Effect of group transitions on magnetic properties of transition metals. I11 By T.1. K a k u s h n d z e With 3 figures Abstract In transition metals and alloys the group isoenergetic transitions account for a new type of paramagnetism. Taking the latter into account enables us to explain the origin of “anomalously” large paramagnetic susceptibilities .and their intricate thermal dependencies. In accordance with the modern theories of magnetism the normal paramagnetic susceptibility of the transition metals xp (determined by the initial non-compensated d-electrons) is given by the formula + (1) The first expression in brackets on the right-hand side of eq. (1)is the observed paramagnetic susceptibility (column 12, table 1) corrected for diamagnetism (column, 14, table 1).~ 8 the , diamagnetic susceptibility of the atomic residues, is estimated by the formula x p = (Xexp - x 8 ) - (Xs xd). is the time average of the square of the elecwhere m is the electron mass, tronic orbit radius ; summation being performed for all electrons of the atom. The magnitudes x 8 evaluated under the approximate method of S l a t e r are listed in table 1 (graph 9), xs and xd are the paramagnetic susceptibilities according to P a u l i for s- and d-electrons with allowances for their diamagnetpic susceptibility; these susceptibilities are found by the formulas In eqs. (3a) and 3b) is the energy of the F e r m i surface when there is one electron wer atom in the s-band where p Bis the Bo h r magneton, n, and n$ are the numbers of s- and d-electrons per atom. For the elements %Ca and ,,Pd, as well as for 771r and 78Pt, cOsis T.I . Kakushadze: Effect of group transitions on magnet& properties of transition metals 367 cvaluated according to S o m m e r f e l d by eq. (4), whereas for the other elcments 5,La to i 6 0 s , a value twice as high as that estimated according to S o m i n e r f e l d l ) is taken as (table 1, column 5). The susceptibilities xs and xa estimated by eqs. (3a) and (3b) are listed in columns 7 and 8 of table 1. The paramagnetic component of susceptibility makes it possible t o determine perf by t.he formula soS where N is A v o g a d r o s number, R the gas constant, T the absolute t.emperature, perf = p Bg V j Y j T 1 , the effective atomic magnetic moment (ps being the B o h r magneton), g the L a n d 6 factor and j the total (1 S) quantum number. For the transition non-ferromagnetic metals, j = S and g = 2 , i. e., + where S is the number of non-compensated electrons per atom. Experimental data are listed in column 1 2 for comparison. The table shows that differences between the predictions of the theory (the sums of quant,ities in columns 7 and 8) and the experimental data corrected for diamagnetism (column 1 2 ) are quite considerable. Besides, we encounter the following diffuculties. As is shown by experiment, the paramagnetic susceptibility of most of transition metals do not, contrary to C u r i e s law, change in inverse ratio to temperature but remain nearly constant or a t any rate weakly dependent on temperature. For the susceptibilit y of the transition elements to be described by eq. (5) therefore it is necessary for p& t o change almost linearly with temperature3). Yet no agreement of the theory with experiment is obtained (table 1) even in the casawhen it is known reliably that a sample possesses, in the absence of a magnetic field, a non-zero atomic magnetic moment (Cr, Mn, Rh, Pd, Ir, Pt4))and its paramagnctic susceptibility should be calculated by eq. (5). Thus, not only the quantitative treatment of susceptibility but also its qualitative dependence on temperature proves t o be in sharp contradiction with the theory of normal paramagnetism, especially for transition metals and alloys. Yet this discrepancy between theoretical results and experimental data cannot point t o the falacy of the L a n g e v i n - P a u l i theory the role of which in the development of magnetism cannot be overestimated. It is common knowledge that the L e n g e v i n - P a u l i theory is valid for bodies with atomic magnetic moments strongly screened against the lattice fields. Eq. (5) holds for substances whose atoms have constants, freely orientating in the external magnetic field, and magnetic moments, independent of temperature. We assume that the principal portion of susceptibility of nearly all transition metals and their alloys are due to group isoenergetic electron umklapprozesse. - l) According to ref.2) fot the elements Ta and W , to, 2 Ry i. e., more than twice aa much aa the values estimated according t o Sommerfeld. 