Vij D.R. Handbook of Applied Solid State Spectroscopy

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6.2 The Crystal Field Interaction

adjustable parameters. For low symmetry sites such as orthorhombic

symmetry with nine independent crystal field parameters (see equation (6.12)),

it is a highly nontrivial task in the least squares fitting procedure to the

experimental data to find a correct set of starting parameters. However, we

can make use of the fact that the crystal field parameters

for a particular

degree n can reasonably be determined by taking account of the geometry of

the nearest neighboring coordination polyhedra (see equation (6.18)):

BB /

nnnn

ff (6.30)

i.e., we are left with only three independent crystal field parameters

(n = 2, 4, 6). Following [12] we introduce a parameterization which covers all

possible ratios of

and B

BF (1 | |)

Wxy



(6.31)

with –1  x, y 1. Equation (6.31) corresponds to the most general

combination of second-, fourth- and sixth-order diagonal crystal field

parameters. Usually only one set of x, y parameters gives a reasonable

agreement with the spectroscopic data, thus we have solved the problem of

assigning the leading diagonal start parameters of the least squares fitting

procedure in a reasonable manner. In a second step, we can then try to achieve

a fine-tuning of the remaining nondiagonal crystal field parameters.

An alternative way to derive a starting set of crystal field parameters is to

consider the extended point-charge model (see Section 6.2.2.2), provided that

the nearest-neighbor point charges, as well as the screening length (in metallic

systems), are known.

6.2.4 Extrapolation Schemes

In many models the structure and charge aspects of the crystal field

interaction can be separated as exemplified for the point-charge model by

equation (6.18) and for the superposition model by equation (6.25). This

offers a convenient extrapolation scheme for the crystal field potential along a

series of rare earth compounds provided that the charge distribution of the

coordinating ligand ions is not affected by the particular rare earth ions. The

procedure for extrapolating the crystal field parameters from one compound

) to another compound (R

) based on equation (6.18) yields:



(R )

B(R) B(R)

(R )

FÃ Ó

(6.32)

267

6. Crystal Field Spectroscopy

respectively; see Section 4.1 for Cs

NaRBr

6.2.5 Calculation of Thermodynamic Magnetic Properties

In order to check the reliability of the crystal field parameters it is

important

to calculate various thermodynamic magnetic properties and

to compare the

results

with experimental data. These properties depend

explicitly upon both

the

crystal field energies E

and the crystal field

wavefunctions |*

Ó .

Based

on general expressions of statistical

mechanics for the Gibbs free energy

F = – k

T ln Z and the internal

energyU = F–T(wF/wT )

, where Z is

function, we obtain

for the magnetization:

1(ln)

M|J|,

ii i



P**



(6.33)

for the single-ion susceptibility:

||J||

M||J||

() ( ),

iji

iij

gppp

kT E E



ÌÜ

Ã* * Ó

Ã**Ó

F P  

ÌÜ



ÌÜ

ÍÝ

ÇÇ

(6.34)

and for the Schottky heat capacity:

ck pp

TkTkT

ËÛ

ÈØ È Ø



ÈØ

ÌÜ



ÉÙ

ÉÙ É Ù

ÊÚ



ÊÚ Ê Ú

ÌÜ

ÍÝ

ÇÇ

(6.35)

with the Boltzmann population factor

exp .

ÎÞ



Ïß

Ðà

(6.36)

6.3 EXPERIMENTAL TECHNIQUES

6.3.1 Introductory Remarks

The crystal field interaction discussed in the preceding section gives rise to

discrete energy levels E

and wavefunctions |

T Ó which can be determined by

a variety of experimental methods. The most powerful techniques are

certainly spectroscopic methods such as inelastic neutron scattering, light

scattering, and point-contact spectroscopy, which allow a direct determination

of the crystal field states. Light scattering offers several methods working

This procedure was successfully applied to a large number of both insulating

and metallic rare earth (R) compounds, e.g., for Cs

NaRBr

[25] and for RPd

[26],

268

the parti-

tion

6.3 Experimental Techniques

with different modes of operation at different wavelengths, e.g., Raman,

infrared, photoemission, X-ray absorption, and luminescence spectroscopy.

