- Email: [email protected]

Contents lists available at ScienceDirect

Physica B journal homepage: www.elsevier.com/locate/physb

Orbital and spin contributions to magnetic hyperﬁne ﬁelds of tri-positive rare earth ions K. Ayuel a,n, P.F. de Châtel b, A. El Hag c a

Department of Physics, Al-Baha University, P.O. Box (1998), Saudi Arabia Institute of Metal Research, Chinese Academy of Sciences, 72 WenhuaRoad, Shenyang 110016, PR China c Physics Department, College of Science, Qassim University, Qassim, Saudi Arabia b

art ic l e i nf o

a b s t r a c t

Article history: Received 11 September 2014 Received in revised form 7 October 2014 Accepted 17 October 2014 Available online 22 October 2014

An alternative approach in the estimation of the magnetic ﬁeld generated by atomic current densities is presented and applied to both the orbital and spin magnetic hyperﬁne ﬁelds of tri-positive rare earth ions. The application of the formulae offered is tested on the free ions Dy3 þ , Er3 þ and Yb3 þ in their ground states. The estimated magnetic hyperﬁne ﬁelds are in full agreement with those found in the literature. Our calculated magnetic hyperﬁne ﬁelds produced in the ground states of Er3 þ ion in Er2Ge2 O7 and Yb3 þ ion in YbNi5 are also in good agreement with experimental and estimated data found in the literature. & 2014 Elsevier B.V. All rights reserved.

Keywords: Crystal ﬁeld eigenstates Magnetic hyperﬁne ﬁeld Rare-earth Tri-positive ions

1. Introduction

where N takes the form [3,4]

For many decades theoretical investigations of magnetic hyperﬁne structures have been performed. By studying the hyperﬁne magnetic structure, one can probe nuclear environment and get some information about the properties of atoms and ions. The source of hyperﬁne magnetic structure is interaction between the magnetic ﬁeld produced by electrons and the magnetic ﬁeld due to the nucleus. The magnetism of 3d ions in solids is mostly determined by the spin moments. The orbital momenta are totally or partially quenched by the crystal ﬁeld and/or the hybridization effects. In contrast, properties of the 4f shell electrons in the free tri-positive ions are relevant in the context of rare earth metals and their compounds, because the 4f states remain well localized in these materials. In fact the Russel–Saunders ground state's orbital and spin momenta are partially quenched to some extent by the crystalline environment. The orbital and spin contributions to the magnetic hyperﬁne ﬁeld for 4f electrons neglecting the core electrons polarization term (the Fermi contact term) are usually expressed [1,2] as

B4f =

n

− 2μ 0 μ B 4π

〈J||N||J〉〈r −3〉J

(1)

Corresponding author. Tel.:+966 59932 8857. E-mail addresses: [email protected], [email protected] (K. Ayuel).

http://dx.doi.org/10.1016/j.physb.2014.10.017 0921-4526/& 2014 Elsevier B.V. All rights reserved.

N=

⎛

∑ ⎜⎜li − si + 3 ⎝

(ri ·si ) ri ⎞ ⎟⎟ ri2 ⎠

(2)

li , si and ri are the orbital, spin angular momenta and position of electrons in the atom, respectively. Eq. (1) is providing good theoretical results that are close to experimental ones (see Table 1 of [3,4]). The main task of estimating the magnetic hyperﬁne ﬁeld from Eq. (1) is the evaluation of the expected value and reduced matrix of the operator of Eq. (2). Despite successful treatment of the magnetic hyperﬁne ﬁeld of tri-positive rare earth ions by Eq. (1), the problem of estimating this ﬁeld when the ions are subjected to crystalline electric ﬁeld has not been exhaustively treated. The aim of this paper is, ﬁrstly, to give an alternative form for estimating magnetic hyperﬁne ﬁeld of free tri-positive rare earth ions generated by orbital and spin current densities [5,6] and secondly, to extend this formula to the ground states of these ions in crystal ﬁeld eigenstates. We hope that this approach will simplify and clarify concepts and methods involved in the calculation of magnetic hyperﬁne ﬁelds by free ions as well as subjects in crystal ﬁelds. The remainder of the paper is organized as follows. In next three sections, we give a brief account of atomic current densities, magnetic ﬁelds and magnetic hyperﬁne ﬁelds. These sections will be followed by a section of the applications that describe how to apply the developed formalism to estimate magnetic hyperﬁne ﬁelds to some particular eigenstates of tripositive rare earth free ions as well as when they are subjected to

246

K. Ayuel et al. / Physica B 457 (2015) 245–250

crystal ﬁeld. Some concluding remarks will be presented in the last section.

