Terahertz Spectroscopy for the Solid StateCharacterisation of Amorphous Systems

Juraj Sibik and Axel Zeitler

Department of Chemical Engineering and Biotechnology, University of Cambridge,Pembroke Street, Cambridge CB2 3RA, UKjaz22@cam.ac.ukhttp://thz.ceb.cam.ac.uk – www.pssrc.org

19 June 2015

Outline

IntroductionDielectric SpectroscopyTerahertz Radiation

Amorphous MaterialsWhat Can be Measured at THz Frequencies?Model System: PolyalcoholsCrystallisationStability Prediction

Summary

Outline

IntroductionDielectric SpectroscopyTerahertz Radiation

Amorphous Materials

Summary

Introduction Dielectric Spectroscopy

Dielectric Spectroscopy

105 106 107 108 109 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021

103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13

Visible Light

Infrared UltravioletRadiowaves Microwaves X-rays Gamma

1 MHz 1 ZHz1 EHz1 PHz1 THz1 GHz

Frequency / Hz

THz

1 nm1 m1 mm1 m1 km

Wavelength / m

Complementary technique to terahertz spectroscopy at lowerfrequencies

2 of 31

Introduction Dielectric Spectroscopy

Absorption Mechanisms

I This technique spansthe frequency rangeover 102 to 1012 Hz

I Dipoles and chargesrespond to theexcitation by anexternal electric fieldand move as a wholeduring relaxation

I ε = ε′ + iε′′ = (n + iκ)2

where α = 4πκ/λ0

Image source: https://commons.wikimedia.org/wiki/file:Dielectric_responses.svg 3 of 31

Introduction Dielectric Spectroscopy

Dielectric Relaxation – Molecular Mobility

α-relaxation

I Structural relaxationprocess

I Relaxation time changesfrom 10−12 to 102 s uponglass transition

I Concept of cooperativelyrearranging regions(CRR)

β-relaxations

I Local motions involvingthe entire molecule orintra-molecularreorientations

I Much faster than αrelaxations

I Commonly observedeither as a separate peakor as a high frequencywing of the α-relaxation.

G. Adam, J.H. Gibbs, The Journal of Chemical Physics. 43, 139 (1965). 4 of 31

Introduction Dielectric Spectroscopy

Dielectric Relaxation in Amorphous Solids

I α and β relaxation process areseparated in frequency (but arevery broad and often overlap)

I The secondary β-relaxationprocesses are typically related tolocal mobility

I It is possible to directly measure therelaxation times using dielectricspectroscopy

H. Wagner, R. Richert, J. Non-Cryst. Sol. 242, 19 (1998).S. Bhattacharya, R. Suryanarayanan, 98, 2935 (2009). 5 of 31

Outline

IntroductionDielectric SpectroscopyTerahertz Radiation

Amorphous Materials

Summary

Introduction Terahertz Radiation

What does Terahertz Radiation Refer to?

1 0 5 1 0 6 1 0 7 1 0 8 1 0 9 1 0 1 0 1 0 1 1 1 0 1 2 1 0 1 3 1 0 1 4 1 0 1 5 1 0 1 6 1 0 1 7 1 0 1 8 1 0 1 9 1 0 2 0 1 0 2 1

1 0 3 1 0 2 1 0 1 1 0 0 1 0 - 1 1 0 - 2 1 0 - 3 1 0 - 4 1 0 - 5 1 0 - 6 1 0 - 7 1 0 - 8 1 0 - 9 1 0 - 1 0 1 0 - 1 1 1 0 - 1 2 1 0 - 1 3

V i s i b l e L i g h t

I o n i s i n g

T r a n s p a r e n c y

I n f r a r e d U l t r a v i o l e t

T r a n s p a r e n c yS p e c t r o s c o p i c I n f o r m a t i o n

T H zR a d i o w a v e s M i c r o w a v e s X - r a y s G a m m a

1 M H z 1 Z H z1 E H z1 P H z1 T H z1 G H z

F r e q u e n c y / H z1 n m1 µm1 m m1 m1 k m

W a v e l e n g t h / m

0 . 0 1 0 . 1 1 1 0 1 0 0F r e q u e n c y / T H z

1 . 0 1 0 . 0 1 0 0 . 0 1 0 0 0 . 0

H y d r o g e n - b o n d i n g s t r e t c h e s a n d t o r s i o n s ( l i q u i d s )