Some information about the crystal field states can also be obtained by

resonance techniques, e.g., in a paramagnetic resonance experiment one is

measuring the Zeeman splitting of a given ground state crystal field level,

which has seen extensive applications in rare earth salts [1]. Sometimes it is

possible to use the Mössbauer effect to observe a zero-field magnetic

hyperfine splitting associated with the rare earth ions yielding valuable

information about the crystal field parameter

[27], which is difficult to

determine by model calculations due to the long-range nature of the 2

order

crystal field potential. Information on the crystal field states is also contained

intrinsically in the thermodynamic magnetic properties (see equations (6.33)–

(6.36)); however, an extraction of crystal field parameters is likely to fail (as

demonstrated in Section 6.4.1) due to the integral nature of these properties.

In this work we will describe only a few spectroscopic techniques that have

been applied to the study of crystal field effects. These include primarily

inelastic neutron scattering and to a lesser extent Raman scattering and point-

contact spectroscopy. Both inelastic neutron and Raman scattering

experiments have their merits and should be considered as complementary

methods. Raman scattering, on the one hand, can be applied to very small

samples of the order of 10 Pm

; it provides highly resolved spectra so that

small line shifts and splittings can be detected [28], and it covers a large

energy range so that intermultiplet transitions can easily be observed [29].

Neutron scattering, on the other hand, is not restricted to particular points in

reciprocal space, i.e., interactions between the rare earth ions can be observed

through the wavevector dependence [30], the intensities of crystal field

transitions can easily be interpreted on the basis of the wavefunctions of the

crystal field states, and data can be taken over a wide temperature range,

which is important when studying linewidths of crystal field transitions.

Inelastic neutron scattering as the most widely used spectroscopic technique is

described below in detail, followed by short descriptions of Raman and point-

contact spectroscopy.

6.3.2 Neutron Spectroscopy

The principal aim of a neutron scattering experiment is the determination of

the probability that a neutron incident on the sample with wavevector k is

scattered into the state with wavevector

'. The intensity of the scattered

neutrons is thus measured as a function of the momentum transfer

(),==

Q= k k

(6.37)

where Q is known as the scattering vector, and the corresponding energy

transfer is given by

269

6. Crystal Field Spectroscopy

2'2

().

Z 

(6.38)

Equations (6.37) and (6.38) describe the momentum and energy conservation

of the neutron scattering process, respectively. The momentum conservation

is schematically sketchedin Figure 6.2. For k =

' we have fromequation (6.38)

0 Z=

, i.e., elastic scattering. Figure 6.2a shows the particular situation

Q = k–k' =

W, which is just the condition known as Bragg’s law. Inelastic

scattering is visualized in Figure 6.2b where the scattering vector can be

decomposed according to Q =

W+ q, with q being the wavevector of an

elementary excitation to be specified. Neutron scattering turns out to be the

only experimental technique that is able to measure the dispersion relation



Z(q)

at any predetermined point in reciprocal space.

Figure 6.2 Visualization of equation (6.37) in reciprocal space for elastic (a) and inelastic (b)

neutron scattering. The lines indicate the boundaries of the Brillouin zone, and the full circles

denote the zone centers.



controlled access to the variables Q and Z. This can be done in various ways,

but by far the most effective experimental method is the triple-axis crystal

spectrometry developed by Brockhouse [31] who received the 1994 physics

Nobel Prize for this achievement. The principle of this method is sketched in

Figure 6.3. An incident beam of neutrons with a well-defined wavevector k is

selected from the white spectrum of the neutron source by the monochromator

crystal and scattered from the sample. The intensity of the scattered beam is

measured as a function of

' by the analyzer crystal and the neutron detector.

The outstanding advantage of the triple-axis spectrometer is that data can

be taken at predetermined points in reciprocal space (which is known as the

“constant-Q” or “constant-Z” method), so that single-crystal measurements of



course, general scans in Q and Z are also possible.

