Lattice Vibrational Spectra and Physical Characteristics of Metals with Cubic Structure

J. N. Plendl; P. J. Gielisse; H. S. Plendl; R. A. Kromhout; L. C. Mansur

doi:10.1364/AO.10.001444

Applied Optics
Vol. 10,
Issue 6,
pp. 1444-1450
(1971)
•https://doi.org/10.1364/AO.10.001444

Lattice Vibrational Spectra and Physical Characteristics of Metals with Cubic Structure

J. N. Plendl, P. J. Gielisse, H. S. Plendl, R. A. Kromhout, and L. C. Mansur

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J. N. Plendl, P. J. Gielisse, H. S. Plendl, R. A. Kromhout, and L. C. Mansur, "Lattice Vibrational Spectra and Physical Characteristics of Metals with Cubic Structure," Appl. Opt. 10, 1444-1450 (1971)

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Abstract

A recently developed method for obtaining the absorbed energy spectrum and calculating its absorption frequencies from elastic constant data has been applied to metals, using cold neutron spectra in absence of infrared data. The emitted spectrum from incoherent neutron scattering becomes, through Kirchhoff’s law, directly related to the absorbed energy spectrum. The cold neutron spectra examined here are for the metals Al, V, Ni, Pd, and Pb. The numerical data of the T and L modes are compared with those calculated from elastic constant data, including their interpretations. The agreement is within experimental error. Interesting differences occur between metals and dielectrics with respect to the intensity of the various T modes. The damping of lattice vibrations in metals is of the same order of magnitude as in dielectrics with ionic binding, i.e., much higher than the damping in dielectrics with covalent binding.

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Compound	ρ (g/cm³)	Interatomic distance (10⁻⁸ cm)	Elastic constants (10¹¹ dynes cm⁻²)			K	Frequencies calculated from elastic constants		Peaks observed

									ir absorption		Cold neutron scattering

			c₁₁	c₁₂	c₄₄		ν (cm⁻¹)	Interpretation	ν(cm⁻¹)	Δν/ν_T	ν(cm⁻¹)	Δν/ν_T
LiF	2.64	2.01	11.77	4.33	6.28	295	310	T(110)	306	0.23	307	0.23
							405	T(100)	396
							555	L(100)	525
							612	L(110)
							632	L(111)	655		640
NaF	2.80	2.31	9.70	2.50	2.80	295	257	T(110)	249	0.11	250
							229	T(100)
							425	L(100)			425
							406	L(110)	400
							402	L(111)	400
NaCl	216	2.815	4.68	1.23	1.19	295	167	T(110)	166.5	0.14	169
							139	T(100)	139
							276	L(100)	290
							258	L(110)	248
							253	L(111)	248
NaI	3.67	3.23	3.80	0.70	0.70	295	112	T(110)	118	0.15	122^b	0.18^b
							75	T(100)0
							175	L(100)			170^b
							155	L(110)			155^b
							147	L(111)
KBr	2.75	3.295	3.49	0.58	0.51	295	117	T(110)	110^c	0.11	121^d	0.14^d
									118
							69	T(100)	60–70
							181	L(100)	180
							155	L(110)	160		164^d
							144	L(111)
MgO	3.59	2.10	28.9	8.80	15.5	295	420	T(110)	422	0.22	400
							522	T(100)
							715	L(100)	515		530
							765	L(110)	725		725
							796	L(111)	750		725

Metal		Sample		Interatomic distance r₀ (10⁻⁸ cm)	Elastic constant 10¹¹ dynes cm⁻²			Frequencies calculated from elastic constants		Peaks observed in inelastic cold neutron scattering^a

Element	Structure	T (K)	ρ (g/cm³)		c₁₁	c₁₂	c₄₄	ν (cm⁻¹)	Interpretation	ν (cm⁻¹)	Previous interpretation	Ref.
Al	fcc	923	2.60	2.83	7.0	—	1.8	126	T(110)	145	(T)	7
								134	T(111)
								150	T(100)
								290	L(100)	300	(L)	7
							—	304	L(110)
								308	L(111)
								—	—	600	(multiphonon scattering)	7
V	bcc	295	6.16	2.62	19.6	13.5	6.7	—	—	64	(electron-phonon scattering)	11
								148	T(110)	~160		14
								173	T(111)	~180		14
								215	T(100)	220		9–10
										240 and above	(partly two-phonon scattering)	13
								368	L(100)	335		20
								400	L(110)
								411	L(111)
Ni	fcc	295	8.90	2.49	25.3	15.2	12.5	163	T(100)	165
								196	T(111)	183
								254	T(100)	260
								357	L(100)	Slight rise from 350 to 400		15
								400	L(110)
								429	L(111)
Pd	fcc	295	12.1	2.75	22.2	17.7	7.15
								32.5	T(110)	80
								109	T(111)	~120
								146	T(100)	~140
								259	L(100)	Broad peak starting at 260		15
								286	L(110)
								295	L(111)
Pb	fcc	295	11.3	3.49	4.76	3.92	1.44	32	T(110)	31
								41	T(111)	42		18
								63	T(100)	66
								106	L(100)	Broad rise above 90
								118	L(110)
								122	L(111)