S e c o n d a r y d i e l e c t r i c r e l a x a t i o n s ( s o l i d )

I n t r a m o l e c u l a r v i b r a t i o n a l m o d e s

C r y s t a l l i n e p h o n o n v i b r a t i o n s ( s o l i d )

W a v e n u m b e r / c m - 1

M o l e c u l a r r o t a t i o n s ( g a s )

6 of 31

Introduction Terahertz Radiation

Vibrational Spectroscopy

Mid-infrared

Intramolecular ModesInformation about the structure of a single

molecule, identification of molecules

Terahertz

Intermolecular ModesInformation about the structure anddynamics of molecular interaction

7 of 31

Introduction Terahertz Radiation

Terahertz Time-Domain Spectroscopy

0 1 0 2 0 3 0 4 0 5 0

- 8

- 6

- 4

- 2

0

2

4

6

8

1 0

1 2

THz e

lectric

field

/ a.u.

t i m e / p s1 2 3 4 5

0 . 1

1

1 0

1 0 0

powe

r / a.u

.f r e q u e n c y / T H z

Typical terahertz pulse in time-domain (left) and frequency components of the pulse (right).Coherent sub-picosecond pulses, bandwidth of 0.1 to 4.0 THz, excellent signal-to-noise

detection

8 of 31

Introduction Terahertz Radiation

Terahertz Time-Domain Technology

I In THz-TDS both amplitude and phase of the electric fieldis measured and not just its intensity

I This means that the complex refractive index can beextracted directly without resorting to Kramer-Kronigrelations:

Esam(ω)

Eref(ω)= T(ω)eiφ(ω)

I In terms of absorption coefficient and refractive index:

α(ω) = −2d

ln[(nm + n)2

4nmnT(ω)

]n(ω) = 1 +

φ(ω)cωd

I This can also directly be expressed in terms of dielectriclosses:

n = n + iκ =√ε =√ε′ + iε′′

9 of 31

Outline

Introduction

Amorphous MaterialsWhat Can be Measured at THz Frequencies?Model System: PolyalcoholsCrystallisationStability Prediction

Summary

Amorphous Materials What Can be Measured at THz Frequencies?

Amorphous Materials

http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Structure/solidstate.htmJ. Bicerano, D. Adler, Pure & Appl. Chem., 59, 101 (1987) 10 of 31

Amorphous Materials What Can be Measured at THz Frequencies?

Disordered Materials – Losses at THz Frequencies

Amorphous Solids and Supercooled Liquids

I Mid-IR: Bond vibrations, slight shift andbroadening compared to crystallinematerials

I THz: No phonon vibrations occur as thereis no long range order

I At lower frequencies molecular rotationsand translations take place

I These molecular motions can be describedby the first order decay of macroscopicpolarisation as proposed by Debye in hisdielectric relaxation theory

11 of 31

Outline

Introduction

Amorphous MaterialsWhat Can be Measured at THz Frequencies?Model System: PolyalcoholsCrystallisationStability Prediction

Summary

Amorphous Materials Model System: Polyalcohols

Dielectric Response of Amorphous Materials

S. Kastner et al., J. Non-Cryst. Sol. 357, 510 (2011). 12 of 31

Amorphous Materials Model System: Polyalcohols

Dielectric Response of Amorphous Materials

S. Kastner et al., J. Non-Cryst. Sol. 357, 510 (2011). 12 of 31

Amorphous Materials Model System: Polyalcohols

Amorphous Sorbitol

100 150 200 250 3000

50

100

150

200

1.5 THz 1.0 THz 0.5 THz

α [c

m-1

]

100wt% sorbitol

T [K]

TgGlass transition

Structural relaxation at Tg leads to increase in absorption

J. Sibik et al., Phys. Chem. Chem. Phys. 15, 11931 (2013). 13 of 31

Amorphous Materials Model System: Polyalcohols

Amorphous Sorbitol

100 150 200 250 3000

50

100

150

200

1.5 THz 1.0 THz 0.5 THz

α [c

m-1

]

100wt% sorbitol

T [K]

TgGlass transition

Subtle but noticeable change in absorption below Tg – origin?