2, M. F. Mannig and M. I. Chodorow, Physic. Rev. 56, 787 (1939). a) The Pauli paramagnetic susceptibility or the diamagnetic susceptibility of the transition elements accounts for only a fraction of that observed. ') C. G . S hu l l and M. K. Rilkintlson, Rev. mod. Physics 25, 100 (1954). 2.5* 368 Annalen der Phyaik. 7. Folge. Band 8. 1961 Table 1 1 ; 2 1 3 / 4 5 6 1 7 1 XS'~@ I I 2OCa 22Ti ' 23V I 24 Cr 25 Mn 40 Zr 41 Nb 42 Mo 44 Ru 46 Rh 46 Pd 67 La 78Pt 1 2 1 5 6 7 4 5 6 8 9 10 3 10 1.4 I 4.6 3.04 3.2 ' 37 5.36 4.1 58.8 5.97 5.0 13.1 6.85 6.1 82 6.79 , 3.5 41 4.43 27.6 4.5 6.90 5.14 5.83 5.5 8.5 7.5 6.38 4.69 6.23 8.5 9.5 10.7 5.89 i 2.5 57.4 7.5 32.1 8.6 14.6 10.6 5.5 11.5 12.1 19 i 1 j 914 1 4.0 1 : 5.96 j 80 1003 1740 393 2240 1067 7 60 112 170 61.8 - 1290 785 1377 440 412 169 51.5 66 6.29 ;33.14 : 3.08 3.85 2.48 2.93 2.69 2.74 2.91 2.6 1.2 1.0 0.87 0.83 0.8 1.62 3.05 i 1 8 Xd'lP 45.2 34.8 33.9 31.4 29.3 43.3 30.2 35.6 30.1 23.2 17.0 26.3 23 20.2 17.6 16.2 12.9 21.1 17.9 1 1 9 Xd.1@ 1i - 11.2 - 7.5 - 6.3 1 - 5.4 ' 1' ' - 4.7 -15.3 -13.3 - 11.7 - 11.9 - 11.0 - 7.42 -26.6 - 19.8 - 17.2 1 fiii - 12.0 - 11.8 If the electron band responsible for the atomic magnetic moment is not ovcrlapped by the conductivity electron band, the group umklapprozesse between the electrons of the conductivity band are impeded. The atomic magnetic moments of such metals and compounds are isolatcd from each othcr and consequently their paramagnetic susceptibility must obey the C u r i e - L e nge v i n law. This is observed in the case of lanthanides, salts and weak solutions of the salts of transition metals whose atomic magnetic moments are actually isolated from each other and do not depend on tcmperature. If on the other hand the electron band respvnsible for the atomic magnetic moment in metals or alloys is overlapped by the band of conductivity clectrons, reciprocal group umklap€ prozesse are possible in them. This accounts forthe changc of atomic moment with temperature. It is generally beiieved that atomic magnetic moments in transition metals are due to the d s ) outer d-electrons. The d-band a UE b UE of transition elements are divided into two halves: for right Fig. 1 and left spins. In ferromagnetic as well a s certain paramagnetic bodies, the right and left halves of d-bands are not equally filled by electrons. As a result of exchange interaction, these bands (d- and d,) shift with respect to each other energetically (fig. l a represents the distribution of electrons at 0 OK and fig. 1b a t T # 0 OK). The number So of the non-compensated d-electrons per atom with the electron configuration T . I . Kakushadze: Effect of group transitions on magnetic properties of transition metals 369 9.5 120 210 47 270 128 91.5 13.4 20.5 7.1 ’i 61.0 158.5 247.3 81..5 302.4 175.1 124.1 51.5 533 33.3 20.0 183.2 118.2 186.2 71.5 65.0 32.7 28.8 28,9 - 154 94 165 53 49 19 6.1 7.9 1 44 153 250 172 532 122 120 54 44 101 558 144 75 163 55 68. 9. 35 189 I 9 8 8 8 ~ I i ti 8 ~ 8 8 8 11 8 8 8 8 8 8 8 I ~ 55.2 160.5 256.3 177.4 536.7 337.3 133.3 65.7 55.9 112.0 565.4 170.7 94.8 170.2 71.2 83.0 23.4 47.0 200.8 0.06 0.17 0.25 0.18 0.48 I 0.14 0.14 0.07 0.06 0.12 0.50 0.18 0.10 0.18 ~ ~ 0.06 0.16 0.25 0.09 0.29 0.17 0.13 0.06 j i 0.19 0.65 0.06 0.04 0.02 0.19 0.12 0.19 shown in fig. l a or 2 b und c is determined by the difference in the number of right and left spins taken a t absolute zero, i. e., 8, = n d + - n d - . \ 7) At non-absolute-zero temperatures, the electrons of the s- and d-bands may, as a result of group umklapprozesse, be excited into the unoccupied part “t “t Y A A A dt dNIEJ b Fig. 2 dl! c UE Fig. 3 of the d-shell. When a sample is placed into an external magnetic field, the electrons excited in the d-band pass into the right half of the same band. Within a spilt second (of an order of the spontaneous return of the electron from the right half t o the initial left semi-band) the equilibrium distribution of electrons is established between the right and left halves of the d-band (fig. 3). As a result we have a new kind of paramagnetism. The transition of the electron from the left half of the d-band to the right one now occurs with insignificant magnetic energy. In this case no energy is spent on dragging 370 Annalen der Physik. 