2. Multipole expansion of the current density

μB ^ s (n) (2l + 1) 〈θJM J | j |θJM J′ 〉 = i ( − 1)l π ⎛ ⎞ l 3′ l ⎟ × ∑ ∑ ⎜⎜ ⎟ 34 3 ′= 3 ± 1 ⎝0 0 0 ⎠ ⎛ ⎞ ⎜ 3 ′ 1 3 ⎟ 23 ′ + 1 ⎜ 0 1 −1⎟ ⎝ ⎠ ⎧ ⎫ ⎪ d 1 1⎪ ⎬ 3 3 3 × ⎨ + + ′ − + [(2 1)( ) 3] ⎪ ⎪ r⎭ 2 ⎩ dr

∑ CM J |θJM J 〉.

′

(1, 3 ) 3 R nl2 〈θJM J |W4 |θJM J′ 〉Y 3 3 − 4.

(3)

MJ

Here CM J are some parameters resulting from crystal ﬁeld mixing of |θJM J 〉 eigenstates and |θ〉 =

|l nαLS〉.

The current density operator

within an {n, l} manifold can be expanded in terms of multipole component [7]:

^ j =

⎛ J J ⎞ 3 ⎜ ⎟ ⎜−M 4 M ′ ⎟ J J⎠ ⎝ of Eq. (7) require that 4 = M ′J − M J . The matrix elements of the spin current density can also take the following form:

It is well known that electric current loops produce magnetic ﬁeld. Electrons that are moving in atoms generate an average very tiny current loops that produce magnetic ﬁelds due to their orbital and spin currents [5,6]. General expressions for multiple expansion of the orbital and spin current densities using the irreducible tensor formalism and Racah algebra were presented for the free ions [7,8]. To generalize these multipole expansions (cf. Eqs. (48) and (49) of [8]) to crystal ﬁeld cases, the eigenvectors of the rare earth tri-positive ion in a crystal ﬁeld can be written in terms of spin–orbit coupled eigenstates |θJM J 〉 as

|ψ 〉 =

Also The 3j symbols

(3 )

∑ ( − 1)4 ^j4

Y3 3 − 4.

′

(1, 3 ) 3 Also the matrix elements 〈θJM J |W 4 |θJM J′ 〉 of the Racah double tensor can be evaluated by using the Winger–Eckart theorem:

^ (1, 3 ′) 3 〈θJM J |W4 |θJM J′ 〉 = ( − 1) J − M J (2J + 1) 23 + 1 ⎛ ⎞ 3 J ⎟ ⎜ J ⎜⎜−M 4 M ′ ⎟⎟ J J ⎝ ⎠

(4)

3 (3 )

^ Each component j4 is an irreducible tensor of rank 3 and order 4 . The expectation value of this current density of the state of Eq. (3) can be written in the form

^ 〈ψ | j |ψ 〉 =

∑ ∑ CM J CM ′J 〈θJM J |^j |θJM J′ 〉. M J M ′J

(5)

^ The matrix elements 〈θJM J | j |θJM J′ 〉 of the orbital current density

(8)

⎧S S 1 ⎫ ⎪ ⎪ ^ (1, 3 ′) × ⎨ L L 3 ⎬ 〈θ ∥ W ∥ θ〉 . ⎪ ⎪ ⎩ J J 3⎭

(9)

The 3j symbols

⎛ l 3′ l ⎞ ⎟ ⎜ ⎝0 0 0 ⎠

operator for n − electron in the ion can take the following form [7]:

^o (n) 〈θJM J | j |θJM J′ 〉 =

2 μB

∑ 34

of Eq. (8) require that 3′ be even and its value has to be within 0 ≤ 3 ′ ≤ 2l i.e. for the f function, 3′ = 0, 2, 4, 6 and 3 = 1, 3, 5, 7. ^ (k1, k1) The reduced matrix elements 〈θ ∥ W ∥ θ〉 can be written in LS terms of the coefﬁcients of fractional parentage G αα¯ LS ¯ ¯ [9,10] as