J. Sibik et al., Phys. Chem. Chem. Phys. 15, 11931 (2013). 13 of 31

Amorphous Materials Model System: Polyalcohols

Secondary Relaxation in Polyalcohols

A. Döss et al., Phys. Rev. Lett. 88 (2002), doi:10.1103/PhysRevLett.88.095701. 14 of 31

Amorphous Materials Model System: Polyalcohols

Terahertz Spectroscopy of Polyalcohols

100

10-1

100

300 K

80 K

120 K 190 K

'' ()

(THz)

(a) glycerol

100

150 K 230 K

240 K

90 K

(THz)

(b) threitol

100

310 K

80 K

180 K 250 K

(THz)

(c) xylitol

100

310 K

(THz)

180 K 260 K

(d) sorbitol

90 K

101 102(cm-1)

101 102(cm-1)

101 102(cm-1)

101 102(cm-1)

The blue and red circles highlight the losses in the proximity of 0.65 Tg and Tg

respectively.

The sample of threitol recrystallised above 250 K – no data above this temperatureare shown.J. Sibik et al., J. Phys. Chem. Lett. 5, 1968 (2014). 15 of 31

Amorphous Materials Model System: Polyalcohols

Terahertz Spectroscopy of Polyalcohols

0.5 1.0 1.50.1

0.3

0.5

0.7

T

T

(iii)(ii)

1.00 Tg

sorbitol(+0.1) xylitol threitol(+0.1) glycerol(-0.1)

''(=

1 TH

z)

T/Tg

0.65 Tg

(i)

The sample of threitol recrystallised above 250 K – no data above this temperatureare shown.J. Sibik et al., J. Phys. Chem. Lett. 5, 1968 (2014). 15 of 31

Amorphous Materials Model System: Polyalcohols

Terahertz Spectroscopy of Polyalcohols

I At temperatures well below Tg, atemperature-independent microscopic peak isobserved, which persists into the liquid phaseand which is identified as being due tolibrational/torsional modes.

I For 0.65 Tg < T < Tg, additional thermallydependent contributions are observed, and wefound strong evidence for its relation to theJohari-Goldstein secondary relaxation process.

I Clear spectroscopic evidence is found for asecondary glass transition at 0.65 Tg, which is notrelated to the fragility of the glasses.

I At temperatures above Tg, the losses becomedominated by primary α-relaxation processes.Our results show that the thermal changes in thelosses seem to be underpinned by a universalchange in the hydrogen bonding structure of thesamples.

0.5 1.0

Molecular relaxations

0.67 Tg

'' THz

T/Tg

Tg

VDOSJG-

Libration-vibration motions

Decoupling (independent of m)

J. Sibik et al., J. Phys. Chem. Lett. 5, 1968 (2014). 16 of 31

Outline

Introduction

Amorphous MaterialsWhat Can be Measured at THz Frequencies?Model System: PolyalcoholsCrystallisationStability Prediction

Summary

Amorphous Materials Crystallisation

Phase Transitions – in situ Spectroscopy

0 . 7 5 0 . 9 0 1 . 0 5 1 . 2 0 1 . 3 5 1 . 5 0 1 . 6 5 1 . 8 0

0 . 5

1 . 0

1 . 5

2 . 0

2 . 52 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0

w a v e n u m b e r / c m - 1

abso

rbanc

e / a.

u.

f r e q u e n c y / T H z

f o r m I I I

f o r m I

Conversion of carbamazepine form III to I at 433 KTerahertz spectroscopy is very sensitive to changes in supramolecular structure

J.A. Zeitler et al., Thermochimica Acta. 436, 71 (2005). 17 of 31

Amorphous Materials Crystallisation

Phase Transitions – Kinetics

Kinetics of the solid state transition. Mechanism occurs assolid-gas-solid transition and can be resolved using THz-TDS.

J.A. Zeitler et al., ChemPhysChem. 8, 1924 (2007). 18 of 31

Amorphous Materials Crystallisation

Amorphous vs. Crystalline Organic Solids

1 0 2 0 3 0 4 0 5 0 6 0 7 00 . 0

0 . 5

1 . 0

1 . 5

2 . 0

2 . 5 a m o r p h o u s c r y s t a l l i n e

Abso

rbanc

e (de

cadic

)

W a v e n u m b e r [ c m - 1 ]

Crystalline vs. amorphous indomethacine.