7 . Fdge. Band 8. 1961 out the elctron into the upper vacant state for right spins. As a result there arises a layer in the right half of the d-band responsible for the atomic magnetic moment (fig. 3). Three cases can be distinguished : 1. Atoms without initial magnetic moments (fig. 2a). Let the number 8, of non-compensated d-electrons a t 0 OK be equal to zero, i. e., So = nd- - nd- = 0. (8) When there are two reciprocally overlapping outer s- and d-bands, the probabilities of the group umklapprozssse of the electron of the s-band into the vacant part of the same band and the d-band will, to a first approximation, be proportional to the respective expressions ns(nOs - ns) and ns - nd)* (9) Similarly, the probabilities of the group umklapprozesse of a d-electron into the vacant part of the same band, or the s-band, will be proportional to the respective expressions nd - nd) and nd s - ns)* (10) The frequency of the group umklapprozesse is approximately proportional t o the electric resistance of the conduct'or R3)4) R=aRoT (3 1 = -, R, is a resistance a t T = To (11) o 0~5)). Then for the number of umklapprozesse for the time dt we obtain the expressions N,, dt = a' R, n,(nos- n,) T dt N,,dt = a' R, n,(nod- nd)T dt N,, dt = a' R,n, (no,- nd)T dt XdSdt = a' (12) R,n,(n,, - n,) T dt where N,,, N,,, N,, and Nds are the numbers of umklapprozesse per unit time of s-electrons to the s-band, of d-electrons to the d-band, of s-electrons t o the s-band, and of d-electrons to the s-band respectively; ns and n, are the numbers of the electrons a t 0 OK, nos and nod are the total numbers of the electron places in the s- and d-bands respectively, a' is the coefficient of proportionality and T is the absolute temperature. The number of electrons Ax, resettled from the d-band into the s-band (at rather low temperatures according to (12) 6)6)) is equal to Ax =a Ro(nosnd -nod n,) T (13) where (y. is a new constant. Then a t the temperature under consideration we have in the s-band TL: = n, + A x (14) j ) T. I. Kakushadee, The Role of Group Transitions in the Production of Certain Sateilites.1 6 ) T. I. Kakushadze, Electron Heat Capacity in Transition Metals and Alloys. I1 T.I . Ka..hlshadze: Ef/ed of group transiticns on magnetic properties of tranailion nndals 37 1 electrons; and in thc d-band n& = nd - A x electrons per atom. When there is no external magnetic field a t T = 0 the vacant part of the d-band 7 for the case of mutually compensated d-electrons (nd+= nd-)will, according to (15), contain = nod- 72; = nod- nd + 7o+ A x (16) electron places. Here qo = nod- n, is the number of free places i n the d-band at 0 O K . The number of the uncompensated electrons S,, due t o group isoenegetic umklapprozesse of d-electrons into the vacant part of the same band are, according to (12) and (15), proportional t o the expression N&j = A Ro(nd- A S ) 7 T. (17) An appreciable contribution can also be made by the electrons excited from the s-band into the vacant part of the d-band. According to (12) and (14), t h i s part of the non-compensated electrons is proportional to the expression 77 N;d = Ax = A R,(n, + Az) 7 T. (18) The electxons excited into the vacant part of the s-band (proportional to the expressions N:, = A Ro(n,+ A s ) (no,- n, -LIZ) T or A'& = A R,,(nd- A x ) (nos- n, - A x ) T) cannot change the number of non-compensated electrons to any appreciable extent. On the strenght of the above said, for the number of non-compensated d-electrons due to the group umklapprozesse we obtain, according to (17) and (18) S,, = Ron 7 T. (19) Here ?i= n, nd. 3. Atoms with initial magnetic moments (both halves of d-band are partially filled). (Fig. 1 and 2b.) As was indicated above, the number So of the noncompensated d-electrons is determined