R nl2 ( − 1)l + 1i (2l + 1) r

( − 1)4

(2l + 3 + 2)(2l − 3 + 1) 4π

⎛ ⎞ l 3 l + 1⎟ (0, 3 ) 3 × ⎜⎜ 〈θJM J |W4 |θJM J′ 〉Y 3 3−4 0 ⎟⎠ ⎝0 0

^ (k1, k1) 〈θ ∥ W |θ〉 = n ( − 1) S + L + s + l + k1+ k 2 (2L + 1)(2S + 1) ×

(6)

∑ (Gα¯αLS¯LS¯ )2 ( − ) L¯ + S¯ ¯¯ α¯ LS

⎧ S k S⎫ ⎧ L k L⎫ 1 1 ⎨ ⎬ ⎨ ⎬ . ⎪ ⎪⎪ ⎪ ⎩ s S¯ s ⎭ ⎩ l L¯ l ⎭

where the matrix element of the Racah double tensor (0, 3 ) 3 〈θJM J |W 4 |θJM J′ 〉 can be evaluated by using the Wingner–Eckart theorem [7]:

⎪

⎪⎪

⎪

(10)

Here k1 = 0 or 1 and k2 = 3 or 3′. Substitution of Eq. (7) into Eq. (6) and Eq. (9) into Eq. (8) enable us to write the multipole ^ expansion of orbital jo = 〈θJM J | j |θJM J′ 〉 and spin current density ^ js = 〈θJM J | j |θJM J′ 〉 by

^ (0, 3 ) 3 〈θJM J |W4 |θJM J′ 〉 = ( − 1) J − M J (2J + 1) 23 + 1 ⎛ ⎞ 3 J ⎟ ⎜ J ⎜⎜−M 4 M ′ ⎟⎟ J J ⎝ ⎠ ⎧S S 0 ⎫ ⎪ ⎪ ^ (0, 3 ) × ⎨ L L 3 ⎬ 〈θ ∥ W ∥ θ〉 . ⎪ ⎪ ⎩ J J 3⎭

(2k1 + 1)(2k2 + 1)

jo (r) =

iμ B π

5

∑

∑

M ′J M J

CM J CM ′J α 3

R 42f (r) r

M J , M ′J 3 (odd) = 1

Y3 3 − 4 (θ , ϕ), (11)

(7)

The 3j symbols

⎛ l 3 l + 1⎞ ⎟ ⎜ ⎝0 0 0 ⎠ of Eq. (6) require that 3 be odd and its value to be within 1 ≤ 3 ≤ 2l − 1 i.e. for the f functions (l ¼3 and 3 = 1, 3 and 5).

and

js (r) =

iμ B π

⎛

7

∑

∑

M ′J M J ⎜ M ′J M J ⎜b 3

CM J CM ′J a 3

M J , M ′J 3 (odd) = 1

R 42f (r) Y 3 3 − 4 (θ , ϕ),

⎜ ⎝

M′ M J ⎞

c3 J d + dr r

⎟ ⎟⎟ ⎠ (12)

K. Ayuel et al. / Physica B 457 (2015) 245–250

Table 1 MJMJ

The coefﬁcients α1

MJMJ

, a1

MJMJ

MJMJ

, b1

and c1

′ are deﬁned [11] as harmonics Y llm of the states of rare earth ′ Y llm =

ions. State

MJ

Ce3 þ

2

7 5/2

MJMJ

7 3/2 7 1/2

Pr3 þ

74

3

H4

73 72 71

Nd3 þ

4

7 9/2

I 9/2

7 7/2 7 5/2 7 3/2 7 1/2

Pm3 þ

74

5

I4

73 72 71

Sm3 þ

6

7 5/2

H5/2

7 3/2 7 1/2

Eu3 þ

7

3þ

8

Gd

F0

MJMJ

MJMJ

(13)