Metal		ρ (g/cm³)	Interatomic distance r₀ (10⁻⁸ cm)	Elastic constants (10¹¹ dynes cm⁻²)			Frequencies calculated from elastic constants		Peaks in frequency spectrum calculated from Krebs’ model ν (cm⁻¹)

Element	Structure			c₁₁	C₁₂	c₄₄	ν (cm⁻¹)	Interpretaion
Li	bcc	0.534	3.04	1.34	1.13	0.96	75.3	T(110)	~75
							134	T(100)	123
							202	T(111)	194
							276	L(100)	265
							351	L(110)	—
							375	L(111)	—
Na	bcc	0.97	3.705	0.739	0.622	0.419	35	T(110)	~30
							61	T(111)	57
							93	T(100)	88
							124	L(100)	121
							151	L(110)	—
							160	L(111)	—
K	bcc	0.862	4.61	0.416	0.333	0.263	25.2	T(110)	34
							39.0	T(111)	50
							63.5	T(100)	62
							80	L(100)	68
							99	L(110)	—
							106	L(111)	—

Compound	ρ (g/cm³)	Interatomic distance (10⁻⁸ cm)	Elastic constants (10¹¹ dynes cm⁻²)			K	Frequencies calculated from elastic constants		Peaks observed

									ir absorption		Cold neutron scattering

			c₁₁	c₁₂	c₄₄		ν (cm⁻¹)	Interpretation	ν(cm⁻¹)	Δν/ν_T	ν(cm⁻¹)	Δν/ν_T
LiF	2.64	2.01	11.77	4.33	6.28	295	310	T(110)	306	0.23	307	0.23
							405	T(100)	396
							555	L(100)	525
							612	L(110)
							632	L(111)	655		640
NaF	2.80	2.31	9.70	2.50	2.80	295	257	T(110)	249	0.11	250
							229	T(100)
							425	L(100)			425
							406	L(110)	400
							402	L(111)	400
NaCl	216	2.815	4.68	1.23	1.19	295	167	T(110)	166.5	0.14	169
							139	T(100)	139
							276	L(100)	290
							258	L(110)	248
							253	L(111)	248
NaI	3.67	3.23	3.80	0.70	0.70	295	112	T(110)	118	0.15	122^b	0.18^b
							75	T(100)0
							175	L(100)			170^b
							155	L(110)			155^b
							147	L(111)
KBr	2.75	3.295	3.49	0.58	0.51	295	117	T(110)	110^c	0.11	121^d	0.14^d
									118
							69	T(100)	60–70
							181	L(100)	180
							155	L(110)	160		164^d
							144	L(111)
MgO	3.59	2.10	28.9	8.80	15.5	295	420	T(110)	422	0.22	400
							522	T(100)
							715	L(100)	515		530
							765	L(110)	725		725
							796	L(111)	750		725

Metal		Sample		Interatomic distance r₀ (10⁻⁸ cm)	Elastic constant 10¹¹ dynes cm⁻²			Frequencies calculated from elastic constants		Peaks observed in inelastic cold neutron scattering^a

Element	Structure	T (K)	ρ (g/cm³)		c₁₁	c₁₂	c₄₄	ν (cm⁻¹)	Interpretation	ν (cm⁻¹)	Previous interpretation	Ref.
Al	fcc	923	2.60	2.83	7.0	—	1.8	126	T(110)	145	(T)	7
								134	T(111)
								150	T(100)
								290	L(100)	300	(L)	7
							—	304	L(110)
								308	L(111)
								—	—	600	(multiphonon scattering)	7
V	bcc	295	6.16	2.62	19.6	13.5	6.7	—	—	64	(electron-phonon scattering)	11
								148	T(110)	~160		14
								173	T(111)	~180		14
								215	T(100)	220		9–10
										240 and above	(partly two-phonon scattering)	13
								368	L(100)	335		20
								400	L(110)
								411	L(111)
Ni	fcc	295	8.90	2.49	25.3	15.2	12.5	163	T(100)	165
								196	T(111)	183
								254	T(100)	260
								357	L(100)	Slight rise from 350 to 400		15
								400	L(110)
								429	L(111)
Pd	fcc	295	12.1	2.75	22.2	17.7	7.15
								32.5	T(110)	80
								109	T(111)	~120
								146	T(100)	~140
								259	L(100)	Broad peak starting at 260		15
								286	L(110)
								295	L(111)
Pb	fcc	295	11.3	3.49	4.76	3.92	1.44	32	T(110)	31
								41	T(111)	42		18
								63	T(100)	66
								106	L(100)	Broad rise above 90
								118	L(110)
								122	L(111)

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Applied Optics