sample synthesis, experimental techniques, and...

Chapter 2

Sample Synthesis, Experimental Techniques, and

Data-analysis Methodologies

This chapter deals with the description of sample preparation and various experi-

mental and computational techniques used to investigate the present thesis work.

The structural properties of the samples have been investigated by various tech-

niques such as X-ray and neutron diffraction, and differerential scanning calorimetry.

Bulk magnetic and magnetocaloric properties have been studied by dc magnetization

as a function of temperature and magnetic field. For microscopic understanding of

magnetic properties, temperature dependent neutron diffraction experiments have

been carried out. Mesoscopic technique neutron depolarization has been used to

study the magnetic properties of domains/clusters. For analyzing the X-ray and

neutron diffraction data, the Reidveld refinement method has been employed.

2.1 Sample synthesis

Polycrystalline samples of all intermetallic compounds investigated in this thesis

work were prepared by an arc-melting method. In this method, constituent elements

of high purity are taken in an appropriate stoichiometric ratio, and kept in a copper

crucible, which has a continuous supply of cold water to prevent melting of the

crucible. Some of the elements (e.g. Tb, Mn, Sb) were taken in excess amount to

compensate the evaporation losses. The furnace chamber is first maintained under

41

Chapter 2. Sample Synthesis, Experimental Techniques...... 42

a high vacuum of 10−3 torr and then flushed with an argon gas. This procedure is

repeated 2-3 times to ensure that no trapped gas molecules are present inside the

chamber. When the thoriated tungsten cathode is brought close to the anode(copper

crucible), an ionized argon gas in the chamber leads to an arc, which is used to

melt the elements under an argon atmosphere. The temperature of the arc can be

controlled by varying the current between two electrodes. In the set-up used for the

present thesis work, the current can be varied up to 270 A. The samples were re-

melted for four times to achieve a better chemical homogeneity. After melting, the

samples were annealed in vacuum-sealed quartz tubes at specific temperature for a

fixed duration. The annealing conditions vary for different intermetallic compounds.

The annealing of the samples was done to eliminate the possibility of any segregation

of other phases, by solid state diffusion.

2.2 Experimental techniques

2.2.1 X-ray diffraction

X-ray diffraction technique has been used to study the structural properties of the

intermetallic compounds. X-ray diffraction is based on the elastic scattering of X-

rays from the electron clouds of the individual atoms in the system, and yields the

atomic structure of materials. The diffraction pattern occurs due to the periodic

arrangements of atoms in space according to the Bragg’s Law

2dhklsinθhkl = nλ (2.1)

Here, θhkl is the Bragg angle, dhkl is the distance between two parallel atomic planes

for a given (hkl), λ is the wavelength of the X-ray, and n is an integer [36]. The

lattice parameters can be calculated from the position of the Bragg peaks [37]. A


Bragg reflection can occur when λ ≤ 2d which means that the wavelength of the X-

ray should be of the order of the interatomic spacing. The X-rays are produced when

the high energy electron beam is bombarded on a metal target. For crystallography,

Cu is the most common metal target(which is also used in the present thesis work)

producing a wavelength i.e Cu (Kα) with λ = 1 .54187 A. The intensity of the

diffracted beam for a Bragg peak with Miller indices (hkl) is proportional to the

square of the structure factor F hkl, where

Fhkl =N∑

n=1

fn e2πi (hun+kvn+lwn) (2.2)

Here, f n is the atomic scattering factor for the nth atom in an unit cell, un, vn, wn

are the fractional atomic coordinates of an atom/ion within an unit cell, N is the

total number of atoms in an unit cell.

2.2.2 Neutron diffraction

Neutron diffraction experiments have been carried out at various temperatures, from

room temperature to down to 5 K, to study the structural and magnetic proper-

ties of the intermetallic compounds. The neutron diffraction patterns were recorded

using the powder diffractrometer-II at the Dhruva reactor, Trombay, Mumbai, In-

dia (Figs. 2.1 and 2.2)[38]. The neutron wavelengths used were either λ = 1.249 or

1.2443 A. The scattering angular range of 5◦−130◦ is scaned by five one-dimentional

position sensitive dectetors with a resolution ( ∆d/d) of ∼ 1 %. The neutron flux

at the sample position is typically of the order of 8 × 105 neutrons cm−2 s−1.

Temperature-dependent neutron diffraction patterns down to 1.5 K for one of the

samples (Cu0.85Ni0.15MnSb) were collected at the DMC cold neutron powder diffrac-

tometer (λ = 2.568 A), Paul Scherrer Institute, Switzerland. All other measurements


were carried out using the powder diffractrometer-II at the Dhruva reactor.