MJMJ

α1

a1

b1

c1

10 6 ± 7 6 6 ± 7 2 6 ± 7

1 6 ± 14 3 6 ± 70 1 6 ± 70

3

4

3

4

3

4

12 6 5 9 6 5 6 6 5 3 6 5

4 6 675 1 6 225 2 6 675 1 6 675

58

39

58

39

58

39

58

39

63 6 22 49 6 ± 22 35 6 ± 22 21 6 ± 22 7 6 ± 22

1 6 242 7 6 ± 2178 5 6 ± 2178 1 6 ± 726 1 6 ± 2178

106

21

l′ and Pm ′ is an associated Legendre polynomial. Phase of the sphel′ rical harmonics Ym ′ is deﬁned as

106

21

l′ m ′ l′ Ym ′ = ( − 1) Y− m′.

106

21

106

21

106

21

14 6 5 21 6 10 7 6 5 7 6 10

2 6 495 1 6 330 1 6 495 1 6 990

125

21

125

21

125

21

125

21

1 6 378 1 6 ± 630 1 6 ± 1890

199

78

199

78

199

78

0

0

0

2 Y10 = − (8π)−1/2 (3 cos θ er − e z ).

7 6 6 5 6 ∓ 6 1 6 ∓ 2 1 6 ∓ 6

1

0

1

0

Here again er , e z , eϕ are the unit vectors in directions r, z and ϕ , respectively.

1

0

1

0

±

15 6 7 9 6 ± 7 3 6 ± 7 ±

0

0

7 7/2

0

7 5/2

0

7 3/2

0

7 1/2

0

S7/2

l′ ∑ Cllm ′m′1μ Y m′ e μ m′,μ

Ion

F 5/2

247

where Cllm ′m′1μ are Clebsch–Gordan coefﬁcient, e μ ( μ = − 1, 0, 1) are covariant spherical base vectors deﬁned in terms of Cartesian base vectors e x , e y and e z as

e−1 = (e x − ie y )/ 2 ;

e0 = ez ;

e1 = (e x + ie y )/ 2 .

(14)

0

The quantities Ylm0 are the spherical harmonics, they may be represented by products of two functions, one of which depends only on ϕ while the other depends on θ

⎡ ⎤1/2 l′ l′ m ′ (2l′ + 1)(l − m′) ! im ′ ϕ , Ym ⎥ Pm ′ (θ , ϕ) = ( − 1) ⎢ ′ (cos θ) e ⎣ 4π (l′ + m′) ! ⎦

(15)

±

⁎

±

(16)

′ In the notation of vector spherical harmonics Y llm , the index l indicates the multi-polarity in accordance with the triangle rule of the Clebsch–Gordan coefﬁcients. Index l′ has three values, l − 1, l and l + 1. This dose not apply to l ¼0, for which there is only one vector spherical harmonics, Y 100 . Note that in deﬁnition Eq. (13) the ﬁrst index of the spherical harmonics matches the upper index of the vector spherical harmonics, not the lower one, which determines its multi-polarity. The covariant spherical base vectors of Eq. (14) are readily transformed to Cartesian or polar ones, which enable easy visualization of lower-rank spherical harmonics vector ﬁelds. Here are the monopole and the dipole functions:

Y100 = − (4π)−1/2er ,

(17)

0 Y10 = (4π)−1/2e z ,

(18)

Y110 = i (3/8π)1/2 sin θ eϕ,

(19)

and

∓

(20)

3. The magnetic ﬁelds generated by current densities Having the angular dependence of the orbital and spin currents in terms of vector spherical harmonics (cf. Eqs. (11) and (12)) and using the expansion of (r − r′)/|r − r′|3 in terms of spherical harmonics, the magnetic ﬁeld generating by the current density j (r′),

M′ M J

M′ M J

M′ M J

M′ M J

where α3 J , a 3 J , b3 J and c3 J are coefﬁcients that are determined in a similar way as coefﬁcients of Tables 1 and 2 (cf. Section (6) [8]) and are tabulated in Tables 1 and 2 for the rare earth ions and for the magnetic hyperﬁne ﬁelds (when 3 = 1, M J = M ′J and 4 = 0) in Section 4. R 4f (r) is the radial wave function of the 4f electrons, Y 3 3 − 4 (θ , ϕ) are the vector spherical harmonics,

i=

−1 and μ B is the Bohr magneton. The vector spherical

B (r) =

μo 4π

′

∫ j (r ′) × |rr −− rr′|3 dr ′,

(21)

is easily calculated, especially due to the selection rules occurring in the integration of products of spherical harmonics. Indeed, the general expression for the magnetic ﬁelds generated by the atomic current densities was presented (see Eq. (21), [8]). In the crystal ﬁeld eigenvectors, this multiple expansion of magnetic ﬁelds can

248

K. Ayuel et al. / Physica B 457 (2015) 245–250

Table 2 Continuation of Table 1.