The determination of Z(q) by neutron scattering techniques requires a

the dispersion relation

Z(Q) can be performed in a controlled manner. Of

270

Figure 6.3 Basic layout of a triple-axis spectrometer. The three axes around which the

respective rest of the spectrometer is rotated are: the monochromator axis (variation of k), the

Figure 6.4 Schematic of a direct time-of-flight spectrometer. The four choppers define the

energy and the pulse width of the incident neutrons. In addition, they suppress higher order

neutrons and frame overlap.

For neutron scattering experiments on polycrystalline, liquid, and

amorphous materials, various types of time-of-flight spectrometers are usually

more appropriate. In the time-of-flight method the neutron beam is

monochromatized by a series of choppers that produce pulses of neutrons with

the desired wavelength and that eliminate higher order neutrons and frame

overlap of pulses from different repetition periods as well (see Figure 6.4).

The monochromatic neutron pulses are then scattered from the sample and are

detected by arrays of neutron counters covering a large solid angle. The

energy transfer

Z= and the modulus of the scattering vector Q are then

determined by the flight time of the neutrons from the sample to the detector

and by the scattering angle at which the detector is positioned, respectively.

sample axis (variation of the scattering angle), and the analyzer axis (variation of k').

6.3 Experimental Techniques

271

6. Crystal Field Spectroscopy

The neutron scattering cross-section corresponds to the number of neutrons

scattered persecondintoa(small) solid angle d: with energy transfers between

Z= and ()dZZ= , divided by the flux of the incident neutrons. Theoretical

expressions for the cross-section usually start from Fermi’s “golden rule”:

|','|U |k, | ( ).

pk EE

dd k

OOO

ÈØ

ÃOOÓGZ

ÉÙ

ÊÚ

:Z S

ÇÇ

(6.39)

Here,

| OÓ denotes the initial state of the scatterer, with energy E

and thermal

population factor

, and its final state is |'O Ó .

is the interaction operator

of the neutron with the sample, which depends on the specific scattering

process. For magnetic scattering the interaction operator Û

of the neutron

with the sample may be described by the interaction of a neutron with a

magnetic field H:

ˆˆ

JP V ¹ ¹H,P + 

(6.40)

where

the

gyromagnetic ratio,

= 0.505·10

–19

erg·T

–1

the nuclear magneton, and

Pauli spin operator. For a large class of magnetic compounds

the magnetic

field

used in equation (6.40) is generated by unpaired

electrons. The

magnetic field due to a single electron moving with

velocity

is given

by:

curl ,

ÎÞ

S 



Ïß

Ðà

RvR

|R| |R|

(6.41)

where R is the distance from the electron to the point at which the field is

measured, e = 1.602·10

–19

C the elementary charge, and c = 3·10

cm·s

–1

the

velocity of light. The magnetic moment operator of an electron is

PP  s

(6.42)

where

= 0.927·10

–16

erg·T

–1

is the Bohr magneton and

the spin

operator

of the electron. The first term of equation (6.41) arises from

the spin of

the

electron, and the second from its orbital motion.

The major task in the evaluation of the magnetic neutron cross-section is

the calculation of the transition matrix element in equation (6.39). This was

first done by Halpern and Johnson [32], and more recently in many excellent

text books on neutron scattering [33, 34]. For unpolarized neutrons, identical

magnetic ions with localized electrons, and spin-only scattering the following

“master formula” is obtained:

() ()exp{2()} S(,),

rF W

dd k

JGZ

ÈØ



ÉÙ

ÊÚ

QQ Q

(6.43a)

where S

(Q, Z) is the magnetic scattering function:

ij i

ˆˆ

S(, )= exp{ ( )} |S|' '|S|

().