by the difference in the numbers of right and left spins taken at the absolute zero, i. e . . So = nd+ - nd-. (20) + 1x1the case of nd+ # nd- the re-settlement of d-electrons into s-band also leads, similarly to the above, to the change in the number of electrons in the d-band. Consequently, the number of vacancies in the d-band increases by Ax (13) a s temperature rises. Of essential importance in the creation of paramagnetism of the new type arc in this case the electrons shifted as it result of group transitions into the vacant part of the d-band situated between the A A and BB levels (figs. 1 and 2). Indeed, when external magnetic field becomes effective, only the electrons excited into the region 7 situated between the A A and B B levels will pass into the right half of the d-band. The number of vacancies in the d-band (at T # 0 OK) is determined by the relation 7 = 70 +Ax. (21) Yet in this case q0 is the number of vacancies situated between the A A and BB levels at 0 "K (figs. 1 and 2h and c). For determining the number of the non- 372 Annahn der Physik. 7. Folge. Band 8. 1961 compensated electrons Ssddue to group umklapprozesse, we shall use eq. (19) in this case as well. For the magnitude S of the t.otal number of noncompensated electrons we have = 'sd (22) where S,, is, according to (19), equal t o B R, n 7 T. 3. Atoms with initial magnetic moments with the right half of the d-band filled (fig. 2 c). h t the right half of the d-band be completely filled (Ni, Pd) and remain filled as temperature increases (fig. 2 c). The umklapprozesse of the d-electron into the vacant part of the left half of the d-band cannot account for the change in the number of non-compensated d-electrons (nd+ - d - remaining constant). Consequently 7, = 0 (the region between the A A and BB levels equals 0) (fig. 2c). As temperature increases the number of d-electrons decreases bjA x ( A x being the numbcr of electrons re-settled from the d-into the s-band); however, 7 will remain equal to zero until the F e r m i surfacc of the left half of the d-band proves below the BB level (fig. 2c). On the other hand, the resettlement of d-electrons into the s-bend, with the right half-band totally filled, decreases the number of electrons in the left half of the band by A x (13). Consequently, the number of non-compensated electrons, will, with the totally filled right half band, increases by the magnitude + (23) Sd = A X . Now the total number of non-compensated d-electrons per atom w511 bc exprcssed by the formula S=S,,+Az (24) where A x is given by (13). 4. Comparison with experiment. According to eqs. (5), (6) and (19) for susceptibility due to group transitions, we find the expression Xsd (T RO 77 (25) under the condition '8, 2. ('26) The condition (26) is fulfilled in reality (column 10 of table 1).For the elements with mutually compensated d-electrons (in the case of there being no outer magnetic field) 7 is determined by eq. (lG), and for the elements wit,h initial atomic magnetic moments by eq. (21). The coefficient a in (25) is obtained through the observed magnitude of susceptibility of some metal with corrections for diamagnetism (column 14). If the magnitude A x , much less than n and 7, is neglected in eqs. (16) and (21), a proves to be of an order of 0.12. The susceptibilities xsd calculated according to (25) are listed in column 10 in table 1. Column 11 lists the theoretical values of paramagnetic susceptibility components < XtlleOr = x s + + Xd Xsd' Columns (15) and (16) contain the numbers of non-compensated d-electrons per atom calculated according to (5) and (6) form xexp-26 (column 14) and from Xtheor (column 11) respectively. As is clear from table 1, the theoretical T. I . Kakushadzs: Effect of group transitions on magnetic properties of transition metals 373 magnitudes are in good agreement with the experimental results for the elements without initial atomic magnetic moments (Ca, Ti, V, Zr, Nb, Mo, Ru, La, Hf, Ta, W, Re, 0 s ) . In the case of Ta and W three bands (6s, ,5d and 4f) overlap'). Therefore in (25) n is assumed equal t o 19 and 20 (the total number of outer electrons for Ta and W respectively. For the elements Cr, Mn, Rh, Pd, Ir and Pt