Table 2 (continued ) Ion

Ion

State

MJ

Tb3 þ

7

76

F6

75 74 73 72 71

Dy3 þ

H15/2

7 15/2

7 11/2 7 9/2 7 7/2 7 5/2 7 3/2 7 1/2

5

I8

78 77 76 75 74 73 72 71

Er3 þ

4

I15/2

7 15/2 7 13/2 7 11/2 7 9/2 7 7/2 7 5/2 7 3/2 7 1/2

Tm3 þ

3 6 2 5 6 4 6 3 6 4 1 6 2 1 6 4

MJMJ

MJMJ

State

MJ

b1

c1

1 6 18 5 6 − 108 1 6 − 27 1 6 − 36 1 6 − 54 1 6 − 108

17

3

17

3

17

3

17

3

17

3

17

3

1 6 18 13 6 ∓ 270 11 6 ∓ 270 1 6 ∓ 30 7 6 ∓ 270 1 6 ∓ 54 1 6 ∓ 90 1 6 ∓ 270

14

3

14

3

14

3

14

3

14

3

be represented as

14

3

J J B 34 (r) =

14

3

14

3

1 6 − 45 7 6 360 1 6 − 60 1 6 − 72 1 6 − 90 1 6 − 120 1 6 − 180 1 6 − 360

29

3

29

3

29

3

29

3

−

MJMJ

72 71

Yb3 þ

7 7/2

2

F 7/2

3

H6

76 75 74

5 6 2 13 6 ± 6 11 6 ± 6 3 6 ± 2 7 6 ± 6 5 6 ± 6 1 6 ± 2 1 6 ± 6 ±

3 6 21 6 8 9 6 4 15 6 8 3 6 2 9 6 8 3 6 4 3 6 8

±3 6 13 6 5 11 6 ± 5 ±

9 6 5 7 6 ± 5 ± 6 ±

3 6 5 1 6 ± 5 ±

5 6 2 25 6 12

±

∓

3 6 2 15 6 ± 14 9 6 ± 14 3 6 ± 14

7 1/2

μ0 ⎡ 1 3 ⎢ i ⎣ 23 + 1 r 3 + 2 3 + 1 3 −1 r 23 + 1

−

∫r

∫0 ∞

r

MJMJ

b1

c1

149

6

149

6

149

6

4

3

4

3

4

3

4

3

−

1 6 18 5 6 ∓ 126 1 6 ∓ 42 1 6 ∓ 126

±

7 5/2

M′ M

1 6 54 1 6 − 72 1 6 − 108 1 6 − 216

5 6 3 5 6 4 5 6 6 5 6 12

73

MJMJ

MJMJ

a1

α1

MJMJ

a1

7 3/2 4

7 13/2

Ho3 þ

MJMJ

α1

∓

M′ M

+1 j34J J (r′) r′3 + 2 dr′Y 3 34 (Ω)

⎤ M′ M −1 j34J J (r′) r′− (3 − 1) dr′Y 3 34 (Ω) ⎥ ⎦ (22)

where M′ M

j34J J (r′) =

iμ B R 42f (r′) π

r′

M ′J M J

CM J CM ′J α 3

(23)

and M′ M j34J J

(r′) =

iμ B π

⎛

M ′J M J ⎜ M ′J M J ⎜b 3

CM J CM ′J a 3

⎜ ⎝

M′ M J ⎞

c3 J d + dr′ r′

⎟ 2 ⎟⎟ R 4f (r′) ⎠

29

3

29

3

29

3

29

3

1 6 90 13 6 ∓ 1350 11 6 ∓ 1350 1 6 ∓ 150 7 6 ∓ 1350 1 6 ∓ 270 1 6 ∓ 451 1 6 ∓ 1350

47

6

47

6

47

6

B hf = B o + B s

47

6

47

6

where Bo and B s are respectively the orbital and spin contributions to the hyperﬁne ﬁeld Bhf . To determine expressions for these ﬁelds, the deﬁnition of the spherical vector waves

47

6

47

6

47

6

1 6 36 5 6 − 216

149

6

∓

−

are the multiple components of the orbital and spin current densities respectively deﬁned in Eqs. (11) and (12).