Figure 2.1: Schematic of the powder diffractometer II at Dhruva reactor, Trombay[38].

Figure 2.2: Photograph of the powder diffractometer II at Dhruva reactor, Trombay[38].

Unlike the X-rays, the neutron interacts with the nucleus of an atom. In case of


magnetic materials, the electronic scattering (due to magnetic interaction) is also

appreciable. The de Broglie wavelength of the thermal neutron is comparable to

the interatomic spacing (∼1 A) in solids, and hence thermal neutron diffraction is

suitable to study the crystal structure of solids. The physical properties of neutron

that make the neutron scattering complementary to X-ray scattering are:

(i) The neutron scattering is isotopic in nature as the nuclear dimension ( 10−15

m) is very small in comparison to wavelength of the thermal neutrons ( 10−10 m).

Therefore, in case of neutron-nuclear scattering, the scattering centers are point

particles. Hence, the neutron-nucleus scattering amplitude does not vary with the

scattering angle, whereas in the case of X-rays, the scattering is from the electron

clouds and the corresponding X-ray scattering amplitute shows a scattering vector

dependency.

(ii) The nuclear scattering cross-section for an atom is given by b2, where b is

nuclear scattering length. As b depends on the nuclear interaction, this scattering

length varies randomly across the periodic table and thus enables us to distinguish

between two nearby atoms of the periodic table. In the present thesis work where

we substitute one element by the nearby element of the periodic table, neutron

diffraction study helps us to confirm whether these atoms occupy the same site or

not. This is not possible in case of X-rays because the scattering cross-section for

X-rays depends on the atomic number (z), hence the variation of X-rays scattering

cross-section for two neighboring atoms in the periodic table is negligible.

(iii) Neutron possesses a magnetic moment of -1.913µN , where µN is the nuclear

magneton, and interacts with the magnetic moment of the atoms/ions. This helps to

determine the magnetic structure of the material by neutron scattering experiments.

In case of unpolarized neutrons, the Bragg diffracted intensity for a ferromagnetic

or a ferrimagnetic material is proportional to the square of the structure factors


F 2hkl, [39] where

F 2hkl = F 2

nuclear + q2F 2magnetic

=

∣∣∣∣∣n∑

j=1

bj exp [2πi (hxj + kyj + lzj)] e−2Wj

∣∣∣∣∣2

+ q2

∣∣∣∣∣m∑j=1

pj exp [2πi (hxj + kyj + lzj)] e−2Wj

∣∣∣∣∣2

(2.3)

Here, bj is the scattering length of the j th nucleus which varies randomly across the

periodic table, n is the total number of atoms in a unit cell. The second term in Eq.

(2.3) gives the magnetic Bragg scattering which occurs if the magnetic moments

have a periodic arrangement. The magnetic interaction vector is given by

q = ϵ(ϵ ·K) −K, (2.4)

where K is a unit vector in the direction of the atomic magnetic moment and ε is

the scattering vector which is perpendicular to the effective ’reflecting’ planes.

pj =

(e2γ

2mc2

)Sjfj (2.5)

where S j is the magnetic moment of the j th atom, f j is the magnetic form factor of

the j th atom which rapidly decreases with (sin θ)/λ, γ is neutron magnetic moment

in nuclear magneton and e2/mc2 is the Bohr radius. x j, y j, z j are the fractional

atomic coordinates in an unit cell. For magnetic scattering, the summation is over

all (m) magnetic atoms/ions in the unit cell. e−2Wj is the Debye-Waller factor

for the j th atom, which is due to thermal fluctuations. For magnetic materials,

neutron diffraction carried out below the magnetic ordering temperature, gives the

information about the magnetic structure of the material, in addition to the crystal


structure. In case of a ferromagnetic or ferrimagnetic material, additional Bragg

intensity to the lower angle fundamental nuclear peaks is observed. In case of

an antiferromagnetic material, where the magnetic unit cell could be larger than

the crystallographic unit cell, additional Bragg peaks appear corresponding to the

magnetic unit cell. To analyze the crystal structure as well as magnetic structure

using the neutron diffraction patterns, the Rietveld refinement technique [40] using

the FULLPROF program [41, 42] has been employed.

2.2.3 dc magnetization

The bulk magnetic properties of the intermetallic compounds under investigation in

this thesis have been investigated using (i) a vibrating sample magnetometer (VSM,

Oxford Instruments), (ii) a superconducting quantum interference device (MPMS

XL magnetometer, Quantum Design) also known as SQUID magnetometer, and

(iii) a Physical Property Measurement System (PPMS, Quantum Design). The

magnetization measurements have been carried out as a function of temperature

and applied magnetic field. The measurements have been carried in the temperature

range of 5-350 K and a magnetic field variation of up to ± 80 kOe.

