagonal elements of the D tensor. D zero field splitting tensor. dD diagonal form of D tensor. e designation of a twofold orbitally degenerate level. e electronic charge. eQ quadrupole coupling. E zero field splitting parameter defined in Eq. (lO-24b). E' E/gf3 E designation of a state with twofold orbital degeneracy. E, amplitude of a microwave electric field. f states of electron orbital angular momentum with 1=3. F states of total orbital angular momentum with L = 3. F quantum number for combined electron and nuclear angular momenta. F noise figure. (See Sec. D-l.) F center electron trapped at a negative ion vacancy in a crystal. g g factor. g' Lande g factor. ge g factor of the free electron. geff effective g factor defined in Eq. (7-3). giJ i, j component of the g tensor. gN nuclear g factor. gxx, gyy, gzz diagonal components of g tensor. gil g component for H parallel to axis of symmetry. gl g component forH perpendicular to axis of symmetry. g g tensor. dg diagonal form of the g tensor. G gauss. G giga = 109• IG) ground state. h Planck's constant. h Planck's constant divided by 27T. Hd hyperfine dipolar field. Heff effective magnetic field experienced by an electron. [See Eq. (7-26).] Hh/ hyperfine field at the electron. [See Eq. (7-26).] Hi resonance field for a shifted line. [See Eq. (6-3).] Hij i = j-coulomb integral; i # j, j adjacent to i-resonance integral. Hk field corresponding to the kth line in a spectrum. Hlocal local magnetic field. [See Eq. (3-2b).] Hm amplitude of modulation field. H min minimum resonant magnetic field. Hr resonant magnetic field. H x, H y, Hz magnetic fields in specified directions. Ho external static magnetic field. HI rf or microwave magnetic field. HII magnetic field parallel to the symmetry axis. 470 TABLE OF SYMBOLS H.L magnetic field perpendicular to the symmetry axis. H' resonant magnetic field in the absence of local fields. H magnetic field vector. Hz hertz = I cycle per second. jjH. separation of two lines in the presence of interconversion of species. jjHo separation of two lines in the absence of interconversion of species. !::.H separation in field. !::.Hpp separation in field of extrema for a first-derivative ESR line. !t hamiltonian operator. !t, perturbation hamiltonian. !ter hamiltonian for a crystalline electric field. !tdjPGlar dipolar hamiltonian defined in Eq. (7-23). !tmag Zeeman hamiltonian defined in Eq. (11-33). !toct hamiltonian for an ion in an octahedral electric field. [See Eq. (11-20).] !t. general spin hamiltonian. !tso hamiltonian operator for spin-orbit coupling. [See Eq. (11-32).] !t.. spin-spin hamiltonian operator. !tUgj hamiltonian for an ion in a tetragonal electric field. [See Eq. (11-21).] !to zero-order hamiltonian. :J'( hamiltonian matrix. current. Y-T. quantum number for nuclear spin angular momentum. moment of inertia. raising and lowering nuclear spin angular momentum operators. intensity. electron exchange interaction constant, MHz. vector sum of angular momenta. fictitious angUlar momentum defined in Sec. 12-1. ith component of the spin-plus-orbital angular momenta; also for general angular momentum. i± raising and lowering spin-plus-orbital angular momentum operators. k Boltzmann's constant. k rate constant. k' orbital reduction factor. [See Eq. (12-27).] K hyperfine parameter defined in Eqs. (11-49). K constant associated with excess charge. [See Eq. (6-20).] K band 12 to 35 GHz microwave region. I quantum number for the electron orbital angular momentum. II direction cosine. Ii; • the i, j element of the :z' matrix. L total orbital angular-momentum quantum number. L inductance. L' fictitious orbital angular momentum quantum number defined in Sec. 12-1. L operator for orbital angular momentum. Li component of the orbital angular-momentum operator. L± raising and lowering orbital angular-momentum operators. ,fi' matrix of direction cosines. electron rest mass. proton rest mass. quantum number for the z component of an angUlar momentum. M[ value for a set of equivalent nuclei such that high-field lines have M[ > O. quantum number for the z component of the nuclear spin angular momentum. TABLE OF SYMBOLS 471 quantum number for the z component of the spin-plus-orbital angular momentum. projection of orbital angular momentum on a space-fixed direction. quantum number for the z component of the orbital angular momentum. quantum number for the z component of the electron spin angular mo• mentum. magnetization (magnetic moment per unit volume). n population difference of two levels. In} ket representation of the function 1/1". (nl bra representation of the function 1/1 •. no equilibrium popUlation difference of two levels. N number of magnetic dipoles per unit volume. N quantum number associated with the rotational angular momentum of a diatomic molecule. N min minimum detectable number of spins. oct octahedral. P states of electron orbital angular momentum with I = 1. P linear momentum. P.. angular momentum. P states of total angular momentum with L = I. Pi probability for species i. Po incident microwave power. q charge. qj total electron charge density at atom i. (See Sec. 6-5.) Q merit factor. [See Eq. (2-2).] Q proportionality constant between a and p. Q band 33 to 50 GHz microwave region. QL Q factor of loaded cavity. Q. Q factor of unloaded cavity. Q' quadrupole interaction parameter, Hz. Q, Q factor assignable to dielectric losses. r radius. ro Bohr radius. Rj vector locating an ion (Sec. 11-3). R resistance in ohms. f!il hyperfine multiplicity factor in Eq. (D-2). states of electron orbital angular momentum with L = O. states of total orbital angular momentum with L = o. symmetric benzene orbital. (See Sec. 5-4.) matrix composed of the elements of Sj. operator for the ith component of the spin angular momentum. overlap integral. fictitious spin such that (2S' + 1) is the multiplicity of the ground state. raising and lowering electron spin angular momentum operators. time. t designation of a threefold orbitally degenerate level. trgl trigonal. ttdl tetrahedral. ttgl tetragonal. T absolute temperature (Kelvin). T designation for states with threefold orbital degeneracy. T purely anisotropic tensor (traceless). IT,} zero field eigenfunction of the zero field operator 1tss. Tx cross-relaxation time. (See Sec. 13-3.) 472 TABLE OF SYMBOLS T, spin-lattice relaxation time. T'e electron spin-lattice relaxation time. T'n nuclear spin-lattice relaxation time. T2 inverse linewidth, Hz-I. T~ spin-spin relaxation time. TEjjk transverse electric designation of a cavity mode. (See Sec. 2-3a.) Tr trace of a matrix. u, U spin hamiltonian parameters defined in Eq. (II-50). v velocity. V potential; voltage. Vc volume of a cavity. Vj electric-field potential (j = x, y, z). [See Eq. (11-5).] VOC! octahedral potential. [See Eq. (11-9).] V, sample volume. VII,I tetragonal potential. [See Eq. (11-10).] V, center defect illustrated in Fig. 7-1. W energy. WdiPolar dipolar energy. [See Eqs. (3-2) and (7-22).] WG energy of a ground state. Wi energy of the ith level. x,y,z laboratory-fixed axes. X band 8.2 to 12.4 GHz microwave region. X, Y,Z molecule-fixed axes. .T, qIj, :l' zero field splitting parameters. [See Eqs. (10-15) and (10-23).] Y amplitude of an absorption line. Y' amplitude of first-derivative of an absorption line. Y" amplitude of second-derivative of an absorption line. a coulomb integral. (See Sec. 5-2.) a coefficient of L'. [See Eq. (12-1).] la} ket representation of the state corresponding to Ms = +! or M/ = +t. a proton proton attached to the carbon atom on which the unpaired electron is pri• marily localized in an alkyl radical. at parameter of the tetragonal component of the crystal-field hamiltonian. f3 Bohr magneton. f3 resonance integral. (See Sec. 5-2.) f3 cavity coupling parameter defined in Eq. (D-8). f3c parameter of the octahedral component of the crystal-field hamiltonian. f3N nuclear magneton. f3 proton proton on a carbon atom adjacent to the carbon atom on which the unpaired electron is primarily localized in an alkyl radical. 'Y magnetogyric ratio. 'Ye magnetogyric ratio of the electron. 'Yp magnetogyric ratio of the proton. [ half the linewidth at half-height in the absence of microwave saturation. [0 linewidth in the absence of exchange processes. 