Closed-Form Solution to Rate Equations for an F + H<sub>2</sub> Laser Oscillator

George Emanuel; James S. Whittier

doi:10.1364/AO.11.002047

Applied Optics
Vol. 11,
Issue 9,
pp. 2047-2056
(1972)
•https://doi.org/10.1364/AO.11.002047

Closed-Form Solution to Rate Equations for an F + H₂ Laser Oscillator

George Emanuel and James S. Whittier

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George Emanuel and James S. Whittier, "Closed-Form Solution to Rate Equations for an F + H₂ Laser Oscillator," Appl. Opt. 11, 2047-2056 (1972)

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Abstract

Chemical lasers pumped by the reaction of atomic fluorine with molecular hydrogen emit power from rotation–vibration transitions of excited HF with upper levels as high as v = 3. Collisional processes compete with stimulated emission for the energy of the excited HF. A simplified analysis is presented here for intensity, energy, and chemical efficiency of a class of such lasers. Results are obtained in closed form. A comparison with more exact computer solutions establishes the validity of the analysis despite its simplifications. A comprehensive parametric study examines the relative importance of initial conditions, optical cavity parameters, and rate coefficients for pumping and deactivation reactions.

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Feature	Present analysis	Computer model
Lasing threshold	Occurs simultaneously for all bands at t = 0	Occurs when Eq. (7) is first satisfied. Individual bands treated separately
Lasing cutoff	Occurs simultaneously for all bands	Individual bands treated separately
Line broadening	Doppler	Simultaneous Doppler and collisional
J	Lasing on single J for whole pulse and for all vibrational bands	Lasing on maximum gain transition for each band; calculates J’s as part of solution
Rotational population distribution	Boltzmann at translational temperature	Boltzmann at translational temperature
Temperature	Constant	Calculated as part of solution
Vibrational energy levels	Harmonic	Anharmonic
Reactions modeled	See Table II; most backward rates neglected	Forward and backward rates for all reactions of Table II. Also other VT and VV and recombination–dissociation processes

Reaction no.	Reaction	Over-all Rate Coefficients^a		Distribution Constants a(v)

		Forward	Backward	v = 0	1	2	3
R-1	F + H₂ ⇌ HF(v) + H	k = 1.62 × 10¹⁴e^−1600/^RT	k_b = 0	0.056	0.111	0.555	0.278
R-2	HF(v) + HF ⇌ HF(v − 1) + HF	k_HF = 6 × 10¹⁶T^−1.43	k_b = 0	0	0.167	0.333	0.500
R-3	HF(v) + H₂(0) ⇌ HF(v − 1) + H₂(1)	k_H₂ = 8.3 × 10⁵T^2.2^e^−562/^RT	k_b = 0	0	0.965	0.035	0
R-4	HF(v) + F ⇌ HF(v − 1) + F	k_F = 5.4 × 10⁹T^1.3	k_b = 0	0	0.167	0.333	0.500
R-5	2HF(v) ⇌ HF(v − 1) + HF(v + 1)	k_f_HF(T)	k_b = 0	0	a_f(1)	a_f(2)	0
R-6	2HF(v) ⇌ HF(v − 1) + HF(v + 1)	k_f = 0	k_b_HF(T)	0	a_b(1)	a_b(2)	0

	Case

	A	B	C	D
Specified parameters
T, K	200	300	400	300
κ₁	0.5	1	2	1
κ₂	0	0	0	0.1
r₁	0.8	0.6	0.4	0.6
Computer model results
l₀, μsec	0.12	0.14	0.20	0.54
(1 → 0)t_c, μsec	21.9	12.9	6.1	11.4
(2 → 1)t_c, μsec	36.1	18.5	7.2	16.8
(3 → 2)t_c, μsec	19.5	6.4	2.4	5.9
Output J	4,5	4,5,6	5,6	5,6
T_c, K	210	311	410	311
η	35.4	27.8	13.8	24.7
Results of theory
t_c, μsec	24.1	12.9	4.9	11.5
Dominant J	5	6	7	6
η	3 6.4	32.1	18.1	29.5