4. The magnetic hyperﬁne ﬁelds In the absence of an applied magnetic ﬁeld, the total ﬁeld exerted at the ion nucleus due to orbital and spin moments of its own electrons can be written as

⎛ 2 ⎞1/2 ′ Y l ′klm = k ⎜ ⎟ bl ′ (kr) Y llm (θ , ϕ) ⎝π ⎠

6

149

6

(25)

(26)

will be employed where bl ′ (kr) is the spherical Bessel function. The power series of the Bessel functions bl begins with the l-th power of the argument, so that

bl (kr) ∝ 149

(24)

(kr)l . (2l + 1) !!

(27)

As kr → 0, then, according to Eqs. (27) and (13) the only component of the magnetic ﬁeld that does not vanish at the origin is the

K. Ayuel et al. / Physica B 457 (2015) 245–250

one that is proportional to Y 10m . Higher-rank multipole ﬁelds go to zero. Also 4 = 0 terms appear in the diagonal terms of the matrix elements of the current (cf. Eqs. (11) and (12)). Hence, from Eq. (22) only the 3 = 1, 4 = 0, and M J = M ′J terms contribute to dipole(3 = 1) part of the magnetic hyperﬁne ﬁeld: M M B10J J (r)

2 3

= iμ 0

∞

∫r′ = 0

M M j10 J J

(r′)

0 dr′Y10 (θ ,

ϕ)

(28)

where according to Eqs. (23) and (24); M MJ

j10 J

(r′) =

iμ B

MJMJ

2 CM J α1

π

R 42f (r′) (29)

r′

(30)

for the spin current density. According to Eq. (28) the component of the multipole expansion of the orbital and spin contributions of the magnetic hyperﬁne ﬁeld can be expressed as MJMJ

=

−μ 0 μ B

2 2 MJMJ CM J α1 3

2π

∞

∫r = 0

R 42f (r′) r′

dr′e z

(31)

and M M Bs J J

=

−μ 0 μ B

2 2 MJMJ C M J a1 3

2π

MJMJ ⎞ ⎛ ⎜b M J M J d + c1 ⎟ 1 r′ = 0 ⎜ dr′ r′ ⎟⎠ ⎝

R 42f (r′) dr′ respectively.

μ B = 9.274 ×

Taking

10−24

A

m2

the

values

of

and a = 5.292 ×

μ0 = 4π × 10−7 T − 10 11 m . Noticing

2 2 MJMJ CM J α1 3

∫r

∞

646.85 902.01 519.40

43.12 20.04 57.71

689.98 881.97 461.69

and

∑ B Ms J M J . (38)

R 42f (r′) r′

dr′e z

5. Application to some rare earth free ions and crystal ﬁeld eigenstates To illustrate application of the formalism which we have developed so far we will estimate the magnetic hyperﬁne ﬁelds for the Mössbauer free ions Dy3 þ , Er3 þ , Yb3 þ and the crystal ﬁeld ground eigenstates of Er3 þ and of Yb3 þ ions in Er2Ge2 O7 and YbNi5 respectively.

(33)

4

R 4f = r 3 ∑ Ci e−Zi r

MJMJ

Bo

= 12.515

2 2 M J M J −3 CM J α1 〈r 〉e z 3

MJMJ

Bs

= 12.515

2 2 MJMJ C M J a1 3

(34)

∫

(35)

or

= 12.515

2 2 M J M J M J M J −3 C M J a1 c1 〈r 〉e z 3

(36)

for spin ﬁeld. The ﬁrst term of Eq. (35) gives a negligible contribution. Having the components of the multipole expansion of the orbital and spin hyperﬁne ﬁelds (cf. Eqs. (33)–(36)), the orbital and spin contribution to magnetic hyperﬁne ﬁelds can be written as