8 separation of orbital energy levels due to the tetragonal component of the crystal field. separation of orbital energy levels in an octahedral or tetrahedral crystal field. splitting parameter defined in Eqs. (12-49) and (12-50). state of a diatomic molecule in which A = ±2. E mixing coefficient. [See Eq. (6-14).] E popUlation difference. (See Sec. 13-3.) TABLE OF SYMBOLS 473 E; excess charge. [See Eq. (6-20).] 1) viscosity. 1) filling factor of a cavity. () angle between Hand r. () dihedral angle. K proportionality factor between T 2-1 and linewidth. A deBroglie wavelength. A spin-orbit coupling parameter. A perturbation mixing coefficient. A; eigenvalue. A general operator. A tensor defined in Eqs. (11-37) and (11-38). A quantum number corresponding to the projection of L on the internuclear axis in a diatomic molecule. Au i, j component of the A tensor. /L magnetic moment. /Le magnetic moment of the electron. /LN magnetic moment of nucleus N. v frequency. Ve electron-resonance frequency. Vn nuclear-resonance frequency. Vo fixed microwave frequency. Vp proton-resonance frequency. 1T molecular orbital comprised of atomic 2pz orbitals. 1TT! atom-atom polarizability between atoms rand t. n state of a diatomic molecule in whichA= ±1. P spin density; also unpaired-electron density. Pi spin density in the ith 2pz atomic orbital. (]' orbital which is cylindrically symmetric about a bond. I state of a diatomic molecule in which A = O. I quantum number corresponding to the projection of S on the internuclear axis of a diatomic molecule. 'T lifetime in a state. 'T time constant of an RC circuit. 'T general variable of integration. f T integration over the full range of variables. cf> atomic wave function. cf> polar angle. cf> Euler angle. (See Fig. A-6.) X magnetic susceptibility. X magnetic susceptibility tensor. 1/1 wave function. w angular frequency. Wm modulation frequency. !1 solid angle. !1 quantum number corresponding to A + I for Hund's case a. Name Index Abragam, A., 135n., 14In., 258n., 314n., 351, Ballhausen, C. J., 341n. 353n.,414n. BanfilJ, D., 165n. Adam, F. C., 204n. Banks, E., 287 n. Adams, R. N., 20In., 203n. Barash, L., 247n. Adrian, F. J., 160, 174n., 175n., 18In., 214n. Barrow, G. M., 156 Alger, R. S., 36, 450n. Bartholomaus, R. C., 389n. AlIendoerfer, R. D., 370n. Bass, A. M., 165n. Amos, T., 123n. Bauld, N. L., 123n. Anderson, D. H., 67n. Bedford, J. A., 81, 103n., 104n. Anderson, E. D., 246n., 247n. Beinert, H., 379n., 388n., 389 Anderson, J. H., 150n. Bennett, J. E., 387 Anderson, J. M., 417 Benson, R. E., 128n. Androes, G. M., 379n., 383n., 384n. Beringer, R., 345n., 442n. Arends, J., 185n. Bernal, I., 105 Armarego, W. L. F., 84 Bernard, H. W., 246n., 247n. Assour, J. M., 288n. Bernheim, R. A., 246n., 247n. Atherton, N. M., 120n. Bernstein, H. J., 198n., 456n. Atkins, P. W., 161, 178n., 180n., 186 Bersohn, R., 97n. Auzins, P., 132n., 280n., 292n., 333n., 335n. Bersuker, I. B., 335n. Azumi, T., 256 Bessent, R. G., 181n. Beveridge, D. L., 121n. Bak, T., 417 Biloen, P., 240, 245n. Baker, J. M., 145n. Bird, G. R., 178n. 475 476 ELECTRON SPIN RESONANCE B1eaney, B., 135n., 141n., 249n., 258n., Coffman, R. E., 33511. 282n., 283n., 324n., 340n., 351, 353n., Cohen, M. H., 18411., 19011. 414n. Cohn, M., 33511. Blinder, S. M., 143n., 144n., 159n., 160n. Cole, T., 2211., 3911., 5211., 72, 11411., 16511., Bloch, F., 193, 365n. 167, 169n., 17211., 173n., 18311., 446n. Blois, M. S., 379 Colpa, J. P., 102, 104n., 123n., 129 Blum, H., 185n., 367n., 369, 370n. Commoner, B., 378n., 383n. Blumberg, W. E., 296, 297n. Coogan, C. K., 18511. Bolton, J. R., 35, 63n., 67n., 69, 80, 81,83, Cook, R. J., 187n. 98n., 101, 102, 103n., 10411., 106n., Coope, J. A. R., 247n. 12311.,12611.,12711., 173n., 20811., 209, Cornell, D. W., 163,217, 218n. 21411.,22011.,37911.,38311.,38911.,39011., Cotton, F. A., 95n., 108, 265n. 46311., 46411. Coulson, C. A., 108, 124 Borg, D. C., 37911., 38011., 38311., 38911., Cramer, R. M. R., 238n., 241 39011., 46311., 46411. Culvahouse, J. W., 296n. Bowers, K. D., 249n., 351 Bowers, V. A., 160, 17411., 17511., 181n., Daniels, J. M., 345n. 183n., 21411. Danon, J., 288n. Bray, R. C., 37911. Das, M. R., 219 Breit, G., 43911. Dauben, H. J., Jr., 5411., 98n. Breslow, R., 24411. Davidson, N., 390n. Briere, R., 25411., 255 Davidson, N. R., 52n. Brivati, J. A., 18011. Davies, E. R., 364n. Broida, H. P., 16511. Davies, J. J., 280n., 319 Brown, I. M., 24211. Davis, E. R., 145n. Brown, L. C., 13611. de Boer, E., 102n., 104n., 114n., 115, 129, Brown, M. S., 12311. 218n. Brown, R. L., 348 de Groot, M. S., 23611., 238n., 242n., 244n. Brown, T. H., 10111., 102 Delbecq, C. J., 132n., 183n. Broze, M., 127n. DerVartanian, D. V., 389n. Brunner, H., 114n. Dessau, R. M., 103n., 104n. Buckman, T., 39011. de Waard, c., 127n. Bulow, G., 380n. Dewar, M. J. S., 108 Dirac, P. A. M., 14, 16, 404 Calvin, M., 37911., 383n., 384n. Dixon, W. T., 52n., 99n. Canters, G. W., 115 Dobosh, P. A., 12111. Carlin, R. L., 25811., 31911., 32411., 336, 344n., Dorain, P. B., 345n. 351 dos Santos Veiga, J., 62, 7In., 72, 74 Carlson, F. F., 18511. Dowsing, R. D., 256n. Carrington, A., 5411., 80, 81, 9811., 9911.,101, Doyle, W. T., 184, 18511., 370n., 373n. 10311.,10411.,11011.,20811.,212,213, Dravnieks, F., 18n., 53, 69 22111.,29711.,34511. Dunn, T. M., 266n. Carter, M. K., 9811. Dupeyre, R. M., 252, 253n., 254n., 255 Castle, J. G., Jr., 345 DuVarney, R. C., 185n. Castner, T. G., 176 Chance, B., 38011. Eaton, D. R., 115n., 128n. Chang, H. W., 24411. Edmonds, A. R., 434 Chantry, G. W., 18011. Ehrenberg, A., 379n., 380n., 38In., 382, Chesnut, D. B., 9711., 11411., 12111. 387n., 388, 389n. Cheston, W., 140n. Eisenhardt, R., 380n. Chien, J. C. W., 37411. Eisinger, J., 296, 297n. Clark, H. M., 30011. Ellis, A. J., 23n. Cochran, E. L., 160, 17411., 17511., 18111., Elmore, J. J., Jr., 38011. 18311.,21411. Elowe, D. G., 388n. tlAME INDEX 477 Emch, G., 300n. Gordy, W., 277n. Eriksson, L. E. G., 381n., 382 Goudsmit, S. A., 1 Estle, T. L., 294n. Gough, T., 80 Eyring, H., 416n., 440n. Gramas, J. Y., 24711. Gray, H. 8., 34111. Falconer, W. E., 5n., 72n., 188n. Greenblatt, M., 287n. Farmer, J. B., 247n. Griffith, J. S., 277n. Feher, G., 183n., 187,353, 354n., 365n., Griffith, O. H., 163,217,21811., 390n. 451n. Griffiths, J. H. E., 13211., 184n., 28011., 29211., Fermi, E., 43 333n., 33511., 34111. Fessenden, R. W., 52n., 53n., 54n., 67n., 68, Gulick, W. M., Jr., 12711. 77n., 78,83, 98n., 120n., 121n., 126n., Gunsalus, I. C., 388n., 389 165n., 167, 169n., 172n., 175n., 214n. Gutowsky, H. S., 6711., 19811., 200n. Fischer, H., 52n., 53, 122n., 204n., 205 Fliigge, S., Table C (see inside back cover) Haarer, D., 23711., 25711. Foner, S., 331n. Hall, G. G., 407n., 417 Foner, S. N., 181n., 183n. Hall, G. R., 340n. Forman, A., 103n., 104n., 125n. Hall, J. L., 18111., 182 Fraenkel, G. K., 36, 53n., 56, 63n., 67n., 84, Hall, T. P. P., 300n. lOin., 102, 104, 106n., 125n., 126n., Ham, F. S., 29711., 30011., 334n. 127n., 20Gn., 205n., 206n., 208n., 210, Ham, N. S., 18511. 218n., 219, 220n., 22In., 465n. Hameka, H. F., 23111. Franck, J., 237n. Hamilton, C. L., 39011. Frank, P. J., 6711. Hansen, R. E., 388n., 389 Frankel, R. B., 364n. Harrison, J. F., 246n. Freed,J. H., 127n., 208n., 210, 220n., 22111., Hausser, K. H., 11411. 370n., 37411. Haven, Y., 23711., 256 Freeman, A. J., 364n. Hayes, R. G., 33511. Freeman, E. S., 411. Hayes, W., 181n., 300n. Frost, A. A., 95n. Heald, M. A., 442n. Fry, D. J. 1.,376 Hecht, H. G., 157n. Heiba, E. I., 10311., 10411. Gardner, C. L., 247n. Heilbronner, E., 108 Garofano, T., 29111. Heise, J. J., 383n. Garrison, A. K., 18511. Heller, C. (see Heller, H. C.) Geiger, W. E., Jr., 12711. Heller, H. C., 22n., 11411., 16511., 166, 167, Gelles, I. L., 183 16911., 170n., 172n., 173n., 446n. Gere, E. A., 365n. Helmholtz, H. L. F., 23 Gerlach, W., 1 Hemmerich, P., :188n., 381 n. Gerritsen, H. J., 296n. Henderson, B., 186 Gerson, F., 84 Henning, J. C. M., 127n. Geschwind, S., 286, 296, 29711., 33511., 364n. Hertel, G. R., 307n. Geske, D. H., 12711. Herzberg, G., 26111., 345n., 346n. Geusic, J. E., 13611. Higuchi, J., 24611. Ghosh, D. K., 167n. Hildebrandt, A. F., 23711. Giacometti, G., 12311. Hinchliffe, A, 12011. Gibson, J. F., 387 Hirota, N., 207, 234n., 23611., 24111. Gibson, Q. H., 380n. Hodby,J. W., 18111. Gillian, O. R., 183n. Hoijtink, G. J., 240, 24511. Gladney, H. M., 75n. Holm, C. H., 11511., 20011. Glarum, S. H., 13011., 250n., 25411. Holmberg, R. W., 14311., 14411. Glasbeek, M., 245n. Holton, W. C., 18511.,29411.,36711.,369, Goodman, 8. A., Table C (see inside back 37011. cover) Hornig, A. W., 22911. 