C.J. Strachan et al., Chem. Phys. Lett. 390, 20 (2004). 19 of 31

Amorphous Materials Crystallisation

Relaxation and Crystallisation

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 002468

1 01 21 4

Abso

rbanc

e (de

cadic

)

W a v e n u m b e r [ c m - 1 ]

654321

330-340 K

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 002468

1 01 21 4 65432

Abso

rbanc

e (de

cadic

)

W a v e n u m b e r [ c m - 1 ]

1

340-356 K

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 002468

1 01 21 4 65432

Abso

rbanc

e (de

cadic

)

W a v e n u m b e r [ c m - 1 ]

1

357-368 K

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 002468

1 01 21 4

Abso

rbanc

e (de

cadic

)

W a v e n u m b e r [ c m - 1 ]

654321

410-440 K

J.A. Zeitler et al., J. Pharm. Sci. 96, 2703 (2007). 20 of 31

Amorphous Materials Crystallisation

Change in Absorbance

2 9 0 3 0 0 3 1 0 3 2 0 3 3 0 3 4 0 3 5 0 3 6 0 3 7 0 3 8 0 3 9 0 4 0 0 4 1 0 4 2 0 4 3 0 4 4 0 4 5 00 . 5

1 . 0

1 . 5

2 . 0

3 0 0 3 1 0 3 2 0 3 3 0 3 4 0 3 5 00 . 9 0

0 . 9 5

1 . 0 0

1 . 0 5

f o r m IT g f o r m I I Ig l a s s ys t a t e

Norm

alise

d abs

orban

ce (d

ecad

ic)

T e m p e r a t u r e [ K ]

f e a t u r e 1 f e a t u r e 2 f e a t u r e 3 f e a t u r e 4 f e a t u r e 5 f e a t u r e 6

r u b b e r ys t a t e

c r y s t a l l -i s a t i o n

p h a s e t r a n s i t i o n

At Tg sample relaxes and crystallises subsequently at higher temperature.

J.A. Zeitler et al., J. Pharm. Sci. 96, 2703 (2007). 21 of 31

Amorphous Materials Crystallisation

Crystallisation Kinetics

0 1 20

40

80

120

160

200

240a )

α (c

m-1)

Frequency (THz)

320 325 330 335 340 345

0.0

0.2

0.4

0.6

0.8

1.0

r a , amorphous fraction

r c , crystalline fraction

Avrami-Erofeev fit

Temperature (K)

r a, r c

b )

a) Terahertz spectra of paracetamol crystallising form the amorphous phase. Asthe crystallisation progresses distinct vibrational modes emerge from the VDOS.

b) Kinetics of the crystallisation process and corresponding fit using theAvrami-Erofeev model.

J. Sibik et al., Molecular Pharmaceutics. 11, 1326 (2014). 22 of 31

Amorphous Materials Crystallisation

Crystallisation of Amorphous Paracetamol

Spectra of the three observed polymorphs

J. Sibik et al., Molecular Pharmaceutics. 11, 1326 (2014). 23 of 31

Amorphous Materials Crystallisation

Crystallisation of Amorphous Paracetamol

0 1 2 30100200300400

300350400450

Temperature (K)

Frequency (THz)

(cm

-1)

Crystallisation and subsequent phase transitions

J. Sibik et al., Molecular Pharmaceutics. 11, 1326 (2014). 23 of 31

Amorphous Materials Crystallisation

Crystallisation of Amorphous Paracetamol

10

30

50

300 350 400 450

10305070

507090110

300 350 400 450

180240300360

T (K)

(cm

-1) (a)

0.7 THz

1.0 THz

LIIIIII

(b)

(cm

-1)A

1.5 THz(c)

(c

m-1)

2.5 THz(d)

T (K)

(cm

-1)

J. Sibik et al., Molecular Pharmaceutics. 11, 1326 (2014). 23 of 31

Amorphous Materials Crystallisation

Crystallisation of Amorphous Paracetamol

0 1 2 30

100

200

300

0.6 0.9

20

40

(cm

-1)

Frequency (THz)

325 K 330 K 335 K

0 25 50 75 100 Wavenumber (cm-1)

1 20

100

200

300 300 K 330 K 335 K 470 K fit

n (c

m-1

)Frequency (THz)

20 40 60Wavenumber (cm-1)