DE D

ZOOOO

¹

¹

ÇÇ

QQRR

(6.43b)

is the magnetic moment operator of the neutron, J = –1.91

272

6.3 Experimental Techniques

= 0.282·10

–12

cm is the classical electron radius, F(Q) the dimensionless

magnetic form factor defined as the Fourier transform of the normalized spin

density associated with the magnetic ions, exp{–2W(Q)} the Debye-Waller

factor, and

(D = x,y,z) the spin operator of the ith ion at site R

. From the

magnitude of r

we expect the magnetic neutron cross-section to be of the

order 10

–24

–2

The essential factor in equation (6.43) is the magnetic scattering function

(Q,Z) which will be discussed in more detail below. There are two further

factors that govern the cross-section for magnetic neutron scattering in a

characteristic way: First, the magnetic form factor F(Q) which usually falls

off with increasing modulus of the scattering vector Q. Second, the

polarization factor (G

–Q

) tells us that neutrons can only couple to

magnetic moments or spin fluctuations perpendicular to Q, which un-

ambiguously allows us to determine moment directions or to distinguish

between different polarizations of spin fluctuations.

Equation (6.43) strictly applies to cases where the orbital angular

momentum of the magnetic ions is either zero or quenched by the crystal field.

A theoretical treatment of the scattering by ions with unquenched orbital

moment was given by Johnston [35], however, the calculation is complicated,

and we simply quote the result for Qo0. In this case the cross-section

measures the magnetization,

P = –P

(L + 2S), i.e., a combination of spin and

orbital moments that does not allow their separation. This clearly contrasts to

magnetic scattering by X-rays. For magnetic neutron scattering an

approximate result can be obtained for modest values of Q. We replace the

spin operator

in equation (6.43) by:

ĮĮ

(6.44)

where

J(J + 1)

L(L + 1) + S(S + 1)

2J(J + 1)

(6.45)

is the Landé splitting factor and

is an effective angular momentum

operator (e.g., for rare earth ions J is the total angular momentum quantum

number resulting from the spin-orbit coupling which combines the spin and

orbital angular momentum L and S, respectively).

If Z is a positive quantity in the scattering function S

(Q,Z) of equation

(6.43), the neutron loses energy in the scattering process and the system is

excited from the initial state

| O which has energy

less than the final state

|'O . Consider now the function S

(Q,–Z) where Zҏ is the same positive

quantity. This represents a process in which the neutron gains energy. The

transitions of the system are between the same states as for the previous

process, but now

|'O is the initial state and | O is the final state. The

probability of the system being initially in the higher state is lower by the

273

6. Crystal Field Spectroscopy

factor

exp{-hȦ/

}

kT than its probability of being in the lower energy state,

hence

Įȕ Įȕ

S(, ) exp S(,) ,

½

Z  Z

®¾

¯¿

(6.46)

which is known as the principle of detailed balance. Equation (6.46) must be

applied in the analysis of experimental data taken in both energy-gain and

energy-loss configurations which correspond to the so-called Stokes and anti-

Stokes processes, respectively.

Using the integral representation of the G-function,

()

()exp exp,

EEt

EE i itdt

GZ Z







ÎÞ

 

Ïß

Ðà

(6.47)

the scattering function S

(Q,Z) (equation (6.43)) transforms into a physically

transparent form:

ˆˆ

S(,) exp ( )S(0)S()exp .

iji

ititdt

DE D





¹ 

QQRR

(6.48)

S(0)S()

is the thermal average of the time-dependent spin operators. It

corresponds to the van Hove pair correlation function [36] and gives

essentially the probability that, if the magnetic moment of the ith ion at site R

has some specified (vector) value at time zero, then the moment of the jth ion

at site R

has some other specified value at time t. A neutron scattering

experiment measures the Fourier transform of the pair correlation function in

space and time, which is clearly just what is needed to describe a magnetic

system on an atomic scale.

The van Hove representation of the cross-section in terms of pair

correlation functions is related to the fluctuation-dissipation theorem [33, 34]:

S(,) 1exp Im (,),

DE DE

ZFZ



ÈØ

ÎÞ



Ïß

ÉÙ

ÊÚ

Ðà

(6.49)

where N is the total number of magnetic ions. Physically speaking, the

neutron may be considered a magnetic probe, which effectively establishes a

frequency- and wavevector-dependent magnetic field H

(Q,Z) in the

scattering sample, and detects its response, M

(Q,Z), to this field by

Q,Z



Q,Z



Q,Z



(6.50)

where F

(Q,Z) is the generalized magnetic susceptibility tensor. This is

really the outstanding property of the neutron in a magnetic scattering

measurement, and no other experimental technique is able to provide such

detailed microscopic information about magnetic compounds.