478 ELECTRON SPIN RESONANCE Horsfield, A., 67n., I65n., I70n., 17 I, I72n., Kowalsky, A., 255n. I80n., I87n. Kramers. H. A., 233 Hoskins, R. H., 296n., 33 In. Kravitz. L. C., 372n. HUckel, E., 87 n. Kuck, V. J., 246n., 247n. Hunter, F. R., 54n., 98n. Kusch. P., 17n., 442n. Hurrell, J. P., 145n. Kushida. T., 22n., 446n. Hutchings, M. T., 266n., 268n. Kuska. H. A., 318n .• 320, 351 Hutchinson, E., 132n. Hutchison, C. A., Jr., 233n., 234n., 235, Lacroix. R., 300n. 236n., 248n., 249, 255n. Lambe, J .• 330n., 33 I. 366n. Hutton, R. S., 246n., 247n. Langenberg, D. N .• Table A (see inside front Hyde, J. S., 4n., 229n., 355, 370, 373n., cover) 374n., 375, 376n., 38 In. Lasher. G. J .• 183 Laurance, N., 366n. Ibers, J. A., 157 Lawler, R. G., lOin .• 102 Ingram, D. J. E., 297n., 387 Lazdins, D .• 121n., 125n. Hoh, K., 167n. Lefebvre, R., 16In .• 242n .• 243 Lemaire, H .• 252, 253n., 254n., 255 Jahn, H. A., 334 Leniart. D. S., 370n. Jayne, J. P., 158 Levanon. H., 3 IOn. Jefferts, K. B., 349n. Levy, D. H., 58n., 60, 96n., 345n. Jen, C. K., 181n., 183n. Levy, P. W .• 183n. Jesse, R. E., 240, 245n. Lewis, G. N., 233 Jochims, J. C., 114n. Lewis. I. C., 97 Jones, M. T., 463n. Lichtenberg. J .• 417 Josey, A. D., 128n. Liebling, G. R .• 55, I74n. Linnett. J. W., 9n., IOn. Kaiser, E. T., 128, 318n., 351 Lipkin, D., 63n. Kiinzig, W., 176, 184n., 186n., 190n. Livingston, R .• 5n., 59, 143n., 144n., 150n .• Kaplan, M., 67n., 75n., 465n. 154n .• 177. 178n., 180n .• 181n. Karplus, M., 121n., 125n., 126n. Llewellyn. P. M., 335n., 340n., 376n. Katz, T. J., 55, 98n. Lloyd, J. P., 201n. Keen, N., 180n. Lonberg-Holm, K. K., 380n. Keller, F. J., 132n. Longuet-Higgins, H. c., 221n. Kellogg, J. M. 8., I7n. Lott, K. A. K.. 297n. Kempf, R. J., 246n., 247n. Low. W., l77n., 187. 297n .• 300n., 326n., Kerkhoff, F., 183n. 327. 33ln., 333n., 335n., 340, 351 Kettle, S. F. A., 108 Luckhurst. G. R .• 25 In. Kevan, L., 128, 318n., 351 Ludwig, G. W., 187, 294n., 297n., 300n. Kholmogorov, V. E., 383n. Lundsford. J. H., 158 Kikuchi, C., 288n., 330n., 331 Luz, Z., 127n .• 3 IOn. Kimball, G., 416n., 440n. Lykos, P. G .• 170n. King, G. J., I85n. Kinoshita, M., 256 McCall, D. W .• 198n. Kirkpatrick, E. S., 305n., 307, 309n. McClure, D. S .• 233n .• 235, 266n .• 275n., Kivelson, D., I60n., 220n., 22In., 222n. 286n. KneubUhl, F. K., I36n., 145n., 157n., 278n. McConnell, H. M., 52n., 54n., 55. 56, 97n., Knolle, W. R., 163, 465n., 466n. 114n., 115n., 142n .• 145n., 163, 165n., Kohl, D. H., 383n. 166.167, 169n., 170n., l72n., 174n .• Kon, H., 181n. 183n .• 217, 218n., 237n .• 255n., 390n. Koski, W. S., 167n. McDowell. C. A .. 247n. Koster, G. F., 283n. McGarvey. B. R .. 25811., 287n., 300n .• 319n., Kottis, P., 242n., 243 324n .• 335n., 336, 344n .• 351 Koultecky, J .• 106n. McGlynn. S. P .• 256 NAME INDEX 479 Mclrvine, E. C., 366n. Orgel, L. E., 103n., l04n., 125n. Mackor, E. L., 218n. Orme-Johnson, W. H., 388n., 389 McLachlan, A. D., 106n., 117n., 119, 170n. Orton,J. W., 132n., 258n., 266n., 272n., MacLean, c., 114n. 280n., 282n., 292n., 324n., 333n., 335n., McMillan, J. A., 289 344n. McMillan, R. C., 185n. Osiecki, J. H., 71 Mahler, H. R., 388n. Ovenall, D. W., 180n. Mahootian, N., 288n. Owen, J., 184n., 262n., 332n., 341n., 344n., Maki, A. H., 75n., 370 351 Malkin, R., 389n. Malmstrom, B. G., 379n., 380n., 389n. Padmanabhan, G. R., 127n. Mangum, B. W., 233n. Pake, G. E., 201n., 378n. Markham, J. J., 186 Palma, M. U., 291n. Marshall, J. H., l30n., 250n., 254n. Palma-Vittorelli, M. 8., 291n. Martienssen, W., 183n. Palmer, G., 379n., 389n. Maruani, J., 161n. Palmer, P., 234n. Mason, H. S., 380 Panepucci, H., 288n. Mathieson, A. McL., 257n. Parker, W. H., Table A (see inside front Memory, J. D., 123n., 128 cover) Merkl, A. W., 237n. Pascal, B., 52 Michaelis, L., 379 Pastor, R. C., 255n., 296n. Miller, B. S., 185n. Patten, F. W., 132n. Miller, T. A., 201n., 203n., 345n. Paul, D. E., 63n. Millman, S., 17 n. Pauling, L., JOn. Miyagawa, I., 167n. Pavan, M. V., 123n. Moore, G. E., 178n. Pearson, G. A., 248n., 249 Morat, C., 254n., 255 Pearson, R. G., 26611. Morrell, M. L., 54n., 98n. Persico, F., 291n. Morton, J. R., 5n., 67n., 68, 72n., 85, 165n., Pescia, J., 221n. 167n., 168, 170n., 171, 172n., 180n., Phillips, W. D., 115n., 128n. 186, 187n., 188n., 189, 191, TableC (see Piette, L. H., 380 inside back cover) Pilbrow, J. R., 185n. Muller, F., 381n., 382 Piper, W. W., 372n. Muller, K. A., 305n., 307, 309n. Pon, N. G., 383n. Muniz, R. P. A., 288n. Poole, C. P., Jr., 36, 450n., 451n., 459n., Murray, R. W., 247n., 248n., 256n. 463n. Murrell, J. N., 108, 125n. Pooley, C., 149n. Musulin, B., 95n. Pople, J. A., 121n., 198n., 456n. Myers, R. J., 58n., 60, 96n., 127n., BOn. Prince, R. H., 81, 103n., 104n. Prins, R., 240 Nafe, J. E., 439n. Pritchard, H. 0., 52n. Neiman, R., 160n. Prokhorov, A. M., 330n. Neiva-Correia, A. F., 7111., 72 Pryce, M. H. L., 314n., 340n. Nelson, E. B., 43911. Nicolau, C., 379n. Rabi, I. I., 17n., 439n. Nordio, P. L., 123n., 390n. Rabinowitz, J. C., 389n. Norman, R. O. C., 52n., 99n. Radford, H. E., 345n., 348 Nyberg, G., 221n. Ramsey, N. F., Jr., 17n., 446n. Randolph, M. L., 463n., 46411. O'Brien, M. C. M., 324n. Rassat, A., 252, 253n., 254n., 255 Offenbacher, E. L., 300n., 351 Rauber, A., 293, 294n. Ogawa, S., 53n., 54n., 98n. Raynor, J. 8., Table C (see inside back cover) O'Mara, W. C., 132n., 280n. Reddy, T. Rs., 364n. Oosterhoff, L. J., 231n. Redfield, A. G., 221n. 480 ELECTRON SPIN RESONANCE Reichenbecher, E. F., 246n., 247n. Smith, I. C. P., 5411., 79, 98n., 9911., 212, 213, Reiger, P. H., 105 390n. Reinmuth, W. H., 105 Smith, J. M., 185n. Reitz, D. c., 69, 25In., 255n. Smith, W. Y., 183,27711. Remeika, J. P., 286n., 335n. Smolinsky, G., 248n., 256n. Rey, P., 254n., 255 Sneed, R. c., Jr., 376n. Rich, A., 390n. Snyder, L. C., 123n., 239n., 240, 242n., 256 Rist, G. H., 373n., 375, 376n. Soffer, B. H., 331n. Robertson, J. M., 257n. Sogo, P. B., 383n. Rogers, M. T., 146, 152, 172n., 173, 318n., Sorokin, P. P., 183 320n., 351 Standley, K. J., 221n. Rose, M. E., 434 Statz, J., 283n. Rowlands, J. R., 172n., 187n., Table C Stein, G., 3 IOn. (see inside back cover) Stern, 0., I Rubins, R. S., 305n., 307, 309n., 333n. Sternlicht, H., 237n. Stevens, K. W. H., 267 Sabisky, E. S., 296n. Stone, E. W., 75n. Salem, L., 108, 118n., 119n. Stone, T. J., 390n. Sander, W., 183n. Strathdee, J., 142n., 169n. San Pietro, A., 389n. Strauss, H. L., 5511., 98n. Sayetta, T. c., 123n. Streitwieser, A., Jr., 9511., 10711., 108 Scheidler, P. J., 83, 173n. Stuart, S. N., 185n. Schlick, S., 165n. Sugano, S., 285, 286n. Schmid, D., 237n., 256n., 257n. Sullivan, P. D., 127n., 128n., 209, 214n. Schneider, J., 293, 294n. Suss, J. T., 333n. Schneider, W. G., 198n., 456n. Swalen, J. D., 157 Schoemaker, D., 132n. Swartz, H. M., 379n., 383n., 38911., 390n., Schoening, F. K. L., 250n. 46311., 464n. Schoffa, G., 379n. Symons, M. C. R., 161, 178n., 180n., 186, Schonland, D. S., 297n. 29711. Schuler, R. H., 52n., 67n., 77n., 78, 120n., 12In., 126n., 175n., 214n. Talcott, C. L., 127n., 130n. Schumacher, R. T., 18In., 182 Tanabe, y., 285, 28611. Schwoerer, M., 237n., 256n. Taub, H., Table C (see inside back cover) Segal, B. G., 53n., 465n. Taylor, B. N., Table A (see inside front Seidel, H., 186, 238n., 256n., 360n., 370n., cover) 371n. Taylor, E. H., 181n. Seitz, F., 132n., 186, 187, 266n., 294n., Tedder, J. M., 108 300n., 327, 351 Teller, E., 237n., 334 Sharma, R. D., 24611. Terenin, A. N., 383n. Sheppard, W. A., 173n. Terhune, R. W., 36611. Shih, S., 10311., 104n. Thomas, D. D., 237n. Shuskus, A. J., 18311.,30711. Thomson, C., 24811. Silver, A. H., 39 Thorland, R. H., 185n. Silver, B. L., 127n. Thornley, J. H. M., 262n., 332n., 344n. Sinclair, Y. C., 257n. Tinkham, M., 22311. Singer, L. S., 97 Title, R. S., 307n. Skell, P. S., 246n., 247n. Todd, P. F., lIOn. Slater, E. C., 382 Townsend, J., 63n., 37811., 383n. Slichter, C. P., 