Deviation from power law: onset of crystallisation

n (ν)α (ν) = A + C (ν − ν0)q

J. Sibik et al., Molecular Pharmaceutics. 11, 1326 (2014). 23 of 31

Amorphous Materials Crystallisation

Crystallisation of Amorphous Paracetamol

0

10

20

30

40290 300 310 320 330 460470

120

130

140

290 300 310 320 330 4604701.0

1.1

1.2

1.3

1.4

1.5

T (K)

A (c

m-1)

(a)

(b)

C (c

m-1TH

z-q)

(c)

T (K)

q

I In paracetamol the crystallisation from the amorphousphase is observed to form III

I Subsequent phase transitions occur to forms II and Ibefore the sample melts

I This observation is in agreement with a previous studyof paracetamol by low frequency Raman scattering

I The featureless spectra of the supercooled liquid andliquid melt can be fitted using a power law model

I The melt spectrum is dominated by the dielectricrelaxation as well as the VDOS, while in thesupercooled liquid the contribution due to the dielectricrelaxation vanishes close to Tg (q changes from 2 in theglassy state to 1 in the liquid melt state)

I Using the simple power law model introduced previously,the onset of crystallisation can be determined precisely

J. Sibik et al., Molecular Pharmaceutics. 11, 1326 (2014). 24 of 31

Amorphous Materials Crystallisation

Crystallisation Below Tg

0.5 1.0 1.5 2.00

100

200

(b)(a)

310 K

Naproxen

() (

cm-1)

(THz)

100 K

0.4 0.6 0.8 1.0 1.250

100

150

200

250 1.2 THz 1.5 THz 1.8 THz

(T/T

g) (cm

-1)

T/Tg

0.67 Tg

~4x faster

Seeded crystallisation: rate increases at ≈ 0.67 Tg!Role of molecular mobility below Tg

J. Sibik et al., Molecular Pharmaceutics, doi:10.1021/acs.molpharmaceut.5b00330(2015). 25 of 31

Outline

Introduction

Amorphous MaterialsWhat Can be Measured at THz Frequencies?Model System: PolyalcoholsCrystallisationStability Prediction

Summary

Amorphous Materials Stability Prediction

Amorphous Drug Stability

J. Sibik et al., Molecular Pharmaceutics, doi:10.1021/acs.molpharmaceut.5b00330(2015). 26 of 31

Amorphous Materials Stability Prediction

Amorphous Drug Stability

0.2 0.4 0.6 0.8 1.0 1.220

25

30

35

40

45

50

0.5 1.0 1.5 2.0 2.50

50

100

150

320 K

Indomethacin

Paracetamol

(cm

-1)

Frequency (THz)

80 K 0.67 Tg

Paracetamol Indomethacin

1.0

THz (c

m-1)

T/Tg

J. Sibik et al., Molecular Pharmaceutics, doi:10.1021/acs.molpharmaceut.5b00330(2015). 26 of 31

Amorphous Materials Stability Prediction

Amorphous Drug Stability

0.2 0.4 0.6 0.8 1.0 1.2

1.0

1.1

1.2

1.3 paracetamol indomethacin flufenamic acid simvastatin

linear fit0

T/Tg

J. Sibik et al., Molecular Pharmaceutics, doi:10.1021/acs.molpharmaceut.5b00330(2015). 26 of 31

Amorphous Materials Stability Prediction

Prediction of Amorphous Stability

J. Sibik et al., Molecular Pharmaceutics, doi:10.1021/acs.molpharmaceut.5b00330(2015). 27 of 31

Summary

Terahertz Spectroscopy

I The terahertz molecular dynamics is strongly related to the molecularmobility governing the stability of amorphous drugs.

I While molecular relaxations are often extracted by dielectric spectroscopy orDSC and used to predict the stability of the amorphous drugs, concerns havebeen raised about the robustness of these methods.

I DSC is useful mainly for measurements of molecular mobility around andabove Tg, but cannot be easily used to measure molecular mobility at lowertemperatures.

I Measurements by dielectric spectroscopy are very useful to measure thelocal mobility in terms of JG-β relaxation, except for cases where thisrelaxation is submerged in the α-relaxation.

I In contrast, terahertz spectroscopy does not suffer from this limitation as itmeasures fast motions and only indirectly resolves the effect of the JG-βrelaxation, which may in principle be observed even when no clear JG-βpeak is present (such as in the case of indomethacin).