∑ MJ

together with the value of the parameters α1 , b1 and c1JJ 15 15 3 + 3 + from Table 2 of ions Dy (M J = J = 2 ), Er (M J = J = 2 ) and JJ

a1JJ,

JJ

7

MJMJ ⎞ ⎛ ∞ ⎜b M J M J d + c1 ⎟ 1 r=0⎜ dr′ r′ ⎟⎠ ⎝

R 42f (r′) dr′e z

MJMJ

(39)

i=1

for orbital ﬁeld and

Bo =

Dy3 þ Er3 þ Yb3 þ

and the values of the coefﬁcients C J are equal to 1. Consequently, again, there is only one term to be considered in the summations of Eqs. (37) and (38), namely BoJJ and B sJJ . These terms correspond to the coefﬁcients α1JJ, a1JJ, b1JJ and c1JJ that are the ﬁrst row entries in Tables 1 and 2. If expressions of the radial wave functions are given, then there is no obstacle to evaluate the integral of Eqs. (33) and (35). As an example the approximate radial wave functions [12]

that

or

Bs

Bhf

m/A ,

terms of Tesla for B and meter for r when considering the most common experimental setting of external magnetic ﬁeld

= 12.515

Bs

For the free ions, there is only one term in the summation of Eq. (3) that corresponds to the full strength state |ΘJM J = J 〉 of an ion,

∞

MJMJ

Bo

(32)

∫r = 0 (R2 (r)/r) dr = 〈r −3〉 then Eqs. (31) and (32) can be rewritten in

Bo

ion

5.1. The magnetic hyperﬁne ﬁelds of the Mössbauer free ions Dy3 þ , Er3 þ and Yb3 þ

∞

∫

wave functions of Eq. (37) or 〈r −3〉 of [12], Eqs. (34) and (36) for rare earth Dy3 þ , Er3 þ and Er3 þ ions.

MJ

M M ⎞ iμ B 2 M J M J ⎛⎜ M J M J d c1 J J ⎟ 2 M M j10 J J (r′) = C M J a1 b + R 4f (r′) 1 ⎜ π dr′ r′ ⎟⎠ ⎝

Bo

Table 3 Hyperﬁne ﬁelds Bo , B s and Bhf in tesla calculated using Eqs. (33), (35) and the radial

Bs =

for the orbital current density and

249

M M Bo J J

(37)

Yb3 +(M J = J = 2 ) can be taken as inputs to Eqs. (33) and (35). The estimated orbital, spin and total magnetic hyperﬁne ﬁelds Bo , B s and Bhf for these ions are presented in Table 3. The same results can also be obtained using Eqs. (34) and (36) together with the values of 〈r −3〉 tabulated in [12]. Evaluating Eqs. (34) and (36) with more accurate values of 〈r −3〉 that are presented in [13] give the orbital, spin and total magnetic hyperﬁne ﬁelds in Table 4. The other estimated magnetic hyperﬁne Bhf [1] in the literature [1] of these ions are also presented in this table for compression. Table 4 Hyperﬁne ﬁelds Bo , B s and Bhf in tesla calculated using Eqs. (34), (36) and 〈r −3〉 of [13] and the estimated magnetic hyperﬁne ﬁeld Bhf of [1] for rare earth Dy3 þ , Er3 þ and Er3 þ ions. Ion

Bo

Bs

Bhf

Bhf [1]

Dy3 þ Er3 þ Yb3 þ

609.61 853.02 492.59

40.64 18.95 55.73

650.25 834.07 437.86

650.0 834 438

250

K. Ayuel et al. / Physica B 457 (2015) 245–250

5.2. The magnetic ﬁelds of Er3 þ ion in Er2Ge2 O7 and of Yb3 þ ion in YbNi5 The ground state of Er3 þ ion in Er2Ge2 O7 is the Kramer's doublets [14]:

|ψ0 〉 = ∓ 0.903| ±

11 〉 2

9

1

± 0.176| ∓ 2 〉 + 0.39| ± 2 〉.