198n. Trambarulo, R. F., 277n. Sloan, G. J., 255n. Trammell, G. T., 143n., 144n., 15011., 154n. Smaller, B., 183n., 289 Treharne, R. W., 38311. Smith, G. W., 453n., 455 Trevalion, P. A., 180n. NAME INDEX 481 Trigger, K. R., 296n. Weger, M., 3261l. Trozzolo, A. M., 24In., 247n., 248n., 256n. Weil, J. A., 150n., 157n. Tsai, R. L., 388n., 389 Weiner, R. F., 167n. Tsibris, J. C. M., 388n., 389 Weissman, S. I., 54n., 63n., 71n., 951l., 165n., Turnbull, D., 186, 187, 266n., 294n., 300n., 203n., 204n., 233n., 241n., 242n., 251n., 327,351 255n., 335n. Tuttle, T. R., Jr., 54n., 71n. Wertz, J. E., 53n., 69, 132n., 186, 280n., 292n., 319, 333n., 3351l. Uhlenbeck, G. E., 1 Wheland, G. W., 255n. Ullman, E. F., 71 Whiffen, D. H., 67n., 146, 149n., 152, 165n., 167n., 170n., 171, 172n., 173, 180n., Van der Waals, J. H., 236n., 238n., 242n., 187n., Table C (see inside back cover) 244n. Wiersema, A. K., 86 van Doorn, C. Z., 237n., 256 Williams, W. L., 184, 185n. Vanngard, T., 379n., 380n., 389n. Wilmshurst, T. H., 450n. van Niekerk, J. N., 250n. Wilson, E. B., IOn. Van Niel, C. B., 383 Wilson, G. V. H., 185n. Vannotti, L. E., 85, 191 Wilson, R., 220n., 221n., 222n. van Voorst, J. D. W., 240, 245n., 288n. Windle, J. J., 86 Vaughan, R. A., 221n. Windsor, C. G., 332n. Vehse, W. E., 185n. Wolf, H. c., 186, 237n., 257n. Veiga, J. dos Santos (see dos Santos Wood, D. E., 54n., 56, 174n. Veiga) Wood, L. S., 246n., 247n. Venkataraman, B., 53n. Woodbury, H. H., 187, 294n., 300n. Viehmann, W., 288n. Woodruff, T. 0., 186n., 190n. Vincow, G., 54n., 98n. Wylie, D. W., 183n. Volland, W. V., 54n., 98n. Yager, W. A., 238n., 239-241, 2441l., 2461l., Waggoner, A. S., 390n. 247n., 248n., 256 Walhurst, H., 453n. Yamazaki, I., 380 Walsh, A. D., 178n. Yasaitis, E. L., 132n. Walter, J., 416n., 440n. Yen, T. F., 287n., 351 Wang, P. S., 246n., 247n. Young, C. G., 183n. Ward, I. M., 184n. Yuster, P. H., 1321l., 183n. Ward, R. L., 7In., 127n., 2031l. Waring, R. K., Jr., 255n. Zacharias, J. R., 17n. Wasserman, E., 238n., 239-241, 246n., Zahlen, A. B., 256 247n., 248n., 256 Zandstra, P. J., 203n. Watkins, G. D., 300n. Zaviosky, E., xi, 13, 17n. Waugh, J. S., 115n., 214n., 221n. Zeldes, H., 5n., 59, 143n., 144n., 150n., Weaver, E. c., 383n., 385n., 386 154n., 177, 1781l., 1801l., 1811l. Weaver, H. E., 4 Zhverev, G. M., 330n. 484 ELECTRON SPIN RESONANCE Anisotropic hyperfine couplings, 138-154 p-Benzosemiquinone anion: for I·C, 171 170 hfs in, 127 definition, 172 p-Benzosemiquinone cation, 80 for I"F, 172 170 hfs in, 127 for IH in CH fragment, 169 Benzyl radical: for "N, 171 hfs,99 for other nuclei, Table C (see inside back versus HMO predictions, 107 cover) unpaired electron densities, 99 Anthracene anion, 63, 64, 75-76, 81, Binomial distribution of intensities, 52, 52 105-106,126-127,467 Binomial expansion, 52, 442n. I·C hfs in, 106, 127 Binomial triangle, 52 Anthracene cation, 105-106, 126-127 Biphenyl anion, 63, 65, 66, 74, 79 13C hfs in, 106, 127 NMR spectrum of, 114,115, 117 Anthracene triplet exciton, 237, 257 Biphenylene anion, 61, 62, 74 Antibonding orbitals, 95, 108 Biphenylene HMO coefficients and energies, Antisymmetric matrix, 404 110 Antisymmetric orbital: cr, a'-Bipyridine anion, 242 benzene, 101 Biradical: H.+,92 definition, 6, 250 Antisymmetric state, 224 exchange interaction in, 250-255 Antisymmetrized wave functions, 398 hfs in, 252-255 Aqueous solution cell, 459 polymer, 255 Archimedean antiprism, 335 Bis-(trifiuoromethyl)nitroxide, (CF 3).NO, 81, Area under an absorption line, 463-464 83 Argand diagram, 391, 392 Bloch equations, modified, 198 Ascorbic acid radical, 380, 380 Bohr magneton, 12 Asymmetric ESR spectra, 73 Bolometer, 30-31 Asymmetric linewidth distributions, 214-220 Boltzmann distribution, 194 Attenuator, microwave, 24,29 Bonding orbitals, 95, 108 Automatic frequency control system, 29 Bra, 14,404 Axial symmetry, 20, 134, 409 Breit-Rabi diagram, 439 definition, 134 Breit-Rabi energies, 439 BrO,349 Backward diobe, 30 Broadening: Bacteriochlorophyll in photosynthetic caused by high microwave power levels, bacteria, 385 456-457,457 Bandwidth, effective, 451, 461 caused by high modulation amplitude, Benzene anion: 452-453,453 alkyl substituted, 103 dipolar, 197-198,203,458 13C hfs in, 68, 69 homogeneous, 196 cyano substituted, 105 inhomogeneous, 196 deuterium substituted, 101, 102 "Burning a hole," 365 effect of substituents on, 99-105 Butadiene anion, 58,60, 79, 96, 129 germanium substituted, 103 HMO energies and wave functions, 95 methyl substituted, 103 silicon substituted, 105 13C hyperfine coupling tensor, 170-171 Benzene cation, alkyl substituted, 103 13C hyperfine splittings, 56, 62, 64, 67-68 Benzene HMO energies, 96, 100 mechanism of, 125-127, 126 Benzene HMO wave functions, 100 Calculation of g components, 135-138 p-Benzosemiquinone: Carbenes, triplet ground state, 246-249 alternating line width effect in, 210-213,212 Catechol anion, 79 neutral,80 Cavity: p-Benzosemiquinone anion, 53,54, 80 coupling factor for, 459 as a g factor standard, 465 cylindrical TEo", 25, 26 SUBJECT INDEX 485 Cavity: Coalescence phenomenon, 200· double, 30, 464 Column vector, 400 modes, 26 Commutation relations for angular rectangular TE"'2' 25,26 momentum operators, 422-423 3 Ce +, free ion spin orbit coupling constant, Commutator, 393 337 Commuting matrices, 402 Cell for aqueous solutions, 459 Commuting operators, 393 CF3 radical, 67, 77, 78 relation to simultaneous determination of l3C hyperfine splitting in, 175 observables, 421 (CF3 )2NO, 81, 83 Compilations of ESR data: C-H fragment, 113,167, 167-169 rare-earth ions, 351 anisotropic hyperfine tensor in, 168-169 transition-metal ions, 351 CHD2 radical, 81, 83, 467 Completely equivalent hyperfine spliUings, CH2D radical, 81,83 207-208 CH20H radical: definition, 207 in aqueous acid, 52, 53 Complex conjugate, 391 in methanol, 58, 59 of a matrix, 403 proton exchange in, 204, 205 Complex forms of wave functions, 264-265 CH3 radical, 52, 53 Complex numbers, 391-392 l3C hfs in, 175 Complex plane, 391, 392 Chemical linebroadening mechanisms, Computer for signal averging, 461 198-204 Computer-averaging techniques, 380 electron spin exchange, 201-203 Computer simulation of ESR spectra, 75 electron transfer, 203-204 Concentration of paramagnetic species, effect general model, 198-201 on sensitivity and resolution, 458 proton exchange, 204 Configuration, definition, 92 Chemical shift, 114-115 Configurational wave function, 118 Chichibabin's hydrocarbon, 255 Constants of motion, 419 Chlamydomonas reinhardi, 386 Coordinate rotation matrix, 402-403, 408 Chlorophyll a in chloroplasts and algae, Coordinate transformation, 402-403, 403 385-386 Coulomb integral a, 93 Chloroplasts, 385-386 Coupled representation of angular momenta, Circulator, 24, 28 429 CI 2 -, V k center, 176, 190 Coupling of angular momenta, 428-433 Clebsch-Gordon coefficients, 429 Coupling parameter, 459 CIO,349 Covalent bonding, 340-344 Cl03 radical, 179, 180 Cr+: Cm3+, 339 in crystal fields, 277 Co+: free ion, 260 in crystal fields, 277 in ZnS, 307 free ion, 260 Cr2+: Co2+: in crystal fields, 277 in CaF2, 300 free ion, 260 in CaO, 333 spin-orbit coupling constant, 277 in CdF2, 301 Cr3+: in crystal fields, 277 in A1 2 0 3, 366 free ion, 260 in CaO, 312 g factors in cobalt phthalocyanine, 288 free ion, 260 in MgO, 18, 333 in MgW04 , 311 ENDOR of, 376 in octahedral fields, 276 spin-orbit coupling constant, 277 spin-orbit coupling constant, 277 in Ti02, 333 Cr4+: in ZnS, 300 in A120 3, 330-331 CO2- radical, 67, 179, 180 in crystal fields, 277 486 ELECTRON SPIN RESONANCE CI-4+: 3d" ions, ground state of, 260 free ion, 260 3d'(oct) ions, 314-3 I 6 in Si, 294 3d1(oct + trgl) ions, 31 I, 311, 324 5 Cr +: 3d 1(oct + ttgl) ions: in Ca2PO.CI, 287 t:.. »3» A, 316-320, 317 in crystal fields, 277 calculation of g factors, 3 17 free ion, 260 t:.. » A - 3: calculation of energy levels, in (oct + ttgl) field, 318 320-322,322 Cr03+,318 calculation of g factors, 322-323, 323 Cross-relaxation processes, 360 3d1(ttdl + ttgl) or (cubal + ttgl) ions, 287-288 Crystal-field eigenfunctions, 264-265, 274 calculation of g factors, 287-288, 3 I I Crystal-field energies: effect of ligands on, 343-344 D-state, octahedral and tetragonal, 273 3d2(oct) ions, 328-329, 329 F-state, octahedral, 