28 of 31

Summary Acknowledgments

Acknowledgments

I Dr Korbinian Löbmann and Professor Thomas Rades (Copenhagen)I U.K. Engineering and Physical Sciences Research Council (EPSRC,

EP/J007803/1)

29 of 31

Summary Acknowledgments

Literature I

I G. Adam and J. H. Gibbs, On the temperature dependence of cooperative relaxation properties in glass-forming liquids,The Journal of Chemical Physics, 43:139, 1965.

I S. Bhattacharya and R. Suryanarayanan, Local Mobility in Amorphous Pharmaceuticals-Characterization andImplications on Stability, 98(9):2935–2953, January 2009, http://dx.doi.org/10.1002/jps.21728.

I A. Döß, M. Paluch, H. Sillescu, and G. Hinze, From Strong to Fragile Glass Formers: Secondary Relaxation inPolyalcohols, Phys. Rev. Lett., 88(9), February 2002, http://dx.doi.org/10.1103/PhysRevLett.88.095701.

I S. Kastner, M. Köhler, Y. Goncharov, P. Lunkenheimer, and A. Loidl, High-frequency dynamics of type B glass formersinvestigated by broadband dielectric spectroscopy, J. Non-Cryst. Sol., 357(2):510–514, January 2011,http://dx.doi.org/10.1016/j.jnoncrysol.2010.06.074.

I J. Sibik, S. R. Elliott, and J. A. Zeitler, Thermal decoupling of molecular-relaxation processes from the vibrationaldensity of states at terahertz frequencies in supercooled hydrogen-bonded liquids, J. Phys. Chem. Lett., 5(11):1968–1972, 2014a, http://dx.doi.org/10.1021/jz5007302.

I J. Sibik, E. Y. Shalaev, and J. A. Zeitler, Glassy dynamics of sorbitol solutions at terahertz frequencies., Phys. Chem.Chem. Phys., 15(28):11931–11942, July 2013, http://dx.doi.org/10.1039/c3cp51936h.

I J. Sibik, M. J. Sargent, M. Franklin, and J. A. Zeitler, Crystallization and Phase Changes in Paracetamol from theAmorphous Solid to the Liquid Phase, Molecular Pharmaceutics, 11(4):1326–1334, March 2014b,http://dx.doi.org/10.1021/mp400768m.

I J. Sibik, K. Loebmann, T. Rades, and J. A. Zeitler, Predicting Crystallisation of Amorphous Drugs With TerahertzSpectroscopy, Molecular Pharmaceutics, page 150619135054002, June 2015,http://dx.doi.org/10.1021/acs.molpharmaceut.5b00330.

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Summary Acknowledgments

Literature II

I C. J. Strachan, T. Rades, D. Newnham, K. C. Gordon, M. Pepper, and P. F. Taday, Using terahertz pulsed spectroscopyto study crystallinity of pharmaceutical materials, Chem. Phys. Lett., 390(1-3):20–24, May 2004,http://dx.doi.org/10.1016/j.cplett.2004.03.117.

I H. Wagner and R. Richert, Spatial uniformity of the β-relaxation in D-sorbitol, J. Non-Cryst. Sol., 242(1):19–24, 1998.

I J. A. Zeitler, P. F. Taday, K. C. Gordon, M. Pepper, and T. Rades, Solid-State Transition Mechanism in CarbamazepinePolymorphs by Time-Resolved Terahertz Spectroscopy, ChemPhysChem, 8(13):1924–1927, 2007a,http://dx.doi.org/10.1002/cphc.200700261.

I J. A. Zeitler, P. F. Taday, M. Pepper, and T. Rades, Relaxation and crystallization of amorphous carbamazepine studiedby terahertz pulsed spectroscopy, J. Pharm. Sci., 96(10):2703–2709, October 2007b, http://dx.doi.org/10.1002/jps.20908.

I J. A. Zeitler, D. A. Newnham, P. F. Taday, C. J. Strachan, M. Pepper, K. C. Gordon, and T. Rades, Temperaturedependent terahertz pulsed spectroscopy of carbamazepine, Thermochimica Acta, 436(1-2):71–77, October 2005,http://dx.doi.org/10.1016/j.tca.2005.07.006.

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