If the coupling between the magnetic ions is weak, we are left with a

single-ion problem, thus the excitation energies will be independent of the

scattering vector Q. Typical examples are rare earth compounds that exhibit

274

6.3 Experimental Techniques

dominant mechanism is the crystal field interaction.

The effect of the crystal field on a rare earth ion is to partially or totally

remove the (2J + 1)-fold degeneracy of the ground state J-multiplet. This is

exemplified at the top of Figure 6.5 for Pr

ions in the hexagonal compound

PrBr

[37]. The crystal field levels resulting from the Hamiltonian (11) are

denoted by the irreducible representations *

, and the corresponding wave-

functions are:

|(M)|M.

(6.51)

From the sequence of the energy levels, properly identified by their

irreducible representations *

, the crystal field potential can be

unambiguously determined.

In evaluating the cross-section for the crystal field transition *

start from the scattering law S

(Q,Z) defined by equation (6.43). Since we

are dealing with single-ion excitations, we have i = j, so that the Q-

dependence vanishes. For N identical magnetic ions we can even drop the

index i. S

(Q,Z) then reduces to:

where N is the total number of magnetic ions and

the Boltzmann

population factor. From the symmetry relations associated with the matrix

elements we find the cross-section





()exp 2 ()

1|||| .

Ngr F W p

dd k

**GZ

ÈØ



ÉÙ

ÊÚ

ÈØ



ÉÙ

ÊÚ

(6.53)

The polarization factor permits discrimination between transverse (D= x,y)

and longitudinal (D = z) crystal field transitions by measuring at different Q.

This is nicely demonstrated in Figure 6.5. For Q||c only transverse transitions

are observed, whereas for QAc the transverse transitions lose half their

intensities, and in addition longitudinal transitions appear. The lines in

Figure 6.5 were calculated without any disposable parameters, i.e., the

intensities of the crystal field transitions are excellently described by equation

(6.53).

For experiments on polycrystalline material, equation (6.53) has to be

averaged in Q space:





()exp 2 () | | | |

Ngr F W p

dd k

J**

ÈØ



ÉÙ

ÊÚ

=

QQ J

(6.54)

very low magnetic-ordering temperatures or do not order at all. In this case the





ˆˆ

S() J J ,

n nm

nmmn

Np E E

αβ α β

ΓΓΓ

ωΓΓΓΓδω= 

(6.52)



275

6. Crystal Field Spectroscopy

where J

= J–(J·Q)Q/Q

is the component of the total angular momentum

perpendicular to the scattering vector Q, and

ˆˆ

|||| ||||.

mn mn

** **

(6.55)

These matrix elements have been tabulated by Birgeneau [38] for cubic

crystal fields.

Figure 6.5 Energy spectra of neutrons scattered from single-crystalline PrBr

at T=1.5 K for Q

parallel and perpendicular to the c-axis. The resulting crystal field level scheme and the

observed transverse (t) and longitudinal (l) ground state transitions are indicated at the top (after

[37]).

6.3.3 Raman Spectroscopy

Raman spectroscopy utilizes light scattering to gain information about

elementary excitations of matter such as phonons, magnons, and electronic

single particle or collective excitations within a sample (for a review and

references see, e.g., [39]). When a photon (usually produced from a laser in

the visible, near infrared, or near ultraviolet range) interacts with a solid it can

scattering); ii) inelastically scattered by quasi-particle excitations (Raman

process), thereby either giving energy to the medium (Stokes scattering) or

removing energy from it (anti-Stokes scattering). A Raman process arises

be: i) elastically scattered and thus retain its incident energy (Rayleigh

276