276 calculation of g factors, 329 Crystal-field operators, 267-269 3d2(oct + trgl) ions, 329-331 Crystal-field potential, 263-267 3d2(ttdl) ions, 292-295 Crystal-field splittings: 3d2(ttdl + ttg\) ions, 297 D-state ions, 270-274, 272 3d3(oct) ions, 298-300 F-state ions, 274-276, 276 in MgO, 298-300, 299 P-state ions, 269-270, 270 3d3(oct + ttgl) ions, 301-303 Crystal-field theory, 263-;277 energies for, 301, 302 Crystalline field: hamiltonian matrix for, 301 octahedral, 269 secular determinant for, 301 tetragonal, 269 3d3(ttdl) ions, 352 tetrahedral, 269 3d4(oct + ttgl) ions, 289 Cu2+: 3d5 (hs)(oct) ions, 303-308 in crystal fields, 277 crystal-field hamiltonian for, 303 free ion, 260 energies for, 305, 306 Jahn-Teller splitting in MgO, 334 hamiltonian matrix for, 304-305 spin-orbit coupling, 277 3d5(hs)(oct + ttgl) ions, 308-310 triplet states of pairs, 249-250 3d5(1s)(oct + ttgl) ions, 324 Cu3+: 3d6 (hs)(oct) ions, 325-327, 325 in A1 20 3 , 296,297,311 calculation of g factors, 326 ENDOR of, 297 3d7(hs)(oct) ions, 331-334, 333 free ion, 260 3d7(hs)(ttdl) ions, 300-301 octahedral, 286-287 3d7(ls)(oct) ions, 335 Cu-a-picolinate in zinc-a-picolinate, 375 3d7(ls)(oct + ttgl) ions, 288-289 Cubal symmetry, 263 3d8(oct) ions, 289-292 Curie's law, 458 calculation of g factors, 291 CuS04 • 5H20 as a standard, 463 3d8(oct + ttgl) ions, 295-297 CycJobutadiene anion, 96 calculation of g factors and zero-field CycJoheptatrienyl (tropyl) dianion, 123 splittings, 295 CycJoheptatrienyl (tropyl) radical, 54,56,96, 3d9(oct) ions, 334-335 98, 123, 174 3d9 (oct + ttg\) ions, 289 CycJooctatetraene anion, 55, 56, 96, 98 3d9 (ttdl + ttg\) ions, 324 CycJopentadienyl radical, 53,' 55, 96, 98, 4d and 5d groups, 335-336 174 4d9 ions (AG++) ions, 289, 289 CycJopropenyl radical, 96 5d5(ls)(oct + ttgl) ions, 344, 352 d xz orbitals, 271, 275 D (zero-field splitting parameter), definition d orbitals, 271 of, 229, 281 3d orbitals, 271 D-state ions, 270-274, 272 combination with ligand p orbitals, 342 orbitally degenerate ground states, 342-343 314-327 SUBJECT INDEX 487 D-state ions: DPPH: orbitally nondegenerate ground states, as a g factor standard, 465 287-289 as an intensity standard, 463 D-state wave functions, 264 Dual cavity, 30,464,465 D tensor: Dy3+, free ion spin-orbit coupling constant, definition, 226 337 relation to A tensor, 279 relation to spin-spin coupling, 226-227 E (asymmetry parameter), definition of, 229, deBroglie wavelength, 9 282 Defect center (see Point defect) Effective field, 136 Degeneracy, 50 Effective g factor, 133, 135, 150,278 Detection system, 30 Effective hyperfine coupling, 146 Detectors, 30 Effective spin, 279 Determinants, 396-398 Eigenfunctions: method of minors, 396 for biradicals, 253 Deuterium atom: crystal field, 264-2"65, 274 energy levels, 46 definition, 14, 394 spectrum, 47 for F-state ions, 290 spin states, 46 of %50' for J. ions, 338 Deuterium substitution, 63, 71, 166-167 of %~, 314-315 Diagonal matrix, 404, 407 for d l ions, 315 Diazafluorene as host for fluorenylidene, 248 for d 2 ions, 328 Dicarbenes, triplet ground state, 247-248 for d 3 ions, 352 cis-1,2-Dichloroethylene anion, 208 for d 6(hs) ions, 326 Dielectric loss, 23, 279,459-460 for d 7(hs) ions, 332 Dihydropyrazine cation, 83, 84 HMO: allyl, 94 Dihydroxydurene cation, 208, 209,213-214 benzene, 100 2,5-Dihydroxy-p-benzosemiquinone, 68, 69 biphenylene, 110 13C hyperfine splittings in, 218, 219 butadiene, 95 Dinitrenes, triplet state, 248 H 2 +, 92 p-Dinitrobenzene anion, 214-215, 216, 222 naphthalene, 110 Dinitrodurene anion, 208, 210, 210 perinaphthenyl, 111 Diphenylmethylene in triplet state, 239, 247 for hydrogen atom, 44, 437-439, 445-446 Dipolar broadening, 197-198,203,458 including electron correlation, 118-119 Dipolar hamiltonian, 140, 142, 143 in ligand field theory, 341-343 Dipolar interaction, 40, 140 nuclear spin, 150 Dirac notation, 14,404-405 orbital angular momentum, real and Direct-product representation of angular complex forms, 264-265 momenta, 428 for particle in a ring, 395 Direction-cosine matrix, 138,411-413 in relation to diagonal matrices, 406 Direction cosines, 135,411,411 of S., 406 Dispersion, 457 of S"" 407 in ENDOR, 361 zero field for S = 1, 231 Distant ENDOR, 366 Eigenvalues: Di-t-butyl nitroxide radical, 69, 70 definition, 394 effect of high viscosity on, 215,217,222 of }2 and }" 423-425 hyperfine and g tensor elements, 163 relation to constants of motion, 419 spin exchange, 201, 202 of S" 14 Double cavity, 30, 464, 465 spin, 44 Double-quantum transition: Eigenvectors, 406 for Fe2+, 327,327 ELDOR (electron-electron double for NiH, 292 resonance), 374, 376 for Ti2+, 293,293 mechanisms, 376 Doublet state, 38 Electric dipole transitions, 347 488 ELECTRON SPIN RESONANCE Electric-field gradient, 370 Ethyl radical, CH3CH2 • , 78 Electron correlation, 113, 1/8-1 2 1 Ethylbenzene anion, NMR spectrum of, INDO method, 121 129 perturbation method, 119-120 Ethylene, 92 Electron paramagnetic resonance, 3 Ethynyl radical, 174 Electron resonance spectroscopy, 349 Eu'+, free ion spin-orbit coupling, 337 Electron spin-electron spin dipolar Euler angles, 242, 416-417,416 broadening, 197 Excess charge density, 123 Electron spin exchange, 201-203 Excess charge effect, 122-123 rate constant, 20 I Exchange broadening, 201-203, 202,458 Electron spin-nuclear spin dipolar Exchange interaction, 250-255 broadening, 197-198 eigenfunctions for S = I, 25 I Electron spin resonance: eigenvalues for S = 1, 252 versus electron paramagnetic resonance, 3 hamiltonian for, 251 versus paramagnetic resonance, 3 Exchange narrowing, 201, 202 Electron transfer, 203-204 Excitons, 236-237, 257 rate constant, 203 Expectation value, 405 ENDOR (electron nuclear double resonance): f orbitals, 265 in alkali halides, 366-370, 371 4f" group, 337-339 cavity, 355 5f" group, 339-340 determination of hyperfine coupling by, 359 4fl ions, 337-338,337 energy levels and transitions, 356-360 calculation of g factors, 338 identification of nuclear type by, 359 spin hamiltonian for, 339 in liquid solutions, 370-372 F center, 184-185 population of levels in, 362-363,363 definition, 6 in powders or nonoriented solids, 373-374, ENDOR of, 185,366-370 375 in KBr, 366-367,368 quadrupole interaction in, 369-370,371 in LiF, 367,369 relaxation processes in, 360-366 in MgO, 185, 189 selection rules for, 357 in NaF, 189 sensitivity, 356 in NaH, 184, 184-185 versus NMR, 360 19F hyperfine splittings, 67 a simple experiment, 354-356 mechanism of, 128 spectra: alkali halides, 368 F-state eigenfunctions: angular dependence, 369 in octahedral fields, 290 definition, 354 real and complex, 265 number of lines in, 354 F-state ions, 274-276, 276 spin hamiltonian, 357 orbitally degenerate ions, 328-334 in study of semiconductors, 183 orbitally nondegenerate ions, 289-303 in study of V OH centers, 184 F 2 - center, 176, 181 Energy levels: F?- center, 190 hydrogen atom, 438-439 FCO radical, 160, 160, 174, 175 'RH 2 radical, 446-449 "Fe, 18 (See also specific entities) Fe+: Equivalent hyperfine splittings, 207-208 in CaO, 333-334 definition, 207 in crystal fields, 277 Equivalent protons: free ion, 260 multiple sets, 57-68 in MgO, 18,18-19,333-334 single set, 50-57 Fe2+: Er'+' free ion spin-orbit coupling, 337 in crystal fields, 277 Errors caused by truncation, 463, 464 free ion, 260 Ethanol radicals, CH.CHOH and in MgO, 326-327,327 CH2 CH2 0H, 5, 83 spin-orbit coupling constant, 277 SUBJECT INDEX 489 Fe3+: Gaussian lines, 34, 34, 35, 36,457 in CaO, 307 K for, 197 in crystal fields, 277 Gaussian lineshape for inhomogeneously in FeCI.-, 307 broadened lines, 196 3 in FeF63-, 310 Gd +, free ion spin-orbit coupling, 337 free ion, 260 Generalized angular momentum, definition of, in K3 Co(CN)6, 324 422 in SrTi03 , 307, 309 Glutaconic acid radical, 173-174 Fe6+, in K2CrO., 297 Glycine radical, NH.+CHC02-, l3C Fe(CN)5NOH, g factors in hyperfine tensor in, 188 NazFe(CN)5NO . 2HzO, 288 Glycolate radical, HOCHCOO-: Fermi (contact) interaction, 43 direction cosine matrix, 149 Ferrihemoglobin. 