2.2.3.1 dc magnetization using a vibrating sample magnetometer

The magnetic moment is measured by the induction technique in a VSM. This

technique was first given by S. Foner [43]. When a magnetic sample is placed in

a magnetic field produced by an electromagnet or by a superconducting magnet,

a dipole moment is induced. When the sample vibrates, an electromotive force

(electrical signal) is induced in the pick-up coil, which is given by

e = −NdϕB

dt(2.6)


Here, e is the induced electromotive force, N is the number of turns in the coil, ϕB

is the magnetic flux in the coil and t is the time. If the sample moves along the

z-axis, then Eq. (2.6) can be written as

e = −NdϕB

dz

dz

dt(2.7)

If the sample vibrates with an amplitude a and frequency ω, then

z = a exp(iωt)

dz

dt= a i ω exp(iωt) (2.8)

Therefore

e = −N a i ωdϕB

dzexp(iωt) (2.9)

The induced emf of an unknown sample is compared with the induced emf of a stan-

dard sample such as nickel. When a standard nickel sample is used for calibration,

the induced emf of Ni (V Ni in volts) with known magnetic moment (MNi in emu)

is stored in calibration file. The magnetic moment of the unknown sample can then

be calculated as

Munknown = VunknownMNi

VNi

(2.10)

Powder samples are compressed in small spherical pellet form and loaded in the

sample holder located at the end of the sample rod. The other end of the sample rod

is connected to the vibrator of the VSM which vibrates vertically with an amplitude

fixed in the range of 0.1-1.5 mm and a fixed frequency of 43-67 Hz. For the present

thesis work, amplitude of 0.5 mm and frequency of 55 Hz have been used. The

ac signal at the pick-up coil is fed to a differential amplifier, and the output of

the amplifier is fed to a tuned amplifier and a lock-in amplifier which also receives a


Figure 2.3: Photograph of the Vibrating Sample Magnetometer (Oxford Instru-ments) set up at Solid State Physics Division, BARC.

reference signal from the oscillator. The output of the lock-in amplifier is a dc signal

which is proportional to the magnetic moment of the sample under investigation.

All the data acquisition is done through computer.

2.2.3.2 dc magnetization using a SQUID magnetometer

SQUID (superconducting quantum interference device) is a very sensitive magne-

tometer. This is used to measure magnetic field or magnetization with the help of

superconducting loops with Josephson junction. The input current I splits into two

equal branches in the superconducting ring as shown in Fig. (2.3). If a magnetic field

B is applied perpendicular to the plane of the ring, a phase difference is produced

between currents in the two branches. The magnetic flux ϕ passing through the

superconducting ring is quantized in terms of h/2e, where h is the Plank’s constant

and 2e is the charge of the Cooper pair of electrons.


Figure 2.4: Schematic diagram of a superconducting ring with Josephson junction ina superconducting quantum interference device (SQUID) as a simple magnetometer.

Some of the magnetization measurements were carried out using the SQUID

magnetometer (MPMS, Quantum Design). Measurement in MPMS is performed by

moving the sample through the superconducting detection coils which are located

at the center of the magnet. It measures the local changes in magnetic flux density

produced by a sample as it moves through the superconducting detection coils. The

sample moves along the symmetry axis of the detection coil and magnet. As the

sample moves through the coils, the magnetic dipole moment of the sample induces

an electric current in the detection coils. The SQUID functions as a highly linear

current-to-voltage convertor, so that variations in the current in the detection coil

circuit produce corresponding variations in the SQUID output voltage. The MPMS

determines the magnetic moment of a sample by measuring the output voltage of

the SQUID detection system as the sample moves through the coil.

2.2.3.3 dc magnetization using a physical property measurement system

dc magnetization measurement technique used in a Physical Property Measurement

System (PPMS) is the extraction magnetometry. In this technique, a magnetic


sample is moved through the detection coils which in turn induces a voltage in

the detection coil set. The amplitude of the signal is proportional to the magnetic

moment as well as the speed of the sample during extraction. The dc servo mo-

tor used to move a sample can extract the sample at a speed of approximately

100 cm per second, thus significantly increasing the signal strength over conven-

tional extraction systems. The greater extraction speed also reduces any error that

may result from non-equilibrium time-dependent effects. The PPMS is provided

with AC/DC Magnetometry System (ACMS) which can perform ac susceptibility

Figure 2.5: Schematic of AC/DC Magnetometry System coil set [42].


as well as dc magnetization measurements in a single sequence [44]. The ACMS

coil set includes the detection coils that measure the sample’s magnetic response.