thc numbers of the initial non-compensated electrons are obtained as the differences Sexr- Stheor = So (column 17) and hence (according t o (6)) the initial atomic magnetic moments po (column 18). In particular, po for Cr and Mn are found equal to 0.43 and 0.66 of B o h r magneton, whereas neutronographic investigations yield, in good agreement with the theory, the magnitudes 0.4 and 0.6 of B o h r magneton pcr atom respectively4). Of special interest is the investigation of the magnetic susceptibility of palladium. The right half of the d-band of Pd is assumed t o be completely filled. Therefore, xsd in palladium is equal to zero, according to (25). The number of the non-compensated electrons Pd is determined by eq. (24) s = s0-+ox where So is the number of initial non-compensated d-electrons and A x is the number of electrons re-settled from the d-band into thc s-band. From the susceptibility of Pd we determine, according to (5) and ( G ) , the number of noncompensated d-electrons; and using (24) we can find A x with the given number So of the initial non-compensated electrons. a t T = 298 "K From the observed susceptibilities of Pd, x = 550. and = 375. a t T = 523 OK7) we obtain, for the number of non-compensated d-electrons, the magnitudes So = 0.44 and A x = 0.09 per atom a t 298 OK. Hence it is possible to determine the coefficient LY by (13) and consequently the number of electrons re-settled from the d-band into the s-band for any metal. The initial effective magnetic moment of Pd proves equal t o ,uu,ff = 1.09 of B o h r magneton (column 18). Thus, t,he theory of group umklapprozesse makes it possible to explain the origin of paramagnetic susceptibilities of transition metals in quantitative terms and from a single point of view. The following points are noteworthy : 3 . The same values of electric resistances were used4) for determining the magnetic susccptibilitics as well as electron heat capacities. 2. The number of non-compensated electrons S,, due to group umklapprozesse is assumed t o be linearly dependent on temperature. Therefore, the susceptibility of a metal must, according to (5) and (6), increase slowly with temperature (in the case of there being no non-compensated electrons) or slowly decrease with temperature (for elements with initial non-compensated electrons), or else have another thermal dependence in accordance with the nature of susecptibility 6). In eq. (13) n, and ndare assumed constant equal to the numbers of s- and d-electrons a t 0 OK. Yet as a result of group transitions these numbers change strongly with temperature. At sufficiently high temperature the factor in brackets in the right-hand part of eq. (13) must tend to zero. In reality A x tends to the final value as T increases beyond bound. x _ 7) . T. I. K a k 11 s h it d z e, Transition Metals. Tbilissi, 1957. 374 A n d e n der Physik. 7’. Folge. Band 8. 1961 Let us designate the numbers of electrons in the s- and d-bands by n: and n& respectively ((14) and (15)). Then from (13) we have n: nos- n6 no, = 0 at T - t o o . Bearing in mind that ni we find + n; = n at T+oo (27) (28) at T - t m . In the case of nickel, for example, a t a sufficiently high temperature n& = 8.33 (instead of 9.4 a t T = 0 OK and n; = 1.67 (instead of 0.6 per atom a t 0 OK). This effect is responsible for a number of anomalous phenomena, anomalously large susceptibilities of Fe, Co and Ni, for example, observable above the C u r i e point. In table 2 comparison is drawn among a number of observed non-compen.sated d-electrons below and above the C u r i e point evaluated by (29) for the elements Fe, Co and Ni. (n = 8.9 and 10 for Fe, Co and Ni respectively.) For the rough approximation used in the present paper, agreement between the theoretical magnitudes and the experimental results should be regarded as quite satisfactory. -. ___ C. J. Kriesman and H. B. Callen, 94, 837 (1964). Pierce W. Selwood, Magnetochemistry, New York, 1943, 1956. 10) M. A. Filyandi and E. I. Semenova, Properties of Itare Elements, Metallurgizdat, Moskva, 1953. l1) F. Trombe, Compl. rend. 198, 1591 (1934). 8) 0) T b i 1i s s i (Georgian SSR),Pedagogical Institute. Bei der Redaktion eingegangen am 4. April 1961.