387 site splitting and hyperfine tensor, 149 Ferrimyoglobin, 387.388 Glycolic acid radical, HOCHCOOH, 57 Fictitious spin, 279 Gradient of the electric field, 370 for 3d5(hs)(oct + ttgl), 308 Graphite, diamagnetic susceptibility in, 410 Filling factor, 460, 462 Group representations, 265 First moment, 463 F1avins,381 H center, 132, 186 Flavoproteins,381-383 H z+: Fluorenylidene, triplet state in, 248-249,249 spectrum, 349 Forbidden hyperfine transitions, 150-153 wave functions, 90 example of, 152 Hamiltonian: intensity of, 15 I for biradicals, 251 Formyl radical, 174, 175 crystal field, 267-269 FP02- radical, 67, 68 dipolar, 140, 142, 143 Free radicals: for electron Zeeman energy, 15, 135,278, definition, 6 281, 314 gas phase, 345-349 axial symmetry, 134 as intermediates in metabolic processes, orthorhombic symmetry, 134 378 in the HMO method, 89 in solids, generation of, 164-165 for hydrogen atom, 436 substrate, 379 for isotropic hyperfine interaction, 43 matrix: D-state ions, 272, 273 F-state ions, 274 g factor: P-state ions, 270 anisotropic, 12, 17, 19,131-138 for systems with S = 1, 227-228, 231 anisotropy, simple example, 131-134 operator, 393, 419 calculation from A tensor elements, 280 definition, 393-394 for D-state ions, 287, 317 spin (see Spin hamiltonian) for F-state ions, 291, 295 for spin-orbit coupling, 278, 314, 316 calculation from mixing of states, 280-281 spin-spin, 224-227 error in determination, 443 for systems with S = 1, 227 measurement of, 464-465 Heisenberg uncertainty principle, 195, 200 standards, 465 Heme proteins, 386-388 g tensor: Hermitian matrix, 404 asymmetric, 136n., 278n. Hermitian operators, 393 experimental determination of, 135-138 complex conjugate of, 403 relation to A tensor, 279 matrix for, 404 sign ambiguity of elements, 138 Heteroatom radicals, 106 Gaseous free radicals, 345-349 High-spin systems, definition, 284, 284 Gaseous ions, states oftransition metals, H0 3+, free ion spin-orbit coupling, 337 259-261 Homogeneous broadening: Gauss in cgs units, 9n. definition of, 196 490 ELECTRON SPIN RESONANCE Homogeneous broadening: Hyperfine splitting constant, 40 effect of microwave power on, 456 hydrogen atom, 442 Hiickel molecular orbital (HMO) approach, relation to coupling constant, 46 87-95 relation to unpaired electron density, 97 Hund's case (a), 345,346 sign of, 114-117 Hund's rules, 108, 114, 284 Hyperfine splittings, 39 Hydrocarbon radical ions, 97 completely equivalent and equivalent, Hydrogen atom: 207-208 calculation by perturbation theory , Hyperfine tensor elements: 443-445 corrections to, 150-154 energy levels, 44, 45, 438-439, 439 relative signs of, 152-154 hamiltonian, 436 sign ambiguity, 149-150 hyperfine coupling constant, 48, 441 Hyperfine transition energy, transition metal hyperfine splitting constant, 442 ions, 282 resonant magnetic fields, 442-443 spectrum, 39 forbidden transitions in, 182 Inhomogeneous line broadening: spin eigenfunctions for, 437-438 definition of, 196 transition frequencies at constant H, 441 effect on ENDOR, 361, 365 effect of microwave power on, 457, 457 trapped in: alkali halides, U 2 center, 183 Inorganic radicals, 175-179 CaF2 , 181-182,182 frozen acids, 181 identification of, 175-178 Hyperconjugation, 124-125, 169-170 Intensities of transitions in the hydrogen Hyperfine anisotropy: atom, 445-446 origin of, 140-144 Intensity measurements: a simple example of, 138-140 absolute, factors affecting measurement of, Hyperfine coupling constants, 441 462 from ENDOR spectra, 359 relative, 462-463 hydrogen atom, 441 Inverse of a matrix, 403 isotropic, 43 Inversion of levels, 361, 365 for f3 protons, 124-125, 169-170 Ion pairing, 71-72 units of, 139-140 Ion quartet, triplet state of, 241-242 Hyperfine coupling tensor: Ir4+, 341, 352 13C, 171 IrCl.--, 341 in the C-H fragment, 169 Iris, 24, 27 experimental determination of, 144-150 Iron-group ions, ground states of, 260 I"F, 172 Iron-sulfur proteins, 388-389 IH, 169 structure of, 389 14N, 171 Isoalloxazine moiety, 381 of nuclei, Table C (see inside back Isolator, 24, 29 cover) Hyperfine effective field, cases of, 141-144, 142 }2 and }z, eigenvalues of, 423-425 Hyperfine interaction, 40 lahn-Teller splitting, 334-335 isotropic, 42 nuclei other than protons with I = 1/2, Ket, 14,404 67-68 Klystron, 21, 24, 28-29 nuclei with I = 1,68-71 mode, 25, 28 nuclei with 1= 3/2,71-73 K 2NO(S03). (see Peroxylamine disulfonate single proton, 49 ion) three equivalent protons, 51 Kramers degeneracy, 223 two equivalent protons, 50 for 3d3(oct + ttgl), 30 I two inequivalent protons, 57 for 3d5(hs)(oct), 305 Hyperfine local field, 141, 143, 145 KrF radical, 188 SUBJECT INDEX 491 Ladder operators, 422 Magnetization: A degeneracy, 347 definition, 8 A tensor, 279-280 relation to T h 193 A-type doubling, 347 units of, 9 Lande splitting factor, 260-261 Magnetogyric ratio: for molecules, 346 definition, 12 Lanthanide ions, 337-339 of 47Ti, 49Ti, 294 Ligand field theory, 340-344 Magnetometer, 22 Ligand orbital participation, 341-343,342 Malonic acid, radicals produced by 'Y Line intensity, relation to amplitude and irradiation, 165 width,33 Malonic acid radical, Line shapes, 32-36 HOOC-CH-COOH, 114 for g anisotopy: axial symmetry, 155, 157 I·C hyperfine tensor in, 172 orthorhombic symmetry, 157,158 Matrices: for hyperfine anisotrophy: axial symmetry, addition and subtraction of, 400 159-160, 159 adjoint of, 403 example of, 160 angular momentum, 427-428 orthorhombic symmetry, 160-161, 161 antisymmetric, 404 in nonoriented systems, 154-161 complex conjugate of, 403 randomly oriented triplet-state molecules, diagonal, 404, 407 238-241 diagonalization of, 405-408 axial symmetry, 238-240 hermitian, 404 non-axial symmetry, 240-241 inverse of, 403 Linear operator, 392 multiplication of, 401 Linewidth: real,404 r, half width at half height, 34, 35 symmetric, 404 IlHpp, peak-to-peak width, 35, 36 transpose of, 403 Linewidth alternation, 208-214 unit, 404 Linewidth variations, 204-221 unitary, 404 Local fields, 41 Matrix elements: Lorentzian lines, 33, 34, 35, 36,456,464 definition of, 400 K for, 197 Dirac notation for, 405 Lorentzian Iineshapes for homogeneously of j +, j x and j y, 426-427 broadened lines, 196 Medium-field case, 261, 262 Low-spin systems, definition, 284, 284 Metal-free flavoproteins, 381 Lumiflavin semiquinone anion, 381, 382 Methyl proton hyperfine splittings, 124- 125 Methylcyclooctatetrene anion, 110 M center, 185,237 Methylene: "IlM = 2" transitions, 242-244 CH2 and CO2, 246-247 Magnet system, 29 substituted D and E values, 247 Magnetic dipole transitions, 347 Metmyoglobin, 387, 387, 388 Magnetic dipoles, interaction with radiation, Microwave attenuator, 24, 29 12 Microwave cavity, 21 Magnetic fields, oscillating, 17 Microwave frequency: Magnetic interaction energy, 8 effect on sensitivity, 460-461 Magnetic moments: Q band, 23 and angular momentum, 11 X band, 23 definition, 8 Microwave power level, effect on sensitivity of nuclei, Table C (see inside back cover) and resolution, 456-457 Magnetic susceptibility, 409 Microwaves, circularly polarized, 138 of 3d" ions, 261, 262 Miller indicies, notation for directions, planes definition, 8 and axes, 132 units of, 9 Minimum detectable number of spins, 451 492 ELECTRON SPIN RESONANCE Mn 2 +: NH,247 in CaO, 307 NH3+ radical, 4,71 in crystal fields, 277, 377 Ni2+: free ion, 260 in AI20 a, 311 in MgO, 307, 308 in crystal fields, 277 in NaCI, 305 free ion, 260 Mn 3+: in MgO, 292, 293 in crystal fields, 277 in Ni(NH3)62+, g factor, 291 free ion, 260 octahedral, 276, 286 spin-orbit coupling constant, 277 in Ti02, 296 Mn4+: in Zn3La2(N03)12 . 24H20, 296 in crystal fields, 277 Ni3+: free ion, 260 in crystal fields, 277, 334 Mn 5 +, in Si, 294 free ion, 260 Mn6+ in (oct + ttg\) field, 318 Jahn-Teller splitting in A1 20 3, 335 MnSO •. H20 as a standard, 463 Ni(NH3).