T, K	r₁	J	ξ_c	t_c, μsec	d_p	−d_H₂	−d_F	−d_HF	E, J/cm³	η
	κ₁ = 0.5		κ₂ = 0
200	0.8	5	0.805	24.15	1.388	1.194⁻³^a	2.653⁻²	2.370⁻¹	8.165⁻⁴	36.39
	0.6	5	0.777	21.75	1.340	1.091⁻³	3.410⁻²	2.531⁻¹	7.641⁻⁴	34.06
	0.4	5	0.741	19.14	1.278	9.847⁻⁴	4.379⁻²	2.677⁻¹	7.012⁻⁴	31.25
300	0.8	7	0.917	15.84	1.550	2.752⁻³	3.963⁻²	1.814⁻¹	6.421⁻⁴	42.92
	0.6	6	0.911	15.35	1.463	2.808⁻³	3.743⁻²	1.728⁻¹	6.053⁻⁴	40.46
	0.4	6	0.887	13.51	1.424	2.546⁻³	5.183⁻²	1.954⁻¹	5.689⁻⁴	38.03
400	0.8	8	0.949	13.41	1.523	4.885⁻³	5.074⁻²	1.322⁻¹	4.849⁻⁴	43.22
	0.6	7	0.945	13.03	1.442	4.808⁻³	4.966⁻²	1.284⁻¹	4.573⁻⁴	40.76
	0.4	7	0.924	11.27	1.410	4.280⁻³	7.368⁻²	1.531⁻¹	4.281⁻⁴	38.16
	κ₁ = 1.0		κ₂ = 0
200	0. 8	6	0.622	17.78	1.124	4.497⁻⁴	4.784⁻²	2.194⁻¹	1.245⁻³	27.74
	0.6	5	0.607	16.66	1.047	4.815⁻⁴	4.335⁻²	1.985⁻¹	1.169⁻³	26.06
	0.4	5	0.582	15.01	1.004	4.480⁻⁴	5.101⁻²	2.040⁻¹	1.087⁻³	24.23
300	0.8	7	0.767	13.90	1.296	1.156⁻³	6.046⁻²	1.681⁻¹	1.033⁻³	34.52
	0.6	6	0.753	12.91	1.209	1.176⁻³	5.880⁻²	1.583⁻¹	9.598⁻⁴	32.08
	0.4	6	0.729	11.40	1.171	1.092⁻³	7.162⁻²	1.662⁻¹	9.028⁻⁴	30.18
400	0.8	8	0.815	12.63	1.307	2.013⁻³	8.333⁻²	1.321⁻¹	7.915⁻⁴	35.28
	0.6	7	0.804	11.84	1.227	1.990⁻³	8.234⁻²	1.258⁻¹	7.387⁻⁴	32.92
	0.4	7	0.776	9.99	1.184	1.799⁻³	1.050⁻¹	1.336⁻¹	6.855⁻⁴	30.55
	κ₁ = 2.0		κ₂ = 0
200	0.8	6	0.399	11.79	0.721	1.380⁻⁴	4.600⁻²	1.091⁻¹	1.645⁻³	18.33
	0.6	5	0.388	10.87	0.670	1.507⁻⁴	4.308⁻²	1.009⁻¹	1.528⁻³	17.03
	0.4	5	0.377	10.04	6.506⁻¹	1.461⁻⁴	4.870⁻²	1.049⁻¹	1.444⁻³	16.09
300	0.8	7	0.454	7.540	7.675⁻¹	2.526⁻⁴	5.692⁻²	5.970⁻²	1.261⁻³	21.07
	0.6	6	0.448	7.062	7.192⁻¹	2.654⁻⁴	5.616⁻²	5.802⁻²	1.172⁻³	19.58
	0.4	6	0.439	6.460	7.049⁻¹	2.613⁻⁴	6.558⁻²	6.276⁻²	1.116⁻³	18.66
400	0.8	8	0.464	5.780	7.447⁻¹	3.699⁻⁴	7.190⁻²	3.601⁻²	9.247⁻⁴	20.61
	0.6	7	0.460	5.490	7.016⁻¹	3.760⁻⁴	7.139⁻²	3.526⁻²	8.639⁻⁴	19.25
	0.4	7	0.449	4.880	6.856⁻¹	3.673⁻⁴	8.727⁻²	3.939⁻²	8.116⁻⁴	18.08
	κ₁ = 1.0		κ₂ = 0.1
200	0.8	6	0.590	15.56	1.067	4.889⁻⁴	4.794⁻²	3.061⁻¹	1.035⁻³	23.06
	0.6	5	0.571	14.36	9.851⁻¹	5.250⁻⁴	4.462⁻²	3.075⁻¹	9.190⁻⁴	20.48
	0.4	5	0.545	12.95	9.410⁻¹	4.881⁻⁴	5.113⁻²	3.069⁻¹	8.464⁻⁴	18.86
300	0.8	7	0.747	12.52	1.263	1.269⁻³	6.199⁻²	2.072⁻¹	9.617⁻⁴	32.15
	0.6	6	0.732	11.54	1.174	1.294⁻³	6.105⁻²	2.023⁻¹	8.814⁻⁴	29.46
	0.4	6	0.708	10.23	1.136	1.204⁻³	7.277⁻²	2.087⁻¹	8.264⁻⁴	27.62
400	0.8	8	0.800	11.52	1.284	2.219⁻³	8.516⁻²	1.558⁻¹	7.561⁻⁴	33.70
	0.6	7	0.788	10.74	1.203	2.202⁻³	8.474⁻²	1.512⁻¹	7.008⁻⁴	31.23
	0.4	7	0.760	9.120	1.159	1.997⁻³	1.060⁻¹	1.584⁻¹	6.486⁻⁴	28.91

Feature	Present analysis	Computer model
Lasing threshold	Occurs simultaneously for all bands at t = 0	Occurs when Eq. (7) is first satisfied. Individual bands treated separately
Lasing cutoff	Occurs simultaneously for all bands	Individual bands treated separately
Line broadening	Doppler	Simultaneous Doppler and collisional
J	Lasing on single J for whole pulse and for all vibrational bands	Lasing on maximum gain transition for each band; calculates J’s as part of solution
Rotational population distribution	Boltzmann at translational temperature	Boltzmann at translational temperature
Temperature	Constant	Calculated as part of solution
Vibrational energy levels	Harmonic	Anharmonic
Reactions modeled	See Table II; most backward rates neglected	Forward and backward rates for all reactions of Table II. Also other VT and VV and recombination–dissociation processes

Abstract

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Figures (5)

Tables (4)

Equations (53)

Applied Optics