The ACMS probe attachment is concentric with the primary dc superconducting

magnet. The detection coils are arranged in a first-order gradiometer configuration

which rejects background signals. The PPMS uses a Digital Signal Processor chip

rather than a lock-in amplifier which takes advantage of digital filtering. This sig-

nificantly improves signal-to-noise performance over analog filters, which have to

sacrifice accuracy in order to perform over a wide frequency band.

2.2.4 ac susceptibility

In ac susceptibility measurements, a small ac magnetic field is applied to the sample

which induces a time dependent magnetic moment in the sample. This induces time

dependent voltage in the pick-up coil. When the ac field is applied, the induced ac

moment of the sample is

Mac =dM

dHHac sin ωt (2.11)

where H ac is the amplitude of the ac field and ω is frequency, and dM /dH is the

slope of the M (H ) curve, called the susceptibility (χac). When the ac frequency is

low, the ac magnetization is similar to that of dc magnetization. But when the fre-

quency is high, the ac magnetization does not follow the dc magnetization curve due

to the dynamic of the sample. In this case, there is a lag between sample magnetiza-

tion and applied ac field. Thus the ac susceptibility is a complex quantity, and the

in and out of phase produce real and imaginary components of χac, usually denoted

as χ’ac and χ”ac, respectively. The frequency dependent ac susceptibility studies

help us to distinguish spin-glass, cluster spin-glass and superparamagnets. The out

of phase susceptibility component indicates a dissipative processes in the sample.

In case of ferromagnetic materials, this component indicates an irreversible domain


wall movement. The relaxation and irreversibility of the magnetization also result in

out of phase susceptibility component. The technique complements dc susceptibility

technique by providing an additional frequency dependent information. Some of the

work in the present thesis has been carried out in a Physical Property Measurement

System with AC/DC Magnetometry System (ACMS) probe. The frequency range

is 10 Hz to 10 kHz and ac field is 5 Oe. The temperature range studied is 5 - 300 K.

2.2.5 Neutron depolarization

Neutron depolarization experiments down to 2 K have been carried out for some of

the intermetallic compounds to study magnetic inhomogenities such as ferromag-

netic clusters in the antiferromagnetic matrix on a mesoscopic length scale. It gives

an estimate of ferromagnetic domain/cluster size. The one-dimensional neutron-

depolarization measurements were carried out using the polarized neutron spec-

trometer (PNS) [45–47] at the Dhruva reactor (λ = 1.205 A). Neutron-depolarization

Figure 2.6: Photograph of the Polarized Neutron Spectrometer installed at theDhruva reactor, Trombay [46].

measurements were carried out by first cooling the sample from room temperature

down to 2 K in the presence of 50 Oe field (this field is required to maintain the

neutron beam polarization at sample position). The sample was placed in front


of the neutron beam such that the propagation direction of the polarized neutron

beam is along the thickness of the flat sample. The beam size was restricted within

the size of the sample. The incident neutron beam was polarized along the -z di-

rection (vertically down) with a beam polarization of 98.60(1) %. In case of an

unsaturated ferromagnet or ferrimagnet, the dipolar field exerted by the magnetic

domains on the neutron polarization depolarizes the neutrons due to Larmor pre-

cession of the neutron spins in the magnetic field of domains. In a ferromagnetic

domain, the spin component parallel to the magnetic induction B does not precess

whereas the component perpendicular to B does. The precession angle is given as

Φδ = (4.63×10−10 G−1 A−2)λδB where λ is the wavelength of the neutrons and

δ is the effective thickness i.e. distance traveled by the neutrons. The value of

average magnetic induction B = 4πM S (in Gauss) was estimated from the bulk

magnetization measurements where M S is the spontaneous magnetization and ρ is

the density of the material in g cm−3. In a pure antiferromagnet, there is no net

magnetization, hence no depolarization occurs. In case of a true spin-glass state,

the spins are randomly frozen in space on a microscopic length scale and hence no

net magnetization is present, resulting in no depolarization. In a paramagnet, the

neutron polarization is unable to follow the variation in the magnetic field as the

temporal spin fluctuation is too fast (10−12 s or faster). Hence, no depolarization is

observed. However, depolarization is expected in case of clusters of spin with a net

magnetization [48]. The neutron beam polarization in a depolarization experiment

can be represented by the following expression

Pf = Pi exp

[−α

(d

δ

)⟨Φδ⟩2

](2.12)


where P i and P f are the initial and final neutron beam polarizations, α is a di-

mensionless parameter ≈ 1/3, d is effective sample thickness, and δ is the average

domain size. Equation (2.12) is valid only when the precession angle Φδ is a small

fraction of 2π over a typical domain/cluster length.