++,291 Mo(CN)83-, 263, 335-336,336 Nitrenes, triplet ground state, 247-248 Mode, klystron, 25, 28 Nitrogen atoms in diamond, 183, 189 Modulation amplitude, 30,31-32,32 Nitroxide radicals, 252, 252-255,255 effect on sensitivity and resolution, NMR: 452-454 for magnetic-field measurement, 30, 465 Modulation frequency, 30 use of, in determining signs of hyperfine effect on sensitivity and resolution, splittings, 114-117 454-456, 455 NO, 345-348,348 Modulation sidebands, 30 energy levels in, 347 Modulation system, 30 N02 radical, 176-178,177,179,180,349 Molecular orbital energy calculations, 88-95 bond angle in, 178 Moment of an absorption curve, 463 NOa radical, 176-178,177 Monocyclic systems, HMO energies, 95-96, N03~ radical, 176-178,177 96 Noise, 1If, 30, 31 Mo03+,318 Noise figure, 451 Multiplicity, 38 Nonbonding orbitals, 95 Mutual atom-atom polarizability, 120 in odd-alternant hydrocarbons, 99 Nonlinear operator, 392 14N hyperfine coupling tensor, 171 Normalization condition, 92, 404 14N hyperfine splitting, 69-71 Np3+, Np4+, Np5+, Np6+, 339 mechanism of, 127 Np02++' 339, 340 23Na hyperfine interaction, 71 NS,349 Naphthalene, HMO orbitals, 109 N(S03)2~ radical, 14N tensor in, 172 Naphthalene anion, 62, 63, 71, 110, 192 Nuclear hyperfine interaction (see Hyperfine as a g factor standard, 465 interaction) ion pair with N a+, 207, 207, 222 Nuclear quadrupole moment, 370 Naphthalene triplet state: Nuclear spins, Table C (see inside back angular dependence of resonant fields, 235 cover) energy levels of, 230, 243 hyperfine splitting in, 234-236 17 0 hyperfine splittings: randomly oriented, 241 in p-benzosemiquinone, anion and cation, resonant fields, 234 127 spectrum of hypothetical rotating, 234 mechanism of, 127 spin densities in, 236 O2, 'd state, 4, 349 triplet exciton in, 237 O 2 - ion: Natural abundances of nuclei, Table C (see in alkali halides, 134-135, 179 inside back cover) principal axes for, 134 N dH , free ion spin-orbit coupling constant, structure of, 184 337 in KCI, 134, 184 SUBJECT INDEX 493 Octahedral crystal field, 263 Perfluorosuccinate radical: Octahedral potential, 266 direction cosine matrix in, 148 Octahedral symmetry, 267 19F hyperfine tensors in, 173 OH,345 spectrum of, 152 ONS, 349 forbidden lines in, 152 Operator algebra, 392-394 Perinaphthenyl radical, 111 Operator equivalents, 267-269 13C hyperfine splittings in, 130 Operators: Peroxidases, 380 angular momentum, 393,420-421 Peroxylamine disulfonate ion, (S03)zNO~, commutation relations, 422-423 85,86 commuting, 15, 393 dipotassium salt as a standard, 463 hermitian, 393 in potassium hydroxylamine disulfonate linear and nonlinear, 392 crystal, 165 nonhermitian, 405 spin exchange, 201 spin, 14 Perturbation theory, 414-416 Optical spectrometer, 21-22 for hydrogen atom, 443-445 Orbital: Perylene anion as a g factor standard, 465 definition, 92 Perylene cation as a g factor standard, 465 degenerate, 100 PF4 radical, 68, 85 Orbital angular momentum wave functions, Phase-sensitive detection, 30-32, 31 real and complex forms, 264, 265 Phenanthrene, triplet state of, 236 Orbital degeneracy: Phenyltrimethylgermane anion, 80-81, 81 removal in electric fields, 261-263 Phosphorus-doped silicon, 365-366 of 3d" ions, 277 Photosynthesis, 383-386 Orbital energies in 7T-electron systems, 88, primary products in, 383 107 Photosynthetic bacteria, 383-385 Orbital reduction factor, 326 effect of growth in D,O, 384 Orthogonal matrix, 404 light-induced ESR signal in, 384 Orthogonal vectors, 399 7T bonds, 88, 341 Orthogonal wave functions, 404 7T-type radical, 88, 165-174 Orthogonality, 94 identification of, 165-167 Orthonormal wave functions, 404 Pm3+ free ion spin-orbit coupling constant, Orthorhombic symmetry, 134-135 337 Outer product of vectors, 399 P03~ radical, 67, 179, 180 Point-charge model, deficiencies in, 340- 31 P hyperfine splittings, 67 344 P-state ions, 269-270, 270 Point defects in solids, 6, 179, 181-186 in tetragonal field, 3 10 generation of, 179, 181 P-state wave functions, 264 substitutional or interstitial defects, Pa3+, Pa4+, 339 181-184 Packet-shifting ENDOR, 365-366 trapped-electron centers, 184-185 Pairing theorem, 106, 122 trapped-hole centers, 185-186 Palladium and platinum groups, 335-336 Polyacene ions, proton splittings, 105 Paramagnetic resonance, 3 Polymer, triplet state in, 248 Particle: Population of triplet states, 249 in a ring, 9 Porphyrin ring system, 386 on a sphere, 10 Potassium nitrate, y irradiated, 176-I 78, 177 Pauli exclusion principle, 107,259,398 Powder spectra, 154-161 Pauli spin matrices, 427 Pr3+, free ion spin-orbit coupling constant, Pentacene anion and cation, 129 337 Pentaphenylcyclopentadienyl cation, 244 Precession frequency, 20 Perfluoro-p-benzosemiquinone, 67 Principal axes, relation to symmetry axes and Perfluorosuccinate radical: planes, 409 analysis of19F hyperfine tensor in, 146- Principal values, 19 149 Proton exchange, 204 494 ELECTRON SPIN RESONANCE f3-Proton hyperfine couplings, 124-125, Resonance equation: 169-170 electronic, 12, 17 Proton hyperfine sp1ittings: nuclear, 17 versus HMO unpaired electron densities, Resonance integral, f3, 93 97 Resonant magnetic fields for the hydrogen origin of isotropic proton hyperfine atom, 442-443 splittings, 112-114 'RH2 radical, 13,48 (See also Hyperfine coupling constant, energy levels of, 446-449 Hyperfine splitting constant) Rhodopseudomonas spheroides, Proton hyperfine tensor in the C-H light-induced ESR signal, 384, 384 fragment, 169 Rhombic symmetry, 410 Pu 3+, PuH , Pu5+, Pu 6 +, 339 Rotation of coordinate system, 411-412,411 Pu02 ++, 339, 340 Row vector, 400 Putidaredoxin, 388-389 Ruby, distant ENDOR in, 366 "Fe hyperfine splitting in, 388-389 Rules for analysis of solution ESR spectra, 33S hyperfine splitting in, 389 74-76 Pyrazine anion, 71, 72, 83,130,467 Pyrene anion, 81, 82 33S hyperfine splittings, mechanism of, 128 Pyridine anion, 130 S-state ions, 276 nondegenerate ground states, 303-310 Q-band frequency range, 23 S-state wave functions, 264 Q factor: S2- radical, 85, 191 of cavity, 25, 27 Sc2+: definition, 459 in crystal fields, 277 due to dielectric loss, 459 free ion, 259, 260 loaded,459 in (oct + ttgl) field, 318 unloaded, 451 Scalar product, 399, 401 Q (parameter for proton hyperfine splittings), Schrodinger equation, 394, 419 112-114, 168 Second-order energies for' RH2 radical, 449 for benzene derivatives, 102 Second-order hyperfine splittings, 75, 77-79, definition, 97 77 negative sign of, 114 Secular determinant: Quadrupole interaction tensor, 368 definition, 407 Quadrupole moment, 449 F-state ions, 275 Quadrupole moment operator, 365, 368, 449 H 2+, 91 Quintet state (S = 2), 256 hydrogen atom, 438-439 7T-electron systems, 93 R center, 185, 256 for triplet states, 228 Raising, lowering operators, J +, J _, 268,423 Secular equations, 91, 398, 406-407 Randomly oriented solids, line shapes in: SeH, 345, 349 doublet states, 154-161 Selection rules for hydrogen atom: triplet states, 238-241 high field, 439-441 H min in, 244 low field, 445-446 flM s = 2 transitions in, 242-244 Sensitivity: Rank of a tensor, 409n., 413 effect of: concentration of paramagnetic Rapid-flow techniques, 380 centers, 458 Rapid passage, inversion, 361, 365 microwave frequency, 460-461 Rare-earth ions, 336-339 microwave power, 456-457, 457 RC filter, 461, 461 modulation amplitude, 452-454, 452, Real forms of wave functions, 264-265 453,455 Real matrix, 404 modulation frequency, 454-456, 455 Relaxation times: Q factor of cavity, 459-460 spin-lattice, 19,193-196,456 signal averaging, 461 spin-spin, 197,456 of an ESR spectrometer, 23,450-452 SUBJECT INDEX 495 SeO: Spin hamiltonian: 3:£, 349 for ENDOR, 357 'd,349 for hydrogen atom, 43, 436 SF,349 including hyperfine terms 282 SH,345 ions with orbitally nondegenerate ground u bonds, 89, 341 states, 278-283 u-type radical, 88, 174-175 for orthorhombic symmetry, 134-136 Signal averaging, 461 for quadrupole interaction, 368 Signal-to-noise ratio, 22 relation to orbital operators, 278-279 improvement by a signal averaging for 'RH2 radical, 447 computer, 461 for S = I systems, 226, 227, 229 Silicon crystal, 30 Spin labels, 390 Site splitting, 148, 149 as standards, 463 Slide-screw tuner, 24, 28 Spin-lattice relaxation time, 193-196, 456 Sm3+, free ion spin-orbit coupling constant, Spin matrices (S = I), 227 337 (See also Angular momentum matrices) SO: Spin-orbit coupling, 11 3:£,345,349 Spin-orbit coupling constants: 'd,349 of 3dn ions, 277 S03- radical, 179, 180 of 4jH ions, 337 (S03)zCH~ radical, 187 Spin-orbit hamiltonian, 278, 290, 314 Sodium atom, 48 Spin packets, 196, 362 Sodium naphthalenide (see Naphthalene Spin quantum number, 10 anion) Spin-rotational interaction, 220-221 Solid angle, definition, 155 Spin-spin hamiltonian, 224-227 Spectral extent, 117 Spin-spin relaxation time, 197, 456 Spectrometer, ESR, 2, 23-32 Square matrix, 400 sensitivity, 