(40)

The values of the coefﬁcients required to evaluate the magnetic MJMJ

hyperﬁne ﬁeld in this state are the values of α1 M M c1 J J

for M J =

11 , 2

−9 2

and

1 2

MJMJ

, a1

MJMJ

, b1

,

in Table 2 for Er3 þ . Substitution of

these values together with the coefﬁcients CM J of Eq. (40) and 〈r −3〉 of [13] into Eqs. (34) and (36) and the resulting values into Eqs. (37) and (38) yields

B o = 477 T,

(41)

B s = − 10.60 T,

(42)

and

B hf = 466.4 T.

(43)

The experimental value for the magnetic hyperﬁne ﬁeld of this compound is 440 T [14], which differs from our result by 6%. Repeating the same procedure for Kramer's doublet [15], the ground state of Yb3 þ ion in YbNi5 is 7

5

|ψ0 〉 = ∓ 0.985| ± 2 〉 + 0.173| ∓ 2 〉.

(44)

The resulting orbital, spin and total magnetic hyperﬁne ﬁeld are

B o = 467.4 T,

(45)

B s = − 51.9 T,

(46)

and

B hf = 415.5 T

(47)

respectively. The experimental reported value for the magnetic hyperﬁne ﬁeld in this state is 399.0 T [15] differing from our result by 4%.

6. Remarks and conclusion

those obtained using the 〈r −3〉 of [13]. The difference between the two results can be attributed to the fact that the estimation of 〈r −3〉 of [13] was based on the accurate, fully relativistic Dirac–Fock approach while the derivation of the radial wave function (39) was based on the non-relativistic Hartree–Fock approach [13]. Our results of total magnetic hyperﬁne ﬁeld Bhf listed in Table 4 are in full agreement with those Bhf of [1]. The results of the last two presented examples show that our results are in good agreement with measured and estimated magnetic hyperﬁne ﬁelds when the rare earth tri-positive ions are subjected to a crystal ﬁeld environment. So apart of producing the same theoretical results of rare earth tri-positive free ions of Eq. (1), our formalism can be extended to give correct predictions of magnetic hyperﬁne ﬁeld produced by the ground state of crystal ﬁeld eigenstates. In conclusion, we have presented an alternative method of estimating the magnetic hyperﬁne ﬁeld of some tri-positive rare earth ions in their free ions and crystal ﬁeld ground eigenstates. The merit of this method is that it can simplify the problem of estimation of the magnetic hyperﬁne ﬁeld drastically for rare earth tri-positive free ions and crystal ﬁeld ground states, by using the M M M M M M M M coefﬁcients α1 J J , a1 J J , b1 J J , and c1 J J of Tables 1 and 2 with Eqs. (34), (36)–(38). The accuracy of the results derived by our method depends on the accuracy of values of 〈r −3〉 and CJ of Eq. (3).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Tables 3 and 4 reveal that the magnetic hyperﬁne ﬁelds obtained using the radial wave functions (39) are less accurate than

G. Netz, Z. Phys. B 63 (1986) 343. M. Forker, Hyperﬁne Interactions 24–26 (1985) 907. R.J. Elliott, K.W.H. Stevens, Proc. Phys. Soc. A 218 (1953) 553. A. Abragam, M.H.L. Pryce, Proc. Phys. Soc. A 205 (1951) 135. W. Gough, Eur. J. Phys. 17 (1996) 208. K. Ayuel, P.F. de Châtel, Eur. J. Phys. 20 (1999) 53. K. Ayuel, Atomic current densities and magnetism (thesis), Amsterdam University, 2000. K. Ayuel, P.F. de Châtel, Physica B 404 (2009) 1209. B.R. Judd, Operator Techniques in Atomic Spectroscopy, Princeton University Press, New Jersey, 1998. I. I Sobeĺman, Introduction to the Theory of Atomic Spectra, Pergamon Press, Oxford, 1972. D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientiﬁc, Singapore, 1988. A.J. Freeman, R.E. Watson, Phys. Rev. 127 (1962) 2058. A.J. Freeman, J.P. Desclaux, J. Magn. Magn. Mater. 12 (1979) 11. A. Chattopadhyay (née Nag), S. Jana, D. Ghosh, E. Gmelin, J. Magn. Magn. Mater. 224 (2001) 153. J.A. Hodes, P. Bonville, M. Ocio, Eur. Phys. J. B 57 (2007) 365.