In a neutron delpolarization study we measure the polarization ratio (R) which

is a measure of the transmitted polarization. The expression for the polarization

ratio is given by

R =1 − PiDPA

1 + (2f − 1)PiDPA

(2.13)

where PA is the efficiency of the analyzer, f is the efficiency of the rf flipper and D

is the depolarization coefficient. P iD is the transmitted beam polarization. When

there is no depolarization, D = 1 and, when there is total depolarization, D = 0.

2.2.6 Differential scanning calorimetry

Differential scanning calorimetry (DSC) measurements were carried out on some of

the intermetallic compounds using the Mettler Toledo DSC 822 set up which is of

heat flux type. This technique can be employed to investigate a system undergoing

structural phase transition. In differential scanning calorimetry, the difference in

the amount of heat required to increase the temperature of a sample with respect to

the reference is measured as a function of temperature. This technique is based on

the fact that when a sample undergoes a physical transformation such as a phase

transition, either heat is required or librated out of the sample with respect to that

of the reference to maintain both at the same temperature. Whether heat flows out

of the sample or into the sample depends on whether the process is exothermic or

endothermic. In the present work, an empty aluminum pan was used as a refer-

ence. The sample was encapsulated in another aluminum pan and then carefully

placed in the DSC cell. The measurements were carried out in the heating as well as


cooling cycles with a scanning rate of 5 K/min. The heat flow versus temperature

was recorded as output. Near the phase transition temperature of a sample, the

absorption or evolution of heat causes a change in the differential heat flow, which

appears as a peak. The peak position of the DSC curve gives the transition temper-

ature of the sample. This curve can be used to calculate enthalpies of transitions.

The integrated area under the peak gives the enthalpy change accompanying the

transition. The nature of the curve also gives the information about the order of

the phase transition. Specific heat capacities and changes in heat capacity can also

be determined from the difference in heat flow.

2.3 Computational techniques

2.3.1 Rietveld refinement

Rietveld refinement technique [40, 49, 50] has been employed to analyze the X-ray

and neutron diffraction patterns. FULLPROF program [41, 42] has been employed

to refine the crystal structure by using the X-ray diffraction patterns. This program

was also employed to refine the crystal structure and to determine the magnetic

structure from neutron diffraction patterns. Rietveld refinement is based on least-

squares minimization technique. This technique minimizes the difference between

experimental powder diffraction pattern and a calculated pattern based on a model

crystal structure and instrumental parameters. This technique can be used to de-

rive certain parameters such as lattice parameters, atomic positions and their oc-

cupancies, thermal parameters, magnetic moment and its direction, phase fraction,

etc. The Rietveld refinement method consists of refining a crystal and/or magnetic

structure by minimizing the weighted squared difference between observed intensity

{y i}i=1...n and calculated intensity { yc,i(α) } i=1...n against a set of parameters (α)


= α1,. . . ,αp. The process involves minimization of

χ2 =n∑

i=1

wi {yi − yc,i(α)}2 (2.14)

where w i = 1/σi2 is the weight factor and σi2 is the variance of the observation y i,

n is the number of data points and p is the number of refined parameters.

The quality of the agreement between observed and calculated profiles are mea-

sured by a set of following conventional factors.

Profile Factor : Rp = 100

∑i=1,n

|yi − yc,i|∑i=1,n

yi(2.15)

Weighted Profile Factor : Rwp = 100

∑i=1,n

wi |yi − yc,i|2∑i=1,n

wi y2i

1/2

(2.16)

Expected Weighted Profile Factor : Rexp = 100

n− p∑i=1,n

wi y2i

1/2

(2.17)

Goodness of fit indicator : S =Rwp

Rexp

(2.18)

Reduced chi − square : χ2 =

[Rwp

Rexp

]2(2.19)

In general, for a sensible refinement, the following sequence is followed of refine-


ment

1. Scale factor

2. Zero shift

3. Background

4. Lattice constants

5. Atomic positions

6. Over all Debye-Waller factor

7. Peak shape and asymmetry

8. Occupancy

9. Isotropic thermal parameter

10. Size and strain

For a ordered magnetic material, once the crystal structure is refined, we proceed

for refinement of the magnetic structure.

sample synthesis, experimental techniques, and...

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