450-452 Standards: signal-to-noise ratio, factors affecting for absolute intensity measurements, 463 (see Sensitivity) for field calibrations, 466 Spherical harmonics, 10 for g factor measurements, 465 Spin coordinate, 420 Stark splitting, 338n. Spin degeneracy of 3dn ions, 277 State, definition, 92 Spin density, 113-114 Steady-state ENDOR: negative, 117 characteristics of, 356-358, 364 normalization of, 113 optimum value of H'e, 362 sign of, 114 population of levels in, 362-365 Spin diffusion, 366 relaxation processes in, 360-365 Spin eigenfunctions (S = I): Stem-Gerlach experiment, 1 intermediate fields, 231 Stevens's theorem, 267 in zero field, 227-228, 231 Stick-plot reconstruction, 75 Spin exchange, 201-203 Strong-field case, 261, 262 Spin-flip transitions, 154 Substitutional defects, 179-184 Spin hamiltonian, 15,278-283 .a-Succinic acid radical, for anisotropic hyperfine interaction, HOOC-CH2-CH-COOH, 166, 187 140-141, 143 Superheterodyne system, 31 for axial symmetry, 134 Susceptibility tensor, 410 for 3d' (oct + trgl) ions, 330 Symmetric matrix, 400, 404 for 3d' (Ud!) ions, 293 Symmetric orbital: for 3d3 (oct) ions, 300 benzene, 10 I for 3d" (hs)(oct) ions, 303 H,+, 92 for 3d' (hs)(oct) ions, 333 Symmetric states, 224 for 3dB (oct) ions, 291 Symmetry: axial, 20, 134, 409 for 3dB (oct + ttgl) ions, 296 definition, 134 496 ELECTRON SPIN RESONANCE Symmetry: relation to angular momentum Ti3+: quantum numbers, 420 spin-orbit coupling constant, 277 Synthesis of spectra, 75 Time averaging computer, 461 Time constant of an RC filter, 461 T I ,193-196 Tm 3+, free ion spin-orbit coupling constant, (S ee also Spin-lattice relaxation) 337 T 2, definition of, 196-197 Toluene anion, 79, 102 (See also Spin-spin relaxation) Trace: T;,197 of D tensor, 226, 282 Tanabe-Sugano diagrams: of hyperfine coupling tensor, 169 d 2 ,328 Transition frequencies in the hydrogen atom, d 3 ,298 441 d., 285 Transpose of a matrix, 403 dO, 304 Tribenzotriptycene in triplet state, 236 d 6 ,325 Triphenylbenzene dianion ground triplet d 1,332 state, 240, 245 dB, 286 Triphenylene dianion ground triplet state, 245 Tb"+, free ion spin-orbit coupling constant, Triplet energies, plots of, versus H, 229, 230, 337 243, 249, 250 Tc4+ in K2PtCl,;, 335 Triplet excited state, 232-238 TeH,345 Triplet exciton, 236-237, 257 Temperature, effect on sensitivity and Triplet ground state, 244-249 resolution, 458 zero-field splittings in, table, 247 Tensor, 408-414 Triplet state population, 249 from outer product of vectors, 399 Triplet states: 1,4,5,8-Tetraaza-naphthalene anion, 83,84 hyperfine splittings in, 234-236, 237 Tetracene cation as a g factor standard, 465 negative spin density in, 247 Tetragonal distortion, effect on: D-state ions, spin densities in, 236 273 Tropyl: P-state ions, 270 dianion, 123 Tetragonal potential, 266 radical, 54,56, 123 Tetragonal splitting: of octahedral field, 272, 273 U 2 center, 183 of P-state ion, 270, 270 U 3+, U'+, U5+, 339 Tetragonal symmetry, 132 Uncoupled representation of angular Tetrahedral crystal field, 263, 274, 275 momenta, 428 Th4+, 339 Unit matrix, 404 Thermally accessible triplet states, 249-250 Unit tensor, 141 niianthrene anion, 128 Unitary matrix, 404, 408 Ti+: Unpaired electron density, 96, 112-113 in crystal fields, 277 normalization of, I 13 free ion, 260 Unpaired electron distribution, 95 Ti2+: Unsaturated organic radicals, 173-174 in crystal fields, 277 U02 ++, 339, 340 free ion, 260 in ZnS, 293-294, 293 VI center in MgO, 132-134,186 Ti3+: angular variation of ESR line positions, 133 in aluminum acetylacetonate, 324 calculation of g factors, 280 in CaO, 318, 351 VI optical band in alkali halides, 132 in crystal fields, 277 V k center, 176, 186 in CsTi(SO.)2 . 12H20, 323-324 VOH center, 184 free ion, 256, 260 definition, 13 8 in (oct + ttgl) field, 3 18 hyperfine coupling for, 139 in octahedral field, 27 I spectrum of, 139 SUBJECT INDEX 497 V2+: Wave functions: in crystal fields, 277 complex forms, 264-265 free ion, 260 configurational, I 18 spin-orbit coupling constant, 277 for hydrogen atom: at low fields, 445-446 V3+: at moderate fields, 42 in Al20 3 , 297, 330-331,331 normalized, 404 in CdS, 297 orthogonal, 404 in crystal fields, 277 for 1T-electron systems, 92, 108 free ion, 259, 260 (See also Eigenfunctions) spin-orbit coupling constant, 277 Wave meter, 24,25,29 in ZnS, 294 W(CN)s3-, 335-336 V4+: Weak field case, 261 hyperfine couplings in CaWO., 288 Wigner coefficients, 429 in (oct + ttgl) field, 3 18 Wurster's blue perchlorate: spin-orbit coupling constant, 277 as a field calibration standard, 466 Vanadyl acetylacetonate, spin rotational as a g factor standard, 465 interaction in, 221, 220 Variational principle, 89 X-band frequency range, 23 Vector-coupling (VC) coefficients, 429 X-band waveguide, 23 Vectors, 398-399 XeF radical, 4,5, 72-73. 188 scalar, vector and outer products of, 399 XeF., 4 Velocity, linear, 12,20 p-Xylene anion, 101, 104 Vinyl radical, 88, 174, 22 I alternating linewidth effect in, 214, 215 Yb3 +, free ion spin-orbit coupling constant, VO++: in crystal fields, 277 337 free ion, 260 in (oct + ttgl) fields, 3 18 Zeeman energies, 12 in toluene, 222 Zeeman hamiltonian, spin-orbital, 278, 338 VO(CN)s"- in KBr, 318,320 Zeeman interaction, 44 Zeeman levels, 12, 17 W center, 257 Zeeman splitting: Wave functions: F-state ions, 291 antisymmetrized, 398 Zero-field splitting in triplet state, 229, 28 I Table C Nuclear spins. abundances. moments. and hyperfine couplings for some common magnetic nucleit Anisotropic Isotropic % Natural Magnetogyric ratio:j: hyperjine coupling hyperjine coupling Nucleus Spin abundance (radG-1s-lx 10--4) B. MHz§ Ao. MHz'; 'H ! 99.985 2.67510 1,420 2H 0.015 0.41064 218 6Li 1 7.42 0.39366 152* 7Li ! 92.58 1.03964 402* (291 calc) "Be ! 100 -0.37594 -358 lOB 3 19.58 0.28748 17.8 672 "B ! 80.42 0.85828 53.1 2,020 13C t 1.108 0.67263 90.8 3,110 14N 99.63 0.19324 47.8 1,540 15N ~2 0.37 -0.27107 -67.1 -2,160 170 ! 0.037 -0.36266 -144 -4,628 '"F t 100 2.51665 1515 47,910 23Na J!2 100 0.70760 886* 25Mg ! 10.13 -0.16370 27AI ! 100 0.69706 59 2,746 29Si ~2 4.70 -0.53141 -86.6 -3,381 3lp t 100 1.08290 287 10,178 33S ! 0.76 0.20517 78 2,715 asCI ! 75.53 0.26212 137 4,664 37CI ! 24.47 0.21818 117 3,880 39K ! 93.10 0.12484 231* "Ca t 0.145 -0.17999 45S C t 100 0.64989 1,833 47Ti ! 7.28 -0.15079 -492 49Ti t 5.51 -0.15083 -492 51V t 99.76 0.70323 2,613 53Cr ! 9.55 -0.15120 -630 55Mn ! 100 0.65980 3,063 57Fe 1- 2.19 0.08644 450 5"CO t 100 0.63171 3,666 6lNi ! 1.19 -0.23905 1,512 63CU ! 69.09 0.70904 4,952 65CU ! 30.91 0.75958 5,305 67Zn ! 4.11 0.16731 1,251 75As ! 100 0.45816 255 9,582 77Se t 7.58 0.51008 376 13,468 79Br ! 50.54 0.67021 646 21,738 81Br ! 49.46 0.72245 696 23,432 83Kr I 11.55 -0.10293 85Rb ! 72.15 0.25829 1,012* 87Rb ! 27.85 0.87533 3,417* "5Mo ! 15.72 0.17428 -3,528 97Mo ! 9.46 -0.17796 -3,601 '07Ag t 51.82 -0.10825 -3,520 '09Ag ! 48.18 -0.12445 -4,044 1271 ! 100 0.53522 129Xe t 26.44 -0.73995 1,052 33,030 13IXe ! 21.18 0.21935 l33CS t 100 0.35089 2,298* I 2·l7Pb "2 22.6 0.55968 t Compiled from data in the following references: I. J. R. Morton, J. R. Rowlands, and D. H. Whiffen, National Physical Laboratory Bul/etin, No. BPR 13, 1962. 2. S. FJiigge (ed.), "Handbook of Chemistry and Physics," 50th ed., p. E75, 1969. 3. J. R. Morton, Chern. Rev., 64:453 (1964). 4. D. H. Whiffen,J. Chirn. Phys., 61:1589 (1964). 5. B. A. Goodman and J. B. Raynor, J. Inorg, Nucl. Chern., 32:3406 (1970). :j: The magnetic moment (erg G-l) can be obtained from the magnetogyric ratio by using the relation P-N = IiIYN § The anisotropic hyperfine couplings are tabulated as B = ih-1gN!3Ng!3(r-3 ) where (r3) is computed for a valence p electron from self-consistent-field wave functions. The couplings are such that the principal values of the traceless tensor are respectively -1, -1, and + 2 times the number quoted. ~ The isotropic hyperfine couplings are tabulated as Ao = 8; h-l gN!3Ng!3 II/Is (0) 12 where 1/18(0) is the value of the valence-shell, self-consistent-field S wave function at the nucleus of the neutral atom. Values indicated with an asterisk are the experimental atomic hyperfine couplings as measured using the atomic-beam technique. [See P. Kusch and H. Taub, Phys. Rev., 75:1477 (1949).]